Q4. The Permanent-Income Hypothesis: Suppose, for simplicity, the interest rate is set to zero. Then an individual who lives for T years will have the following lifetime budget constraint: (1) Ct t=1 T A0 + Yt t=1 T , where Ct is consumption in year t, A0 is initial wealth, Yt is income in year t, and the "Sigma-operator" provides compact notation for a summation. That is, Ct t=1 T =C1 +C2 + CT-1 +CT . 2 Suppose the individual's preferences are such that they would like to smooth consumption. A simple example would be the following utility function: (2) U = lnCt t=1 T In particular, maximizing (2) for Ct for t =1,...,T subject to the constraint (1) implies a constant marginal utility of consumption over time, which in turn implies a constant consumption over time: i.e., Ct =C . Maximizing this utility function also implies that the budget constraint will hold as a strict equality: (1') Ct t=1 T = A0 + Yt t=1 T , (a) Substitute the constant consumption into the lifetime budget constraint (1') and solve for C. (b) What does your solution for C imply about the relevance for consumption of whether all lifetime income is received in one year or is spread out evenly over all years? (c) Suppose for simplicity that initial wealth is zero (i.e., A0 = 0 ) and note that the individual's saving in year t is St Yt -Ct . What will happen to the individual's saving if their income is above average this year? What if it is below average? Can you relate the answers here to the predictions of the lifecycle model in Figure 11.5 of the Jones textbook? (d) Suppose that in addition to the lifetime budget constraint, the individual faces an additional constraint on borrowing (which economists refer to as a "liquidity constraint") such that consumption cannot be greater than the sum of current income and past accumulated savings: i.e., Ct Yt + St t =1 t-1 . Will the timing of when income is received matter for consumption (compare to part (b))? In terms of the life-cycle model Figure 11.5, what will happen to consumption early in life? Will the individual's consumption be higher or lower at the end of their life? Will the individual start saving sooner or later compared to Figure 11.5?