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Suppose the total cost of a representative perfectly competitive apple producer is given as TC = 12 +64 + q2

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Suppose the total cost of a representative perfectly competitive apple producer is given as TC = 12 +64 + q2. All apple producers in the market are assumed to be identical. Suppose further that the demand for apples is estimated as Qd = 18,000 – 500P and market supply is Qs = 2,000+ 500P. a. (2 points) Find the equilibrium market price and total supply of apples in the market. b. (4 points) What is the profit maximizing quantity of apples each company would produce? Find the total revenue, total cost and profits associated with the profit maximizing quantity. C. (4 points) Comment on whether this is an equilibrium in the short-run or in the long-run. Which assumption of perfectly competitive markets do you base your response on? d. (3 points) What is the short-run supply function of this apple producer? e. (2 points) What is the number of companies in the market in the short run? f. (5 points) Using the assumptions of the perfectly competitive model, comment on what will happen in the market in the long run. What will be the new equilibrium price? What will be the number of companies? Assume input prices will remain the same, no matter what, regardless of the number of apple producers in the market.

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Solution :

(a) : Market equilibrium occurs when market demand = market supply.

18,000 - 500P = 2,000 + 500P

1,000P = 16,000

P = 16

Q = 2,000 + 500 x 16 = 2,000 + 8,000 = 10,000

(b) : For individual Firms,

Marginal cost (MC) = dTC/dq = 6 + 2q

Profit is maximized when firms equate Market Price with MC.

6 + 2q = 16

2q = 10

q = 5

TR = P x q = 16 x 5 = 80

TC = 12 + 6 x 5 + 5 x 5 = 12 + 30 + 25 = 67

Firm Profit = TR - TC = 80 - 67 = 13

(c) : The Perfect competitive model's one of the assumptions is that entry and exit are free in long run (So in long run equilibrium, each firm earns zero profit and zero loss, since Price = MC = ATC).

Here, typical firm is earning positive profit, hence this is a short run equilibrium.

(d) : Firm's Short run supply function is its Marginal cost.

Therefore:

Firm's short run supply function: P = MC = 6 + 2q

(e) : Let n be the number of firm, so n firms producing q = (p-6)/2 units, have total supply of nq = n(p-6)/2

given that market supply Q = 2000+500P, we can say that Q = nq

thus, 2000 + 500P = n(p-6)/2

also, with Qd = 18000 - 500p, market price is determined where Qd = Q

18000 - 500p = 2000 + 500P

18000 - 2000 = 500p + 500p

16000 = 1000p

p = 16.

using p = 16 in 2000 + 500P = n(p-6)/2 to find n

2000 + 500*16 = n(16-6)/2

2000 + 500*6 = 5n

n = 2000/5+ (500/5)*6

n = 400 + 600 = 1000

There are 1000 firms

(f) : At p = 16, a firm's supply = q = (p-6)/2 = (16-6)/2= 5

At q = 5, TC = 12 + 6*5 + (5)²

= 12 + 30 + 25 = 67

TR = pq = 16*5 = 80

Since TR>TC, profit of each firm = TR - TC = 80 - 67 = 13

In short run, each firm is earning profit, so over the long run, more firms will enter the industry such that profits are wiped out.

Thus, in long run, each firm produces q such that profits = 0

This occurs when P = AC = MC

AC (= TC/q = 12/q + 6 + q) = P

and AC = MC is achieved at the minimum point of AC.

minimising Ac with respect to q, by FOC, dAC/dq = -12/(q)² + 1 = 0

12/(q)² = 1

12 = q²

q = $\sqrt{}$ 12 = 3.46

Long run price p = AC at q = 3.46, AC = 12/3.46 + 6 + 3.46 = 3.46 + + 3.46 = 12.92

Thus in long run price = 12.92

Qd = 18000 - 500*12.92 = 11540

and industry supply with N firms = Nq = 3.46N

Thus, at Qs = Qd, we have 3.46N = 11540

N = 11540/3.46 = 3335.2

There are 3335.2 firms in long run.

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Suppose the total cost of a representative perfectly competitive apple producer is given as TC = 12 +64 + q2

Economics