The cost function for a firm is given by TC = 500 + Q2. The firm sells output in a
perfectly competitive market and other firms in the industry sell at a price of $100.
a) What price should the manger of this firm put on its product?
b) What level of output should be produced to maximize profits?
c) How much profit will be earned?
To solve these questions, we need to understand the concept of profit maximization in a perfectly competitive market. In such a market, firms are price takers, meaning they cannot influence the market price. They simply choose the output level that maximizes their profit based on the market price.
a) The manager of this firm should set the price equal to the market price, which is $100. This is because in a perfectly competitive market, firms have no control over the price and must accept the prevailing market price.
b) To determine the level of output that maximizes profits, we need to find the quantity at which the marginal cost (MC) equals the market price (P). The marginal cost is the derivative of the total cost with respect to the quantity, so let's find the derivative of the cost function:
TC = 500 + Q^2
Taking the derivative with respect to Q:
d(TC)/dQ = 2Q
Now, equate the marginal cost to the market price:
MC = P
2Q = 100
Q = 50
Therefore, the level of output that should be produced to maximize profits is 50 units.
c) To calculate the profit, we need to find the total revenue (TR) and subtract the total cost (TC). The total revenue is the product of the price and the quantity sold:
TR = P * Q
TR = 100 * 50 = 5000
Now, substitute the value of Q into the cost function to find the total cost:
TC = 500 + Q^2
TC = 500 + 50^2 = 500 + 2500 = 3000
Finally, the profit is calculated as:
Profit = TR - TC
Profit = 5000 - 3000 = 2000
Therefore, the firm will earn a profit of $2000.
a) In a perfectly competitive market, the firm should set the price equal to the market price. Therefore, the manager should put a price of $100 on its product.
b) To maximize profits, we need to find the quantity at which marginal cost equals marginal revenue. In a perfectly competitive market, the price is equal to marginal revenue. The marginal cost (MC) for this firm is the derivative of the total cost (TC) function with respect to quantity (Q):
MC = d(TC)/dQ = d(500 + Q^2)/dQ = 2Q
So, we need to find the quantity at which MC = MR = price.
2Q = $100
Q = $50
Therefore, the firm should produce a quantity of 50 units to maximize profits.
c) To calculate the profit, we need to subtract the total cost from the total revenue. The total revenue (TR) is given by price (P) multiplied by quantity (Q):
TR = P * Q = $100 * 50 = $5000
The total cost (TC) is given by the cost function:
TC = 500 + Q^2 = 500 + 50^2 = 500 + 2500 = $3000
Profit (π) is calculated as:
π = TR - TC = $5000 - $3000 = $2000
Therefore, the firm will earn a profit of $2000.