Question

Suppose a representative firm with total cost given by TC=50+2q+2q^2 operates in a competitive industry where the short-run market demand and supply curves are given by Q_D =1410−40P and Q_S =−390+20P a. Compute the equilibrium price and market quantity. P1=__$ and Q1=___ b. Calculate its short-run profit-maximizing level of output of the individual firm. q_SR= ____

c. Find the number of firms operating in the short run. N=___ d. At this output level, what is its short-run profit? Profit= ____$ e. If it continues to operate, calculate its long-run equilibrium level of output of the individual firm. q_LR = ____
f. Compute the long-run equilibrium price and market output level Q at the price. P2=____$ and Q2=_____ g. Compute the number of firms operating in the long run. N2=___

Answer 1

a. To find the equilibrium price and market quantity, we need to set the quantity demanded equal to the quantity supplied and solve for P:

1410 - 40P = -390 + 20P

Adding 40P and 390 to both sides:

1410 + 390 = 20P + 40P

1800 = 60P

Dividing both sides by 60:

P = 30

Substituting P = 30 into either the demand or supply equation to find the market quantity:

Q = 1410 - 40(30)

Q = 1410 - 1200

Q = 210

Therefore, the equilibrium price is P1 = $30 and the market quantity is Q1 = 210.

b. The short-run profit-maximizing level of output for the individual firm can be found by setting marginal revenue (MR) equal to marginal cost (MC). In a competitive market, MR is equal to the market price, which we found to be $30. The marginal cost for the representative firm is given by the coefficient on Q^2 in the total cost equation, which is 2. Setting MR = MC:

30 = 2 + 4q

Subtracting 2 from both sides and dividing by 4:

4q = 28

q = 7

Therefore, the short-run profit-maximizing level of output for the individual firm is q_SR = 7.

c. The number of firms operating in the short run can be determined by dividing the market quantity (Q1) by the individual firm's quantity (q_SR):

N = Q1 / q_SR

N = 210 / 7

N = 30

Therefore, there are 30 firms operating in the short run.

d. To find the short-run profit, we need to calculate total revenue (TR) and subtract total cost (TC):

TR = P1 * Q1

TR = 30 * 210

TR = 6300

Profit = TR - TC

Profit = 6300 - (50 + 2q_SR + 2q_SR^2)

Substituting q_SR = 7:

Profit = 6300 - (50 + 2(7) + 2(7)^2)

Profit = 6300 - (50 + 14 + 98)

Profit = 6300 - 162

Profit = $6138

Therefore, the short-run profit is $6138.

e. In the long run, firms in a competitive market will adjust their output levels until they are operating at the minimum point of their average cost curve. The average cost for the representative firm is given by the total cost divided by the quantity:

AC = (50 + 2q + 2q^2) / q

Taking the derivative with respect to q and setting it equal to zero, we can find the minimum point:

dAC/dq = 2 + 4q

2 + 4q = 0

4q = -2

q = -1/2

Therefore, the long-run equilibrium level of output for the individual firm is q_LR = -1/2.

f. To find the long-run equilibrium price and market output level, we need to substitute q_LR into the demand or supply equation. Let's use the demand equation:

Q2 = 1410 - 40P2

Substituting q_LR = -1/2:

Q2 = 1410 - 40P2

1410 - 40P2 = -1/2

Adding 40P2 to both sides:

1410 = 40P2 - 1/2

Multiplying both sides by 2:

2820 = 80P2 - 1

Adding 1 to both sides:

2821 = 80P2

Dividing both sides by 80:

P2 = 35.2625

Substituting P2 = 35.2625 into the demand or supply equation to find the market output level:

Q2 = 1410 - 40(35.2625)

Q2 = 1410 - 1410.5

Q2 = -0.5

Therefore, the long-run equilibrium price is P2 = $35.2625 and the market output level is Q2 = -0.5.

g. The number of firms operating in the long run can be determined by dividing the market output level (Q2) by the individual firm's quantity (q_LR):

N2 = Q2 / q_LR

N2 = -0.5 / -1/2

N2 = 1

Therefore, there is 1 firm operating in the long run.