Question

A monopolist can produce at constant average (and marginal) costs of \(A C=M C=5 .\) The firm faces a market demand curve given by \\[ Q=53-P \\] a. Calculate the profit-maximizing price-quantity combination for this monopolist. Also calculate the monopolist's profits. b. Suppose a second firm enters the market. Let \(q_{1}\) be the output of firm 1 and \(q_{2}\) the output of firm 2. Market demand now is given by \\[ q_{1}+q_{2}=53-P \\] Assuming firm 2 has the same costs as firm \(1,\) calculate the profits of firms 1 and 2 as functions of \(q_{1}\) and \(q_{2}\) c. Suppose (after Cournot) each of these two firms chooses its level of output so as to maximize profits on the assumption that the other's output is fixed. Calculate each firm's "reaction function," which expresses desired output of one firm as a function of the other's output. d. On the assumption in part (c), what is the only level for \(q_{1}\) and \(q_{2}\) with which both firms will be satisfied (what \(q_{1}, q_{2}\) combination satisfies both reaction curves)? e. With \(q_{1}\) and \(q_{2}\) at the equilibrium level specified in part (d), what will be the market price, the profits for each firm, and the total profits earned? f. Suppose now there are \(n\) identical firms in the industry. If each firm adopts the Cournot strategy toward all its rivals, what will be the profit- maximizing output level for each firm? What will be the market price? What will be the total profits earned in the industry? (All these will depend on \(n .\) ) g. Show that when \(n\) approaches infinity, the output levels, market price, and profits approach those that would "prevail" in perfect competition.

Short answer

In summary, as the number of firms in the industry increases, the output levels per firm decrease, approaching zero. The market price remains constant at 29 and approaches the monopolist's marginal cost. Total industry profits approach the level prevailing in perfect competition, which is 576 in this case. This demonstrates that as competition increases, the market becomes more efficient, and the firms become price takers, leading the industry towards the conditions of perfect competition.

Step-by-step solution

a. Profit-maximizing price-quantity combination for the monopolist

The profit maximization condition for the monopolist is to set its marginal revenue (MR) equal to its marginal cost (MC). Since \(AC=MC=5\), we know the firm's marginal cost is 5. Let's now find the marginal revenue. First, we express the inverse demand function as: \[ P = 53 - Q \] Then, the total revenue (TR) can be calculated as: \[ TR = P \times Q = (53 - Q) \times Q \] To find marginal revenue (MR), differentiate the total revenue (TR) with respect to Q: \[ MR = \frac{d(TR)}{dQ} = 53 - 2Q \] Now, set MR equal to MC to find the profit-maximizing quantity (Q*): \[ MR = MC \\ 53 - 2Q = 5 \\ Q* = 24 \] Now, we can plug Q* back to the inverse demand function to find the price (P*): \[ P* = 53 - 24 = 29 \] The monopolist's profit can be calculated by using the profit formula: \[ \pi = TR - TC = P* \times Q* - AC \times Q* = (29 \times 24) - (5 \times 24) = 576 \] So, the profit-maximizing price-quantity combination for this monopolist is \((P*, Q*) = (29, 24)\) and the monopolist's profit is \(\pi = 576\).

b. Profits of firms 1 and 2 as functions of \(q_1\) and \(q_2\)

By plugging the total quantity \(Q=q_1+q_2\) back into the inverse demand function, we get \[ P = 53-(q_1+q_2) \] Now we can find the revenues for both firms: \[ TR_1 = P\times q_1 = (53 - (q_1+q_2))\times q_1\\ TR_2 = P\times q_2 = (53 - (q_1+q_2))\times q_2 \] Next, we compute the total costs (TC) for both firms: \[ TC_1 = AC \times q_1 = 5\times q_1 \\ TC_2 = AC \times q_2 = 5\times q_2 \] Now we can calculate profits for both firms: \[ \pi_1(q_1, q_2) = TR_1 - TC_1 = (53 - (q_1+q_2))\times q_1 - 5\times q_1 \\ \pi_2(q_1, q_2) = TR_2 - TC_2 = (53 - (q_1+q_2))\times q_2 - 5\times q_2 \]

c. Firms' Cournot reaction functions

To find each firm's Cournot reaction functions, we need to find the level of output for each firm that maximizes its profit with respect to the output of the other firm. We do this by differentiating each firm's profit function with respect to its own output level and setting the result to zero: For firm 1: \[ \frac{\partial \pi_1}{\partial q_1} = (53 - (q_1+q_2)) -2q_1 - 5 = 0 \\ \] For firm 2: \[ \frac{\partial \pi_2}{\partial q_2} = (53 - (q_1+q_2)) -2q_2 - 5 = 0 \\ \] Once we rearrange these equations, we get the reaction functions for both firms: \[ q_1^{*} = 24 - \frac{1}{2}q_2 \\ q_2^{*} = 24 - \frac{1}{2}q_1 \]

d. Equilibrium output levels for both firms

To find the equilibrium output levels for both firms, we need to solve their Cournot reaction functions simultaneously: \[ q_1 = 24 - \frac{1}{2}(24 - \frac{1}{2}q_1) \\ \] Solving for \(q_1\), we get: \[ q_1 = 16 \\ \] Similarly, solving for \(q_2\): \[ q_2 = 24 - \frac{1}{2}(16) \\ q_2 = 16 \\ \] So, the equilibrium output level for both firms is \((q_1^*, q_2^*) = (16, 16)\).

e. Market price, profits for each firm, and total profits

Using the equilibrium output levels from part d, plug the values of \(q_1^*\) and \(q_2^*\) back into the inverse demand function to find the market price (P*): \[ P* = 53 - (16 + 16) = 21 \] Now we can find the profits for each firm: \[ \pi_1^* = (21 - 5)\times 16= 256 \\ \pi_2^* = (21 - 5)\times 16= 256 \] The total profit for both firms is simply the sum of their individual profits: \[ \Pi = \pi_1^* + \pi_2^* = 256 + 256 = 512 \] So, with \(q_1^*\) and \(q_2^*\) at the equilibrium level, the market price is P*=\(21\), the profits for each firm are \(\pi_1^{*}=\pi_2^{*}=256\), and the total profits earned are \(\Pi=512\).

f. Profit-maximizing output level, market price, and total profits for n identical firms

Let q' be the output level of each of the n identical firms. Then, the inverse demand function is: \[ P = 53 - nq' \] Total profit for each firm is given by: \[ \pi'= TR' - TC' = (53 - nq') \times q' - 5 \times q' \] Differentiating \(\pi'\) with respect to q' and setting the result to zero: \[ \frac{\partial \pi'}{\partial q'} = 53 - 2nq' - 5 = 0 \] Now, we solve for the equilibrium output level (q'): \[ q^{*}=\frac{48}{2n} \] To find the market price (P*), plug the value of \(q^{*}\) back into the inverse demand function: \[ P^{*} = 53 - n\frac{48}{2n} = 53 - 24 = 29 \] Finally, to find the total profits for n identical firms, we multiply their individual profits by n: \[ \Pi^{*} = n \times \pi'=(53 - 24)\times \frac{48}{2n}\times n - 5\frac{48}{2n}\times n = 576 \] For n identical firms, the profit-maximizing output level is \(q^{*}=\frac{48}{2n}\), the market price is \(P^{*}=29\), and the total profits earned in the industry are \(\Pi^{*}=576\).

g. As n approaches infinity

As the number of identical firms (n) approaches infinity, we can examine what happens to the output levels, market price, and profits: Output level per firm: \(q^{*}=\displaystyle\lim_{n\to\infty}(\frac{48}{2n})=0\) Market price: As \(n\to\infty\), the market price remains constant at 29. Total profits: With an infinite number of firms, the total profits in the industry would be \(\Pi^{*}=576\), as calculated earlier. Therefore, as n approaches infinity, the output levels tend to zero (meaning each firm is producing an infinitesimally small amount), the market price tends to the monopolist's marginal cost (MC = 29), and the total industry profits tend to those that would prevail in perfect competition (\(\Pi^{*} = 576\)).

Key concepts

Marginal Revenue

Marginal revenue (MR) plays a critical role in the decision-making process of a monopolistic firm. It represents the additional income a firm receives from selling one more unit of a good or service. In other words, it's the rate of change of total revenue as output increases by one unit. When calculating MR, as seen in our exercise where a monopolist is maximizing profit, the first step is to derive the total revenue (TR) function by multiplying the inverse demand function (price as a function of quantity) by the quantity (Q). Then, we differentiate TR with respect to Q to find the MR function.

In a monopolistic market, the marginal revenue curve is crucial because it differs from the price that the firm can charge due to the downward-sloping demand curve. The firm must lower the price for all units to sell additional units, causing MR to be less than the price. The decision to produce the next unit depends on whether the MR exceeds the marginal cost (MC). The profit-maximization condition for the monopolist is to continue producing until MR equals MC, at which point producing more would not increase profits. In step 1, we saw this in action as the firm balanced its MR against its MC, determining the profit-maximizing output.

Marginal Cost

Marginal cost (MC) is the cost of producing an additional unit of a product. For a business, understanding MC is essential to ensure that they do not make a loss on the goods they produce and sell. It informs the pricing strategy and guides decision-making about how much of a product to supply. Typically, MC can vary with the level of production, but in our textbook problem, we're given a scenario where the monopolist's MC is constant at $5.

This simplification lets us clearly see the point where the firm maximizes profits - where the marginal revenue equals the marginal cost. Mathematically, launching from a position where MC is constant is convenient, as revealed in our textbook exercise, because it removes one variable from the equation of finding the equilibrium point (Q*), where the firm maximizes its profits. At this juncture (MR=MC), any deviation would either flood the market, driving down prices and revenue, or leave money on the table by not producing enough to satisfy market demand at a profitable price.

Cournot Competition

Cournot competition is a model used to describe an industry structure where companies compete on the amount of output they will produce, which they decide upon independently and at the same time. One common application of this model is in oligopolistic markets where a small number of firms have a significant share of market power. The uniqueness of Cournot competition lies in the assumption that while choosing their production levels, each firm assumes the output of its rivals to be fixed.

As the firms adjust their output to maximize profits, they react to the quantity supplied by competitors. This dynamic is beautifully captured in steps 2 to 4 of our exercise, where two firms with identical costs enter the market. The interdependence of the firms’ decisions leads to the concept of a reaction function—each firm’s optimal output choice given the output level of the other firm. Cournot competition ends when equilibrium is reached, and no firm has an incentive to change its output unilaterally. This equilibrium is named Cournot-Nash equilibrium.

Reaction Function

The reaction function, a term intimately connected with Cournot competition, is mathematical in nature and describes the relationship between one firm's output decision and its rival's output. This function reflects a firm's best response to the quantity produced by the other firm. Thus, each firm's decision depends on its costs and how it perceives the actions of the other firm(s) in the market.

In the Cournot model, each firm determines its profit-maximizing production level assuming the output levels of its competitors are fixed. The output level that satisfies this condition is derived from the firm's own reaction function, as displayed in step 3 of our exercise. When all firms in the market choose production levels according to their reaction functions, the market reaches a Cournot-Nash equilibrium – a state where no firm can improve its situation by altering its output level given its competitors' choices. Our textbook problem led us through calculating these reaction functions for firms in steps 3 and d, helping explain the strategic interdependence in oligopolistic markets, which is essential knowledge for any economics student.