Suppose that you can sell as much of a product as you want for$100 per unit. You

Question

Suppose that you can sell as much of a product as you want for$100 per unit. Your marginal cost is: MC=2Q. Your fixedcost is $50. What is the optimal output level? I am looking for help with how to set this problem up. Iknow the MC=C/Q, but the way I read this is that theprice is $100 and the fixed cost is $50. The textbook is notreal clear on how to calculate optimal level. Thanks Suppose that you can sell as much of a product as you want for$100 per unit. Your marginal cost is: MC=2Q. Your fixedcost is $50. What is the optimal output level? I am looking for help with how to set this problem up. Iknow the MC=C/Q, but the way I read this is that theprice is $100 and the fixed cost is $50. The textbook is notreal clear on how to calculate optimal level. Thanks

Explanation / Answer

When you want to find the optimal output level, simply set priceequal to marginal cost, and solve. P = MC 100 = 2Q Q = 50. This is an easy rule to follow, but it does not fully explain thelogic behind it, without more thinking. If you would like anelaboration, read below. The idea here is that, to choose optimal output, a firm wants tomaximize its profits. Profits are defined as Revenue -Costs. Revenue can be defined as price times quantity (PQ),and costs can be defined as variable costs plus fixed costs. Variable costs can be calculated by integrating the marginalcost. In this situation, MC = 2Q, and the integral of 2Q isQ2/2 + C. That C is the fixed cost, which is givenas 50. So, total costs are Q2/2 + 50. We nowhave profits equal to PQ - (Q2/2 + 50). Tomaximize profits, we need to take the first derivative of theprofit equation (with respect to Q, because that is what we areoptimizing), and set it equal to zero (this is the definition offinding extrema from Calculus). Differentiating and settingequal to zero yields P - 2Q = 0, or P = 2Q. This is exactlyprice equals marginal cost, as stated above.

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