Download Q1: What assumptions do we make about perfectly competitive firms

ECON 303
Spring 2005
Exam 1
Dr. Cary Deck
This exam consists of 4 written problems worth 25 points each. Your exam should
contain 5 pages. Please write your name on the top of each page. Answer each question
as best you can. Where appropriate you must show work in order to receive full credit.
The exam is closed book. If you have any questions please raise your hand and someone
will come to you. There is no talking allowed during the exam. The use of electronic
devices other than approved calculators is prohibited. You have one hour and twenty
minutes to complete this exam. Exams will not be accepted after the end of the exam has
been announced.
Name:___________
Score:____________
Q1: What assumptions do we make about perfectly competitive firms and what are the
implications of these assumptions? (6 points)
1) Lots of buyers and sellers with homogenous products. This implies that everyone
is a price taker so MR=P.
2) There is free entry and exit in the market. This implies that profits are 0 in the
long run.
If a competitive firm has TC=2q3-q2+50, what are its AVC and AFC? (4 points)
FC= TC when q =0, so FC=0-0+50=50. Thus AFC =50/q. VC=TC-FC=2q3-q2 and
thus AVC = 2q2-q.
At what profit level would this firm prefer to simply shut down? Explain (4 points)
If the firm shuts down it will have no revenues but it will still have to pay its fixed cost of
50. Thus shutting down will earn a profit of -50. Therefore, the firm would shutdown at
any profit level below -50.
If the market price is 140, what quantity should this firm produce and what profit will it
earn? (8 points)
The firm should set MR=MC. By perfect competition P=MR so MR=140. MC =dTC/dq
=6q2-q. Thus at the optimal quantity, 140= 6q2-q or 6q2-q-140=0. Using the quadratic
formula q=5. The firm’s profits are 5*140- 2(5)3+(5)2-50= $425.
What would we expect to see happen to the long run price in this market?
Explain (3 points)
Since the firm is earning positive profits, other firms will enter, pushing supply out and
price down.
Q2: You manage a firm that can produce output according to the following production
function: f(L,K)=L1/2K1/2. What kind of returns to scale does your production function
exhibit? (3 points). What does this imply about your firm’s long run ATC curve?
Explain (4 points)
Since 2*f(1,1)=2 *1*1=2=21/2*21/2 = f(2,2), this firm has constant returns to scale. The
ATC will be constant. This is true since ATC=TC/q and doubling the inputs will both
double the cost and the output leaving the ratio unchanged.
If the wage rate is w=1 and the rental rate of capital is v=4, what is the equation for your
ATC? (6 points)
The RTS=K/L. Setting this equal to w/v we have K/L=1/4 or 4K=L. q= L1/2K1/2 =
(4K)1/2K1/2=2K so K=q/2 and L=4(q/2)=2q. TC=wL+vk=1*2q+4*q/2=4q. Therefore
ATC=4.
An alternative technology exists that enables your firm to produce output according to the
following production function: f(L,K)=LK. What will be your MC if you use this new
technology? (6 points)
With this production function RTS=K/L and thus RTS=w/v again gives 4K=L. q= LK
=4K2 so K2=q/4 or K=q1/2/2. L=4K=2 q1/2 and TC= wL+vK=1* 2 q1/2 +4*q1/2 /2 = 4
q1/2. MC=dTC/dQ=2 q-1/2.
Unfortunately, this new technology requires that your labor force to be retrained. How
much would you be willing to pay to switch to this new technology if you need to
produce 100 units? (6 points)
With the old technology, TC for 100 units is 4(100)=400. With the new technology TC
for 100 units would be 4(100)1/2=40. So you would be willing to adopt the technology as
long as the training costs were less than 400-40=$360.
Q3: A firm produces hog ties, ties with a Razorback holding a noose with a Longhorn in
it. Given the production process, the number of hog ties is determined by f(L,K) =
3L+5K. The current wage rate is w=10 and the rental rate of capital is v=15. Sketch a
graph of the output expansion path for this firm. (4 points)
With a linear production curve, the firm will either use only K or only L. MPL/w=3/10
and MPK/v=5/15 so the firm would prefer to use only capital. The output expansion path
is just the K axis.
Assuming that this firm is competitive, what will be the market price of hog ties?
Explain (4 points).
We know that P=MR by perfect competition and MR=MC by profit maximization. Given
the output expansion path q=0+5K so K=q/5 and TC=0+15*q/5=3q. So MC (and thus
P)=3.
If instead, the inventor gets a patent so the firm is a monopolist and demand is given by
P=99-2Q, what price will we observe in the market? (6 points)
MC is still 3. TR=(99-2Q)*q so MR=99-4Q. Setting MR=MC gives 96=4q so q=24.
From demand we know that P=99-2q=51.
Calculate the deadweight loss that would result from this monopolist. (6 points)
We know the MC of the last unit the monopolist produces is 3 and its value (price) is 51.
Since everything is linear we know we are dealing with a triangle so all we need to do is
calculate the difference between the monopolist’s quantity and the efficient quantity
where P=MC. At the efficient quantity we have 3 =99-2Q so qe=48. DWL=.5*(513)*(48-24)=$576.
Using what we know about how marginal revenue and price are related by elasticity,
explain why the hog ties manufacturer should not operate along the portion of demand
that is inelastic. (5 points)
MR=P(1+1/e). If demand is inelastic then e>-1 so 1/e is less than -1 and 1+1/e is
negative implying that MR is negative since we assume that P is positive. As MC is
positive, it must be that MC>MR at this point and thus not the optimal quantity. You can
also see this be realizing that with an inelastic demand lowering the quantity and raising
the price will increase TR. At the same time lowering the quantity will lower TC so
profits would have to increase by lowering the quantity.
Q4: A typical firm in a competitive market has production given by f(L,K)= L1/2K1/3.
The wage rate is w=2 and the rental rate is v=5. If this market is under a union contract
to hire 25 workers, what is the MC for the firm? (5 points)
This is a short run problem as labor is fixed. q=5K1/3 so K=(q/5)3=q3/125.
TC=wL+vK=2*25+5* q3/125=50+ q3/25. Thus MC=3 q2/25.
Explain why MC is increasing in a short run problem such as this? (4 points)
Since labor is fixed the only way to produce more output is by hiring more K. But due to
the diminishing marginal product of capital, you must hire ever greater amounts of
additional capital to generate the same additional amount of output. Since you are
having to buy increasing amounts of capital the marginal costs are growing.
Suppose this firm is representative of the 5 firms in this industry. What is the equation
for the market supply curve? (6 points)
Since P=MR and MR=MC we have P=3 q2/25. The market supply curve tells us the total
number of units firm’s will supply at a given price. From the individual firm’s supply
curve we know one firm will supply q=(25P/3)1/2. Hence 5 firms will supply Q = 5*q =
5*(25P/3)1/2. One can rewrite this as P=3Q2/252.
If the market demand is given by P=600/Q, what will be the market price and quantity?
(4 points)
Setting supply equal to demand we have 600/Q=3Q2/252
Or Q3=200*252=8*25*252=23*253. So Q=2*25=50. The price will be 600/50=12.
What will be the profits to each firm? (6 points)
One firm will sell 50/5=10 units. The profit will be 12*10-50-103/25=$30