Problem Set 1 – Consumption
Due 11:59 PM Sunday 24th February 2013
Problem 1
Solve for ct, ct+1 and st the following two-period model:

max U (ct,ct+1) = −
1
α
e−αct −

β

α
e−αct+1

subject to:

ct +
(

1

1 + r

)
ct+1 = Yt +

(
1

1 + r

)
Yt+1

Problem 2
Solve the following problem:

max
Ct,Ct+1

U =
(
Ct − aC2t

)
+ β

(
Ct+1 − aC2t+1

)
subject to:

i) Ct + bt+1 + Kt+1 = Yt
ii) Ct+1 = Yt+1 + (1 + r) bt+1 + (1 − δ) Kt+1
iii) Yt+1 = A ln (1 + Kt+1)

where A > r + δ.

a) Find the optimal values of Kt+1, Yt+1, Ct, Ct+1, and bt+1;

b) Given the optimal value of Kt+1, find ∂Kt+1/∂r. Does this make sense?
Why?

c) For Yt = 1, β (1 + r) = 1, and A = 2 (r + δ), find ∂bt+1/∂r. Does the
income or substitution effect dominate?

Problem 3
Solve the following optimization problem for a three-period lived individual:

max
Ct,Ct+1,Ct+2

U = ln Ct + β ln Ct+1 + β2 ln Ct+2

subject to:

i) Ct + bt+1 = Yt
ii) Ct+1 + bt+2 = Yt+1 + (1 + r) bt+1
iii) Ct+2 = Yt+2 + (1 + r) bt+2
a) Find the optimal values of C∗t , C∗t+1 and C∗t+2;

b) Find the value function V = U(C∗t ,C∗t+1,C∗t+2), i.e. evaluate the total utility
function at the optimal values. What form does V have compared to U?