Question

A firm in a perfectly competitive market has the following cost curve: TC = 200 + Q + 2Q^2 and The market demand is: Qd = 121 - P. There are 20 identical firms in the market (N =20) in the short-run.

What is the price of the product in the long-run?

Answer 1

In order to maximize profit a firm produces that quantity at which P = MC -------------(1)

where P = Price and MC = Marginal cost = d(TC)/dQ
In a long run a perfect competitive firm earns 0 profit because if they starts earning positive profit then new firms enters which increases supply and results in decrease in price till each firm starts earning 0 profits. And if they earn losses firm will exit the market till each firm earns 0 profit.

Profit = 0 => Total revenue = Total Cost => PQ = ATC*Q => P = ATC ------------------(2)

Note TC = ATC*Q , here ATC = average total cost

From (1) and (2) we get P = MC = ATC

Now, MC = d(TC)/dQ = 1 + 4Q

ATC = TC/Q = 200/Q + 1 + 2Q

So, MC = ATC => 1 + 4Q = 200/Q +1 + 2Q

=> 2Q² = 200

=> Q = 10

Hence This firm will produce Q = 10 units

So, P = MC = 1 + 4Q = 1 + 4*10 = 41

Thus P = 41

Hence, the price of the product in the long-run = 41