$($

$24\%$

$)$ Farmer Boris lives for two periods. In the $$rst period, he has $120$

bushels of wheat. If Boris consumes only a portion of the wheat in period $1,$

he can use the rest as seeds so he can harvest and consume in period $2.$ Each

bushel of seeds yield

$2$

bushels of wheat in period $2.$ Boris$$utility function

is

u

$($

c

$1$

; c

$2$

$)=$

p

c

$1$

$+0$

:

$5$

p

c

$2$

;

where

c

$1$

and

c

$2$

are the amount of wheat $($in

bushels$)$ he consumes in periods $1$ and $2$ respectively.

$(1)$ How much wheat will Boris consume in the $$rst period and in the

second period?

Suppose that a new irrigation system can double the yields $($i$.$e$.,$ each

bushel of seeds planted in period $1$ yield

$4$

bushels of wheat in period $2).$

$(2)$ What is the maximum amount of wheat Boris is willing to give up in

period $1$ in exchange for the new irrigation system?

$(3)$ Boris$$brother Christo$$ is considering giving Boris

one

of the two gifts:

$1$

d

p

x

dx

$=$

$1$

$2$

$1$

p

x