ANSWERS TO HOMEWORK QUESTIONS Chapter 3Review Questions 1. A production function shows how much output can be produced with a given amount of capital and labor. The production function can shift due to supply shocks, which affect overall productivity. Examples include changes in energy supplies, technological breakthroughs, and management practices. Besides knowing the production function, you must also know the quantities of capital and labor the economy has. 2. The upward slope of the production function means that any additional inputs of capital or labor produce more output. The fact that the slope declines as we move from left to right illustrates the idea of diminishing marginal productivity. For a fixed amount of capital, additional workers each add less additional output as the number of workers increases. For a fixed number of workers, additional capital adds less additional output as the amount of capital increases. 3. The marginal product of capital (MPK) is the output produced per unit of additional capital. The MPK can be shown graphically using the production function. For a fixed level of labor, plot the output provided by different levels of capital; this is the production function. The MPK is just the slope of the production function. 4. The marginal revenue product of labor represents the benefit to a firm of hiring an additional worker, while the nominal wage is the cost. Comparing the benefit to the cost, the firm will hire additional workers as long as the marginal revenue product of labor exceeds the nominal wage, since doing so increases profits. Profits will be at their highest when the marginal revenue product of labor just equals the nominal wage. The same condition can be expressed in real terms by dividing through by the price of the good. The resulting equation shows that the real wage (W/P) equals the marginal product of labor.
12. Frictional unemployment arises as workers and firms search to find matches. A certain amount of frictional unemployment is necessary, because it is not always possible to find the right match right away. For example, an unemployed banker may not want to take a job flipping hamburgers if she cannot find another banking job right away, because the match would be very poor. By remaining unemployed and continuing to search for a more suitable job, the banker is likely to make a better match. That will be better both for the banker (because the salary is likely to be higher) and for society as a whole (because the better match means greater productivity in the economy). 13. Structural unemployment occurs when people suffer long spells of unemployment or are chronically unemployed. Structural unemployment arises when the number of potential workers with low skill levels exceeds the number of jobs requiring low skill levels, or when the economy undergoes structural change. When there is structural change, workers who lose their jobs in shrinking industries may have to move to find new jobs. 14. The natural rate of unemployment is the rate of unemployment that prevails when output and employment are at their fullemployment levels. The natural rate of unemployment is equal to the amount of frictional unemployment plus structural unemployment. Cyclical unemployment is the difference between the actual rate of unemployment and the natural rate of unemployment. When cyclical unemployment is negative, output and employment exceed their full-employment levels. 15. Okuns Law is a rule of thumb that tells how much output falls when the unemployment rate rises. Specifically, it says that the percentage change in GDP equals 3 minus 2 times the change in the unemployment rate. If the unemployment rate increases by two percentage points, the percentage change in GDP will be 3 2(2) = -1.
Numerical problems 3. (a)N 1 2 3 4 5 6 Y 8 15 21 26 30 33 MPN 8 7 6 5 4 3 MRPN (P = 5) 40 35 30 25 20 15 MRPN (P = 10) 80 70 60 50 40 30
(b)P = $5. (1)W = $38. Hire one worker, since MRPN ($40) is greater than W ($38) at N = 1. Do not hire two workers, since MRPN ($35) is less than W ($38) at N = 2. (2)W = $27. Hire three workers, since MRPN ($30) is greater than W ($27) at N = 3. Do not hire four workers, since MRPN ($25) is less than W ($27) at N = 4. (3) W = $22. Hire four workers, since MRPN ($25) is greater than W ($22) at N = 4. Do not hire five workers, since MRPN ($20) is less than W ($22) at N = 5.
(c) Figure 1 plots the relationship between labor demand and the nominal wage. This graph is different from a labor demand curve because a labor demand curve shows the relationship between labor demand and the real wage. Figure 2 shows the labor demand curve.
(d) P = $10. The table in part (a) shows the MRPN for each N. At W = $38, the firm should hire five workers. MRPN ($40) is greater than W ($38) at N = 5. The firm shouldnt hire six workers, since MRPN ($30) is less than W ($38) at N = 6. With five workers, output is 30 widgets, compared to 8 widgets in part (a) when the firm hired only one worker. So the increase in the price of the product increases the firms labor demand and output.
(e) If output doubles, MPN doubles, so MRPN doubles. The MRPN is the same as it was in part (d) when the price doubled. So labor demand is the same as it was in part (d). The output produced by five workers now doubles to 60 widgets. (f) Since MRPN = P MPN, then a doubling of either P or MPN leads to a doubling of MRPN. Since labor demand is chosen by setting MRPN equal to W, the choice is the same, whether P doubles or MPN doubles. Analytical problems 2. (a) An increase in the number of immigrants increases the labor force, increasing employment and increasing full-employment output. (b)If energy supplies become depleted, it is likely to reduce productivity, because energy is a factor of production. So the reduction in energy supplies reduces full-employment output. (c) Better education raises future productivity and output, but has no effect on current full-employment output. (d)The reduction in the capital stock reduces full-employment output (although it may very well increase overall welfare).
Chapter 4Review Questions 1. Saving is current income minus consumption. For a given income, any increase in consumption means an equal decrease in saving, so consumption and saving are inversely related. The basic motivation for saving is to provide for future consumption. 2. When a consumer gets an increase in current income, both current consumption and future consumption increase. That means part of the additional current income will be spent and part will be saved. When the consumer gets an increase in expected future income, again both current and future consumption increase. Because current income does not increase, but current consumption does, saving decreases. When the consumer gets an increase in wealth, both current and future consumption again rise. Again, there has been no increase in current income, so saving decreases.
3. An increase in the real interest rate has two effects on desired saving: (1) The substitution effect increases saving, because the amount of future consumption that can be obtained in exchange for giving up a unit of current consumption rises. (2) The income effect may increase or reduce saving. The income effect reduces saving for a lender, because a person who saves is better off as a result of having a higher real interest rate, so he or she increases current consumption. However, for a borrower, the income effect increases saving, because the borrower is worse off with a higher real interest rate. So the income effect works in different directions depending on whether a person is a lender or a borrower. For a borrower, then, both the income and substitution effects work in the same direction, and saving definitely increases. For a lender, the income and substitution effects work in opposite directions, so the result on desired saving is ambiguous. 4. The expected after-tax real interest rate equals (1 t)i e , where t is the tax rate, i is the nominal interest rate, and e is the expected inflation rate. If the tax rate declines, then the expected after-tax real interest rate will rise. 6. The two components of the user cost of capital are the interest cost and the depreciation cost. The depreciation cost is the value lost as the capital wears out during the period. The interest cost represents the opportunity cost of using the funds to purchase capital instead of another asset, such as a bond 7. The desired capital stock is the amount of capital that allows the firm to earn the largest possible profit. The higher the expected future marginal product of capital, the higher the desired capital stock, because any given amount of capital will be more productive in the future. The higher the user cost of capital, the lower the desired capital stock, because a higher user cost yields lower profits on each unit of capital. The higher the effective tax rate, the lower the desired capital stock, because the firm gets lower profits on each unit of capital. 8. Gross investment is the total purchase of new capital goods during a specified period. Net investment is gross investment minus the depreciation on existing capital. Net investment is the overall increase in the capital stock. Yes, it is possible for gross investment to be positive when net investment is negative. This occurs whenever gross investment is less than the amount of depreciation (as happened in the United States during World War II).
9. Equilibrium in the goods market occurs when the aggregate supply of goods (Y) equals the aggregate demand for goods (Cd + Id + G). Since desired national saving (Sd) is Y Cd G, an equivalent condition is Sd = Id. Equilibrium is achieved by the adjustment of the real interest rate to make the desired level of saving equal to the desired level of investment. The relevant