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econ412_lecture10_handout.pdf

ECON 412 - Penn State University

Labor Economics and Labor Markets

Lecture 10

Ewout Verriest

Spring 2020

Last lecture Last lecture, we covered the “long-run” firm’s problem when both labor and capital can be freely chosen by the firm: • Isoquants: set of (E,K ) such that f (E,K ) = q0.

• Slope of an isoquant = MRTS = MPE MPK

• Isocost lines: set of (E,K ) such that wE + rK = C0. • Slope of isocost lines = relative price ratio w

r .

• Firm’s optimization problem: • First step: Cost minimization - produce q0 at minimal cost:

MRTS = w

r (or

MPE =

MPK )

• Second step: Profit maximization - choose optimal level of q0:

p = w

MPE =

MPK

Solve system for (E∗, K∗).

• Scale effect and substitution effect • Labor supply elasticity in the short vs. long run

2 / 26

Today’s Lecture

Today:

• Input price changes: scale vs. substitution effects • Labor demand elasticity: determinants • Elasticity of substitution • Application: unions • Labor market equilibrium • Application: minimum wage

Read: Borjas, Chapter 3.3, 3.4, 3.5, 3.6, 3.8, 3.10.

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Total effect of a wage decrease on firm behavior If the wage rate decreases from w0 to w1, the firm’s marginal cost of producing an additional unit of output decreases as well (from, say, MC0 to MC1). Therefore, the firm wants to expand, i.e. move to a higher isoquant where q1 > q0. Given the flatter isocost line, the optimum shifts from P to R.

Note: This left graph plots the market demand for the firm’s product as fully elastic, and hence does not take into account the fact that p might also decrease due to the wage decrease.

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Scale vs. Substitution Effects

Now let’s decompose this total effect.

When the wage drops, two effects arise in the long run:

1. The firm takes advantage of the lower price of labor by expanding production (the scale effect).

• Graphically: we move to a new isoquant, and draw a hypothetical isocost line which is parallel to the original one, and tangent to the new isoquant.

2. The firm takes advantage of the wage change by rearranging its mix of inputs, by employing more labor and less other inputs, even if output is held constant (the substitution effect).

• Graphically: we stay on the new isoquant, and rotate along it to reflect the change in the slope of the isocost line, from w0

r to w1

r .

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Example: Scale effect Suppose the wage rate drops from w0 to w1. In this graph, the firm chooses to move from point P to point R, which lies on a higher isoquant.

The scale effect implies that due to the wage cut, the firm now wants to expand by producing more units. This requires increasing both E and K . This is represented by the move from P to Q, which requires drawing a hypothetical isocost-line which is parallel to the original one and tangent to the new isoquant.

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Example: Substitution effect Suppose the wage rate drops from w0 to w1. In this graph, the firm chooses to move from point P to point R, which lies on a higher isoquant.

The substitution effect implies that due to the wage cut, the firm wants to become more labor-intensive, for any fixed level of output. If the firm wants to produce q1 = 150 units, this means it is optimal to increase E and reducing K , by moving from Q to R.

As it is drawn, the total effect on capital is positive, although we could draw it the other way around too. However, E must unambiguously increase! 7 / 26

Summary: Scale vs. Substitution Effects (1)

More generally, if the price of an input (e.g. w or r) decreases, then

• the marginal cost of production decreases (i.e. the product supply curve shifts to the right/down)

• the firm will grow by moving to a higher isoquant (output level), which raises demand for all inputs (scale effect)

• given the change in slope of isocost line w r

, it is also optimal to substitute towards the cheaper input (substitution effect)

• the demand for the affected input (which became cheaper) unambiguously increases

• the demand for the other input may increase or decrease, depending on the shape of the isoquants (i.e. the technology f (K,E ))

• total output and profits should both increase Note: whether the “demand for the other input” increases or decreases crucially depends on whether the inputs are complements or substitutes in production. See discussion below on the elasticity of substitution.

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Summary: Scale vs. Substitution Effects (2)

Similarly, if the price of an input (e.g. w or r) increases, then

• the marginal cost of production increases (i.e. the product supply curve shifts to the left/upwards)

• the firm will shrink by producing less output, which reduces demand for all inputs (scale effect)

• given the change in slope of isocost line w r

, it is also optimal to substitute away from the more expensive input (substitution effect)

• the demand for the affected input (which became more expensive) unambiguously decreases

• the demand for the other input may increase or decrease • total output and profits should both decrease

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Summary: Scale vs. Substitution Effects (3)

In summary: What happens to the demand for labor E and capital K in each of these 4 scenarios?

Increase in w Decrease in w Increase in r Decrease in r

Scale Effects

Substitution Effects

Total Effect on Inputs

Effect on Output

Effect on Profits

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Summary: Scale vs. Substitution Effects (3)

In summary: What happens to the demand for labor E and capital K in each of these 4 scenarios?

Increase in w Decrease in w Increase in r Decrease in r

Scale Effects E and K ↓ E and K ↑ E and K ↓ E and K ↑

Substitution Effects E ↓, K ↑ E ↑, K ↓ E ↑, K ↓ E ↓, K ↑

Total Effect on Inputs E ↓, K ?? E ↑, K ?? K ↓, E ?? K ↑, E ??

Effect on Output q ↓ q ↑ q ↓ q ↑

Effect on Profits π ↓ π ↑ π ↓ π ↑

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Labor demand elasticity To study the effect of a wage change on long-run labor demand, we introduce the long-run elasticity of labor demand:

δLR = ∆ELR ELR ∆w w

= ∂ELR ∂w

ELR

How can we estimate labor demand elasticities from data?

Same as before: using regression analysis. Collect data on many firms’ payrolls (E ), their employees’ salaries or wages (w), and include some control variables such as the firm’s size, industry, age, etc., to reduce omitted variables bias as much as possible.

Based on a variety of studies, empirical evidence suggests that • the short-run elasticity (δSR ) lies around −0.5 to −0.4 (i.e.

demand is relatively inelastic), whereas • the long-run elasticity (δLR ) lies around −1, • which means that in the (E,w) two-dimensional space, the short-run

demand curve is steeper, as we would expect.

Question: How do you think this affects unions fighting for higher wages? 12 / 26

Different technologies: Special cases of isoquants

Question: What would the production function q = f (E,K ) look like in either of these two extreme cases?

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Different technologies: Elasticity of Substitution

The elasticity of substitution (σ) measures

• the “curvature” of the isoquants, or • the extent to which firms can substitute capital for labor as the

relative cost of the two factors changes:

σ = %∆ K

%∆ w r

Hence, σ measures the percentage change in the capital-labor ratio given a 1 percent change in the relative price of labor to capital.

See Borjas, Chapter 3.5.

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Examples (1)

• It is easy to see that as σ becomes infinitely large, the isoquants lose almost all their curvature and become almost entirely flat, i.e. E and K are perfect substitutes).

Example: a computer or robot which can replace the work of 4 people and vice versa. Technology can be described as q = 4K + E . Isoquants are just straight lines. Optimum will usually be a corner solution!

• Similarly, as σ approaches 0, the isoquants become right-angled, i.e. E and K are perfect complements.

Example: shovels and diggers; we need them in equal proportions to produce holes. Technology can be described as q = min(E,K ). Isoquants are straight angles. Optimum will be somewhere on the 45-degree line!

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Examples(2)

• Intermediate cases: for example, if σ = 5 (i.e. inputs are highly substitutable, but not perfectly so), then a 1 percent increase in the relative cost of labor (or equivalently, a 1% increase in w keeping r fixed) would result in the firm increasing its capital-to-labor ratio by 5%. Because the isoquant has some curvature, a change in the slope of the isocost curve ( w

r ) makes us rotate along the isoquant, towards

using more labor and less capital. The higher σ, the larger the fluctuations.

• If the production function is Cobb-Douglas (q = AEαK 1−α), then σ = 1, i.e. E and K are neither substitutes nor complements. (Don’t worry about the proof.)

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Determinants of labor demand elasticity (1) In general, labor demand becomes more elastic (δ ↑) when • the elasticity of substitution (σ) is larger, i.e. the firm becomes

more responsive to wage changes because workers can more easily be replaced by machines.

• the elasticity of demand for the firm’s output product is larger. Suppose there is a wage change which shifts the product supply (or marginal cost) curve. Then, if the product demand curve is flatter (i.e. more elastic), there will be a smaller effect on the output price p and a larger effect on total quantity q. This means there will also be a larger effect on labor demand.

• the production process is more labor-intensive, i.e. if the firm’s costs are mostly coming from labor, not capital. In a very labor-intensive sector, even a small increase in the wage rate could dramatically increase the marginal cost, which, in turn, leads to a larger reduction in total output and labor demand. This makes labor demand more sensitive to wage shocks, i.e. more elastic.

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Determinants of labor demand elasticity (2)

Question: Think back to our old example of a car manufacturer hiring assembly line workers versus a specialized hospital hiring surgeons. How do you think the labor supply elasticity (and each of its determinants on the previous slide) would differ between these two sectors?

econ412_lecture11_handout.pdf

ECON 412 - Penn State University

Labor Economics and Labor Markets

Lecture 11

Ewout Verriest

Spring 2020

Last lecture

Last lecture:

2 / 26

Today’s Lecture

Today: application of labor market equilibrium to minimum wage

• History of the minimum wage • Basic model with full coverage • Extended model with 2 sectors: covered vs. uncovered • Empirical evidence + Controversy + Example • Empirical methods to estimate minimum wage effects

1. Regression analysis 2. Before vs. after analysis 3. Diff-in-diff analysis

Read: Borjas, Chapter 3.8, 3.10.

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Labor Market Equilibrium The equilibrium wage is determined by both sides of the labor market:

If the market is competitive, equilibrium occurs where demand and supply intersect, such that E∗ workers are employed at a wage of w∗.

Note: in this world, the only unemployed people are voluntarily unemployed, i.e. those who have a reservation wage w̃ < w∗.

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Policy Application: The Minimum Wage

1. History

2. Model: basic version + extended version

3. Empirical evidence + controversy

4. How to estimate minimum wage effects?

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Minimum wage: History

• Minimum wage was introduced in 1938, as part of the “Fair Labor Standards Act” (which also regulated the length of the work week, child labor, overtime pay, etc.).

• Initially non-universal, covering only about half of all workers (e.g. not the agricultural sector).

• At first it was set at $0.25 per hour, and most recently raised to $7.25 per hour by Congress in 2010.

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Minimum wage: History

Orange line: plots the nominal minimum wage (left vertical axis).

Black line: plots the ratio of the minimum wage to the average manufacturing wage (right vertical axis).

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Minimum wage: Some statistics

See article on Canvas: “BLS Report - minimum wage statistics 2018”:

• In 2018, 81.9 million workers age 16 and older in the United States were paid at hourly rates, representing 58.5 percent of all wage and salary workers.

• Among those paid by the hour, 434,000 workers (or about 0.5 percent) earned exactly the prevailing federal minimum wage of $7.25 per hour.

• About 1.3 million had wages below the federal minimum. Together, these 1.7 million workers with wages at or below the federal minimum made up 2.1 percent of all hourly paid workers.

• This remains well below the percentage of 13.4 recorded in 1979, when data were first collected on a regular basis.

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Minimum wage: Exceptions

Note: Federal law does not require you to pay the following workers the federal minimum wage:

• independent contractors (only employees are entitled to the minimum wage)

• Example: Uber and Lyft drivers • outside salespeople (a salesperson who works a route, for example) • workers on small farms • employees of seasonal amusement or recreational businesses • newspaper deliverers, • apprentices, students, and learners, as defined by federal law, • employees of local newspapers having a circulation of less than 4,000

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Minimum wage: The case of independent contractors • Drivers for ride-hailing company Uber Technologies Inc are independent

contractors and not employees, the general counsel of a U.S. labor agency has concluded.

• Uber, its top rival Lyft Inc, and many other “gig economy” companies have faced dozens of lawsuits accusing them of misclassifying workers as independent contractors under federal and state wage laws.

• Employees are significantly more costly because they are entitled to the minimum wage, overtime pay and reimbursements for work-related expenses under those laws.

Source: Reuters, May 2019 10 / 26

https://www.reuters.com/article/us-uber-contractors/uber-drivers-are-contractors-not-employees-us-labor-agency-says-idUSKCN1SK2FY

Minimum wage: Model

For now, let’s make two assumptions for simplicity:

• Full (universal) coverage: all workers are affected by the law. • Full compliance: all employers adhere to the law.

If the minimum wage w is larger than the free market wage w∗, we say it is binding. In that case, employment falls to a level E < E∗.

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Minimum wage: Model

Recall: the unemployment rate is defined as the fraction of labor force participants who are unemployed. Every person older than 16 can either be

• employed (E ): have a paying job for at least 1 hour • unemployed (U): actively looking for a job but don’t currently have

a job, or on temporary layoff.

• out of the labor force (OLF ): neither unemployed nor employed, e.g. retirees, stay-at-home parents.

Before the minimum wage was introduced, the number of employed was E∗, and everyone else was “OLF ” because their reservation wage exceeded w∗ and they were not looking for jobs. Hence, the unemployment rate in this model (defined as U

E +U ) was 0!

After the (binding) minimum wage is introduced, the number of employed is given by E , and the number of (involuntary) unemployed is

U = ES −E . Hence, the unemployment rate is now ES −EES > 0.

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Minimum wage: Model

Questions: If the minimum wage is binding, then what happens to

1. employment (E )?

2. involuntary unemployment (U)(i.e. people who are willing to work at the prevailing wage rate, but can’t find a job)?

3. voluntary unemployment (OLF ) (i.e. people who choose not to work at the going wage rate)?

4. employment in sectors with less vs. more elastic labor demand?

5. employment in the short vs. long run?

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Trade-offs of minimum wage laws

According to our simple equilibrium model, what happens if the government imposes a (binding) minimum wage?

• On the one hand, the income of the least skilled workers who are able to maintain their job goes up.

• On the other hand, the probability of being laid off also goes up for people at the lower end of the income (or wage) distribution.

• In general, the negative effect on employment is larger if the labor demand curve is flatter (e.g. in more elastic labor markets, in the long run, etc.).

Note: Although the minimum wage is defined as an hourly wage, this doesn’t mean that you have to pay employees by the hour. You may pay a salary, commission, wages plus tips, or piece rate, as long as the total amount paid divided by the total number of hours worked is equal to at least the minimum wage.

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Extension of the minimum wage model

Now, let us consider an extension of the model, by relaxing the “universal coverage” assumption.

Consider a world with two different sectors:

1. A covered sector, where a minimum wage needs to be paid if it exceeds the market wage in that sector (e.g. McDonald’s)

2. An uncovered sector, which is not subject to the minimum wage law (e.g. Uber drivers)

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Extension of the minimum wage model

What is the impact of minimum wages on each sector? Important assumptions:

1. The demand for labor does not change after the minimum wage.

2. Prior to the minimum wage law, there exists a single equilibrium wage in both sectors, w∗. That is, all workers in both sectors make the same wage.

3. Workers can freely migrate between sectors, i.e. there is no cost to switching occupations.

(Are these assumptions realistic? Why or why not?)

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Extension of the minimum wage model

Left graph shows the covered sector where the minimum wage is binding.

Right graph shows the uncovered sector.

In equilibrium, two opposing migrations happen after the minimum wage is in effect:

1. Migration towards the uncovered sector

2. Migration away from the uncovered sector

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1. Migration towards the uncovered sector

Why is there migration from covered to uncovered sector?

• Some workers in the covered sector lose their jobs. • Since they can freely migrate, some move to the uncovered sector

(e.g. some ex-McDonald’s workers now become Uber drivers).

• There is no minimum wage here, so they all immediately find jobs. • However, this migration increases labor supply in the covered sector

(i.e. downward shift of SU in the graph), which puts downward pressure on the equilibrium wage w∗ in the uncovered sector.

• Note: now everyone in the uncovered sector would earn less, including all the workers who never switched occupations.

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2. Migration away from the uncovered sector

Conversely, why is there migration from uncovered to covered sector?

• Some workers in the uncovered sector quit their current job which pays less than the minimum wage, and migrate to the covered sector (e.g. some low-paid Uber drivers now want to work at McDonald’s).

• Since there is excess labor supply in the covered sector, not all of these workers find a job immediately (i.e. McDonald’s doesn’t have enough jobs for everyone who applies).

• However, the prospect of possibly finding a minimum wage job after some time might make it worth it to remain (involuntarily) unemployed in the covered sector for a while.

• This migration reduces labor supply in the covered sector (i.e. upward shift of SU in the graph), which puts upward pressure on the equilibrium wage w∗.

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Equilibrium wage and employment in each sector When does this back-and-forth migration stop? In other words, what is the new labor market equilibrium in each sector?

Under our assumption that migration is costless, (net) migration will be 0 if workers are, in expectation, equally well off in each sector (i.e. indifferent).

Assume workers only care about the wage (and not about other aspects of the type of job they perform).

In the uncovered sector (e.g. Uber): • All workers are employed, i.e. Prob(employed) = 1. • All workers make the same market wage, w∗. • Therefore, the expected wage in this sector is w∗.

In the covered sector (e.g. McDonald’s): • Only some workers are employed, i.e. Prob(employed) = π < 1. • All employed workers make the same minimum wage, w. • Assume unemployed workers earn nothing. • Therefore, the expected wage in this sector is πw + (1 −π)0.

In equilibrium, w∗ and π will adjust up to the point where w∗ = πw. 20 / 26

Empirical evidence on minimum wages What does the data say about the effects of minimum wage changes?

Many studies find that the elasticity of (teenage) employment with respect to the minimum wage is around −0.3 to −0.1. Interpretation? The estimated effects of a minimum wage increase depend on many factors, including • the elasticity of the labor demand curve and its determinants, which

vary across sectors (see previous lectures) • the size of the minimum wage increase, as well as the starting point • timing: whether the effects are measured in short vs. long run • whether the minimum wage increase is federally mandated or only in

some states (which also affects migration!) • the type of data used (e.g. survey data vs. aggregate city-level

data), and the sample composition (age, sector, time period). • the methodology used (e.g. regression vs. diff-in-diff), including

what control variables were included

Recommended reading: the final subsection in Borjas Chapter 3.10 on “The Seattle Minimum wage debate” shows how economists and policy makers heavily disagree on the costs and benefits of minimum wages.

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Recent controversy: Wall Street Journal survey

Source: https://www.wsj.com/articles/raising-minimum-wage-would-cost-jobs-say- economists-in-wsj-survey-11554991320

22 / 26

https://www.wsj.com/articles/raising-minimum-wage-would-cost-jobs-say-economists-in-wsj-survey-11554991320

Recent controversy: Wall Street Journal survey The current minimum wage is $7.25 per hour. California wants to gradually raise it to $15 per hour.

The survey (among business and financial economists) showed that there is a lot of disagreement about at which point a minimum wage increase would reduce employment:

• 33% says increasing the minimum wage to between $7.25 and $10 would be bad for employment (i.e. any increase is bad).

• 26% says increasing the minimum wage to between $10.01 and $13 would be bad for employment.

• 12% says increasing the minimum wage to between $13.01 and $15 would be bad for employment.

• 28% says increasing the minimum wage to above $15 would be bad for employment.

• On average, they favored a minimum wage of $10.83 per hour. • However, 33% said the minimum wage should ideally be $0.

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Recent controversy: Wall Street Journal survey

Some more quotes from the WSJ article:

• “Just under half of respondents said raising the minimum wage would help lower-income households by putting more disposable income in their pockets.”

• “Just over half of respondents, 53%, said a higher minimum wage would hurt lower-income households by reducing employment.”

• “Six states—California, New York, Massachusetts, New Jersey, Illinois and Maryland—have already set into motion increases that will bring their minimum wage to $15 an hour in the coming years. There are 29 states with a minimum wage above the federal level.”

• “The impact on payroll costs of raising the minimum wage could be disproportionately felt by Southern states.” ⇒ perhaps a state-specific minimum wage would be better than a federally mandated one?

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Minimum wage change: Example In 1989, the minimum wage was $3.35 per hour. By April 1991, it was raised to $4.25 per hour. Based on our “consensus” elasticity estimates of −0.3 to −0.1, how would you estimate the number of teenagers who lost their jobs because of this?

Note: in 1990, the U.S. civilian labor force was around 125 million, around 8 million of which were teenagers.

• The wage change corresponds to a 100 ∗ 4.25−3.35 3.35

≈ 27% increase.

• “Conservative” estimate: Based on an elasticity of −0.1, this would imply a 2.7% decrease in employment, or 216000 teens.

• “Average” estimate: Based on an elasticity of −0.2, this would imply a 5.4% decrease in employment, or 432000 teens.

• “Worst case” estimate: Based on an elasticity of −0.3, this would imply an 8.1% decrease in employment, or 648000 teens.

Even “small” uncertainty about elasticity can lead to huge aggregate effects (and political debate)!

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Empirical methods

What empirical methods or techniques are used in applied work to estimate minimum wage impact?

1. Regression analysis, e.g. collecting panel data from various states and multiple points in time.

2. Look at “before” and “after” effect of a specific minimum wage increase (e.g. in 1991 from $3.35 to $4.25) in a particular industry.

3. Difference-in-difference analysis.

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econ412_lecture12_handout.pdf

ECON 412 - Penn State University

Labor Economics and Labor Markets

Lecture 12

Ewout Verriest

Spring 2020

Last lecture

Last lecture: application of labor market equilibrium to minimum wage

• History of the minimum wage • Basic model with full coverage • Extended model with 2 sectors: covered vs. uncovered • Empirical evidence + Controversy + Example

2 / 20

Today’s Lecture

• Empirical methods to estimate minimum wage effects 1. Regression analysis 2. Before vs. after analysis 3. Diff-in-diff analysis

• Chapter 4: Labor Market Equilibrium • Worker’s surplus • Firm’s surplus • Total surplus, welfare, gains from trade • Invisible hand theorem

3 / 20

Empirical methods

What empirical methods or techniques are used in applied work to estimate minimum wage impact?

1. Regression analysis, e.g. collecting panel data from various states and multiple points in time.

2. Look at “before” and “after” effect of a specific minimum wage increase (e.g. in 1991 from $3.35 to $4.25) in a particular industry.

3. Difference-in-difference analysis.

4 / 20

1. Regression analysis • Regress “(teenage) employment” on a constant, “minimum wage”

and additional macroeconomic control variables. • Often done in a panel data context:

Eit = β0 + β1MinWageit + γMacroVariablesit + δt + δi + �it

where i = 1, ..., N denotes U.S. states and t = 1, ..., T denotes time periods. δt and δi denote time fixed effects and state fixed effects, respectively. (Why do we need them? What do they capture?)

• Findings: β̂1 is often extremely sensitive to the choice of time window, e.g. is t a week, a month, a quarter, a year?

• Possible explanations: • What mostly drives employment are the macro effects, and the true effect

of minimum wage is small. Disentangling the small effect from the more important macro effects is hard. E.g. what if the minimum wage was increased in response to a (local or global) recession? What if there are delayed responses? Maybe it takes some time for employers to respond to minimum wage increases.

• Variables in above regression are also prone to measurement error (e.g. employers misreporting their number of employees at each point in time).

5 / 20

2. Before vs. after analysis

• Alternatively, we could look at employment “before” and “after” a specific minimum wage increase.

• For example, Katz and Krueger (1992) looked at the fast-food industry data from Texas, from December 1990 and July 1991.

• They find little impact of the minimum wage on employment. • Discussion: trade-off in the timing of the “after” data:

• Taking data right after the increase of the minimum wage may not capture the longer term impact, e.g. in fast food restaurants employment may remain roughly constant in the short run. Remember: capital is “fixed” in the short run, so employers often cannot fully adjust employment right away!

• Taking data from a long period after the increase makes it harder to disentangle the impact of the minimum wage change from the impact of changes in other macroeconomic variables.

6 / 20

https://www.nber.org/papers/w3997.pdf

3. Difference-in-difference analysis

• Compared to the simple “before vs. after” analysis, how could we try to control for changes in macro-economic variables?

• Ideally, we’d want something similar to a “randomized controlled trial” or RCT, the gold standard for causal inference in medicine.

• Intuitively: do a before-and-after analysis in two different locations: 1. a “treatment” site, e.g. a state which increased the minimum wage

at some fixed point in time. 2. a “control” site, e.g. a very similar and/or nearby state which did

not increase the minimum wage at the same time.

• This is what Card and Krueger (1994) did: • Collect data from the fast food industry in New Jersey and

Pennsylvania • In 1992, NJ raised its minimum wage from $4.25 to $5.05. • However, PA kept its minimum wage at $4.25. • Hence, PA is our control, NJ is the treatment.

7 / 20

http://davidcard.berkeley.edu/papers/njmin-aer.pdf

3. Difference-in-difference analysis New Jersey Pennsylvania

Before EB,NJ EB,PA After EA,NJ EA,PA Differences ∆NJ ∆PA ∆DID = ∆NJ − ∆PA • The treatment group difference ∆NJ = EA,NJ − EB,NJ captures

the change in employment in NJ due to changes in the minimum wage AND any changes in macro variables over time (i.e. “time fixed effects”).

• The control group difference ∆PA = EA,PA − EB,PA captures the change in employment in PA due to ONLY changes in macro variables over time (since the minimum wage didn’t change in PA).

• In theory, our diff-in-diff estimator, ∆DID = ∆NJ − ∆PA isolates the pure minimum wage effect, assuming that NJ and PA experienced similar changes in macro-economic variables during that time period (e.g. because they are neighboring states and relatively similar in composition). So the time FEs drop out.

• Controversially, the authors found that ∆DID = +2.7, i.e. the minimum wage increase actually increased labor demand.

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3. Difference-in-difference analysis

New Jersey Pennsylvania Differences Before EB,NJ EB,PA ∆B After EA,NJ EA,PA ∆A Differences ∆NJ ∆PA ∆DID = ∆NJ − ∆PA

Note: alternatively, we could rewrite ∆DID = ∆A − ∆B , where • ∆A = EA,NJ − EA,PA captures the “after” difference in employment

between NJ and PA after the minimum wage change, due to different minimum wages AND systematic differences in employment between those 2 states (i.e. “state fixed effects”).

• ∆B = EB,NJ − EB,PA captures the “before” difference in employment between NJ and PA before the minimum wage change, due to ONLY systematic differences between those 2 states.

• Our diff-in-diff estimator, ∆DID = ∆A − ∆B isolates the pure minimum wage effect, assuming that the systematic differences between NJ and PA did not change before vs. after the minimum wage change in NJ (i.e. the state FEs drop out).

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Summary (1)

In summary, what are the main challenges when estimating minimum wage effects?

• If the true causal effect is small, it is hard to isolate it in the presence of other important macro variables which we need to control for.

• Measurement error in the data. • The timing of “after” data often matters a lot!

1. If you collect data only one week after the minimum wage changed, there is likely not much change in macro variables, BUT also firms barely had a chance to respond...

2. Conversely, if you collect data long after the minimum wage changed, firms have had time to react, BUT employment may have also changed because of other factors that have changed over time (e.g. macro variables).

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Summary (2)

How effective is the minimum wage as an anti-poverty tool?

• Depends on the trade-off between employment vs. wage/welfare effects. As always, there will be winners and losers.

• Unemployment typically goes up among low-skilled workers. • On the other hand, some low-skilled workers (and/or teenagers of

poor and middle-class families) make more money if they keep their jobs.

• Moreover, increases in the minimum wage raise firms’ marginal cost of production (MC ), which may lead to an industry-wide decrease in supply (q) and an increase in output price (p). Essentially, consumers end up paying part of the increased labor costs for firms.

• Finally, wages and employment in “uncovered” sectors may also be affected due to migration (e.g. Uber).

• Often hard to separate objective empirical findings from subjective political motives (e.g. attracting voters).

11 / 20

Chapter 4: Labor Market Equilibrium

12 / 20

Introduction

In a market with perfect competition, Adam Smith’s invisible hand theorem implies that without any form of outside intervention,

• firms (“buyers” of labor) maximize their profits, • workers (“suppliers” of labor) maximize their utility, • the economy will naturally reach an equilibrium (e.g. a pair (w ∗, E ∗)

that is efficient,

• where total gains (defined as the sum of worker surplus and producer surplus are maximized.

13 / 20

Producer surplus

What is the producer’s (or firm’s) surplus in this context?

Assume for simplicity that the firm only uses one input, labor.

In essence, the producer surplus (P ) is given by the profits, which are given by the revenues minus the input costs. How can we see this graphically and mathematically?

• Revenue is the sum of the value of marginal product of all hired workers: from the first one who was hired (who was very productive, i.e. had high VMP ) to the last one who was hired (whose VMP is exactly equal to the wage, w ∗). Therefore, revenues are given by the entire (trapezoidal) surface underneath the labor demand curve, from E = 0 up to E = E ∗. Or mathematically:

Revenue =

∫ E∗ 0

p ∗ MPE dE = ∫ E∗

p ∂f

∂E dE

= p ( f (E ∗) − f (0)

) = p ∗ f (E ∗) = pq∗

14 / 20

Producer surplus

• Secondly, Costs are simply given by the firm’s input expenditures:

Costs = w ∗E ∗

Graphically, this corresponds to the rectangular area between E = 0 and E ∗ on the horizontal axis, and between w = 0 and w = w ∗ on the vertical axis.

• Therefore, Profits (or the firm’s surplus/welfare) are given by the triangular area underneath the labor demand curve, but above the flat line where w = w ∗. Mathematically, this corresponds to the sum of all the net contributions to profit made by every individual worker:

Profits =

∫ E∗ 0

(p ∗ MPE − w ∗) dE = pq∗ − w ∗E ∗

15 / 20

Worker surplus

The worker surplus (or welfare) is given by the difference between

1. the labor income received: w ∗E ∗. This is the same as the rectangular area depicting “Costs” for the employer.

2. the value (or cost) of the worker’s time (spent at work) if spent outside of the labor market, i.e. the opportunity cost of time. This is given by the area underneath the labor supply curve from E = 0 to E = E ∗.

Graphically, the resulting worker surplus corresponds to the triangular area (denoted by Q ) above the labor supply curve and below the flat line where w = w ∗.

16 / 20

Labor market equilibrium

In equilibrium (without any government intervention), total surplus is given by the sum of

• producer surplus: area P , which “integrates out” the difference between their willingness to pay for labor (i.e. the VMPE , denoted by the demand curve) and what they actually pay workers (w ∗).

• worker surplus: area Q , which “integrates out” the difference between what workers receive (w ∗) and their reservation wage (i.e. the value of their time outside of the labor market, given by the supply curve).

17 / 20

Equilibrium forces

Question: Why is (w ∗, E ∗) an equilibrium?

• At any higher wage w H > w ∗, there is an excess supply of workers who are (involuntarily) unemployed in the market. Competition between workers puts downward pressure on their wage.

• Similarly, at any lower wage w L < w ∗, there is an excess demand for workers, so labor is scarce. Competiting firms can attract workers by offering higher wages, until the point where it is no longer profitable for them to do so (i.e. at w = w ∗).

Importantly: In this competitive equilibrium,

• there is no involuntary unemployment, and • total gains are maximized by letting the market “choose” the

optimal wage rate w ∗.

econ412_lecture13_handout.pdf

ECON 412 - Penn State University

Labor Economics and Labor Markets

Lecture 13

Ewout Verriest

Spring 2020

Last Lecture

• Empirical methods to estimate minimum wage effects 1. Regression analysis 2. Before vs. after analysis 3. Diff-in-diff analysis

• Chapter 4: Labor Market Equilibrium • Worker’s surplus • Firm’s surplus • Total surplus, welfare, gains from trade • Invisible hand theorem

2 / 22

Today’s Lecture

Chapter 4: Labor Market Equilibrium

• Single wage property • Wage convergence within the U.S. • Wage convergence across countries: the case of NAFTA • Free trade and global efficiency • Policy application:

1. Payroll taxes on firms 2. Payroll taxes on workers 3. Deadweight loss, excess burden, welfare loss

Read: Borjas Chapter 4.2, 4.3.

3 / 22

Many labor markets

Before: Equilibrium in a single competitive labor market.

Now: deal with many labor markets within a single economy.

Example: Consider 2 labor markets in the U.S.; North and South

• For simplicity, assume supply of labor is fully inelastic (i.e. vertical). • Labor supply in each region: SN and SS • Labor demand in each region: DN and DS • Equilibrium wages in each region: wN and wS • Suppose wN > wS . Can this wage gap persist in equilibrium?

No. But only if we are willing to make some assumptions:

1. workers in the North and South are perfect substitutes, i.e. have identical skill levels.

2. workers (or firms) can move freely from one market to the other one.

4 / 22

Competitive eqm. in 2 labor markets Question: Make a graph of what happens in the labor market in (a) the North and (b) the South.

Under these assumptions: • some workers from the South will migrate North • labor supply in the North increases from SN to S′, pushing down wage wN to w∗ • labor supply in the South decreases, pushing up wS to w∗ • (net) migration stops when the two wages are equalized:

wN = wS = w ∗

• This is called the Single Wage Property. 5 / 22

Single Wage Property

Note that the single wage property generalizes to any number of labor markets, as long as we have (1) perfect mobility between them and (2) perfect substitutability between all workers.

Recall that in equilibrium, the wage equals the value of marginal product (VMP) of labor. Therefore, since wages are equalized, it must also be the case that all workers (of the same skill level) have the same VMP of labor across all markets.

Claim: Migration leads to an allocation that maximizes the total value of output in all labor markets combined. In other words: the single wage property is efficient.

Proof: Recall that the value of output in a single market is given by the trapezoid area under the labor demand curve. It is easy to see in our previous figure that due to migration, the loss of value in the South is strictly smaller than the gain in value in the North. The net gain in total value of output in the economy is given by the triangle ABC. (Verify this.)

6 / 22

Single Wage Property

Questions:

1. How do free migration and the single wage property relate to the “invisible hand” theorem?

2. If this model is indeed a good approximation of reality, then what should we observe in the data?

Answer:

1. If workers (or firms) are freely mobile, so that they can selfishly pursue their own self-interests and move to wherever they prefer, then that achieves an efficient allocation of the economy’s resources, i.e. aggregate output is maximized, as if by the invisible hand of a benevolent social planner who had that exact efficiency goal in mind.

2. We should observe that wages across regions (within a given sector or skill level, say, manufacturing) should converge to a “national average”.

7 / 22

Convergence of regional wages in the U.S.: 1950-1990

We see a clear negative correlation, i.e. places with low wages in 1950 experienced high average wage growth after that, and vice versa. Hence, wage gaps are shrinking over time, in line with what our model predicts. Convergence happens, BUT it may take a lot of time.

8 / 22

Convergence across countries: The case of NAFTA

What about convergence across wages in different countries?

The wage differential between the U.S. and Mexico is huge: wUS > wMEX .

Why do these income and wage gaps persist?

What are the effects of international trade policies such as the North American Free Trade Agreement (NAFTA)?

9 / 22

Convergence across countries: The case of NAFTA

NAFTA permits free movement of goods (but not workers) across U.S., Canadian and Mexican borders. What does our theory say about this?

• Since firms are mobile, U.S. firms should relocate to Mexico to find cheaper labor there.

• Labor demand in the U.S. decreases (shifts left), pushing down wUS . • Labor demand in Mexico increases (shifts right), pushing up wMEX . • Hence, we should see some convergence between wMEX and wUS ,

especially among groups of workers who are very “substitutable”.

• Who should like/dislike NAFTA in the U.S.? • Some U.S. workers suffer wage cuts (e.g. in manufacturing sectors),

but firms benefit from cheaper products.

10 / 22

Free Trade and Global Efficiency

Overall, our theory implies that free trade (i.e. open markets across all countries) maximizes efficiency and total income across all countries.

If workers and/or firms could freely move to any region or country in the world, then the “invisible hand” theorem implies that

1. eventually, wages would equalize across all regions and countries, and

2. total economic surplus or output in the world would be maximized.

11 / 22

Free Trade and Global Efficiency

Note: the model also implies that if there were a hypothetical “global” and benevolent government which could

1. remove all existing trade barriers in the world, AND

2. redistribute part of the additionally created wealth towards the individuals who are affected negatively by free trade (e.g. through various transfers within and across nations)

then everyone could be made better off by free trade.

This is one of the reasons why many economists are in favor of free trade and open markets, and not in favor of tariffs, import restrictions, immigration restrictions, etc.

12 / 22

Policy application: Payroll Taxes and Subsidies

Consider a government policy which taxes firms’ payrolls, i.e. a tax on their employees’ earnings which is not paid by the worker but by the firm.

In the U.S.: firms pay a 6.2% payroll tax on their employees’ gross earnings to fund Social Security, and another 1.45% payroll tax to fund Medicare.

Question: For simplicity, assume that for every hour of labor that the firm hires, the firm has to pay 1 dollar to the government. What is the impact of this payroll tax on equilibrium employment and wages, (E∗, w∗)? What are the welfare effects?

Note: This tax effectively reduces the value of marginal product of labor by 1 dollar everywhere, i.e. it causes a parallel downward shift by 1 dollar in the labor demand curve. Since the tax is levied on firms, the workers’ problem (and therefore labor supply) are not affected!

13 / 22

Payroll tax imposed on firms Before the tax: labor demand is equal to D0; equilibrium is (E0, w0).

After the tax: labor demand shifts down to D1; new equilibrium is (E1, w1). As we move from point A to point B, the equilibrium wage decreases by less than 1 dollar, but employment decreases as well. The firm essentially passes on part of the tax onto their workers.

14 / 22

Payroll tax imposed on firms

We have shown that the payroll tax on firms leads to

• a decrease in employment from E0 to E1 • a decrease in the equilibrium wage received by workers from w0 to w1 • an increase in the total wage cost paid by the firm: from w0 to

(w1 + 1)

• a government revenue of E1 ∗ 1 dollars • a decrease in total welfare, “gains” or surplus in the economy (i.e.

workers + firms + government)

15 / 22

Payroll tax imposed on firms

Questions:

1. Show graphically what happens to the workers’ surplus, the firms’ surplus and government surplus before and after the payroll tax on firms.

2. Show the welfare loss caused by the decrease in employment.

3. What happens if labor supply is more inelastic (i.e. steeper)?

16 / 22

Payroll tax if labor supply is inelastic

In the extreme case where labor supply is fully inelastic (i.e. vertical), employment remains the same, but workers pay the entire tax burden: E1 = E0 and w1 = w0 − 1. There is no welfare loss: firms’ surplus remains the same, and government revenue is equal to the loss in worker surplus.

Implication: if certain subgroups supply labor more inelastically (e.g. adult men always working full-time), they will pay for a higher fraction of their employers’ payroll taxes through lower competitive wages.

17 / 22

Payroll tax imposed on workers

Now consider the alternative scenario where the tax is not imposed on firms, but on workers instead.

In particular: for every hour a worker works at a wage rate of w, (s)he now has to pay a labor income tax of 1 dollar to the government.

Questions:

1. What is the impact of this payroll tax on equilibrium employment and wages, (E∗, w∗)? What are the welfare effects?

2. How do the welfare effects differ from the case where the government imposed the tax on firms instead?

18 / 22

Payroll tax imposed on workers Before the tax: labor demand is equal to D0; equilibrium is (E0, w0).

After the tax: labor supply decreases (i.e. shifts up) from S0 to S1; new equilibrium is (E1, w1). As we move from point A to point B, the equilibrium wage increases by less than 1 dollar, and employment decreases. The workers essentially pass on part of the tax onto the firms.

19 / 22

Payroll tax imposed on workers

We have shown that the payroll tax on workers leads to

• a decrease in employment, from E0 to E1 • an increase in the equilibrium gross wage (or market wage) paid by

firms, from w0 to w1 • a decrease in the equilibrium net wage received by workers, from w0

to w1 − 1 • a government revenue of E1 ∗ 1 dollars • a decrease in total welfare, “gains” or surplus in the economy (i.e.

workers + firms + government)

20 / 22

Payroll tax imposed on workers

Questions:

1. Show graphically what happens to the workers’ surplus, the firms’ surplus and government surplus before and after the payroll tax on workers.

2. Show the welfare loss caused by the decrease in employment.

3. What happens if labor supply is more inelastic (i.e. steeper)?

Crucial findings:

• It does not matter whether the tax is imposed on workers or on firms! The welfare consequences (i.e. the workers’ surplus, firms’ surplus, government revenue and deadweight loss) are all exactly identical!

• According to this simple model of a competitive labor market, the political debate about whether payroll taxes should be levied on firms or on workers is moot.

21 / 22

Payroll taxes: Summary

22 / 22

econ412_lecture2_handout.pdf

ECON 412 - Penn State University

Labor Economics and Labor Markets

Lecture 2

Ewout Verriest

Spring 2020

Today’s Lecture

Today:

• Math review • How do we measure unemployment? • What do (un)employment rates look like in the US? • How do we model workers’ choice of whether and how much to

work?

Reading: Borjas, Chapter 2

2 / 30

Today’s Lecture

Key words:

− (un)employment rates, labor force participation − model of labor/leisure choice − utility function, budget constraint, indifference curves − optimality conditions − marginal utility of consumption/leisure

3 / 30

Math review: Linear functions

Consider a linear function y = b + m x

• b is the intercept: if x = 0, then y = b. • m is the slope: if x increases by one unit, then y increases by m

units.

• Plot the graph for b = 100 and m = −5. • Relevant examples: budget constraints, linear regressions

4 / 30

Math review: Derivatives Consider a utility function U(C, L) = 75 ln(C ) + 300 ln(L) • If C = 1 and L = 50, then U = 300 ln(50) = 1173.607. • Assuming we keep consumption fixed at 1 unit, what happens to

utility if we increase leisure by a tiny amount, say 0.1 hours? • Two possible ways to calculate this:

1. Calculate the changes manually:

∆U

∆L =

U(1, 50.1) − U(1, 50) 50.1 − 50

= 1174.206 − 1173.607

0.1 = 5.994

2. Simpler and faster: use the partial derivative method:

∂ U(C, L)

∂ L =

300

= 300

50 = 6

• The two approaches should always give nearly identical results! 5 / 30

Math review: Optimization

• In many cases, we want to optimize a function, e.g. maximize utility or profits, minimize costs, etc.

• Example: U(C ) = 24C − 4C2

• In order to find the maximum, we need to derive the first order condition:

• dU(C) dC

= 0

• or: 24 − 8C = 0 • so C∗ = 3 units. • Note: How do we know this is a maximum and not a minimum? • By also checking the second derivative:

• For f (x) to reach a maximum, we need d 2f (x)

dx2 < 0.

• For a minimum, we need d 2f (x)

dx2 > 0.

• Here, the second derivative is d 2U(C) dC2

= −8, so we have a maximum at C∗ = 3.

6 / 30

Math review: Optimization

• Similarly, for functions with multiple arguments, optimization requires taking multiple first order conditions.

• Example: f (x, y) = 24x − 4x2 + 16y − 2y2 − 2xy • First, take the FOC with respect to x: • 24 − 8x − 2y = 0, or y + 4x = 12 • Second, take the FOC with respect to y: • 16 − 4y − 2x = 0 or 2y + x = 8 • This gives a system of 2 unknowns and 2 equations. • Verify that the solution is x∗ = 16

7 and y∗ = 20

7 / 30

Math review: Random variables

• Consider a discrete random variable X , which can take values {x1, ..., xn} with corresponding probabilities {p1, ..., pn}.

• Then, the expected value of X is

E (X ) = p1x1 + p2x2 + ... + pnxn

= Σni=1pixi

8 / 30

Math review: Random variables

• Example: expected payoff of a coin flip. • Suppose I win 6 dollars for Heads (H), and 2 dollars for Tails (T) • If the game costs 1 dollar to participate, what is my expected payoff? • E (X ) = p(H)(6 − 1) + p(T )(2 − 1) = 0.5 ∗ 5 + 0.5 ∗ 1 = 3. • Now suppose my utility function is U(x) = ln(x). What is my

expected utility from playing the game?

• E [U(X )] = p(H)U(5) + p(T )U(1) = 0.5 ln(5) + 0.5 ln(1) = 0.805

9 / 30

Measuring the Labor Force (1)

• Three types: employed (E ), unemployed (U), out of labor force (OLF )

• Empirical definitions: • E : Job with pay for at least 1 hour or worked at least 15 hours on a

non-paid job (e.g. a family farm) • U: Temporary layoff from a job, or have no job and actively looking

for work in the past 4 weeks • OLF : Residual group (i.e. stay-at-home parent, retiree, student, etc.)

• Measured based on survey data. In the US: based on Current Population Survey, which interviews about 50.000 households per month. All individuals above 16 years of age are considered.

• Alternative method/data: Unemployment register

10 / 30

Measuring the Labor Force (2)

• Labor Force LF : LF = E + U

• Labor Force Participation Rate LFP:

LFP = LF

Population

• Employment rate:

Employment Rate = E

Population

• Unemployment rate:

Unemployment rate = U

Note: Employment rate and Unemployment rate do NOT sum up to 1!

11 / 30

Measuring the Labor Force (3)

• Hidden unemployment: Calculation is based on a subjective measure of unemployment

• What does it mean to be “actively looking for work in the past 4 weeks”?

• People who gave up looking for a job are not considered unemployed. • Conversely, everyone claiming to be “actively looking” is considered

unemployed.

• Having many “discouraged workers” in the economy leads to an understatement of the true unemployment rate.

• Should we therefore prefer the Employment Rate?

12 / 30

Basic Facts about Male Labor Supply

13 / 30

Basic Facts about Female Labor Supply

14 / 30

CPS Charts (1)

Figure: Civilian Labor Force Participation, 1990-2019 (seasonally adjusted)

Source: CPS Charts on Employment and Unemployment, see https://www.bls.gov/web/empsit/cps_charts.pdf

15 / 30

https://www.bls.gov/web/empsit/cps_charts.pdf

CPS Charts (2)

Figure: Employment Rate, 1990-2019 (seasonally adjusted)

Source: CPS Charts on Employment and Unemployment, see https://www.bls.gov/web/empsit/cps_charts.pdf

16 / 30

https://www.bls.gov/web/empsit/cps_charts.pdf

CPS Charts (3)

Figure: Civilian Unemployment Rate, 1990-2019 (seasonally adjusted)

Source: CPS Charts on Employment and Unemployment, see https://www.bls.gov/web/empsit/cps_charts.pdf

17 / 30

https://www.bls.gov/web/empsit/cps_charts.pdf

CPS Charts (4)

Figure: Duration of Unemployment, 1990-2019 (seasonally adjusted)

Source: CPS Charts on Employment and Unemployment, see https://www.bls.gov/web/empsit/cps_charts.pdf

18 / 30

https://www.bls.gov/web/empsit/cps_charts.pdf

CPS Charts (5)

Figure: Long-term Unemployed as a percent of Total Unemployed, 1990-2019 (seasonally adjusted)

Source: CPS Charts on Employment and Unemployment, see https://www.bls.gov/web/empsit/cps_charts.pdf

19 / 30

https://www.bls.gov/web/empsit/cps_charts.pdf

CPS Charts (6)

Figure: Unemployment Rates for adult men, adult women, and teenagers, 1990-2019 (seasonally adjusted)

Source: CPS Charts on Employment and Unemployment, see https://www.bls.gov/web/empsit/cps_charts.pdf

20 / 30

https://www.bls.gov/web/empsit/cps_charts.pdf

CPS Charts (7)

Figure: Unemployment Rates by Ethnicity, 1990-2019 (seasonally adjusted)

Source: CPS Charts on Employment and Unemployment, see https://www.bls.gov/web/empsit/cps_charts.pdf

21 / 30

https://www.bls.gov/web/empsit/cps_charts.pdf

CPS Charts (8)

Figure: Unemployment Rates by Educational Attainment, 1990-2019 (seasonally adjusted, and for ages 25 and above)

Source: CPS Charts on Employment and Unemployment, see https://www.bls.gov/web/empsit/cps_charts.pdf

22 / 30

https://www.bls.gov/web/empsit/cps_charts.pdf

CPS Charts (9)

Figure: Alternative measures of Unemployment, 1990-2019 (seasonally adjusted)

Note: Discouraged workers (U-4, U-5, and U-6 measures) are persons who are not in the labor force, want and are available for work, and had looked for a job sometime in the prior 12 months.

23 / 30

Historical Changes in Weekly Work Hours

Source: Our World In Data, see https://ourworldindata.org/working-hours

24 / 30

https://ourworldindata.org/working-hours

Historical Changes in Weekly Home Production (US)

Source: Our World In Data, see https://ourworldindata.org/working-hours

25 / 30

https://ourworldindata.org/working-hours

Historical Changes in Weekly Housework (US)

Source: Our World In Data, see https://ourworldindata.org/working-hours

26 / 30

https://ourworldindata.org/working-hours

The Worker’s Problem

How can we model an individual’s choice of time allocation?

• Preferences (e.g. leisure vs. consumption vs. ...) • Constraints (e.g. time constraint, budget constraint) • Different model settings: static vs. dynamic, single parent vs.

married couple, etc.

• For now: consider a simple neoclassical model of labor-leisure choice.

27 / 30

The Worker’s Preferences

• Consumption of goods: C • Consumption of leisure: L • Measure of well-being through a utility function:

U = f (C, L)

• Indifference Curve IC : • Visualize the tradeoff between C and L, keeping the utility level fixed. • Indifference curves are downward sloping. • Higher indifference curve indicates higher level of utility. • Indifference curves do not intersect (for a given person). • Indifference curves are convex to the origin (implies diminishing

marginal rate of substitution; see below). • Exception: if two goods are perfect substitutes: U(C, L) = C + L

28 / 30

Indifference Curves

Points X and Y lie on the same IC and yield the same level of utility (25000 utils); point Z lies on a higher IC and yields more utility. Question: Explain the convex shape in words.

29 / 30

Indifference Curves do not intersect

Question: Explain why this contradicts individual rationality. Points X and Y yield the same utility because they are on the same IC. Points Y and Z should also yield the same utility. Point Z, however, is clearly preferable to point X, since it offers more of everything.

30 / 30

econ412_lecture3_handout.pdf

ECON 412 - Penn State University

Labor Economics and Labor Markets

Lecture 3

Ewout Verriest

Spring 2020

Last week

− key players in the labor market: workers, firms, government − economic models: what are they and why do we need them? − measures of (un)employment − what do unemployment and participation rates look like in the U.S.? − the worker’s problem: neoclassical model of labor-leisure choice

2 / 23

Today’s Lecture

Today:

• How do we model workers’ choice of whether and how much to work?

• How do we assess the effects of a change in non-labor income or hourly wage on labor supply?

• What are income and substitution effects? Reading: Borjas, Chapter 2

3 / 23

Today’s Lecture

Key words:

− model of labor/leisure choice − utility function, budget constraint, indifference curves − optimality conditions − marginal utility of consumption/leisure − marginal rate of substitution − opportunity costs (e.g. wage), opportunity set, budget line − comparative statics − income effect, substitution effect

4 / 23

The Worker’s Preferences • Consumption of goods: C • Consumption of leisure: L • Measure of well-being through a utility function:

U = f (C, L)

• Indifference Curves (IC ): Summary • Visualize the tradeoff between C and L, keeping the utility level fixed. • Indifference curves are downward sloping: because both C and L are

“desirable”. • Higher indifference curve indicates higher level of utility: because

more of everything is always better. • For a given person, indifference curves do not intersect: preferences

are rational and transitive (see below). • Indifference curves are convex to the origin: because marginal rate of

substitution is decreasing (see below). • Exception: if two goods are perfect substitutes (e.g.

U(C, L) = C + L).

5 / 23

Indifference Curves

Points X and Y lie on the same IC and yield the same level of utility (25000 utils); point Z lies on a higher IC and yields more utility. Questions:

• Explain why the IC is decreasing. • Explain why the IC is convex to the origin.

6 / 23

Indifference Curves do not intersect

Question: Explain why this would contradict individual rationality.

Points X and Y yield the same utility level because they are on the same IC: U(X ) = U(Y ) = U0. Similarly, points Y and Z also yield the same utility level: U(Y ) = U(Z ) = U1. Transitivity implies that U(X ) = U(Z ). Point Z, however, is clearly preferable to point X, since it offers more of everything. Contradiction.

7 / 23

Slope of an Indifference Curve What happens to a person’s utility as she allocates one more hour to leisure, or buys an additional dollar’s worth of goods?

• Marginal utility of leisure (MUL): the change in utility resulting from an additional unit (hour) devoted to leisure activities, holding constant the amount of goods consumed.

MUL = ∂U(C, L)

∂L • Marginal utility of consumption (MUC ): the change in utility resulting from an

additional unit (dollar) of consumption goods, holding constant the amount of leisure consumed.

MUC = ∂U(C, L)

∂C • (Absolute value of the) slope of the IC (with L on the horizontal axis, C on the

vertical axis) is equal to the marginal rate of substitution (MRS):

MRSL,C = MUL

MUC =

∂U(C,L) ∂L

∂U(C,L) ∂C

• If L is small and C is large, the IC is steep, i.e. MRSL,C is large (MRS >> 0). • If L is large and C is small, the IC is flat, i.e. MRSL,C is small (MRS >≈ 0). • In other words: MRSL,C shows how much consumption I’m willing to give up for

an additional unit of leisure, in order to remain indifferent.

8 / 23

Differences in preferences across individuals Different workers will typically view this trade-off differently, and will have different indifference curves.

Question: Which person exhibits the strongest “taste for working”? In other words, who values consumption (resp. leisure) the most (resp. the least)?

Answer: Mindy. Cindy’s ICs are steeper, indicating that she requires a larger increase in consumption to be willing to give up an additional hour of leisure. Mindy’s ICs are relatively flat, indicating that she attaches a much lower value to her leisure time.

Note: this intuition will become clearer once you also plot the budget constraint for a given wage, and confirm that Mindy will work the most (all ellse equal).

9 / 23

The Budget Constraint

Consumption of goods and leisure is constrained by the time and budget constraints.

• Weekly non-labor income (e.g. property income, dividends): V • Working hours (usually per week): h • Hourly wage rate: w • Budget constraint:

C = V + w ∗ h • In this static model, we abstract from savings. • Further, we assume that w is constant (across h).

10 / 23

The Budget Constraint

Allocation of time:

• A person has two alternative uses for her time: working hours (h) or leisure (L) • Time constraint (per week):

T = L + h

• Exogenously set T = 16 ∗ 7 = 112 (or T = 24 ∗ 7 = 168). • Combining the constraints:

C = V + w ∗ h C = V + w ∗ (T − L) C = (V + w ∗ T ) − w ∗ L

• Intercept of the budget constraint: V + w ∗ T . This is maximal consumption if you consume no leisure.

• Slope of the budget constraint: −w. Every hour of L consumed comes at a price (or “opportunity cost”), which is w.

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The Budget Line is the boundary of the worker’s opportunity set

Point E is the endowment point, telling the person how much she can consume if she does not enter the labor market. The worker moves up the budget line as she trades off an hour of leisure for additional consumption. The absolute value of the slope of the budget line is the wage rate, w.

12 / 23

Cindy vs. Mindy Questions:

• What is the optimal bundle of (C, L) chosen by each person? • How can we describe this optimality condition mathematically? • Who is more likely to choose point E (i.e. not to work)?

Recall their preferences led to the following indifference curves:

Answer: Plot the budget constraint, and solve each worker’s individual utility maximization problem. All else equal (i.e. keeping w and V fixed for both people), Cindy is more likely to choose E .

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The worker’s optimal decision

The worker solves the following problem, taking her hourly wage w and non-labor income V as given:

max (C,L)

U(C, L)

subject to C = (V + wT ) − wL

Solution:

• Choose the point on the budget constraint (BC) that lies on the highest possible indifference curve (IC)

• In the case of an interior solution (where 0 < L∗ < T ), the BC and the IC must be tangent in the optimum, i.e. have identical slopes.

• Mathematically: w = MUL MUC

(= MRSL,C ), or w ∗ MUC = MUL, or MUC =

MUL w

• Intuition?

14 / 23

Intuition

Question: How can we interpret this optimality condition intuitively?

Optimality condition: w ∗ MUC = MUL. • If I work one more hour, I make w dollars. That buys me w units of

consumption, giving my w ∗ MUC extra utils. • If I work one more hour, I also lose one hour of leisure, which costs

me 1 ∗ MUL utils. • In equilibrium, I must be indifferent.

Alternatively: MUC = MUL w

• If I consume an extra dollar, I get MUC extra utils. • In order to make that dollar, I have to work for 1

w more hours, which

costs me 1 w ∗ MUL utils.

• In equilibrium, I must be indifferent.

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Optimal solution of the worker’s problem Example: let T = 110, V = 100, w = 10.

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Counterfactual Analysis

What happens to the optimal bundle (L, C ) of leisure and consumption as the wage rate (w) or non-labor income (V ) change?

In economics, this is often called a comparative statics exercise: change something exogenous (like w or V ) and see how the endogenous variables of interest (C , L, h) change as a result.

Two experiments to consider:

1. An exogenous increase in non-labor income, keeping wage constant.

2. An exogenous increase in wage, keeping non-labor income constant.

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(1) The effects of a change in non-labor income Example: let T = 110, initial V = 100, new V = 200. Key insights:

• An increase in non-labor income leads to a parallel upward shift of the budget line.

• If leisure if a normal good, the demand for leisure would increase, and hours of work would fall.

• Intuitively: I am now wealthier, so I take more vacation (and more consumption).

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(1) The effects of a change in non-labor income Note: for some people, leisure might be an inferior good, i.e. when their income increases (and wage is kept constant), they would consume less leisure.

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(2) The effects of a change in the wage

Example: let T = 110, V = 100, initial w = 10, new w = 20.

20 / 23

(2) The effects of a change in the wage

Key insights:

• An increase in the wage rotates the budget line outward, around the endowment point E (which has fixed coordinates (T, V )).

• The substitution effect (SE ) implies that because the opportunity cost of leisure (w) is now higher, I should consume less leisure, i.e. work more.

• The income effect (IE ) implies that because I now make more money, I should consume more leisure (assuming it is a normal good), i.e. work less.

• The total effect of the wage change on leisure (TE = IE + SE ) is ambiguous, and depends on the shape of the indifference curves.

21 / 23

Example where the Income Effect dominates

Key insights:

• Graphically, we can plot the IE as the change in L resulting purely from the income increase, i.e. keeping the slope w the same, but through a parallel upward shift in the IC from U0 to U1.

• In the above example, the IE = 85 − 70 = +15. Shift from P to Q. • Similarly, we can plot the SE as the change in L resulting purely from the wage

increase, i.e. by rotating along the new IC (U1).

• In the above example: the SE = 75 − 85 = −10. Shift from Q to R. • Since the total effect TE = 15 − 10 = +5, we say the income effect dominates

the substitution effect: IE > SE .

22 / 23

Example where the Substitution Effect dominates

Key insights:

• In the above example, the IE = 80 − 70 = +10. Shift from P to Q. • In the above example: the SE = 65 − 80 = −15. Shift from Q to R. • Since the total effect TE = 10 − 15 = −5, we say the substitution effect

dominates the income effect: SE > IE .

23 / 23

econ412_lecture4_handout.pdf

ECON 412 - Penn State University

Labor Economics and Labor Markets

Lecture 4

Ewout Verriest

Spring 2020

Last lecture

Last lecture, we covered:

• How do we model workers’ choice of whether and how much to work?

• How do we assess the effects of a change in non-labor income or hourly wage on labor supply?

• What are income and substitution effects?

2 / 32

Today’s Lecture

Today:

• Numerical example on basic worker’s problem • Reservation wage • Policy applications

1. Take-it-or-leave-it cash grant 2. Welfare program with phase-out 3. Earned Income Tax Credit (EITC)

Reading: Borjas, Chapter 2

3 / 32

Recap: Counterfactual Analysis What happens to the optimal bundle (L, C ) of leisure and consumption as the wage rate (w) or non-labor income (V ) changes?

We have seen last time that the effects are ambiguous:

• When V increases, then we only have an income effect: L goes up if L is a normal good. We typically assume L to be a normal good.

• When w increases, we can decompose the overall effect on L into an income effect and a substitution effect. See Lecture 3 for graphical analysis.

1. The income effect (IE) increases L. My income grows, so I want to consume more of the things I like (assuming both C and L are normal goods).

2. The substitution effect (SE) decreases L, because leisure has now become more expensive in terms of opportunity cost. The worker “loses” an amount of w in income for every hour not spent working.

• If |IE| > |SE| (“income effect dominates substitution effect”), then L will move in the same direction as w.

• If |IE| < |SE| (“substitution effect dominates income effect”), then L will move in the opposite direction as w.

• Note: In the case of a wage decrease, the IE < 0 and the SE > 0.

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Thought experiment: SE versus IE at low and high wages

Question: Suppose you are currently working 20 hours per week at McDonalds at a minimum wage of 8 dollars per hour, and then the minimum wage goes up to 10 dollars per hour. If given the choice, would you work more, less, or the same amount of hours?

Try to verbalize your thought process using the IE/SE terminology we just used. Which effect do you think would dominate for most people?

Presumably, many minimum-wage employees would take advantage of the wage hike to start working more hours, e.g. by picking up an extra shift or by starting to work full-time. Since they are employed at the minimum wage, their income is relatively low, so the marginal utility of earning an extra dollar should be large. In other words, the income effect (i.e. wanting to consume more leisure just because you got “wealthier”) should be relatively small. The SE likely dominates the IE , so leisure L should go down, and labor h should go up.

Remember: when w is relatively low, SE is likely to dominate the IE , so that labor supply typically INCREASES with w.

5 / 32

Thought experiment: SE versus IE at low and high wages

Question: Now imagine you are a 55-year-old consultant at McKinsey, currently working 60 hours per week at a wage rate of 250 dollars per hour. Suppose your hourly wage goes up to 255 dollars. If given the choice, would you work more, less, or the same amount of hours?

Answer: Presumably, many people in this situation would prefer to “take it easy” and start working a bit less. Since they already have a large income (and therefore a lot of consumption, but not much leisure), their marginal utility of making an extra dollar is relatively small. In other words, we would expect |IE| > |SE| for many people, so they would consume more leisure L and reduce their labor supply h.

Remember: when w is relatively high, IE is likely to dominate the SE , so that labor supply typically DECREASES with w.

Note: this thought process will come back later, when we derive the labor supply curve (Borjas Chapter 2.7).

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Numerical Example

Suppose the following utility function: u(C,L) = (C − 100)(2L− 40).

Let T = 112 hours. Suppose the non-labor income is V = 100. Tip: when solving, might be best to only plug in known parameter values (i.e. V or w) at the very end.

How do the optimal choices (C∗,L∗,h∗) depend on the wage w?

Step 1: Calculate the marginal utility of leisure and the marginal utility of consumption.

MUL = δu(C,L)

δL = (C − 100) ∗ 2 = 2C − 200

MUC = δu(C,L)

δC = 2L− 40

7 / 32

Numerical Example (continued)

Step 2: Exploit the optimality condition (marginal rate of substitution is equal to the wage rate):

MUL MUC

= w

⇐⇒ 2C − 200 2L− 40

= w

⇐⇒ 2C − 200 = w(2L− 40) ⇐⇒ C − 100 = w(L− 20) ⇐⇒ C = wL− 20w + 100

This is NOT the end. We still have 2 unknowns and only 1 equation.

8 / 32

Numerical Example (continued)

Step 3: Exploit the budget restriction: consumption must equal total (labor + non-labor) income:

C = w(T −L) + V ⇐⇒ C = wT −wL + V ⇐⇒ C = 112w −wL + V

9 / 32

Numerical Example (continued) Step 4: Now we have a system of 2 equations and 2 unknowns (L,C ). We can solve the system by e.g. plugging the second one into the first one, and solving for L as a function of w:

C = wL− 20w + 100 and C = 112w −wL + V ⇐⇒ wL− 20w + 100 = 112w −wL + V ⇐⇒ 2wL = 112w + V + 20w − 100 ⇐⇒ 2wL = 132w + (V − 100)

⇐⇒ L = 66 + V − 100

So we get the following optimal choice functions:

L∗(w,V ) = 66 + V − 100

h∗(w,V ) = 112 −L∗ = 46 − V − 100

C∗(w,V ) = V + wh∗ = 46w + V

2 + 50

10 / 32

Numerical Example (continued)

When we plug in V = 100, we get that L∗ = 66.

(Question: Is L a normal good here?)

We notice that at V = 100, L∗ is independent of the wage rate w! This is because for the utility function we have assumed, the income effect is exactly offset by the substitution effect (IE = SE ), such that the optimal choice of leisure does not depend on w.

Note: the other optimal choices are:

• Labor supply h∗ = 112 −L∗ = 112 − 66 = 46 hours. • Consumption C∗ = wh∗ + 100 = 46w + 100 dollars.

Consumption increases linearly with the wage.

11 / 32

Numerical Example (continued) Question: Now assume my wage is fixed at 10 dollars. If my non-labor income would increase by 10 dollars, how would my choices change?

Answer: Consider what happens to optimal choices when we fix w = 10, and vary V :

After plugging in w = 10, optimal choices now become:

L∗ = 66 + V − 100

2w = 61 +

20 ⇒ if V + 10, then L + 0.5 hours

h∗ = 112 −L∗ = 51 − V

20 ⇒ if V + 10, then h − 0.5 hours

C∗ = V + wh∗ = 510 + V

2 ⇒ if V + 10, then C + 5 dollars

I would spend the extra 10 dollars on consumption. However, since I optimally reduce my labor supply by 0.5 hours, I also lose 5 dollars (i.e. half my hourly wage) in labor income. In total, my consumption only increases by 5 dollars. 12 / 32

Reservation Wage

Given w, V and U(C,L), should a person work or not?

See Borjas Chapter 2.6.

The person will work if, for some point on her budget constraint, she generates a utility level that exceeds U0, her reservation utility, which is generated by consuming her endowment point where h = 0, L = T and C = V .

So U0 = U(C = V ,L = T ) = U(V ,T ).

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Reservation Wage

14 / 32

Reservation Wage

Case 1: The wage rate is low: w = wlow .

In this case, the budget line is given by the vertical line from T to E, and the sloped line from E to G.

Note: the intersection with the vertical axis is at G = V + wlow ∗T .

All indifference curves that intersect the budget line lead to utility levels smaller than U0. In the graph, point X yields strictly less utility than point E . So the utility-maximizing choice here is point E , which involves not working.

In this case, we know that wlow lies below the reservation wage.

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Reservation Wage

Case 2: The wage rate is high: w = whigh.

In this case, the budget line is given by the vertical line from T to E, and the sloped line from E to H.

Note: the intersection with the vertical axis is at H = V + whigh ∗T .

In the graph, point Y yields strictly more utility than point E (or any other point on the budget line). So the utility-maximizing choice here is point Y , which involves working positive hours.

In this case, we know that whigh lies above the reservation wage.

16 / 32

Reservation Wage

Case 3: The wage rate is such that the person is exactly indifferent between not working, or working a tiny but positive amount.

This is exactly how we define the reservation wage, denoted by w̃.

We know that w̃ lies somewhere between wlow and whigh.

In the graph, the reservation wage is equal to the slope of the budget line such that the optimal tangency point between the budget line and the (highest) indifference curve corresponds exactly to point E , which yields a utility level U0. Otherwise, the person would not be indifferent!

In other words: the reservation wage is equal to the MRS in the endowment point: w̃ = MUL

MUC when evaluated at (L = T,C = V ).

17 / 32

Reservation Wage If the market wage (how much an employer is willing to pay the worker for one hour of work) is larger than the reservation wage (the smallest possible wage which makes the worker willing to give up some leisure and start working) i.e. if w > w̃, then the person will choose to an interior solution where h > 0 (or 0 < L∗ < T ).

Otherwise, if w ≤ w̃, then she will choose not to work, and we obtain the boundary solution given by the point E , where labor h = 0, leisure L∗ = T and consumption C∗ = V .

Since people have different preferences (as captured by the MRS), they will also have different reservation wages.

Thus: if the market wage (or the minimum wage) w increases, some people who were not working previously will now want to start working. More specifically, if w goes up from w1 to w2, then people whose reservation wage w̃ lies in between w1 and w2 will now start working. They move from “OLF” into the labor force (E + U)!

Note: do NOT confuse “reservation wage” with “minimum wage”. 18 / 32

Reservation Wage How come we obtain an unambiguous effect of wage on labor force participation (LFP)? We have just argued that as w goes up, more people will start working, i.e. leisure will go down.

However, in our previous comparative statics analysis, we found that the effect of the wage on leisure can be ambiguous, and depends on the relative size of the income and substitution effects.

The solution is that here there is NO income effect! If a person does not work (at point E ), her income is unaffected by a marginal change in the wage rate (e.g. an increase from w = wlow to w = wlow + 0.01). So we only have a substitution effect that always leads to a reduction in leisure.

Given the unambiguous effect of w on LFP, does this also mean that a minimum wage increase necessarily implies an increase in LFP? NO! We have simply shown that for a higher wage, more workers would be willing to work (i.e. be willing to supply labor to the market). But of course firms, who demand labor, may not be willing to employ as many workers at a higher wage. (See also Chapter 3 and 4!)

19 / 32

Policy Application

On the one hand, as a government or social planner, we probably want to help people in need.

On the other hand, we want to make sure the system of support does not foster dependency.

Let’s look at three different types of welfare support and look at what our model predicts in terms of the each policy’s effects on (1) labor force participation and (2) hours worked among working people.

Example 1: Consider a welfare program which pays $500 to eligible people as long as they remain outside the labor force (i.e. work h = 0 hours), and which pays $0 otherwise.

For simplicity, assume that non-labor income V = 0.

How does this take-it-or-leave-it cash grant affect the budget set?

20 / 32

Example 1: Take-it-or-leave-it Cash grant

21 / 32

Example 1: Take-it-or-leave-it Cash grant

The budget set is given by the downward sloping line AND the single point, G .

If P is the optimal allocation without the welfare program, then the introduction of the welfare program will induce this person to move from point P to point G , since that provides higher utility than P.

We may end up at the corner solution G , where labor supply h = 0, leisure L = T , and consumption C = 500.

Question: According to this model, which “types” of workers are more likely to choose to leave the work force and “live on welfare”? Explain graphically and intuitively. Do you agree with the model’s prediction?

Answer: Intuitively: workers with relatively low “tastes for work” and/or relatively low wages.

22 / 32

Example 1: Take-it-or-leave-it Cash grant

Comparative Statics 1: What happens if the wage w increases (keeping preferences u(C,L) fixed)?

If w increases, the budget constraint now becomes steeper, so it should become less likely that dropping out of the labor market and living off welfare is optimal.

Draw a figure such that a worker with a low wage (wlow ) prefers to live off of welfare, whereas a different worker with the same preferences but a high wage (whigh) prefers to keep working.

Hint: plot the indifference curve which goes exactly through the point G .

23 / 32

Example 1: Take-it-or-leave-it Cash grant

Comparative Statics 2: What happens if preferences change by exhibiting a larger “taste for work” (keeping the wage w fixed)?

Consider Person 1 who really likes leisure, to Person 2 who really likes consumption. Assume both have the same wage w = 10.

Draw a figure such that Person 1 prefers to live off of welfare, whereas Person 2 prefers to keep working.

Hint: the “workaholic” cares mostly about consumption, so her indifference curves are relatively FLAT. By contrast, the “lazy” person cares mostly about leisure, so her indifference curves are relatively STEEP.

24 / 32

Example 2: Welfare program with phase-out Now consider a more sophisticated welfare program which pays $500 to people who do not work, but which reduces this cash grant by 50 cents for every dollar earned in the labor market. For simplicity, assume that non-labor income V = 0.

What this means for someone with a wage rate of w = 10:

• If I work 0 hours, I receive 500 dollars. • If I work 1 hour, my labor income is 10 dollars, but my cash grant

drops by 0.50∗10 = 5 dollars, so my total income is now 505 dollars. • If I work 20 hours, my labor income is 200 dollars, but my cash

grant is only 500 − 0.50 ∗ 200 = 400 dollars, so my total income is now 600 dollars.

This is essentially the same as giving everyone a 500 dollar cash grant, while imposing a 50 percent tax rate on labor income.

How does this cash grant affect the budget set?

25 / 32

Example 2: Welfare program with phase-out

The budget constraints mathematically, assuming V = 0 and w = 10: Before the policy:

C = wh = w(T −L) = 10(110 −L) = 1100 − 10L

After the policy:

C = wh + (500 − 0.50wh) = 0.5wh + 500

= 5(110 −L) + 500 = 1050 − 5L

26 / 32

Example 2: Welfare program with phase-out

27 / 32

Example 2: Welfare program with phase-out

The new budget set is given by the following fragments:

• the vertical part from E to G (i.e. the cash grant part) • the sloped part from G to H (i.e. the labor income tax part)

If P is the optimal allocation without the welfare program, then the introduction of the welfare program will induce this person to move from point P to the new tangency point R, since that provides higher utility than P.

Question: What is the Income Effect, the Substitution Effect and the Total Effect of this policy on leisure and work?

Spoiler: This type of welfare program which combines a cash grant with a labor tax unambiguously leads to a reduction in hours of work, assuming that leisure is a normal good.

The reason is that here, both the income effect and the substitution effect go in the same direction!

28 / 32

Example 2: Welfare program with phase-out

• Income effect: since income goes up due to the cash grant (which shifts out the budget constraint, keeping the slope fixed), I want to consume more leisure, under the assumption that leisure is a normal good. In the graph, the IE can be depicted as the “parallel” shift from point P to point Q.

• Substitution effect: since my effective wage rate decreases due to the labor tax (which rotates the budget constraint, keeping the indifference curve fixed at U1), the opportunity cost of leisure goes down, so I want to consume more leisure. In the graph, the SE can be depicted as the “rotational” shift from point Q to point R.

• Both the IE (P to Q) and the SE (Q to R) are positive, so the total effect of the policy on leisure is unambiguously positive.

29 / 32

Example 3: The Earned Income Tax Credit This is a famous policy implemented in 1975, which has since been expanded, and has been the focus of a lot of economic studies.

30 / 32

Example 3: The Earned Income Tax Credit

This is how the EITC alters the budget constraint:

31 / 32

Example 3: The Earned Income Tax Credit

Recommended reading: Borjas, Chapter 2-12: Policy Application on the Earned Income Tax Credit (EITC).

What to remember about the EITC:

• The EITC is a large cash-benefit program for poor households with children.

• It introduces a number of “concave kinks” in the budget constraint, similar to what a progressive labor income tax system would look like.

• Main positive effect: it increases the labor force participation rates among the targeted groups (i.e. the “extensive margin”).

• Possible negative effect: it decreases the number of hours worked among working people (i.e. the “intensive margin”).

32 / 32

econ412_lecture5_handout.pdf

ECON 412 - Penn State University

Labor Economics and Labor Markets

Lecture 5

Ewout Verriest

Spring 2020

Last lecture

Last lecture, we covered:

• Comparative statics: How does (C∗,L∗) change as we vary w or V ?

• Reservation wage: wage rate at which individual is indifferent between working and not working.

• Policy applications: comparing the effects of welfare programs, in terms of whether and how much they reduce labor supply (assuming leisure is a normal good).

1. Take-it-or-leave-it cash payment 2. Welfare program with phase-out 3. Earned Income Tax Credit (EITC)

2 / 27

Today’s Lecture

Today:

• Policy application: the Earned Income Tax Credit (EITC) • Labor supply curve • Labor supply elasticity

Reading: Borjas, Chapter 2.7, 2.8, 2.12.

3 / 27

Example 3: The Earned Income Tax Credit This is a famous policy implemented in 1975, which has since been expanded, and has been the focus of a lot of economic studies.

4 / 27

Example 3: The Earned Income Tax Credit

This is how the EITC alters the budget constraint:

5 / 27

Example 3: The Earned Income Tax Credit What to remember about the EITC:

• The EITC is a large cash-benefit program for poor households with children.

• It introduces a number of “concave kinks” in the budget constraint, similar to what a progressive labor income tax system would look like. (Why?)

• Main positive effect: it increases the labor force participation rates among the targeted groups (i.e. the “extensive margin”).

• Possible negative effect: it may decrease the number of hours worked among working people (i.e. the “intensive margin”). • Between J and E: IE and SE cause ambiguous effect on h. • Between H and J: no SE, only IE. Negative effect on h. • Between G and H: IE and SE have same sign. Negative effect on h. • Total effect depends on the population distribution of preferences!

Recommended reading: Borjas, Chapter 2-12: Policy Application on the Earned Income Tax Credit (EITC).

econ412_lecture6_handout.pdf

ECON 412 - Penn State University

Labor Economics and Labor Markets

Lecture 6

Ewout Verriest

Spring 2020

Last lecture

Last lecture, we covered:

• Policy application: the Earned Income Tax Credit (EITC) • Labor supply curve • Labor supply elasticity: concept + estimation through regression • Stats review

2 / 24

Today

Today’s lecture:

• Finalize stats review • Linear regression • Estimating the labor supply elasticity

See also Borjas, Chapter 1 (appendix), Chapter 2.7, 2.8.

3 / 24

Stats Review

Let (X,Y ) be a discrete random vector, with possible realizations {(X1,Y1), ..., (XK,YK )}. • Joint probability distribution:

P(X,Y ) = (Xk,Yk ), k = 1, ...,K

• Marginal distributions: • P(X = Xj ) = ΣKk=1P(X = Xj & Y = Yk ) • P(Y = Yj ) = ΣKk=1P(X = Xk & Y = Yj )

• Conditional distribution of Y given (i.e. “conditional on”) X = Xj :

P(Y = Yk|X = Xj ) = P(Y = Yk & X = Xj )

P(X = Xk )

4 / 24

Stats Review

Let (X,Y ) be a discrete random vector, with possible realizations {(X1,Y1), ..., (XK,YK )}. • Expectation of Y :

E (Y ) = ΣKk=1P(Y = Yk )Yk

• Conditional expectation of Y given X = Xj :

E (Y |X = Xj ) = ΣKk=1P(Y = Yk|X = Xj )Yk

This conditional expectation is the the object of a linear regression model! What is the impact of a given X on average Y ?

Note: E (Y |X ), which denotes the conditional expectation of Y given X , is also a random variable. More specifically, it takes on a realization of E (Y |X = Xj ) with probability P(X = Xj ).

5 / 24

Simple Linear Regression Model

Data: observations of (Xi,Yi ) for i = 1, ...,N. We postulate a linear relationship between X and Y :

Yi = β0 + β1Xi + ui

where ui denotes the error term.

Important assumption: E (u|X ) = 0. In words: knowing your observable characteristics (X ) does not give me any information about your unobservable characteristics (u).

Note: E (u|X ) = 0 implies corr(X,u) = 0. And vice versa: if corr(X,u) 6= 0, then E (u|X ) 6= 0.

6 / 24

Simple Linear Regression Model

How do we check the assumption that E (u|X ) = 0? 1. Think about all possible variables which might have an effect on Y ,

but which we are not including in the regression (e.g. because we do not have data on them). Call these variables u1, u2, etc.

2. Then we can rewrite the error term

u = γ1u1 + γ2u2 + ...

where γ1 6= 0, γ2 6= 0, etc. 3. Think about whether or why we expect any of these variables to be

correlated with X .

4. If even one of them might be positively or negatively correlated with X (e.g. corr(X,u1) 6= 0), then our “conditional independence” assumption that E (u|X = 0) fails!

5. In that case, our OLS estimate of our coefficient of interest (β1) would be biased, and cannot be trusted.

7 / 24

Example: test scores vs. STR

Example: Yi = test score of students in school district i, and Xi = the student-to-teacher ratio (STR) in district i.

In this case, ui encompasses everything that determines Y which is not explained by X , such as

• u1 = the average income in district i • u2 = the percentage of English learners in district i • other examples?

Is it plausible that u and X are correlated in this case? Why?

What relationship do you expect between Y and X in this case? Make a scatterplot.

8 / 24

Conditional Expectation

The assumption that E (u|X ) = 0 is very strong, but delivers exactly what we desire! To see why, take the conditional expectation of the regression equation given X = Xi , and note that (β0,β1) are population constants:

Y = β0 + β1X + u

⇒ E (Y |X = Xi ) = E (β0 + β1X + u|X = Xi ) = β0 + β1E (X|X = Xi ) + E (u|X = Xi ) = β0 + β1Xi + 0 (under cond. independence!)

= β0 + β1Xi

Then, we can perfectly predict Y for a given value of X (i.e. the left-hand side) by estimating β0 and β1.

Problem: β0 and β1 are unknown.

9 / 24

How to estimate regression coefficients

Problem: β0 and β1 are unknown.

Solution: use data on (Xi,Yi )i=1,...,N to estimate them. This is exactly

what linear regression does, by finding the parameters (β̂0, β̂1) which “best” fit the scatterplot of (Xi,Yi ) data points.

In theory, we find the estimators by solving the following “ordinary least squares” (OLS) problem:

(β̂0, β̂1) = arg min b0,b1

ΣNi=1(Yi − (b0 + b1Xi )) 2

In practice, we use statistical software to do the work for us.

E.g. in STATA, we can

• Go to Statistics → Linear models and related → Linear regression, then select Y as Dependent Variable and X as Independent variable.

• Or just simply type the command “regress Y X” in the command window.

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OLS Estimator

“Law of large numbers”: As the sample size N grows large, the OLS estimator (β̂0, β̂1) converges to the “true” values, (β0,β1).

Recall that E (u|X ) = 0 may be a strong assumption, especially if we only have 1 single X variable, X1. One common solution to make this assumption more credible is to condition on multiple variables.

Idea: there are variables in the error term u (call them X2,X3, ...,XM ) which may be correlated with X1. If we can observe those variables, then we can just add them as additional regressors:

Yi = β0 + β1X1i + β2X2i + β3X3i + ... + βMXMi + vi

where we now require the new error term, vi , to satisfy conditional independence with respect to all regressors:

E (v|X1,X2,X3, ...,XM ) = 0

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OLS Estimator

Then we obtain the OLS estimator of the conditional mean (or the predicted value) of Y for an individual i with observable characteristics (X1i, ...,XMi ):

E (Yi|X1i, ...,XMi ) = β0 + β1X1i + β2X2i + β3X3i + ... + βMXMi

The interpretation of the slope parameter β1 is as follows: If we increase X1i by 1 unit and keep (X2i, ...,XMi ) fixed (all else equal), then the average or expected value of Yi changes by β1.

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OLS Estimator

Using our sample of N observations given by (X1i, ...,XMi,Yi )i=1,...,N , we estimate the M + 1 unknown “population” quantities (β0, ...,βM ) by OLS:

(β̂0, ..., β̂M ) = arg min b0,...,bM

ΣNi=1(Yi − (b0 + b1X1i + ... + bMXMi )) 2

Outline of solution:

• Take first order conditions with respect to every parameter, b0,...,bM . • This yields a system of M + 1 equations and M + 1 unknowns. • Solving the system by hand is too difficult whenever M > 1; use

software!

• In STATA, type “regress Y X1 X2 ... XM ” in the command window. • The output will contain the OLS estimator given by (β̂0, ..., β̂M ) (as

well as the standard errors, t-stats and p-values for each slope coefficient).

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OLS: Example

Suppose we want to estimate the effect of a one unit increase in the student-to-teacher ratio on average test scores, keeping other variables (such as e.g. the average income in a district) fixed.

Run the following regression:

scoresi = β0 + β1STRi + β2incomei + ... + ui

Then β1 is the unknown population parameter we would like to know, and our estimate (which we obtain by running OLS in STATA) is denoted by β̂1.

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Estimating elasticities using OLS Suppose we are interested in estimating the elasticity of labor supply, which is the percentage change in hours worked caused by a 1% wage increase.

How can we go from absolute effects to proportional (%) effects in the linear regression model?

Take the logarithmic transformation. Property of natural logarithms:

ln(x + ∆x) − ln(x) = ln( x + ∆x

x ) = ln(1 +

∆x

x ) ≈

∆x

where the approximation holds if ∆x is small enough.

This implies that the first derivative of ln(x) can be interpreted as a percentage change of x. This is exactly what we need to calculate our elasticities!

Example: let x = 100 and ∆x = 5. Then ln(105) − ln(100) = 0.0487 ≈ 0.05, or 5%. Similarly, if ∆x = −5, then ln(95) − ln(100) = −0.0513 ≈−0.05 or −5%.

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Estimating elasticities using OLS Suppose we would estimate OLS using Y = hours worked and X1 = hourly wage in levels :

hoursi = β0 + β1wagei + β2X2i + ... + ui

Here, notice that β1 can only be interpreted in terms of absolute changes, so NOT as an elasticity:

β1 = ∂ hoursi ∂ wagei

Now, suppose we first take logs of Y and X1 and run this new regression:

ln(hoursi ) = β0 + β1 ln(wagei ) + β2X2i + ... + ui

Here, β1 can be interpreted in terms of relative or percentage changes:

β1 = ∂ ln(hoursi )

∂ ln(wagei ) =

d hoursi hoursi d wagei wagei

= %∆hoursi %∆wagei

≡ σ

where σ denotes the wage elasticity of labor supply we defined in the previous lecture.

Therefore, we can estimate elasticities by running OLS on logs! 16 / 24

Estimating elasticities using OLS

Key thing to remember: if we are running a regression where the dependent variable Y and some independent variable Xk are BOTH expressed in natural logs, then we can interpret the corresponding slope βk as the elasticity of Y with respect to Xk .

We say: Keeping all else equal, a 1 percent increase in Xk is associated with a β1 percent change in Y .

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Testing hypotheses

Often, we are interested in testing certain hypotheses, e.g. testing whether some variable Xk has a significant impact on the outcome variable Y or not.

This essentially corresponds to testing whether the “true” population parameter is either zero or different from zero:

H0 : βk = 0 versus H1 : βk 6= 0.

If we run the regression in STATA, it will provide not only the point estimates for our intercept and each slope (i.e. β̂0, β̂1, ..., β̂K ), but also

• standard errors (SEs), • t-statistics, and • the p-values associated with each slope coefficient either being 0

(H0) or different from 0 (H1). Note: p-values always lie between 0 and 1.

18 / 24

Testing hypotheses

Key thing to remember: the smaller the p-value corresponding to our estimate β̂k ), the stronger the evidence from the data that the true (unknown) coefficient βk is different from 0.

More precisely: if we test the null hypothesis H0 : βk = 0 versus the alternative hypothesis H1 : βk 6= 0, then the p-value is the probability that, if H0 is actually true, we would still estimate a slope parameter β̂k as “extreme” (or even more extreme) than the one we got based on our data.

So if p is “small”, that means it is very unlikely that our observed data were generated in a world where H0 is true. In that case, we should probably conclude that the true (unknown) slope βk is NOT equal to 0.

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Interpreting p-values

How “small” does the p-value have to be before we rule in favor of the alternative hypothesis?

In practice, we say that we reject the null hypothesis whenever the p-value is less than some threshold value of α, where we typically choose α = 0.05 or 0.01.

So if some variable Xk has an associated p-value p < α, then we are fairly confident that Xk is a strong predictor of Y .

Alternatively, if p > α, then we fail to reject the null hypothesis that βk = 0. That means that Xk is not a strong predictor of Y .

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Multiple Linear Regression Model: Recap

Consider the application where we regress weekly labor hours (h) on hourly wage (w) and weekly non-labor income (V ):

hi = β0 + β1wi + β2Vi + ui

Questions:

• When can we say that the substitution effect dominates the income effect? Answer: if β1 > 0 (keeping V constant)

• When can we say that the income effect dominates the substitution effect? Answer: if β1 < 0 (keeping V constant)

• When can we say that leisure is a normal good? Answer: if β2 < 0 (keeping w constant)

• When can we say that leisure is an inferior good? Answer: if β2 > 0 (keeping w constant)

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Multiple Linear Regression Model: Recap

Recall the often-observed backward-bending labor supply curve (with h on the horizontal axis, w on the vertical axis).

The problem with the linear equation specified on the previous slide is that we are restricting the relationship between h and w to be LINEAR, i.e. we can only have either β1 < 0 (IE always dominates), or β1 = 0 (IE always exactly offsets SE), or β1 > 0 (SE always dominates).

Question: How could we capture the possible backward-bendingness (or non-linearity) of the labor supply curve in our regression setup?

Answer: we could consider specifying h as a quadratic in w instead:

hi = β0 + β1wi + β2(wi ) 2 + β3Vi + ui

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SE vs. IE: Is there consensus? Consider the application where we regress LOG weekly labor hours (h) on LOG hourly wage (w) and weekly non-labor income (V ):

ln(hi ) = β0 + β1 ln(wi ) + β2Vi + ui

The empirical literature running this type of regression is vast, and produces very different findings regarding the sign and magnitude of the β1. There is some consensus that β̂1 ≈−0.1 is a reasonable point estimate.

Questions: • How do we interpret this slope estimate?

All else equal, a 1 percent increase in wage is associated with a 0.1 percent decrease in hours worked. On average, the IE slightly dominates the SE, and labor supply seems to be relatively inelastic (i.e. almost vertical).

• What are possible reasons for the vastly different results across empirical studies?

Different data sets (countries, sectors, cohorts, gender), different control variables, ... Important to compare apples to apples!

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Problems with regression analyses

There are several issues one should be mindful of when doing regression analyses. Two of the most important ones are:

1. Measurement error in the data (e.g. hourly wage, annual work hours)

2. Selection bias: our observed sample consists only of people who choose to work, who may be systematically different from people choosing not to work.

24 / 24

econ412_lecture7_handout.pdf

ECON 412 - Penn State University

Labor Economics and Labor Markets

Lecture 7

Ewout Verriest

Spring 2020

Last lecture

Last lecture, we covered:

• Stats review • Linear regression • Estimating elasticities • Problems with regression analysis (intro)

2 / 24

Today’s lecture

Today:

• Problems with regression analysis • Measurement error • Selection bias • Labor demand: introduction (Borjas Chapter 3)

3 / 24

Problems with regression analyses

There are several issues one should be mindful of when doing regression analyses. Two of the most important ones are:

1. Measurement error in the data (e.g. hourly wage, annual work hours)

2. Selection bias: our observed sample consists only of people who choose to work, who may be systematically different from people choosing not to work.

4 / 24

Problem 1: Measurement Error

Question: How should we measure the wage? Compensation on the margin (for one additional hour of work, e.g. overtime), or on average? What about when a worker is paid a fixed yearly salary?

Often, we define the wage as the ratio of yearly income divided by the hours worked during the whole year, to get an average hourly rate.

This average wage rate is likely measured with error, because workers don’t always recall how many hours they work per year.

If yearly labor supply h is estimated too high, then the estimated hourly wage w will be lower than what it actually is (or vice versa). This creates a negative correlation between h and w!

So assuming the IE dominates the SE (so an increase in w causes a decrease in h), then the above type of measurement error would amplify (i.e. overestimate) the IE. In other words, measurement error causes a downward bias in our slope or elasticity estimates.

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Problem 2: Selection Bias

A second common problem is selection bias, which results from the fact that researchers don’t observe wages for people who are currently unemployed.

We know from theory that the unobserved wages for these people are not just “randomly” missing or uncorrelated with everything else, since it must be the case that their (unobserved) w is less than their reservation wage. In other words, non-workers are NOT a random sample of the US workforce; they are the people who get low wage offers and/or have high reservation wages.

So if we simply eliminate “incomplete” observations from our sample resulting from unemployed workers for whom h = 0 and w is unobserved, then that will create non-random bias in our regression! In particular, we only observe wages for people who choose to work, i.e. those who got a wage offer strictly larger than their reservation wage.

6 / 24

Problem 2: Selection Bias

Consider the simple regression of labor supply on wage and non-labor income:

hi = β0 + β1wi + β2Vi + ... + ui

Question: How would selection bias translate into a violation of our typical OLS assumption that the error term (u) must be conditionally independent from all X -variables (i.e. wage, non-labor income, etc.)? Is E (u|X ) = 0 satisfied here?

Hint: can you think of a possible omitted variable in u which may be correlated with w and/or V , as well as correlated with our dependent variable, h?

7 / 24

Problem 2: Selection Bias Possible answer: consider u1 = “taste for work” as omitted variable (i.e. people with a higher value of u1 have “flatter” indifference curves, and lower reservation wages). Do you think u1 is correlated with any of the independent variables, i.e. w and/or V ?

• If u1 was high in the past, then today’s V may be high, because I’ve worked a lot and saved/invested a lot of money, which now yields dividends or interest. So corr(u,V ) > 0.

• If u1 is persistent over time (e.g. “stable preferences”), then having a high u1 would be associated with high work hours (h) today, and possibly having high w as well. So corr(u,h) > 0 and (maybe) corr(u,w) > 0.

• This leads to a violation of E (u|w,V ) = 0, and as a result, our estimate of β2 (and maybe β1 too) would be biased.

• In this case, we would likely over-estimate the true effect of non-labor income on work hours (i.e. β̂2 > β2), because u1 is positively associated with both V and h.

You could repeat the same logic for u2 = innate ability, u3 = parental income, etc.

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Multiple Linear Regression Model: Recap To summarize: if we want to explain or predict wages, hours worked, earnings etc. using OLS, we should:

1. Make sure our collected sample of individuals is representative, random, large enough (e.g. N > 30) and i.i.d.

2. Collect and include enough control variables which are possibly correlated both with our outcome variable (Y ) and our explanatory variable(s) of interest (X1, X2 etc.), because otherwise our slope estimates could be biased. Common variables of interest and/or control variables in labor economics: gender, ethnicity, age, non-labor income, years of education, years of experience, marital status, parental education, presence and number of children, standardized test scores, etc.

3. Express X and Y variables in natural logs instead of in levels, if we want to estimate an elasticity.

4. Be mindful of measurement error and selection bias. 5. Remember that many studies find that the labor supply elasticity σ ≈−0.1: negative (i.e. income effects slightly dominate) but small (i.e. inelastic). But there is a LOT of variance across samples/studies!

9 / 24

Chapter 3: Labor Demand

10 / 24

Chapter 3: Labor Demand

In the previous chapter, we have derived

• the optimal labor supply and consumption decision for the individual. • the labor supply curve: relationship between “wage” and “hours

worked” for workers.

Who pays my wage? Who produces what I consume? Firms!

Central themes for today and next couple of classes:

• introducing the new key players: (small) firms • how many workers do firms hire? • the labor demand curve: relationship between “wage” and

“number of labor hours (or employees) hired” by firms.

• how much are workers paid?

Reading: Borjas, Chapter 3.

11 / 24

The Production Function Describe the technology that the firm uses to produce goods and services:

q = f (E,K )

where

• q denotes Output, e.g. number of units produced of a certain good, • E is Hours Worked by all employees (or simply the number of

workers hired by the firm),

• K is Capital, e.g. machines, land, buildings.

Key insight: a given output level can (typically) be produced by a variety of capital and labor combinations.

Strong assumption: employees are all seemingly “identical”; output q only depends on E =

∑ i≤I hi where I denotes the number of workers in

the firm. This ignores the fact that workers may vary in their education, experience, skill etc., such that f only depends on E and not on (h1, ...,hI ). Same goes for capital K . We could relax this by having multiple types of laborers, e.g. blue and white collar workers: q = f (Eb,Ew,K ).

12 / 24

Marginal Products

The marginal product of labor (denoted MPL) is the change in output resulting from hiring an additional worker, holding capital (K) constant:

MPL = ∆q

∆E = ∂f (E,K )

∂E

The marginal product of capital (denoted MPK ) is the change in output resulting from using an additional unit of capital, holding labor (L) constant:

MPK = ∆q

∆K = ∂f (E,K )

∂K

The value of the marginal product is given by the marginal product multiplied by the price per unit of output (p):

VMPL = p ∗MPL, VMPK = p ∗MPK

13 / 24

Average Product

The average product of labor (denoted APL) is the output produced by the “typical” worker:

APL = q

E =

f (E,K )

Typically, APL 6= MPL. Why?

Not every employee affects output in the same way (even though they may be identical in every respect)! The first ones you hire may be very productive, but as you hire more, they eventually become less productive (e.g. due to spatial constraints, redundancy etc.).

14 / 24

Example: Marginal vs. Average Product of Labor

15 / 24

Example: Marginal vs. Average Product of Labor

Law of Diminishing Returns: The Marginal Product of Labor eventually has to decline.

16 / 24

The Production Function: Example

Consider a simple echnology which requires only labor to produce output:

q = f (E ) = AEα

where A and α denote two “technology parameters”. Assume A > 0 and α ∈ (0, 1).

Assume that the firm is “small”, and takes the output price (p) and the hourly wage (w) as given (as opposed to e.g. a monopolist).

Questions:

1. What is the marginal product of labor here?

2. What is the firm’s profit function?

3. Write down the firm’s first order condition, and solve for the firm’s labor demand (E ) as a function of known parameters (p,w,A,α).

4. All else equal, what happens to labor demand if the wage goes up?

17 / 24

Example: Solution

1. MPL = dq dE

= AαEα−1 = Aα E 1−α

. Since α ∈ (0, 1), MPL is everywhere decreasing in E .

2. Profits are given by revenues minus input costs (note: there is only one input in this simple example):

π(E ) = pq − wE = pAEα − wE

3. The firm’s FOC with respect to labor is:

dπ(E )

dE = 0 ⇐⇒ p ∗ MPL − w = 0

⇐⇒ pAαEα−1 = w

⇐⇒ Eα−1 = w

pAα

⇐⇒ E∗ = ( w pAα

) 1 α−1

or E∗ = (pAα

) 1 1−α

where 1 1−α > 1.

4. From the closed form solution in the last line, we know that the optimal labor demand function (E∗(w,p,A,α)) is decreasing (and convex) in w. Makes sense! How about other comparative statics?

18 / 24

The Firm’s Problem Now consider the more general case where capital is also a productive input. The firm optimally chooses inputs to maximize profits, π:

max E,K

π(E,K ) = max E,K

p ∗ f (E,K ) −w ∗E − r ∗K

where

• p = selling price per unit of output • w = wage rate = price of labor • r = (rental) price of capital

Note: we are making a very important market structure assumption: Assumption: All markets (for inputs and outputs) are perfectly competitive. All prices (p,r,w) are exogenous parameters, and not affected by the firm’s actions.

When would this assumption be violated?

19 / 24

The Firm’s Problem: Long vs. Short Run

We consider two scenarios of the firm’s problem:

1. Short run problem: consider capital to be fixed at K = K0. The firm chooses only E to maximize profits π(E,K0). Capital cannot easily be changed in the short run (e.g. build a new factory, buy or sell specialized equipment).

2. Long run problem: choose E and K simultaneously to maximize profits, π(E,K ). All inputs can now be freely chosen.

20 / 24

Scenario 1: Profit Maximization in the Short Run

21 / 24

The Firm’s Problem: Short Run Solution In the short run, there’s only a hiring decision, since capital remains fixed at its short-run level, K = K0. The firm thus maximizes

π(E,K0) = p f (E,K0) −w ∗E − r ∗K0 We solve the problem by first taking the first order condition (FOC) of the profit function with respect to labor:

∂π(E,K0)

E = 0

⇐⇒ p ∂f (E,K0)

∂E = w

⇐⇒ p ∗MPE = w ⇐⇒ VMPE = w

Intuition: the firm keeps hiring labor until the point where the marginal value created by hiring an extra unit (e.g. one worker for one hour) becomes equal to the marginal cost of that unit.

That is why this condition is also called the marginal productivity condition.

22 / 24

Other conditions for optimality

Important: the First Order Condition (FOC) derived on the previous slide is only necessary for profit maximization, but not sufficient! We also need to check the following two conditions:

• whether we are at a profit maximum and not at a minimum ⇒ Second Order Condition.

• whether profits are actually higher at the optimum than if the firm would not be active.

To verify that we have a profit maximum and not a minimum, we also need the Second Order Condition (SOC) to hold:

∂2π(E,K )

∂2E < 0 or, equivalently:

∂MPE ∂E

< 0

In practice, this sufficient condition implies that we only care about the decreasing part of the MPE (or VMPE ) curve.

Example: take f (x) = (x − 1)2. The first order condition is satisfied at x = 1, but this yields a minimum, not a maximum!

23 / 24

Other conditions for optimality Finally, we have to verify whether the firm actually prefers operating at all, or prefers inactivity (i.e. choosing E = 0) instead.

In the latter case, we would be at a corner solution for labor demand, since E = 0.

The interior solution (where E = E∗ > 0 and q = q∗ > 0) is better than the corner where E = q = 0 if:

π(E∗,K0) > π(0,K0)

⇐⇒ pq∗ −wE∗ − rK0 > −rK0 ⇐⇒ pq∗ −wE∗ > 0

⇐⇒ p q∗

E∗ > w

⇐⇒ VAPE > w

So the firm will be “active” if the value of the average product of labor (VAPE ) exceeds the wage level. Intuition: rK0 can be treated as a “sunk cost”. If pq < wE , then the firm is better off picking E = 0, i.e. the boundary solution.

24 / 24

econ412_lecture8_handout.pdf

ECON 412 - Penn State University

Labor Economics and Labor Markets

Lecture 8

Ewout Verriest

Spring 2020

Last lecture

Last lecture, we covered:

• Issues with linear regression: measurement error and selection bias • Labor demand: introduction of the firm as a new player • Marginal vs. average product of inputs • Scenario 1: Short-run profit maximization

2 / 20

Today’s lecture

Today:

• Recap of short-run firm’s problem • Labor demand elasticity • Issues with aggregation • Comparative statics • Numerical example: Problem 3.10 from Borjas • Long-run firm’s problem

3 / 20

The Firm’s Problem in the Short Run In the short run, the firm takes capital K = K0 as fixed and only chooses the optimal amount of labor E to solve

max E π(E,K0) = p f (E,K0) − wE − rK0

Recap of how we find the solution:

1. Marginal productivity condition: find all values of E that solve the FOC of profits with respect to labor:

VMPE = w or equivalently: p ∂f (E,K0)

∂E = w

2. Second order condition (SOC): check whether we are at a maximum, not a minimum. Verify that profits are concave in E , or equivalently, verify that MPE is decreasing in E :

∂2π(E,K0)

∂E 2 < 0 or equivalently:

∂MPE

∂E < 0

(Note: eventually, MPE always starts to decrease because of congestion, redundancy, etc.)

3. Activity condition: Check that the solution from Steps 1 and 2 gives larger profits than the boundary solution where E = 0: verify whether π(E∗,K0) > π(0,K0), or equivalently whether VAPE > w. Treat the cost of financing capital (rK0) as “sunk”, since we are paying rent for our (fixed) stock of capital irrespective of how many units of output we produce.

4 / 20

The Firm’s Problem in the Short Run Do NOT confuse marginal and average products!

• If APE is increasing, it means that MPE > APE , and vice versa. • The profit maximizing level of E has to satisfy VMPE = w (due to the FOC). • The profit maximizing level of E has to be on the decreasing part of the VMPE

curve (due to the SOC).

• Case 1: Suppose w = 22. The FOC is satisfied at E = 1 and at E = 8. However, the SOC is violated at E = 1, but is satisfied at E = 8. We can see in the graph that VAPE > w at E = 8, so the activity condition is also satisfied. The firm optimally chooses to be active and hire 8 workers.

• Case 2: Now suppose w = 38. Choosing E = 4 would satisfy the FOC and SOC, but not the activity condition since VAPE < w! The firm optimally chooses not to be active and hire 0 workers.

5 / 20

Short-run Labor Demand

Note: we assume that the firm is a price-taker in both the output market as well as the input markets (e.g. the labor market), such that it takes all prices (w,p,r) as exogenous.

Moreover, in the short run, we can also treat the capital stock K0 as a fixed parameter!

By solving the firm’s problem for varying levels of the wage w, we can derive the firm’s short-run labor demand function:

E∗(w; p,r,K0, f )

where f denotes the (parameters of the) production function, q = f (E,K ).

6 / 20

Comparative Statics in the Short Run

After having derived the firm’s short-run labor demand function, we can consider various comparative statics exercises:

• Assume the output price p increases. How does labor demand change?

If p goes up, the VMPE = p ∗ MPE increases for every level of w, i.e. the labor demand curve shifts to the right. At any given wage w, the firm now wants to hire more workers. In exercises, you could verify mathematically that E∗ is increasing in p.

7 / 20

Comparative Statics in the Short Run

• Assume the capital stock K0 goes up. How does labor demand change?

It depends on the production technology (i.e. the shape of f (K,E ))! See also next classes.

If labor and capital are complements in production, then the MPE (and therefore VMPE ) increases in K0. E.g. if there are two machines in the factory, then an additional worker could increase output by more than she would if there were only one machine present. In that case, an increase in K0 would cause labor demand (VMPE ) to shift to the right. For any given wage w, the firm would hire more workers, because E and K can “reinforce” each other.

Conversely, if labor and capital are substitutes in production, the opposite would happen. If the capital stock K0 increases (e.g. the firm buys robots who can do the same tasks as human workers), then the firm would reduce their demand for labor, i.e. labor demand shifts to the left. For any given wage w, the firm would hire fewer workers, because E and K can “replace” each other.

8 / 20

Comparative Statics in the Short Run Finally, we can also look at the effects of a change in the production function, f . Consider a production function of the Cobb-Douglas form:

q = f (E,K ) = AEαK 1−α

In this popular specification, the first parameter A > 0 is called “total factor productivity” (TFP), and the second parameter α ∈ (0, 1) captures how important labor is in production, relative to capital. For example, if A = 1 and α = 1

2 , then

q = √ EK .

• Consider an increase in A. Why would TFP ever increase? Think of any technological advance which increases output (q) even while keeping all input levels (L,K ) constant, e.g. the invention of the internet, streamlining the production process, using efficiency-enhancing software, hiring consultants to improve your factory, etc.

Clearly, an increase in A increases MPE , and shifts labor demand to the right.

• Similarly, the relative productivity of labor (α) could increase by having a more educated or stronger workforce, setting up a more labor-intensive (and less capital-intensive) production, having a longer (mandatory) work week, etc.

• Verify for yourself that under a Cobb-Douglas technology, E∗ is increasing in A and α, as we would expect.

• Similarly, output and profits are also increasing in TFP, A. This is why technological progress spurs economic growth!

9 / 20

Aggregation of labor demand Now that we can solve each individual firm’s problem, how can we aggregate them up and find e.g. the whole industry’s labor demand?

Recall that in Chapter 2, we could find total labor supply H(w) just by “horizontally” adding up the individual labor supply curves.

Aggregating labor demand curves is trickier! Example: see Figure 3-4 in Borjas.

Industry demand curve (TT) is steeper than the sum of the individual firm demand curves (DD). Why? The problem is that we cannot “keep all else equal”! If the market wage (w) goes down, then every firm would hire more workers (E ) and produce more output (q). This increase in aggregate product supply would lead to a decrease in the market price p (Micro 101!). This would in turn lead to a (small) drop in every firm’s labor demand, i.e. a shift to the left of VMPE . This is a “general equilibrium” effect!

10 / 20

Wage elasticity of labor demand To characterize the effect of a wage change on labor demand, we define the elasticity of labor demand in the short run (SR) as

δSR = %∆ESR %∆w

= ∆ESR ESR ∆w w

= ∂ESR ∂w

ESR

where ESR denotes the short run employment of an industry.

In words: δSR denotes the percentage change in employment in the short run when the wage changes by 1 percent. For example, if 30 workers are employed at w = 20, and 56 workers are employed at w = 10, then

δSR = 56−30

30 10−20

= +86.6%

−50% = −1.73

If |δSR| > 1, then we say that labor demand is “elastic”. If −1 ≤ δSR ≤ 1, then labor demand is “inelastic”. Fun fact: for any Cobb-Douglas production function of the form q = AEαK 1−α, the wage elasticity of labor demand is exactly 1, i.e. neither elastic nor inelastic.

11 / 20

Wage elasticity of labor demand

Question: Consider a car manufacturer (hiring assembly line workers) versus a specialized hospital (hiring surgeons). Where would labor demand be more elastic? Why? How does the production technology differ in both sectors?

(Possible) answer: The car manufacturer probably finds it easier to replace workers by machines (capital) in the event of a wage increase, or might even outsource production entirely to another (cheaper) country. However, the hospital’s technology of production is more labor-intensive and cannot easily substitute surgeons by robots or machines. Therefore, we would expect the demand for manual laborers to be more elastic (i.e. the VMPE curve to be relatively flatter) than the demand for surgeons (which would be relatively steep).

12 / 20

Firm’s Short-Run Problem: Numerical Example Let f (E,K ) =

√ EK , w = 10, r = 25, p = 0.50, and K0 = 1600.

Question: Find the optimal choice of E in the short run, and the resulting profits. Show your work (including verifying all conditions for optimality).

Note: if f (x) = √ x = x

1 2 , then f ′(x) = 1

2 x−

1 2 = 1

2 √

x , and

f ′′(x) = −1 4 x

−3 2 = −1

4x √ x < 0.

More generally, if f (x,y) = xαyβ, then ∂f ∂x

= αxα−1yβ = α yβ

x1−α . It is easy to see

that if α ∈ (0, 1), then ∂f ∂x

is decreasing in x, so ∂ 2f ∂2x

< 0.

Solution:

• Write down profit function: π(E,K ) = p

√ EK − wE − rK

Note: Best to plug in parameter values for (w,r,p,K0) only at the end, especially if there are some “comparative statics” (sub)questions.

• First, have to derive and solve FOC with respect to labor: ∂π(E,K0)

∂E = 0 ⇐⇒ VMPE = w ⇐⇒

(K E

) 1 2

= w

⇐⇒ √ E =

p √ K0

2w ⇐⇒ E∗ =

p2K0

4w 2

where plugging in w = 10, p = 0.5 and K0 = 1600 yields E ∗ = 1.

• Note: In the short run, there is no FOC with respect to capital, which is fixed! 13 / 20

Firm’s Short-Run Problem: Numerical Example

• Second, we have to verify that profit is maximized (and not minimized) at E∗ = 1. This means checking the Second Order Condition (SOC): is the VMPE decreasing at E = 1?

Why do we need to check this condition? For certain functional forms of f (E,K ), you may find multiple values of E which satisfy the first order condition (see also Figure 3-2 in Borjas). In that case, check the SOC at every value of E which solves the FOC.

Since VMPE = p 2

( K E

) 1 2

is always decreasing in E (since E is in the

denominator), the SOC is satisfied in this case.

14 / 20

Firm’s Short-Run Problem: Numerical Example • Third, we have to verify the activity condition: choosing E = E∗ = 1 (i.e. the

value that solves the FOC) must yield higher profits than choosing the corner solution where E = 0.

Note that the portion rK0 of the profit expression is sunk, and does not matter in the short run, since capital (and therefore all costs connected to owning that capital) are fixed (i.e. even if you don’t produce, you have to pay the rental cost of your machines, buildings etc.).

At the corner where E = 0, we get an output q = √

0 ∗ 1600 = 0 (Note: this is not always the case; e.g. consider q = f (E,K ) =

√ E + √ K ).

In this case, profits at the corner where E = 0 are

π(0, 1600) = 0 − 0 − rK0 = −rK0 = −40000

Conversely, if we choose E = E∗ = 1, we produce q = √

1 ∗ 1600 = 40 units, and get a profit of

π(1, 1600) = 0.50 ∗ 40 − 10 ∗ 1 − 40000 = −39990

Therefore, the interior solution is strictly better than the corner, so we choose to hire 1 worker and produce 40 units.

• In conclusion, the firm hires E = 1 workers and makes a loss of π(1, 1600) = −39990 dollars. This loss is still better than the loss if they would not produce at all, so it is in fact optimal to be active in the short run.

15 / 20

Scenario 2: Profit Maximization in the Long Run

Read Borjas, Chapter 3.3, 3.4.

16 / 20

The Firm’s Problem: Long Run Solution

In the long run, the firm maximizes profits by choosing how many workers to hire AND how much to invest in capital goods (plants, equipment, etc.).

Mathematically, the firm’s problem now becomes:

max E,K

π(E,K ) = p f (E,K ) −wE − rK

subject to the obvious constraints that E ≥ 0 and K ≥ 0. As before, we assume the firm is a price taker in all markets, such that prices (p,w,r) are exogenously given and not affected by the firm’s individual decisions.

17 / 20

The Firm’s Problem: Long Run Solution

We can solve the long-run problem by taking first order conditions of π(E,K ) with respect to the two choice variables, E and K , which yields a system of 2 equations:

∂π(E,K )

∂E = 0 ⇐⇒ p ∗MPL = w ⇐⇒ p =

MPL ∂π(E,K )

∂K = 0 ⇐⇒ p ∗MPK = r ⇐⇒ p =

MPK

Intuitively: hire workers until the marginal cost of an additional worker (w) is equal to the value of their marginal product, VMPL = p ∗MPL.

Similarly: hire capital until the marginal cost of an additional unit (r) is equal to the value of its marginal product, VMPK = p ∗MPK .

18 / 20

The Firm’s Problem: Long Run Solution

By combining the FOCs on the previous slide (and equating to p), we find that

MPL =

MPK ⇐⇒

MPL MPK

= w

We define the marginal rate of technical substitution as

MRTS = MPL MPK

which is equal to the slope of an isoquant: a line giving all combinations of (E,K ) which produce the same level of output.

19 / 20

The Firm’s Problem: Long Run Solution

Preview for next class:

To analyze the long-run firm’s problem economically, we can provide an analogous derivation to the one when studying the individual labor supply problem, except

• putting employment E and capital K on the axes instead of leisure L and consumption C ,

• replacing “utility maximization” by “cost minimization”, • replacing “indifference curves” by “isoquants”, • replacing “budget constraint” by “isocost line”. • replacing “marginal rate of substitution” (MRS) by “marginal rate of technical substitution” (MRTS).

20 / 20

econ412_lecture9_handout.pdf

ECON 412 - Penn State University

Labor Economics and Labor Markets

Lecture 9

Ewout Verriest

Spring 2020

Last lecture

Last lecture, we covered the “short-run” analysis where the firm’s capital stock is fixed:

• The firm’s problem • Labor demand • Comparative statics • Aggregation of labor demand • Wage elasticity of labor demand • Numerical example

2 / 23

Today’s Lecture

Today, we cover the “long-run” firm’s problem when capital is not fixed:

• Isoquants • Isocost lines • Long-run profit maximization

Read: Borjas, Chapter 3.3, 3.4.

3 / 23

Review: Short-run Analysis Short-run analysis: Keep capital fixed at K = K0.

Firm solves: max E

p f (E,K0) −wE − rK0

1. Necessary condition for maximum = the “marginal productivity condition” or FOC:

VMPE = w

where VMPE ≡ MPE ∗p denotes the value of marginal product of labor.

2. Second order condition: verify VMPE is decreasing in E .

3. Activity condition: Check whether the interior solution (E∗ > 0) is preferred over the corner solution where E = 0.

4 / 23

Review: Short-run Analysis By varying the wage rate (w) and re-solving the firm’s optimization problem, we can trace out the labor demand curve of the firm:

Notes:

• Any change of a parameter other than w may cause a shift of the VMPE curve. • Comparative statics: output price p, initial stock K0, technology parameters. • Aggregation issues: To obtain labor demand of the entire industry for a given w,

one cannot simply “horizontally” add up labor demand of each firm. If we did, then we would miss the implication that a decrease in w might lead to a decrease in the market price, p, because all firms are now hiring and producing more.

5 / 23

The Firm’s Problem: Long Run Solution

In the long run, the firm maximizes profits by choosing how many workers to hire AND how much to invest in capital goods (plants, equipment, etc.).

Mathematically, the firm’s problem now becomes two-dimensional:

max E,K

π(E,K ) = p f (E,K ) −wE − rK

6 / 23

Isoquants An isoquant is a line representing all factor combinations which yield a given, constant output level.

In the simple twodimensional space with only two inputs, E and K : Definition: An isoquant at level q0 is the set of all combinations of (E,K ) such that f (E,K ) = q0.

where q1 > q0, i.e. higher isoquants are associated with higher output. 7 / 23

Isoquants

In general, isoquants have the following properties:

• downward sloping. This is because MPE > 0 and MPK > 0: a reduction in K requires an increase in E to compensate for lost production.

• they do not intersect (similar to the argument we made for indifference curves).

• higher isoquants correspond to higher output levels. • continuous: labor and capital can be “smoothly” adjusted. • convex relative to the origin: the marginal productivity of each

factor declines. (i.e. ∂MPE ∂E

< 0 and ∂MPK ∂K

< 0).

8 / 23

Slope of an isoquant In the previous graph, moving from point X to point Y requires

• an increase in labor by some amount, ∆E > 0 • a decrease in capital by some amount, ∆K < 0

Since we are staying on the same isoquant, it must be the case that the total change in output is zero:

∆q = MPE ∗ ∆E + MPK ∗ ∆K = 0

⇐⇒ ∆K

∆E = −

MPE MPK

In other words, in any given point (E,K ), the slope of the isoquant through that point is given by the ratio of the marginal products!

Definition: The absolute value of the slope of an isoquant is also called the marginal rate of technical substitution, or MRTS:

MRTS = MPE MPK

9 / 23

Marginal Rate of Technical Substitution

Important remarks:

• The MRTS is the firm’s equivalent of the MRS we saw in chapter 2!

• The convexity condition (which stated that MPE decreases with E ) implies that the MRTS is decreasing with E . In words: as the firm replaces more capital by labor, the isoquant becomes flatter and the MRTS becomes smaller (i.e. gets closer to 0).

10 / 23

Isocost Lines

On the one hand, isoquants represent what the firm can produce given the technology it has.

On the other hand, the firm must also take into account the factor prices of each input, in our case the wage w and the rental rate of capital r.

At a given combination of inputs (E,K ), the firm’s costs are given by

C0 = wE + rK

We can rewrite this to

K = C0 r −

r E

which represents a line in the twodimensional (E,K ) space, with

• intercept on the vertical (K ) axis: C0 r

• intercept on the horizontal (E ) axis: C0 w

• slope equal to (minus) the ratio of input prices: −w r

11 / 23

Isocost Lines

K = C0 r −

r E

This is what we call an isocost line: all input combinations (E,K ) which yield the same total cost level, C0. Higher lines correspond to higher costs.

12 / 23

The Firm’s Problem: Long Run Solution

Recall that in the long run, the firm solves

max E,K

π(E,K ) = p f (E,K ) −wE − rK

We can solve the long-run problem by taking first order conditions of π(E,K ) with respect to the two choice variables, E and K , which yields a system of 2 equations and two unknowns:

∂π(E,K )

∂E = 0 ⇐⇒ p ∗MPE = w ⇐⇒ p =

MPE ∂π(E,K )

∂K = 0 ⇐⇒ p ∗MPK = r ⇐⇒ p =

MPK

Intuitively: hire workers until the marginal cost of an additional worker (w) is equal to the value of their marginal product, VMPE = p ∗MPE .

Similarly: hire capital until the marginal cost of an additional unit (r) is equal to the value of its marginal product, VMPK = p ∗MPK .

13 / 23

Optimality condition

By combining the FOCs on the previous slide (and equating to p), we find the following necessary condition for optimality:

MPE =

MPK ⇐⇒

MPE MPK

= w

Given our previous definitions, this means that, at the optimum, we need the marginal rate of technical substitution (MRTS = MPE

MPK ) to be equal

to the slope of the isocost line ( w r

14 / 23

Cost Minimization In words, this necessary condition requires that at whatever output level the firm wants to produce (say, q0 units), the isoquant corresponding to that level of production should be tangent to the isocost line. This implies that the firm produces those q0 units at the lowest cost possible:

If the firm wants to produce q0 units, then point P is optimal, since it produces the desired output level at lower cost than points A or B.

15 / 23

Profit maximization However, this optimality condition (MRTS = w

r ) just tells us how to

minimize costs at a given output level q0. It does not inform us what that output level should be! Which isoquant should the firm produce at?

Let us (again) rewrite the two first-order conditions:

p = w

MPE and p =

MPK

Intuitively, the firm should aim to equalize these three things:

1. marginal increase in revenues when producing an additional unit: p

2. marginal increase in labor costs needed to produce that unit: w MPE

3. marginal increase in capital costs needed to produce that unit: r MPK

Taken together, the firm should hire labor and capital up to the point where the marginal revenue of selling an additional unit of output (p) is equal to the marginal cost of producing that unit. In order to produce that unit, we should either hire w

MPE extra workers, or hire r

MPK extra

capital units.

16 / 23

Counterfactual analysis

We can also perform comparative statics exercises in the long run. We look at how changes in exogenous parameters (i.e. w,r,p or technology) affect endogenous quantities, i.e. the demand for labor E∗, the demand for capital K∗, the optimal output q = f (E∗,K∗), and optimal profits π(E∗,K∗).

Suppose the wage rate w decreases. What happens to the optimal allocation of labor and capital?

Recall from labor supply analysis in Chapter 2, where we decomposed the overal effect of a wage change into an income effect and a substitution effect.

For firms, we will analogously decompose the total effect of a wage change into two parts:

• Scale effect • Substitution effect

econ412_midterm2_examplequestions_solutions.pdf

ECON 412 - Penn State University

Labor Economics and Labor Markets

Midterm 2 - Example Questions

Ewout Verriest

Spring 2020

Question 1

Assume the labor supply curve is given by w = E 2

+ 1 and the labor demand curve by w = −E 2

+ 4, where E stands for employee-hours (or number of workers) and w is the wage rate.

a) Calculate the competitive equilibrium (E∗,w∗), the worker surplus and the producer surplus. Make a graph, clearly labeling all axes and relevant points.

b) Now assume the government assesses a $1 payroll tax on firms for every employee-hour. Calculate the worker surplus, the producer surplus, and the government’s tax revenue. How large is the deadweight loss caused by the tax? Make a graph.

c) Now assume instead that the government makes firms provide a mandated benefit to workers that costs firms $1 per worker and that workers value at $1. What is the resulting equilibrium? How does it differ from the one you found in part a)?

d) Suppose there is an economic downturn, such that unemployment rate is deemed too high. De- scribe how the government could raise employment to a level that exceeds the one reached in the competitive equilibrium.

Answer:

a) • The intersection between supply and demand (found by equating the right-hand sides of each equation) yields an equilibrium employment level of E∗ = 3, and hence an equilibrium wage of w∗ = 2.5.

• The intercept on the vertical axis of the labor demand curve (where E = 0) is 4, such that the producer surplus is equal to a triangle with area P =

(4−2.5)∗3 2

= 2.25.

• The intercept on the vertical axis of the labor demand curve is 1, such that the worker surplus is equal to a triangle with area Q =

(2.5−1)∗3 2

= 2.25.

• Total welfare gains are therefore 2.25 + 2.25 = 4.5.

b) • Graph: similar to Figure 4-4 (or Figure 4-7) in Borjas. Because of the tax on firms, labor demand shifts down vertically by $1 everywhere, such that the new demand curve is given by w = −E

2 + 3 (i.e. at every level of E, the VMP of labor ( or equivalently, the wage rate

w) is now 1 dollar lower than before because of the payroll tax).

• The new equilibrium is therefore given by the intersection of the new demand curve and the old labor supply curve, which occurs at Etax = 2. The resulting equilibrium wage received by workers is wnet =

Etax 2

+ 1 = 2, which is less than before. However, the total cost of labor for firms is now wtot = wnet + 1 = 3, which is higher than before.

• The new producer surplus is now equal to a triangle with area P ′ = (4−wtot)∗Etax 2

= 1. Hence, the firm loses 1.25 in profits because of the tax. Note: this corresponds to the surface area underneath the original labor demand curve, and above the horizontal line given by the total labor cost, w = wtot = 3.

• The new worker surplus is now equal to a triangle with area Q′ = (wnet−1)∗Etax 2

= 1. Hence, the workers lose 1.25 in surplus because of the tax. Note: this corresponds to the surface area above the labor supply curve, and below the horizontal line given by the worker’s received wage, w = wnet = 2.

• The government’s tax revenue is equal to a rectangle with area T = Etax ∗ 1 = 2. Note: the height of this rectangle is equal to the “wedge” between the total firm cost wtot = 3 and the worker’s net wage wnet = 2.

• Total gains after the payroll tax are P ′ + Q′ + T = 4. Total gains before the tax were 4.5, so the deadweight loss is 0.5. In other words: both firms and workers are losing 1.25 in surplus (so 2.5 total), but the government gets 2 units in tax revenue. The remaining surplus loss (or the DWL) caused by the tax is 0.5. Hence, payroll taxes reduce the total “size of the pie” in this case.

• Alternatively, we could calculate the deadweight loss as the size of a triangle with “base” equal to the size of the wedge (i.e. the tax, equal to 1) and a “height” equal to the change in employment (i.e. E∗ −Etax = 3 − 2 = 1, which gives a surface area DWL = 1∗12 = 0.5.

c) • The mandated benefit costs the firm C = 1 and is valued by the worker at B = 1. This benefit reduces (shifts down) labor demand by C = 1, and increases (shifts down) labor supply by B = 1. The new labor supply curve is then w = E

2 + 1 − 1 = E

2 , and the new labor demand

curve is then w = −E 2

+ 4 − 1 = −E 2

+ 3.

• The new equilibrium in the presence of a Mandated Benefit (MB) is then given by EMB = 3, which is the same as in part a). The new equilibrium wage rate is wMB = 1.5, which is exactly one dollar less than the wage we found in part a).

• The worker’s total compensation package is given by wall = wMB + B = 1.5 + 1 = 2.5, which is the same as in part a). Since workers get a pay cut that’s equal to the benefit they receive, their welfare is not affected by the MB.

• Similarly, the firm’s total labor cost is given by wtot = wMB + C = 1.5 + 1 = 2.5, which is the same as in part a). Therefore, the equilibrium has essentially not changed; the only difference is that now workers get part of their salary in the form of a mandated benefit, which they value as much as if it had been paid out in money. Similarly, firms now have lower wage costs, but their total costs (wages + benefits) remain the same. Worker surplus, firm surplus and total welfare does not change compared to part a).

d) The government could provide a subsidy for firms, i.e. a negative payroll tax on firms. This would keep labor supply unchanged, and would shift the labor demand curve upwards by the amount of the subsidy. Unless labor supply is fully inelastic (i.e. vertical), this would lead to an increase in employment. Alternatively: the government could force firms to offer a mandated benefit for which B > C (see also Homework 3, Problem 1). If you believe e.g. “health insurance” is an example of a

mandated benefit for which the cost for firms is less than how much the workers value it, then mandated health insurance could also increase employment.

Question 2

Consider the market for risky labor. Denote the wage in safe jobs by w0, and the wage in risky jobs by w1. Answer the following questions in detail, and add a graph if needed.

a) Explain the concept of “reservation price”, and how it relates to risk aversion.

b) Discuss the intuition behind why the labor supply curve for risky labor is upward sloping. What is the interpretation of the intercept?

c) Discuss the intuition behind why the labor demand curve for risky labor is downward sloping. What is the interpretation of the intercept?

d) Make a graph of the worker’s surplus in the market for risky labor, and explain what it means.

e) Could it ever be the case that, in equilibrium, the compensating wage differential for riskiness (w1 −w0)∗ is negative? Why? Give an example. (See also below: Question 4, part b).

f) Explain the concept of the “hedonic wage function”, and how we could estimate it using data for a large sample of individuals in various sectors or professions. What kinds of variables do we need to collect? Write down the regression you would run. What is the main coefficient of interest, and what sign would you expect it to be? Note: this last question relates to material we will see after the 2nd Midterm (Lectures 18 and 19), so you don’t have to know the answer to this yet. However, it will become relevant for the final exam.

Answer: For more information: most of the answers can be found in Lecture 17.

a) A worker’s reservation price corresponds to the “bribe” required to make a worker exactly indif- ferent between taking the safe job and taking the risky job. As workers get more risk averse (i.e. have steeper indifference curves in the two-dimensional (ρ,w)-space), their reservation price goes up, as they require higher bribes to make them willing to accept more risk. Graph: See Borjas Figure 5-1, or Lecture 17.

b) If we put (w1−w0) on the vertical axis and employment in the risky sector on the horizontal axis, the intercept of the labor supply curve corresponds to the reservation price of the least risk-averse (or most risk-loving) worker. As the compensating differential goes up, more and more workers (who become increasingly risk-averse) are willing to take the bribe and work in the risky sector, which causes labor supply to be upward sloping.

c) If the compensating wage differential for risk (w1 −w0) is very high, then none (or very few) firms would find it optimal to provide a risky work environment; they would instead prefer to create a safe work environment and pay their employees a lower wage, w0. However, as (w1 −w0) goes down, then more and more firms will find it optimal not to invest in safety, but instead to provide a risky work environment and pay their employees a higher wage w1. Hence, the demand for risky labor is downward sloping.

d) The worker’s surplus is given by the triangle “above” the labor supply curve and “below” the equilibrium wage differential, (w1−w0)∗, where supply meets demand. Each employee in the risky sector gets a welfare surplus equal to the difference between the compensating wage differential that the market is willing to pay them (i.e. (w1 − w0)∗), and their individual reservation price (given by their respective point on the supply curve). In equilibrium, only the marginal worker is indifferent between the safe and the risky job; everyone else is strictly better off in utility terms.

e) In theory: yes; if there is a fraction of risk-loving (as opposed to risk-averse) workers for whom the reservation price is negative (i.e. they actually strictly prefer the risky job even if it paid the same wage as the safe job, i.e. they have downward sloping indifference curves in the (ρ,w) space), and if the demand for risky labor is sufficiently small, then the equilibrium risk premium could be negative (see also Borjas, Figure 5-3 for an example on how to draw this).

f) See Borjas chapter 5.2 and Figure 5-6. The hedonic wage function (in the context of “risk” as the relevant job characteristic) captures the correlation between wages and injury rates, which should be positive given our model on compensating differentials. In equilibrium, risk-averse people choose safe firms who pay less, whereas less risk-averse people optimally choose to work for risky firms, who optimally pay more in exchange for not having to invest in safety measures. Estimation: If we regress wages (or annual salaries) on injury rates (and add some controls for education, age, gender etc.), we would expect a positive slope coefficient, capturing the positive compensating wage differential for riskiness. See also Lecture 18 for more specifics on how to write down the regression equation.

Question 3

Discuss the impact of immigration on the aggregate wealth (or surplus) of native workers and firms, (i) in the short run, and (ii) in the longer run after native workers and firms have had time to respond.

Answer: See also our discussion on immigration in Lectures 14 and 15.

• For simplicity, consider the case where immigrant workers are perfect substitutes for native workers. In the short run, immigration shifts out labor supply in the host country, leading to lower wages (from w0 to w1), and increased total employment (from N0 to E1). Employment among native workers goes down from N0 to N1 < N0, and hence their welfare surplus decreases drastically due to the drop in their wages and employment.

• In the short run, firms hire more workers and get cheaper labor, so their surplus goes up.

• In the longer run, firms respond to cheaper labor costs by expanding (scale effect) and by switching their input mix towards more labor (substitution effect), both of which increase the demand for labor. Moreover, “mobile” firms from other regions may move to take into account of cheaper labor costs in the region where immigrants arrived, which further pushes up labor demand. At the same time, “mobile” workers may respond by moving away to other (higher-paying) regions, which would reduce labor supply in the region or market where the immigrants first entered. In equilibrium, all of these forces would cause wages (in all markets) to return to their original levels.

• Hence, according to our model, immigration only has a temporary impact on the welfare of native workers. After firms and workers adjust, native workers’ welfare is the same as before, and firms’ welfare is permanently higher (since they produce more output but face the same wage costs as before).

Question 4

Suppose there are 100 workers in the economy in which all workers must choose to work a risky or a safe job. Workers have heterogeneous preferences over risk and wages, such that worker 1’s reser- vation price for accepting the risky job is $1; worker 2’s reservation price is $2, and so on. Because of technological reasons, there are only 10 risky job positions in the economy, which are demanded inelastically.

a) Make a graph of the supply and demand for risky labor in this world? What is the equilibrium wage differential between safe and risky jobs? (Note: there could be a range of possible answers here). Describe the equilibrium forces at work here. Which workers will be employed at the risky firm?

b) Suppose now that an advertising campaign, paid for by the employers who offer risky jobs, stresses the excitement associated with “the thrill of injury”, and this campaign changes the attitudes of the work force toward being employed in a risky job. Worker 1 now has a reservation price of −10 (that is, she is willing to pay $10 for the right to work in the risky job); worker 2’s reservation price is −9, and so on. There are still only 10 risky jobs. What is the new equilibrium wage differential?

Answer:

a) The supply curve to the risky job is given by the fact that worker 1 has a reservation price of $1, worker 2 has a reservation price of $2, and so on. As the figure below illustrates, this supply curve (given by S) is upward sloping, and has a slope of 1. The demand curve (D) for risky jobs is perfectly inelastic at 10 jobs. Market equilibrium is attained where supply equals demand so that 10 workers are employed in risky jobs; the market compensating wage differential is $10 (or, some number weakly larger than $10 and strictly smaller than $11) since this is what it takes to entice the marginal (tenth) worker to accept a job offer from a risky firm. Equilibrium forces: If the compensating wage differential were less than $10, then only 9 people would apply for 10 jobs, creating upward pressure on the risky wage. If the wage differential were $11 or above, then 11 workers would be applying for only 10 jobs, creating downward pressure on the wage. The firm employs those workers who least mind being exposed to risk, i.e. workers 1,2,...,10 whose reservation prices are all less than or equal to $10.

b) Now, workers’ preferences have been altered in such a way that everyone’s reservation price (RP) has decreased by $11. Now worker 1’s RP is -10, worker 2’s RP is -9, worker 10’s RP is -1, worker 11’s RP is 0, etc. Because tastes towards risk change, the supply curve shifts downdwards (by 11 units) to S′ and the market equilibrium is attained when the compensating wage differential is -$1 (or any number between -1 and 0). This is the compensating differential required to hire the marginal worker (that is, the 10th worker whose reservation price is equal to −1). Note that this compensating differential implies that even though most workers (from worker 12 onwards) still dislike risk (i.e. have a positive RP), the market now determines that risky jobs will actually pay less than safe jobs. If the number of available risky jobs were any higher than 10, then the equilibrium compensating wage differential would no longer be negative. Note: This is an example of a scenario where the compensating wage differential for risk may actually be negative, which was asked above in Question 2, part e.

Question 5

Suppose all workers have identical preferences represented by

U = √ w − 2x

where w is the wage and x is the proportion of the firm’s air that is composed of toxic pollutants. There are only two types of jobs in the economy: clean jobs (x = 0) and dirty jobs (x = 1). Let w0 be the wage paid by the clean job and w1 be the wage paid for doing the dirty job. If the clean job pays $16 per hour, then what is the wage in dirty jobs? What is the compensating wage differential?

(Hint: since all workers are identical, labor supply will be perfectly elastic!) Answer: If all persons have the same preferences regarding working in a job with polluted air,

then everyone has the same reservation price, which we defined as the wage increase that makes people indifferent between the clean and the dirty job. Market equilibrium requires that the equilibrium wage differential is equal to the reservation price of the “marginal” individual. Since everyone is identical here, this means that equilibrium requires that the utility offered by the clean job be the same as the utility offered by the dirty job, otherwise all workers would move to the job that offers the higher utility. This implies that:

U(safe) = √ w0 − 2 ∗ 0 = U(dirty) =

√ w1 − 2 ∗ 1 ⇒

√ w1 = 4 + 2 = 6

Solving for w1 implies that w1 = 36. The compensating wage differential, therefore, is $20 as the risky job pays $36 per hour and the clean job pays $16 per hour. In equilibrium, every (identical) worker is indifferent between clean and dirty jobs. Irrespective of what the labor demand curve looks like, the equilibrium compensating wage differential will be set at $20, since labor supply is flat.