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Section 14.10 d Consumers and Producers Surplus i E 2 Lectures a l l u d b Dr. Abdulla Eid A . College of Science r D MATHS 104: Mathematics for Business II Dr. Abdulla Eid (University of Bahrain) Surplus 1 / 20

  1. Section 14.10 d Consumers’ and Producers’ Surplus i E 2 Lectures a l l u d b Dr. Abdulla Eid A . College of Science r D MATHS 104: Mathematics for Business II Dr. Abdulla Eid (University of Bahrain) Surplus 1 / 20

  2. Motivation Example Example Scenario: In the supermarket, you find that candy bar are on sale for 250 Fils. This is good news for you, because you were prepared to pay 300 Fils d i for them. E The supermarket manager is happy to see you loading them into the a l basket, because he knows the store’s wholesale cost is just 180 Fils a bar. l u You are thinking: “Lucky me! I’m paying 250 Fils for a 300 Fils item, so d b I’m coming out 50 Fils ahead on each bar¡‘ A he’s thinking: “Lucky me! I’m selling these 180-Fils bars for 250 Fils, so . I’m coming out 70 Fils ahead on each bar¡‘ r D Both parties feel like they are coming out ahead on the transaction. The term surplus is used to describe how much each party gains from the transaction Before we study the surplus, we need to recall the definition of equilibrium . Dr. Abdulla Eid (University of Bahrain) Surplus 2 / 20

  3. Consumer Surplus Recall: The Equilibrium point in the market is the point where the demand d function intersect with the supply function. We find that point by solving i E the equation a Demand = Supply l l u Question: What does the point of equilibrium in term of economics mean? d b Answer: The equilibrium point ( q 0 , p 0 ) is the point that represent where A the stability in the producer–consumer relationship occurs. In short, the . price p 0 is the price at which the consumers will purchase the same r D quantity of a product that producers wish to sell at that price. Sometimes, it best to think about it as the “fair price“ or “best price“. Dr. Abdulla Eid (University of Bahrain) Surplus 3 / 20

  4. Example Graph the demand and supply curves, find the equilibrium point and analyze the graph. d Demand: p = − 50 q + 2000 i E Supply: p = 10 q + 500 a l l Solution: We find first the equilibrium point ( q 0 , p 0 ) . u d b Demand = Supply A − 50 q + 2000 = 10 q + 500 . r D 2000 − 500 = 10 q + 50 q 1500 = 60 q q = 25 So the equilibrium point is ( q 0 , p 0 ) = ( 25, 750 ) . Dr. Abdulla Eid (University of Bahrain) Surplus 4 / 20

  5. Preparing to find the Surplus Draw a horizontal line at the p 0 = 750. We will analyze the suplus with q = 10. d Example i E 1 demand=1500. a l 2 Supply=600. l u d 3 Equilibrium Price =750 (Fair price). b A The consumers are thinking: “We would have bought 10 units for 1500, . but instead we got them for the equilibrium price of 750! What a deal! r D We came out 750 ahead on each of these items¡‘ The producers are thinking: “We would have sold 10 units for 600, but instead we sold them for the equilibrium price of 750. What a deal! We made an extra 150 on each of these items¡‘ Dr. Abdulla Eid (University of Bahrain) Surplus 5 / 20

  6. Definition 1 Consumer surplus is the total amount by which the consumers came d out ahead. It’s equal to the area between equilibrium and demand. i E � q 0 a CS = 0 ( Demand − p 0 ) dq l l u d 2 Producer surplus is the total amount by which the producers came b A out ahead. It’s equal to the area between equilibrium and supply. . r � q 0 D PS = 0 ( p 0 − Supply ) dq Dr. Abdulla Eid (University of Bahrain) Surplus 6 / 20

  7. � q 0 CS = 0 ( Demand − p 0 ) dq � 25 � 25 = ( − 50 q + 2000 − 750 ) dx = ( − 50 q + 1250 ) dx d 0 0 i � − 25 q 2 + 1250 q � 25 E = 0 a = 15625 l l u d b A � q 0 PS = 0 ( p 0 − Supply ) dq . r D � 25 � 25 = ( 750 − ( 10 q + 500 )) dx = ( 250 − 10 q ) dx 0 0 250 q − 5 q 2 � 25 � = 0 = 3125 Dr. Abdulla Eid (University of Bahrain) Surplus 7 / 20

  8. Example (Old Final Exam Question) The demand and supply equations for a product are: p = 30 − 0.05 q 2 Demand: d p = 21 + 0.04 q 2 Supply: i E a Determine the consumers’ and producers’ surplus at the market l l u equilibrium. d b Solution: We find first the equilibrium point ( q 0 , p 0 ) . A Demand = Supply . r D 30 − 0.05 q 2 = 21 + 0.04 q 2 9 = 0.09 q 2 0 = 0.09 q 2 − 9 q = 10 or q = − 10 ( rejected! ) So the equilibrium point is ( q 0 , p 0 ) = ( 10, 25 ) . Dr. Abdulla Eid (University of Bahrain) Surplus 8 / 20

  9. � q 0 CS = 0 ( Demand − p 0 ) dq � 10 � 10 ( 30 − 0.05 q 2 − 25 ) dx = ( 5 − 0.05 q 2 ) dx = 0 0 � 10 d � 5 q − 0.05 3 q 3 i = E 0 a = 100 l l u 3 d b � q 0 A PS = 0 ( p 0 − Supply ) dq . r � 10 � 10 D ( 25 − ( 21 + 0.04 q 2 )) dx = ( 4 − 0.04 q 2 ) dx = 0 0 � 10 � 4 q − 0.04 3 q 3 = 0 = 80 3 Dr. Abdulla Eid (University of Bahrain) Surplus 9 / 20

  10. Example (Old Final Exam Question) The demand and supply equations for a product are: p = 75 − q 2 Demand: d p = 27 + 2 q 2 Supply: i E a Determine the consumers’ and producers’ surplus at the market l l u equilibrium. d b Solution: We find first the equilibrium point ( q 0 , p 0 ) . A Demand = Supply . r D 75 − q 2 = 27 + 2 q 2 48 = 3 q 2 0 = 3 q 2 − 48 q = 4 or q = − 4 ( rejected! ) So the equilibrium point is ( q 0 , p 0 ) = ( 4, 59 ) . Dr. Abdulla Eid (University of Bahrain) Surplus 10 / 20

  11. � q 0 CS = 0 ( Demand − p 0 ) dq � 4 � 4 0 ( 75 − q 2 − 59 ) dx = 0 ( 16 − q 2 ) dx = � 4 d � 16 q − 1 3 q 3 i = E 0 a = 128 l l u 3 d b � q 0 A PS = 0 ( p 0 − Supply ) dq . r � 4 � 4 D 0 ( 59 − ( 27 + 2 q 2 )) dx = 0 ( 32 − 2 q 2 ) dx = � 4 � 32 q − 2 3 q 3 = 0 = 256 3 Dr. Abdulla Eid (University of Bahrain) Surplus 11 / 20

  12. Exercise (Old Final Exam Question) The demand and supply equations for a product are: p = 20 − 0.06 q 2 Demand: d p = 11 + 0.03 q 2 Supply: i E a Determine the consumers’ and producers’ surplus at the market l l u equilibrium. d b Solution: We find first the equilibrium point ( q 0 , p 0 ) . A Demand = Supply . r D 20 − 0.06 q 2 = 11 + 0.03 q 2 9 = 0.09 q 2 0 = 0.09 q 2 − 9 q = 10 or q = − 10 ( rejected! ) So the equilibrium point is ( q 0 , p 0 ) = ( 10, 14 ) . Dr. Abdulla Eid (University of Bahrain) Surplus 12 / 20

  13. � q 0 CS = 0 ( Demand − p 0 ) dq � 10 � 10 ( 20 − 0.06 q 2 − 14 ) dx = ( 6 − 0.06 q 2 ) dx = 0 0 d � 10 � 6 q − 0.06 i 3 q 3 = E 0 a l = 40 l u d b A � q 0 PS = 0 ( p 0 − Supply ) dq . r D � 10 � 10 ( 14 − ( 11 + 0.03 q 2 )) dx = ( 3 − 0.03 q 2 ) dx = 0 0 � 10 � 3 q − 0.03 3 q 3 = 0 = 20 Dr. Abdulla Eid (University of Bahrain) Surplus 13 / 20

  14. Example (Old Final Exam Question) The demand and supply equations for a product are: p = 25 − 0.2 q 2 Demand: d i p = 5 + 3 q Supply: E a Determine the consumers’ and producers’ surplus at the market l l u equilibrium. d b A Solution: We find first the equilibrium point ( q 0 , p 0 ) . . r Demand = Supply D 25 − 0.2 q 2 = 5 + 3 q 0 = 0.2 q 2 + 3 q − 20 q = 5 or q = − 20 ( rejected! ) So the equilibrium point is ( q 0 , p 0 ) = ( 5, 20 ) . Dr. Abdulla Eid (University of Bahrain) Surplus 14 / 20

  15. � q 0 CS = 0 ( Demand − p 0 ) dq � 5 � 5 0 ( 25 − 0.2 q 2 − 20 ) dx = 0 ( 5 − 0.2 q 2 ) dx = � 5 d � 5 q − 0.2 3 q 3 i = E 0 a = 50 l l u 3 d b � q 0 A PS = 0 ( p 0 − Supply ) dq . r � 5 � 5 D = 0 ( 20 − ( 5 + 3 q )) dx = 0 ( 15 − 3 q ) dx � 5 � 15 q − 3 2 q 2 = 0 = 75 2 Dr. Abdulla Eid (University of Bahrain) Surplus 15 / 20

  16. Example (Old Final Exam Question) The demand of a product is q = 10 √ Demand: 100 − p d i E Determine the consumers’ surplus at the market equilibrium which occurs a when p = 84. l l u Solution: We find first the equilibrium point ( q 0 , p 0 ) . Now d b q 0 = 10 √ 100 − 84 = 40. Moreover, we re–write the function to get A q 2 = 100 ( 100 − p ) . r D q 2 100 = 100 − p 1 100 q 2 p = 100 − Dr. Abdulla Eid (University of Bahrain) Surplus 16 / 20

  17. � q 0 d CS = 0 ( Demand − p 0 ) dq i E � 40 � 40 1 1 a 100 q 2 dq 100 q 2 − 84 ) dx = = ( 100 − ( 16 − l l 0 0 u � 40 d � 1 300 q 3 b = 16 q − A 0 = 1280 . r D 3 Dr. Abdulla Eid (University of Bahrain) Surplus 17 / 20

  18. Significance of Consumer Surplus d 1- When Consumer Surplus is high: i E This means that consumers have more money ’left over’ to spend than they a were expecting. They were prepared to pay higher prices for items they l l needed, and because they didn’t have to, they have money left over (more u d saving in hand). Some of the effects of a high consumer surplus are: b A Lower crime rate . Increased sales of non-necessities: vacation travel goes up, movie r D attendance goes up, etc. Dr. Abdulla Eid (University of Bahrain) Surplus 18 / 20

  19. Significance of Producer Surplus 2- When Producer Surplus is high: d i This means that producers have more profits than they were expecting. E They were able to sell their products for a higher price than they were a l willing to sell them for. Some of the effects of a high producer surplus are: l u d More money invested in product research (resulting in better, safer, b A more efficient products.) . More money for company growth and expansion (resulting in more r D jobs and lower unemployment.) Dr. Abdulla Eid (University of Bahrain) Surplus 19 / 20

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