Economics 210B
Due: Tues., Sept. 5, 2006
Problem Set 5.
Problem 1. Consumption versus leisure, part 3.
Consider again the optimization problem we started studying in PS3:
max
c, h
U
(
c, h
)
≡
ln(
c
) +
α
ln(
T

h
)
subject to:
c
=
wh
This time we will use MATLAB to study the problem numerically.
a.
Assume
T
= 24,
α
= 2, and
w
= 3. Use MATLAB to plot several indifference
curves, and the budget line, using the explicit formula for the indifference curves
that you calculated in PS3.
To use the plot command, you will want to type something like:
plot(labvec,budget,‘’);
hold on;
plot(labvec,indiff,‘ ’);
title(‘my plot’);
where “labvec” is a vector of values of
h
, and “budget” and “indiff” are vectors of
corresponding values of
c
along the budget line and the indifference curve.
b.
Calculate numerically the optimal
h
and
c
and the maximum achievable
U
.
(You can do this by looking for an indifference curve that is approximately tangent
to the budget line, or by searching for a pair (
n, c
) which approximately solve the
two necessary conditions.) Also calculate the gradient vector at the optimum and
show it on your graph.
c.
Using your comparative statics solution, write a short program involving
matrices to calculate how
c
and
h
change as
w
and
T
change. Estimate how much
c
and
h
would change if
T
were raised to 25, and check the accuracy of the firstorder
approximation by computing the true solution at
T
= 25.
Problem 2. Duality in consumption choice.
Consider the utility maximization problem:
max
x, y
U
(
x, y
)
≡
1
α
x
α
+
1
β
y
β
subject to:
px
+
qy
=
W
a.
Find the firstorder conditions that characterize an optimum.
b.
Calculate the comparative statics for
x
and
y
with respect to
p
and
W
. Are
the signs of the effects sensible?
1
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c.
Use the envelope theorem to calculate the effect of a change in
W
on the
value function.
d.
Suppose the amount of utility achieved by optimal choice of consumption
in this problem is
U
*
.
Notice that utility cannot be maximized in this problem
unless the total amount spent to achieve
U
*
is
minimized
. Rewrite the problem as
an expenditure minimization problem subject to the constraint that utility must be
at least
U
*
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 Spring '11
 PP
 Economics, Utility, firstorder conditions

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