Economics 2020a / HBS 4010 / Kennedy School API-111
Microeconomic Theory
Midterm Examination
October 29, 2008
This exam contains three questions for a total of 80 points.
Please complete all three questions.
The exam is open book, so it is acceptable to use notes, books, calculators and laptop computers
(assuming that the laptop computer is not connected to the internet).
1.
(15 points) A consumer has Cobb-Douglas preferences corresponding to the (ordinal) utility
function
u(x, y)
=
x
1/2
y
1/2
, so that
u(x, x)
=
x
.
Your friend says that he wishes to construct a different utility function
v
to represent these same
preferences based on the rule that
v
(
x
, 4
x
) =
x
in place of the rule
u
(
x
,
x
) =
x
.
So for example,
his utility representation satisfies
v
(1, 4) = 1;
v
(2, 8) = 2, and so on.
(a) Graph two indifference curves labeled with corresponding utility values for his utility
function.
Note that his indifference curves are the same as those for u(x, y)
=
x
1/2
y
1/2
, with the only
difference that the utility values associated with each indifference curve have been slightly
changed.
(Graph Omitted).
(b) Find the utility function
v
resulting from his constructive method.
By definition of v, v(x, 4x) = x.
But u(x, 4x) = x
1/2
4x
1/2
= 2x.
That is, for any given
indifference curve, the utility value corresponding to function v is half the utility value
corresponding to function u.
That is, v(x, y) = x
1/2
y
1/2
/ 2.
An alternate approach would be to determine the indifference relationship between points
of form (x, x), which determine the value of u, and points of form (x’, 4x’), which determine
the value of v.
Specifically, u(x, x) = x and u(x’, 4x’) = 2x’, so (x’, 4x’) ~ (x, x) if x = 2x’.
That is (2x’, 2x’) ~ (x’, 4x’), so u(x’, 4x’) = u(2x’, 2x’) = 2x’.
This implies in turn that
v(x’, 4x’) = x’ = ½ u(x’, 4x’), so in general, v(x, y) = ½ u(x, y).
(c) Identify a transformation function
f
so that
v(x, y) = f(u(x, y))
.
As shown in (b), v(x, y) = u(x, y) / 2, corresponding to transformation function f(u) = u / 2.