Econ 101A — Problem Set 2
Due in class on Th February 19. No late Problem Sets accepted, sorry!
This Problem set tests the knowledge that you accumulated in lectures 4 to 8. It is focused on preferences,
utility functions, and utility maximization. General rules for problem sets: show your work, write down the
steps that you use to get a solution (no credit for right solutions without explanation), write legibly. If you
cannot solve a problem fully, write down a partial solution. We give partial credit for partial solutions that
are correct. Do not forget to write your name on the problem set!
Problem 1.
Addictive goods.
(23 points) In this exercise, we propose a generalization of Cobb
Douglas preferences that incorporates the concept of reference point. We use it to model the consumption
of addictive goods. Consider the following utility function:
u
(
x
1
, x
2
;
r
1
) = (
x
1
−
r
1
)
α
x
2
β
with
α
+
β
= 1
,
0
< α <
1
,
0
< β <
1
,
and
r
1
>
0
.
Notice that the above utility is only de
fi
ned for
x
1
≥
r
1
and
x
2
≥
0
.
Assume that for
x
1
< r
1
or
x
2
<
0
the utility is zero. Good
x
1
is an addictive good
with addiction level
r
1
. Examples of addictive goods are alcohol, drugs or... chocolate. The more you have
consumed of these goods in the past, the higher the addiction level
r
1
.
1. Draw an approximate map of indi
ff
erence curves for the case
α
=
β
=
.
5
.
(2 points)
2. How does the utility function change as
r
1
changes? In other words, compute
∂u
(
x
1
, x
2
;
r
1
)
/∂r
1
.
Why
is this term negative? [Hint: If I have gotten used to drinking a lot of alcohol, my utility of drinking
three bottles of beer...] (3 points)
3. Compute now the marginal utility with respect to
x
1
.
In other words, compute
∂u
(
x
1
, x
2
;
r
1
)
/∂x
1
for
x
1
> r
1
.
How does this marginal utility change as
r
1
changes? In other words, compute
∂
2
u
(
x
1
, x
2
;
r
1
)
/∂x
1
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 Spring '08
 Staff
 Utility, X1, Hicksian

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