5. A market contains a group of identical price-taking firms. Each firm has a marginal cost
curve MC(Q) = 2Q, where Q is the annual output of each firm. A study reveals that each firm
will produce if the price exceeds $20 per unit and will shut down if the price is less than $20.
The market demand curve for the industry is D(P) = 240- P/2. At the equilibrium market
price, each firm produces 20 units. What is the equilibrium market price, and how many firms
are in this industry?
Shut Up
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To find the equilibrium market price and the number of firms in this industry, we need to determine the price at which the quantity demanded equals the quantity supplied.
1. Start by determining the equilibrium quantity:
- Each firm produces 20 units, and there are "N" firms in the industry.
- Therefore, the total quantity supplied in the market is 20N.
2. Now, determine the equilibrium price:
- The market demand curve is given by D(P) = 240 - P/2.
- At equilibrium, the quantity demanded equals the quantity supplied, so 240 - P/2 = 20N.
3. Solve for the equilibrium price:
- Subtract 240 from both sides: -P/2 = 20N - 240.
- Multiply both sides by -2: P = 480 - 40N.
4. Since each firm produces 20 units, the total quantity supplied (20N) is equal to the total quantity demanded (D(P)):
- D(P) = 20N = 240 - P/2.
- Substitute P = 480 - 40N: 20N = 240 - (480 - 40N)/2.
- Simplify the equation: 20N = 240 - 240 + 20N/2.
- Multiply by 2 to eliminate the fraction: 40N = 480 - 240 + 20N.
- Combine like terms: 20N = 240.
5. Solve for N:
- Divide both sides by 20: N = 240/20 = 12.
Therefore, there are 12 firms in this industry at equilibrium. To find the equilibrium market price, substitute N = 12 into the price equation: P = 480 - 40N = 480 - 40(12) = 480 - 480 = 0.
The equilibrium market price is $0, and there are 12 firms in this industry.