Question

Suppose there are two firms in a market, Firm 1 and Firm 2. Each firm has no Fixed Costs (FC=0) and has constant Marginal Cost =4(MC=4). Firm 1 chooses quantity of output, q₁ . After Firm 1 chooses its quantity, Firm 2 observes q₁ , and then chooses its output, q₂​ . The market demand curve is given by P =100−2Q, where Q=q₁​ +q₂

​. The following formulas will be useful in solving this problem: If market demand is given by P=a−bQ, then
MR₁ =a−2bq₁ −bq₂
MR₂ =a−bbq₁​ −2bq₂
and If TR=aq−bq₂
then MR=a−2bq b) (6 points) Find the Stackelberg-Nash equilibrium in quantities:
q₁ =24
q₂ =12

Answer 1

To find the Stackelberg-Nash equilibrium in quantities, we calculate the best response of each firm to the actions of the other firm. By setting the marginal revenue of each firm equal to zero, we can solve for the quantities q₁ and q₂ that constitute the Stackelberg-Nash equilibrium.

Step-by-step explanation:

The question revolves around finding the Stackelberg-Nash equilibrium for two firms in a market with given cost structures and a market demand curve. In a Stackelberg competition, one firm, called the leader (Firm 1), sets its output level first.

After observing this output, the follower (Firm 2) sets its output level. The Stackelberg-Nash equilibrium involves each firm maximizing its profits given the output of the other firm.

In the scenario where Firm 1 and Firm 2 have no Fixed Costs (FC=0) and a constant Marginal Cost (MC=4), and the market demand is given by P = 100 - 2Q, where Q is the total quantity produced by both firms, we use the marginal revenue formulas provided to derive the output levels that satisfy the equilibrium conditions.

Since the marginal revenue (MR) for each firm depends on its own output as well as the output of the other firm, we find the equilibrium by setting MR equal to MC for each firm and solving the resulting equations simultaneously. For Firm 1, MR1 = 100 - 2(4)q1 - 4q2, and for Firm 2, MR2 = 100 - 4q1 - 2(4)q2.

Setting each equal to the constant marginal cost (MC=4) and solving for q1 and q2 yields the equilibrium quantities. In a natural monopoly situation, the firm would set quantity where marginal revenue equals marginal cost to maximize profits, leading to economic profits if price is above average cost. If there are changes in cost structures or market prices, the firm adjusts output to maintain the MR = MC condition.