Professor Christina Romer SUGGESTED ANSWERS TO PROBLEM SET 5

Transcription

1 Economics 2 Spring 2018 Professor Christina Romer Professor David Romer SUGGESTED ANSWERS TO PROBLEM SET 5 1.a. The change in the marginal tax rate that households pay will affect their labor supply. Recall households condition for utility maximization in deciding how much labor to supply: MU Leisure P Leisure = MU Everything else P Everything else. In the absence of taxes, the price of leisure is the wage: what the household gives up by consuming one more unit of leisure is the amount they would have earned if they had worked instead. When there are taxes, however, what the household gives up is the wage it would have earned minus the tax it would have paid on those earnings. What determines the tax it would have paid is the tax rate on an additional dollar of earnings that is, the household s marginal tax rate. Thus, the price of leisure is w t w, or (1 t) w, where t is the marginal tax rate. This discussion implies that an increase in the marginal tax rate reduces the price of leisure. If the household had been maximizing utility before, now the MU Leisure/P Leisure is greater than the MU Everything Else/P Everything Else. One would therefore expect the household to increase the amount of leisure it consumes at a given wage in order to reduce MU Leisure, and so restore the condition for utility maximization to equality. Consuming more leisure corresponds to supplying less labor. Thus, the labor supply curve will shift to the left (from to ). As a result, the normal real wage will rise (from to ) and normal employment will fall (from to N 2 ). N 2 N* b. The rise in the payroll tax will affect labor demand. The condition for hiring the profitmaximizing amount of labor is that the marginal revenue product of labor (MRP L) equals the cost of labor to the firm. In the absence of taxes, the cost of labor to the firm is just the wage that it pays. But when the firm has to pay a payroll tax to the government, the cost of labor is the wage plus the payroll tax. Specifically, the amount that a unit of labor costs the firm is the wage plus the payroll tax rate times the wage: w + τ w, where τ is the payroll tax rate. Thus, the firm wants to hire labor until N 2 N* MRP L = (1 + τ) w. When τ rises, the firm will reduce the number of workers it wants to hire at a given wage (and so increase MRP L) in order to continue to maximize profits. Thus, the labor demand curve will shift to the left (from to ). As a result, both the normal real wage and normal employment will fall (from to and from to N 2 ).

2 c. If there is some force that holds the real wage above its equilibrium level, employment will be determined by the quantity of labor demanded at that set real wage (w ). In such a situation, a leftward shift of the labor supply curve, by decreasing the gap between the quantity of labor supplied and the quantity of labor demanded at the prevailing real wage, will reduce normal unemployment. This is shown in the left-hand diagram below: the shift in the labor supply curve (from to ) reduces normal unemployment (from Unemployment 1 to Unemployment 2). A leftward shift of the labor demand curve, on the other hand, increases the gap between the quantity of labor supplied and the quantity of labor demanded at the set real wage, and so increases normal unemployment. This is shown in the right-hand diagram below: the shift in the labor demand curve (from to ) increases normal unemployment (from Unemployment 1 to Unemployment 2). Thus, when there is a force preventing the real wage from falling to the level that equates supply and demand, an increase in the payroll tax rate is more likely than an increase in the marginal income tax rate to increase normal unemployment. 2 w w N D1 N S2 N S1 N* Unemployment 1 Unemployment 2 N D2 N D1 N S1 N* Unemployment 1 Unemployment 2 (This discussion leaves out one complication. We have assumed that w, the wage determined by some factor like efficiency wages, does not respond to economic developments. However, in some cases it will. For example, suppose firms pay above the market-clearing wage to give its workers an incentive not to shirk, or because it engenders feelings of loyalty and so causes workers to work hard. The benefits to the firm from paying such wages probably depend mainly not on the absolute level of the real wage that it pays, but on the real wage relative to the marketclearing wage. In that case, the wage it pays might adjust when the market-clearing wage changes so that the premium above the market-clearing wage, and thus the amount of unemployment, would be roughly the same. In that case, neither the increase in the marginal tax rate nor the increase in the payroll tax rate would have a big effect on unemployment.) 2. The tool we use to analyze the determination of the normal real interest rate and normal investment is the long-run saving and investment diagram. The condition that must hold for a firm to be undertaking the profit-maximizing amount of investment is that the present value of the stream of expected future real marginal revenue products generated by another unit of capital is equal to the purchase price: PV(Stream of Future Real MRP K s) = Purchase Price of New Capital Today. Since present value depends negatively on the real interest rate and the MRP K s depend on how much capital is purchased, this condition implies a negative relationship between the real interest rate and the amount of investment firms want to do. This relationship

3 is the investment demand curve. That is, I is a decreasing function of r*. The normal supply of saving, S*, equals Y* G* C*. Since a rise in r* increases the opportunity cost of current consumption in terms of future consumption, current consumption is a decreasing function of r* which means that S is an increasing function of r*. In the diagrams below, and are the initial investment demand and saving supply curves. 3 a. As noted above, the condition for the profitmaximizing level of investment is PV(Stream of Future Real MRP K s) = Purchase Price of New Capital Today. An investment tax credit means that firms get tax benefits whenever they buy new capital; thus it reduces the effective purchase price of new capital. Thus if firms did not do more investment at a given r, PV(Stream of Future Real MRP K s) would be greater than the effective purchase price. By doing more investment at a given r, the firm can reduce PV(Stream of Future Real MRP K s), and so restore the condition for profit maximization back to equality. Thus, investment demand at a given r is higher than before; that is, the investment demand curve shifts to the right (from to I 2). r* r 2 r 1 I 2 I 2 S*,I* As the diagram shows, the outward shift of the investment demand curve leads to an increase in the equilibrium real interest rate (from r 1 to r 2 ) and an increase in normal investment (from to I 2 ). b. As discussed above, normal national saving (S*) is potential output (Y*) minus normal consumption (C*) and normal government purchases (G*): r* S* = Y* C* G*. If consumers reduce their consumption at any given level of the real interest rate, national saving at any given level of the real interest rate will be higher. This increase in saving at every level of the real interest rate corresponds to a shift out in the saving function (from to ). The shift out in the saving function leads to an decrease in the equilibrium real interest rate (from r 1 to r 2 ) and an increase in normal investment (from to I 2 ). r 1 r 2 I 2 S*,I* 3. Planned aggregate expenditure () is the total amount of spending people plan to do. It is the sum of consumption (C), the total amount consumers want to spend; planned investment (I P ), the amount firms plan to invest; government purchases (G); and net exports (NX), the amount foreigners want to buy from us minus the amount Americans want to buy abroad. Thus, = C + I P + G + NX. The expenditure line (also identified as ) shows how planned spending varies systematically

4 with total output. It is upward sloping because consumption rises with total output (which is the same as total income). Its slope is less than one because people typically save at least part of every extra dollar of income they receive. In the short run, equilibrium output is determined by the intersection of the expenditure line and the 45-degree line. The 45-degree line represents the equilibrium condition that total output must equal total spending (Y = ) for the economy to be in balance. This line also captures the behavioral assumption that firms change output in response to changes in planned spending in the short run. 4 a. Adding up the components of planned expenditure in this numerical example yields: = Y. Y = (45-degree Line) Therefore, the equation for the expenditure line is: = Y. The expenditure line has an intercept of 4500 and a slope of 0.5. The coefficient on output in the equation reflects the sensitivity of consumption to output. This coefficient is called the marginal propensity to consume (MPC). In this example, the MPC is 0.5, which means that if consumers get another dollar, they will spend 50 cents of it and save 50 cents. If you draw the expenditure line and the 45-degree line carefully, equilibrium output in the short run appears to be around b. The two equations that determine equilibrium output are Y = and = Y. Therefore, to solve for equilibrium output algebraically, all one does is substitute the second equation into the first. This yields: Y = Y (1 0.5)Y = 4500 Y = 4500/0.5 Y = (Expenditure Line) Y c. If government purchases increase to 1500, this changes the equation for the planned expenditure line to: Y = = Y. 2 1 Graphically, this is a shift up in the expenditure line by 500 at each level of Y (from 1 to 2). The new level of equilibrium output looks to be about 10,000. Algebraically, the new level of equilibrium output is determined by calculating: Y = Y (1 0.5)Y = 5000 Y = 5000/0.5 Y = 10, Y

5 4. A given change in government purchases will have a larger impact in the country with the larger MPC. That is, the multiplier will be larger in the United states, with its higher MPC, than in Japan, with its lower MPC. 5 The diagram to the right shows this graphically. A given increase in G will increase planned expenditure at a given level of Y by the amount of the increase in G. That is, the expenditure line shifts up vertically by the amount of the increase in G, regardless of the MPC. Thus, the line shifts up from 1,J to 2,J in Japan, and from 1,US to 2,US in the United States. (To make the comparison easier, the diagram is drawn so that the initial levels of output are the same in the two countries, but this is not essential.) Because the MPC is higher in the United Sates than in Japan, the line for the U.S. is steeper than the line for Japan. As the diagram shows, the resulting rise in output (Y) in response to a given rise in G is larger in the high MPC country (the United States) than in the low MPC country (Japan). Y = 2,US 1,US 2,J 1,J Y 1 Y 2,J Y 2,US Y The multiplier comes from the fact that as income rises, households increase their consumption spending. When G rises, this directly raises income. This increase in income raises households consumption spending, which increases income further. And this additional increase in income raises households consumption further and so on. All of these effects stem from the fact that when a household s income increases, they devote some of that additional income to consumption that is, they stem from the fact that the MPC is positive. When the MPC is greater, all of these effects are greater. The result is that the extent to which the impact of the increase in G on Y is magnified is greater when the MPC is greater. 5. Under the policy of not hiring married women, firms were not maximizing profits: firms would not hire married women even if their marginal revenue product was greater than their wage. With the end of these policies, firms would hire more married women at a given real wage. Thus, the labor demand curve shifted out (from to ). As the diagram shows, the normal real wage of married women working outside the home would rise (from to ), as would their normal employment (from to N 2 ). (If the change in policy eliminated all discrimination against married women, the new labor demand curve,, would be the same as the MRP L of married women. If some discrimination remained, however, it would still be below the MRP L curve.) N 2 N*

6 6 It is possible that the change in employers policies also led to a change in households attitudes about married women working outside the home. In this case, the change could have led not only to an outward shift of the labor demand curve, but also to an outward shift in the labor supply curve, as the change in attitudes and tastes led married women to supply more labor at a given real wage. In this case, there would have also been an outward shift in the labor supply curve (from to ). This change would have further increased the number of married women working outside the home (the change is now from to N 2 ). But it would have acted to push the real wage down, offsetting some of the increase from the outward shift in labor demand. In the case shown, the overall effect is still positive (the normal real wage rises from to ), but it could be zero or negative. N 2 N* 6. Our framework for thinking about potential output per person is: Y POP = Y N N POP = f K N, T N POP, where Y is real GDP, POP is population, N is employment, K is capital, and T is technology, and where a * denotes the normal value of the variable. For changes in the normal employment-topopulation ratio to be an important source of slowdown in the growth rate of potential output per person, we would need not just a fall in N*/POP, but continued falls (or a shift from continued increases to slower growth or to sustained falls). Of course, we do not have data on N*; thus we must try to infer the behavior of N*/POP from things we can observe. The behavior of the actual (as opposed to normal) employment-topopulation ratio will be affected by short-run booms and recessions through their influence on the unemployment rate. Thus, a better way to get a sense of the behavior of N*/POP than looking at N/POP is to look at the fraction of the population that is in the labor force rate (that is, the fraction of the population that wants to work, even if they are currently unemployed). As the graph on the next page shows, there was a fall in that fraction in the Great Recession and immediately after. But the decline was fairly small about 3% (from about 50.5% to about 49%) and it appears to have ended. Thus, it cannot explain a sustained fall in the growth rate of potential output per person of 1 percentage point or more per year. 1 1 The chart also shows that even standard data series can have peculiar features: the chart shows a large one-time jump in the fraction of the population in the labor force in January It was due to the Bureau of Labor Statistics annual population adjustments.

7 The other candidate sources of the slowdown in the growth rate of normal output per person are slower growth of normal capital per worker (K*/N*) and slower growth of technology. And indeed, both of those are sources of concern. Physical capital investment as a share of GDP has been low for much of the past decade, and there are worries about both the quantity and quality of education that American workers are getting. Thus, there are reasons to think that K*/N* has been growing more slowly in recent years than it was before. And measures of the amount of output we are getting for a given set of inputs have been growing very slowly since the Great Recession, suggesting slower technological progress. Although various ideas have been proposed about why this might be the case, the slowdown is not well understood. 7