Question

3. Consider a intertemporal choice problem for a saver. Say the saver has some amount of wealth today given by w > 0, can consumer or save this wealth, and if they save the wealth, on savings they receive a return of r> 0 on every unit of wealth invested. Preference are U(C,C2) = u(c) + Bu(C2) where the discount factor is B E (0, 1) and the utility function u(c) = c. a. If you over this consumer 1 unit of wealth today vs 1 unit of wealth tomorrow, which does she prefer? b. What is this consumers 2 period consumption-savings problem? c. Explain how this problem is related to a standard "static" consumer choice problem for "two goods". d. What is the single first order condition for this problem?

Answer 1

Intertemporal choice can be defined as the choice of consumer to distribute his/ her wealth between today's and future consumption.

Here, the consumer has wealth w1 >0. He can consume all wealth today or can save whole wealth for future or can distribute it between today's and future consumption. This decision depends on the interest rate r and his willngness to substitute that is given by Beta(B).

The utility function for the consumer is given below:

U(c1, c2)=u(c1)+Bu(c2)

u(c)=c^1/2

U(c1, c2)=c1^1/2+B*c2​​​​​^1/2

The Marginal substitution between c1 and c2 is given below:

MRS=MU1/MU2

=c2​​​​​^1/2/B*c1^1/2

The price ratio given price for futures consumption is 1 and price for today's sconumption will be 1+r is 1+r

For optimal bundle equating MRs to price ratio

c2​​​​​^1/2/B*c1​​​​​​​^1/2=1+r

c2=B^2*(1+r)^2*c1

The budget line for the consumer is

(1+r)c1+c2=(1+r)w1

(1+r)c1+B^2*(1+r)^2*c1=(1+r)w1

c1=w1/(1+B^2*(1+r))

Higher interest rate and higher rate of willingess to substiture that is B leads to lower consumption in today's time period.

a. She prefers today's consumption if interest rate and B is zero and if interest rate and B value is really high she will prefer future's consumption

b. The consumer's 2 period consumption saving problem is given below:

(1+r)c1+c2=(1+r)w1

c2=(1+r)w1-(1+r)c1

c2=(1+r)*(w1-c1)

c. The problem is related to standard "staticconsumer choice problem in a way "that in intertemporal choice problem also the price ratio is equated to the marginal rate of substituion to get the optimal bundle.

d. The first order condition to get the optimal bundle is given below:

max U(c1, c2)=u(c1​​​​​​​)+Bu(c2) subject to (1+r)c1+c2=(1+r)w1

=c1​​​​​​​^1/2+B*c2​​​​​^1/2+z((1+r)w1-((1+r)c1+c2)

dU/dc1=1/(2(c1​​​​​​​^1/2))-z(1+r)

dU/dc2=B/(2(c1​​​​​​​^1/2))-z

The single first order condition is

MRS=MU1/MU2=P1/P2

c2​​​​​^1/2/B*c1​​​​​​​^1/2=1+r