Paper, Order, or Assignment Requirements
Take Home Assignment for Eco460
Due at 6:10pm on November 25, 2014
No late work will be accepted
Note: Please hand in your assignment in class.
(40 points) Consider a two-period extension of the model in Chapter 4 of
the Lecture Notes. The two periods are indexed by t = 0 and 1. There is no
uncertainty in period 0 and let c(0) be the household’s consumption in period
There is uncertainly in the second period and the household’s consumption
plan is denoted by c(1) = (c1, c2, …, cN ), where N is the number of states. The
household’s preferences for consumption in two periods are summarized by the
following utility function:
U(c(0), c(1)) = u(c(0)) + β
X
N
i=1
πiu(ci)
Here u(c) = 1 −
1
1−γ
(c
1−γ − 1) and β is the time discount factor, 0 < β < 1.
That is, household value the utility from future consumption at a discount.
The household is endowed with a certain income y(0) in period 0, which the
household can be used for (1) consumption in period 0, (2) invest in a risk-free
bond that yields a risk-free return of e
rf
in period 1 or (3) invest in a risky asset
that yields a risky return of e
er = (e
r1
, …, erN ) in period 1. So the household’s
budget constraint in period 0 is:
c(0) + b + s = y0
where b is the amount invested in the risk-free bond and s is the amount invested
in the risky asset (stock). There is no endowment income in period 1. Since
this is a two-period economy, there would be no investment in period 1 either.
So the household’s budget constraint in period 1 is
ci = berf + seri
, i = 1, …, N
(10 points) The household’s optimal investment problem is to choose c(0),
b and s to maximize U(c(0), c(1)) subject to the budget constraints in both
periods. Use the Lagrangian method to derive the first-order conditions
for optimal choices of c(0), b and s. [Hint: you can substitute the budget
constraints for ci
into the expected utility function.]
(10 points) Let gi = ln [ci/c(0)] be the consumption growth rate in state
i, and g = (g1, …, gN ). Use the first-order conditions to prove that
1 = βE
e
rf −γg
,
and
1 = βE
e
r˜−γg
.
1
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