Economics 2 Spring 2016 Professor Christina Romer Professor David Romer SUGGESTED ANSWERS TO PROBLEM SET 4 1. The demand curve for auto orkers is derived from profit maximization on the part of automobile manufacturing firms. It is the donard-sloping marginal revenue product of labor (MRP L) curve because profit-maximizing firms ant to hire labor up to the point here the MRP L of another orker is just equal to the going age. The supply curve of auto orkers is derived from utility maximization on the part of households. As discussed in class, the income and substitution effects of a age change on labor supply go in opposite directions. Hoever, as an empirical matter, the substitution effect is typically stronger than the income effect, so the labor supply curve is upard sloping. The equilibrium age and level of employment of auto orkers are determined by the intersection of the labor demand curve and the labor supply curve in this market. a. Widespread movement of orkers out of Michigan ill decrease the number of people illing to ork in the auto industry at a given age. Thus, it ill shift the supply curve of autoorkers to the left (from to S 2). This shift back in the labor supply curve ill increase the equilibrium age of auto orkers (from 1 to 2) and decrease the equilibrium level of employment (from L 1 to L 2). Ho much the outmigration is likely to affect the age in this industry depends on many factors. For example, the more elastic labor demand is, the less the age rises and the more employment falls. 2 1 L 2 L 1 S 2 L b. The left-hand diagram on the next page shos the effect of the fall in the orld price of cars. The quantity of cars produced in the U.S. falls (from Q US S1 to Q US S2 ), the quantity of cars purchased US in the U.S. rises (from Q D1 to Q US D2 ), and the quantity of imports rises (from Imports 1 to Imports 2). In terms of the market for U.S. autoorkers, the key effect comes directly from the fall in the orld price of cars. The fall loers the marginal revenue product of auto orkers. The MRP L is the extra revenue generated by another orker. It is equal to the marginal (physical) product of another orker times the marginal revenue associated ith selling one more unit of the good. Assuming that the market for cars is competitive, the marginal revenue associated ith selling one more unit is just the going market price for cars. Because the price of the output has fallen, the MRP L of U.S. autoorkers has fallen. Thus, this development ill shift back the labor demand curve for U.S. autoorkers (from to D 2 in the right-hand diagram on the next page). As a result, the age and employment of U.S. autoorkers ill decrease (from 1 to 2 and from L 1 to L 2, respectively). Since the first diagram shoed that output of the U.S. auto industry fell, the fact that the second diagram shos that employment also falls means that the to diagrams are telling a consistent story. (Note: We have dran D 2 as asymmetrically belo in the labor market diagram. This makes sense because the MRP L is marginal product times price. Therefore, if e reduce the price, the ne MRP L is loer by a fixed proportion, not by a constant vertical distance.)
This problem shos that increased foreign supplies of a good can indeed reduce employment in the industry producing that good in the United States. But, it also makes clear that American consumers gain from this development because the price of something that e buy falls. 2 Price of Cars S US 1 2 P W1 P W2 D US D 2 Q US US S2 Q S1 US US Q D1 Q D2 Q of Cars L 2 L 1 L Imports 1 Imports 2 c. The neer, more efficient machines ill make autoorkers able to produce more in a given amount of time. This ill increase the marginal (physical) product of autoorkers the amount produced by another orker ill no be greater than before at every level of employment. This increase in the marginal product of labor ill tend to increase the marginal revenue product of labor (hich is marginal product times marginal revenue). The only complication is that the increase in productivity ill tend to shift out the supply curve of cars and hence loer the price. For a competitive industry, marginal revenue is just equal to the prevailing price of the good. Therefore, this reduction in the price ill tend to loer the MRP L. Theoretically, this negative effect on marginal revenue product could counteract the positive effect of the increase in the marginal product. Hoever, demand for cars ould have to be quite inelastic for the shift out in supply to decrease price by as much or more than the marginal product of labor increased. For this reason, it is reasonable to suppose that the marginal revenue product of autoorkers ill increase because of the more efficient machines. Since the demand curve for labor is the marginal revenue product of labor, the demand curve for 2 autoorkers ill shift out (from to D 2). The shift 1 out in the demand curve ill result in a rise in both the equilibrium age of autoorkers (from 1 to 2) and the equilibrium level of employment (from D 2 L 1 to L 2). This problem illustrates the important point that technological progress and investment L 1 L 2 L are typically beneficial to orkers.
3 d. When there is a union that negotiates a age that is above the age that equates the supply and demand for labor, the market is not in equilibrium. At the initial negotiated age ( N1), the supply of labor (L S1) exceeds the demand (L D1). The number of orkers hired is L D1, and there is unemployment among autoorkers. N2 N1 If the UAW is able to negotiate a age, N2, that is higher than before, automakers ill hire feer autoorkers than before. A profitmaximizing firm only hires labor up to the point here the marginal revenue product of another orker (MRP L) is equal to the prevailing age. If L D2 L D1 L S1 L S2 L the union succeeds in raising the prevailing age, Unemployment 1 firms ill respond by cutting back on employment until the MRP L of another orker is equal to the Unemployment 2 higher age. The employment of autoorkers ill fall (from L D1 to L D2). Ho much employment falls ill depend on the elasticity of labor demand. For example, if demand is quite inelastic, the fall in employment ill be small. The number of autoorkers ho ant to ork ill rise (from L S1 to L s2), but that has no impact on employment, since there ere already more autoorkers ho anted jobs than ere able to get them. 2. The condition that must hold for a firm to be purchasing the profit-maximizing amount of capital is that the present value of the stream of expected future marginal revenue products generated by another machine is equal to the purchase price. Or, in symbols: PV(Stream of Future MRP K s) = Purchase Price of the Machine Today. Since present value depends on the interest rate and the MRP K depends on ho much capital is purchased, this condition implies a negative relationship beteen the interest rate and the amount of ne capital demanded. We call this relationship the investment demand curve. ( is just the name e give to purchases of ne capital.) Such a relationship exists for each individual firm and for all of the firms in the economy aggregated together. a. To see hat the increase in the effective price of capital goods does to the relationship beteen the interest rate and the amount of investment firms ant to do, e need to look at the purchase condition. If a firm as buying the optimal amount of capital at a given interest rate before the change in the tax code, it no ants to buy less capital. This is true because the rise in the purchase price makes the purchase price greater than the present value of future marginal revenue products. Only by moving up the declining marginal revenue I 2 I 1 product of capital curve can the firm raise the present value of future marginal revenue products and restore the condition for profit maximization to equality. This decrease in ne capital demanded at a given interest rate corresponds to a shift in of the investment demand curve (from I 2 to I 1).
4 b. If the Federal Reserve raises interest rates, this ill loer the present value of the future marginal revenue products of capital. This is true because present value depends negatively on the interest rate. If firms had been purchasing the optimal amount of capital before the rise in interest rates, they no ant to purchase less capital. This is precisely the behavior that is captured by the investment demand curve. A rise in interest rates therefore results in a movement along the investment demand curve (from a point such as A to a point such as B). B A I 1 c. Because firms do not kno for sure hat the future marginal revenue products of a machine ill be, they have to form expectations. A sitch to less optimistic expectations means that firms loer their guesses about the future MRP K s. This ill loer the present value of the stream of future MRP K s at a given interest rate. If a firm had been buying the optimal amount of capital at a given interest rate before the change in expectations, it no ants to buy less at the same interest rate. By doing so, it ill move up the declining marginal revenue product of capital curve and restore the profit-maximization condition to equality. This corresponds to a shift back in the investment demand curve (from I 1 to I 2). I 2 I 1 d. Technological progress that makes each machine able to produce more output than before raises the future MRP K s. If a firm had been buying the optimal amount of capital at a given interest rate before the increase in MRP K s, it no ants to buy more at the same interest rate. By doing so, it ill move don the declining marginal revenue product of capital curve and restore the profit-maximization condition to equality. This increase in ne capital demanded at a given interest rate corresponds to a shift out in the investment demand curve (from I 1 to I 2). I 1 I 2 3.a. To convert the 1973 price into today s dollars, e need to multiply the 1973 price by the ratio of the CPI today to the CPI in 1973: The 1973 price in today s dollars is given by: Price in 1973 * CPI today / CPI 1973. The problem states that the price in 1973 as $2500, that the CPI today is 236.9, and that the CPI in 1973 as 44.4. Thus, the 1973 price in today s dollars is $13,339. Since the price of a Beetle today is about $20,000, this means that relative to a basket of other goods, a Beetle as cheaper in 1973 than today.
(Of course, a 1973 Beetle and a 2016 Beetle are not identical. For all its charms, the 1973 Beetle as much less reliable, safe, environmentally-friendly, and comfortable than today s Beetle.) b. When the percent changes in the numerator and denominator of a ratio are small, the percent change in the ratio is roughly equal to the percent change of the numerator minus the percent change of the denominator. Thus if real GDP gros by 4% and population increases by 1%, then the ratio of the to, real GDP per capita, rises by about 3%. With the numbers given, real GDP rose from 100 billion divided by 100 million, hich is exactly 1000, to 104 billion divided by 101 million, hich is 1029.70297. Thus to three decimal places, the increase in real GDP per capita is 2.970% very close but not identical to the approximate anser of 3%. 1 c. The correct anser is ii. A fall in the unemployment rate from 6% to 5% is a fall of 1 percentage point (and a fall of 17 percent). d. The correct anser is i. Prices rose 2% from 2013 to 2014, and by an additional 1% from 2014 to 2015, for a total increase of approximately 3%. (More precisely, prices rose by a factor of 1.02 from 2013 to 2014 and by a factor of 1.01 from 2014 to 2015. Thus prices in 2015 ere higher than prices in 2014 by a factor of 1.02 1.01 = 1.0302, so prices rose by 3.02%.) 4.a. False. In the absence of the tax, the labor demand curve is the marginal revenue product of labor curve. Firms ant to hire orkers up to the tax point here the marginal revenue product of labor is just equal to the age. Another ay to think about the labor demand curve is that it shos hat firms are illing to pay for a given quantity of labor. Without the tax, firms are illing to pay the 2 +tax 1 2 marginal revenue product of labor. With the tax, hoever, firms are illing to pay a loer age at each level of employment because they have to pay D 2 a tax in addition to the age. The total amount they are illing to pay (the age plus the tax) ill be equal to the marginal revenue product of labor. L 2 L 1 L This implies that the labor demand curve shifts don (from to D 2). (In fact, e can describe the amount that it shifts by: it shifts don by the amount of the tax.) The equilibrium age and employment fall (from 1 to 2 and from L 1 to L 2). Notice that at the ne equilibrium, the total amount firms are paying ( 2 plus the tax) is equal to the marginal revenue product of another orker. b. False. The attractiveness of college depends on a comparison of the upfront cost and the present value of the higher future earnings. A rise in interest rates has no effect on the cost 5 1 For those of you ho ould like to kno: To see hy the approximation orks: First, rite 1.04/1.01 as 1.04 times 1/1.01. No notice that 1/1.01 is very close to 0.99; it is 0.99 + (0.0001/1.01). So e can rite 1.04/1.01 as the product of 1 + 0.04 and 1 0.01 + (0.0001/1.01). When e do this multiplication, e get 1 + 0.04 0.01 and three very small terms: 0.04 0.01, 1 (0.0001/1.01), and 0.04 (0.0001/1.01). As a result, the anser is very close to 1 + 0.04 0.01, hich is 1.03 that is, a rise of 3 percent. (Finally, for those of you ho have encountered Taylor series approximations in math: a first-order Taylor series approximation of (1 + a)/(1 + b) around a = b = 0 is 1 + a b.)
(hich by assumption is paid today), but reduces the present value of the higher future earnings. Thus, a rise in interest rates tends to make going to college less attractive. Notice that the condition for hether buying a college education makes sense is very similar to the condition for hether buying a ne machine makes sense. As described in the anser to Problem 2, a firm should buy a ne machine if the present value of the stream of the machine s future marginal revenue products is greater than the purchase price. A rise in the interest rate reduces the present value of the stream of future marginal revenue products, and so makes buying the machine less attractive. That is hy the investment demand curve slopes don. Likeise, going to college hich is a purchase of human capital makes sense financially if the present value of the stream of future higher earnings hich e can think of as the marginal revenue products of the human capital is greater than the price of college today. A rise in the interest rate reduces the present value of the stream of future higher earnings, and so makes buying a college education less attractive. To see this more formally, suppose the cost of college is C and is paid immediately; that college raises one s earnings by $E 5 5 years from no, by $E 6 6 years from no, by $E 7 7 years from no, ; and that the interest rate is i. Then the present value of the benefits of going to college is ($E 5 [1 + i] 5 ) + ($E 6 [1 + i] 6 ) + ($E 7 [1 + i] 7 ) +. A higher value of i makes this present value loer, and so makes going to college less attractive. One final comment: it is a simplification to assume that the full cost of going to college is paid right aay. In practice, the cost (in the form of tuition and foregone earnings) is likely to be paid over the 4 years the student is in college. As a result, a rise in i ould loer the present value of the cost somehat. But because the cost is paid relatively soon, the effect ould be small. For the higher earnings, hich are earned many years into the future, the effect of a rise in i ould be much larger. c. True. We can see this using the to-panel diagram shoing the labor markets for lo-skill and high-skill orkers. Technological progress that raises the productivity of lo-skill orkers means that the marginal revenue products of lo-skill orkers are higher than before. Thus, the demand for lo-skill orkers shifts out (from D L1 to D L2). The problem says that the productivity of high-skill orkers is unaffected. Thus, the supply and demand for high-skill orkers do not change. 6 As the diagrams belo sho, the ages and employment of lo-skill orkers rise (from L1 to L2 and from L L1 to L L2), hile the ages and employment of high-skill orkers are unaffected. By raising the ages of lo-skill orkers and not affecting the ages of high-skill orkers, the technological change reduces income inequality. Lo-Skill High-Skill L H S L S H W H1 L2 L1 D L2 D H D L1 L H1 L H2 L L L H1 L H