Problem Set 6 Solutions
C(q) = 200 + 2q2
MC(q)=4q
1)
a) If P = 100, how many watches should be produced?
For competitive firms profits will be maximized by producing up until the point
where P = MC(q):
100 = 4q
so q = 25
b) What are profits?
The profit from producing 25 units is equal to the total revenue earned on these 25
units minus the total cost of producing these 25 units:
profits =TR – TC
= P*q – C(25)
= (100)*25 – (200 + 2*(252))
= $1050
c) At what minimum price will the firm produce a positive output?
In the short run, firms will produce as long as they can cover their variable cost.
This means we must have that TR(q) > VC(q). This condition can also be written as: P >
AVC (q). With a competitive firm, we know that P = MC(q). Thus as long as MC(q) >
AVC (q), firms will be covering their variable cost.
Recall that MC (q) =4q and AVC(q) =2q. Thus MC is greater than AVC for any
q >0. This means that P must be greater than zero, and thus a firm will produce as long
as price is positive.
2) C(q) = 4q2 + 16.
a)
VC (q) = 4q2
FC = 16
ATC (q) = 4q + 16/q
AVC (q) = 4q
AFC(q) = 16/q
cost ($/unit of output)
b)
45
40
35
30
25
atc
avc
20
15
10
mc
5
0
1
2
3
4
5
quantity
c) Find the output that minimizes average cost.
The output level that minimizes average cost occurs where the ATC curve is at its
minimum. Since we know the MC curve intersects the ATC curve at its minimum, we
can just set these equal and solve for q:
MC (q) = ATC (q)
8q = 4q + 16/q
4q = 16/q
q2 = 4
so q = 2
d) At what range of prices will the firm produce a positive output?
From 1c, we know the firm will produce a positive output as long as MC (q) >
AVC (q). This will happen as long as P > 0.
e) At what range of prices will the firm earn a negative profit?
Firms earn a negative profit whenever P < ATC (q). Since firms set P = MC (q),
they will earn negative returns whenever MC (q) < ATC (q). We know MC (q) = ATC
(q) when q=2. Since MC =16 when q=2, profits will be negative whenever P < $16.
f) At what range of prices will the firm earn a positive profit?
Similarly, firms will earn a positive profit whenever P > $16.
3) Can a tax that raises no revenue for the government have a deadweight loss?
The following diagram shows an example of a tax that is imposed on the seller.
This tax effectively shifts the supply curve up and the new equilibrium occurs where the
after-tax supply curve intersects the demand curve. Note that this occurs at q=0, and thus
this tax will generate no revenue. Essentially this tax is so large that it eliminates the
market for this good. The deadweight loss of this tax is represented by A.
S + tax
P
S
A
D
Q
4) The diagram on the following page shows the market for grain. The middle supply
curve (S) is what the supply curve would be in normal times, the SB curve represents the
supply curve during the years of bountiful harvests, and the SF curve represents the
supply curve during the years of crop failures. If a tax was collected on crop yields
during the times of bountiful harvests, this would act to shift the supply curve from SB to
S. If this stored grain was sold during the times of crop failures, this would shift the
supply curve from SF to S. Thus this policy acts to keep the supply curve at S no matter
what, which will keep the price of grain stable.
SF
S
P
SB
D
Q
5) a) Section 9.6 in the text provides a formula for the pass-through fraction, i.e., the
fraction of the tax borne by the consumer. This fraction is ES/( ES – ED), where ES is the
own-price elasticity of supply and ED is the own-price elasticity of demand. Substituting
for ES and ED, the pass-through fraction is:
4/(4-(-.2)) =.95
Therefore 95% of the tax is passed through to the consumers because supply is relatively
elastic and demand is relatively inelastic.
b) With an increase in the price of liquor (from the large pass-through of the liquor tax),
some consumers will substitute away from liquor to beer, shifting the demand curve for
beer outward. With an infinitely elastic supply for beer (a perfectly flat supply curve),
there will be no change in the equilibrium price for beer.