Department of Economics Shanghai University of Finance and Economics Intermediate Macroeconomics

Transcription

1 Department of Economics Shanghai University of Finance and Economics Intermediate Macroeconomics Instructor Min Zhang Answer 3 1. Answer: When the government imposes a proportional tax on wage income, the consumer s budget constraint in period t is now given by: c + s = w(h l)(1 t): where t is the tax rate on wage income. The real wage rate is the relative price of ;eisure in terms of consumption goods, A proportional tax on wage income makes eisure cheaper, thus the (intratemporal) substitution e ect is that the consumer will consume less while enjoy more leisure. The tax also has income e ect which makes the consumer worse o, in that he or she has less consumption and less leisure than before, because both consumption and leisure are normal goods. The net e ect of the tax is to reduce consumption, but the direction of the net e ect on leisure is ambiguous. Although consumption must fall, hours worked may rise, fall, or remain the same. 2. Answer: Lump-sum Tax vs. Proportional Tax. For the proportional tax, the substitutional e ect tells us that the increase in marginal tax rate will lead to more leisure and less consumption, since the relative price of leisure decreases. The income e ect shows that both leisure and consumption will decrease. For the lump-sum tax, there is no substitution e ect because the relative price does not change. The income e ect shows that both consumption and leisure will be lower than before. As a result, since the two types of tax yields the same revenue, consumption will decrease under both of the taxes, but will decrease more under the proportional tax. 3. Leisure represents all time used for nonmarket activities. If the government is now providing for some of those, like providing free child care, households will take advantage of such a program, thereby allowing more time for other activities, including market work. Concretely, this translates in a change of preferences for households. For the same amount of consumption, they are now willing to work more, or in other words, they are willing to forego some 1

2 additional leisure. The increase in labor supply implies increased consumption given xed real wage rate. 4. The rm chooses its labor input, N d, so as to maximize pro ts. When there is no tax, pro ts for the rm are given by = zf (K; N d ) wn d : That is, pro ts are the di erence between revenue and costs. In the following top gure, the revenue function is zf (K; N d ) and the cost function is the straight line, wn d. The rm maximizes pro ts by choosing the quantity of labor such that MP L = w: With a tax that is proportional to the rm s output, the rm s pro ts are given by: = zf (K; N d ) wn d tzf (K; N d ) = (1 t)zf (K; N d ) wn d : where the term (1 t)zf (K; N d ) is the after-tax revenue function, and as before, wn d is the cost function. As before, the rm will maximize pro ts by equating marginal product of labor to marginal cost of labor, which becomes (1 t)mp L = w: So the tax acts to reduce the after-tax marginal product of labor, and the rm will hire less labor at any given real wage. 5. With an employment subsidy, the rm s pro ts are given by: = zf (K; N d ) (w s)n d : where the term zf (K; N d ) is unchanged the revenue function, and as before, (w s)n d is the after-subsidy cost function. The subsidy acts to reduce the cost of each unit of labor by the amount of the subsidy, s. As before, the rm will maximize pro ts by equating marginal product of labor to marginal cost of labor, which becomes MP L = w s: Since the subsidy acts to reduce the marginal cost of labor, the rm will hire more labor at any given real wage. 6. As the rm has to internalize the pollution, it realizes that labor is less e ective than it previously thought. It now needs to hire N(1+x) workers where N were previously su cient. This is qualitatively equivalent to a reduction of z, total factor productivity. So the rm now hires fewer people for a given wage and thus its labor demand is reduced. 2

3 7. There are three e ects at work in this example. First, there is a negative income e ect from the increase in taxes needed to pay for the increased government spending. This e ect tends to lower both consumption and leisure. Second, there is a substitution e ect due to the productive e ect of the increase in G. Since the rise in G is similar to a rise in K; the marginal product of labor rises, which leads to a higher real wage rate in the labor market equilibrium. The increase in real wage induces substitution e ect, which leads to an increase in consumption and a decrease in leisure. Third, there is a positive income e ect from the increase in G on productivity. Since the rise in G is similar to a rise in K; the output produced rises as a result. This e ect tends to increase both consumption and leisure. In general, consumption may rise or fall, and leisure may rise or fall. The overall e ect on output is the same as in any increase in total factor productivity. Output surely rises. 8. Given information: y = 100 y0 = 120 t = 20 t0 = 10 r = 0:1 (a) To calculate wealth, we compute: w = y t + y0 t0 1 + r (b) The consumer s budget constraint is = :1 = 180: c + c0 = c + 0:91c0 = lifetime wealth = 180: 1 + r Since the current and future consumption are perfect complements for the consumer and he or she always wants to have equal consumption in the current and future periods, we have c = c0: simultaneously solves the above equations, we have c = c0 = 94:2: 3

4 Savings is therefore given by: s = y t c = 80 94:2 = 14:2: We can conclude that the consumer is a borrower. (c) First-period income rises from 100 to 140. We now recompute w = 220. Solving as in part (b), we nd that c = c0 = 115:2, and s = 4:8. (d) In part (b), the consumer is a borrower. In part (c), rst-period income increases and savings has consequently increased enough that the consumer is now a lender. 9. In this problem, there is a simultaneous increase in both future income and the real interest rate. The increase in future income is a positive income e ect for both borrowers and lenders. The increase in the real interest rate includes a pure substitution e ect and a pure income e ect. The substitution e ect induces the consumer to consume less in the current period and more in the second period. The direction of the pure income e ect part of the real interest rate change depends on whether the consumer is a borrower or a lender. Lenders are better o with a higher real interest rate and borrowers are worse o with a higher real interest rate. 10. (a) This problem involves a rm s o er to provide an interest-free advance on the consumer s income. If the consumer takes the advance, then his lifetime wealth (we) is given by: we = y + y0 1 + r + x( r ): Therefore, for any positive r, provided that the consumer should take the advance, as any increase in his lifetime wealth makes him better o. (b) Since the interest rate does not change, there is no intertemporal substitution e ect. The income e ect implies that the consumption this year and next year will both increases. Saving this year will increase in order to transfer part of the income this year to the consumption next year. 11. Temporary and Permanent Tax Increases. (a) The increase in rst-period taxes induces a parallel leftward shift in the budget line of the rst-period. Due to the income e ect, the consumer reduces both current and future consumption. That is, the consumer will spread the decrease in current period income to both consumption of current and future periods. First-period consumption falls by less than the increase in taxes and so savings falls. 4

5 (b) Now consider a permanent increase in taxes. A permanent tax increase also induces a parallel leftward shift in the budget line of both periods. Due to the income e ect, the consumer reduces both current and future consumption. The di erence is that the First-period consumption falls by more or less compared with the increase in taxes and so savings may either rise, fall, or remain unchanged. 12. A tax on interest income. (a) Initially, denote AB in the rst gure below as the consumer s budget constraint. Point E is called endowment point. At E, consumers do not save or borrow. That is, they eat up all they have in each period, c and c are respective consumptions in current and future periods. Any point along line AE represents budget constraint for consumers who save, since they consume less than what they have in the current period, and carry forward what is left as savings into the future period to support higher consumption. Likewise, any point along BE represents the budget constraint for consumers who borrow. The introduction of the tax results in a kink in the budget constraint, since the interest rate at which the consumer can lend, is now smaller than the interest rate at which the consumer borrows, r. The kink occurs at the endowment, E. (b) In the case that a consumer who was a borrower before the imposition of the tax, this consumer is una ected by the introduction of the tax. In another case that a consumer who was a lender before the imposition of the tax, there are both income and substitution e ect. The substitution e ect results in an increase in rst-period consumption and a reduction in second-period consumption. The income e ect reduces both rst-period and second-period consumption, and increases savings. On net, consumption must fall in period 2, but in period 1, consumption may rise or fall. 5

6 13. Assume that the time path of government spending G t is given. Benchmark case: no tax cut, but positive initial government debt of one unit. Denote B g 0 =p as initial government debt and Bg 0 =p: At period 0: G 0 = T 0 + B g 0 =p: = T 0 + 1: At period 1: the government repays the interest and principle of the debt issued in period 0 by taxation. G 1 + (1 + r) = T 1 : From period 2: the government keeps balanced budget and the government debt goes to zero. G t = T t for t = 2; 3; 4::: Case 2: tax cut in period 1 and B g 0 =p = 1: 6

7 At period 0: G 0 = T 0 + B g 0 =p: = T 0 + 1: At period 1: tax cut of one unit, the government repays the interest and principle of the debt issued in period 0 by taxation and issuing debt. Since And T 1 drops by one unit, one has B g 1 =p = 1 to keep the budget balanced: G 1 + (1 + r) = T 1 + B g 1 =p: At period 2: the government repays the interest and principle of the debt issued in period 1 by taxation. G 2 + (1 + r) = T 2 : From period 3: the government keeps balanced budget and the government debt goes to zero. G t = T t for t = 3; 4::: Comparing the benchmark case with case 1, one can compute the present value of changes in taxes r 1 + r = 0: Thus, we can conclude that the Ricardian Equivalence Theorem still holds when initial government debt is positive. The intuition is the same as the simple case. In face of a tax cut, households anticipate that for given government spendings and balanced budget strategy of the government, they would have to pay higher taxes in future period. In response, the households spend the extra income on purchase of the bonds which yield them (1 + r) units of income in future period, exactly the amount of extra taxes they are required to pay in the future period. Reacting in this way leaves no changes in the time path of consumption. More generally, since the present value of changes in taxes is zero, which implies there is no change in the disposable incomes faced by households, therefore, no income e ect on consumption. 7