Download Chapter 8 3.14 Each firm maximizes profit by producing where price

Chapter 8
3.14
Each firm maximizes profit by producing where price equals marginal cost. The
marginal cost of production is
C
 2q .
q
Setting price equal to marginal cost (MC) and solving for q,
p = MC
p = 2q
q = 0.5p.
Given 10 firms, market supply is
Q = 10(0.5p)
Q = 5p.
Setting market supply equal to market demand and solving for p, the equilibrium price is
5p = 120 – p
6p = 120
p = $20.
If the equilibrium price is $20, then each firm supplies
q = 0.5(20)
q = 10 units.
Substitute the market equilibrium price into the market demand curve and solve for Q to find
the market quantity:
Q = 120 – p
Q = 120 – 20
Q = 100 units.
4.3 Assume the cost function is C = q – q2 + q3. The long-run equilibrium price will be
where the long-run average cost curve is at a minimum. Average costs (AC) are
AC = –1q + q2.
To find where average costs reach a minimum, take the derivative of the average cost
function with respect to q:
C
= –1 + 2q.
q
Setting this equal to zero and solving for q, long-run average costs reach a minimum at a
quantity of
–1 + 2q = 0
1 = 2q
q = 0.5 units.
At this quantity, the long-run average cost of production and hence the long-run
equilibrium price is
AC = 1 – q + q2
AC = 1 – (0.5) + (0.25)
AC = p = $0.75.
Using the market demand function, the market quantity is
Q = 24 – p
Q = 23.25 units.
Each firm produces 0.5 units of output, and the total market quantity is 23.25 units.
Therefore, there must be 46.5 firms, from 23.25/0.5.
If the government imposes a $1.00 tax, the minimum average cost of production and hence
the equilibrium price increase by $1.00 (to $1.75), but the quantity produced by the firm
does not. Market quantity is
Q = 24 – p
Q = 22.25 units.
Each firm produces 0.5 units of output, and the total market quantity is 22.25. Therefore,
there must be 44.5 firms, from 22.25/0.5.
Chapter 9
2.2
Producer surplus (PS) is the difference between the amount for which a good sells and the
minimum amount necessary for the seller to be willing to produce the good. This is
equal to the area under the price and above the firm's marginal cost curve (which is the
firm's supply curve). When the price is $10, the quantity supplied is
q = 2 + 2p
q = 2 + 2(10)
q = 22 units.
The area under price is 220, from 22 multiplied by $10. The area under the supply
curve is the area of a triangle with a height equal to the price ($10) and a base equal to
the difference in the 22 units supplied at the $10 price and the 2 units supplied when
price equals zero:
0.5(10)20 = 100.
The area under price and above marginal cost is equal to the difference in the area under
price (220) and the area under the supply curve (100):
PS = $120.
5.1
The initial equilibrium is Q*  30, p*  30. The tax reduces output to 29. Consumers pay $31
and producers receive $29. Tax revenue is $58, and deadweight loss is $1.
Chapter 11
1.8 The firm's profit function π is (where R is total revenue and C is total cost):
π = R(Q) – C(Q)
π = pQ – [25 + Q + 0.5Q2]
π = [13 – Q]Q – [25 + Q + 0.5Q2].
Maximizing this with respect to Q to find the profit – maximizing quantity:

= 13 – Q – Q – 1 – Q
Q
13 – Q – Q – 1 – Q = 0
12 – 3Q = 0
Q = 4 units.
Using the demand equation, the profit – maximizing price is
p = 13 – Q
p = 13 – (4)
p = 9.
Profit per unit is the $9 price minus the average total cost of production. The average
total cost (ATC) of production is
ATC =
ATC =
C
Q
25  Q  0.5Q 2
Q
ATC = (25/Q) + 1 + 0.5Q.
When producing four units of output, this cost is
ATC = (25/4) + 1 + 0.5(4)
ATC = 6.25 + 1 + 2
ATC = $9.25.
Therefore, the profit margin is –$0.25, from the $9 price minus the $9.25 average total
cost of production.
Total profit is –$1, from the profit margin (–$0.25) multiplied by the profit-maximizing
quantity (4 units).
If the monopoly shuts down, then its losses equal fixed costs ($25). If fixed costs are
greater than the monopoly's losses, then the monopoly should operate in the short run.
Since profit is –$1 and fixed costs are $25, the monopoly should operate.