Example (2)
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The document uses calculus to determine the profit-maximizing level of output for a firm in perfect competition facing a price of $4 and total costs of Q3 – 7Q2 + 12Q+5. It finds the marginal revenue is $4 and sets marginal cost equal to marginal revenue. This yields two possible output levels: 2/3 and 4. However, only at 4 is the slope of the marginal cost curve positive, indicating profit maximization. The total profit at this optimal output of 4 is calculated to be $11.
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Example (2)
- 1. Example A perfect competitive firm faces P=4 and TC = Q3 – 7Q2 + 12Q+5 Determine by using calculus the best level of out put of the firm by the marginal approach. TR = PQ = 4Q MR = d(TR) = 4 = P dq MC = d(TC) = 3Q2 – 14Q +12 dq
- 2. 4 = 3Q2 – 14Q +12 0 = 3Q2 – 14Q +8 0 = (3Q – 2)(Q-4) So that , Q = 2/3 and Q = 4 Thus, MR =MC at Q =2/3 and Q=4 In order for profit maximized rather than minimized, the MC curve should be rising(slope must be positive) at the point where MR =MC. Slope of the MC curve = d(MC) = 6Q -14 dq At Q=2/3, the slope is -10(the firm minimizes total profit) At Q=4, the slope is 10 so the firm maximizes its total profits.
- 3. • Find the total profit of the firm at this level of out put Profit = TR –TC = 4Q - 3Q2 – 14Q +12 = - Q3 – 7Q2 +8Q-5 = -64+112-32-5 = 11