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?
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To determine the optimal price, output level, and profit, we need to understand a few economic concepts and use mathematical calculations. Let's break down each question step by step:
a) What price should the manager of this firm put on its product?
In a perfectly competitive market, each firm is a price-taker, meaning they have no control over the market price. Therefore, the firm should set its price equal to the market price.
Given that other firms in the industry sell at a price of $100, the manager should also set the price of their product at $100.
b) What level of output should be produced to maximize profits?
To maximize profits, the firm needs to find the output level where marginal revenue (MR) equals marginal cost (MC). In perfect competition, the marginal revenue is equal to the market price, as each additional unit sold brings in the same amount of revenue.
To find the output level, we need to first find the marginal cost (MC) function from the total cost (TC) function.
The given total cost function is TC = 500 + Q^2.
To find the marginal cost function (MC), we need to take the derivative of the total cost function with respect to quantity (Q):
MC = d(TC)/dQ.
In this case, the derivative of TC = 500 + Q^2 with respect to Q is:
MC = d(500 + Q^2)/dQ = 2Q.
Now, we set the marginal cost (MC) equal to the market price ($100) to find the optimal output level:
2Q = 100.
Solving for Q:
Q = 50.
Therefore, the firm should produce an output level of 50 units to maximize profits.
c) How much profit will be earned?
To calculate the profit, we need to find the difference between total revenue (TR) and total cost (TC) at the optimal output level.
Total revenue (TR) is given by the formula:
TR = Price * Quantity.
Using the given price of $100 and the optimal output level of 50 units, we can calculate total revenue (TR):
TR = $100 * 50 = $5000.
Total cost (TC) is given by the function:
TC = 500 + Q^2.
Plugging in the optimal output level of 50 units, we can calculate total cost (TC):
TC = 500 + 50^2 = 500 + 2500 = $3000.
Profit (π) is calculated as:
Profit (π) = Total Revenue (TR) - Total Cost (TC).
Plugging in the calculated values:
Profit (π) = $5000 - $3000 = $2000.
Therefore, the firm will earn a profit of $2000.