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The Standard of Living over Time and Across Countries

Describe the aggregate production function.

Explain how real GDP is determined in the long run.

Understand why the standard of living varies across countries.

Understand why labor productivity varies across countries.

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Learning Objectives
After studying this chapter, you should be able to:
5.1
5.2
5.3
5.4

Who’s number one?

What is the leading economy in the world today?

The table shows two measures, with different results.

Both measures have some value.

Second table shows a wide disparity in standard of living between countries with highest GDP.

What explains differences, and changes, in standards of living?

Rank	GDP	GDP per capita
1	United States	Qatar
2	China	Singapore
3	India	Norway
4	Japan	Hong Kong
5	Germany	United Arab Emirates
6	Russia	United States
7	Brazil	Switzerland
8	United Kingdom	Netherlands
9	France	Austria
10	Italy	Australia

Country	GDP	GDP per capita
United States	$15.0 trillion	$48,100
China	11.3 trillion	8,400
India	4.5 trillion	3,700

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Some countries have experienced rapid rates of long-run economic growth, while other countries have grown slowly, if at all.

Why isn’t the whole world rich?

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Key Issue and Question

Issue:

Question:

Describe the aggregate production function.

5.1

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Learning Objective

The aggregate production function

Firms combine land, labor, natural resources, and capital to produce goods and services.

Technology represents the processes used to turn inputs into outputs.

Can include new products.

Improved management and/or worker skills.

Improved speed and efficiency in production.

A production function is a microeconomic concept applied to a firm that can be expanded to the macroeconomic level.

Aggregate production function An equation that shows the relationship between the inputs employed by firms and the maximum output firms can produce with those inputs.

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General aggregate production function

At the macroeconomic level, output is measured as real GDP (Y).

Include only labor and capital as inputs.

Y = real GDP, K = capital stock, L = labor, A = index of technology

Y = A × F(K,L), or Y = AF(K,L)

The higher the value of A, the more efficient is the economy and the higher is real GDP.

“A” measures the influence of all factors of production other than labor or capital used in production.

Factors that can affect the value of A:

Technology

Government regulations and institutions

Quality of the labor force

Geography

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Land/natural resources incorporated into A; changes relatively unimportant for a given economy.

The Cobb-Douglas production function

Cobb-Douglas production function A widely-used macroeconomic production function that takes the form Y = AKαL1-α.

Notice that the sum of the exponents on capital and labor is 1.

Suppose α = 1/3, so that 1 – α = 2/3. Then we would write:

Y = AK1/3L2/3

Suppose A = 1,627; K = $40,000 billion; and L = 140 million workers. Then

Y = 1,627 ×($40,000 billion)1/3 × (0.140 billion workers)2/3

= $15.002 billion

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Properties of Cobb-Douglas production function

Cobb-Douglas production function is just one possible production function we could use to model output.

But it is commonly used, because it is simple, and seems to do a good job of explaining changes in GDP over time within a country.

Cobb-Douglas production function has several important properties:

The function exhibits constant returns to scale.

The function exhibits diminishing returns.

Capital and labor both earn shares of total income equal to the value of their exponents in the production function.

Next slides will discuss 1. and 2.; 3. left until later.

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Constant returns to scale

Constant returns to scale A property of a production function such that if all inputs increase by the same percentage, real GDP increases by the same percentage.

Under constant returns to scale, if the quantities of capital and labor both double, real GDP will also double:

2Y = AF(2K,2L)

The Cobb-Douglas production function has constant returns to scale, because the exponents (α and 1-α) add up to 1.

Example: Multiplying both capital and labor by 5, we obtain:

Y = A(5K)1/3(5L)2/3 = 5(1/3+2/3)AK1/3L2/3 = 5AK1/3L2/3

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A microeconomic example might help here; like “suppose a pizza parlor uses 20 workers and 2 pizza ovens to produce 200 pizzas per day. If the owner of pizza parlor expands her business to using 40 workers and 4 pizza ovens and her production function has constant returns to scale, she will now be able to produce 400 pizzas.”

Diminishing marginal returns

Diminishing marginal returns refers to the idea of the effect of increasing one factor being smaller for greater levels of that factor, keeping the other factors constant.

We can look at the marginal returns to capital and labor by considering the marginal product of capital and the marginal product of labor.

Marginal product of capital (MPK) The extra output a firm receives from adding one more unit of capital, holding all other inputs and efficiency constant.

Marginal product of labor (MPL) The extra output a firm receives from adding one more unit of labor, holding all other inputs and efficiency constant.

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Diminishing marginal returns to capital

This panel shows the aggregate production function, holding labor and efficiency constant.

It allows us to show the marginal product of capital:

As we increase capital (moving along the K axis), MPK is falling: diminishing marginal returns to capital.

Aggregate production functions

Figure 5.1a

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Diminishing marginal returns to labor

This panel shows the aggregate production function, holding capital and efficiency constant.

It allows us to show the marginal product of labor:

As we increase capital (moving along the L axis), MPL is falling: diminishing marginal returns to labor.

Aggregate production functions

Figure 5.1b

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Why diminishing marginal returns?

Diminishing marginal returns are easiest to understand in a microeconomic context.

Suppose you have two administrative assistants creating an accounting report, but they have only one computer.

Adding a second computer increases output a lot.

Adding a third computer increases output much less.

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The demand for labor and the demand for capital

Graphing MPK and MPL, we have:

Downward sloping MPK and MPL curves (diminishing marginal returns)

Both are always positive

The MPK and MPL curves are the demand curves for capital and labor respectively.

Hire capital or labor until price equals marginal productivity.

The marginal product of capital and marginal product of labor curves

Figure 5.2

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Changes in capital or labor

As we increase labor or capital, diminishing marginal returns means that the same increases in labor or capital will produce progressively smaller increases in real GDP.

The figure shows this for increases in labor.

The effect of an increase in labor in the aggregate production function

Figure 5.3a

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Changes in total factor productivity

The A in the Cobb-Douglas production function is also called total factor productivity (TFP).

Total factor productivity An index of the overall level of efficiency of transforming capital and labor into real GDP.

TFP does not experience diminishing marginal returns.

The effect of an increase in productivity in the aggregate production function

Figure 5.3b

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Foreign direct investment and real GDP in China

Foreign direct investment (FDI) refers to the purchase or building of capital goods by foreign firms.

2011: FDI in China = $116 billion

FDI in China increases Chinese stock of capital goods

Causes movement along production function

Also causes technology transfer: increase in TFP for China

U.S. companies reconsidering investing in China:

FDI in China largely from other Asian countries; 25% decrease from U.S.

Why? Partly sluggish growth in U.S.

Also Chinese government restrictions on FDI, and poor intellectual property controls

FDI from U.S. companies has been good for Chinese real GDP growth; but continued growth may require forward-thinking government policy.

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Making the Connection

Summary of an aggregate production graph

Summary of aggregate production function graph with capital on horizontal axis

Table 5.1

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Results would be similar with labor on horizontal axis

Explain how real GDP is determined in the long run.

5.2

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Learning Objective

A model of real GDP in the long run

To examine aggregate real GDP, we first explain how firms choose quantities of capital and labor to maximize profit. Economists assume firms are profit maximizers, such that:

Firms purchase capital and hire labor only to maximize profits.

Firms operate in a perfectly competitive market (i.e., firms are price takers).

Firms take the price of capital and labor inputs as given.

Firms decide how much capital and labor to hire using available technology based on prices of output and inputs.

We consider the behavior of a single representative firm, and assume all firms act in a similar way.

Profit = Revenue – Cost

Profit = PY – (WL + RK)

Profit = PY – WL – RK

Profit Total revenue minus total cost.

Note: P = Nominal output price W = Nominal wage R = Nominal rental cost of capital

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The market demand for capital

Firms are price takers in input markets and will hire more capital as long as the revenue produced from the next unit of capital exceeds the rental rate.

Δ Profit = P ×MPK – R

So hire capital if

P ×MPK > R

Firms continue to hire until

MPK = R/P

= r

= real rental cost of capital

So the marginal product of capital curve is the demand curve for capital.

Aggregate capital and labor markets

Figure 5.4a

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Note standard assumption that the supply of capital is fixed (in short run)

The market demand for labor

Since firms are price takers in input markets, firms will hire more workers as long as the revenue produced from the next worker exceeds the wage.

Δ Profit = P ×MPL – W

So hire workers if

P ×MPL > W

Firms continue to hire until

MPL = W/P

= w

= real wage

So the marginal product of labor curve is the demand curve for labor.

Aggregate capital and labor markets

Figure 5.4b

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Note standard assumption that the supply of labor is fixed (in short run)

Combining the factor markets with aggregate production

Factor markets determine equilibrium prices and quantities of capital and labor.

In this panel, real GDP is determined with capital stock on the horizontal axis.

The lower graph shows how the equilibrium quantity of capital is determined.

The upper graph uses the production function and the level of capital to determine the level of real GDP.

Determination of real GDP

Figure 5.5a

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Combining the factor markets with aggregate production

Factor markets determine equilibrium prices and quantities of capital and labor.

In this panel, real GDP is determined with the quantity of labor on the horizontal axis.

The lower graph shows how the equilibrium quantity of labor is determined.

The upper graph uses the production function and the level of labor to determine the level of real GDP.

Determination of real GDP

Figure 5.5b

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The division of total income

What determined how much of total income is received by labor, and how much is received by capital?

Labor:

By definition, Total labor income = w ×L

But w = MPL; so Total labor income = MPL ×L

Assuming Cobb-Douglas production, we can derive that MPL = (1-α)(Y/L); so

Total labor income = (1-α)(Y/L) ×L

Labor’s share of total income = (1-α)Y

Capital:

By definition, Total capital income = r ×K

But r = MPK; so Total capital income = MPK ×K

Assuming Cobb-Douglas production, we can derive that MPK = α(Y/K); so

Total capital income = α(Y/K) ×K

Capital’s share of total income = αY

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The division of total income

What determined how much of total income is received by labor, and how much is received by capital?

Labor’s share of total income = (1-α)Y

Capital’s share of total income = αY

This result implies that over time, the shares of labor and capital in total income should be roughly constant.

Over time, in the U.S. and other high-income countries, labor’s share of total income has typically been about two-thirds, and capital’s share has been about one-third.

This fact gives some justification for using the Cobb-Douglas production function, since it generates this property.

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Calculating MPL and MPK

Suppose that the production function for the economy is:

Y = AK1/2L1/2

Assume that real GDP is $12,000 billion, capital stock is $40,000 billion, and the labor supply is 0.150 billion workers.

Calculate the value of the marginal product of capital. Given this value, if the capital stock increases by $1 billion, by how much will real GDP increase?

Calculate the value for the marginal product of labor. Given this value, if the labor supply increases by one worker, by how much will real GDP increase?

What fraction of total income is received by labor, and what fraction is received by capital?

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Solved Problem

Calculating MPL and MPK

Step 1 Review the chapter material.

Step 2 Answer part (a) by calculating the value of the marginal product of capital, and use the value to determine how much real GDP will increase if the capital stock increases by $1 billion.

The marginal product of capital is equal to the exponent on the capital term, multiplied by (Y/K):

MPK = ½ ($12,000 billion / $40,000 billion)

= $0.15 per dollar of capital

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Solved Problem

Calculating MPL and MPK

Step 3 Answer part (b) by calculating the value of the marginal product of labor and use the value to determine how much real GDP will increase if the labor supply increases by one hour.

The marginal product of labor is equal to the exponent on the labor term, multiplied by (Y/L):

MPL = ½ ($12,000 billion / 0.150 billion workers)

= $40,000 per worker

Step 4 Answer part (c) by determining labor and capital’s shares of total income.

In a Cobb-Douglas production function, the shares of labor and capital in total income both equal the value of their exponents.

So labor and capital each obtain ½ of total income.

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Solved Problem

What determines levels of real GDP across countries?

Countries such as the U.S., China, and India have high levels of real GDP.

Possible reasons:

Large quantities of capital and labor

High levels of total factor productivity

These three countries have the world’s three largest labor forces.

But of these three, only the U.S. has a high level of real GDP per capita. Why?

Labor productivity is the critical determinant of real GDP per capita

Labor productivity is much higher in the U.S. than in China or India

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Understand why the standard of living varies across countries.

5.3

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Learning Objective

The per worker production function

To explain labor productivity, we modify the Cobb-Douglas production function to the per worker production function, by multiplying each input by a factor of (1/L).

Because of constant returns to scale, real GDP is also multiplied by (1/L)

Y = AF(K,L)

Y (1/L) = AF(K (1/L), L (1/L))

Y/L = AF(K/L,1)

y = Af(k)

y is “output per worker”, or labor productivity.

k is the capital-labor ratio, K/L.

f(k) = F(k,1) is a convenient way to rewrite the function F.

Capital-labor ratio The dollar value of capital goods per unit of labor; measured as the dollar value of capital divided by the total number of workers.

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The per worker production function

The per worker production function is very similar to the production function we used earlier except that the capital–labor ratio is on the horizontal axis, and real GDP per worker is on the vertical axis.

Diminishing marginal returns are demonstrated by the flattening slope.

The per worker production function

Figure 5.6

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What determines labor productivity?

Per worker production function: y = Af(k)

Which is more important, the capital-labor ratio or total factor productivity?

Capital experiences diminishing returns, but total factor productivity does not; so generally, total factor productivity matters more for labor productivity.

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What determines real GDP per capita?

Labor productivity measure how much output, or real GDP, the economy can produce per worker:

Standard of living can be measured by real GDP per capita:

So both labor productivity and the fraction of the population working affect real GDP per capita.

Labor productivity is more important, because labor input cannot change very much.

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Capital allocation across countries

How well do international capital markets allocate capital?

If there are no barriers to capital mobility, and capital markets are functioning well, then capital should move to countries with the highest rates of return.

This should make rate of return to capital in each country equal.

The graph shows MPK varies relatively little, and is mostly unrelated to real GDP per worker.

Estimate: reallocating capital from low- to high-rate of return countries would increase real GDP per worker by only 0.1%.

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Macro Data

Understand why labor productivity varies across countries.

5.4

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Learning Objective

What explains total factor productivity?

Capital accumulation experiences diminishing marginal returns, so increases in labor productivity generally come from total factor productivity.

Research and development, and the level of technology

Development of computers, assembly line technology, etc. has raised productivity of workers

Investment in R&D critical

In 2011, U.S. spent about $405 billion on R&D, more than either China or India

Quality of labor

Human capital improvements make workers more productive.

Average years of education for adults in U.S.: 13.3 years; China: 7.5 years; India: 4.4 years

Learning by doing is also critical.

Human capital The accumulated knowledge and skills that workers acquire from education and training or from life experiences.

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What explains total factor productivity?

Government and social institutions

Markets, secure property rights, and effective legal systems are critical for making workers more productive

Example: 20th century experience of North Korea (communist dictatorship, no strong markets, insecure property rights) vs. South Korea

South Korean real GDP per capita estimated at 17 times greater than in North Korea

Geography

Navigable rivers and coastline facilitate trade

Tropical climates experience higher rates of infectious disease such as malaria, reducing labor productivity

Agricultural productivity also important

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What explains total factor productivity?

The financial system

Well-functioning financial systems:

Households and firms can obtain capital

R&D investments are more likely

Capital flows to its most profitable use

Stock market liquidity also matters for economic growth

Investors are more likely to purchase stocks that they know will be easy to sell

This raises stock prices (value of firms)

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R&D spending and labor productivity

Why is labor productivity higher in the U.S. than in China?

One reason is the difference in research and development spending:

2.7% of GDP in U.S. vs. 1.6% in China

As recently as 1996, Chinese R&D spending: just 0.6% of GDP

Why can’t the Chinese government encourage more R&D spending?

Chinese economic freedom still low (138th out of 179 countries in 2012)

So Chinese firms still have little incentive to invest

Foreign firms investing in China are also constrained by government

Much land still collectively owned, reducing incentives to invest

While China has made remarkable progress in economic liberalization since the 1970s, investment in R&D is still low, hindering labor productivity growth.

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Making the Connection

How important were Chinese reforms of 1978

Republic of China established in 1911, ending hereditary rule.

But Republic of China had difficulty extending authority over country

People’s Republic of China established in 1949 created socialist economy based on state ownership.

Without secure property rights, markets limited and unimportant

In 1978, Deng Xiaoping began many economic reforms:

Allowed private ownership of farms and businesses

Allowed some crops to be sold in markets

Established special economic zones for foreign/domestic joint ventures

Opened country to international trade and finance

Reforms have allowed total factor productivity and real GDP per capita to increase rapidly in China. But continued improvement relies on continued commitment by Chinese government to economic liberalization.

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Making the Connection

Answering the key question

“Why isn’t the whole world rich?”

Countries with a low standard of living have low levels of total factor productivity, due to a combination of:

Lack of investment in R&D

Low quality of labor from low investment in education

Government institutions that do not protect private property and that discourage investment

Geography that makes trade difficult or makes diseases more prevalent

Lack of financial institutions that allow funds to flow to firms with profitable investment projects

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