INCOME AND SUBSTITUTION EFFECTS. Two Demand Functions CHANGES IN INCOME. [See Chapter 5 and 6]

Transcription

1 INCOME AND SUBSTITUTION EFFECTS [See Chater 5 and 6] Two Deand Functions Marshallian deand i ( n describes how consution varies with rices and incoe. Obtained by aiizing utility subject to the budget constraint. Hicksian deand h i ( n u describes how consution varies with rices and utility. Obtained by iniizing eenditure subject to the utility constraint. CHANGES IN INCOME 3

2 Changes in Incoe An increase in incoe shifts the budget constraint out in a arallel fashion Since / does not change the otial MRS will stay constant as the worker oves to higher levels of utility. 4 Increase in Incoe If both and increase as incoe rises and are noral goods Quantity of As incoe rises the individual chooses to consue ore and C B A U 3 U U Quantity of 5 Increase in Incoe If decreases as incoe rises is an inferior good Quantity of C B As incoe rises the individual chooses to consue less and ore U 3 Note that the indifference curves do not have to be oddly shaed. The references are conve A U U Quantity of 6

3 Changes in Incoe The change in consution caused by a change in incoe fro to can be couted using the Marshallian deands: ( ' ( If ( is increasing in i.e. / 0 then good is noral. If ( is decreasing in i.e. / < 0 then good is inferior. 7 Engel Curves The Engel Curve lots deand for i against incoe. 8 OWN PRICE EFFECTS 9 3

4 Changes in a Good s Price A change in the rice of a good alters the sloe of the budget constraint When the rice changes two effects coe into lay substitution effect incoe effect We searate these effects using the Slutsky equation. 0 Quantity of Changes in a Good s Price A Suose the consuer is aiizing utility at oint A. B If falls the consuer will aiize utility at oint B. U U Quantity of Total increase in Deand Curves The Deand Curve lots deand for i against i holding incoe and other rices constant. 4

5 Changes in a Good s Price The total change in caused by a change in its rice fro to ' can be couted using Marshallian deand: ( ' ( 3 Suose falls. Two Effects. Substitution Effect The relative rice of good falls. Fiing utility buy ore (and less.. Incoe Effect Purchasing ower also increases. Agent can achieve higher utility. Will buy ore/less of if noral/inferior. 4 Substitution Effect Quantity of Let s forget that with a fall in rice we can ove to a higher indifference curve. A C The substitution effect is the oveent fro oint A to oint C U The individual substitutes good for good because it is now relatively cheaer Substitution effect Quantity of 5 5

6 Substitution Effect The substitution effect caused by a change in rice fro to ' can be couted using the Hicksian deand function: Sub. Effect h ( ' U h ( U 6 Quantity of Incoe Effect Now let s kee the relative rices constant at the new level. We want to deterine the change in consution due to the shift to a higher curve A C B The incoe effect is the oveent fro oint C to oint B U U If is a noral good the individual will buy ore because real incoe increased Incoe effect Quantity of 7 Incoe Effect The incoe effect caused by a change in rice fro to ' is the difference between the total change and the substitution effect: Incoe Effect [ ( ' ( ] [ h ( ' U h ( U ] 8 6

7 Increase in a Good s Price Quantity of B C An increase in the rice of good eans that the budget constraint gets steeer A The substitution effect is the oveent fro oint A to oint C U U The incoe effect is the oveent fro oint C to oint B Substitution effect Incoe effect Quantity of 9 Hicksian & Marshallian Deand Marshallian deand Fi rices ( and incoe. Induces utility u v( When we vary we can trace out Marshallian deand for good Hicksian deand (or coensated deand Fi rices ( and utility u By construction h ( u ( When we vary we can trace out Hicksian deand for good. 0 Hicksian & Marshallian Deand For a noral good the Hicksian deand curve is less resonsive to rice changes than is the uncoensated deand curve the uncoensated deand curve reflects both incoe and substitution effects the coensated deand curve reflects only substitution effects 7

8 Hicksian & Marshallian Deand At the curves intersect because the individual s incoe is just sufficient to attain the given utility level U h Quantity of Hicksian & Marshallian Deand At rices above incoe coensation is ositive because the individual needs soe hel to reain on U h ' Quantity of 3 Hicksian & Marshallian Deand At rices below incoe coensation is negative to revent an increase in utility fro a lower rice h Quantity of 4 8

9 Noral Goods Picture shows rice rise. SE and IE go in sae direction 5 Inferior Good Picture shows rice rise. SE and IE go in oosite directions. 6 Inferior Good (Giffen Good Picture shows rice rise IE oosite to SE and bigger than SE 7 9

10 SLUTSKY EQUATION 8 Slutsky Equation Suose increase by.. Substitution Effect. Holding utility constant relative rices change. h Increases deand for by. Incoe Effect Agent s incoe falls by. Reduces deand by 9 Slutsky Equation Fi rices ( and incoe. Let u v(. Then h ( u ( ( ( SE always negative since h decreasing in. IE deends on whether noral/inferior. 30 0

11 Eale: u( Fro UMP ( and v( 4 Fro EMP / h ( u u and e( u ( u LHS of Slutsky: ( RHS of Slutsky: / / 3/ / h u CROSS PRICE EFFECTS 3 Changes in a Good s Price The total change in caused by a change in the rice fro to ' can be couted using the Marshallian deand function: ( ' ( 33

12 Substitutes and Coleents Let s start with the two-good case Two goods are substitutes if one good ay relace the other in use eales: tea & coffee butter & argarine Two goods are coleents if they are used together eales: coffee & crea fish & chis 34 Gross Subs/Cos Goods and are gross substitutes if > 0 and > 0 They are gross coleents if < 0 and < 0 35 Gross Coleents Quantity of When the rice of falls the substitution effect ay be so sall that the consuer urchases ore and ore In this case we call and gross coleents U0 U / < 0 Quantity of 36

13 Gross Substitutes Quantity of When the rice of falls the substitution effect ay be so large that the consuer urchases less and ore In this case we call and gross substitutes U / > 0 U0 Quantity of 37 Gross Substitutes: Asyetry Partial derivatives ay have oosite signs: Let foreign flights and doestic flights. An increase in ay increase (sub effect An increase in ay reduce (inc effect Quasilinear Eale: U(y ln + y Fro the UMP deands are / and ( / We therefore have / > 0 and / 0 38 Net Subs/Cos Goods and are net substitutes if h h > 0 and > 0 They are net coleents if h h < 0 and < 0 Partial derivatives cannot have oosite signs Follows fro Sheard s Lea (see EMP notes Two goods are always net substitutes. Moving round indifference curve. 39 3

14 Gross Cos & Net Subs Quantity of In this eale and are gross coleents but net substitute U0 U If the rice of increases and we want to find the iniu eenditure that achieves U we buy less of and hence ore of. Quantity of 40 Substitution and Incoe Effect Suose rises.. Substitution Effect The relative rice of good falls. Fiing utility buy ore (and less. Incoe Effect Purchasing ower decreases. Agent can achieve lower utility. Will buy ore/less of if inferior/noral. 4 Increase in a Good s Price Quantity of B C An increase in the rice of good eans that the budget constraint gets steeer A The substitution effect is the oveent fro oint A to oint C U U The incoe effect is the oveent fro oint C to oint B Substitution effect Incoe effect Quantity of 4 4

15 5 43 Slutsky Equation Suose increase by.. Substitution Effect. Holding utility constant relative rices change. Increases deand for by. Incoe Effect Agent s incoe falls by. Reduces deand by h 44 Slutsky Equation Fi rices ( and incoe. Let u v(. Then SE deends on net coleents or substitutes IE deends on whether is noral/inferior. ( ( ( ( u h 45 Eale: u( Fro UMP Fro EMP LHS of Slutsky: RHS of Slutsky: / / / u h 4 v( and ( / / ( e( and ( u u u u h 0 (

16 DEMAND ELASTICITIES 46 Deand Elasticities So far we have used artial derivatives to deterine how individuals resond to changes in incoe and rices. The size of the derivative deends on how the variables are easured (e.g. currency unit size Makes coarisons across goods eriods and countries very difficult. Elasticities look at ercentage changes. Indeendent of units. 47 Incoe Elasticities The incoe elasticity equals the ercentage change in caused by a % increase in incoe. / d ln / d ln e Noral good: e > 0 Inferior good: e < 0 Luury good: e > Necessary good: e < 48 6

17 7 49 Marshallian Deand Elasticities The own rice elasticity of deand e is If e < - deand is elastic If e > - deand is inelastic If e > 0 deand is Giffen ln ln / / e 50 Marshallian Deand Elasticities The cross-rice elasticity of deand (e is ln ln / / e 5 Elasticities: Interesting Facts If deand is elastic a rice rise leads to an increase in sending: 0 ] [ ] [ < + + e

18 Elasticities: Interesting Facts Deand is hooegenous of degree zero. ( k k k ( Differentiating with resect to k Letting k and dividing by e e + e 0 + A % change in all rices and incoe will not change deand for. 5 Elasticities: Engel Aggregation Take the budget constraint + Differentiating + Divide and ultily by and + se + s e where s / is eenditure share. Food is necessity (incoe elasticity< Hence incoe elasticity for nonfood> 53 Soe Price Elasticities Secific Brands: Coke -.7 Pesi -.08 Tide Detergent

19 Soe Price Elasticities Narrow Categories: Transatlantic Air Travel -.30 Touris in Thailand -.0 Ground Beef -.0 Pork Milk Eggs Soe Price Elasticities Broad Categories: Recreation -.30 Clothing Food Iorts Transortation CONSUMER SURPLUS 57 9

20 Consuer Surlus How do we deterine how our utility changes when there is a change in rices. What affect would a carbon ta have on welfare? Cannot look at utilities directly (ordinal easure Need onetary easure. 58 Consuer Surlus One way to evaluate the welfare cost of a rice increase (fro to would be to coare the eenditures required to achieve a given level of utilities U under these two situations Initial eenditure e( U Eenditure after rice rise e( U 59 Consuer Surlus Clearly if > the eenditure has to increase to aintain the sae level of utility: e( U > e( U The difference between the new and old eenditures is called the coensating variation (CV: where U v(. CV e( U - e( U 60 0

21 Quantity of Consuer Surlus Suose the consuer is aiizing utility at oint A. If the rice of good rises the consuer will aiize utility at oint B. B A U The consuer s utility falls fro U to U U Quantity of 6 Consuer Surlus Quantity of The consuer could be coensated so that he can afford to reain on U B C A CV is the increase in incoe (eenditure that the individual would need to be achieve U. U U Quantity of 6 Consuer Surlus Fro Sheard s Lea: e( U h ( U CV equals the integral of the Hicksian deand CV e( U ' - e( U E(z U dz h ( z U dz This integral is the area to the left of the Hicksian deand curve between and ' 63

22 Consuer Surlus When the rice rises fro to the consuer suffers a loss in welfare P welfare loss h ( U Quantity of 64 Consuer Surlus Consuer surlus equals the area under the Hicksian deand curve above the current rice. CS equals welfare gain fro reducing rice fro to current arket rice. That is CS equals the aount the erson would be willing to ay for the right to consue the good at the current arket rice. 65 A Proble Proble: Hicksian deand deends on the utility level which is not observed. Answer: Aroiate with Marchallisn deand. Fro the Slutsky equation we know the Hicksian and Marshallian deand functions have aroiately the sae sloe when the good fors only a sall art of the consution bundle (i.e. when incoe effects are sall 66

23 Quasilinear Utility Suose u( v( + Fro UMP Marshallian deand for v ( / Fro EMP Hicksian deand for v (h / Hence ( h ( u. And ' ' CV h ( z U dz ( z dz 67 Consuer Surlus We will define consuer surlus as the area below the Marshallian deand curve and above rice It shows what an individual would ay for the right to ake voluntary transactions at this rice Changes in consuer surlus easure the welfare effects of rice changes 68 3