3 Consumption over time (Adapted from Silberberg, 1990) The world lasts T > 2 periods. A consumer has income wt in any period t. The real interest rate is r 2 0, and the consumer can borrow and lend as much as he/she wants at this rate. The consumer consumes ct in each period and has utility function U(c). 3a. Write down the consumer's budget constraint. 3b. Write down the consumeris utility maximization problem. T I The consumer has utility function: U (c) = Z [1;] CUE {=0 + :0 3c. Show that for the optimal consumption stream, consumption in any time period after period 0 can be expressed as a constant times the consumption in the previous period. 3d. Under what conditions on the exogenous variables will consumption be increasing over time\"? Under what conditions will consumption be decreasing over time\"? * **Optional Extra Practice* * * 1 t Repeat problem #3 for U(C) = 231:0(3) in ct