solution manual to macroeconomics by mankiw 8th edition, Summaries of Macroeconomics

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Answers to Textbook Questions

and Problems

QQuueessttiioonnss ffoorr RReevviieeww

1. Microeconomics is the study of how individual firms and households make decisions, and how they interact with one another. Microeconomic models of firms and households are based on principles of optimization—firms and households do the best they can given the constraints they face. For example, households choose which goods to pur- chase in order to maximize their utility, whereas firms decide how much to produce in order to maximize profits. In contrast, macroeconomics is the study of the economy as a whole; it focuses on issues such as how total output, total employment, and the overall price level are determined. These economy-wide variables are based on the interaction of many households and many firms; therefore, microeconomics forms the basis for macroeconomics.
2. Economists build models as a means of summarizing the relationships among economic variables. Models are useful because they abstract from the many details in the econo- my and allow one to focus on the most important economic connections.
3. A market-clearing model is one in which prices adjust to equilibrate supply and demand. Market-clearing models are useful in situations where prices are flexible. Yet in many situations, flexible prices may not be a realistic assumption. For example, labor contracts often set wages for up to three years. Or, firms such as magazine pub- lishers change their prices only every three to four years. Most macroeconomists believe that price flexibility is a reasonable assumption for studying long-run issues. Over the long run, prices respond to changes in demand or supply, even though in the short run they may be slow to adjust.

PPrroobblleemmss aanndd AApppplliiccaattiioonnss

1. The many recent macroeconomic issues that have been in the news lately (early 2002) include the recession that began in March 2001, sharp reductions in the Federal Reserve’s target interest rate (the so-called Federal Funds rate ) in 2001, whether the government should implement tax cuts or spending increases to stimulate the economy, and a financial crisis in Argentina.
2. Many philosophers of science believe that the defining characteristic of a science is the use of the scientific method of inquiry to establish stable relationships. Scientists exam- ine data, often provided by controlled experiments, to support or disprove a hypothesis. Economists are more limited in their use of experiments. They cannot conduct con- trolled experiments on the economy; they must rely on the natural course of develop- ments in the economy to collect data. To the extent that economists use the scientific method of inquiry, that is, developing hypotheses and testing them, economics has the characteristics of a science.
3. We can use a simple variant of the supply-and-demand model for pizza to answer this question. Assume that the quantity of ice cream demanded depends not only on the price of ice cream and income, but also on the price of frozen yogurt:

Q d^ = D ( P IC, P FY, Y ). We expect that demand for ice cream rises when the price of frozen yogurt rises, because ice cream and frozen yogurt are substitutes. That is, when the price of frozen yogurt goes up, I consume less of it and, instead, fulfill more of my frozen dessert urges through the consumption of ice cream.

3

C H A P T E R 1 The Science of Macroeconomics

QQuueessttiioonnss ffoorr RReevviieeww

1. GDP measures both the total income of everyone in the economy and the total expendi- ture on the economy’s output of goods and services. GDP can measure two things at once because both are really the same thing: for an economy as a whole, income must equal expenditure. As the circular flow diagram in the text illustrates, these are alter- native, equivalent ways of measuring the flow of dollars in the economy.
2. The consumer price index measures the overall level of prices in the economy. It tells us the price of a fixed basket of goods relative to the price of the same basket in the base year.
3. The Bureau of Labor Statistics classifies each person into one of the following three cat- egories: employed, unemployed, or not in the labor force. The unemployment rate, which is the percentage of the labor force that is unemployed, is computed as follows:

Unemployment Rate =.

Note that the labor force is the number of people employed plus the number of people unemployed.

4. Okun’s law refers to the negative relationship that exists between unemployment and real GDP. Employed workers help produce goods and services whereas unemployed workers do not. Increases in the unemployment rate are therefore associated with decreases in real GDP. Okun’s law can be summarized by the equation:

%∆Real GDP = 3% – 2 × (∆Unemployment Rate). That is, if unemployment does not change, the growth rate of real GDP is 3 percent. For every percentage-point change in unemployment (for example, a fall from 6 percent to 5 percent, or an increase from 6 percent to 7 percent), output changes by 2 percent in the opposite direction.

PPrroobblleemmss aanndd AApppplliiccaattiioonnss

1. A large number of economic statistics are released regularly. These include the follow- ing: GGrroossss DDoommeessttiicc PPrroodduucctt—the market value of all final goods and services produced in a year. TThhee UUnneemmppllooyymmeenntt RRaattee—the percentage of the civilian labor force who do not have a job. CCoorrppoorraattee PPrrooffiittss—the accounting profits remaining after taxes of all manufacturing corporations. It gives an indication of the general financial health of the corporate sec- tor. TThhee CCoonnssuummeerr PPrriiccee IInnddeexx ((CCPPII))—a measure of the average price that consumers pay for the goods they buy; changes in the CPI are a measure of inflation. TThhee TTrraaddee BBaallaannccee—the difference between the value of goods exported abroad and the value of goods imported from abroad.

Number of Unemployed × 100 Labor Force

C H A P T E R 2 The Data of Macroeconomics

2. Value added by each person is the value of the good produced minus the amount the person paid for the materials necessary to make the good. Therefore, the value added by the farmer is $1.00 ($1 – 0 = $1). The value added by the miller is $2: she sells the flour to the baker for $3 but paid $1 for the flour. The value added by the baker is $3: she sells the bread to the engineer for $6 but paid the miller $3 for the flour. GDP is the total value added, or $1 + $2 + $3 = $6. Note that GDP equals the value of the final good (the bread).
3. When a woman marries her butler, GDP falls by the amount of the butler’s salary. This happens because measured total income, and therefore measured GDP, falls by the amount of the butler’s loss in salary. If GDP truly measured the value of all goods and services, then the marriage would not affect GDP since the total amount of economic activity is unchanged. Actual GDP, however, is an imperfect measure of economic activ- ity because the value of some goods and services is left out. Once the butler’s work becomes part of his household chores, his services are no longer counted in GDP. As this example illustrates, GDP does not include the value of any output produced in the home. Similarly, GDP does not include other goods and services, such as the imputed rent on durable goods (e.g., cars and refrigerators) and any illegal trade.
4. a. government purchases b. investment c. net exports d. consumption e. investment
5. Data on parts (a) to (g) can be downloaded from the Bureau of Economic Analysis (www.bea.doc.gov—follow the links to GDP and related data). Most of the data (not necessarily the earliest year) can also be found in the Economic Report of the President. By dividing each component (a) to (g) by nominal GDP and multiplying by 100, we obtain the following percentages: 1950 1975 2000

a. Personal consumption expenditures 65.5% 63.0% 68.2% b. Gross private domestic investment 18.4% 14.1% 17.9% c. Government consumption purchases 15.9% 22.1% 17.6% d. Net exports 0.2% 0.8% –3.7% e. National defense purchases 6.7% 6.6% 3.8% f. State and local purchases 7.1% 12.8% 11.7% g. Imports 3.9% 7.5% 14.9%

(Note: These data were downloaded February 5, 2002 from the BEA web site.)

Among other things, we observe the following trends in the economy over the period 1950–2000: (a) Personal consumption expenditures have been around two-thirds of GDP, although the share increased about 5 percentage points between 1975 and 2000. (b) The share of GDP going to gross private domestic investment fell from 1950 to 1975 but then rebounded. (c) The share going to government consumption purchases rose more than 6 percentage points from 1950 to 1975 but has receded somewhat since then. (d) Net exports, which were positive in 1950 and 1975, were substantially negative in

(e) The share going to national defense purchases fell from 1975 to 2000. (f) The share going to state and local purchases rose from 1950 to 1975. (g) Imports have grown rapidly relative to GDP.

6 Answers to Textbook Questions and Problems

This calculation shows that the price of goods purchased in 2010 increased by 60 percent compared to the prices these goods would have sold for in 2000. The CPI for 2000, the base year, equals 1.0. b. The implicit price deflator is a Paasche index because it is computed with a chang- ing basket of goods; the CPI is a Laspeyres index because it is computed with a fixed basket of goods. From (5.a.iii), the implicit price deflator for the year 2010 is 1.52, which indicates that prices rose by 52 percent from what they were in the year 2000. From (5.a.iv.), the CPI for the year 2010 is 1.6, which indicates that prices rose by 60 percent from what they were in the year 2000. If prices of all goods rose by, say, 50 percent, then one could say unam- biguously that the price level rose by 50 percent. Yet, in our example, relative prices have changed. The price of cars rose by 20 percent; the price of bread rose by 100 percent, making bread relatively more expensive. As the discrepancy between the CPI and the implicit price deflator illus- trates, the change in the price level depends on how the goods’ prices are weight- ed. The CPI weights the price of goods by the quantities purchased in the year

2000. The implicit price deflator weights the price of goods by the quantities pur- chased in the year 2010. The quantity of bread consumed was higher in 2000 than in 2010, so the CPI places a higher weight on bread. Since the price of bread increased relatively more than the price of cars, the CPI shows a larger increase in the price level. c. There is no clear-cut answer to this question. Ideally, one wants a measure of the price level that accurately captures the cost of living. As a good becomes relatively more expensive, people buy less of it and more of other goods. In this example, consumers bought less bread and more cars. An index with fixed weights, such as the CPI, overestimates the change in the cost of living because it does not take into account that people can substitute less expensive goods for the ones that become more expensive. On the other hand, an index with changing weights, such as the GDP deflator, underestimates the change in the cost of living because it does not take into account that these induced substitutions make people less well off.
2001. a. The consumer price index uses the consumption bundle in year 1 to figure out how much weight to put on the price of a given good:

CPI^2 =

According to the CPI, prices have doubled. b. Nominal spending is the total value of output produced in each year. In year 1 and year 2, Abby buys 10 apples for $1 each, so her nominal spending remains con- stant at $10. For example,

Nominal Spending 2 = ( P × Q ) + ( P × Q )

= ($2 × 0 ) + ($1 × 10 )

8 Answers to Textbook Questions and Problems

($2 × 10 ) + ($1 × 0 )

($1 × 10 ) + ($2 × 0 )

2 red 2 red 2 green 2 green

P Q P Q

P Q P Q

red red green green red red green gre

2 1 2 1 1 1 1

× + ×

× + × (^1) een)

c. Real spending is the total value of output produced in each year valued at the prices prevailing in year 1. In year 1, the base year, her real spending equals her nominal spending of $10. In year 2, she consumes 10 green apples that are each valued at their year 1 price of $2, so her real spending is $20. That is,

Real Spending 2 = ( P × Q ) + ( P × Q )

= ($1 × 0 ) + ($2 × 10 )

Hence, Abby’s real spending rises from $10 to $20. d. The implicit price deflator is calculated by dividing Abby’s nominal spending in year 2 by her real spending that year:

Implicit Price Deflator 2 =

Thus, the implicit price deflator suggests that prices have fallen by half. The rea- son for this is that the deflator estimates how much Abby values her apples using prices prevailing in year 1. From this perspective green apples appear very valu- able. In year 2, when Abby consumes 10 green apples, it appears that her con- sumption has increased because the deflator values green apples more highly than red apples. The only way she could still be spending $10 on a higher consumption bundle is if the price of the good she was consuming feel. e. If Abby thinks of red apples and green apples as perfect substitutes, then the cost of living in this economy has not changed—in either year it costs $10 to consume 10 apples. According to the CPI, however, the cost of living has doubled. This is because the CPI only takes into account the fact that the red apple price has dou- bled; the CPI ignores the fall in the price of green apples because they were not in the consumption bundle in year 1. In contrast to the CPI, the implicit price defla- tor estimates the cost of living has halved. Thus, the CPI, a Laspeyres index, over- states the increase in the cost of living and the deflator, a Paasche index, under- states it. This chapter of the text discusses the difference between Laspeyres and Paasche indices in more detail.

8. a. Real GDP falls because Disney does not produce any services while it is closed. This corresponds to a decrease in economic well-being because the income of work- ers and shareholders of Disney falls (the income side of the national accounts), and people’s consumption of Disney falls (the expenditure side of the national accounts). b. Real GDP rises because the original capital and labor in farm production now pro- duce more wheat. This corresponds to an increase in the economic well-being of society, since people can now consume more wheat. (If people do not want to con- sume more wheat, then farmers and farmland can be shifted to producing other goods that society values.) c. Real GDP falls because with fewer workers on the job, firms produce less. This accurately reflects a fall in economic well-being. d. Real GDP falls because the firms that lay off workers produce less. This decreases economic well-being because workers’ incomes fall (the income side), and there are fewer goods for people to buy (the expenditure side). e. Real GDP is likely to fall, as firms shift toward production methods that produce fewer goods but emit less pollution. Economic well-being, however, may rise. The economy now produces less measured output but more clean air; clean air is not

C h a p t e r 2 The Data of Macroeconomics 9

1 red 2 red 1 green 2 green

Nominal Spending 2 Real Spending 2 $ $

QQuueessttiioonnss ffoorr RReevviieeww

1. Factors of production and the production technology determine the amount of output an economy can produce. Factors of production are the inputs used to produce goods and services: the most important factors are capital and labor. The production technology determines how much output can be produced from any given amounts of these inputs. An increase in one of the factors of production or an improvement in technology leads to an increase in the economy’s output.
2. When a firm decides how much of a factor of production to hire, it considers how this decision affects profits. For example, hiring an extra unit of labor increases output and therefore increases revenue; the firm compares this additional revenue to the addition- al cost from the higher wage bill. The additional revenue the firm receives depends on the marginal product of labor ( MPL ) and the price of the good produced ( P ). An addi- tional unit of labor produces MPL units of additional output, which sells for P dollars. Therefore, the additional revenue to the firm is P × MPL. The cost of hiring the addi- tional unit of labor is the wage W. Thus, this hiring decision has the following effect on profits: ∆Profit = ∆Revenue – ∆Cost = ( P × MPL ) – W. If the additional revenue, P × MPL , exceeds the cost ( W ) of hiring the additional unit of labor, then profit increases. The firm will hire labor until it is no longer profitable to do so—that is, until the MPL falls to the point where the change in profit is zero. In the equation above, the firm hires labor until ∆profit = 0, which is when ( P × MPL ) = W. This condition can be rewritten as: MPL = W/P. Therefore, a competitive profit-maximizing firm hires labor until the marginal product of labor equals the real wage. The same logic applies to the firm’s decision to hire capi- tal: the firm will hire capital until the marginal product of capital equals the real rental price.
3. A production function has constant returns to scale if an equal percentage increase in all factors of production causes an increase in output of the same percentage. For exam- ple, if a firm increases its use of capital and labor by 50 percent, and output increases by 50 percent, then the production function has constant returns to scale. If the production function has constant returns to scale, then total income (or equivalently, total output) in an economy of competitive profit-maximizing firms is divided between the return to labor, MPL × L , and the return to capital, MPK × K. That is, under constant returns to scale, economic profit is zero.
4. Consumption depends positively on disposable income—the amount of income after all taxes have been paid. The higher disposable income is, the greater consumption is. The quantity of investment goods demanded depends negatively on the real inter- est rate. For an investment to be profitable, its return must be greater than its cost. Because the real interest rate measures the cost of funds, a higher real interest rate makes it more costly to invest, so the demand for investment goods falls.
5. Government purchases are those goods and services purchased directly by the govern- ment. For example, the government buys missiles and tanks, builds roads, and provides

C H A P T E R 3 National Income: Where It Comes From

and Where It Goes

services such as air traffic control. All of these activities are part of GDP. Transfer pay- ments are government payments to individuals that are not in exchange for goods or services. They are the opposite of taxes: taxes reduce household disposable income, whereas transfer payments increase it. Examples of transfer payments include Social Security payments to the elderly, unemployment insurance, and veterans’ benefits.

6. Consumption, investment, and government purchases determine demand for the econo- my’s output, whereas the factors of production and the production function determine the supply of output. The real interest rate adjusts to ensure that the demand for the economy’s goods equals the supply. At the equilibrium interest rate, the demand for goods and services equals the supply.
7. When the government increases taxes, disposable income falls, and therefore consump- tion falls as well. The decrease in consumption equals the amount that taxes increase multiplied by the marginal propensity to consume ( MPC ). The higher the MPC is, the greater is the negative effect of the tax increase on consumption. Because output is fixed by the factors of production and the production technology, and government pur- chases have not changed, the decrease in consumption must be offset by an increase in investment. For investment to rise, the real interest rate must fall. Therefore, a tax increase leads to a decrease in consumption, an increase in investment, and a fall in the real interest rate.

PPrroobblleemmss aanndd AApppplliiccaattiioonnss

1. a. According to the neoclassical theory of distribution, the real wage equals the mar- ginal product of labor. Because of diminishing returns to labor, an increase in the labor force causes the marginal product of labor to fall. Hence, the real wage falls. b. The real rental price equals the marginal product of capital. If an earthquake destroys some of the capital stock (yet miraculously does not kill anyone and lower the labor force), the marginal product of capital rises and, hence, the real rental price rises. c. If a technological advance improves the production function, this is likely to increase the marginal products of both capital and labor. Hence, the real wage and the real rental price both increase.
2. A production function has decreasing returns to scale if an equal percentage increase in all factors of production leads to a smaller percentage increase in output. For example, if we double the amounts of capital and labor, and output less than doubles, then the production function has decreasing returns to capital and labor. This may happen if there is a fixed factor such as land in the production function, and this fixed factor becomes scarce as the economy grows larger. A production function has increasing returns to scale if an equal percentage increase in all factors of production leads to a larger percentage increase in output. For example, if doubling inputs of capital and labor more than doubles output, then the pro- duction function has increasing returns to scale. This may happen if specialization of labor becomes greater as population grows. For example, if one worker builds a car, then it takes him a long time because he has to learn many different skills, and he must constantly change tasks and tools; all of this is fairly slow. But if many workers build a car, then each one can specialize in a particular task and become very fast at it.
3. a. According to the neoclassical theory, technical progress that increases the margin- al product of farmers causes their real wage to rise. b. The real wage in (a) is measured in terms of farm goods. That is, if the nominal wage is in dollars, then the real wage is W/PF , where PF is the dollar price of farm goods.

12 Answers to Textbook Questions and Problems

How does this increase in investment take place? We know that investment depends on the real interest rate. For investment to rise, the real interest rate must fall. Figure 3–1 graphs saving and investment as a function of the real inter- est rate.

The tax increase causes national saving to rise, so the supply curve for loan- able funds shifts to the right. The equilibrium real interest rate falls, and invest- ment rises.

5. If consumers increase the amount that they consume today, then private saving and, therefore, national saving will fall. We know this from the definition of national saving: National Saving = [Private Saving] + [Public Saving] = [ Y – T – C ( Y – T )] + [ T – G ]. An increase in consumption decreases private saving, so national saving falls. Figure 3–2 graphs saving and investment as a function of the real interest rate. If national saving decreases, the supply curve for loanable funds shifts to the left, thereby raising the real interest rate and reducing investment.

14 Answers to Textbook Questions and Problems

S 1 S 2

I ( r )

I, S Investment, Saving

r 1

r 2

r

Real interest rate

FFiigguur Figure 3–1 ree 3 3––1 1

S 2 S 1

I ( r )

I, S Investment, Saving

r 1

r 2

r

Real interest rate

Figure 3– FFiigguurree 3 3––2 2

6. a. Private saving is the amount of disposable income, Y – T , that is not consumed: S private^ = Y – T – C = 5,000 – 1,000 – (250 + 0.75(5,000 – 1,000)) = 750. Public saving is the amount of taxes the government has left over after it makes its purchases: S public^ = T – G = 1,000 – 1, = 0. Total saving is the sum of private saving and public saving: S = S private^ + S public = 750 + 0 = 750. b. The equilibrium interest rate is the value of r that clears the market for loanable funds. We already know that national saving is 750, so we just need to set it equal to investment: S = I 750 = 1,000 – 50 r Solving this equation for r , we find: r = 5%. c. When the government increases its spending, private saving remains the same as before (notice that G does not appear in the S private^ above) while government saving decreases. Putting the new G into the equations above: S private^ = 750 S public^ = T – G = 1,000 – 1, = –250. Thus, S = S private^ + S public = 750 + (–250) = 500. d. Once again the equilibrium interest rate clears the market for loanable funds: S = I 500 = 1,000 – 50 r Solving this equation for r , we find: r = 10%.
7. To determine the effect on investment of an equal increase in both taxes and govern- ment spending, consider the national income accounts identity for national saving: National Saving = [Private Saving] + [Public Saving] = [ Y – T – C ( Y – T )] + [ T – G ]. We know that Y is fixed by the factors of production. We also know that the change in consumption equals the marginal propensity to consume ( MPC ) times the change in disposable income. This tells us that ∆National Saving = [– ∆ T – ( MPC × ( – ∆ T ))] + [∆ T – ∆ G ] = [– ∆ T + ( MPC × ∆ T )] + 0 = ( MPC – 1) ∆ T.

C h a p t e r 3 National Income: Where It Comes From and Where It Goes 15

c. The total quantity of investment does not change because it is constrained by the inelastic supply of savings. The investment tax credit leads to a rise in business investment, but an offsetting fall in residential investment. That is, the higher interest rate means that residential investment falls (a shift along the curve), whereas the outward shift of the business investment curve leads business invest- ment to rise by an equal amount. Figure 3–5 shows this change. Note that

9. In this chapter, we concluded that an increase in government expenditures reduces national saving and raises the interest rate; it therefore crowds out investment by the full amount of the increase in government expenditure. Similarly, a tax cut increases disposable income and hence consumption; this increase in consumption translates into a fall in national saving—again, it crowds out investment.

C h a p t e r 3 National Income: Where It Comes From and Where It Goes 17

S

I 2

I 1 I, S

A

B

1. An increase in desired investment... 2.... raises the interest rate.

Investment, Saving

Real interest rate

r (^) Figure 3– 4

Business Investment

I 1 B I 2 B Residential Investment

I 1 R I 2 R

r

r 1

r 2

r

r 1

r 2

FFiigguurree 3 3––5 5

I 1^ B + I 1 R = I 2 B + I 2 R= S

Business investment

Residential investment

FFiigguurree 3 3––4 4

If consumption depends on the interest rate, then these conclusions about fiscal policy are modified somewhat. If consumption depends on the interest rate, then so does saving. The higher the interest rate, the greater the return to saving. Hence, it seems reasonable to think that an increase in the interest rate might increase saving and reduce consumption. Figure 3–6 shows saving as an increasing function of the interest rate.

Consider what happens when government purchases increase. At any given level of the interest rate, national saving falls by the change in government purchases, as shown in Figure 3–7. The figure shows that if the saving function slopes upward, investment falls by less than the amount that government purchases rise; this happens because consumption falls and saving increases in response to the higher interest rate. Hence, the more responsive consumption is to the interest rate, the less government purchases crowd out investment.

18 Answers to Textbook Questions and Problems

S 2 ( r ) (^) S 1 ( r ) Figure 3– 7

I(r )

I 1 I I, S Investment, Saving

∆ G

r 1

r

r

Real interest rate

S ( r )

S Saving

Real interest rate

r^ Figure 3–

FFiigguurree 3 3––6 6

FFiigguurree 3 3––7 7

The rental price increases by the ratio

=

= 1.069. So the rental price increases by 6.9 percent. To determine how the increase in the labor force affects the real wage, con- sider the formula for the real wage W/P : W/P = MPL = (1 – α) AK α L

• α . We know that α = 0.3. We also know that labor ( L ) increases by 10 percent. Let ( W/P ) 1 equal the initial value of the real wage and ( W/P ) 2 equal the final value of the real wage. To find ( W/P ) 2 , multiply L by 1.1 to reflect the 10-percent increase in the labor force: ( W/P ) 1 = (1 – 0.3) AK

L

. ( W/P ) 2 = (1 – 0.3) AK

(1.1 L )

. To calculate the percentage change in the real wage, divide ( W/P ) 2 by ( W/P ) 1 :

= 0.972. That is, the real wage falls by 2.8 percent. c. We can use the same logic as (b) to set Y 1 = AK 0.3 L 0.7.

Y 2 = A (1.1 K )0.3 L 0.7.

Therefore, we have:

=

This equation shows that output increases by 2 percent. Notice that α < 0.5 means that proportional increases to capital will increase output by less than the same proportional increase to labor. Again using the same logic as (b) for the change in the real rental price of capital:

=

The real rental price of capital falls by 6.5 percent because there are diminishing returns to capital; that is, when capital increases, its marginal product falls.

20 Answers to Textbook Questions and Problems

0.3 AK

• (1.1 L ) 0.

0.3 AK

• L 0.

( R / P ) 2

( R / P ) 1

A (1.1 K )

L

AK

L

Y 2

Y 1

0.3 A (1.1 K )

–0. L

0.3 AK

• L 0.

( R / P ) 2

( R / P ) 1

.. ..

W/P)

W/P

AK L

AK L

2 1

0 3 0 3 0 3 0 3

− −

Finally, the change in the real wage is:

=

(1.1)0.

= 1.029. Hence, real wages increase by 2.9 percent because the added capital increases the marginal productivity of the existing workers. (Notice that the wage and output have both increased by the same amount, leaving the labor share unchanged—a feature of Cobb–Douglas technologies.) d. Using the same formula, we find that the change in output is:

=

This equation shows that output increases by 10 percent. Similarly, the rental price of capital and the real wage also increase by 10 percent:

=

= 1.1.

2. a. The marginal product of labor MPL is found by differentiating the production function with respect to labor:

MPL =

= K 1/3 H 1/3 L –2/3.

This equation is increasing in human capital because more human capital makes all the existing labor more productive. b. The marginal product of human capital MPH is found by differentiating the pro- duction function with respect to human capital:

MPH =

= K 1/3 L 1/3 H –2/3.

This equation is decreasing in human capital because there are diminishing returns. c. The labor share of output is the proportion of output that goes to labor. The total amount of output that goes to labor is the real wage (which, under perfect compe- tition, equals the marginal product of labor) times the quantity of labor. This quantity is divided by the total amount of output to compute the labor share:

Labor Share =

C h a p t e r 3 National Income: Where It Comes From and Where It Goes 21

0.7 A (1.1 K )

–0. L

0.7 AK

• L 0.

( W / P ) 2

( W / P ) 1

0.7(1.1 A ) K

L –0.

0.7 AK

L

(1.1 A ) K

L

AK

L

Y 2

Y 1

0.3(1.1 A ) K

–0. L

0.3 AK

• L 0.

( R / P ) 2

( R / P ) 1

dY dL 1 3

dY dH 1 3

( W / P ) 2

( W / P ) 1

( 13 1 3^ 1 3^ 2 3)

1 3 1 3 1 3

K H L L

K H L

-