Let us have an economy that lives only for 2 periods. The representative household has a utility function that values consumption goods for any period as follows

u(c) = ln(c).

The valuation throughout both periods can be represented as a total utility function that discounts future utility with a β discount parameter as follows,

U(c1, c2) = u(c1) + βu(c2) = ln(c1) + β ln(c2),

Where ct represent consumption good in period t for every period t = 1, 2. The budget constraint for this person in every period is expressed as

(1+τtc)ct + xt =(1−τtI)(wtnt +rtkt),   t=1,2.

where xt represent investment in period t, wt the wages earned by hours worked, rt the return of capital in period t, nt the hours worked supplied by the household in every period t, kt the stock of capital that the household has in period t, τtc the tax rate imposed by the government in consumption goods in every period t, and τtI the tax rate imposed by the government to total household income in every period t. In every period, the total amount of hours that the representative household is of H ¯ = 1. Investment in every period can be defined by the following law of motion of capital

kt+1 = xt +(1−δ)kt,   t=1,2.

Acknowledge that k3 = 0 because there is no period 3 in this economy. The household starts with a given amount of capital k ¯1 > 0. The representative firm produces good by the following production function in every period t,

Yt =ztKtαNt1−α, t=1,2.

There is a government that covers government expenditures in period 1 and period 2 by taxing capital rent and labor. Hence, their budget can be expressed as

τtcct +τtI(wtnt +rtkt)=gt, t=1,2.

a) Before writing any competitive equilibrium definition or the Social Planner Problem, can we modify the budget constraint in order to get rid a variable in every period? Explain.

b) Define the Social Planner Problem.

c) Solve the Social Planner Problem. This is, find the Euler equation in terms of exogenous variables and the only endogenous variable k2.

d) Define the tax distorted competitive equilibrium of this economy.

e) Solve the representative household problem.

f) Solve the representative firm problem in period 1 and 2.

g) Find the optimal Euler equation in terms of exogenous variables and the only endogenous variable k2.

h) Imagine that the government can only implement income taxes. This is τtc = 0 for every period t = 1, 2. Can the government implement a tax plan that achieves the Pareto allocation reached by the Social Planner Problem?

i) Imagine that the government can only implement consumption taxes. This is τtI = 0 for every period t = 1,2. Can the government implement a tax plan that achieves the Pareto allocation reached by the Social Planner Problem?