Suppose the two firms in a duopoly pursue Cournot competition as described in Equation 19.10. Suppose each firm operates under conditions of increasing marginal cost but that firm \(A\) has a larger scale of operations than does firm \(B\) in the sense that \(M C_{A} < M C_{B}\) for any given output level. In a Nash equilibrium, will marginal cost necessarily be equalized across the two firms? Will total output be produced as cheaply as possible?

In the Cournot model of duopoly, each firm chooses its production level, assuming that its rival has already chosen its own production level. The crucial characteristic of this model is simultaneous decision-making, where each firm reacts to the anticipated production of the other.

A Nash equilibrium occurs when each firm chooses the best possible output level given its rival's choice, such that neither firm has an incentive to deviate from its chosen output level. In this case, each firm's output choice will depend on the other's marginal cost, and we can describe this equilibrium as a set of output levels \((Q_A^*, Q_B^*)\) that satisfy the following conditions: Firm A: \(\frac{\partial \pi_A}{\partial Q_A} = 0\) Firm B: \(\frac{\partial \pi_B}{\partial Q_B} = 0\) Where \(\pi_A\) and \(\pi_B\) are the profit functions for firm A and B, respectively.

In Cournot competition framework, the marginal cost equalization is not a necessary condition for Nash equilibrium. This is because each firm chooses its production based on the anticipated production of the other firm and doesn't take into account the industry marginal cost. At Nash equilibrium, the profit-maximization condition of each firm is given by: Firm A: \(P(Q_A^* + Q_B^*) - MC_A(Q_A^*) = 0\) Firm B: \(P(Q_A^* + Q_B^*) - MC_B(Q_B^*) = 0\) Thus, it is possible that at equilibrium, the marginal costs of the two firms are not equal i.e., \(MC_A(Q_A^*) \neq MC_B(Q_B^*)\)

In the case of Cournot competition and Nash equilibrium, the total output will not necessarily be produced as cheaply as possible. This is because each firm is only considering their rival's output decisions and trying to maximize their individual profits, not the total profit of the industry. In this exercise, since firm A has a lower marginal cost at any given output level, the total output would be produced more cheaply if firm A produced more output and firm B produced less. However, the Cournot competition does not guarantee that allocation because each firm's primary focus is their own profit maximization, resulting in a less efficient allocation of resources in the industry. In conclusion, under Cournot competition and the described conditions of the firms, marginal cost will not necessarily be equalized across the two firms in a Nash equilibrium, and the total output will not be produced as cheaply as possible.