Slutsky vs. Hicks

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Economics II - Exercise Session # 3, October 8, 2008 - Suggested Solution Problem 1: Assume a person has a utility function U = XY, and money income of $10,000, facing an initial price of X of $10 and price of Y of $15. If the price of X increases to $15, answer the following questions: (a) What was the initial utility maximizing quantity of X and Y? (b) What is the new utility maximizing quantity of X and Y following the increase in the price of X? (c) What is the Hicks compensating variation i
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  Economics II - Exercise Session # 3, October 8, 2008 - Suggested Solution Problem 1: Assume a person has a utility function U = XY, and money income of $10,000, facing an initial price of X of $10 and price of Y of $15. If the price of Xincreases to $15, answer the following questions:(a) What was the initial utility maximizing quantity of X and Y?(b) What is the new utility maximizing quantity of X and Y following the increase inthe price of X?(c) What is the Hicks compensating variation in income that would leave this personequally well o ff  following the price increase? What is the Slutsky compensatingvariation in income?(d) Calculate the pure substitution e ff  ect and the real income e ff  ect on X of this increasein the price of X. Distinguish between the calculation of these e ff  ects using theHicksian analysis vs. the Slutsky analysis. Solution: In this exercise, we will seek to understand the concept of Hicks’ method of decomposing a demand curve’s slope into its component income and substitution e ff  ects.The Hicksian substitution e ff  ect is the change in quantity demanded that occurs whenthe price of the good changes, but the consumer is given enough additional income sothat her total utility remains the same. The income e ff  ect is the change in the quantitydemanded of a good due to the change in real income caused by the price change. Theincome e ff  ect is measured as the di ff  erence between the quantity demanded after the pricechange and the quantity demanded after the price change when income is adjusted toreturn the consumer to the srcinal level of utility.The Slutsky substitution e ff  ect is the change in quantity demanded that occurs when theprice of the good changes, but the consumer is given enough additional income so that shecan just a ff  ord her initial consumption. The income e ff  ect is the change in the quantitydemanded of a good due to the change in real income caused by the price change. Theincome e ff  ect is measured as the di ff  erence between the quantity demanded after the pricechange and the quantity demanded after the price change when income is adjusted tokeep consumer’s purchasing power the same.1  (a) We use optimality condition and budget constraint: MU  X MU  Y   = P  X P  Y   ⇒ Y X  =1015 ⇒ X  = 1 . 5 Y P  X X  + P  Y   Y  = I  ⇒ 10 X  + 15 Y  = 1000010(1 . 5 Y  ) + 15 Y  = 10000 ⇒ 30 Y  = 10000 ⇒ Y  =10003 . = 333 . 3 X  = 1 . 5 Y  =15003= 500(b) MU  X MU  Y   = P  X P  Y   ⇒ Y X  =1515 ⇒ X  = Y P  X X  + P  Y   Y  = I  ⇒ 15 X  + 15 Y  = 1000015 X  + 15 X  = 10000 ⇒ X  =1000030 . = 333 . 3 X  = Y . = 333 . 3(c) The Hicksian compensating variation in income is that amount of money, holdingthe price of X constant at its higher level of $15, that will allow the person to be aswell o ff  as they were before the price increase. In Hicksian terms of course, beingequally well o ff  means having the same level of utility. Original utility was U  = XY  = (333 . 33)(500) = 166667When the price ratio of X to Y had been 10/15, we observed above that Y = .67 Xand X = 1.5 Y. Now, with both prices equal to $15, the price ratio is 1:1, and Y =X, meaning that the utility maximizing bundle at the new relative price ratio on thesrcinal indi ff  erence curve must have the same quantity of X and Y. Substitutionallows the following result: U  = XY  ; but if  X  = Y,U  = X  2 = 166667 (srcinal utility) ⇒ X  = Y  = 408 . 25Note that on a graph, these would be the quantities at the tangency point of theshifted budget line (with the higher relative price of X) and the srcinal indi ff  erencecurve.Final step is to calculate the amount of money that must be spent to achieve X =Y = 408.25, which is$15(408 . 25) + $15(408 . 25) = $12247 . 502  Original income was $10,000, so the Hicksian compensating variation of income is$12 247.50 - $10 000 = $2 247.50The Slutsky compensating variation is much easier to calculate: At the new pricesthe money income required to consume the srcinal X,Y bundle of X = 500, Y =333.33 is simply: I = $15 (500) + $15 (333.33) = $12 500. This is the moneyincome required to allow a budget line at the new slope (with higher price of X) togo through the srcinal consumption point. Since $12 500 - $10,000 = $2 500, thatis the Slutsky compensating variation.(d) Finally, if no compensating variation is actually paid, the full reduction in theconsumption of X is (500-333.33) = 167.67. How much of this 167.67 reduction isdue to a pure substitution e ff  ect and how much is due to a real income e ff  ect. Wecan rely on the analysis in (c) to derive the results for both the Slutsky and theHicksian analysis.Hicksian analysis: We found above that if the real income e ff  ect is eliminated by”hypothetically” (in this case) giving the person another $2 247.50, the new point onthe srcinal indi ff  erence curve is X = 408.25, Y = 408.25. Therefore, the movementalong that srcinal indi ff  erence curve representing the pure substitution e ff  ect is 500- 408.25 = 91.75. Then, the remaining change in X of 408.25 - 333.33 = 74.92 is thereal income e ff  ect (the result of now taking that $2 247.50 away from the person, sothere is a parallel shift to the left to the lower indi ff  erence curve at X = 333.33 andY = 333.33).Slutsky derivation of substitution and income e ff  ects: We found above that theelimination of the real income e ff  ect as defined by Slutsky would require a ”hypothetical”increase in I of $2 500 to I = $12 500. We also know as stated earlier that with P  X = P  Y   , the first order condition requires that Y = X. Thus, we can calculate thepoint on the higher utility indi ff  erence curve that can be achieved with $12,499.95(and the more steeply sloped budget line incorporating the higher price of X) as$12 500 = $15 X + $15 Y, or since X = Y, $12 500 = 15 X + 15 X, so X = 416.67,and Y = 416.67. Therefore, the pure substitution e ff  ect related to X is 500 - 416.67= 83.33 and the real income e ff  ect is then the ”residual” of 416.67 - 333.33 = 83.34(essentially equal, just a rounding di ff  erence).3  . Problem 2: Suppose that the price elasticity, ￿ , for cigarettes is 4, the price of cigarettesis $3 per pack and we want to reduce smoking by 20%. What should we do? Solution: First, recognize that we need to raise the price. Then, figure out by how much:4 = ￿ = ∆ Q/Q ∆ P/P  =0 . 2 ∆ P/ 3 ⇒ ∆ P  =0 . 2 ∗ 34= 0 . 15In order to decrease smoking by 20% we need to increase price by 15 cents. Problem 3: Consumer consumes two goods with their prices P  X = 10, P  Y   = 80 and hasincome I  = 5000 CZK  . The demand function is given by X  = 80 − 0 . 8 P  2 X − 0 . 5 P  Y   +0 . 04 I  .(a) Are X and Y substitutes or complements?(b) Is X normal or inferior good?(c) What is price elasticity of demand for good X? What information does this give tothe producer of good X?4
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