Question

Consider two firms facing the demand curve \(P=50-5 Q,\) where \(Q=Q_{1}+Q_{2}\). The firms' cost functions are \(C_{1}\left(Q_{1}\right)=20+10 Q_{1}\) and \(C_{2}\left(Q_{2}\right)=10+12 Q_{2}\) a. Suppose both firms have entered the industry. What is the joint profit- maximizing level of output? How much will each firm produce? How would your answer change if the firms have not yet entered the industry? b. What is each firm's equilibrium output and profit if they behave noncooperatively? Use the Cournot model. Draw the firms' reaction curves and show the equilibrium. c. How much should Firm 1 be willing to pay to purchase Firm 2 if collusion is illegal but a takeover is not?

Short answer

Firm 1's willingness to pay to acquire Firm 2 would be up to the difference between its monopolistic and non-cooperative Cournot profits over an appropriate time horizon. The exact values of the output and profit levels would depend on the specifics of the given demand and cost functions and require further calculation.

Step-by-step solution

Joint Profit Maximization

First, calculate the total cost \(C(Q) = C_1(Q_1) + C_2(Q_2)\). Then form the profit function \(\Pi(Q) = P(Q)\times Q - C(Q)\). Assume the two firms are a single entity and calculate the maximization of this function by taking the derivative with respect to \(Q\) and setting it equal to zero. Then solve for \(Q_1\) and \(Q_2\) as follow: If the firms have not entered the industry, the analysis would be similar, but we additionally consider the firms' initial costs to enter the industry.

Output and Profit in a Non-cooperative Scenario

Assume the firms behave non-cooperatively and each aims to maximize its own profit based on the expected output of the other firm. Derive each firm's reaction function as the output level that maximizes its own profit given the output level of the other firm, again using calculus by taking the derivative of each firm's profit function with respect to its own output, setting this equal to zero, and solving for the output level. The intersection of these two reaction functions is the Cournot equilibrium, which is a Nash equilibrium in quantities.

Willingness to Pay for a Takeover

Calculate how much the first firm should be willing to pay to acquire the second firm. It should be willing to pay up to the difference between the profit it would earn as a monopolist (Step 1) and the profit it earns in the non-cooperative Cournot equilibrium (Step 2) over an appropriate time horizon, taking into account any discounting. This will be the case if collusion is illegal, but a takeover is not.

Key concepts

Joint Profit Maximization

Understanding the principle of joint profit maximization involves recognizing a scenario where two or more entities combine their production strategies to maximize total profits. In the exercise scenario, two firms with different cost functions are considering this strategy. To find this joint profit-maximizing output level, we sum the firms' costs and set up a profit function (Profit = Total Revenue - Total Costs). Calculating the maximum profit is a straightforward exercise in calculus: taking the derivative of the profit function with respect to the total quantity (Q), and solving for (Q) when the derivative equals zero.
In situations where the firms haven't entered the market yet, we must also factor in entry costs. This changes the cost functions, adding a fixed cost to each firm's existing variable cost structure. Keep in mind that while joint profit maximization proposes an appealing strategy for firms, real-world market regulations and competition laws may restrict this level of cooperation.

Cournot Model

The Cournot model offers a classic framework for understanding how firms compete on quantity. Instead of banding together to maximize profits jointly, each firm selects its output independently with the goal of maximizing its own profit, considering the output choice of its rival. The Cournot model assumes that each firm has some degree of market power, and their products are substitutes.
When applying this model to the exercise, we assume that each firm decides its production level by predicting the other's output. We then determine each firm's 'reaction function', which is essentially a best-response output level given what they believe their competitor will produce. This reaction is based on individual profit maximization calculus. In the Cournot equilibrium, no firm has an incentive to unilaterally change its output, because they are best responding to each other—reaching what is known as a Nash equilibrium in quantities.

Reaction Curves

When analyzing market competition through quantities, reaction curves are indispensable tools. These curves graphically represent the best response of one firm to the quantity produced by another firm. In the context of our exercise, each firm's reaction curve delineates its optimal output level (Q_1 or Q_2) for each possible output level of the competitor, assuming the rival's output remains fixed.
The intersection point of Firm 1's and Firm 2's reaction curves represents a state where both firms are best responding to the other's quantity. This intersection is crucial: it indicates the Cournot Nash equilibrium we're seeking. In the exercise, plotting these curves helps to visually identify equilibrium outputs for both firms without relying solely on algebraic methods.

Nash Equilibrium

The concept of Nash equilibrium is central to game theory and economic modeling. It is a situation in which each player, or firm in our case, chooses the best strategy given the strategies chosen by other players. In the Cournot model's context, it's a state where both firms' outputs are set in such a way that neither firm can increase profit by unilaterally altering its own production level.
For our exercise, finding the Nash equilibrium involves solving the reaction functions simultaneously. It's the core of the Cournot model, enabling us to predict each firm's equilibrium output and profit in a non-cooperative setting. This condition of mutual best responses is necessary for equilibrium; if either firm deviates, it can only do so to its detriment. This framework guides firms in strategic decision-making, as any deviation from the Nash equilibrium signifies missed profit opportunities.