410 Midterm 2 Solution, Summer 2013
June 15, 2013
Name:
PID:
• You have 90 minutes to complete this exam.
•
You MUST put your name and PID on the scantron. Failure to do so will result in a loss
of 1 point.
• You must mark your answers to the multiple choice questions on your scantron.
• You must work alone and may not consult any books or notes, nor may you use any electronic devices.
•
As always, the honor code is in eect for this exam.
• Each problem is worth 4 points, for a total of 100 points.
• You receive no penalty for a wrong answer, so when you don't know the answer you should guess.
• There are 4 sections, with easier denition and concept questions at the beginning and harder problem
solving questions at the end.
Section I
Problem 1
For a given price, the market demand function is___
A. The sum of the individual demand functions of all consumers at that price.
B. The product of the individual demand functions of all consumers at that price.
C. The value of the highest individual demand function at that price.
D. The average demand of all consumers at that price.
E. None of the above.
Problem 2
M RT S12 =_____.
A. f2 /f1 .
B. p2 /p1
f1 /f2
D. p1 /p2
C.
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Problem 3
F actorP riceRatio12 =_____.
A. f2 /f1 .
B. p2 /p1
C. f1 /f2
D.
p1 /p2
Problem 4
At an interior optimum, and assuming there are no kinks in the utility function,
A. M RT S12 < F actorP riceRatio12 .
B. M RT S12 = F actorP riceRatio12 .
C. M RT S12 > F actorP riceRatio12 .
D. None of the above.
Problem 5
Mathematically, the Isocost and Isoquant curves are, respectively, equivalent to
A. The indierence curve and budget line, but in the Isoquant utility takes the place of quantity.
B. The budget line and indierence curve, but in the Isoquant utility takes the place of quantity.
C. The indierence curve and budget line, but in the Isoquant quantity takes the place of utility.
D. The budget line and indierence curve, but in the Isoquant quantity takes the place of
utility.
E. None of the abovel.
Problem 6
The only major dierence(s) between the rm's cost minimization problem and the consumer's utility maximization problem is(are):
A. The names of the curves are dierent.
B. We are minimizing for the rm instead of maximizing.
C. With the rm's problem we care about cardinal values of the objective function (cost).
D. We switch the roles of the budget line and the output equation in the Lagrangian,
E. All of the above.
Problem 7
Suppose you run a rm that produces lawn chairs from plastic. You've found that if you double the amount
of plastic you use, you only end up with 50% more chairs. Your rm displays
A. Increasing returns to scale.
B. Constant returns to scale.
C. Decreasing returns to scale.
D. Increasing, then decreasing returns to scale.
E. Decreasing, then increasing returns to scale.
Problem 8
In the cost minimization problem, moving in which direction along a graph of bundles of input goods x and
y ensures that we move to a more preferred isocost curve, given our standard assumptions?
A. Up and to the left.
B. Up and to the right.
C. Down and to the left.
D. Down and to the right.
E. None of the above.
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Problem 9
A production function allows us to plug in ____ and tells us _______.
A. A bundle of input goods, the minimum amount of the output good that can be produced, given the
production technology.
B. A bundle of input goods, the unique amount of the output good that the rm can produce.
C. A bundle of input goods, the average amount of the output good that the rm can produce.
D. A bundle of input goods, the maximum amount of the output good that can be produced,
given the production technology.
E. None of the above.
Problem 10
The rst order condition for maximizing prot for a rm can be broken into the following factors, assuming
you're considering raising the price a bit:
A. Marginal cost of selling a little more of the good, change in revenue due to selling a little more of the
good, change in revenue from lowering the price of the units you were already going to sell.
B. Marginal cost of selling a little more of the good, change in cost due to selling a little more of the
good, change in cost from lowering the price of the units you were already going to sell.
C. Marginal revenue of selling a little more of the good, change in revenue due to selling a little more of
the good, change in revenue from lowering the price of the units you were already going to sell.
D. Marginal revenue of selling a little more of the good, change in cost due to selling a little more of the
good, change in cost from lowering the price of the units you were already going to sell.
E. None of the above.
Problem 11
A cost function for a rm is
A. The sum of the costs of all input goods, given their prices and the quantities used
B. The cost of a unit of the output good.
C. A vector giving the price of each input good.
D. The cost to the consumer of purchasing the rm's good.
E. None of the above.
Problem 12
In a perfectly competitive market, rms _____.
A. Can choose whatever price they want, since consumers will always buy from them.
B. Can only sell at the market price, because increasing price will lead all customers to buy
from a rival rm.
C. Can choose any quantity they want, because the price is always the same.
D. Can only produce at the quantity where their price is strictly lower than any competitor's.
E. None of the above.
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Section II
Problem 13
For which of the following functions do we care about cardinal properties?
I. Utility functions.
II. Production functions.
III. Prot functions.
A. I
B. II
C. III
D. I and II
E. II and III
Problem 14
The rst order condition for maximizing prot for a rm in a perfectly competitive market is:
I. p'(q)q+p(q)-c'(q)=0
II.p-c'(q)=0
A. I
B. II
C. I and II
D. Neither
Problem 15
If average costs are increasing, we say
A. The rm could have any sort of returns to scale.
B. The rm has constant returns to scale.
C. The rm has increasing returns to scale.
D. The rm has decreasing returns to scale.
E. None of the above.
Problem 16
Given U (x, y) = ln(x) + y , nd the demand for x given a xed budget w. Assume an interior solution.
A. w/px
B. py /px
C. px /py
D. px /w
E. None of the above.
Solution:
U1 = 1/x and U2 = 1. MRS=MRT so1/x = px /py . Thenx =py /px .
Problem 17
For a perfectly competitive market with market price p=12, Solve for the optimal production quantity for a
rm with cost function C(q) = q 3
A. 1
B. 2
C. 3
D. 4
E. 8
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Solution:
C 0 (q) = 3q 2 and P=MC, so
12 = 3q 2
4 = q2
2=q
Problem 18
For a monopolistically competitive market with demand D(p)=7-p and cost C(q)=q, solve for the prot
maximizing price and quantity.
A. p=1, q=2
B. p=4, q=3
C. p=2, q=5
D. p=3, q=4
E. p=5, q=2
Solution:. To maximize prot, we must nd q such that
∂Π
= D0 (p)p + D(p) − C 0 (D(p))D0 (p) = 0
∂p
∂Π
= −p + 7 − p − 1 ∗ −1 = 0
∂p
8 − 2p = 0
4=p
Then we can use the demand function to solve for q:
q = D(3) = 7 − 4 = 3
.
Problem 19
Given f (x, y) = xy , nd the Hicksian demand for x given prices px and py . That is, the demand for x given
a xedp
quantity to be produced q.
A.
qp /p
p y x
B. p
q px /py
C. pqpx /py
D. q py /px
E. None of the above.
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Solution:. We can solve this problem using the interior optimum condition fx /fy = px /py and the
isoquant equation q = xy . From the rst equation, we have
y/x = px /py
y = px /py ∗ x
Substituting into our isoquant equation, we have
q = x ∗ px /py ∗ x
q = x2 ∗ px /py
q ∗ py /px = x2
q
qpy /px = x
Section III
Problem 20
For a monopolistically competitive market with demand D(p) = 6 − p/2 and cost C(q) = 4q 2 , solve for the
prot maximizing price and quantity.
A. p=2, q=5
B. p=4, q=4
C. p=6, q=3
D. p=8, q=2
E. p=10, q=1
Solution:. To maximize prot, we must nd q such that
∂Π
= D0 (p)p + D(p) − C 0 (D(p))D0 (p) = 0
∂p
∂Π
= −p/2 + 6 − p/2 − 4 ∗ 2(6 − p/2) ∗ −1/2 = 0
∂p
6 + 24 − 3p = 0
10 = p
Then we can use the demand function to solve for q:
q = D(10) = 6 − 10/2 = 1
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Problem 21
Given f (x, y) = xy + y , px = 2, py = 6,what is the minimum cost that 3 units can be produced at? (Hint:
solve for x and y, then plug them into the cost function).
A. 2
B. 6
C. 10
D. 18
E. 25
Solution:. First, we solve for x and y, using fx /fy = px /py and the isoquant equation q = xy + y
y/(x + 1) = 2/6
y = (x + 1)/3
We can substitute into the isoquant equation:
q = x(x + 1)/3 + (x + 1)/3 = (x + 1)2 /3
p
3q = x + 1
p
3q − 1 = x
Quantity is 3, so
√
9−1=x
x=2
Now we can solve for y:
y = (2 + 1)/3 = 1
Cost is then
C(3) = px x + py y = 2 ∗ 2 + 6 ∗ 1 = 10
.
Problem 22
What sort of returns to scale does f (x, y) = (x2 + 2xy + y 2 )1/2 have?
A. Increasing .
B. Constant.
C. Decreasing.
D. Decreasing, then increasing.
E. Increasing, then decreasing.
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Solution:. Note that x2 + 2xy + y2 = (x + y)2 . Then (x2 + 2xy + y2 )1/2 = x + y. Now we must compare
tf(x,y) to f(tx,ty)
tf (x, y) = t(x + y) = tx + ty
f (tx, ty) = tx + ty
Then tf(x,y) = f(tx,ty), so we have CRS.
Problem 23
1/2
Given f (x1 , x2 ) = 4x1/2
and D(p)=8-p (monopolisticly competitive market), nd the rm's cost function
1 x2
C(q, p1 , p2 ), where p is the price of the output good and p1 and p2 are the prices of the input goods.
√
A. q 2√ p1 p2 /2
B. q 2√ p1 p2 /4
C. q√ p1 p2 /2
D. q p1 p2 /4
E. None of the above.
1/2
Solution:. First, we solve for x and y, using f1 /f2 = p1 /p2 and the isoquant equation q = 4x1/2
1 x2 :
1/2
1/2
1/2
1/2
4(x2 /2x1 )/4(x1 /2x2 ) = x2 /x1 = p1 /p2
x2 = p1 /p2 ∗ x1
We can substitute into the isoquant equation:
1/2
q = 4x1 (p1 /p2 ∗ x1 )1/2
q = 4x1 (p1 /p2 )1/2
q(p2 /p1 )1/2 /4 = x1
Now we can solve for y
x2 = p1 /p2 ∗ q(p2 /p1 )1/2 /4
x2 = q(p1 /p2 )1/2 /4
Cost is then
√
C(q, p1 , p2 ) = p1 x1 + p2 x2 = p1 q(p2 /p1 )1/2 /4 + p2 q(p1 /p2 )1/2 /4 = q(p2 p1 )1/2 /4 + q(p2 p1 )1/2 /4 = q p1 p2 /2
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Section IV
Problem 24
1/2
Consider the same market as in problem 23. f (x1 , x2 ) = 4x1/2
and D(p)=8-p (monopolisticly competitive
1 x2
market). Also assume that the prices of the two input goods are the same. Denote the price of both p*.
That is, p1 = p2 = p∗. Find the maximum prot (plug the prot maximizing price and quantity into the
prot function).
Hint: you just computed C(q) in the previous problem.
A. 32 − p ∗ − 87 p∗2
B.24 − 3p ∗ − 14 p ∗ 2
C. 8 − p ∗ − 34 p ∗ 2
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D. 64 − 4p ∗ − 32
p∗2
E. 16 − 2p ∗ − 163 p∗2
Solution:. We know C(q, p1 , p2 ) = q√p1 p2 /2 = q√p ∗ p∗/2 = qp ∗ /2.
To maximize prot, we must nd q such that
∂Π
= D0 (p)p + D(p) − C 0 (D(p))D0 (p) = 0
∂p
∂Π
= −p + 8 − p − p ∗ /2 ∗ −1 = 0
∂p
8 + p ∗ /2 = 2p
p = 4 + p ∗ /4
Then we can use the demand function to solve for q:
q = D(4 + p ∗ /4) = 8 − 4 − p ∗ /4 = 4 − p ∗ /4
Finally, we nd the maximum prot by plugging p and q into the prot function.
Π = qp − C(q) = (4 − p ∗ /4)(4 + p ∗ /4) − (4 − p ∗ /4)p ∗ /2
= 16 − p ∗2 /16 − 2p ∗ −p ∗2 /8
= 16 − 2p ∗ −3p ∗2 /16
Problem 25
Given U (x, y) = x2 y, py = 1,pxold = 1, pxnew = 8, and w=6, the substitution and income eects for x are,
respectively,
A. 0, −3.5
B. −1, −2.5
C. -2,-1.5
D. −1.5, −2
E. −3.5, 0
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Solution:. First, we solve for xorig and yorig , using U1 /U2
w = pxorig x + py y :
= pxorig /py and the budget line equation
Using the optimality condition,
2xy/x2 = 2y/x = 1
y = x/2
Plugging this into the budget constraint, we have
12 = x + x/2
6 = 3/2 ∗ x
4 = xorig
We can nd y easily:
yorig = 4/2 = 2
Second, we solve for xnew , using U1 /U2 = pnew /py and the budget line equation w = pnew x + py y :
Using the optimality condition,
2xy/x2 = 2y/x = 4
y = 4x
Plugging this into the budget constraint, we have
6 = 8x + 4x
6 = 12x
1/2 = xnew
Finally, we nd xsubs using the optimality condition with the new price and the utility from the old price:
Uorig = x2orig yorig = 42 ∗ 2 = 32
32 = x2 y
32 = x2 ∗ 4x
8 = x3
2 = xsubs
We can now compute the substitution eect xsubs − xorig = −2 and income eect xnew − xsubs = −1.5.
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