max{log c1 + log c2} C1 + a1 = Y1 – Tị + (1+r)ao C2 = Y2 – T2 + (1+r)a1 where Y1 and y2 denote income levels in period 1 and period 2, T1 and T2 are taxes in the two periods, and ao and a, denote the assets of the households in each period. ao is exogenously given. Assume the interest rate r = 0, and the government can borrow or save at the same interest rate so that its present-value budget constraint is given by 9i + 92 = T1 + T2 where 9i and 92 are exogenous government expenditures in the two periods.
Consider the two-period household-maximization model discussed in class. The model is modified in order to look at applications including credit constraints, interest-rate markups, and taxation. A representative household lives for two periods and maximizes utility of consumption in period 1 and in period 2. The utility is represented by log(c) where c denotes consumption. Assuming no discounting between period 1 and period 2. The maximization problem for the representative household can be written as
Show consumption c1 and c2 (you can use algebraic or graphical methods). In the answer, you should discuss whether a1 ≥ 0 or a1 < 0 and provide an economic interpretation. What determine(s) the sign of a1 and why?
Explain how a credit constraint can be modeled by a1 ≥ 0 and under this constraint how does your answer above change? How does it change if there is a banking markup such that the borrowing interest rate is above the interest rate of r = 0 assumed earlier? Hint: try to relate your reasoning to the permanent income hypothesis
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