Chater 5 nome and Substitution Effets Effets of Changes in nome and Pries on Otimum Consumer Choies
As shown earlier for utilit maimization, (otimal ) is a funtion of ries and inome: (,,...,, ); i 1...n i i For two 1 goods, 2 it beomes n (,, ); (,, ) This hater deals with this relationshi. These funtions an be derived from the onstrained utilit maimization roblem from the last hater. These are demand funtions for goods for an individual onsumer! f we know,, and, we an alulate, the otimal quantit of demanded, for a given individual s utilit funtion.
ndividual demand funtions are homogeneous of degree zero in all ries and. That is, if all ries and inome double, will not be affeted. For eamle, if = +, then 2 = 2 +2. On the grah of the budget onstraint and will not hange if both numerator and denominator are multilied b 2 or an other onstant (t). Preferenes do not hange, ie., indifferene urves do not hange with hanges in ries or inome. So if all ries and inome move together, no hange in and. 2 2 2 2 Pure inflation does not affet hoies among goods.
Changes in nome Ceteris Paribus Pries and referenes (utilit funtion) Changes in inome shift the budget onstraint arallel beause ries are assumed not to hange. 2 1 1 2 > 1 2 Suessive otima when hanges will have same MRS beause the sloe of the budget onstraint remains the same.
Engel Curves show how or hanges as hanges, Ceteris Paribus. 1 3 > 2 > 1 2 3 1 2 3 f() or E
A Line tangent to f() at oint B. Line tangent to f() at oint A. f() f() Line tangent to f() at oint A. B A or E or E f a line tangent to f() at oint A has a negative interet, eenditures on the good ( ) derease as a erentage of total eenditures (E) as inreases around oint A. f a line tangent to f() at oint A has a ositive interet, eenditures on the good ( ) inrease as a erentage of total eenditures (E) as inreases around oint A. f a line tangent to f() at oint B has a zero interet, eenditures on the good ( ) remain onstant as a erentage of total eenditures (E) as inreases around oint B.
Normal and nferior Goods Normal good Curves ma bend u for luur goods (f > ) and down for neessities (f < ). =f() Mostl we talk about. Luur, e.g., eating out, boats Neessit, e.g., underwear Both luuries and neessities are Normal Goods.
nferior good e.g., maaroni & heese X X=f() 1 > 1 1 X Whether a good is normal or inferior ma var along the Engle Curve. The good ma hange from Normal to nferior. Norm nfer
Changes in Pries Ceteris Paribus. nvolves a hange in the rie ratio ( / ) and thus a hange in the sloe of the budget onstraint and a hange in the MRS at the new otimal oint, beause at otimal, MRS. Derease in nominal inome () required to sta on same indifferene urve. A C B U 2 dereases to ' ausing otimum to move from A to B (Δ is total effet). The total effet is divided into two effets, the substitution effet and the inome effet. The substitution effet is the hange in in going from A to C, while the inome effet is the hange in in going from C to B. U 1 To find C, use the original indifferene urve and find the oint of tangen S with a fititious budget onstraint that n has the new rie ratio. Tot What kind of good is with regard to inome? t is a normal good beause.
Summar of substitution and inome effets The movement from A to B is omosed of two effets: Substitution effet - Caused b hange in / U = U1. Beause is lower, the rie ratio is smaller and the new tangen oint must be at a smaller MRS (smaller d /d). Can measure the substitution effet b holding real inome onstant (hold U onstant), remaining on the same ndifferene Curve, but using the new rie ratio to find oint C. The hange in in going from A to C measures the substitution effet. For a given rie hange ( ), the magnitude of the substitution effet deends on the availabilit of substitute goods. Large substitution effet No substitution effet U = U 1 U = U 1 U = U U = U
nome Effet Caused b an inrease in real inome reresented b an inrease in utilit, or movement to a higher indifferene urve. A rie derease brings about an inrease in real inome; urhasing ower. The inome effet is the hange in in going from C to B. The magnitude of the inome effet deends on the ortion of inome sent on. The sum of the nome and Substitution Effets is the total effet of a rie hange (total hange in ). Could show a similar analsis for a rie inrease (tet. 127). n most situations, the two effets are omlementar, in that the move in the same diretion and reinfore eah other as in the ase of Normal Goods. U For a derease in, the substitution effet auses to inrease holding real inome onstant (have to redue nominal inome to kee real inome onstant). The inrease in nominal inome required to reah the higher indifferene urve (higher real inome resulting from the lower rie), auses to inrease further.
nome and Substitution Effets for nferior Goods nreasing inome auses a deline in urhases of, The inome effet is erverse for an inferior good.. A B C Tot n S U 1 A B Gives Total effet on. A C Gives Substitution U 2 effet. U C B Gives nome effet. n this ase, a derease in auses an inrease in, but onl beause the substitution effet is large enough to offset the erverse inome effet. U 1
The substitution effet is alwas (for tial ndifferene Curves) oosite the rie movement! The inome effet generall annot be lassified. The inome effet is oosite the rie movement for a normal good and in the same diretion as the rie movement for an inferior good. Giffen Goods When the erverse inome effet for an inferior good is large enough to overwhelm the substitution effet (ver unusual). Probabl requires the inferior good to make u a ver large ortion of total eenditures (see tet.128) and have no lose substitutes.!! Uward sloing demand urve (,)
ndividual Demand Curve (single onsumer) Demand funtions 1 1(,,...,, ) 1 2 n from earlier n a two-good world: n Demand funtions (,, ) Traditional or Marshallian (,,) demand funtions n (,,..., 1 2 n,) Could hold, referenes, and (nominal) onstant and var to get tial demand urve in and sae. ( (,),)
Per unit of time One erson, so indifferene urves an t ross. Shows that suessive rie delines result in larger otimal quantities of (eet for Giffen goods). ' ( /unit of time,) (,) /unit of time Shows how hanges as hanges when nominal inome () and ries of all other goods ( ) are onstant and when the individual s referene sstem onstant. This is a Marshallian demand urve (unomensated demand urve).
Shifts in the demand urve are aused b shifts in the individual s referenes or utilit funtion (shifts in the indifferene urves imling hanges in the MRS), the ries of other goods, and nominal inome. nrease, the demand urve for shifts u for normal good (down for inferior good). nrease, the demand urve for shifts u for substitute good (down for a omlement good). nrease MU relative to MU, the demand urve for shifts u. At an, MRS is larger. The absolute sloe of the indifferene urve is larger (steeer indifferene urve at an level of ). Changes in ause movements along the urve, i.e., hanges in the Quantit Demanded rather than shifts in demand.
Comensated Demand Curves (Hiksian Demand Curve) We ould eliminate the inome effets of hanges in and show the effets on, holding utilit or real inome onstant. ' '' ''' U1 /u.t. (,U 1) nome omensated demand urve (Hiksian) shows onl the substitution effets of hanges in, while, referenes and utilit (real inome) are held onstant. Means real inome onstant! Comensated demand urve alwas has a negative sloe beause the substitution effet is alwas negative if MRS is diminishing. /u.t. ( / alwasif MRS is diminishing)
Comensated demand urve is steeer (flatter) than the Marshallian demand urve for a normal (inferior) good. For >, the inome effet auses less (more) onsumtion for a normal good (an inferior good), so dereases more (less) for than for. Beginning otimal oint Marshallian Hiksian X is a normal good on this grah. For <, the inome effet auses more (less) onsumtion for a normal good (an inferior good), so inreases more (less) for than for.
Mathematial Aroah to Resonse to Prie Changes nome and Substitution Effets (two-good world) Use the omensated demand funtion (, real inome, U) and the ordinar demand funtion. ( nominal inome,, ) Hiksian or Comensated Marshallian, Ordinar, or Unomensated These two demand funtions are equal at the beginning oint (U ma oint) where the ross. (,,U) (,,)
Net, remember the Eenditure Funtion E E(,, U) Substitute the Eenditure Funtion into above and get the following (beause = E at U ma). (,,U) (,,E(,,U)) Partiall differentiate with reset to to get Chain rule Rearrange terms to get E E E E We will elore this relationshi on net age.
(-) eet for Giffen good Marshallian (total effet) This is not the sloe of the Marshallian demand urve; it is the inverse of the sloe. The more negative /, the flatter the demand urve. f were on the vertial ais, this would be the sloe of the demand urve. Alwas (-) Hiksian (sub effet) Minus hanges sign. E (-) for normal goods (+) for inferior goods nome effet E Usuall (+); for normal (+), for inferior (-) E E Alwas (+ or ) beause a derease (inrease) in imlies a derease (inrease) in E to maintain same utilit level as before the rie hange. Also, the Eenditure Funtion is nondereasing in ries.
Summar of Equation on Previous Page is the gross hange in in resonse to a hange in (total effet). t is the inverse of the sloe of the Marshallian demand urve. The first term on the right-hand side is the hange in in resonse to a hange in holding utilit (real inome) onstant. t is the substitution effet! t is alwas negative beause MRS is diminishing. t is the inverse of the sloe of the omensated (Hiksian) demand urve. E The seond term E shows the resonse of to a hange in through the effet of on inome (=E). t is the inome effet! For a normal good, the substitution and inome effets have negative signs; the reinfore eah other.
The Slutsk Equation Sub. effet = U (real inome) onstant n. effet = E E E E= Differentiate the eenditure funtion with reset to to get (,,U), whih is the omensated demand funtion. See Sheard s Lemma and Enveloe Theorem on age 137 of tet.
The Slutsk Equation Combine substitution and inome effets. (-) usuall, eet Giffen (-) alwas U (-) normal good (+) inferior good (-) alwas (+) normal good (-) inferior good
Elastiit - General definition The elastiit is the erentage hange in Y for a 1% hange in X. f Y = f(x,.) is some general funtion, e Y X X Y f X Y ΔY/Y ΔX/X ΔY ΔX Y, X X Elastiit at a oint on the funtion. Elastiit ma be different at eah oint on the funtion. The artial shows how Y hanges as X hanges. The artial is eressed in units of Y er unit of X. To remove the units, multil b X/Y to get a ure erentage hange. Units are gone. X Y Average elastiit over a segment of the funtion.
Marshallian Demand Elastiities Prie elastiit of demand e, f e 1,elasti;e 1, unit elasti;e f f e,,,, nome elastiit of demand e, e,, normal good; e,, inferior good Cross- rie elastiit of demand e, e, gross substitutes; e, gross omlements;,, indeendent The use of artial derivatives indiates that all other demand determinants are held onstant., 1, inelasti
Effet of a Prie Change on Total Eenditures for a Good f total eenditures on good =, then eenditures will reat to a rie hange as follows: e, Elasti (< -1) E. E. Unit elasti (= -1) Con. E. Con. E. nelasti (> -1) E. E. Beause % inrease in is > than % derease in. Beause % inrease in is < than % derease in.
1). (e TE : Multil this equation b / to get ) ( TE, Produt Rule: First ( ) times the derivative of the seond () lus the seond () times the derivative of the first ( ). f elasti (e, < -1), then (ΔTE is oosite the Δ ). f unit elasti (e, = -1), then (ΔTE = ). f inelasti e, > -1, then (ΔTE is same diretion as Δ ). TE TE TE Mathematiall: So the hange in total eenditures on the good is determined b the rie elastiit of demand as follows:
Comensated Demand Elastiities f elastiit is : and e, the omensated demand funtion is given b (,, U), then the omensated own rie e the omensated ross -., rie elastiit is : The relationshi between these omensated rie elastiities and Marshallian rie elastiities an be shown b utting the Slutsk Equation in elastiit form.
Relationshis Among Elastiities Slutsk Equation in Elastiit Form onst U Original Slutsk Multil b /. 1 onst U e, s e, Substitution Elastiit. The elastiit of the omensated demand urve. onst U Multil the seond term on the right-hand side b /. e,
e e s e,,, (-) eet Giffen (-) alwas (-) for normal or (+) for inferior good. The Slutsk Equation in elastiit form shows how the rie elastiit of demand an be disaggregated into the substitution elastiit lus the eenditure roortion times the inome elastiit of demand. 1) f substitution effet is or lose to ( e ), is, roortional to e,. 2) f the eenditure share is small ( small), is almost equal s e, e, e,. 3) f ou an estimate, e,, and, ou an derive, s e, e,.
Euler s Theorem and the Homogeneit Condition f 1 f(,,...,,) is homogeneous of 1 2 n degree m, then f n f1 f2... fn f mf(,,...,,) m 1. 1 2 n 1 2 n When m = (beause f is homogeneous of degree zero in ries and inome), then f f2... fn f ()1 1 1 2 n. f 1 f Degree of homogeneit (m=).
Homogeneit Condition Beause demand funtions are homogeneous of degree zero, we an use Euler s theorem on the demand funtion for (unomensated) to get the following for a two-good world: Dividing through b gives: e e,, (-) eet Giffen (+) for gross substitutes; (-) for gross omlements e, (+) for normal good; (-) for inferior good Conlusion The sum of own-rie, ross-rie, and inome elastiities equals zero for an seifi good. This reaffirms that demand is homogeneous of degree zero (equal erentage hanges in ries and inome leave the quantit demanded unaffeted). f ou know or an estimate two of the three terms, ou an alulate the third. f ou had more than two goods, ou would have several rossrie elastiities, some of whih ould be <.
Engel Aggregation Assuming a tial onsumer (diminishing MRS) and two goods, the budget onstraint is: at otimallit. For utilit maimization, an inrease in must be aomanied b an inrease in total eenditures beause = E. The demand funtions are: ( (,,,),)
Differentiate the budget onstraint with reset to assuming otimalit: 1 Multil the first term on the left-hand side b and and multil the seond term b s e, s For n goods e s, and 1,. where 1 s e, s e,... s e, 1. 1 1 2 2 n n is the roortion of sent on and s is the roortion of sent on. Thus, the roortion of inome sent on eah good times its inome elastiit of demand summed over all goods is 1. f inome inreases b 1%, erteris aribus, total urhases must inrease b 1% (beause of the budget onstraint), whih imlies that goods whose e, < 1 must be offset b others whose e, > 1 (assumes savings is a good). f ou know e, and s and s in a two-good world, ou an alulate e,. Engle Aggregation alies to market demand as well as individual demand.
Cournot Aggregation This onet deals with what haens to the demand for all goods when the rie of a single good hanges. The budget onstraint is. Differentiate the budget onstraint with reset to to get : beause nominal inome Multil eah term of this equation b /, the first term b /, and the last term b / to get : s e s, e, s and s e, do s e s not hange,. to give when the Cournot hanges. result The ross-rie effet of a hange in on demanded is restrited b the budget onstraint. :
Linear Demand Funtions a b d (-) if not Giffen (+) normal (-) inferior (+) Gross substitute (-) Gross omlement Emirial (not theoretial) demand funtion, so it ma not be homogeneous of degree zero beause a and b,, and d are not indeed. Pries and inome should be deflated b CP or other inde. e, b e, b, e,, e, d. Elastiities are tiall evaluated at the means of,,, and. is not onstant along the demand urve. is onstant, but / is not onstant!
= e, e 1 (more negative than 1) Elasti large, e 1, 1 (between 1 and ) nelasti small =
Constant Elastiit Demand Funtions a b d Cobb-Douglas te b1 d e, ab b d a Or, ln ln a b ln ln d ln b So e everwhere on the urve. The, b same roert holds for other elastiities (inome and ross-rie elastiities).
Consumer Surlus Ch market rie Use individual s urve --- hoke rie Area 1 A A Ch CS Consumer surlus is the area under the urve, above. t is the amount of etra eenditures an individual would be willing to make above what he/she has to make to get eah unit of the good.
For linear demand funtion The onet of onsumer surlus is used to evaluate the effets of rie hanges on onsumers. f the rie goes from to 1, the onsumer will suffer a net loss of 1 onsumer surlus equal to the area 1 ( 1 B ) 1 ( 2 A 1 )( P ) ABP net in onsumer surlus aused b the rie of inreasing from to 1.. loss
We will use the omensated demand urve and the eenditure funtion to illustrate the measurement of onsumer surlus. A hange in onsumer surlus an be measured b eenditure differenes to maintain a fied level of utilit, U=U. Thus we an use the omensated demand funtion where utilit is fied.
E E E( 1,,U 1 E(,,U ) ) 1 sloe E E1 1 E 1 E sloe U Eenditures at, given U and. Eenditures at 1, given U and. The hange in onsumer surlus is E E 1, whih is the negative of the hange in eenditures required to maintain the same level of utilit when the rie hanges. For an inrease in, E E 1 <, meaning real inome has fallen and nominal inome has to inrease to maintain U. For a derease in, E E 1 >, meaning real inome has inreased and nominal inome has to derease to maintain U.
We an differentiate the eenditure funtion to get the omensated demand funtion. E E (,,U ) is the hange in onsumer surlus for a ver small hange in. As shown earlier b enveloe theorem, Shehard s Lemma. The instantaneous hange in E resulting from a ver small hange in is equal to (quantit demanded on the omensated demand urve). Beause the hange from to 1 overs some distane, must integrate ( to get,,u) the hange in onsumer surlus.
ΔCS 1 (,, U )d Gives the area to left of the omensated demand urve between and. 1 1 B CS A 1
The omensated demand funtion gives the true estimate of the hange in onsumer surlus resulting from a rie hange, but it is diffiult to estimate, and one estimated, the estimate of onsumer surlus is usuall quite similar to the estimate obtained from the Marshallian demand urve, so using the Marshallian demand urve is more ratial in the real world.