Question

An Individual lives for two periods, 1 and 2. In the first he works and earn an income of M. In the second he is retired and has no income.

His/her life time utility is a function of how much he consumes in the two periods. C1 denotes consumption in period 1 and C2 consumption in period 2. (Hint: If you want to, you can view and treat C1 and C2 as any pair of “goods”, e.g. good x and y).

He/her uses one part of M to buy C1 in period 1 and saves the rest to buy C2 in period 2.

This means that C1=M-S, where S denotes how much the individual saves of his earnings M.

The price of both C1 and C2 is equal to 1.

The amount he saves earns an interest rate of r implying that C2=(1+r)S The individuals budget restriction could for instance hence be written as: M = C1+S or M=C1+C2/(1+r).

a) Draw a graph showing the individual’s budget line (i.e. combinations of C₁ and C₂ where the individual uses all of M for consumption in the two periods). What is the slope of the budget line?

Now, the individual has an utility function U=lnC₁+(1-d)lnC₂ where 0<d<1. The term d denotes a discount factor, i.e. that the individual values consumption in period 2 lower than consumption in period 1.

b) Derive the demand functions for C₁ and C₂ as a function of M, r and d.

c) Assume that the interest rate is 5% (r=0.05), the discount rate is 10% (d=0.1) and that the individual has an earning in period 1 of 380.000$. How much will he consume in period 1 and how much will he save?

d) Assume that the interest rate increases to 10% (r=0.1). How will this affect his savings and total utility?

Answer 1