ECO 310  Fall 2007
Microeconomic Theory  A Mathematical Approach
Problem Set 2  Answer Key
Question 1:
Since you are being asked to do essentially the same calculations for two diﬀerent values of the net earnings
per hour, one strategy is to ﬁnd a general formula, denoting the net hourly earnings rate by say
w
, and then
substitute the two particular values
w
= $50 and $40 into the formula. (Of course it is perfectly ﬁne if you
do two separate numerical calculations.)
(a) The budget constraint is
I
=
w H

25000.
(b) and (c) : The utility when you work
H
hours is
U
= ln(
w H

25000) + 2 ln(5000

H
)
.
(You could use Lagrange, but in this case substitution from the constraint to reduce the problem to a
onevariable optimization is clearly simpler.) Utility is a concave function of hours, so the FONC gives the
unique max. The only possible problems are (1) an endpoint solution at
H
= 0, but that turns out not to
be relevant. (2) the FONC yields
H >
5000, this is examined below.
Setting
dU
dH
≡
w
w H

25000

2
1
5000

H
= 0
gives
5000
w

w H
= 2
w H

50000
,
or
H
= 5000
10 +
w
3
w
.
You were not asked to do this, but here is some extra information: The formula gives
H >
5000 if
10 +
w >
3
w
, or
w <
5. If
w <
5, the problem is that even working for 5000 hours is not enough to service
the debt. But no selfrespecting doctor would even contemplate earning less than $5 per hour.
Now we can answer the numerical questions:
When
w
= 50,
H
= 5000
×
60
/
150 = 2000.
When
w
= 40,
H
= 5000
×
50
/
120 = 2083.
So you work more when the net earnings rate goes down. The intuition is that the income eﬀect of the
net earnings rate decrease (which leaves you poorer and therefore reduces your leisure choice) exceeds the
substitution eﬀect (which makes you want to work less).
Question 2:
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 Fall '08
 StephenE.Morris
 FONC

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