AP Calculus AB Worksheet: Consumers' and Producers' Surplus

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1 WorksheetConsumersProducersSurlus.nb 1 AP Calculus AB Worksheet: Consumers' and Producers' Surlus Finding consumers' and roducers' surlus requires a knowledge of rice equations; = D() is the rice-demand equation for a roduct, where is the number of units of the roduct that consumers will urchase at a rice of $ er unit and = S() is the rice-suly equation for a roduct where is the number of units of the roduct that roducers will suly at a rice of $ er unit. Let's look at some eamles. Eamle 1: Let = D() = (a) Find the rice er item if the demand for the item is 100 units and (b) find the number of units consumers will urchase if the rice is $8. Solution: (a) = D(100) = (100) = 20-5 = 15. So if the demand is 100 units the rice will be $15. (b) We wish to find so that = D() = = = = = -12 = 240 So, if the rice is $8, 240 consumers will urchase the item. Eamle 2: Let = S() = (a) Find the rice at which the roducers would be willing to rovide 550 units and (b) find the number of units the roducers will suly if the rice is $6.50. Solution: (a) = S(550) = H550L 2 = 62.5 So, the roducers would be willing to rovide 550 units if the rice were $62.50 er unit. (b) We wish to find so that = S() = = = = 22500

2 WorksheetConsumersProducersSurlus.nb 2 = 150 So, at a rice of $6.50, the roducers would be willing to rovide 150 units. Enough review. Let's get on to the new stuff! Look at the grah of the rice-demand equation below. reresents the current rice and reresents the number of units that can be sold at that rice. = DHL Now look at the same grah below. Notice that if the rice were higher than, the demand is less than, but some consumers are still willing to ay the higher rice. = DHL This means that consumers who were willing to ay more for the item ($) but were able to buy the roduct for less ($ ) have saved money. Let's ut this into ersective. You have saved your money to buy a new CD layer. You walk into the store and find out it's on sale for half off! You were erfectly willing to ay the full rice but didn't have to. So you saved

3 WorksheetConsumersProducersSurlus.nb 3 money. The amount consumers save on an item when they are able to ay less than they are willing to ay is called consumers' surlus. We can use definite integrals to find the total amount all of the consumers saved who were willing to ay more for an item. The consumers' surlus is reresented by the area between the rice-demand equation and the rice-level equation. Look at the definition below. Consumers' Surlus If (, ) is a oint on the grah of the rice-demand equation = D() for a articular roduct, then the consumers' surlus, CS, at a rice level is CS = Ÿ - D which is the area between = and = D() from = 0 to =, as shown below. The consumers' surlus reresents the total savings to consumers who are willing to ay more than for the roduct but are still able to buy the roduct for. = DHL CS Similarly, if = S() is the rice-suly equation for a roduct, is the current rice, and is the current suly, some suliers are still willing to suly some units at a lower rice than. The additional money that these suliers gain from the higher rice is called roduc-

4 WorksheetConsumersProducersSurlus.nb 4 ers' surlus and can be eressed in terms of a definite integral. See the definition below. Producers' Surlus If (, ) is a oint on the grah of the rice-suly equation = S(), then the roducers' surlus, PS, at a rice level is PS =Ÿ - SHLD which is the area between = and = S() from = 0 to =, as shown below. The roducers' surlus reresents the total gain to roducers who are willing to suly units at a lower rice than but are still able to suly units at. = SHL PS In a free cometitive market, the rice of a roduct is determined by the relationshi between suly and demand. If = D() and = S() are the rice-demand and rice-suly equations, resectively, for a roduct and if (, ) is the oint of intersection of these equations, is called the equilibrium rice and is called the equilibrium quantity. This is the rice level that often determines both the consumers' surlus and the roducers' surlus. Look at the eamle below. Eamle 3: Find the equilibrium rice and then find the consumers'surlus and the roducers' surlus at the equilibrium rice level if

5 WorksheetConsumersProducersSurlus.nb 5 = D() = and = S() = Solution: First find the equilibrium oint by setting the rice-demand and rice-suly equations equal to each other. Enter both equations into your calculator as Y 1 and Y 2. Set the Xmin and Xma to 0 and 400 resectively (notice that the rice-demand equation is 0 at = 400). Now ZoomFit the curve. Use your calculator to find the oint of intersection. You should find the intersection to be (200, 10). The -value in your calculator is the equilibrium quantity and the y-value is the equilibrium rice. Look at the grah below. 20 = SHL = CS PS H, L = DHL = The consumers' surlus is the area between the grah of = D() = and the grah of = 10 from = 0 to = 200. This results in the following definite integral CS = Ÿ H L Evaluating on our calculator gives us 1,000. So the consumers' surlus is $1,000. The roducers' surlus is the area between the grah of = 10 and the grah of = S() = from = 0 to = 200. PS = Ÿ 0 - H LD = $1,067 Rounded to the nearest dollar.