Question

The raspberry growing industry in the U.S. is perfectly competitive and constant-cost, and each producer has a long-run marginal cost curve given by MC(Q) = 20+2Q. The corresponding long-run average cost function is given by AC(Q) = 20 + Q +144/Q. The market demand curve is D(P) = 2488 − 2P . (a) What is the long-run equilibrium quantity for each individual firm? (b) What is the long-run price? (c) How many active producers are in the raspberry growing industry in a long-run competitive equilibrium?

Answer 1

The long-run equilibrium quantity for each individual firm is 12 packs of raspberries, at a long-run price of $44. There will be 200 active producers in the raspberry growing industry in a long-run competitive equilibrium.

Step-by-step explanation:

The student's question relates to finding the long-run equilibrium in a perfectly competitive and constant-cost raspberry growing industry. Given the marginal cost (MC) and average cost (AC) functions, along with the market demand curve, we proceed to find the equilibrium.

In a perfectly competitive market, long-run equilibrium is established when economic profit is zero, which happens where Price (P) = Marginal Cost (MC) = Average Cost (AC). This is because firms will enter or exit the market until there are no economic profits left, leading to the price equaling average cost.

Given the firm's MC function MC(Q) = 20 + 2Q, and the AC function AC(Q) = 20 + Q +144/Q, we equate MC to AC to find the equilibrium quantity Q. Solving 20 + 2Q = 20 + Q + 144/Q, we find that Q = 12 packs of raspberries.

Substitute Q into the MC or AC function to find the long-run price. Thus, the long-run price (P) is P = 20 + 2(12) = $44.

With the demand curve D(P) = 2488 - 2P and the long-run price of $44, we substitute P into the demand equation to solve for the total quantity demanded in the market: Qd = 2488 - 2(44) = 2400 packs of raspberries. Dividing total quantity demanded by quantity per firm (2400/12), we get the number of active producers, which is 200 producers.

Answer 2

The long-run equilibrium quantity is the output level where marginal cost equals average cost, and the long-run price is the cost at which the market supply and demand balance. The number of active producers is the total market quantity divided by the quantity each firm produces at the equilibrium price.

Step-by-step explanation:

To find the long-run equilibrium quantity and price in a perfectly competitive market, the following steps must be taken:

1. First, set the marginal cost equal to the price, since a perfectly competitive firm will continue to produce up to the point where the price (P) is equal to the marginal cost (MC), as per the profit-maximization rule (P = MC).
2. To find the long-run equilibrium price, we recognize that in the long run, firms will break even, meaning that price will also equal average cost (AC). Therefore, we set MC equal to AC and solve for Q to find the equilibrium quantity for the individual firm.
3. Having found the quantity, substitute it back into MC or AC (since they're equal at equilibrium) to find the equilibrium price.
4. To determine the number of active producers, divide the total quantity demanded at the equilibrium price by the individual firm's equilibrium quantity.

In the given example, at the break-even point: MC(Q) = AC(Q). Thus, setting 20+2Q = 20 + Q + 144/Q, we solve for the firm's equilibrium quantity Q. To solve for the equilibrium price P, we substitute Q into either MC(Q) or AC(Q), and then use the market demand curve D(P) to find the total market quantity and divide by the firm quantity Q to get the number of firms.

In mathematical terms, the equilibrium quantity Q is found by solving 20+2Q = 20 + Q + 144/Q. For the long-run price, we solve the equation 2488 - 2P = Q, which gives us the equilibrium price P. Lastly, the number of active producers can be determined by dividing the total market quantity at equilibrium price by the equilibrium quantity per firm.