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Profit, Cost, and Revenue October 28, 2013 Profit, Cost, and Revenue Maximizing profit A fundamental issue for a producer is how to maximizing profit. Profit, Cost, and Revenue Maximizing profit A fundamental issue for a producer is how to maximizing profit. π(q) = R(q) − C (q) Profit, Cost, and Revenue Maximizing profit A fundamental issue for a producer is how to maximizing profit. π(q) = R(q) − C (q) MC = C 0 : Marginal cost; MR = R 0 : Marginal revenue Profit, Cost, and Revenue Example Estimating maximum profit if the revenue and cost are given by the curves R and C , respectively, in the figure. C $ (thousands) R 150 95 20 40 60 80 100 q (quantity) Profit, Cost, and Revenue Example Profit =Revenue − Cost C $ (thousands) R 150 95 20 40 60 80 100 q (quantity) Profit, Cost, and Revenue Example Profit =Revenue − Cost Profit is represented by the the vertical distance from curve C to curve R, marked by vertical arrow. C $ (thousands) R 150 95 20 40 60 80 100 q (quantity) Profit, Cost, and Revenue Example Profit =Revenue − Cost Profit is represented by the the vertical distance from curve C to curve R, marked by vertical arrow. Arrow is going down =⇒ No profit C $ (thousands) R 150 95 20 40 60 80 100 q (quantity) Profit, Cost, and Revenue Example Profit =Revenue − Cost Profit is represented by the the vertical distance from curve C to curve R, marked by vertical arrow. Arrow is going down =⇒ No profit Arrow is going up =⇒ Making profit C $ (thousands) R 150 95 20 40 60 80 100 q (quantity) Profit, Cost, and Revenue Example Profit =Revenue − Cost Profit is represented by the the vertical distance from curve C to curve R, marked by vertical arrow. Arrow is going down =⇒ No profit Arrow is going up =⇒ Making profit Profit is maximized if the arrow is going up and has the largest distance. C $ (thousands) R 150 95 20 40 60 80 100 q (quantity) Profit, Cost, and Revenue Example We now analyze the marginal costs and marginal revenues near the optimal point. MC<MR MC>MR R C MC=MR Profit, Cost, and Revenue Example We now analyze the marginal costs and marginal revenues near the optimal point. Global maxima and minima occur at critical points or at the endpoints of the interval MC<MR MC>MR R C MC=MR Profit, Cost, and Revenue Example We now analyze the marginal costs and marginal revenues near the optimal point. Global maxima and minima occur at critical points or at the endpoints of the interval π 0 (q) = R 0 (q) − C 0 (q) = 0 MC<MR MC>MR R C MC=MR Profit, Cost, and Revenue Example We now analyze the marginal costs and marginal revenues near the optimal point. Global maxima and minima occur at critical points or at the endpoints of the interval π 0 (q) = R 0 (q) − C 0 (q) = 0 MR = R 0 = C 0 = MC . MC<MR MC>MR R C MC=MR Profit, Cost, and Revenue Conclusion The maximum or minimum profit can occur where marginal profit=0. That is where marginal revenue=marginal cost. Profit, Cost, and Revenue Example The (total) revenue and (total) cost curves for a product are given in the Figure. (a) Sketch (roughly) the marginal cost and revenue. (b) Graph the profit function π(q). Profit, Cost, and Revenue Example Profit, Cost, and Revenue Example Find the quantity which maximizes the profit if the total revenue and total cost (in dollars) are given by R(q) = 5q − 0.003q 2 C (q) = 300 + 1.1q where q is quantity and 0 ≤ q ≤ 1000 units. What production level gives the maximize profit? Profit, Cost, and Revenue Maximize Revenue At a price of $80 for a half-day trip, a white-water rafting company attracts 300 customers. Every $5 decrease in price attracts an additional 30 customers. Find the demand equation. Profit, Cost, and Revenue Maximize Revenue At a price of $80 for a half-day trip, a white-water rafting company attracts 300 customers. Every $5 decrease in price attracts an additional 30 customers. Find the demand equation. Express revenue as a function of price Profit, Cost, and Revenue Maximize Revenue At a price of $80 for a half-day trip, a white-water rafting company attracts 300 customers. Every $5 decrease in price attracts an additional 30 customers. Find the demand equation. Express revenue as a function of price What price should the company charge per trip to maximize revenue? Profit, Cost, and Revenue Maximize Revenue At a price of $80 for a half-day trip, a white-water rafting company attracts 300 customers. Every $5 decrease in price attracts an additional 30 customers. Profit, Cost, and Revenue