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Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Macroeconomic Theory Lecture Notes

II. A One-Period Model of the Macroeconomy

Chunzan Wu University of Miami

October 4, 2017

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Outline

The Representative Household

The Representative Firm

Competitive Equilibrium

Applying the Theory: Income Tax and Labor Supply

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

A One-Period Model of the Macroeconomy

In this part of the course, we construct a one-period model of the market economy that can be used to analyze some key macroeconomic issues.

There are three types of economic agents in this economy: households, firms and a government. Households are workers, consumers, and also the owners of firms. Firms hire workers for production, sell the output to consumers, and transfer all the profits to their owners. The government taxes households to finance certain government expenditures.

For simplicity, we assume all the households and firms are identical, so we can analyze the economy as if there are only one representative household and one representative firm. The representative household and the representative firm behave competitively, i,e, they are price-takers. They treat market prices as given and acts as if their actions have no effect on those prices.

Following the approach of modern macroeconomics, we first state explicitly the choice problems faced by the representative household and the representative firm, and derive their behavior for given prices and other economic conditions by solving their optimization problems. We then impose the equilibrium conditions to find the prices such that every market clears (i.e., supply equals demand). Finally, we discuss the effects of government tax policy using our model.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Outline

The Representative Household

The Representative Firm

Competitive Equilibrium

Applying the Theory: Income Tax and Labor Supply

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Modeling the Representative Household

How many hours a week should a household work? If the household works more, then: • it earns more income and can purchase more goods and services (sweaters, tv sets,

hair cuts, etc.); • it has less time for leisurely activities (hanging out with friends, playing with the kids,

watching TV, etc.).

In the one-period model, we focus on the basic trade-off between consumption and leisure, which determines the labor supply in the economy. In short, the representative household decides how much to consume and how much to work by choosing the best feasible option according to its preferences.

• How to represent the household’s preferences over consumption and leisure? • How to represent the household’s budget constraint which defines the set of feasible

options? • Given market prices, how the household’s consumption and labor supply are

determined? • How does the household respond to a change in nonwage income and to a change in

the market wage rate?

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

The Representative Household’s Preferences

It is simple to suppose that there are two goods that the household desire:

1. consumption good: an aggregation of all consumer goods in the economy.

2. leisure: any time spent not working in the market.

The household’s preferences over consumption goods and leisure can be represented by a utility function written as

U(C,l)

where C is the quantity of consumption and l is the quantity of leisure. We say a consumption bundle (C1, l1) is strictly preferred by the household to (C2, l2) if

U(C1, l1) > U(C2, l2);

(C2, l2) is strictly preferred to (C1, l1) if

U(C1, l1) < U(C2, l2);

and the household is indifferent between the two bundles if

U(C1, l1) = U(C2, l2).

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Assumptions on the Household’s Utility Function

We assume that the utility function is monotonic and concave. • Monotonicity: For all (C2, l2) and all (C1, l1) such that C2 ≥ C1, l2 ≥ l1, and either C2 > C1 or l2 > l1, we have

U(C2, l2) > U(C1, l1)

When the utility function is monotonic, the household prefers more to less of either good (consumption and leisure).

• Concavity: For all (C2, l2), (C1, l1) and all 0 < α < 1, we have

U(αC1 + (1 −α)C2,αl1 + (1 −α)l2) > αU(C1, l1) + (1 −α)U(C2, l2).

When the utility function is concave, the household prefers consuming a little bit of both goods rather than consuming a lot of one and very little of the other.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Indifference Curves

It is helpful to consider the household’s preferences using a graphical representation of the utility function, called indifference curves. An indifference curve connects a set of points with these points representing consumption bundles among which the household is indifferent.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Note that the indifference curves in the figure have two key properties:

1. The indifference curves are downward sloping.

2. The indifference curves are convex (i.e., bowed-in toward the origin).

These two properties of indifference curves are directly linked to the two assumptions we made on the utility function.

1. The negative slope of indifference curves is due to the monotonicity of utility function. When leisure increases (x-axis), the household is strictly better off if consumption (y-axis) remains the same due to the monotonicity assumption. Hence, to keep the household’s utility level constant as required by the definition of the indifference curve, consumption must fall.

2. The convexity of indifference curves is due to the concavity of utility function. The concavity of utility function tells us that any consumption bundle that is within the line segment between two different points on the same indifference curve must make the household strictly better off, i.e., be on an indifference curve with higher utility level. Hence, the shape of the indifference curve must be convex.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Marginal Rate of Substitution

We define the marginal rate of substitution of leisure for consumption, denoted MRSl,c as the rate at which the consumer is just willing to substitute leisure for consumption goods, i.e., the number of units of consumption goods that just compensate the household for losing one unit of leisure. We have

MRSl,C = −[the slope of the indifference curve passing through (l,C)]

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Note: The convexity of the indifference curve implies that MRSl,c is decreasing as leisure increases.

Deriving the formula for MRSl,c: 1. The change in the household’s utility from losing ∆l units of leisure is

∆Ul = U(C,l− ∆l) −U(C,l) ≈−U2(C,l)∆l;

where U2(C,l) = ∂U(C,l)

∂l is the partial derivative of utility function with respect to

the second argument, i.e., leisure. 2. The change in the household’s utility from increasing ∆C units of consumption is

∆UC = U(C + ∆C,l) −U(C,l) ≈ U1(C,l)∆C;

where U1(C,l) = ∂U(C,l)

∂C is the partial derivative of utility function with respect to

the first argument, i.e., consumption. 3. To keep the household’s utility unchanged, we must set ∆l and ∆C such that

∆Ul + ∆UC = −U2(C,l)∆l + U1(C,l)∆C = 0

⇒ ∆C

∆l = U2(C,l)

U1(C,l) = MRSl,C

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Set of Feasible Choices

We say that a consumption bundle (C,l) belongs to the set of feasible choices (or, simply that it is feasible) if the household can afford it given its endowment and the market prices.

• The household is assumed to have h hours of time available, which can be allocated between leisure time l and time spent working (or labor supply), denoted by Ns. So the time constraint for the household is

l + N s

= h

• There is no money in this economy, so we choose the consumption good as numeraire, i.e., the good in which all prices and quantities are denominated. By this choice, the price of the consumption good is always 1. On the other hand, labor time is traded in the labor market at a price w (i.e., the real wage, or the wage rate in units of the consumption good). So if the household sells Ns working time in the labor market, it receives wNs wage income.

• The representative household is the owner of the representative firm in this economy, so it also receives dividend income (i.e., the profits earned by the firm) denoted by π.

• Finally, the household pays taxes to the government. We assume a lump-sum tax, denoted by T , which is a tax that does not depend in any way on the actions of the economic agent who is being taxed.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Now we can write down the budget constraint faced by the representative household. Because this is a one-period economy, the household has no motive to save. The household’s disposable income (i.e., labor income+dividend income - taxes) must be greater or equal to the total expenditures (i.e. consumption). Hence we have

C ≤ wNs + π −T.

From the time constraint of the household, we know Ns = h− l. Substituting it for Ns, we get

C ≤ w(h− l) + π −T.

Alternatively, we can rewrite the budget constraint as

C + wl ≤ wh + π −T.

An interpretation to this form of the budget constraint is that the right-hand side is the implicit disposable income (i.e., the disposable income the household would receive by selling all of its endowment), and the left-hand side is implicit expenditures on the two goods, consumption and leisure. (wl is what is implicitly “spent” on leisure.)

Note: Because time cannot be negative, for a consumption bundle (C,l) to be feasible, it must satisfy the constraint 0 ≤ l ≤ h in addition to the budget constraint. Similarly, because consumption cannot be negative, we also need C ≥ 0.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

We can graph the set of feasible consumption bundles in the (l,C) space. When the budget constraint holds with equality, we have

C = w(h− l) + π −T = (−w)l + (wh + π −T). This is a straight line in the (l,C) space with slope −w, vertical intercept wh + π −T , and horizontal intercept, h + π−T

w . Any point that is on or below this straight line and

satisfies C ≥ 0 and 0 ≤ l ≤ h is feasible. • When T > π, the set of feasible (C,l) bundles is the shaded area below.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

• When T < π, the set of feasible (C,l) bundles is the shaded area below.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Household Optimization

We have described the representative household’s preferences over consumption bundles (consumption and leisure) and the set of feasible bundles (determined by the budget constraint and nonnegative constraints). Now we need to model how the household makes choices given these.

We assume that the household will choose the most preferred consumption bundle that is feasible, i.e., the feasible consumption bundle that maximizes the household’s utility. Formally, the household’s choice (C∗, l∗) is the solution to the following maximization problem

max (C,l)

U(C,l)

s.t. C ≤ w(h− l) + π −T C ≥ 0, 0 ≤ l ≤ h

Graphically, this is saying the household will choose the consumption bundle that is on the highest possible indifference curve and is within the feasible set.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Graphical Solution to the Household’s Problem

Given our assumptions on the household’s preferences (monotonicity and concavity), if there exists a point where an indifference curve is tangent to the budget constraint, and it satisfies all the nonnegative constraints (i.e., C ≥ 0, 0 ≤ l ≤ h), it is then an interior solution to the household’s optimization problem.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Note that at such tangent point, i.e., the optimal consumption bundle (C∗, l∗), two optimality conditions must hold:

1. The budget constraint holds with equality:

C ∗

= w(h− l∗) + π −T.

This means the household must spent all of its disposable income with nothing left.

2. The marginal rate of substitution of leisure for consumption (the slope of the indifference curve in absolute value) must equal the real wage (the slope of the budget constraint in absolute value):

MRSl,C(C ∗ , l ∗ ) = w,

where MRSl,C(C∗, l∗) is the marginal rate of substitution at the consumption bundle (C∗, l∗). MRSl,C represents the household’s private price of leisure, i.e., how many units of the consumption good the household is willing to give away in exchange for one unit of leisure, and the real wage w is the market price of leisure, i.e., how many units of the consumption good the household needs to pay for one unit of leisure. This equation states that at the optimum, the household’s private price of leisure must equal the market price of leisure.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

If the tangent point violates some nonnegative constraints, then it is not a solution to the household’s optimization problem, and we will have a corner solution. For example, the household may choose not to work at all (or not to have leisure at all).

At such corner solution, the first optimality condition about the budget constraint still holds, but the second optimality condition about MRSl,C is no longer true.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Analytic Solution to the Household’s Problem

First, because we assume that the utility function U(C,l) is monotonic, the household’s optimal choice (C∗, l∗) must satisfy the budget constraint with equality. That is

C ∗

= w(h− l∗) + π −T.

Proof. Suppose this is not the case, then it must be

C ∗ < w(h− l∗) + π −T,

and the household has unspent disposable income. Now let

C ∗′

= w(h− l∗) + π −T > C∗,

and compare the bundle (C∗′, l∗) with (C∗, l∗). Both bundles are feasible and have the same level of leisure l∗. Since C∗′ > C∗, by the monotonicity of utility function, we have U(C∗′, l∗) > U(C∗, l∗). Hence we have find a feasible bundle (C∗′, l∗) that is strictly preferred by the household to the optimal bundle (C∗, l∗), and this is a contradiction.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Because we know at the solution to the household’s problem, the budget constraint will hold with equality, we can replace the inequality constraint with the equality constraint.

max (C,l)

U(C,l)

s.t. C = w(h− l) + π −T C ≥ 0, 0 ≤ l ≤ h

If we ignore the nonnegative constraints for now, this is a constrained maximization problem with one equality constraint, which can be solved by the method of Lagrange multipliers. The associated Lagrangian is

L = U(C,l) −λ [C −w(h− l) −π + T]

where λ is the Lagrange multiplier. The necessary conditions for the solution are

∂L

∂C = U1(C,l) −λ = 0, (1.1)

∂L

∂l = U2(C,l) −λw = 0, (1.2)

∂L

∂λ = − [C −w(h− l) −π + T] = 0. (1.3)

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

With some additional technical conditions, these necessary conditions are also sufficient. Now we have three equations and three unknowns (C,l,λ), so we can simply solve this system of equations for the solution to the household’s problem.

One way to proceed is to simplify the 3 × 3 system to a 2 × 2 system first by eliminating the λ’s. Notice that Equation (1.3) is just the budget constraint, and it has no λ in it. Combining Equation (1.1) and (1.2), we have the second equation

U1(C,l)w = U2(C,l) (1.4)

The left-hand side of Equation (1.4) is the marginal benefit of one extra unit of labor supply: one extra unit of labor supply earns the household w extra labor income which can be consumed to generate U1(C,l)w extra utility. The right-hand side of Equation (1.4) is the marginal cost of one extra unit of labor supply: one extra unit of labor supply reduces the leisure by one unit which lowers the utility by U2(C,l). Equation (1.4) simply states that at the optimum the marginal benefit of labor supply must equal the marginal cost.

Using our previous formula for MRSl,C, we can rewrite Equation (1.4) as

MRSl,C = U2(C,l)

U1(C,l) = w

And this is the same optimality condition we get from the graphical solution.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Now we can find a potential solution (C∗, l∗) to the household’s optimization problem by searching for the value of (C,l) that satisfies Equation (1.3) and (1.4) simultaneously.

However, remember that we have so far ignored all the nonnegative constraints, so the last step is to check if this potential solution (C∗, l∗) satisfies them as well. If the answer is yes, we have an interior solution; if the answer is no, it is then not a solution, and we must search for corner solutions.

Remarks: 1. Once we have the necessary conditions, it is a matter of taste how to solve the system

of equations. In some cases and for some people, it may be easier to solve for the λ first rather than eliminating it first as in the notes.

2. Since we only have one equality constraint here, it is possible and also not too difficult to solve the household’s problem with the method of substitution rather than the method of Lagrange multipliers. For example, we can use the budget constraint to substitute for C in the utility function, and transform the problem into an unconstrained maximization problem with respect to only l. However, the method of substitution is less applicable when we have more constraints and choice variables.

3. By making additional assumptions on the utility function (and the production function in the future), we can make sure that in the end, there is an interior solution to the household’s problem.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Comparative Statics

Now we know how the representative household chooses the level of consumption and leisure. We can do comparative statics and investigate how the household’s choice is affected by changes in economic environment.

Question 1: How does the representative household respond to a change in nonwage income?

To answer this question, we need some additional assumption on the household’s preferences: that is, both the consumption good and leisure are normal goods. A good is normal (or inferior) if the quantity of the good purchased by a household increases (or decreases) with the household’s income.

By the definition of normal goods, we know that both consumption and leisure should increase when the household receives more nonwage income such as an increase of π −T in our model.

The change of household behavior resulting from a pure income change is called income effect.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

The effects of an increase in real dividend income minus taxes (π −T ) is illustrated in the figure below. Note that an increase of leisure means a decrease of labor supply, so a higher income reduces the labor supply of the household.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Question 2: How does the representative household respond to a change in the real wage?

The effects of an increase in the real wage on the household’s behavior can be broken down into two components:

• Substitution effect: With a higher real wage, the household tends to choose more consumption and less leisure because leisure is more expensive.

• Income effect: With a higher real wage, the household tends to choose more consumption and more leisure because the household’s time endowment is worth more. (Note that we assume here both consumption and leisure are normal goods.)

Therefore, the household with a higher real wage certainly chooses more consumption. It also chooses more leisure if the income effect dominates the substitution effect. It chooses less leisure if the opposite is true.

Formally, consider an increase of real wage from w1 to w2. Consequently, the household’s choice changes from (C1, l1) to (C2, l2). We can decompose the whole change into two steps:

1. Let wage increase from w1 to w2, but simultaneously decrease the household’s nonwage income π −T to π̂ −T such that the household will choose a consumption bundle (Ĉ, l̂) that is on the same indifference curve as the original bundle (C1, l1), i.e., U(Ĉ, l̂) = U(C1, l1).

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

2. Keep the new wage w2, but restore the nonwage income to the original level π −T . By definition, the household will choose (C2, l2).

The change from (C1, l1) to (Ĉ, l̂) is the substitution effect, and the change from (Ĉ, l̂) to (C2, l2) is the income effect.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Labor Supply Curve

The labor supply curve describes how the representative household’s labor supply changes with the real wage, holding everything else constant (in particular, the nonwage income).

N s (w) = h− l(w)

Assuming that the substitution effect is larger than the income effect, labor supply increases with the real wage. Also, from the income effect, we know an increase of nonwage income shifts the labor supply curve to the left.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Outline

The Representative Household

The Representative Firm

Competitive Equilibrium

Applying the Theory: Income Tax and Labor Supply

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

The Representative Firm’s Technology

Now we turn to the behavior of the representative firm, which hires labor and produces the consumption good. The choice of the firm is determined by the available technology and by profit maximization.

The technology available to the representative firm is like the budget constraint to the representative household: it determines the set of feasible choices. The technological process of transforming inputs into outputs can be represented by a production function:

Y = zF(K,N d )

where z is total factor productivity (a measure of overall level of technology), Y is output of the consumption good, K is the quantity of capital input (plant and equipment) in the production process, Nd is the quantity of labor input measured as total time worked by employees of the firm, and F is a function.

In this economy, the representative firm owns productive capital. Because this is a one-period (or static) model, we treat capital K as being a fixed input (or exogenous), and labor Nd as being a variable input (or endogenous).

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Marginal Product

We define the marginal product (denoted by MP ) of a factor of production as the additional output that can be produced with one additional unit of that factor input, holding constant the quantities of the other factor inputs. Formally, the marginal product of labor is

MPN = ∂Y /∂N d

= zF2(K,N d )

and the marginal product of capital is

MPK = ∂Y /∂K = zF1(K,N d )

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Assumptions on the Production Function

1. Constant returns to scale (CRS): for all (K,Nd) and given any constant x > 0,

zF(xK,xN d ) = xzF(K,N

d )

This assumption means that the output of a firm with two workers and two computers is the same as the output of two firms with one worker and one computer each. Alternatively, a production function exhibits increasing returns to scale (IRS) if

zF(xK,xN d ) > xzF(K,N

d ), for all x > 1;

or decreasing returns to scale (DRS) if

zF(xK,xN d ) < xzF(K,N

d ), for all x > 1;

The constant-returns-to-scale assumption allows us to analyze the economy as if there is only one representative firm because the size distribution of firms no longer matters, provided that all firms behave competitively.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

2. Monotonicity: for all (K2,Nd2 ) and (K1,N d 1 ) such that K2 ≥ K1, Nd2 ≥ Nd1 , and

either K2 > K1 or Nd2 > N d 1 , we have

zF(K2,N d 2 ) > zF(K1,N

d 1 ).

This assumption means output increases when either the capital input or the labor input increases. In other words, the marginal products of labor and capital are both positive:

MPN = zF2(K,N d ) > 0 and MPK = zF1(K,N

d ) > 0.

3. Decreasing marginal product: for all Nd1 , N d 2 and K with N

d 2 > N

d 1 , we have

MPN(K,N d 2 ) < MPN(K,N

d 1 ).

Similarly, for all K1, K2 and Nd with K2 > K1, we have

MPK(K2,N d ) < MPK(K1,N

d ).

This assumption means the marginal product of a factor input (labor or capital) decreases as the quantity of the factor input (labor or capital) increases:

∂MPN/∂N d

= zF22(K,N d ) < 0 and ∂MPK/∂K = zF11(K,N

d ) < 0.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

4. The marginal product of labor increases as the quantity of the capital input increases (and vice versa): for all K1, K2 and Nd with K2 > K1, we have

MPN(K2,N d ) > MPN(K1,N

d ).

Similarly, for all Nd1 , N d 2 and K with N

d 2 > N

d 1 , we have

MPK(K,N d 2 ) > MPK(K,N

d 1 ).

This assumption means the marginal product of a factor input (labor or capital) increases as the quantity of the other factor input (capital or labor) increases:

∂MPN/∂K = zF21(K,N d ) > 0 and ∂MPK/∂N

d = zF12(K,N

d ) > 0.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Our assumptions on the production function imply that the marginal product of labor is always positive, decreasing with the quantity of labor input, and increasing with the quantity of capital input.

The most common representation of technology in macroeconomics is the Cobb-Douglas production function:

Y = z(K) α (N

d ) 1−α

, 0 < α < 1.

The marginal products are then

MPN = (1 −α)z(K/Nd)α, MPK = αz(Nd/K)1−α.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

The Effect of a Change in Total Factor Productivity

Changes in total factor productivity (TFP), z, are critical to our understanding of the causes of economic growth and business cycles, and so we must understand how a change in z alters the production technology.

• An increase in TFP shifts the production function up: Y = zF(K,Nd); • An increase in TFP increases the marginal product of labor and capital: MPN = zF2(K,N

d) and MPK = zF1(K,Nd).

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Profit Maximization Problem of the Representative Firm

The goal of the representative firm is to maximize its profits, given by

Y −wNd,

where Y is the total revenue that the firm receives from selling its output, in units of the consumption good, and wNd is the total real cost of the labor input. Then, substituting for Y using the production function, the firms problem is

max Nd

zF(K,N d ) −wNd, s.t. Nd ≥ 0.

Ignoring the nonnegative constraint on labor input (i.e., assuming an interior solution), this is an unconstrained maximization problem. The necessary condition for the solution is

zF2(K,N d ) −w = 0,

i.e., MPN = w.

Hence at the optimum, the firm chooses the quantity of labor input such that the marginal product of labor equals the real wage.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Graphical Solution to the Profit Maximization Problem

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Labor Demand Curve

Our analysis tells us that, given a real wage w, the representative firm optimally hires the quantity of labor input such that MPN = w. Because the marginal product of labor schedule tells us how much labor the firm needs to hire such that MPN = w, the marginal product of labor schedule and the firm’s demand curve for labor are the same thing.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Outline

The Representative Household

The Representative Firm

Competitive Equilibrium

Applying the Theory: Income Tax and Labor Supply

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Government

We have already described the behavior of the representative household and the representative firm, and now we explain what the government does in this economy.

The behavior of the government is quite simple in this model: it wishes to purchase a given quantity of consumption goods, G, and finances these purchases by taxing the representative household through a lump-sum tax, T .

The government must abide by the government budget constraint, which we write as

G = T.

Introducing the government in this way allows us to study some basic effects of fiscal policy. In general, fiscal policy refers to the governments choices over its expenditures, taxes, transfers, and borrowing. Because this is a one-period economic environment, the governments choices are very limited. For example, the government cannot borrow to finance government expenditures and repay its debt later. The only element of fiscal policy in this model is the size of government expenditures, G.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Exogenous and Endogenous Variables

An exogenous variable is determined outside the model, while an endogenous variable is determined by the model itself.

In the model we are working with here, the exogenous variables are G, z, K, and h—that is, government spending, total factor productivity, the economy’s capital stock, and the household’s time endowment, respectively. The endogenous variables are C, l, Ns, Nd, T , Y , π, and w–that is, consumption, leisure, labor supply, labor demand, taxes, aggregate output, dividend income, and the market real wage, respectively.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Competitive Equilibrium

We model the behavior of aggregate economy as a competitive equilibrium at which: • Households make choices to maximize utility subject to their budget constraints

taking prices as given. • Firms make choices to maximize profits subject to their technology constraints taking

prices as given. • Government collects taxes and purchases goods subject to the government budget

constraint taking prices as given. • Prices are such that all the markets clear, i.e., demand equals supply in all markets.

Remarks: We can think that equilibrium prices are determined as in an auction. The auctioneer names a price for a certain good or service (labor, consumption, etc.). Given the price, buyers report to the auctioneer the amount of good they would like to buy. Given the price, sellers report to the auctioneers the amount of good they would like to sell. If aggregate demand is greater than aggregate supply, the auctioneer increases the price. If aggregate supply is greater than aggregate demand, the auctioneer reduces the price. When aggregate supply and demand are equal, the price clears the market and all the transactions are executed at that price.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Definition of Competitive Equilibrium for the One-Period Economy

Given the fundamentals of the economy, i.e. utility function U, production function F , and exogenous variables G (government spending), z (total factor productivity), K (capital stock), and h (time endowment), a competitive equilibrium is a set of the representative household’s consumption and leisure choice (C∗, l∗), the representative firm’s labor demand Nd∗ and profits π∗, the government’s lump-sum tax T∗, and the real wage w∗

such that 1. Given the real wage w∗, profits π∗, and lump-sum tax T∗, (C∗, l∗) solves the

representative household’s problem:

max (C,l)

U(C,l)

s.t. C ≤ w∗(h− l) + π∗ −T∗

C ≥ 0, 0 ≤ l ≤ h; 2. Given the real wage w∗, Nd∗ solves the representative firm’s problem:

max Nd

zF(K,N d ) −w∗Nd, s.t. Nd ≥ 0,

and π ∗

= zF(K,N d∗

) −w∗Nd∗;

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

3. Government budget constraint holds:

G = T ∗ ;

4. Markets clear: • The labor market:

h − l∗ = Nd ∗

• The consumption good market:

zF(K, Nd ∗ ) = C∗ + G

Remarks:

1. Note that in the household and firm’s problems, the prices and other endogenous variables that are not chosen by the current optimizing agent are set at the equilibrium values (i.e., with the ∗ superscript).

2. For simplicity, we have substitute zF(K,Nd) and h− l for the endogenous variables Y (output) and Ns (labor supply) in the definition of competitive equilibrium.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Solving for the Competitive Equilibrium

Given the definition of competitive equilibrium in the one-period economy, solving for the competitive equilibrium is equivalent to finding the values of (C,l,Nd,π,T,w) that satisfy all the equilibrium conditions.

From our previous analysis of the representative household’s problem, we conclude that under certain conditions, the following two conditions are necessary and sufficient for the solution to the household’s problem (i.e., the optimality conditions for the representative household):

MRSl,C = U2(C,l)

U1(C,l) = w, (3.1)

C = w(h− l) + π −T. (3.2)

Similarly, from the representative firm’s problem, under certain conditions, the necessary and sufficient condition for the solution (i.e., the optimality condition for the representative firm) is

MPN = zF2(K,N d ) = w. (3.3)

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Because at the competitive equilibrium, both the household and the firm should be optimizing, Equation (3.1), (3.2) and (3.3) must hold at the competitive equilibrium. Combining these optimality conditions with the definition of profits

π = zF(K,N d ) −wNd, (3.4)

the government budget constraint G = T, (3.5)

and the market clearing conditions

h− l = Nd, (3.6) zF(K,N

d ) = C + G, (3.7)

we have 7 equations ((3.1) to (3.7)) that must be satisfied at the competitive equilibrium in 6 unknown endogenous variables (C,l,Nd,π,T,w). In general, there could be no solution to such system of equations. But it turns out that one of these 7 equations is redundant:

Walras’ Law: Consider an economy with N markets. If N − 1 markets clear and all agents satisfy their budget constraints, then the Nth market clears as well.

So we can drop one market clearing condition, and solve the remaining 6 × 6 system of equations to find the competitive equilibrium.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Which market clearing condition to drop and how to proceed with solving the 6×6 system of equations is a matter of taste. Without loss of generality, let us drop the market clearing condition for the consumption good market here, i.e., Equation (3.7). It is easy to verify that Equation (3.7) is implied by Equation (3.1) to (3.6) as suggested by Walras’ Law.

To simplify the system of equations, let us first use Equation (3.5) and (3.6) to substitute for all the T ’s and Nd’s in the equations. Then we have the firm’s profits

π = zF(K,h− l) −w(h− l).

Substituting π in Equation (3.2) with this formula, we have

C = w(h− l) + zF(K,h− l) −w(h− l) −G = zF(K,h− l) −G, ⇒ C + G = zF(K,h− l). (3.8)

Combining Equation (3.1) with Equation (3.3) to eliminate the wage w, we have

U2(C,l)

U1(C,l) = zF2(K,h− l),

⇔ MRSl,C = MPN. (3.9)

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Now we have simplified the 6 × 6 system into a 2 × 2 system: Equation (3.8) and (3.9) in (C,l). The solution to this 2 × 2 system tells us the equilibrium level of consumption C∗ and leisure l∗. The equilibrium values of other endogenous variables can then be recovered as the following:

N d∗

= N s∗

= h− l∗, T ∗

= G,

w ∗

= zF2(K,h− l∗), π ∗

= zF(K,h− l∗) −w∗(h− l∗) = F(K,h− l∗) −zF2(K,h− l∗)(h− l∗),

Y ∗

= zF(K,h− l∗).

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Graphical Representation of the Competitive Equilibrium

We have derived that the competitive equilibrium is eventually pinned down by Equation (3.8) and (3.9) together.

Equation (3.8) is simply a resource constraint and we can draw it in the (l,C) space. The resulting curve is called a production possibilities frontier (PPF) because it describes what the technological possibilities are for the economy as a whole, in terms of the production of the consumption good and leisure.

From Equation (3.8), the production possibilities frontier is

C = zF(K,h− l) −G,

and we can derive the slope of PPF as

dl = −zF2(K,h− l) = −MPN.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

The negative of the slope of the PPF is called the marginal rate of transformation (MRT), which is the rate at which one good can be converted technologically into another. In this economy, it is the same as the marginal product of labor:

MRTl,C = MPN.

Equation (3.8) tells us that at the competitive equilibrium, the economy must be on the production possibilities frontier.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Equation (3.9) tells us that at the competitive equilibrium,

MRSl,C = MPN.

Recall that MRSl,C is the negative of the slope of the indifference curve, and MPN = MRTl,C is the negative of the slope of the production possibilities frontier. So the competitive equilibrium must be a tangent point between the indifference curve and the production possibilities frontier.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Outline

The Representative Household

The Representative Firm

Competitive Equilibrium

Applying the Theory: Income Tax and Labor Supply

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Labor Income Tax and Labor Supply

We have developed a general equilibrium model of the aggregate economy, and we can now apply this model to study practical issues.

Given that our model is featured with the endogenous labor supply of households, we focus on two interesting questions related to the labor income tax:

• Why do Americans work so much more than Europeans? • How much tax revenue the government can generate through labor income tax?

Labor income is the dominating source of household income, so we can roughly equate the labor income tax with the income tax in general. (This may not be true for households at the very top of the income distribution.)

We explore these questions in a modified version of our one-period model. In particular, we introduce a proportional labor income tax, and assume this is the only source of income for the government. The government transfers all the tax revenue back to the representative household in a lump-sum fashion, and there are no other government expenditures. For simplicity, we also assume away capital in this economy.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

A Modified One-Period Model with Labor Income Tax

We assume household’s utility function takes the following form:

U(C,l) = log C −ψ (h− l)1+

1 η

1 + 1 η

where ψ is a parameter controlling the disutility of supplying labor, and η is a parameter controlling the elasticity of labor supply to wage changes (i.e., the Frisch elasticity of labor supply).

The household’s problem is:

max (C,l)

log C −ψ (h− l)1+

1 η

1 + 1 η

s.t. C ≤ (1 − t)w(h− l) + π + T C ≥ 0, 0 ≤ l ≤ h

where t is the labor income tax rate. Note that T is now a positive lump-sum transfer to the household.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Assume that output is produced only with labor as an input, and the production function is

Y = zF(K,N d ) = N

d .

The firm’s problem is: max Nd

N d −wNd, s.t. Nd ≥ 0.

The government’s budget constraint is:

tw(h− l) = T.

We can define a competitive equilibrium for this modified economy in a similar way as we did for the standard one-period economy. The exogenous variables are (h,t), and the endogenous variables are (C,l,Nd,π,T,w). A competitive equilibrium is then a set of values for (C,l,Nd,π,T,w) such that:

• Given (w,π,T), (C,l) solves the household’s problem; • Given w, Nd solves the firm’s problem, and π equals the profit of the firm; • Given w,l, T balances the government’s budget; • The consumption good and labor markets clear.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

The equilibrium conditions are then:

• The household’s optimality conditions:

(1 − t)w C

= ψ(h− l) 1 η ,

C = (1 − t)w(h− l) + π + T;

• The firm’s optimality condition:

1 −w = 0, π = N

d −wNd;

• The government budget constraint:

tw(h− l) = T;

• The market clearing conditions:

h− l = Nd, N d

= C.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

We can drop one market clearing condition by Walras’ Law and solve the remaining 6 × 6 system of equation for the competitive equilibrium:

C =

( 1 − t ψ

) η 1+η

l = h− (

1 − t ψ

) η 1+η

N d

( 1 − t ψ

) η 1+η

π = 0,

T = t

( 1 − t ψ

) η 1+η

w = 1

We can see that the equilibrium allocations such as household consumption and labor supply depend on the tax policy, which is represented by the labor income tax rate t in the model.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Why Do Americans Work So Much More than Europeans?

This is the question discussed by Edward C. Prescott in an influential paper published in 2004.

Fact: There is a large difference between the U.S. and Europe with respect to the average number of hours worked by people aged 15-64:

• U.S.: 25.9 hours per week; • Germany: 19.3 hours per week; • France: 17.5 hours per week; • Italy: 16.5 hours per week.

What does our modified one-period model say about the household’s hours worked? We have derived that at the equilibrium, the household’s working time is

h− l = (

1 − t ψ

) η 1+η

Therefore, based on our model, hours worked h− l decreases with the labor income tax rate t, and the differences in hours worked between the U.S. and Europe can potentially be explained by differences in labor income tax rates.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Note that how much differences in hours worked can be explained by differences in income tax rates depends crucially on the parameter value of η, ie., the Frisch’s elasticity of labor supply. The elastiticy of hours worked h− l with repect to 1 − t is simply

d log(h− l) d log(1 − t)

= η

1 + η .

If η is zero, hours worked does not respond to changes in labor income tax rate at all.

Prescott (2004) shows that most of the differences in hours worked between the U.S. and Europe can be accounted for by differences in tax rates alone. (Prescott did not use the exact same model as we have here, but the intuition is the same.)

Hours Worked

Actual Predicted Tax Rate

U.S. 25.9 24.6 0.40 Germany 19.3 19.5 0.59 France 17.5 19.5 0.59 Italy 16.5 18.5 0.64

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

The Laffer Curve

The Laffer Curve is a curve that shows the quantity of tax revenue generated by the government as a function of a tax rate. It is named after the economist Arthur Laffer.

Based on our modifed model, the tax revenue that the government can collect with a labor income tax rate t is

REV = tw(h− l) = t (

1 − t ψ

) η 1+η

where REV is total revenue from the labor income tax, and w(h− l) is the tax base.

It is clear that the tax revenue is zero if the tax rate t is either 0 or 1. Intuitively, if t = 0, the government does not impose a tax; if t = 1, the household will not work and the tax base is zero.

To figure out the shape of the Laffer curve when 0 < t < 1, we can take the derivative of REV with respect to t:

dREV

dt =

( 1 − t ψ

) η 1+η

( 1 −

1 − t η

1 + η

) (4.1)

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

From Equation (4.1), we have • dREV

dt = 0 when 1 − t

1−t η

1+η = 0, i.e., when t = 1+η

1+2η .

• dREV dt

> 0 when 1 − t 1−t

1+η > 0, i.e., when 0 < t < 1+η

1+2η .

• dREV dt

< 0 when 1 − t 1−t

1+η < 0, i.e., when 1+η

1+2η < t < 1.

So the Laffer curve should be hump shaped, and there is a maximum amount of tax revenue REV ∗ that the government can generate with the corresponding tax rate t∗.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Note that from the shape of the Laffer curve, for a given amount of government expenditures G that is lower than REV ∗, there are two different tax rates that can balance the government’s budget: one lower than t∗ and one higher than t∗. Let us denote the two tax rates generating the same tax revenue G by t1 and t2 and t1 < t∗ < t2.

A sensible government would never choose the high tax rate t2 since it could collect the same quantity of tax revenue with the low tax rate t1 and make the representative household better off. However, a less-than-sensible government could get stuck in a bad equilibrium with the high tax rate t2, and thus be on the wrong side of the Laffer curve where an increase in the tax rate will reduce tax revenue.

Remarks:

1. In our modified model, a higher tax rate leads to both higher consumption and lower labor supply (i.e., more leisure), and they move household utility in the opposite directions. However, it is easy to verify that the consumption effect is stronger than the leisure effect, and hence household’s utility at equilibrium is strictly decreasing in tax rate.

2. A change in labor income tax rate has both substitution and income effects on labor supply, and the shape of the Laffer curve depends on the fundamentals of the economy and how the tax revenue is used.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

During the 1980 Presidential election, Ronald Reagan, supported by the reasoning of so-called supply-side economics, put forward an economic program including reductions in income tax rates. Reagan argued that tax rates could be reduced without sacrificing any tax revenue, everyone would work harder as a result, GDP would be higher, and everyone would be better off. Reagans arguments can be interpreted as being that the U.S. economy in 1980 was on the wrong side of the Laffer curve.

Reagan’s views are consistent with theory, but the empirical question is whether the U.S. economy was operating in 1980 on the good side of the Laffer curve (the upward-sloping portion) or the bad side (the downward-sloping portion). The general consensus among economists concerning this debate is that the U.S. economy is typically on the good side, rather than the bad side of the Laffer curve.

Lecture Notes: A One-Period Model of the Macroeconomy Chunzan Wu

Readings

• Chapter 4 and 5 in Williamson. • Prescott, E. 2004. ”Why Do Americans Work So Much More than Europeans?”

Federal Reserve Bank of Minneapolis Quarterly Review

The Representative Household
The Representative Firm
Competitive Equilibrium
Applying the Theory: Income Tax and Labor Supply