Advanced Macroeconomics
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?