2.2 BUDGET-CONSTRAINED CHOICE WITH TWO COMMODITIES

Transcription

1 Essential Miroeonomis BUDGET-CONSTRAINED CHOICE WITH TWO COMMODITIES Continuity of demand 2 Inome effets 6 Quasi-linear, Cobb-Douglas and CES referenes 9 Eenditure funtion 4 Substitution effets and Comensated demand 7 Elastiity of substitution 9 Determinants of demand elastiity 25 John Riley Otober 4, 204

2 Essential Miroeonomis -2- Consider the hoie roblem of a onsumer with inome I faing a rie vetor n Ma{ U( ) I, } (22-) We assume that the loal non-satiation assumtion holds and that the utility funtion is ontinuous and stritly quasi-onave on 2 The loal non-satiation assumtion ensures that the onsumer sends all his or her inome and strit quasi-onavity ensures that there is a unique solution 0 (, I) Suose 0 and are both solutions Then that for any onve ombination 0 U( ) U( ) and beause U is stritly quasi-onave it follows 0 Finally, beause, U( ) U( ) I Thus is feasible and stritly referred 0 I and I, then John Riley Otober 4, 204

3 Essential Miroeonomis -3- Given uniqueness and ontinuity of referenes, it is intuitively lausible that the imlied demand funtion (, I) Aendi C) must be ontinuous This follows from the Theorem of the Maimum (See EM Proosition 22-: Theorem of the Maimum (I) Consider the maimization roblem Ma{ f (, ) X ( ), A} where X ( ) {, h (, ) 0, i,, m} n i If f and X ( ) are ontinuous and, for all, there is a unique solution ( ), then ( ) is ontinuous For the most general roositions about onsumers (and firms as well) ontinuity is all that we need However, to simlify modeling, it is onvenient to assume some degree of differentiability as well Then the neessary onditions for a maimum are restritions on the gradient vetor of the maimand and onstraint funtions The maing from a arameter to a set is alled a orresondene For a formal definition of a ontinuous orresondene, see EM Aendi C John Riley Otober 4, 204

4 Essential Miroeonomis -4- Assumtions Differentiability: The utility funtion is ontinuously differentiable on 2 n U Stritly inreasing referenes: For all, ( ) 0 Whenever we wish to avoid orner solutions we also assume that U lim,,, n 0 Forming the Lagrangian for the maimization roblem (22-), L U ( I ) FOC L U ( ) 0,,, n, with equality if 0 Note that beause the marginal utility of at least one ommodity is stritly ositive, the shadow rie (or marginal utility of inome) must be stritly ositive John Riley Otober 4, 204

5 Essential Miroeonomis -5- Heneforth we will fous rimarily on the simlest 2 ommodity ase If both ommodities are onsumed, the FOC an be rewritten as follows U U (22-2) 2 2 One etra dollar sent on ommodity yields sent on ommodity is U additional units thus the marginal value er dollar Sending on ommodity is inreased until this is equal to the marginal value of sending an additional dollar on ommodity 2 We an also rewrite the FOC as follows: MRS( ) U U 2 2 That is, the sloe of the indifferene urve must equal the sloe of the budget line in Figure 22- Figure 22-: Budget onstrained hoie John Riley Otober 4, 204

6 Essential Miroeonomis -6- Inome Effets Figure 22-2 shows the ath of the onsumer s hoie (, I) as his inome inreases As shown, this Inome Eansion Path is initially ositively sloed ie I 2 and are both ositive I In this ase the ommodities are said to be normal in the Figure 22-2: Inome Eansion Path neighborhood of the otimum However, as deited, for higher inomes onsumtion of ommodity delines as inome inreases In this ase ommodity is inferior in the neighborhood of the otimum To failitate omarisons aross ommodities, it is helful to onsider the roortional effets on demand as inome hanges, in other words the inome elastiity of demand (, I) I I John Riley Otober 4, 204

7 Essential Miroeonomis -7- In Figure 22-2 the sloe of the Inome Eansion ath at (, I) is steeer than the line oining and the origin, that is d d I IEP I Rearranging this inequality, I I (, I) (, I) I I Figure 22-2: Inome Eansion Path Aealing to the following lemma, the inome elastiities weighted by their eenditure shares must sum to Thus, in Figure 22-2, the inome elastiity of demand for ommodity 2 eeeds and for ommodity is less than Lemma 22-2: Inome Elastiities Weighted by Eenditure Shares Sum to 2 2 k (, I) k (, I) where k is the eenditure share for ommodity I John Riley Otober 4, 204

8 Essential Miroeonomis -8- Proof: To establish this roosition, we substitute the onsumer s hoie into the budget onstraint and differentiate by I 2 2 I I Rearranging the left-hand side, I 2 2 I 2 I I I 2 I ( ) ( ) k (, I) k (, I) 2 2 QED John Riley Otober 4, 204

9 Essential Miroeonomis -9- We now eamine inome elastiities for three ommonly used utility funtions Eamle : Quasi-Linear Preferenes Definition: Preferenes are quasi-linear if they an be reresented by the utility funtion U( ) v( ) 2 For suh a utility funtion the marginal rate of substitution (MRS) at is U d2 v( ) d U U U ( ) 2 Inome Eansion Path The MRS is indeendent of ommodity 2 whih means that the indifferene urves are vertially arallel As deited in Figure 22-3, it follows that the inome eansion ath is first horizontal and then vertial Figure 2-2-3: Quasi-linear referenes Over the range in whih both ommodities are onsumed, it follows that the inome elastiity of ommodity is zero Given Lemma 22-2, the inome elastiity of ommodity 2 is the inverse of the eenditure share John Riley Otober 4, 204

10 Essential Miroeonomis Eamle 2: Cobb-Douglas Preferenes 2 U( ),, 0 Differentiating by U 2, 2 U U 2U Similarly, 2 2 At the maimum the FOC must be satisfied, hene U U 2 2 Substituting and then dividing by U, 2, U 2 2 hene Sine u( ) ln U( ) ln 2 ln 2, U( ) is stritly quasi-onave and so the FOC are suffiient John Riley Otober 4, 204

11 Essential Miroeonomis -- We then solve for by substituting bak into the budget onstraint 2, hene I 2 22 I Demand for ommodity is therefore (, I) I 2 Finally, substituting bak into the utility funtion, maimized utility is I ( (, )) ( ) ( ) ( ) U I 2 2 John Riley Otober 4, 204

12 Essential Miroeonomis -2- Eamle 3: CES Preferenes U( ) ( ),,, 0, From the definition of U, 2 2 U( ) Differentiating by, U ( ) U ( ) Hene U U We follow the same stes as for Eamle 2 Substituting into the FOC and then dividing by U U, (22-3) Hene ( ) and so John Riley Otober 4, 204

13 Essential Miroeonomis -3- We then solve for by substituting bak into the budget onstraint 2 2 ( 2 I 2 ) I, hene Demand for ommodity is therefore I (, I) ( ) (22-4) Substituting (, I ) into the utility funtion and olleting terms, U( (, I)) 2 2 ( ) I Note that the demand funtions given by (22-4) redue to the Cobb-Douglas demand funtions if Note: For, maimizing U( ) is equivalent to minimizing equivalent to maimizing the onave funtion suffiient 2 2 v( ) whih, in turn, is 2 2 u( ) Thus the neessary onditions are John Riley Otober 4, 204

14 Essential Miroeonomis -4- Dual otimization roblem: eenditure minimization Given the very weak assumtion of loal non-satiation, for any budget onstrained utility maimization roblem there is a dual otimization roblem As we shall see, this dual roblem is very useful in understanding the determinants of demand Suose that is a solution to a onsumer s maimization roblem That is, arg Ma{ U( ) 0, I} Suh a onsumtion bundle is deited in Figure 22-4 Figure 22-4: Eenditure Minimization John Riley Otober 4, 204

15 Essential Miroeonomis -5- Consider any onsumtion bundle ˆ suh that ˆ I If is suffiiently small the neighborhood N ( ˆ, ) lies in the budget set If the loal non-satiation roerty holds, then there eists some neighborhood that is stritly referred to ˆ Then ˆ annot be otimal Hene ˆ in this I U( ) U( ) Equivalently, U( ) U( ) Thus, is eenditure minimizing, among all onsumtion bundles that are referred to summarize this result as follows We Lemma 22-3: Duality Lemma If the loal non-satiation assumtion holds and arg Ma{ U( ) 0, I}, then U( ) U( ) and so arg Min{ 0, U( ) U( )} John Riley Otober 4, 204

16 Essential Miroeonomis -6- For any level of utility, U and rie vetor we define the eenditure funtion M (, U ) to be the minimum eenditure needed to ahieve the utility level U Definition: Eenditure Funtion M (, U) Min{ U( ) U} While it is not diffiult to solve for the eenditure funtion, it is often more onvenient to solve first for the indiret utility funtion V (, I) Ma{ U( ) I} Given loal non-satiation, the onsumer sends his entire inome Figure 22-5: Maimized utility as a funtion of inome Moreover the higher his inome the greater is his utility Thus maimized utility V (, I ) is a stritly inreasing funtion of inome This is deited in Figure 22-5 The eenditure funtion is then the inverse maing from U to I John Riley Otober 4, 204

17 Essential Miroeonomis -7- Comensated demand Let (, U ) be the solution to the dual roblem, that is (, U ) solves M (, U) Min{ U( ) U} This is known as the onsumer s omensated demand Consider the effet on omensated demand of an inrease in the rie of ommodity This is deited in Figure 22-6 for rie vetors 0 and As the rie of ommodity rises, the onsumer is omensated so that he is ust able to maintain the utility level 0 U Figure 22-6: Substitution effet John Riley Otober 4, 204

18 Essential Miroeonomis -8- The following useful roerty of the eenditure funtion is an immediate imliation of the Enveloe Theorem M 0 (, U ) Informally, if the rie of ommodity rises and the onsumer maintains his onsumtion lan, his etra eenditure is hange in the rie is of seond order This is the diret effet The indiret effet assoiated with adusting to the so See Eerise 22-2 Converting the eenditure minimization roblem to a maimization roblem, L ( U( ) U) M L John Riley Otober 4, 204

19 Essential Miroeonomis -9- Substitution Effet The effet on demand of a omensated rie hange is alled the substitution effet As Figure 22-6 illustrates, the size of this effet deends ritially on the urvature of the indifferene urve In the left diagram, as the rie ratio hanges, the onsumtion ratio 2 0 (, U ) 0 (, U ) hanges a lot That is, the substitution effet is large In the right diagram a rie hange has a small effet on the onsumtion ratio so the substitution effet is small As we shall see, the elastiity of the onsumtion ratio with reset to the rie ratio is a very useful measure of rie sensitivity 2 Definition: Elastiity of substitution (, ) Eamle: CES utility funtion 2 2 From equation (22-3), Taking the logarithm, ln( ) ln( ) 2 onstant John Riley Otober 4, 204

20 Essential Miroeonomis -20- d 2 As is readily onfirmed, ( y, ) ln y Hene, (, ) d 2 Thus for the CES utility funtion, the arameter is the elastiity of substitution Quik review of elastiity For any funtion y f ( ) let y be the hange in y when the hange in is A measure of the sensitivity of y with reset to is the erentage hange in y divided by the erentage hange in y y 00 /00 y y An advantage of this measure is that it is indeendent of the units in whih the two variable are measured To see this suose we resale units so that ˆ and ŷ y Then ˆ yˆ y yˆ ˆ y John Riley Otober 4, 204

21 Essential Miroeonomis -2- (Point) Elastiity We define the oint elastiity to the limit as 0 y dy ( y, ) lim y 0 y d Then from the above observation ( y, ) ( y, ) Note that d ln y d dy Therefore the oint elastiity an be rewritten as follows: y d dy d ( y, ) ln y y d d The following lemma is an immediate imliation Lemma 22-4: 2 (, ) ( 2, ) (, ) 2 We aeal to this lemma in roving the following result John Riley Otober 4, 204

22 Essential Miroeonomis -22- Proosition 22-5: Elastiity of Substitution and Comensated Own Prie Elastiity and ( 2, ) where k k (, ) ( k ) Proof: To demonstrate equivalene, first note that around the indifferene urve as rises we have U 2 U 2 0 Also, from the first order ondition, the marginal utility of eah ommodity is roortional to its rie Hene (22-5) Dividing by and rearranging this equation, ( ) k k, where k / John Riley Otober 4, 204

23 Essential Miroeonomis -23- Therefore k (, ) (, ) (22-6) 2 2 k From Lemma 22-4, 2 (, ) (, ) k2 From (22-6), (, ) ( 2, ) k Substituting for the seond term on the right hand side k (, ) (, ) (, ) k k Substituting this eression into (22-6), (, ) k ( k ) 2 QED John Riley Otober 4, 204

24 Essential Miroeonomis -24- Remark: Note that the omensated own rie elastiity, (, ) ( k), is bounded from below by the elastiity of substitution Moreover, if the eenditure share is small, the elastiity of substitution is a good aroimation for the omensated own rie elastiity Deomosition of rie effets To understand the imat of a rie hange it roves helful to deomose this into two arts: a omensated rie effet B A B C A and an inome effet Consider the figure The left hand diagram illustrates the effet O O of an inrease in the rie of ommodity Figure 22-7: Deomosition of the rie effet into inome and substitution effets Suose net that the individual is fully omensated as the rie rises In the right hand diagram in Figure 22-7 the onsumer moves along his indifferene urve from A to C substituting ommodity 2 for ommodity This is the substitution John Riley Otober 4, 204

25 Essential Miroeonomis -25- effet of the rie inrease In the seond ste, the etra omensation is taken away and the budget line is ulled in towards the origin The onsumer then moves from C to B Note that if, as deited, ommodity is a normal good, both the substitution and inome effets are negative That is, the bigger (ie the more negative) the substitution effet and the bigger the inome effet, the greater will be the total effet on demand for ommodity Slutsky Equation We now onsider the deomosition of the rie effet in mathematial terms If M (, U) is minimized total eenditure at utility level U, and (, I ) is the onsumer s demand for ommodity, the omensated demand is (, M (, U )) Differentiating by, the sloe of the omensated demand urve is M I M Yet we have seen that Substituting into the above eression and rearranging, John Riley Otober 4, 204

26 Essential Miroeonomis -26- I total rie omensated inome effet rie effet effet Slutsky equation In artiular, the Slutsky deomosition of the "own rie effet" is as follows: I (22-7) Determinants of Demand Prie Elastiity Using the Slutsky equation, we an develo insights into the determinants of demand elastiity Converting (22-7) into elastiity form, I I I where (, U ) is the omensated demand for ommodity Hene (, ) (, ) k (, I) (22-8) From Proosition 22-5, (, ) ( k) John Riley Otober 4, 204

27 Essential Miroeonomis -27- Substituting for (, ) in equation (22-8) we have the following roosition Proosition 22-6: Deomosition of Own Prie Elastiity (, ) ( k ) k (, I) It follows that the absolute value of the own rie elastiity must lie between the inome elastiity and the elastiity of substitution Holding the eenditure share onstant, the higher the elastiity of substitution or the inome elastiity, the more negative is the own rie elastiity Moreover, the higher the eenditure share of ommodity, the greater the weight on the inome elastiity Intuitively, a higher share means that a rie rise requires a bigger hange in inome for the individual to be omensated Thus the inome effet on the hange in demand is greater John Riley Otober 4, 204