Description Marginal Revenue, Marginal Cost and Profit. Assignement contains five parts (a-e). UNF ...

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Description Marginal Revenue, Marginal Cost and Profit. Assignement contains five parts (a-e). UNFORMATTED ATTACHMENT PREVIEW University of Virginia BUSN 201: Business Mathematics Calculus Application: Marginal Revenue, Marginal Cost & Profit A firm faces the following demand (price) function for its product: p(q) = 6000 ? 3q where • p(q) is the price per unit (in US dollars), • q is the quantity demanded and sold (units per month). The firm’s total cost function is: C(q) = 200,000 + 500q + 0.5q 2 , where all monetary values are in US dollars. (a) Revenue and Profit Functions 1. Write the total revenue function R(q) in terms of q. 2. Write the profit function ?(q) in terms of q, where ?(q) = R(q) ? C(q). (b) Marginal Revenue and Marginal Cost 1. Find the marginal revenue function M R(q) = dR . dq M C(q) = dC . dq 2. Find the marginal cost function 1 (c) Profit Maximization 1. Find the quantity q that maximizes profit by solving the condition M R(q) = M C(q). 2. Find the corresponding selling price p (in dollars) at that optimal quantity. 3. Compute the maximum profit (in dollars) at this quantity. 4. Verify that the solution gives a maximum (not a minimum) using the second derivative of the profit function ?(q). (d) Economic Interpretation Answer the following questions in words: 1. Explain what it means when M R > M C at some quantity q. 2. Explain what it means when M R < M C at some quantity q. 3. Explain why the condition M R = M C is used to determine the profit-maximizing output level. (e) Sensitivity Analysis (Optional) Suppose due to market conditions, the demand function changes to p(q) = 5500 ? 2.5q. 1. Write the new revenue function R?(q) and the corresponding profit function ??(q). 2. Find the new profit-maximizing quantity under this modified demand. 3. Compare the optimal quantity and maximum profit in this new case with the original case. Briefly explain how and why the optimal decision changes. 2