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).

1. 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.

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

3. 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?

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

Answer 1

Solution:

a)

b)

differentialy

For maximum utitlity

demand functions

c)

Save = 380-200 = 180

d)

will remain same so, saving will also remain same i.e, $180 since is independent of "r"

Earlier when r = 0.05

U=ln200+(1-0.1)ln(189)

U=10.01

Total utility will increase from 10.01 to 10.05

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