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Multiple choice question: A monopolist has the total cost function 𝑐(𝑞) = 800 + ...

Apr 26, 2024

Multiple choice question: A monopolist has the total cost function 𝑐(𝑞) = 800 + 8𝑞. The inverse demandfunction is 𝑝 = 80 – 6𝑞, where prices and costs are measured in pounds. If the firmis required by law to price its commodity at its marginal cost, Which of the following statements is correct? a. the firm’s loss is £400. b. the firm’s profits are zero. c. the firm’s loss is £800. d. the firm is making positive profit but not as much profit as it would make if itwere allowed to choose its own price.

Key Concept

Pricing at Marginal Cost in Perfect Competition

Explanation

In perfect competition, firms price their goods equal to the marginal cost. Since the monopolist is required to price at marginal cost, it behaves like a perfectly competitive firm, breaking even with zero economic profit.

Analysical question: Consider a monopolist that has a production function 𝑦 = 𝑓(𝑙, 𝑘) = 𝐴𝑙^𝛼𝑘^𝛽. The prices of labour and capital are denoted by 𝑤 and 𝑣 respectively. i. Determine the long run conditional demand and marginal cost functions. Doyou need to impose any restrictions on 𝛼 and 𝛽 for these functions to bevalid? Explain your answer.(20 marks). ii. Assume that 𝛼 = 𝛽 = 0.5, 𝐴 = 1 and 𝑤 = 𝑣 = 0.25. The market consists oftwo consumer groups. Each group consists of 100 identical consumers. Thedemand function of a consumer in Group 1 is 𝑦 = 4 − 𝑃 and of a consumerin Group 2 is 𝑦 = 2 − 𝑃. The monopolist charges a two-part tariff where aconsumer must pay a lump sum 𝐾 to be able to buy at price 𝑃 per unit. Determine the market demand function. If the monopolist serves only Group1, determine the profit maximising values of 𝐾 and 𝑃. If the monopolistserves both groups, determine the values of 𝐾 and 𝑃 that maximises themonopolist’s profit. Determine the group(s) that the monopolist would wantto serve.(25 marks) b. An incumbent faces the challenge of an entrant. The entrant decides whether toenter or stay out. Upon entry, the incumbent decides whether to accommodate entryor fight. If the incumbent accommodates the entry then they compete a la Cournot.The monopolist can also threaten to be aggressive and fight upon entry and producethe perfectly competitive output. If the new firm stays out, the incumbent earns themonopoly profit. Each of the two firms has zero marginal cost but the entrant hasfixed entry cost of 𝐹 = 0.1. The market demand function is 𝑄 = 1 – 𝑃. i. Determine the payoffs of both firms (i.e., profits) in each possible outcome(market structure) and draw the extensive form representation of thegame.(15 marks) ii. Is the threat of aggressive behaviour by the monopolist credible? Explainyour answer.(10 marks). iii. Determine all the Nash equilibria of the game that are not credible. Explainwhich parts of the Nash equilibrium strategies are not credible.(15 marks) iv. Assume that the incumbent serves 3 different markets and faces the threatof entry in each market sequentially. First, it faces the threat of entry inmarket 1 then in market 2 and finally in market 3. Determine the credibleNash equilibria of the dynamic game. Can the incumbent build reputationof being an aggressive player? Explain your answer

Answer

For part i, the long-run conditional demand functions for labor and capital are derived from the cost minimization problem, and the marginal cost function is derived from these demand functions. Restrictions on α and β are necessary to ensure diminishing marginal returns. For part ii, the market demand function is the horizontal sum of the individual demand functions of the consumers in both groups. The profit-maximizing values of K and P depend on the monopolist's ability to perfectly price discriminate between the two groups.

Solution

To determine the long-run conditional demand functions for labor (l) and capital (k), we need to minimize the total cost, which is the sum of the cost of labor (wl) and the cost of capital (vk), subject to the production function y = Al^αk^β

The Lagrangian for this minimization problem is given by:

\mathcal{L} = wl + vk + \lambda(y - Al^αk^β)

where λ is the Lagrange multiplier

Taking the partial derivatives of the Lagrangian with respect to l, k, and λ, and setting them to zero gives us the first-order conditions for a minimum

Solving the system of first-order conditions will yield the conditional demand functions for labor and capital

The marginal cost (MC) function can be derived by differentiating the total cost function with respect to output y, using the conditional demand functions for l and k

For the functions to be valid, we need to impose the restriction that 0 < α < 1 and 0 < β < 1 to ensure diminishing marginal returns

For part ii, the market demand function is obtained by summing the individual demand functions of all consumers in both groups

The profit-maximizing values of K and P for serving only Group 1 can be found by maximizing the monopolist's profit function, which includes the lump sum K and the price P per unit, taking into account the demand function of Group 1

If the monopolist serves both groups, the profit-maximizing values of K and P are determined by considering the combined market demand and maximizing the profit function that includes the lump sum K and the price P per unit for both groups

The monopolist will want to serve the group(s) that maximize its profit, which may involve serving one or both groups depending on the relative demand and willingness to pay

Key Concept

Long-run conditional demand functions and marginal cost function in the context of a monopolist's production function

Explanation

The long-run conditional demand functions for labor and capital are derived from the cost minimization problem given the production function, and the marginal cost function is obtained from these demand functions. Restrictions on α and β ensure the production function exhibits diminishing marginal returns.

Answer

For part b, the payoffs of both firms in each possible outcome are determined by the market demand function and the firms' cost structures. The extensive form representation of the game illustrates the strategic decisions and payoffs. The credibility of the monopolist's threat and the Nash equilibria depend on the strategic interactions and the potential for reputation building in a dynamic setting.

Solution

To determine the payoffs of both firms, we need to consider the market demand function Q = 1 – P and the fact that both firms have zero marginal cost

The incumbent's monopoly profit is the area under the demand curve above the price level when it is the sole producer in the market

If the entrant decides to enter and the incumbent accommodates, they compete à la Cournot, and the profits depend on the Cournot equilibrium quantities and prices

If the incumbent fights, it produces the perfectly competitive output, which may result in zero economic profit for both firms

The extensive form representation of the game includes the entrant's decision to enter or stay out and the incumbent's subsequent decision to accommodate or fight

The credibility of the monopolist's threat to fight is determined by comparing the monopolist's payoffs in the case of fighting versus accommodating

A Nash equilibrium is credible if the strategies are best responses to each other and if no player has an incentive to deviate given the other player's strategy

In the dynamic game with sequential threats of entry in three different markets, the incumbent's ability to build a reputation as an aggressive player depends on the outcomes of the early stages of the game and the observability of actions

Key Concept

Game theory and strategic interactions between incumbent and entrant firms

Explanation

The payoffs and credibility of threats in a strategic game are determined by the firms' cost structures, market demand, and the potential for reputation effects in a dynamic setting. The extensive form representation helps visualize the strategic decisions and payoffs. Nash equilibria that are not credible fail to satisfy the condition that no player has an incentive to deviate given the other player's strategy.