A firm's demand curve is Q = 2 - 0.01P, where Q is measured in millions. The firm's output when marginal revenue is -$50 is ____.

Question

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To find the firm's output when marginal revenue (MR) is -$50, we first need to derive the marginal revenue curve from the demand curve. The demand curve is given as Q = 2 - 0.01P. To find the marginal revenue curve, we need to double the slope of the demand curve because the slope of the MR curve is twice the slope of the demand curve when dealing with linear demand. Thus, the MR curve is MR = 2 - 0.02P. Setting MR = -$50, we get 2 - 0.02P = -50. Solving for P gives P = 2600. Substituting P = 2600 back into the demand equation, Q = 2 - 0.01(2600) = 2 - 26 = -24. However, this approach seems to have a mistake in interpreting the MR and its relation to the given information. Let's correct the approach:Given the demand curve Q = 2 - 0.01P, to find the marginal revenue, we first express P in terms of Q and then differentiate the revenue (R = P*Q) with respect to Q.First, express P in terms of Q: Q = 2 - 0.01P implies P = (2 - Q)/0.01 = 200 - 100Q.Revenue, R = P*Q = (200Q - 100Q^2). The marginal revenue (MR) is the derivative of R with respect to Q, MR = dR/dQ = 200 - 200Q.Setting MR = -$50 (or -50) and solving for Q: -50 = 200 - 200Q implies 200Q = 250, so Q = 250/200 = 1.25 million.The correct interpretation involves understanding the relationship between the demand curve, marginal revenue, and the calculation error in the initial explanation.

A Firm's Demand Curve Is Q = 2 - 0