Econ 101A — Midterm 1
Th 28 February 2008.
You have approximately 1 hour and 20 minutes to answer the questions in the midterm. Dan and Mariana
will collect the exams at 11.00 sharp. Show your work, and good luck!
Problem 1. Consumption and Leisure Decision.
(58 points) In class, we considered separately
a consumption decision between goods
x
1
and
x
2
and a lesisure decision between consumption good
x
and
leisure
l
. Now we consider those together. Yingyi likes three goods: consumption goods
x
1
,
x
2
, and leisure
l.
He maximizes the utility function
u
(
x
1
,x
2
3
)=
x
α
1
1
x
α
2
2
l
γ
,
with
0
<α
i
<
1
for
i
=1
,
2
and
0
<γ<
1
. The consumption good
x
i
has price
p
i
(for
i
,
2
), the hourly
wage is
w
and the individual has total income
M
.
1. Write down the budget constraint. Consider that Yingyi has
H
hours to work and, if he does not
work, he takes leisure. For example,
H
could be 24 hours if the time period is a day. Hence, the hours
worked
h
equal
H
−
l.
There are no sources of income other than income from hours worked. Write
down the budget constraint as a function of
x
1
2
,
and
l
. [Hint: Money spent on goods has to be
smaller than or equal to money earned] (5 points)
2. Write down the maximization problem of the worker with respect to
x
1
2
,
and
l
with
all
the relevant
constraints Assume that the budget constraint is satis
f
ed with equality. Why can we assume that the
budget constraint is satis
f
ed with equality? Provide as complete an explanation as you can. (5 points)
3. Write down the Lagrangean and derive the
f
rst order conditions with respect to
x
1
2
,l,
and
λ.
(4
points)
4. Solve for
x
∗
1
as a function of the prices
p
1
,p
2
,w,
the total number of hours
H,
and the parameters
α
1
,α
2
,
and
γ
. [Hint: combine the
f
rst and second
f
rst-order condition, then combine the
f
rst and
third
f
rst-order condition, and
f
nally plug in budget constraint] Similarly solve for
x
∗
2
and
l
∗
.(
6
points)
5. Plot the Engel function relating the demand for good 1
x
∗
1
(
H
)
to the number of hours available
H.
(Plot
x
1
in the x axis and
H
intheyax
is
)Inwha
tsense
H
plays the role of income? Explain. (5
points)
6. Plot the demand function for good 1
x
∗
1
(
p
1
)
as a function of
p
1
.(Pu
t
x
1
in the x axis and price
p
1
in
the y axis) Is the demand function downward sloping? Interpret. (5 points)
7. Are goods
x
1
and
x
2
gross complements, gross substitutes, or neither? De
f
ne and answer. (5 points)
8. Plot the demand function for leisure
l
∗
(
w
)
as a function of its (shadow) price
w
t
l
in the x axis
and price
w
in the y axis) Is the demand function downward sloping? Interpret. (6 points)
9
. Re
latetheresponsetoquest
ion8(
le
isure
l
and price
w
) to substitution and income e
f
ects. Be careful
here, this is not exactly the case we saw in class. (6 points)
10. Using the envelope theorem, compute how the indirect utility
v
(
p
1
2
,w,α
1
2
,γ,H
)