The economy has 100 identical consumers. Each consumer has income of y = 100 in the current period, and y’ = 325 in the future period. Each consumer pays lump-sum taxes t = 10 in the current period and t’ = 10 in the future period. Each consumer also has a house with value pH = 105. Housing is illiquid and can only be sold in the future period. The real interest rate is 0.05 per period. A consumer’s preference over current and future consumption is represented as u(c, c’) = log(c) + log(c’).

(a) Suppose there is perfect contract commitment. Determine a consumer’s optimal current and future consumption and optimal saving.

(b) In equilibrium, what has to be the life-time present value of government expenditure G + G’/(1 + r)?

(c) Suppose there is limited commitment in the credit market, and consumers face the following collateral constraint −(1 + r)s ≤ pH. Plot a consumer’s budget constraint, label the intersection with x- and y-axis. Determine the optimal current and future consumption. Does the consumer have a higher or lower welfare than case (b)? Explain briefly.

(d) Under the limited commitment in (c), how can the government change its tax policy to restore a consumer’s welfare back to (b) without changing the government expenditures? (You need to write down the exact numbers for current and future tax.)

Note that MRCc,c’ = MUc/MUc’ = (1/c) / (1/c’) . I really need detail solution for this question to comprehend it.