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ECON 102 Tutorial: Week 3 Shane Murphy www.lancaster.ac.uk/postgrad/murphys4/ [email protected] Elasticity Questions 1-3 deal with elasticity, specifically, we are looking at the price elasticity of demand. The price elasticity of demand is the percentage change in quantity demanded given a percentage change in price. We can write this mathematically as: πππππππ‘πππ πβππππ ππ ππ’πππ‘ππ‘π¦ ππππππππ π= πππππππ‘πππ πβππππ ππ πππππ In this course, weβll use two methods for elasticity of demand. The Arc Elasticity, and the Point Elasticity. Arc Elasticity: %β π π= %β π When we use this equation to find the elasticity between two points, we call this the arc elasticity. Point Elasticity: π= Where π ππππ = 1 π ππππ πππ π ππ’π = π π × , βπ βπ = βπ βπ . We can use this to find the elasticity at any particular point. Question 1(i) 1 2 The demand curve for potatoes is given by: P = 10 β Q Where p is the price of a pound of potatoes. Calculate the arc elasticity of demand if the price of potatoes increases from £2 to £4 Letβs start with our equation for the arc elasticity: %β π π= %β π To fill in our numerator, we need to know the quantity demanded for each given price. We find that by plugging in a price value into the demand equation, and solving for Q. So, At a price of £2, 1 2 2 = 10 β Q 1 2 -8 = β Q 16 = Q quantity demanded is 16. At a price of £4, quantity demanded is 12. Question 1(i) The demand curve for potatoes is given by: P = 10 β 1 Q 2 Calculate the arc elasticity of demand if the price of potatoes increases from £2 to £4 We found that At a price of £2, quantity demanded is 16. At a price of £4, quantity demanded is 12. We now have enough information to fill in our equation for the arc elasticity: π= π= %β π %β π = (12 β16)/16 (4β2)/2 (π2 βπ1 /π1 (π2 βπ1 )/π1 1 = β4 1 = .25 So the arc elasticity of demand is |-1/4| = ¼ or .25. Question 1(ii) The demand curve for potatoes is given by: 1 P = 10 β Q 2 Calculate the point price elasticity of demand when p = £3. Ok, letβs start with our equation for point price elasticity of demand: 1 π π= × π ππππ π There are three things we need to know to be able to solve this equation: P, Q, and the slope of the demand curve. We know that P = £3. We can solve for Q by plugging P into the demand curve equation: 1 2 3 = 10 β Q 1 2 -7 = β Q 14 = Q So, Q = 14. Now, all we need to do is find the slope. How can we find the slope? Question 1(ii) The demand curve for potatoes is given by: 1 2 P = 10 β Q Calculate the point price elasticity of demand when p = £3. When P = 3, we found Q = 14. We have 3 different ways we can find the slope. One is to utilize the fact that our demand curve is linear. So that means the equation of the demand curve can be written in slope-intercept form. Slope-intercept form is written as: Y = mX + c, where m is the slope, and c is the Y-intercept. So, if we re-write our demand curve in Y = mX + c, format, we will have: 1 1 P = β 2 Q + 10, so our slope is β 2. We now have enough information to fill in our equation for point price elasticity of demand: π= 1 π ππππ × π π Plugging in, we get: π= π= 1 3 1× 14 β2 3 7 The two alternative methods of calculating slope, would be: 1) to use calculus, where the derivative of any function is itβs slope, or 2) to calculate slope using the equation: π ππππ = πππ π ππ’π = βπ βπ = βπ . βπ Question 1(iii) Suppose the price of potatoes is currently £2. If the price of potatoes were to increase slightly, would total revenue increase or decrease? How do you know? 1 2 First, letβs calculate elasticity when P = 2. We know that P = 10 β Q, 1 so Q = 16 when P = 2. From part (ii), we know our slope is β . 2 1 π 1 2 1 π= × = × = 1 π ππππ π 16 4 β 2 Elasticity and the Effect of a Price Change on Total Expenditure β When price elasticity is greater than 1, changes in price and changes in total expenditures always move in opposite directions. β When price elasticity is less than 1, changes in price and changes in total expenditures always move in the same direction. β Another way to say this is: that a price increase will increase total revenue when the % change in P (price) > than the % change in Q (quantity). Question 2 The demand curve for some good is given by: P = a β bQ Find the price (in terms of the parameters in the problem) where the point price elasticity of demand is 1. For what prices will demand be elastic? For what prices will demand be inelastic? Question 2 The demand curve for some good is given by: P = a β bQ Find the price (in terms of the parameters in the problem) where the point price elasticity of demand is 1. For what prices will demand be elastic? For what prices will demand be inelastic? The point price elasticity at a point (P, Q) is (1/b)*(P/Q). Setting this equal to 1 we obtain: P/Q = b, or equivalently, Q = P/b. Plugging this quantity into the demand curve we obtain: P = a β b*(P/b). Which means, P = a β P, and hence, P = a/2. Question 2 The demand curve for some good is given by: P = a β bQ Find the price (in terms of the parameters in the problem) where the point price elasticity of demand is 1. For what prices will demand be elastic? For what prices will demand be inelastic? As P increases, the ratio P/Q will also increase. This means that for P > a/2, elasticity will be greater than 1 (ie demand is elastic), and for P < a/2, elasticity will be less than 1 (demand is inelastic). Question 3 For each of the following pairs of goods, would you expect the cross-price elasticities to be positive or negative? Why? First, letβs define the Cross-Price Elasticity of Demand: The percentage change in quantity demanded of one good in response to a 1 percent change in the price of another good. If the other good is a Substitute Good: Cross-Price Elasticity of demand is positive If the other good is a Complement Good: Cross-Price Elasticity of demand is negative Question 3 For each of the following pairs of goods, would you expect the cross-price elasticities to be positive or negative? Why? a) Almonds and Peanuts Almonds and peanuts are substitutes. Therefore, the cross price elasticity of demand is positive. b) Tortilla chips and Salsa Tortilla chips and salsa are complements. Therefore, the cross price elasticity of demand is negative. c) iPhones and iPhone apps iPhones and iPhone apps are complements. Therefore, the cross price elasticity of demand is negative. Question 4: Ch6 Q1(a) Suppose the weekly demand and supply curves for used DVDs in Brussels, are as shown in the graph below. Consumer surplus is the triangular area Calculate the weekly consumer surplus. between the demand curve and the price line. How do we measure Consumer Surplus? We measure the area of that particular triangular area, using the equation for the area of a triangle. 1 π΄ = ×π×β 2 where b is the base of the triangle and h is the height. In this case, b = 6 units and h = 1.5 units, measured in euros. Therefore, consumer surplus is 1 πΆπ = × 6 π’πππ‘π π€πππ × 1.5 β¬ π’πππ‘ 2 πΆπ = β¬4.50 per week. Question 4: Ch6 Q1(b) Suppose the weekly demand and supply curves for used DVDs in Brussels, are as shown in the graph below. Calculate the weekly producer surplus. Producer surplus is the triangular area between the supply curve and the price line. Using the base-height formula, it is (0.5)(β¬4.50/unit)(6 units/wk), or β¬13.50 per week. Question 5: Ch6 Q1(c) Calculate the maximum weekly amount that producers and consumers in Brussels would be willing to pay to be able to buy and sell used DVDs in any given week. The maximum weekly amount that consumers and producers together would be willing to pay to trade in used DVDs is the sum of gains from trading in used DVDsβnamely, the total economic surplus generated per week. The total economic surplus, in this case is PS + CS, which is β¬18 per week. Question 5: Ch 6 Q2(a) Refer to Problem 1. Suppose a coalition of students from a Brussels secondary school succeeds in persuading the local government to impose a price ceiling of β¬ 7.50 on used DVDs, on the grounds that local suppliers are taking advantage of teenagers by charging exorbitant prices. Calculate the weekly shortage of used DVDs that will result from this policy. Question 5: Ch 6 Q2(a) Refer to Problem 1. Suppose a coalition of students from a Brussels secondary school succeeds in persuading the local government to impose a price ceiling of β¬ 7.50 on used DVDs, on the grounds that local suppliers are taking advantage of teenagers by charging exorbitant prices. Calculate the weekly shortage of used DVDs that will result from this policy. At a price of β¬7.50, the quantity supplied per week = 2. The quantity demanded at this price is 18 per week, which implies a weekly shortage of 16 used DVDs. B 1 C64 Price 3 P=6+0.75Q D 12 6 P=12-0.25Q 2 6 48 Quantity A Question 5: Ch 6 Q2(b) Calculate the total economic surplus lost every week as a result of the price ceiling. Question 5: Ch 6 Q2(b) Calculate the total economic surplus lost every week as a result of the price ceiling. The weekly economic surplus lost as a result of the price ceiling (also called deadweight loss) is the area of the dark-shaded triangle in the diagram, or the sum of the areas of the two triangles ABC and ACD. Using the information given in the graph, this amount is calculated as (0.5)(4)(1) + (0.5)(4)(3) = β¬8/wk. B 1 C64 Price 3 P=6+0.75Q D 12 6 P=12-0.25Q 2 6 48 Quantity A Textbook example of consumer surplus and producer surplus when a price ceiling is imposed The textbook uses an example of a market for home heating oil. With no price controls the market equilibrium price is β¬1.40/litre and the equil. quantity is 3,000 litres/day. We can see what happens when a price ceiling of β¬1 is imposed. With no price controls: With a price ceiling of β¬1.00 per litre: Question 6: Ch 6 Q4(a) Suppose the weekly demand for a certain good, in thousands of units, is given by the equation P = 8 β Q, and the weekly supply of the good is given by the equation P = 2 + Q, where P is the price in euros. Calculate the total weekly economic surplus generated at the market equilibrium. Question 6: Ch 6 Q4(a) Suppose the weekly demand for a certain good, in thousands of units, is given by the equation P = 8 β Q, and the weekly supply of the good is given by the equation P = 2 + Q, where P is the price in euros. Calculate the total weekly economic surplus generated at the market equilibrium. The total economic surplus = PS + CS To find PS & CS, we need to find the Equilibrium price and equilibrium quantity β This is where S = D, where the two curves cross. We can find this 2 ways: graphically, or algebraically. If we graph, we should get the figure on the right: Algebraically, we set S = D 2+Q=8βQ 2Q = 6 Q=3 Plugging Q = 3 in to either equation, we get P = 5 So, the equilibrium price is β¬5 and the equilibrium quantity is 3,000 units per week. The consumer surplus is the area between the demand curve and the price lineβ triangle ABC in the diagramβwhich is β¬4,500/wk. The producer surplus generated is the area of triangle ABD, which is β¬4,500/wk. We find these using the A = ½bh formula. Adding PS + CS, we get the total economic surplus is β¬9,000/wk. Question 6: Ch 6 Q4(b) Demand: P = 8 β Q, Supply: P = 2 + Q, where P is the price in euros. Suppose a per-unit tax of β¬2, to be collected from sellers, is imposed in this market. Calculate the direct loss in economic surplus experienced by participants in this market as a result of the tax. Question 6: Ch 6 Q4(b) Demand: P = 8 β Q, Supply: P = 2 + Q, where P is the price in euros. Suppose a per-unit tax of β¬2, to be collected from sellers, is imposed in this market. Calculate the direct loss in economic surplus experienced by participants in this market as a result of the tax. The tax shifts the vertical intercept of the supply curve up by β¬2 to β¬4. The new equilibrium price and quantity are β¬6 and 2,000 respectively. The tax revenue is β¬2(2,000), or β¬4,000/wk. Consumer surplus is now the area of the triangle AβBβC, which is β¬2,000/wk. Net of the β¬2 tax, sellers receive a price of β¬4 per unit. Their surplus is the area of the triangle DβED, which is β¬2,000/wk. The direct loss in economic surplus, or the deadweight loss, is given by the triangle AβAE, and is equal to β¬1,000/wk. Question 6: Ch 6 Q4(c) Demand: P = 8 β Q, Supply: P = 2 + Q, where P is the price in euros. How much government revenue will this tax generate each week? If the revenue is used to offset other taxes paid by participants in this market, what will be their net reduction in total economic surplus? Question 6: Ch 6 Q4(c) Demand: P = 8 β Q, Supply: P = 2 + Q, where P is the price in euros. How much government revenue will this tax generate each week? If the revenue is used to offset other taxes paid by participants in this market, what will be their net reduction in total economic surplus? The tax revenue collected is (β¬2/unit)(2,000 units/wk) = β¬4,000/wk. If we count the revenue from the tax as part of total economic surplus, the new total economic surplus is thus β¬2,000/wk + β¬2,000/wk + β¬4,000/wk = β¬8,000/wk, or β¬1,000/wk less than without the tax. Textbook example of a market with a tax The market for potatoes with tax The market for potatoes without tax Deadweight loss tax revenue = (price paid by consumers β price kept by sellers ) *Q Question 7(a) As in prob. 4 in ch. 6, suppose the supply curve for some good is given by p = 2 + Q, and the demand curve for the good is given by p = 8 β Q. Suppose the government implements a β¬2 per-unit subsidy for suppliers. What is the equation of the new supply curve under the subsidy? p = 2 +Q 8 A 6 B E 4 F P=8-Q 2 4 Question 7(a) As in prob. 4 in ch. 6, suppose the supply curve for some good is given by p = 2 + Q, and the demand curve for the good is given by p = 8 β Q. Suppose the government implements a β¬2 per-unit subsidy for suppliers. What is the equation of the new supply curve under the subsidy? The equation of the new supply curve is p = Q. The supply curve shifts to the right, and the vertical distance between the old and new supply curves is equal to the amount of the subsidy, β¬2. Question 7(b) Supply Curve: p = Q Demand curve: p = 8 β Q. Find the market equilibrium price and quantity under the subsidy. In this equilibrium what is the price paid by buyers, and what is the price received by sellers? p = 2 +Q 8 A 6 C B E 4 D F P=8-Q 2 4 Question 7(b) Supply Curve with subsidy: p = Q Demand curve: p = 8 β Q. Find the market equilibrium price and quantity under the subsidy. In this equilibrium what is the price paid by buyers, and what is the price received by sellers? p = 2 +Q 8 The market equilibrium price and A 6 quantity are found by equating the C B D E new supply curve with the demand curve. 4 F 2 P=8-Q So, Q = 8 β Q 4 Q = 4, which means p = 4. Buyers pay the price of 4, and sellers receive a price of 4 + subsidy, which is: 4 + 2 = 6. Maths Questions Simplify the following: 23 ( i) 4 2 2 32 ο1 (ii) 4 ο2 22 2 -2 3ο1 (iii) -4 ο2 2 3 ο3 a (i) ο4 a 3b ο1 a (ii) 4 ο2 a b -2 b ο1 3ο1 a (iii) -4 ο2 ο2 a b 3 Maths Questions Solve for x: (i) y = x2 (ii) y2 ο½ x4 (iii) ya ο½ x b (i) y = 9x2 (ii) y2 ο½ 9x4 (iii) ya ο½ 9x b Next Week ο§ Please check Moodle for next weekβs worksheet from Prof. Rietzke (on Wednesdays), and for maths questions from Prof. Peel (on Thursdays).
