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Economics 2020a / HBS 4010 / Kennedy School API111
Final Examination
Fall 2008
Instructions:
You have 180 minutes to complete the following examination.
The exam
has five questions, each with multiple parts.
Answer all parts of all questions in the exam
books provided.
Partial credit will be awarded on the basis of (partially correct) work
shown, so
write legibly
and
show your work.
If you believe a question is ambiguous,
clearly state any assumptions you are making.
You may use a calculator on this exam.
This exam has 5 pages, including this one.
The exam has 150 points, with point
distribution shown below.
Question
Points
1
30
2
20
3
35
4
35
5
30
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View Full Document 1. A consumer has a quasilinear utility function with one modification:
u(x, y) = x + 40 ln y if x < 100
u(x, y) = 2x + 40 ln y if x >
100.
Assume that the consumer is restricted to bundles (x, y) with x >
0, y >
0.
(a) Suppose that p
x
= 1 and p
y
= 1.
Solve for the consumer’s optimal bundle for all values
w of wealth in the following ranges: (1) 0 < w < 100; (2) w >
140.
Here, it is necessary to check for optimal solutions for (1) x < 100, (2) x >
100.
Within each region, the utility function is quasilinear, so an interior solution to the
consumer problem satisfies (du / dx) / p
x
= (du / dy) / p
y
, or du / dx = du / dy since
both prices are equal to 1.
For x < 100, du / dx = 1, du / dy = 40 / y, so an interior solution satisfies y
1
* = 40.
By
Walras’ Law, x
1
* = w – 40 at this interior solution with x < 100.
For x >
100, du / dx = 2, du / dy = 40 / y, so an interior solution satisfies y
2
* = 20.
By
Walras’ Law, x
2
* = w – 20 at this interior solution with x >
100.
Now specialize analysis to different ranges of wealth.
RANGE 1: For w < 40, neither of these interior solutions is feasible.
Since du / dx is
constant while du / dy is decreasing, the consumer maximizes utility at a boundary
solution where she only consumes y.
The optimal bundle is (0, w).
RANGE 2: For 40 <
w < 100, the first interior solution is feasible, but it is not
possible to reach the point x = 100 where the utility function changes.
Thus, the
optimal bundle is (w – 40, 40).
RANGE 3: For w >
140, the first interior solution is not feasible (it would leave x >
100), but the second interior solution is feasible.
Since the change from region 1 (x <
100) to region 2 (x >
100) only increases utility, it will not be optimal to choose a
boundary solution in region 1.
The optimal bundle is (w – 20, 20).
(b) Is good x inferior or normal?
Explain your answer in words.
Good x is normal.
As wealth increases, the optimal choice of x increases.
At some
point, there may be a discontinuous jump in x (e.g. from w – 40 to w – 20), but this
does not cause x to be inferior.
(c) Is good y inferior or normal?
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This note was uploaded on 04/12/2009 for the course HKS API111 taught by Professor Avery during the Fall '08 term at Harvard.
 Fall '08
 Avery

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