Questions on microeconomics, provide solutions for the same
Consider an economy with two types of individuals: skilled and unskilled workers. The only differ- ence between the two is that the skilled have a higher hourly wage w. = 40 than the unskilled do, wu = 10. Suppose that there are 400 unskilled and 100 skilled workers in this economy. Suppose that each individual has a utility function over consumption (c) and leisure (1) of the following form: U(c, 1) = In(c) + 2In(1) where I E [0, 24]. 1. (a) (3 minutes) Write down the individual's budget constraint in terms of consumption and leisure. Draw the budget constraints for the skilled and unskilled workers in the same graph with leisure on the r-axis. (b) (7 minutes) Solve for each individual's optimal leisure, labor, and consumption choice. 2. (10 minutes) Now suppose that the government wants to redistribute from the skilled to the unskilled workers. It levies an income tax which collects 20% of each skilled worker's earnings and then uses the tax revenue to give an equal amount (lump-sum transfer) T to each unskilled worker. So only the skilled workers are taxed and only the unskilled workers receive the transfer. (a) (4 minutes) On the same set of axes, draw the new budget constraints faced by the two types of individuals. (b) (4 minutes) Solve for each skilled individual's new optimal leisure, labor, and consump- tion. Does labor supply change? What is the intuition behind this result? (c) (2 minutes) Compute the total tax revenue collected by the government from taxing the skilled individuals. 3. (a) (4 minutes) Suppose that for every tax dollar collected, 6.25 cents are lost due to admin- istrative costs. Suppose that the government sets 7 so that it spends what it collects (the government balances its budget). How large is T"?Suppose that households live for two periods t = 1,2 and die at the end of period 2. They have wealth in the rst period W 2- [l but no wealth in the second period. Their utility over consumption in the first and semnd period is given by Ui1521= EE'I-v where c1 is consumption in period 1, cg is consumption in period 2 and 13 E [i]. l] is a preference parameter. Buying 1 unit of consumption costs $1 in both periods. The households can save by investing in a safe asset at the market interest rate 1*. Assume that the household gets no utility from leaving any money behind after death. 1. {4 minutes) Write down the household budget constraints for periods 1. 2. Then, write an expression for the household's intertemporal budget constraint in terms of today's dollars. 2. {3 minutm} How much of its immme will the household consume in each of the two periods and how much will it save given the interest rate r? 3. (4 rninutm} Does increasing ,5 increase or decrease savings? What is the intuition behind this result? cl. {4 minutes} Does a higher interest rate increase or decrease savings? Provide the intuition for this result. 5. {20 minutes} Now suppose that 13 = 1 and that there are two types of households: rich and poor. They differ only in their rst period wealth which is W = Jr for the rich and W = for the poor for some It 2: [1. Now. the households can only save by investing in a risky asset {the safe asset option of the previous questions is not available now]. This asset requires an investment of exactly '3' and will yield gross second period income equal to l: with probability US and gross second period income oil] with probability 2H. Note that households can only buy exactly one unit of the risky asset. Households seek to maximise their expected utility of consumption. {a} {3 minutes] Will the poor households choose to invest in the risky asset in order to transfer resources to the next period? Why or why not? (b) {12 minutes] 1Will the rich households choose to invat in the risky asset? Is your answer different from the previous subqution? Provide an intuition for why or why not. There are three periods, t = 0, 1,2. In t = 1 Mary maximizes her utility over leisure and consumption given the following function: UI(M, CI) = NICE subject to the following budget constraint: Ci + wiN = 24w1 where wi = 10. Note the price of the consumption good is assumed to be one in all periods. After she has made this decision, in t = 2 she maximizes this utility function: U2(N2, C2) = Nici subject to the following budget constraint: C2 + w2 N2 = 24w2 where w2 = 20. (a) (6 points) For t = 1, 2 calculate Mary's choice of leisure and consumption in each period. (b) (6 points ) For t = 1, provide economic intuition for the income and substitution effects of a wage increase on leisure. Can you say anything about the relative magnitudes of these income and substitution effects? (c) (7 points) Go back to your solution in part (a). If the interest rate is 10% per period, what is the present value of her consumption in t = 0? Please use 0.9 and 0.8 as approximations for 1/(1.1) and 1/(1.1) respectively. (d) (7 points) Mary now has the option of obtaining additional job training in t = 0 at an investment cost of $200. As a result, her wage rate increases in t = 1 to wj = 20 and in t = 2 to w2 = 30. Calculate the net present value of this investment on consumption. Consider only the value of consumption (and not the value of leisure). (e) (7 points ) For more general utility functions, when will the net present value of the investment on consumption from part (d) likely be negative? Use income and substi tution effects in your explanation. (f) (7 points) Does Mary have a Laffer curve for income taxes (as opposed to consumption taxes)
