10. Suppose an industry has a duopoly structure. Duopolist 1 has a cost function given by: C₁(y₁) = (y₁)² for output y₁20. Duopolist 2 has a cost function given by: C2(y2) = 12y2 for output y2 20. Denoting total output produced in the industry by y = (y₁ + y₂) the inverse demand function for the good produced in the industry is given by: p = 100 - y (a) Find the reaction function of each duopolist. (b) Using (a), obtain the output levels that will be produced in a Cournot-Nash equilibrium, and the price level in such an equilibrium. Illustrate your solution in (b) above in a diagram.

Problem 2.3. Monopoly with increasing marginal cost (15 points) A firm with cost function 2 CQ Q () 0.50 = is a monopoly in a market where the inverse demand function is pQ Q ( ) 120 2 = - . (a) Find the monopolist's marginal revenue and marginal cost. (b) Find the monopolist's profit- maximizing quantity and price Update: C(q)=.5q^2 and P(q)=120-2q

The demand function in a duopoly market is P(Q) = 300-0.3Q, where P is the price and Q is the total quantity demanded. Both companies have the same constant marginal cost, What is the deadweight loss if both companies maximize their profits and they are not allowed to cooperate? (MR is not MC since its not monopoly)

Lagatt Green is a monopoly beer producer and distributor operating in the hypothetical economy of Lightington. Assume that Lagatt Green is not able price discriminate, and so it sells its beer to all customers at the same price per bottle. The following graph gives the marginal cost (MC), marginal revenue (MR), average total cost (ATC), and demand (D) curves that Lagatt Green faces for beer in Lightington. Place the black point (plus symbol) on the graph to indicate the profit-maximizing price and quantity for Lagatt Green. If Lagatt Green is making a profit, use the green rectangle (triangle symbols) to shade in the area representing its profit. On the other hand, if Lagatt Green is suffering a loss, use the purple rectangle (diamond symbols) to shade in the area representing its loss. PRICE (Dollars per bottle) 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 D MC D 15 20 25 30 3.5 QUANTITY (Thousands of bottles of beer) 45 ATC MR Price (Dollars per bottle) 2.00 2.25 40 Monopoly Outcome…

4. A monopolist produces exactly the same product at three plants with respective costs C = Q1 + Qỉ C2 = 3Q2 C3 = 3Q3 – Q3 Ci If the price for the product is P; = 40 – Qi Rị = P;Qi i = 1,2,3 → TR = E{ R{ Find the critical value (Qj, Q2, Q3) of the firm's profit ( = TR – TC) and use the Hessian to determine if it is a relative minimum or maximum.

You are the manager of a monopoly. Your analytics department estimates that a typical consumer's inverse demand function for your firm's product is P = 350-20Q, and your cost function is C(Q) = 70Q. a. Determine the optimal two - part pricing strategy. Per-unit fee: $ Fixed fee: $ b. How much additional profit do you earn using a two-part pricing strategy compared with charging this consumer a per- unit price? $

Assume that one of the hot dog vendors successfully lobbies the city council to obtain the exclusive right to sell hot dogs within the city limits. This firm buys up all the rest of the hot dog vendors in the city and operates as a monopoly. Assume that this change doesn't affect demand and that the new monopoly's marginal cost curve corresponds exactly to the supply curve on the previous graph. Under this assumption, the following graph shows the demand (D), marginal revenue (MR), and marginal cost (MC) curves for the monopoly firm. Place the black point (plus symbol) on the following graph to indicate the profit-maximizing price and quantity of a monopolist. PRICE (Dollars per hot dog) 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 0 45 Monopoly MC MR 90 135 180 225 270 315 QUANTITY (Hot dogs) D 360 405 450 Monopoly Outcome Deadweight Loss ?

Assume that a local train station has a monopoly over train rides. The firm has estimated that its market demand curve can be drawn from the following equation P = 60 - (1/2) Q. Let MC AC = 15. The firm has determined that younger consumers (under 50) are willing to pay up to $30 per ticket for a train ride and that older consumers (50 or older) are willing to pay up to $50 per ticket for a train ride. If the firm price discriminates what is their profit?

3. Suppose a software monopolist faces two markets for its software, students and professionals. The demand curve of professionals is given by Qp = 200-2Pp and the demand curve by students is given by Qs = 150 - 3Ps. The firm's cost function is C(Q) = 400+5Q. (a) If the firm can price discriminate, what price should it charge in each market to maximize profits? How much profit does it earn? (b) If the firm cannot price discriminate, what price should it charge? Verify that it sells to both markets at this price. How much profit does it earn? Hint: If the firm cannot price discriminate, this means it must treat the two markets as a single combined market.

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