Consider a competitive, constant-cost industry. Firms operating in this market are identical and each firm's total cost is given by TC = 180 + 43q + 5q2. If the market demand curve is P = 331 - 2Q, what is the number of firms in this market in the long run?

To find the number of firms in this market in the long run, we first need to find the equilibrium price and quantity in the market.

Step 1: Find the Marginal Cost (MC) The marginal cost is the derivative of the total cost function with respect to quantity. The total cost function is TC = 180 + 43q + 5q^2. The derivative of this function is MC = d(TC)/dq = 43 + 10q.

Step 2: Find the equilibrium price In a competitive market, the price is equal to the marginal cost. So, we set MC = P and solve for q: 43 + 10q = 331 - 2Q This simplifies to 10q = 288 - 2Q And further simplifies to q = 28.8 - 0.2Q

Step 3: Find the equilibrium quantity Substitute q = 28.8 - 0.2Q into the demand function P = 331 - 2Q: 331 - 2Q = 43 + 10(28.8 - 0.2Q) This simplifies to 288 = 2Q So, Q = 144

Step 4: Find the number of firms The number of firms is the total quantity divided by the quantity produced by each firm. From step 2, we know that each firm produces q = 28.8 - 0.2Q = 28.8 - 0.2*144 = 1 unit. So, the number of firms is Q/q = 144/1 = 144.

Therefore, there are 144 firms in this market in the long run.

To find the number of firms in this market in the long run, we first need to find the equilibrium price and quantity in the market.

Step 1: Find the Marginal Cost (MC) of a firm The marginal cost of a firm is the derivative of the total cost function with respect to quantity. The total cost function is given by TC = 180 + 43q + 5q^2. The derivative of this function is MC = d(TC)/dq = 43 + 10q.

Step 2: Find the equilibrium price In a competitive market, the price is equal to the marginal cost. So, we set MC = P to find the equilibrium price. 43 + 10q = 331 - 2Q Since q (quantity per firm) times N (number of firms) equals Q (total quantity), we can substitute q = Q/N into the equation: 43 + 10(Q/N) = 331 - 2Q Solving this equation for Q gives us Q = 24N.

Step 3: Find the equilibrium quantity Substitute Q = 24N into the demand function P = 331 - 2Q to find the equilibrium quantity: P = 331 - 2(24N) = 331 - 48N

Step 4: Find the number of firms In the long run, the number of firms in the market is such that price equals average cost (AC). The average cost is total cost (TC) divided by quantity (q), or AC = TC/q. Substituting the given total cost function and q = Q/N gives us AC = (180 + 43Q/N + 5(Q/N)^2)/Q/N = 180N/Q + 43 + 5Q/N^2. Setting this equal to the price and solving for N gives us the number of firms in the market in the long run.

Please note that the final step involves solving a quadratic equation, which may have two solutions. In the context of this problem, we would take the positive solution as the number of firms cannot be negative.