Consumer Welfare Econ 3010 PDF

Consumer Welfare Econ 3010 Introduction Thus far, we have studied: I How consumers make choices Introduction Thus far, we have studied: I How consumers make choices I How those choices depend on parameters (i.e., demand functions) Introduction Thus far, we have st...

Consumer Welfare Econ 3010 Introduction Thus far, we have studied: I How consumers make choices Introduction Thus far, we have studied: I How consumers make choices I How those choices depend on parameters (i.e., demand functions) Introduction Thus far, we have studied: I How consumers make choices I How those choices depend on parameters (i.e., demand functions) Economists don’t just want to know what happens when market conditions and/or policies are changed - we also want to know whether these changes are “good” or “bad” We need measures of consumer welfare A useful policy tool: when deciding whether a policy should be implemented, we can ask whether it will raise or lower welfare Outline 1. Consumer welfare: how do we measure the e↵ect of a price change on a consumer? Outline 1. Consumer welfare: how do we measure the e↵ect of a price change on a consumer? 2. How much money would we need to give a consumer to & big question compensate him for a price change? Outline 1. Consumer welfare: how do we measure the e↵ect of a price change on a consumer? 2. How much money would we need to give a consumer to compensate him for a price change? 3. E↵ects of government policies on consumer welfare: consumer welfare lets us measure the e↵ects of various policies (e.g., taxes, subsidies, quotas, etc.). Consumer welfare In some sense, this is simple: a price increase puts a consumer on a lower indi↵erence curve =) welfare decreases Consumer welfare In some sense, this is simple: a price increase puts a consumer on a lower indi↵erence curve =) welfare decreases To measure the di↵erence: just calculate U (new) U (old) numerical a measure Consumer welfare In some sense, this is simple: a price increase puts a consumer on a lower indi↵erence curve =) welfare decreases To measure the di↵erence: just calculate U (new) U (old) Two problems: Consumer welfare In some sense, this is simple: a price increase puts a consumer on a lower indi↵erence curve =) welfare decreases To measure the di↵erence: just calculate U (new) U (old) Two problems: we don't know sictitous are 1. We don’t actually know utility functions. (they Consumer welfare In some sense, this is simple: a price increase puts a consumer on a lower indi↵erence curve =) welfare decreases To measure the di↵erence: just calculate U (new) U (old) Two problems: 1. We don’t actually know utility functions. utils 2. Say U (new) U (old) = 1000 900 = 100. What does this 100 mean? z Consumer welfare In some sense, this is simple: a price increase puts a consumer on a lower indi↵erence curve =) welfare decreases To measure the di↵erence: just calculate U (new) U (old) Two problems: 1. We don’t actually know utility functions. 2. Say U (new) U (old) = 1000 900 = 100. What does this 100 mean? numbers utils fictional = Answer: Nothing Consumer welfare How can we actually make meaningful comparisons? Consumer welfare How can we actually make meaningful comparisons? One way: measure consumer welfare in willingness to pay (i.e., in $). basically consumer surplus Consumer welfare How can we actually make meaningful comparisons? One way: measure consumer welfare in willingness to pay (i.e., in $). We can understand what it means if you’re willing to give up $1,000 for a bundle that I’m only willing to give up $900. In this sense, it makes sense that you value it more than I do. Consumer welfare Our first measure of welfare is consumer surplus: the benefit from consuming the good beyond what it cost. This should be familiar from Econ 2010, but let’s review. Consumer welfare Our first measure of welfare is consumer surplus: the benefit from consuming the good beyond what it cost. This should be familiar from Econ 2010, but let’s review. Say you are willing to pay up to $10 for a pizza. If you buy a pizza at Domino’s for $10, what is the change in your consumer surplus if you buy the pizza? Domino sells it for $10 Consumer welfare Our first measure of welfare is consumer surplus: the benefit from consuming the good beyond what it cost. This should be familiar from Econ 2010, but let’s review. Say you are willing to pay up to $10 for a pizza. If you buy a pizza at Domino’s for $10, what is the change in your consumer surplus if you buy the pizza? CS = 0. You just as well o↵ buying the pizza vs. having $10. Consumer welfare Our first measure of welfare is consumer surplus: the benefit from consuming the good beyond what it cost. This should be familiar from Econ 2010, but let’s review. Say you are willing to pay up to $10 for a pizza. If you buy a pizza at Domino’s for $10, what is the change in your consumer surplus if you buy the pizza? CS = 0. You just as well o↵ buying the pizza vs. having $10. What if instead Domino’s charged $8? CS = $2 Consumer welfare Our first measure of welfare is consumer surplus: the benefit from consuming the good beyond what it cost. This should be familiar from Econ 2010, but let’s review. Say you are willing to pay up to $10 for a pizza. If you buy a pizza at Domino’s for $10, what is the change in your consumer surplus if you buy the pizza? CS = 0. You just as well o↵ buying the pizza vs. having $10. What if instead Domino’s charged $8? CS = $2. You are $2 better o↵ having bought the pizza. Willingness to pay (WTP) To measure WTP, we use the inverse demand function: p = p(Q). Take (Q = D(p)) and invert to get p(Q). in demand price get put , P t the opposite go - way d Willingness to pay (WTP) To measure WTP, we use the inverse demand function: p = p(Q). Take (Q = D(p)) and invert to get p(Q). p(q) = a - Q Linear demand example: invert Q = a bp to get p = p(Q) = a/b Q/b instead right side of Q p pur on Willingness to pay (WTP) To measure WTP, we use the inverse demand function: p = p(Q). Take (Q = D(p)) and invert to get p(Q). Linear demand example: invert Q = a bp to get p = p(Q) = a/b Q/b Say p(1) = 5 and p(2) = 4. You are willing to pay $5 for the first pizza, $4 for the second pizza, etc. ↳ Your marginal willingness to pay for the second unit is $4. Willingness to pay (WTP) gettingo, price p $5 is 3 $4 3 demand or , $3 price $2 D $1 q1 1 2 3 4 5 Start with this demand curve: marginal value of first unit is $5, second unit is $4, and so on. Willingness to pay (WTP) p $5 $4 $3 $2 D $1 q1 1 2 3 4 5 So at price $5, you buy one unit. At price $4, you buy two units. And so on. Willingness to pay (WTP) p $5 $4 $3 $2 D $1 q1 1 2 3 4 5 Consumer surplus, CS=maximum amount you’re willing to pay - what you pay for it. Willingness to pay(WTP) p $5 S $4 a $3 3 $2 2 D $1 I q1 1 2 3 4 5 The most you’re willing to pay for one unit is the shaded area ($(5 ⇥ 1)). Willingness to pay (WTP) p $5 S $4 u a $3 38 $2 z T D $1 I C q1 1 2 3 4 5 The most you’re willing to pay for two units is the shaded area ($(5 ⇥ 1) + (4 ⇥ 1)). = 9 Willingness to pay (WTP) p $5 S $4 Y 9 $3 3 8 12 $2 27 Ih D $1 16 10 q1 1 2 3 4 5 The most you’re willing to pay for three units is the shaded area ($(5 ⇥ 1) + (4 ⇥ 1) + (3 ⇥ 1)). = 12 Calculating consumer surplus Suppose that the price is $3. Calculating consumer surplus Suppose that the price is $3. You buy three pizzas. This costs you 3 ⇥ $3 = $9. Calculating consumer surplus Suppose that the price is $3. You buy three pizzas. This costs you 3 ⇥ $3 = $9. Why 3 pizzas? Calculating consumer surplus Suppose that the price is $3. You buy three pizzas. This costs you 3 ⇥ $3 = $9. Why 3 pizzas? Because your marginal WTP for pizza #4 is only $2. Calculating consumer surplus is area shaded p $5 or spend what y $4 x quantity price $3 p = $3 ↓ $9 $2 D $1 q1 1 2 3 4 5 The area of the green shaded area is what you pay for three units if the price is $3. Calculating consumer surplus p $5 $4 $3 p = $3 $2 D $1 q1 1 2 3 4 5 Consumer surplus (red area) = maximum that you’re willing to pay (the previous blue area) - what you paid (the green area) Consumer surplus of a smooth demand curve p -Spurple rea CS p1 E q1 CS is the area under the demand curve, above the price. Consumer surplus of a smooth demand curve p CS p1 E q1 CS is the area under the demand curve, above the price. The area under the demand curve is still the willingness to pay and you get CS by subtracting expenditure (E). E↵ect of a price change on consumer surplus When the price of a good goes up, the consumer is worse o↵. A loss in welfare : do you're morefor units still paying M purchasing · B : loss from buying fewer units units buying Still have a 1-2 still , in CS woss 7 10 E↵ect of a price change on consumer surplus When the price of a good goes up, the consumer is worse o↵. But how much? E↵ect of a price change on consumer surplus When the price of a good goes up, the consumer is worse o↵. But how much? Change in CS is a way to quantify this. E↵ect of a price change on consumer surplus p p01 A B p1 q1 q10 q1 The change in CS is the sum the areas of regions A and B. E↵ect of a price change on consumer surplus p p01 A B p1 q1 q10 q1 The change in CS is the sum the areas of regions A and B. A is the loss of welfare from paying more for the units you’re still buying. E↵ect of a price change on consumer surplus p p01 not dryingunitse A B any p1 q1 q10 q1 The change in CS is the sum the areas of regions A and B. A is the loss of welfare from paying more for the units you’re still buying. B is the loss from buying fewer units. Textbook exercise 1.1 Inverse demand: p = 60 q. What is the consumer surplus if the price is 30? = 30 find CS when p Textbook exercise 1.1 Inverse demand: p = 60 q. What is the consumer surplus if the price is 30? How much is demanded at the price 30? Solve 30 = 60 q for q we find that q = 30. 30 = 60-a - 60 - 60 - 50- Fa-E 30 9 = Textbook exercise 1.1 Inverse demand: p = 60 q. What is the consumer surplus if the price is 30? How much is demanded at the price 30? Solve 30 = 60 q for q we find that q = 30. Let’s draw a picture so that we can figure out what CS is. Textbook exercise 1.1 p1 60 = 900 30 30. 30 - CS=$450 = P 30 Expenditure =$900 D q1 30 60 a 30 = CS is the area of the pink triangle: base = 30, height = 30. area of a triange = 5. % h - CS=Area= base⇥height 2 = 30⇥30 2 = 450. Change in CS from a tax Say that the government imposed a tax of $10 per unit. What is the change in consumer surplus due to the tax? Recall that our inverse demand function was p = 60 q. 40 = 60-9 - 60 - 60 - 20 = -9 - - - - 4 20 G = Change in CS from a tax Say that the government imposed a tax of $10 per unit. What is the change in consumer surplus due to the tax? Recall that our inverse demand function was p = 60 q. Under the tax, the new price is p = 30 + 10 = 40. To find the new demand, set: = 40 p = 20 a Change in CS from a tax Say that the government imposed a tax of $10 per unit. What is the change in consumer surplus due to the tax? Recall that our inverse demand function was p = 60 q. Under the tax, the new price is p = 30 + 10 = 40. To find the new demand, set: 40 = 60 q =) q = 20 Change in CS from a tax p1 60 CS CONSUME 40 = 40 p Expenditure D 20 60 q1 a= 20 1 CS = ⇥ 20 ⇥ 20 = 200 2 400 = 200 to Change in CS from a tax p1 60 CS 40 Expenditure D 20 60 q1 1 CS = ⇥ 20 ⇥ 20 = 200 2 The change in CS from the tax is 200 450 = 250. The consumer is $250 worse o↵ due to the tax. Two reasons: (1) purchases less and (2) pays more for what she does purchase CS and Demand Elasticity Two linear demand curves go through an initial equilibrium e1. One demand curve is less elastic than the other at e1. For which demand curve will a price increase cause the larger consumer surplus loss? CS and Demand Elasticity Two linear demand curves go through an initial equilibrium e1. One demand curve is less elastic than the other at e1. For which demand curve will a price increase cause the larger consumer surplus loss? Recall that dQ p "Q,p = ⇥ dp q CS and Demand Elasticity Two linear demand curves go through an initial equilibrium e1. One demand curve is less elastic than the other at e1. For which demand curve will a price increase cause the larger consumer surplus loss? Recall that dQ p "Q,p = ⇥ dp q Since they both pass through the same point e1 , the more elastic demand curve is flatter than the less elastic one. dQ That’s because dp is larger (in absolute value) for the more elastic one. CS and Demand Elasticity p1 flatter-mareeasta D1 —less elastic at e1 e1 p D2 —more elastic at e1 q1 q CS and Demand Elasticity p1 D1 —less elastic at e1 elastic p0 = more ↑ e1 D p & - D2 —more elastic at e1 q2 q1 q q1 If price increase to p0 , quantities change to q 1 and q 2. CS and Demand Elasticity p1 D1 —less elastic at e1 p0 DI e1 p D2 —more elastic at e1 q2 q1 q q1 The green shaded area is the loss of CS for D1 CS and Demand Elasticity p1 D1 —less elastic at e1 p0 D2 e1 p D2 —more elastic at e1 q2 q1 q q1 The yellow shaded area is the loss of CS for D2. CS and Demand Elasticity In general, the less elastic (smaller |"Q,p |) a demand curve is, the greater the loss in CS. Why? extra loss undera the more less the loss in CS because greater 're buying more they  big point CS and Demand Elasticity [ J In general, the less elastic (smaller |"Q,p |) a demand curve is, the greater the loss in CS. Why? Less elastic =) demand falls less (for a given price change) CS and Demand Elasticity In general, the less elastic (smaller |"Q,p |) a demand curve is, the greater the loss in CS. Why? Less elastic =) demand falls less (for a given price change) =) consumers purchase more units at the higher price p0. CS and Demand Elasticity In general, the less elastic (smaller |"Q,p |) a demand curve is, the greater the loss in CS. Why? Less elastic =) demand falls less (for a given price change) =) consumers purchase more units at the higher price p0. They lose extra surplus on these additional purchased units Problems with consumer surplus CS is OK as a welfare measure, but has some problems Main problem: considers only demand function for one good at at time - ignores that consumer may substitute to another good Problems with consumer surplus CS is OK as a welfare measure, but has some problems Main problem: considers only demand function for one good at at time - ignores that consumer may substitute to another good Example: price of pizza rises, and you just substitute to beer. Aren’t necessarily worse o↵ in utility terms. Problems with consumer surplus CS is OK as a welfare measure, but has some problems Main problem: considers only demand function for one good at at time - ignores that consumer may substitute to another good Example: price of pizza rises, and you just substitute to beer. Aren’t necessarily worse o↵ in utility terms. In other words: “WTP for 2nd pizza” is not really the same as “loss in utility when you consume 1 pizza instead of 2” Compensating a consumer for the change in price Closely related to “income e↵ects” and “substitution e↵ects” Compensating a consumer for the change in price Closely related to “income e↵ects” and “substitution e↵ects” When price rises, substitution e↵ect does not make you worse o↵ - you just substitute Compensating a consumer for the change in price Closely related to “income e↵ects” and “substitution e↵ects” When price rises, substitution e↵ect does not make you worse o↵ - you just substitute To get a more accurate measure, need to “remove” the substitution e↵ect from our analysis Compensating a consumer for the change in price Closely related to “income e↵ects” and “substitution e↵ects” When price rises, substitution e↵ect does not make you worse o↵ - you just substitute To get a more accurate measure, need to “remove” the substitution e↵ect from our analysis What we really want to know is: “How much money would I need to give you to make you as happy as you were before the price increase?” Compensating a consumer for the change in price Closely related to “income e↵ects” and “substitution e↵ects” When price rises, substitution e↵ect does not make you worse o↵ - you just substitute To get a more accurate measure, need to “remove” the substitution e↵ect from our analysis What we really want to know is: not tp thesam A “How much money would I need to give you to make you as happy as you were before the price increase?” Puts a “dollar value” on utility changes - easier to interpret How much to compensate? Consider your grandma, who gets all of her income from Social Security She cares about two things: yarn for knitting (q1 ) and seeds for gardening (q2 ) Start at p1 = p2 = 1. How much to compensate? Originally, she purchases bundle a: q2 I1 a L1 q1 How much to compensate? Say the price of yarn rises to p1 = 1.5 (50% yarn inflation). Budget set shrinks, and grandma is (obviously) worse o↵ q2 I1 I2 from set gues budget 12 v, & a b & L2 L1 q1 The Social Security administration wants to raise grandma’s check because of inflation - question is “How much?” How much to compensate? Proposal 1: Restore her old purchasing power - raise income so she can continue to a↵ord a (budget set shifts out). This is called Slutsky compensation. q2 I1 I2 shiftdosame parallelbuch get I a b L2 L1 q1 What will Grandma do? How much to compensate? Proposal 1: Restore her old purchasing power - raise income so she can continue to a↵ord a (budget set shifts out) q2 I1 so too nice I2 I3 grandman = O c Slutsky for $ b a more ow L2 L1 q1 What will Grandma do? Chooses c. She’s better o↵ - we gave her too much! How much to compensate? Raising income by same amount as inflation makes grandma strictly better o↵ (because she substitutes) Overestimates necessary income adjustments (assuming government’s goal is to keep people at same utility in the cheapest way possible) How much to compensate? Raising income by same amount as inflation makes grandma strictly better o↵ (because she substitutes) Overestimates necessary income adjustments (assuming government’s goal is to keep people at same utility in the cheapest way possible) Proposal 2: Raise income so grandma can get back to old utility (or indi↵erence curve). This is called the Hicks proportical compensation. so How much to compensate? Give her just enough money to get back to old utility (IC I1 ) q2 I1 I2 to In back get norande to enough c soundle a a b L2 L1 q1 This will be less than needed to get back to bundle a Recap: Slutsky vs. Hicks Compensation When prices rise (due to inflation, taxes, or other things), a consumer will be worse o↵ (on a lower IC) There are two ways to “compensate” people for price rises: 1. Slutsky compensation: return to original bundle I Pro: easy to calculate I Con: su↵ers from “subsitution bias” - people will be strictly better o↵ because once compensated, may substitute to chepaer goods Recap: Slutsky vs. Hicks Compensation RECAP When prices rise (due to inflation, taxes, or other things), a consumer will be worse o↵ (on a lower IC) There are two ways to “compensate” people for price rises: 1. Slutsky compensation: return to original bundle I Pro: easy to calculate too good for gua I Con: su↵ers from “subsitution bias” - people will be strictly better o↵ because once compensated, may substitute to chepaer goods more expensive for gou 2. Hicks compensation: return to original IC/utility level I Pro: corrects for substitution bias by restoring a consumer to her original indi↵erence curve (not bundle) I Con: harder to calculate (requires us to know consumer’s ICs/utility function) less expensive for you Policy implications U.S. measures inflation using the Consumer Price Index (CPI). Calculate cost of fixed basket of goods in base year, C2000 , and current year 20XX, C20XX , and calculate: C20XX CP I20XX = C2000 Used to adjust social security checks, tax brackets, etc. - a↵ects a item trilions of dollars in spending somerando +dundle random Some in 2000 Policy implications U.S. measures inflation using the Consumer Price Index (CPI). Calculate cost of fixed basket of goods in base year, C2000 , and current year 20XX, C20XX , and calculate: C20XX CP I20XX = C2000 Used to adjust social security checks, tax brackets, etc. - a↵ects trilions of dollars in spending from offer This is basically Slutsky compensation: substitution bias. To an economist, measured inflation is higher than real inflation (estimates of bias ⇡ 1 percentage point) " "It's 5 % 4 measured > true actually 1% its Interpreting compensation as a welfare measure Can use this thought experiment to measure the welfare e↵ects of a price change zo duy yarr The larger the compensation required, the more Grandma was hurt by the price change (larger decrease in her welfare) Interpreting compensation as a welfare measure Can use this thought experiment to measure the welfare e↵ects of a price change The larger the compensation required, the more Grandma was hurt by the price change (larger decrease in her welfare) U (prices don’t change) = U (prices change but you get $x dollars) * before price change =- if = 0 then X , higher would be to get back trying equation to make hold Interpreting compensation as a welfare measure Can use this thought experiment to measure the welfare e↵ects of a price change The larger the compensation required, the more Grandma was hurt by the price change (larger decrease in her welfare) U (prices don’t change) = U (prices change but you get $x dollars) x that satisfies this equation is called the compensating variation (CV) Interpreting compensation as a welfare measure Can use this thought experiment to measure the welfare e↵ects of a price change The larger the compensation required, the more Grandma was hurt by the price change (larger decrease in her welfare) U (prices don’t change) = U (prices change but you get $x dollars) x that satisfies this equation is called the compensating variation (CV) For consistency, usually reported as negative for a price increase (welfare decreases), and positive for price decrease (welfare increases) Compensating variation If original income was Y , the necessary new income is Y + x. Since p2 = 1, this is the vertical distance between the intercepts. a q2 I1 u amounthepush I2 Y +x CV Y c a b L2 L1 q1 Grandma’s welfare decreased by CV = $x dollars Example: Calculating CV Consider perfect substitutes preferences. U (q1 , q2 ) = 2q1 + 2q2 Let p1 = 1, p2 = 2, Y = 200. What are the welfare costs of a price rise of p1 from 1 ! 4? Answer by calculating CV. First step: Finding the old and new utilities What is initial demand? First step: Finding the old and new utilities What is initial demand? Y q1 = = 200 q2 = 0 p1 First step: Finding the old and new utilities What is initial demand? Y q1 = = 200 q2 = 0 p1 old Initial utility: U = 2 ⇥ 200 + 2 ⇥ 0 = 400 First step: Finding the old and new utilities What is initial demand? Y q1 = = 200 q2 = 0 p1 old Initial utility: U = 2 ⇥ 200 + 2 ⇥ 0 = 400 What is final demand (after p1 rises to 4)? First step: Finding the old and new utilities What is initial demand? Y q1 = = 200 q2 = 0 p1 old Initial utility: U = 2 ⇥ 200 + 2 ⇥ 0 = 400 What is final demand (after p1 rises to 4)? Y q1 = 0 q2 = = 100 p2 new Final utility: U = 2 ⇥ 0 + 2 ⇥ 100 = 200 First step: Finding the old and new utilities What is initial demand? Y q1 = = 200 q2 = 0 p1 old Initial utility: U = 2 ⇥ 200 + 2 ⇥ 0 = 400 What is final demand (after p1 rises to 4)? Y q1 = 0 q2 = = 100 p2 new Final utility: U = 2 ⇥ 0 + 2 ⇥ 100 = 200 Prices increase ! utility decreases. We want to calculate CV from this price change. Example: Calculating CV Y Example: Calculating CV Say I compensate you by giving you $x at the new prices (so total income is $Y + x dollars). Your demand is: & the pack is what will get y that so old utility Example: Calculating CV Say I compensate you by giving you $x at the new prices (so total income is $Y + x dollars). Your demand is: Y +x 200 + x q1 = 0 q2 = = = 100 + x/2 2 2 Example: Calculating CV Say I compensate you by giving you $x at the new prices (so total income is $Y + x dollars). Your demand is: Y +x 200 + x q1 = 0 q2 = = = 100 + x/2 2 2 Utility is: 2 ⇥ 0 + 2 ⇥ (100 + x/2) = 200 + x Example: Calculating CV Say I compensate you by giving you $x at the new prices (so total income is $Y + x dollars). Your demand is: Y +x 200 + x q1 = 0 q2 = = = 100 + x/2 2 2 Utility is: 2 ⇥ 0 + 2 ⇥ (100 + x/2) = 200 + x old We want the x such that this is equal to U = 400: Example: Calculating CV Say I compensate you by giving you $x at the new prices (so total income is $Y + x dollars). Your demand is: Y +x 200 + x q1 = 0 q2 = = = 100 + x/2 2 2 Utility is: 2 ⇥ 0 + 2 ⇥ (100 + x/2) = 200 + x old We want the x such that this is equal to U = 400: 200 + x = 400 =) x = 200 The consumer needs $200 as compensation for the price change. This is the magnitude. Because price increased, the sign is negative: CV = 200 (she is $200 worse o↵). Discussion To repeat: CV = 200 means consumer indi↵erent between 1. Prices (1, 2) and original income $200 2. Prices (1, 4) and income $400(= 200 + |CV |) Discussion To repeat: CV = 200 means consumer indi↵erent between 1. Prices (1, 2) and original income $200 2. Prices (1, 4) and income $400(= 200 + |CV |) Say we had a second consumer (with di↵erent prefences) who needed an extra $500 to be indi↵erent. Since this consumer needs more money to compensate her for the price change, she is more worse o↵ than the first consumer: CV = 500 < 200. Discussion To repeat: CV = 200 means consumer indi↵erent between 1. Prices (1, 2) and original income $200 2. Prices (1, 4) and income $400(= 200 + |CV |) Say we had a second consumer (with di↵erent prefences) who needed an extra $500 to be indi↵erent. Since this consumer needs more money to compensate her for the price change, she is more worse o↵ than the first consumer: CV = 500 < 200. For a price decrease: conceptually the same, but signs reversed. Discussion hurder to measure RECAP To repeat: CV = 200 means consumer indi↵erent between 1. Prices (1, 2) and original income $200 2. Prices (1, 4) and income $400(= 200 + |CV |) Say we had a second consumer (with di↵erent prefences) who needed an extra $500 to be indi↵erent. Since this consumer needs more money to compensate her for the price change, she is more worse o↵ than the first consumer: CV = 500 < 200. For a price decrease: conceptually the same, but signs reversed. CV gives more accurate framework for thinking about welfare, but harder to actually measure Tradeo↵ between a theoretically “justified” concept (CV) and an easily implementable concept in practice (CS); fortunately, CS is often a good approximation of CV E↵ects of government policies on consumer welfare The compensating variation method can be used to measure the welfare cost of many other policy changes as well. We’ll do an example with a quota to illustrate. Quotas q2 L1 q1 You start with this budget set (both prices are $1). Quotas q2 L1 q1 q Suppose you have a quota of q. Quotas q2 q1 q Now, you can only select from this set. Quotas q2 a I1 q1 q You pick your favorite bundle. Quotas q2 a b I2 I1 q1 q You’re worse o↵ than without the quota. Quotas q2 a b I2 I1 q1 q You’re worse o↵ than without the quota. Just as for price changes, we can ask: what is the compensating variation to imposing the quota? Quotas q2 CV c a b I2 I1 q1 q CV defined by: U (no quota) = U (imposing the quota but giving you CV dollars) Quotas Note: Quotas do not necessarily make you worse o↵. When will it not? Quotas Note: Quotas do not necessarily make you worse o↵. When will it not? When you would have consumed less than the quota anyway q2 a q1 q Numerical example Jon spends all of his income on two things: data for his cellphone (d) and gas for his car (g). His utility function is p U (d, g) = d + 10 g. Gas costs $5 per gallon, and data costs $2 per gigabyte. His total monthly income is $120. (i) What is his optimal consumption bundle? (ii) Say Jon is on a family plan, and his parents decide to impose a hard cap on his data usage at 5 GB per month. What is the welfare cost of this policy to Jon, as measured by the compensating variation? Solution p U (d, g) = d + 10 g pd Part (i): Begin by setting M RS = pg p g 2 = =) g ⇤ = 4 5 5 Solution p U (d, g) = d + 10 g pd Part (i): Begin by setting M RS = pg p g 2 = =) g ⇤ = 4 5 5 Plug into the budget constraint: 2 ⇥ d + 5 ⇥ 4 = 120 =) d⇤ = 50 Jon’s initial consumption bundle is (50, 4). Notice that this is above the data cap of 5 GB per month, so this cap is binding. Solution Part (ii): What is the welfare cost of a hard cap (quota) of 5 GB per month, as measured by the compensating variation? Strategy is the same as for a price increase: find the x such that U (old (no quota) regime) = U (new (5GB quota) regime +x) Solution Part (ii): What is the welfare cost of a hard cap (quota) of 5 GB per month, as measured by the compensating variation? Strategy is the same as for a price increase: find the x such that U (old (no quota) regime) = U (new (5GB quota) regime +x) Note that if original d⇤  5, then the quota is non-binding, and x = 0; there is zero welfare cost. Solution Part (ii): What is the welfare cost of a hard cap (quota) of 5 GB per month, as measured by the compensating variation? Strategy is the same as for a price increase: find the x such that U (old (no quota) regime) = U (new (5GB quota) regime +x) Note that if original d⇤  5, then the quota is non-binding, and x = 0; there is zero welfare cost. In our case, d⇤ = 50 > 5, so there is a welfare cost. Solution Part (ii): What is the welfare cost of a hard cap (quota) of 5 GB per month, as measured by the compensating variation? Strategy is the same as for a price increase: find the x such that U (old (no quota) regime) = U (new (5GB quota) regime +x) Note that if original d⇤  5, then the quota is non-binding, and x = 0; there is zero welfare cost. In our case, d⇤ = 50 > 5, so there is a welfare cost. We can measure it by: (1) finding his original utility, U old Solution Part (ii): What is the welfare cost of a hard cap (quota) of 5 GB per month, as measured by the compensating variation? Strategy is the same as for a price increase: find the x such that U (old (no quota) regime) = U (new (5GB quota) regime +x) Note that if original d⇤  5, then the quota is non-binding, and x = 0; there is zero welfare cost. In our case, d⇤ = 50 > 5, so there is a welfare cost. We can measure it by: (1) finding his original utility, U old (2) calculating how much money ($x) he needs to get back to U old if he can only choose d⇤ = 5 (which he will do by purchasing additional g) Solution p Part (ii): Initial utility is U old = 50 + 10 4 = 70 After the new data plan, Jon consumes d⇤ = 5 GB of data. Solution p Part (ii): Initial utility is U old = 50 + 10 4 = 70 After the new data plan, Jon consumes d⇤ = 5 GB of data. Find how much g he needs to get back to (old) utility of 70: p 5 + 10 g = 70 =) g ⇤ = (65/10)2 = 42.25 Solution p Part (ii): Initial utility is U old = 50 + 10 4 = 70 After the new data plan, Jon consumes d⇤ = 5 GB of data. Find how much g he needs to get back to (old) utility of 70: p 5 + 10 g = 70 =) g ⇤ = (65/10)2 = 42.25 To get U old = 70 under the quota, he consumes the bundle (5, 42.25). The total income required is 2 ⇥ 5 + 5 ⇥ 42.25 = 221.25 Solution p Part (ii): Initial utility is U old = 50 + 10 4 = 70 After the new data plan, Jon consumes d⇤ = 5 GB of data. Find how much g he needs to get back to (old) utility of 70: p 5 + 10 g = 70 =) g ⇤ = (65/10)2 = 42.25 To get U old = 70 under the quota, he consumes the bundle (5, 42.25). The total income required is 2 ⇥ 5 + 5 ⇥ 42.25 = 221.25 Jon starts with $120, so with the quota in place, he would need an extra $101.25 as compensation to get back to his old utility. Thus, the welfare cost of the data cap (quota) is CV = $101.25 Subsidized goods Quotas (weakly) lower utility, so have a (negative) welfare cost Subsidized goods Quotas (weakly) lower utility, so have a (negative) welfare cost Subsidies (weakly) raise utility, so will have a (postive) welfare benefit Subsidized goods Quotas (weakly) lower utility, so have a (negative) welfare cost Subsidies (weakly) raise utility, so will have a (postive) welfare benefit We can also use CV to measure the welfare gain of policies like subsidies, and compare them to other policies Subsidized goods Quotas (weakly) lower utility, so have a (negative) welfare cost Subsidies (weakly) raise utility, so will have a (postive) welfare benefit We can also use CV to measure the welfare gain of policies like subsidies, and compare them to other policies With a subsidy, the consumer is given a certain amount of the good “for free”, and only pays for any of the good that he buys above that amount. Subsidized goods Quotas (weakly) lower utility, so have a (negative) welfare cost Subsidies (weakly) raise utility, so will have a (postive) welfare benefit We can also use CV to measure the welfare gain of policies like subsidies, and compare them to other policies With a subsidy, the consumer is given a certain amount of the good “for free”, and only pays for any of the good that he buys above that amount. Examples: daycare in Canada, food stamps in the US. Subsidized goods q2 Y q1 Without a subsidy this is your budget line (both prices are $1). Subsidized goods q2 Y q1 q = 100 Suppose the first q units are free. Subsidized goods q2 Y q1 q = 100 Your now pick from this set. Subsidized goods q2 Y a I1 q1 q = 100 You might choose bundle a. Subsidized goods q2 Y a I1 b I2 q1 q = 100 Without the subsidy, you would have chosen b. So, the subsidy makes you better o↵. Subsidized goods We can use CV to quantify how much better o↵ the subsidy makes you. In this case, we want to find the x such that: Subsidized goods We can use CV to quantify how much better o↵ the subsidy makes you. In this case, we want to find the x such that: U (no subsidy) = U (subsidy $x) We define CV = +x. Subsidized goods We can use CV to quantify how much better o↵ the subsidy makes you. In this case, we want to find the x such that: U (no subsidy) = U (subsidy $x) We define CV = +x. Note that in this case, $x is negative compensation Subsidized goods We can use CV to quantify how much better o↵ the subsidy makes you. In this case, we want to find the x such that: U (no subsidy) = U (subsidy $x) We define CV = +x. Note that in this case, $x is negative compensation This is because the policy under consideration (a subsidy) is a utility gain, so to bring you back to your original utility, we have to take money away We don’t do this transfer - its just a thought experiment to find x. We then use +$x as a measure of the welfare gain Subsidized goods We can use CV to quantify how much better o↵ the subsidy makes you. In this case, we want to find the x such that: U (no subsidy) = U (subsidy $x) We define CV = +x. Note that in this case, $x is negative compensation This is because the policy under consideration (a subsidy) is a utility gain, so to bring you back to your original utility, we have to take money away We don’t do this transfer - its just a thought experiment to find x. We then use +$x as a measure of the welfare gain That is, you are indi↵erent between (1) “no subsidy” and (2) “subsidy $x” So, the subsidy made you +$x better o↵ Subsidized goods q2 Y a CV I1 b d I2 q1 q = 100 Subsidized goods The subsidy (100 free units of good 1) makes you better o↵. But what if we gave the consumer $100 in cash instead? q2 Y + 100 c Y I3 a I1 b I2 q1 q = 100 The consumer is better o↵ with $100 in cash to 100 free units of good 1. Q: What is the CV of $100 cash? How will it compare to the CV of the subsidy? A word of caution The welfare exercises we have been studying are a way to quantify losses from certain policies in a systematic way However, the exercises quantifying these losses should not be interpreted as saying that there should never be such policies The framework we have developed thus far looks at the impact on the consumption of a single consumer We have ignored potential benefits that may accrue outside of our model (e.g., war time rationing, limits on carbon emissions) Ultimately, when making decisions, both the costs and benefits must be weighed. The framework we have developed is a rigorous way to begin to conduct such analysis