Final answer:
The equilibrium market output rate is approximately 0.994 and the equilibrium price is approximately $5.0497. The optimal output for a profit-maximizing firm is approximately 6.262425. The rate of profit (or loss) earned by the typical firm is approximately $32.793.
Step-by-step explanation:
Given the market demand function P = 15 - 10Q and the market supply function P = 5 + 0.05Q, we can determine the equilibrium market output rate and price by setting the quantity demanded equal to the quantity supplied. Thus:
15 - 10Q = 5 + 0.05Q
Combine like terms:
10.05Q = 10
Divide by 10.05:
Q = 0.994
Substitute this value of Q into either the demand or supply function to find the equilibrium price. Using the supply function:
P = 5 + 0.05(0.994) = 5.0497
So, the equilibrium market output rate is approximately 0.994 and the equilibrium price is approximately $5.0497.
To determine the optimal output for a profit-maximizing firm, we need to find the quantity of output at which marginal revenue (which is equal to the market price for a perfectly competitive firm) equals marginal cost. The marginal cost function can be found by taking the derivative of the total cost function C = 100 - 20q + 2q^2 with respect to q. Thus:
MC = -20 + 4q
Setting MC equal to the market price, which is $5.0497, we have:
-20 + 4q = 5.0497
Combine like terms:
4q = 25.0497
Divide by 4:
q = 6.262425
So, the optimal output for a profit-maximizing firm is approximately 6.262425.
To determine the rate of profit (or loss) earned by the typical firm, we need to calculate total revenue, total cost, and profit at the optimal output. Total revenue can be found by multiplying the market price by the quantity of output:
Total revenue = (5.0497)(6.262425) = 31.6604
The total cost can be found by substituting the optimal output into the total cost function:
Total cost = 100 - 20(6.262425) + 2(6.262425)^2 = 64.4534
Profit can be calculated by subtracting total cost from total revenue:
Profit = 31.6604 - 64.4534 = -32.793
So, the typical firm is experiencing a rate of loss of approximately $32.793.