Answer Key
ECON 400
Practice Midterm Exam
Answer Key
(15 points) Question 1:
The following function is not a cost function because it violates properties of cost functions.
Prove one such violation without calculating the Hessian.
α
⎛ α ⎞ ⎛ 1− α ⎞
f ( p1 , p2 ,u) = u ⎜ ⎟ ⎜
!
⎝ p1 ⎠ ⎝ p2 ⎟⎠
1−α
where 0 < α < 1
This function violates homogeneity of degree 1 in prices, that the cost function is nondecreasing in p and increasing in at least one price, and concavity.
To prove that concavity is violated you’d need to use the Hession, so instead you should either
show a violation of homogeneity of degree 1 in prices or that the function is not increasing in
either price.
To prove a violation of homogeneity of degree 1,
α
⎛ α ⎞ ⎛ 1− α ⎞
f (θ p1 ,θ p2 ,u) = u ⎜
!
⎝ θ p1 ⎟⎠ ⎜⎝ θ p2 ⎟⎠
1−α
α
⎛ α ⎞ ⎛ 1− α ⎞
= θ u⎜ ⎟ ⎜
⎝ p1 ⎠ ⎝ p2 ⎟⎠
1−α
−1
f (θ p1 ,θ p2 ,u) = θ −1 f ( p1 , p2 ,u)
So the function is homogeneous of degree negative 1, not 1.
Alternatively, you can show that the derivative with respect to either p1 or p2 is negative and
hence a violation of non-negativity in prices and increasing ness in at least one.
!
⎛ 1− α ⎞
∂ f ( p1 , p2 ,u)
= −α p1−α −1uα α ⎜
∂ p1
⎝ p2 ⎟⎠
1−α
<0
α
⎛α⎞
∂ f ( p1 , p2 ,u)
= − (1− α ) p2−α u ⎜ ⎟ 1− α 1−α < 0
∂ p2
⎝ p1 ⎠
So the function is not increasing in at least one price and not decreasing in the other.
Question 2:
Is the following utility function strictly quasi-concave? Prove your answer.
! u ( q1 ,q2 ) =
q1 + 2 q2
There are two ways to prove that this function is strictly quasi-concave. You can either show that
the function is strictly concave and hence strictly quasi-concave or you can show that the
indifference curves are strictly convex.
For the first method, derive the Hession of the utility function and then show that it is negative
definite.
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Answer Key
⎡
1
⎢ − 3/2
⎢ 4q1
!H =
⎢
0
⎢
⎢⎣
0
−
1
2q23/2
⎤
⎥
⎥
⎥
⎥
⎥⎦
Both elements on the main diagonal (left upper corner to bottom right corner) are negative for
any positive amounts of the two goods and the determinant is equal to the product of the
elements of the main diagonal which is also strictly positive for any positive amounts of both
goods 1 and 2. So the Hessian is negative definite for any positive amounts of both goods 1 and
2 and hence the utility function is strictly concave. This implies strict quasi-concavity of the utility
function for any positive amounts of both goods 1 and 2.
Alternatively, you could ask if the ICs are strictly convex. This is also equivalent to asking if the
MRS along an IC is decreasing.
Proofing that the IC is strictly convex:
u = q1 + 2 q2
(u − q )
q (q ) =
2
1
!
2
1
4
(
q2' = − u − q1
q2'' =
) 4 1q
=−
1
u
1
+
4 q1 4
u
>0
8q13/2
Proofing that the MRS is decreasing:
MRS =
1 q2
2 q1
u = q1 + 2 q2
(u − q )
! q (q ) =
2
1
2
1
4
(u − q )
2
1
MRS =
1
2
4
q1
=
(u − q ) =
1
4 q1
u
1
−
4 q1 4
∂MRS
u
= − 3/2 < 0
∂q1
8q1
Question 3:
Use the utility function from question 2 and derive the Marshallian demand.
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Answer Key
Set up the Lagrange function or go straight to the tangency condition. We know the function is
strictly quasi-concave, so the first order conditions are necessary and sufficient.
I set up the Lagrange function below.
max L ( q1 ,q2 , λ ) = q1 + 2 q2 + λ ( x − p1q1 − p2 q2 )
x1 ,x2 ,λ
!
∂L
1
=
− λ p1 = 0
∂q1 2 q1
∂L
1
=
− λ p2 = 0
∂q2
q2
∂L
= x − p1q1 − p2 q2 = 0
∂λ
The first order conditions above can be rewritten in form of the tangency condition and the
budget constraint
MRS =
1⎛ q ⎞
! ⎜ 2⎟
2 ⎝ q1 ⎠
p1
p2
1/2
=
p1
p2
p1q1 + p2 q2 = x
Using these two equations in two variables, we can express q2 as a function of q1.
2
⎛p ⎞
q2 = ⎜ 1 ⎟ 4q1
⎝ p2 ⎠
2
!
⎛p ⎞
p1q1 + 4 p2 ⎜ 1 ⎟ q1 = x
⎝ p2 ⎠
⎛
⎞
p2
g1 ( p1 , p2 , x ) = x ⎜
2⎟
⎝ p1 p2 + 4 p1 ⎠
⎛
⎞
4 p1
g2 ( p1 , p2 , x ) = x ⎜ 2
⎝ p2 + 4 p1 p2 ⎟⎠
Inspection of this interior solution shows that for any combination of x, p1, and p2, the terms will
be positive. This means we have found our Marshallian demand given by the interior solution
and do not need to worry about corner solutions.
Question 4:
Consider the following two budget constraints.
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Answer Key
( p , p , x ) = (1, 3, 4)
( p , p , x ) = (1,1 / 3, 4 / 3)
!
(1)
(2)
0
1
0
2
0
1
1
1
2
1
a) (10 points) Draw the budget constraints.
The budget constraints can be written as
! B.C.0 : q2 =
4 1
− q1 , B.C.1 : q2 = 4 − 3q1
3 3
Note that the two budget constraints are symmetric. Below is a sketch.
Good 2
4
4/3
4/3
4
Good 1
b) (10 points) Can you tell whether a consumer with strongly monotone preferences and a
strictly quasi-concave utility function is better or worse off under the first budget constraint
compared to the second? Give a detailed explanation.
Without further information we can’t tell if the person is better off or worse off under the old
budget set. Neither budget set is a subset of the other one. So if the person consumed a
consumption bundle under the old budget constraint that is no longer available under the new
one, it is possible that the person is better off under the old budget set, worse off, or the same
off. If we knew that the person consumed a consumption bundle on the old budget line that’s
below the new budget line or at the intersection with the new line, we could say that the person
is better off under the new budget set due to strong monotonicity in the first case and strict
convexity in addition to strong monotonicity in the second. Similarly, if we knew that the person
consumed a consumption bundle on the new budget line that’s below the old budget line or at
the intersection with the old line, we could say that the person is better off under the old budget
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Answer Key
set due to strong monotonicity in the first case and strict convexity in addition to strong
monotonicity in the second. But we don’t have this information.
c)
.
Suppose in addition to the assumptions in b) you’re also told that the utility
function is symmetric. That is, for any bundle with ! q10 ≠ q20 ,
u ( q10 ,q20 ) = u ( q20 ,q10 ) . This implies
that, for example, a consumption bundle like (5,7) would be as good as a consumption bundle
(7,5). Can you tell now whether the person is better off or worse off under the first budget
constraint than under the second? Give a detailed explanation. The person is exactly the same off on both budget lines because the utility function is
symmetric. Any point the person could choose on the first budget line has a mirror image on the
second budget line that yields the same utility.
d) Given the same assumptions as in c) prove that the consumer would never consume at (1,1)
given our budget constraints. (1,1) is not optimal on either budget line because the utility function is symmetric. Suppose (1,1)
is optimal and given strict quasi-concavity of the utility function it would therefore imply that the
tangency condition holds for one of the two budget constraints. However, the (absolute values of
the) slopes of the budget constraints are 1/3 and 3 respectively, but the (absolute value of the)
slope of the IC at (1,1) given symmetry is 1. Thus the IC going through (1,1) is neither tangent to
the first nor the second budget constraint and hence (1,1) is not an optimal choice on either
budget line.
e) Given the same assumptions as in c) is it possible that the consumer chose a consumption
bundle with a higher amount of good 2 than good 1? Prove your answer. There are several ways to answer this question. For, example, you can talk about a specific
budget constraint and whether the choice is optimal and then move on to the other budget
constraint.
Given that our ICs are symmetric and we know that they have a slope equal to 1 at their
intersection with the 45 degree line, it must be the case that the slope of an IC is more than 1 for
bundles with a higher amount of good 2 than good 1 and that the slope is less than 1 for
bundles with a higher amount of good 1 than good 2. The slope of the first budget line is 1/3 and
the slope of the second is 3, so this means that the person would optimally consume more of
good 2 than good 1 when faced with the second budget constraint and more of good 1 and less
of good 2 under the first budget constraint.
Note that this conclusion holds even if we end up with a corner solution. Our utility function is
strictly quasi-concave and so inspecting the tangency condition will always guide us to the
correct solution.
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Answer Key
f) (10 points) Given the same assumptions as in c), suppose the consumer consumed good 1
only under the first budget constraint, be as specific as possible about the information you have
on the consumer’s optimal consumption bundle under the second budget constraint. Consuming good 1 only under the first budget constraint leads to a consumption bundle of (4,
0). Since the budget lines are symmetric and the utility function is symmetric this means that the
person consumes (0, 4) on the second budget line.
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