1. Consider a consumer who begins in period 0, with a stock of non-human wealth ko...

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1. Consider a consumer who begins in period 0, with a stock of non-human wealth ko (ownership of physical capital). The consumer-known as the representative consumer in macroeconomics - is assumed to live forever. In each period t, t = 0, 1, ..., she has 1 unit of time that she supplies in-elastically on the labor market at the wage rate wt. The market rate of interest in period t is rt. The present value of the stream of labor income (wt), which is the stock of her human wealth, is ho=wo+ W] 1+11 ' (1+11) (1+12) Im (1+1) (1+/2)... (1+1+) The sum of her human wealth and non-human wealth is then given by ko+ho. Let c 0 be her consumption in period t, t = 0, 1, ... The present value of the stream of consumption (Ct) is Ct Co+ +... 1+1 (1+1) (1+12) (1+11) (1+12)... (1+r) Let u[c] denote her utility in a period, when her consumption in that period is c. Suppose that 6, 0 < <1, is the factor the consumer uses to discount her future utilities. The problem faced by the consumer is max(t)=6u[cr] subject to the inter-temporal budget constraint C2 ko+ho = Co+ 1+1+(1+1) (1+2) +... (1+1) (1+12)... (1+r) Solve the representative consumer's problem. In answering this question, assume the following functional form for the single-period utility function: u[c] = Log[c]. 1. Consider a consumer who begins in period 0, with a stock of non-human wealth ko (ownership of physical capital). The consumer-known as the representative consumer in macroeconomics - is assumed to live forever. In each period t, t = 0, 1, ..., she has 1 unit of time that she supplies in-elastically on the labor market at the wage rate wt. The market rate of interest in period t is rt. The present value of the stream of labor income (wt), which is the stock of her human wealth, is ho=wo+ W] 1+11 ' (1+11) (1+12) Im (1+1) (1+/2)... (1+1+) The sum of her human wealth and non-human wealth is then given by ko+ho. Let c 0 be her consumption in period t, t = 0, 1, ... The present value of the stream of consumption (Ct) is Ct Co+ +... 1+1 (1+1) (1+12) (1+11) (1+12)... (1+r) Let u[c] denote her utility in a period, when her consumption in that period is c. Suppose that 6, 0 < <1, is the factor the consumer uses to discount her future utilities. The problem faced by the consumer is max(t)=6u[cr] subject to the inter-temporal budget constraint C2 ko+ho = Co+ 1+1+(1+1) (1+2) +... (1+1) (1+12)... (1+r) Solve the representative consumer's problem. In answering this question, assume the following functional form for the single-period utility function: u[c] = Log[c].