Gamerix is a monopolist in the industry of video game consoles. Itsmarket demand is given byQ = 600 − 10P ,where P and Q are, respectively, the market price and quantity of consoles. The total cost ofproducing Q units is given by the cost function C(Q) = 100 + 2Q. Gamerix practises simplemonopoly pricing.(a) Find the Marginal Revenue function and the marginal cost function. Find the profit-maximisingquantity. (3 marks)(b) Find the profit-maximising price. Calculate the corresponding profit and consumer surplus.(3 marks)(c) Calculate the dead-weight loss associated with the monopoly power of Gamerix. (2 marks)(d) A potential competitor, Game List, arrives in the country. Game List has total costsCGL(Q) = α · Q,where α is less than the price you found in part (b), but also satisfies α > 2. (i) Whatprice can Gamerix charge in order to retain the market? (ii) What would be the profits ofGamerix in that scenario? (iii) Find the value of α such that, if the marginal cost of GameList is below that value, it is not profitable for Gamerix to retain the market. (5 marks)
Question
Gamerix is a monopolist in the industry of video game consoles. Itsmarket demand is given byQ = 600 − 10P ,where P and Q are, respectively, the market price and quantity of consoles. The total cost ofproducing Q units is given by the cost function C(Q) = 100 + 2Q. Gamerix practises simplemonopoly pricing.(a) Find the Marginal Revenue function and the marginal cost function. Find the profit-maximisingquantity. (3 marks)(b) Find the profit-maximising price. Calculate the corresponding profit and consumer surplus.(3 marks)(c) Calculate the dead-weight loss associated with the monopoly power of Gamerix. (2 marks)(d) A potential competitor, Game List, arrives in the country. Game List has total costsCGL(Q) = α · Q,where α is less than the price you found in part (b), but also satisfies α > 2. (i) Whatprice can Gamerix charge in order to retain the market? (ii) What would be the profits ofGamerix in that scenario? (iii) Find the value of α such that, if the marginal cost of GameList is below that value, it is not profitable for Gamerix to retain the market. (5 marks)
Solution
(a) The demand function is Q = 600 - 10P. We can rewrite this in terms of P to get P = 60 - 0.1Q. The revenue function is R = PQ = Q(60 - 0.1Q) = 60Q - 0.1Q^2. The marginal revenue (MR) is the derivative of the revenue function, which is MR = 60 - 0.2Q. The total cost function is C(Q) = 100 + 2Q. The marginal cost (MC) is the derivative of the cost function, which is MC = 2. To find the profit-maximising quantity, we set MR = MC, so 60 - 0.2Q = 2, which gives Q = 290.
(b) Substituting Q = 290 into the price function gives P = 60 - 0.1290 = 31. The profit is R - C = PQ - C(Q) = 31290 - (100 + 2290) = 8480. The consumer surplus is the area under the demand curve and above the price, which is (1/2)(600 - 290)*(60 - 31) = 5015.
(c) The deadweight loss is the area of the triangle formed by the demand curve, the marginal cost curve, and the vertical line at the profit-maximising quantity, which is (1/2)(600 - 290)(2 - 31) = 4060.
(d) (i) To retain the market, Gamerix must charge a price less than or equal to α. (ii) The profits of Gamerix in this scenario would be PQ - C(Q) = α290 - (100 + 2290). (iii) The value of α such that it is not profitable for Gamerix to retain the market is when the profit is zero, so α290 - (100 + 2*290) = 0, which gives α = 2.69.
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