Question

An industry consists of two firms with identical costs C(q) = 5q + q^2/2. The market demand is Q = 125 − p.

What is the equilibrium price, quantity per firm, and profit per firm, if we use the Bertrand Model to analyze this market? Assume that consumers can notice infinitesimally small differences in price.

Answer 1

Answer

As consumer can notice infinitesimally small differences in price implies that whatever produces charges lower price will \only sell its output and other firm will sell 0 units.

Thus If a firm charges price greater than its Marginal cost(MC) then the other will charge just lesser than MC and will have all the market. So no firm will charge Price greater than its MC.

Also No firm will charge Price lesser than its MC because this will result in the loss for that Firm.

Hence , each firm will charge Price equal to their MC and this is what we called a Bertrand Equilibrium.

Thus each firm is charge price equal to MC and share Market equally.

MC = d(C)/dQ = 5 + q and p = 125 - Q where Q = q₁ + q₂ and as q₁ + q₂ = q => Q = 2q

=> MC = 2q => 5 + q = 2q => q = 5.

=> p = 125 - Q = 125 - 10 = 115.

Hence Price = 115.

Thus equilibrium Price = 115 , Quantity of firm 1 = 5 , quantity of firm 2 = 5.

Profit = TR - TC = pq - C = 115*5 - (5*5 + 5²/2) = 115*5 - 37.5 = 537.5

Hence, Profit of each firm = $537.5.