Please help 3 (a)

3. (85 points) In class, we talked a bit about taxation. The following model is based on the paper by Stokey and Lucas (1983). There are infinite periods and there is no capital in this economy. All agents are identical. A representative agent has a quadratic utility u(c, m) = atr - [c + 2;] where c, is the private consumption of produced good in period t and r, be the private consumption of leisure in period t. The agent will maximize the discounted utility max SBu( C , It ). (1) 1 0 The household's intertemporal budget constraint is given by E Pela - (1 - 14)] =0 (2) where pe is the price at time t and 1 - r, is the amount of labor put in by the household (wage is normalized to 1). Note that the consumer earns income from working only. The resource constraint is given by "t Iitg =1 Vt (3) where the output y (and the technology) is normalized to 1. The government spending is defined as follows 9 = 9 > 0 Vt = 0,2,4,... 4, 9> 0 Vt = 1,3,5, ... Now, we are ready to do our analysis. (a) (20 points) Given equations (1) and (3), formulate and solve the social plan- ner's problem. Solve for the efficient c; and r; for all t as a function of 9. (Remark: the planner's problem is the most efficient result you can get. The social planner will maximize the agent's utility function subjected to the resource constraint).(b) (15 points) Now, consider the case where the government will start levying taxes on households. First, the government levies a lump sum tax of the amount 7'. In this case, the household budget constraint is E Pela + T - (1 - 14)] = 0. (4) 1 0 Using equations (1) and (4), formulate and solve the household's maximiza- tion problem. Show that the optimal consumption in lump-sum tax case c and r, are the same as part (a) so there is no deadweight loss. Also, solve for the interest rates 1 + 7:+1 defined by - = 1 + re+1 as a function of g and B for all t. (c) (20 points) Now, consider the case where the government decides to levy a tax 7 on labor income for all periods (not lump-sum anymore). The consumer's budget constraint then becomes E DelG - (1- 7) (1 - 14)] = 0. (5) Using equations (1) and (5), formulate and solve the household's maximiza- tion problem. Show that the optimal consumption in this labor tax case of and r, are different from part (a) so there is a deadweight loss. Calculate c and r, for all t as a function of T and g. (d) (20 points) Use your result in part (c) to calculate for the welfare loss in consumption and leisure from imposing the labor income tax instead of lump-sum tax (or planner's problem solution) for both odd and even periods. The welfare loss is defined as wc = -9 wx = #-z (e) (10 points) Given the results in part (d), what can you conclude about the efficiency of lump-sum tax and marginal tax? What's the economic intuition about why the efficiency plays out as such