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Question 1 (25 points) Suppose there are 6 firms competing in a perfectly compe...

Jan 23, 2025

Solution

To find the equilibrium price and quantity in a perfectly competitive market, we first need to determine the marginal cost (MC) from the cost function

C(q)=\frac{3}{2} q^{2}+3 q+\frac{27}{2}

. The marginal cost is given by the derivative of the cost function:

MC(q) = C'(q) = 3q + 3

. In equilibrium, price

P

equals marginal cost

MC

. Setting

D(p) = 144 - 3P

equal to

MC

, we can find the equilibrium price. First, we set

P = MC

P = 3q + 3

. Now, substituting this into the demand equation:

144 - 3(3q + 3) = 0

. Solving for

q

gives us the equilibrium quantity. Finally, substituting

q

back into the price equation gives us the equilibrium price

In the long run, firms can enter or exit the market until economic profits are zero. The long-run equilibrium occurs where price equals the minimum average cost (AC). The average cost is given by

AC(q) = \frac{C(q)}{q} = \frac{\frac{3}{2} q^{2}+3 q+\frac{27}{2}}{q} = \frac{3}{2} q + 3 + \frac{27}{2q}

. To find the minimum AC, we take the derivative of

AC(q)

and set it to zero. Solving for

q

gives us the quantity at which AC is minimized. We then substitute this quantity back into the demand function to find the long-run equilibrium price

Answer

Equilibrium Price:

P = 21

, Equilibrium Quantity:

q = 3

; Long-run Equilibrium Price:

P = 12

, Quantity:

q = 6

Key Concept

In a perfectly competitive market, equilibrium occurs where supply equals demand, and in the long run, firms enter or exit until economic profits are zero.

Explanation

The equilibrium price and quantity are determined by the intersection of the demand curve and the marginal cost curve, while the long-run equilibrium is characterized by zero economic profit, leading to price equaling the minimum average cost.