In class, we solved a two$-$period savings model where a consumer allocates income across two

periods. We assumed the consumer$$s intertemporal utility function was given by: U $($c$1,$ c$2)=$

log$($c$1)+$ log$($c$2)$ and that their intertemporal budget constraint was M$1+$ M$2$

$1+$r $=$ c$1+$ c$2$

$1+$r $.$

Along the way to solving that problem, we found that consumers should select their consumption

in each period so that:

u$($c$1)=$ $(1+$ r$)$u$($c$2),$

where is the exponential discount rate and r is the interest rate.

In this problem, we will extend this problem from two to three periods. We will solve it with

exponential discounting and quasi$-$hyperbolic discounting.

$1$

$1\mathrm{.}1.$ Assume the consumer$$s $$flow$$ utility is given by log$($ct$)$ in each period and that the consumer

has an exponential discount rate $.$ What is the intertemporal utility function with three

periods $($instead of two$)?(10$ points$)$

$1\mathrm{.}2.$ Assume the consumer receives income M$1$ in period $1,$ M$2$ in period $2,$ and M$3$ in period $3$

and that the interest rate is still r$.$ What is the intertemporal budget constraint with three

periods $($instead of two$)?(10$ points$)$

$1\mathrm{.}3.$ We normally solve utility maximization problems where the consumer only chooses two things.

We use two equations to solve for these two unknowns: $(1)$ the marginal rate of substitution

equals the price ratio and $(2)$ the budget constraint. When solving for more than two things,

we can replace the MRS$=$price ratio equation with the requirement that the marginal utility

per dollar for each good must be the same. If it is helpful, you can write for each good, g$,$

M Ug$/$pg $=$ where $$ is sometimes called the $$Lagrange Multiplier.$$ With only two goods,

this gives you exactly the same information as MRS$=$price ratio, but with three goods it gives

you more equations.

Use this to extend the condition that u$($c$1)=$ $(1+$ r$)$u$($c$2)$ to include utility in the third

period. $(5$ points$)$

$1\mathrm{.}4.$ Solve for consumption in each period assuming u$($ct$)=$ log$($ct$)$ in every period, $=0\mathrm{.}95,$

r $=0\mathrm{.}05,$ and M$1=$ M$2=$ M$3=100.(25$ points$)$

$1\mathrm{.}5.$ Now, assume the consumer has quasi$-$hyperbolic time preferences with additional parameter

$=0\mathrm{.}9.$ Solve for consumption in each period assuming u$($ct$)=$ log$($ct$)$ in every period,

$=0\mathrm{.}95,$ r $=0\mathrm{.}05,$ and M$1=$ M$2=$ M$3=100.(25$ points