1Separating Income and Substitution EffectsECON 370: Microeconomic TheorySummer 2004 – Rice UniversityStanley GilbertEffects of a Price Decrease• Can be broken down into two components– Income effect• When the price of one goods falls, w/ other constant;• Effectively like increase in consumer’s real income• Since it unambiguously expands the budget set• Income effect on demand is positive, if normal good– Substitution effect• Measures the effect of the change in the price ratio;• Holding some measure of ‘income’ or well being constant• Consumers substitute it for other now relatively more expensive commodities• That is, Substitution effect is always negative– Two decompositions: Hicks, SlutskyEcon 370 - Ordinal Utility 3Hicks and Slutsky Decompositions•Hicks– Substitution Effect: change in demand, holding utility constant– Income Effect: Remaining change in demand, due to mchange• Slutsky– Substitution Effect: change in demand, holding real income constant– Income Effect: Remaining change in demand, due to mchangeEcon 370 - Ordinal Utility 4Hicks Substitution and Income Effects• Due to Sir John Hicks (1904-1989; Nobel 1972)– To get Substitution Effect: Hold utility constant and find bundle that reflects new price ratio– Substitution Effect = change in demand due only to this change in price ratio (movement along IC)– Income Effect = remaining change in demand to get back to new budget constraint (parallel shift)2Econ 370 - Ordinal Utility 5Hicks Decomposition Graphicallyx2x1x2´x1´Given a drop in Price:Econ 370 - Ordinal Utility 6Hicks Decomposition Graphically 2• Insert an “imaginary” budget linetangent to original IC and parallel to new budget linex2x1x2´x1´Given a drop in Price:Substitution EffectIncome EffectEcon 370 - Ordinal Utility 7Slutsky Substitution and Income Effects• Due to Eugene Slutsky (1880-1948)– To get Substitution Effect: Hold purchasing power constant • (that is, adjust income so that the consumer can exactly afford the original bundle) – and find bundle that reflects new price ratio– Substitution Effect = change in demand due only to this change in price ratio (movement along IC)– Income Effect = remaining change in demand to get back to new budget constraint (parallel shift)Econ 370 - Ordinal Utility 8Slutsky Decomposition Graphicallyx2x1x2´x1´Given a drop in Price:3Econ 370 - Ordinal Utility 9Slutsky Decomposition Graphically 2• Insert an “imaginary” budget linethrough the original bundle…x2x1x2´x1´Given a drop in Price:Substitution EffectIncome EffectEcon 370 - Ordinal Utility 10Signs of Substitution and Income Effects• Sign of Substitution Effect is unambiguously negative as long as Indifference Curves are convex• Income effect may be positive or negative– That is, the good may be either normal or inferior• For Normal goods, the income effect reinforces the substitution effect• For Inferior goods, the two effects offset•ForGiffen Goods– Remember, the Income effect is Negative– And the income effect is greater than the substitution effectEcon 370 - Ordinal Utility 11Slutsky’s Effects for Normal Goodsx2x1x2´x1´From Before…Substitution EffectIncome Effect• Since Substitution Effect and Income Effect reinforce each other…• This is a Normal GoodEcon 370 - Ordinal Utility 12Slutsky’s Effects for Inferior Goodsx2x1In this case:x2´x1´Substitution Effect• Since Substitution Effect and Income Effect offset each other…• This is an Inferior GoodIncome Effect4Econ 370 - Ordinal Utility 13Slutsky’s Effects for Giffen Goodsx2x1In this case:x2´x1´Substitution Effect• Since Income Effect completely cancels the Substitution Effect• This is a Giffen GoodIncome EffectEcon 370 - Ordinal Utility 14Mathematics of Slutsky Decomposition• We seek a way to calculate mathematically the Income and Substitution Effects• Assume:– Income: m– Initial prices: p10, p2– Final prices: p11, p2– Note that the price of good two, here, does not change• Given the demand functions, demands can be readily calculated as:– Initial demands: xi0= xi( p10, p2, m)– Final demands: xi1= xi( p11, p2, m)Econ 370 - Ordinal Utility 15Slutsky Mathematics (cont)• We need to calculate an intermediate demand that holds buying power constant•Let msthe income that provides exactly the same buying power as before at the new price– Thus: ms= p11x10+ p2x20• The demand associated with this income is:– xis= xi( p11, p2, ms) = xis( p11, p2, x10, x20)• Finally we have:– Substitution Effect: SE = xis– xi0– Income Effect: IE = xi1– xisEcon 370 - Ordinal Utility 16Hicks’ Mathematics• The only difference is between Hicks’ and Slutsky is in the calculation of the intermediate demand•Let mhthe income that provides exactly the same utility as before at the new price–If u0is initial utility level, then– Thus: mhsolves u0= u( x1(p11, p2, mh), x2(p11, p2,mh))• The demand associated with this income is:– xih= xi( p11, p2, mh) = xih( p11, p2, u0)• Finally we have:– Substitution Effect: SE = xih– xi0– Income Effect: IE = xi1– xih5Econ 370 - Ordinal Utility 17Calculating the Slutsky DecompositionAssume that u = xαy1 – αSo the demand functions are:xpmxα=()ypmyα−= 1Initial Price is px0Final Price is px100xpmxα=ypmyyyα===1011xpmxα=()mppxx⎥⎦⎤⎢⎣⎡−+=αα101()yyxxpmppmpαα−+= 101ypxpmyxs+=01Econ 370 - Ordinal Utility 18Calculating the Slutsky Decomposition 2()⎥⎦⎤⎢⎣⎡−+==αααα10111xxxxsspppmpmx()mppmxxs⎥⎦⎤⎢⎣⎡−+=αα101SinceWe get:()1021xxpmpmααα−+=()101 xxxsαα−+=orFinally, we get:()()()01010011 xxxxxxxSEs−−=−−+=−=ααα()[]()0110111 xxxxxxxIEs−=−+−=−=αααEcon 370 - Ordinal Utility 19Calculating the Hicks DecompositionWe need to calculate mh, soSubstituting our demand functions back into utility we get:()mpppmpmyxuyxyxαααααααααα−−−⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎠⎞⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎠⎞⎜⎝⎛==11111Then mhsolves:mppmppyxhyxαααααααα−−⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛101111ormppmxxhα⎟⎟⎠⎞⎜⎜⎝⎛=01Econ 370 - Ordinal Utility 20Calculating the Hicks Decomposition 2()()αααααα−=⎟⎟⎠⎞⎜⎜⎝⎛==1100111xxxxxxhhppmpppmpmxSinceWe get:Finally, we