INCOME AND SUBSTITUTION EFFECTS

Transcription

1 3076-C5.df 3/12/04 10:56 AM Page Chater 5 INCOME AND SUBSTITUTION EFFECTS In this hater we will use the utility-maimization model to study how the quantity of a good that an individual hooses is affeted by a hange in that good s rie. This eamination will allow us to onstrut the individual s demand urve for the good. In the roess we will rovide a number of insights into the nature of this rie resonse and into the kinds of assumtions that lie behind most analyses of demand. Demand funtions As we ointed out in Chater 4, in rinile it will usually be ossible to solve the neessary onditions of a utility maimum for the otimal levels of 1, 2,..., n (and λ, the Lagrangian multilier) as funtions of all ries and inome. Mathematially, this an be eressed as n demand funtions of the form ô 1 * = 1 ( 1, 2,... n, I) 2 * = 2 ( 1, 2,... n, I) * n = n ( 1, 2,... n, I). (5.1) If there are only two goods ( and y the ase we will usually be onerned with), this notation an be simlified a bit as * = (, y, I) y* = y(, y, I). (5.2) One we know the form of these demand funtions and the values of all ries and inome, these an be used to redit how muh of eah good this erson will hoose to buy. The notation stresses that ries and inome are eogenous to this roess that is, these are arameters over whih the individual has no ontrol at this stage of the analysis. Changes in the arameters will, of ourse, shift the budget onstraint and ause this erson to make different hoies. That is the question that is the fous of this hater and the net. Seifially, in this hater we will be looking at the artial derivatives / I and / for any arbitrary good,. Chater 6 will arry the disussion further by looking at ross-rie effets of the form / y for any arbitrary air of goods, and y.

2 3076-C5.df 3/12/04 10:56 AM Page Part 2 Choie and Demand Homogeneity A first roerty of demand funtions requires little mathematis. If we were to double all ries and inome (indeed, if we were to multily them all by any ositive onstant), the otimal quantities demanded would remain unhanged. Doubling all ries and inome hanges only the units by whih we ount, not the real quantity of goods demanded. This result an be seen in a number of ways, although erhas the easiest is through a grahi aroah. Referring bak to Figures 4.1 and 4.2, it is lear that if we double, y, and I, we do not affet the grah of the budget onstraint. Hene, *, y* will still be the ombination that is hosen. + y y = I is the same onstraint as y y = 2I. Somewhat more tehnially, we an write this result as saying that for any good i, * i = i ( 1, 2,... n, I) = i (t 1, t 2,... t n, ti) (5.3) for any t > 0. Funtions that obey the roerty illustrated in Equation 5.3 are said to be homogeneous of degree zero. 1 Hene, we have shown that individual demand funtions are homogeneous of degree zero in all ries and inome. Changing all ries and inome in the same roortions will not affet the hysial quantities of goods demanded. This result shows that (in theory) individuals demands will not be affeted by a ure inflation during whih all ries and inomes rise roortionally. They will ontinue to demand the same bundle of goods. Of ourse, if an inflation were not ure (that is, if some ries rose more raidly than others), this would not be the ase. EXAMPLE 5.1 Homogeneity Homogeneity of demand is a diret result of the utility-maimization assumtion. Demand funtions derived from utility maimization will be homogeneous and, onversely, demand funtions that are not homogeneous annot reflet utility maimization (unless ries enter direty into the utility funtion itself, as they might for goods with snob aeal). If, for eamle, an individual s utility for food () and housing (y) is given by utility = U(, y) =.3 y.7, (5.4) it is a simle matter (following the roedure used in Eamle 4.1) to derive the demand funtions * = y* =. 3I. 7I. (5.5) These funtions obviously ehibit homogeneity a doubling of all ries and inome would leave * and y* unaffeted. If the individual s referenes for and y were refleted instead by the CES funtion: U(, y) =.5 + y.5, (5.6) y 1 More generally, as we saw in Chater 2, a funtion f ( 1, 2,..., n ) is said to be homogeneous of degree k if f (t 1, t 2...,t n ) = t k f ( 1, 2,..., n ) for any t > 0. The most ommon ases of homogeneous funtions are k = 0 and k = 1. If f is homogeneous of degree 0, doubling all of its arguments leaves f unhanged in value. If f is homogeneous of degree 1, doubling all its arguments will double the value of f.

3 3076-C5.df 3/12/04 10:56 AM Page 123 Chater 5 Inome and Substitution Effets 123 we showed in Eamle 4.2 that the demand funtions are given by 1 * = 1 + / 1 y* = 1 + / I y I y y. (5.7) As before, both these demand funtions are homogeneous of degree zero a doubling of, y, and I would leave * and y* unaffeted. Query: Do the demand funtions derived in this eamle ensure that total sending on and y will ehaust the individual s inome for any ombination of, y, and I? Can you rove that this is the ase? DEFINITION Changes in inome As a erson s urhasing ower rises, it is natural to eet that the quantity of eah good urhased will also inrease. This situation is illustrated in Figure 5.1. As eenditures inrease from I 1 to I 2 to I 3, the quantity of demanded inreases from 1 to 2 to 3. Also, the quantity of y inreases from y 1 to y 2 to y 3. Notie that the budget lines I 1, I 2, and I 3 are all arallel, refleting the fat that only inome is hanging, not the relative ries of and y. Beause the ratio / y stays onstant, the utility-maimizing onditions also require that the MRS stay onstant as the individual moves to higher levels of satisfation. The MRS is therefore the same at oint ( 3, y 3 ) as at ( 1, y 1 ). Normal and inferior goods In Figure 5.1, both and y inrease as inome inreases / I and y/ I are both ositive. This might be onsidered the usual situation, and goods that this roerty are alled normal goods over the range of inome hange being observed. For some goods, however, the quantity hosen may derease as inome inreases in some ranges. Some eamles of these goods might be rotgut whiskey, otatoes, and seondhand lothing. A good z for whih z/ I is negative is alled an inferior good. This henomenon is illustrated in Figure 5.2 on age 125. In this diagram the good z is inferior beause for inreases in inome in the range shown, less of z is atually hosen. Notie that indifferene urves do not have to be oddly shaed to ehibit inferiority; the urves orresonding to goods y and z in Figure 5.2 ontinue to obey the assumtion of a diminishing MRS. Good z is inferior beause of the way it relates to the other goods available (good y here), not beause of a euliarity unique to it. Hene, we have develoed the following definitions: Inferior and normal goods. A good i for whih i / I < 0 over some range of inome hanges is an inferior good in that range. If i / I 0 over some range of inome variation, the good is a normal, or noninferior, good in that range.

4 3076-C5.df 3/12/04 10:56 AM Page Part 2 Choie and Demand FIGURE 5.1 Effet of an Inrease in Inome on the Quantities of and y Chosen As inome inreases from I 1 to I 2 to I 3, the otimal (utility-maimizing) hoies of and y are shown by the suessively higher oints of tangeny. Notie that the budget onstraint shifts in a arallel way beause its sloe (given by / y ) does not hange. Quantity of y U 1 U 2 U 3 y 3 y 2 y 1 I 2 I 3 U 3 U 2 I U 1 Quantity of Changes in a good s rie The effet of a rie hange on the quantity of a good demanded is more omle to analyze than is the effet of a hange in inome. Geometrially, this is beause hanging a rie involves hanging not only the interets of the budget onstraint, but also its sloe. Consequently, moving to the new utility-maimizing hoie entails not only moving to another indifferene urve, but also hanging the MRS. When a rie hanges, therefore, two analytially different effets ome into lay. One of these is a substitution effet even if the individual were to stay on the same indifferene urve, onsumtion atterns would be alloated so as to equate the MRS to the new rie ratio. A seond effet, the inome effet, arises beause a rie hange neessarily hanges an individual s real inome the individual annot stay on the initial indifferene urve, but must move to a new one. We begin by analyzing these effets grahially. Then we will rovide a mathematial develoment. Grahial analysis of a fall in rie Inome and substitution effets are illustrated in Figure 5.3 on age 126. This individual is initially maimizing utility (subjet to total eenditures, I) by onsuming the ombination *, y*. The initial budget onstraint is I = 1 + y y. Now suose that the rie of falls to. 2 The new budget onstraint is given by the equation I = 2 + y y in Figure 5.3.

5 3076-C5.df 3/12/04 10:56 AM Page 125 Chater 5 Inome and Substitution Effets 125 FIGURE 5.2 An Indifferene Curve Ma Ehibiting Inferiority In this diagram, good z is inferior beause the quantity urhased atually delines as inome inreases. y is a normal good (as it must be if there are only two goods available), and urhases of y inrease as total eenditures inrease. Quantity of y y 3 y 2 U 3 y 1 U 2 I 1 I 2 z 3 z 2 z 1 U I 1 3 Quantity of z It is lear that the new osition of maimum utility is at **, y**, where the new budget line is tangent to the indifferene urve U 2. The movement to this new oint an be viewed as being omosed of two effets. First, the hange in the sloe of the budget onstraint would have motivated a move to oint B, even if hoies had been onfined to those on the original indifferene urve U 1. The dashed line in Figure 5.3 has the same sloe as the new budget onstraint (I = 2 + y y), but is drawn to be tangent to U 1 beause we are onetually holding real inome (that is, utility) onstant. A relatively lower rie for auses a move from *, y* to B if we do not allow this individual to be made better off as a result of the lower rie. This movement is a grahi demonstration of the substitution effet. The further move from B to the otimal oint **, y** is analytially idential to the kind of hange ehibited earlier for hanges in inome. Beause the rie of has fallen, this erson has a greater real inome and an afford a utility level (U 2 ) that is greater than that whih ould reviously be attained. If is a normal good, more of it will be hosen in resonse to this inrease in urhasing ower. This observation elains the origin of the term inome effet for the movement. Overall then, the result of the rie deline is to ause more to be demanded. It is imortant to reognize that this erson does not atually make a series of hoies from *, y* to B and then to **, y**. We never observe oint B; only the two otimal ositions are refleted in observed behavior. However, the notion of inome and substitution effets is analytially valuable beause it shows that a rie hange affets the quantity of that is demanded in two onetually different ways. We will see how this searation offers major insights in the theory of demand.

6 3076-C5.df 3/12/04 10:56 AM Page Part 2 Choie and Demand FIGURE 5.3 Demonstration of the Inome and Substitution Effets of a Fall in the Prie of When the rie of falls from 1 to, 2 the utility-maimizing hoie shifts from *, y* to **, y**. This movement an be broken down into two analytially different effets: first, the substitution effet, involving a movement along the initial indifferene urve to oint B, where the MRS is equal to the new rie ratio; and seondly, the inome effet, entailing a movement to a higher level of utility, beause real inome has inreased. In the diagram, both the substitution and inome effets ause more to be bought when its rie delines. Notie that oint I/ y is the same as before the rie hange. This is beause y has not hanged. Point I/ y therefore aears on both the old and new budget onstraints. Quantity of y U 1 U 2 I y y** y* 1 I y y 2 I y y B U 2 U 1 * B ** Quantity of Substitution effet Inome effet Total inrease in Grahial analysis of an inrease in rie If the rie of good were to inrease, a similar analysis would be used. In Figure 5.4 the budget line has been shifted inward beause of an inrease in the rie of from 1 to 2. The movement from the initial oint of utility maimization (*, y*) to the new oint (**, y**) an be deomosed into two effets. First, even if this erson ould stay on the initial indifferene urve (U 2 ), there would still be an inentive to substitute y for and move along U 2 to oint B. However, beause urhasing ower has been redued by the rise in the rie of, he or she must move to a lower level of utility. This movement is again alled the inome effet. Notie in Figure 5.4 that both the inome and substitution effets work in the same diretion and ause the quantity of demanded to be redued in resonse to an inrease in its rie.

7 3076-C5.df 3/12/04 10:56 AM Page 127 Chater 5 Inome and Substitution Effets 127 FIGURE 5.4 Demonstration of the Inome and Substitution Effets of an Inrease the Prie of When the rie of inreases, the budget onstraint shifts inward. The movement from the initial utility-maimizing oint (*, y*) to the new oint (**, y**) an be analyzed as two searate effets. The substitution effet would be deited as a movement to oint B on the initial indifferene urve (U 2 ). The rie inrease, however, would reate a loss of urhasing ower and a onsequent movement to a lower indifferene urve. This is the inome effet. In the diagram, both the inome and substitution effets ause the quantity of to fall as a result of the inrease in its rie. Again, the oint I/ y is not affeted by the hange in the rie of. Quantity of y I y U 1 U 2 B y** y* 2 I y y 1 I y y U 2 U 1 ** Inome effet B Substitution effet * Quantity of Total redution in Effets of rie hanges for inferior goods So far we have shown that substitution and inome effets tend to reinfore one another. For a rie deline, both ause more of the good to be demanded, whereas for a rie inrease, both ause less to be demanded. Although this analysis is aurate for the ase of normal (noninferior) goods, the ossibility of inferior goods omliates the story. In this ase, inome and substitution effets work in oosite diretions, and the ombined result of a rie hange is indeterminate. A fall rie, for eamle, will always ause an individual to tend to onsume more of a good beause of the substitution effet. But if the good is inferior, the inrease in urhasing ower aused by the rie deline may ause less of the good to be bought. The result is therefore indeterminate the substitution effet tends to

8 3076-C5.df 3/12/04 10:56 AM Page Part 2 Choie and Demand inrease the quantity of the inferior good bought, whereas the (erverse) inome effet tends to redue this quantity. Unlike the situation for normal goods, it is not ossible here to redit even the diretion of the effet of a hange in on the quantity of onsumed. Giffen s arado If the inome effet of a rie hange is strong enough, the hange in rie and the resulting hange in the quantity demanded ould atually move in the same diretion. Legend has it that the English eonomist Robert Giffen observed this arado in nineteenthentury Ireland when the rie of otatoes rose, eole reortedly onsumed more of them. This euliar result an be elained by looking at the size of the inome effet of a hange in the rie of otatoes. Potatoes were not only inferior goods, but also used u a large ortion of the Irish eole s inome. An inrease in the rie of otatoes therefore redued real inome substantially. The Irish were fored to ut bak on other luury food onsumtion in order to buy more otatoes. Even though this rendering of events is historially imlausible, the ossibility of an inrease in the quantity demanded in resonse to an inrease in the rie of a good has ome to be known as Giffen s arado. 2 Later we will rovide a mathematial analysis of how Giffen s arado an our. A summary Hene, our grahial analysis leads to the following onlusions: OPTIMIZATION PRINCIPLE Substitution and inome effets. The utility-maimization hyothesis suggests that, for normal goods, a fall in the rie of a good leads to an inrease in quantity urhased beause (1) the substitution effet auses more to be urhased as the individual moves along an indifferene urve; and (2) the inome effet auses more to be urhased beause the rie deline has inreased urhasing ower, thereby ermitting movement to a higher indifferene urve. When the rie of a normal good rises, similar reasoning redits a deline in the quantity urhased. For inferior goods, substitution and inome effets work in oosite diretions, and no definite reditions an be made. The individual s demand urve Frequently eonomists wish to grah demand funtions. It will ome as no surrise to you that these grahs are alled demand urves. Understanding how suh widely used urves relate to underlying demand funtions rovides additional insights to even the most fundamental of eonomi arguments. To simlify the develoment, assume there are only two goods and that, as before, the demand funtion for good is given by * = (, y, I). The demand urve derived from this funtion looks at the relationshi between and while holding y, Ī, and referenes onstant. That is, it shows the relationshi * = (, y, Ī ), (5.8) where the bars over y and I indiate that these determinants of demand are being held onstant. This onstrution is shown in Figure 5.5. The grah shows utility-maimizing hoies of and y as this individual is resented with suessively lower ries of good 2 A major roblem with this elanation is that it disregards Marshall s observation that both suly and demand fators must be taken into aount when analyzing rie hanges. If otato ries inreased beause of the otato blight in Ireland, then suly should have beome smaller, so how ould more otatoes ossibly have been onsumed? Also, sine many Irish eole were otato farmers, the otato rie inrease should have inreased real inome for them. For a detailed disussion of these and other fasinating bits of otato lore, see G. P. Dwyer and C. M. Lindsey, Robert Giffen and the Irish Potato, Amerian Eonomi Review (Marh 1984):

9 3076-C5.df 3/12/04 10:56 AM Page 129 Chater 5 Inome and Substitution Effets 129 FIGURE 5.5 Constrution of an Individual s Demand Curve In (a) the individual s utility-maimizing hoies of and y are shown for three different ries of (,, and ). In (b) this relationshi between and is used to onstrut the demand urve for. The demand urve is drawn on the assumtion that y, I, and referenes remain onstant as varies. Quantity of y er eriod I / y I y y I y y I y y U 3 U 2 U 1 Quantity of er eriod (a) Individual's indifferene urve ma ( y I) Quantity of er eriod (b) Demand urve

10 3076-C5.df 3/12/04 10:56 AM Page Part 2 Choie and Demand DEFINITION (while holding y and I onstant). We assume that the quantities of hosen inrease from to to as that good s rie falls from to to. Suh an assumtion is in aord with our general onlusion that, eet in the unusual ase of Giffen s arado, / is negative. In Figure 5.5b information about the utility-maimizing hoies of good is transferred to a demand urve, having on the vertial ais and sharing the same horizontal ais as the figure above it. The negative sloe of the urve again reflets the assumtion that / is negative. Hene, we may define an individual demand urve as follows: Individual demand urve. An individual demand urve shows the relationshi between the rie of a good and the quantity of that good urhased by an individual assuming that all other determinants of demand are held onstant. The demand urve illustrated in Figure 5.5 stays in a fied osition only so long as all other determinants of demand remain unhanged. If one of these other fators were to hange, the urve might shift to a new osition, as we now desribe. Shifts in the demand urve Three fators were held onstant in deriving this demand urve: (1) inome; (2) ries of other goods (say, y ); and (3) the individual s referenes. If any of these were to hange, the entire demand urve might shift to a new osition. For eamle, if I were to inrease, the urve would shift outward (rovided that / I > 0; that is, that the good is a normal good over this inome range). More would be demanded at eah rie. If another rie, say, y, were to hange, the urve would shift inward or outward, deending reisely on how and y are related. In the net hater we will eamine that relationshi in detail. Finally, the urve would shift if the individual s referenes for good were to hange. A sudden advertising blitz by the MDonald s Cororation might shift the demand for hamburgers outward, for eamle. As this disussion makes lear, one must remember that the demand urve is only a two-dimensional reresentation of the true demand funtion (Equation 5.8) and that it is stable only if other things in fat stay onstant. It is imortant to kee learly in mind the differene between a movement along a given demand urve aused by a hange in and a shift in the entire urve aused by a hange in inome, in one of the other ries, or in referenes. Traditionally, the term an inrease in demand is reserved for an outward shift in the demand urve, whereas the term an inrease in the quantity demanded refers to a movement along a given urve aused by a hange in. EXAMPLE 5.2 Demand Funtions and Demand Curves To be able to grah a demand urve from a given demand funtion, we must assume that the referenes that generated the funtion remain stable and that we know the values of inome and other relevant ries. In the first ase studied in Eamle 5.1, we found that I =.3 (5.9) and y I =. 7. y

11 3076-C5.df 3/12/04 10:56 AM Page 131 Chater 5 Inome and Substitution Effets 131 If referenes do not hange and if this individual s inome is $100, these funtions beome y = = y (5.10) or = 30 y y = 70, whih makes lear that the demand urves for these two goods are simle hyerbolas. A rise in inome would shift both of the demand urves outward. Notie also, in this ase, that the demand urve for is not shifted by hanges in y and vie versa. For the seond ase eamined in Eamle 5.1, the analysis is more omle. For good, say, we know that 1 = 1 + / (5.11) so to grah this in the lane we must know both I and y. If we again assume I = 100 and let y = 1, Equation 5.11 beomes 100 =, (5.12) 2 + whih, when grahed, would also show a general hyerboli relationshi between rie and quantity onsumed. In this ase the urve would be relatively flatter beause substitution effets are larger than in the Cobb-Douglas ase. From Equation 5.11 we also know that I y ì 1 1 = > ìi / y (5.13) and ì ì = I ( + ) y y 2 > 0, so inreases in I or y would shift the demand urve for good outward. Query: How would the demand funtions in Equations 5.10 hange if this erson sent half of inome on eah good? Show that these demand funtions redit the same onsumtion at the oint = 1, y = 1, I = 100 as does Equation Use a numerial eamle to show that the CES demand funtion is more resonsive to an inrease in than is the Cobb-Douglas demand funtion. Comensated demand urves In Figure 5.5, the level of utility this erson gets varies along the demand urve. As falls, he or she is made inreasingly better off, as shown by the inrease in utility from U 1 to U 2 to U 3. The reason this haens is that the demand urve is drawn on the assumtion that nominal inome and other ries are held onstant; hene, a deline in makes

12 3076-C5.df 3/12/04 10:56 AM Page Part 2 Choie and Demand this erson better off by inreasing his or her real urhasing ower. Although this is the most ommon way to imose the eteris aribus assumtion in develoing a demand urve, it is not the only way. An alternative aroah holds real inome (or utility) onstant while eamining reations to hanges in. The derivation is illustrated in Figure 5.6. There we hold utility onstant (at U 2 ) while suessively reduing. As falls, the individual s nominal inome is effetively redued, thus reventing any inrease in utility. In other words, the effets of the rie hange on urhasing ower are omensated so as to onstrain the individual to remain on U 2. Reations to hanging ries inlude only substitution effets. If we were instead to eamine effets of inreases in, inome omensation would be ositive: This individual s inome would have to be in- FIGURE 5.6 Constrution of a Comensated Demand Curve The urve h shows how the quantity of demanded hanges when hanges, holding y and utility onstant. That is, the individual s inome is omensated so as to kee utility onstant. Hene, reflets only substitution effets of hanging ries. Quantity of y Sloe y Sloe y Sloe y U 2 * (a) Individual's indifferene urve ma Quantity of (, y,u) * ** (b) Comensated demand urve Quantity of

13 3076-C5.df 3/12/04 10:56 AM Page 133 Chater 5 Inome and Substitution Effets reased to ermit him or her to stay on the U 2 indifferene urve in resonse to the rie rises. We an summarize these results as follows: 133 DEFINITION Comensated demand urve. A omensated demand urve shows the relationshi between the rie of a good and the quantity urhased on the assumtion that other ries and utility are held onstant. The urve (whih is sometimes termed a Hiksian demand urve after the British eonomist John Hiks) therefore illustrates only substitution effets. Mathematially, the urve is a two-dimensional reresentation of the omensated demand funtion * = (, y, U). (5.14) Relationshi between omensated and unomensated demand urves This relationshi between the two demand urve onets we have develoed is illustrated in Figure 5.7. At the urves interset, beause at that rie the individual s inome is just suffiient to attain utility level U 2 (omare Figures 5.5 and 5.6). Hene, is demanded under either demand onet. For ries below, however, the individual suffers a omensating redution in inome on the urve to revent an inrease in utility from the lower rie. Hene, assuming is a normal good, less is demanded at FIGURE 5.7 Comarison of Comensated and Unomensated Demand Curves The omensated ( ) and unomensated () demand urves interset at beause is demanded under eah onet. For ries above, the individual s inome is inreased with the omensated demand urve, so more is demanded than with the unomensated urve. For ries below, inome is redued for the omensated urve, so less is demanded than with the unomensated urve. The standard demand urve is flatter beause it inororates both substitution and inome effets whereas the urve reflets only substitution effets. (, y,i) (, y,u) * ** Quantity of

14 3076-C5.df 3/12/04 10:56 AM Page Part 2 Choie and Demand along than along the unomensated urve. Alternatively, for a rie above (suh as ), inome omensation is ositive, beause the individual needs some hel to remain on U 2. Hene, again assuming is a normal good, at more is demanded along than along. In general then, for a normal good, the omensated demand urve is somewhat less resonsive to rie hanges than is the unomensated urve, beause the latter reflets both substitution and inome effets of rie hanges whereas the omensated urve reflets only substitution effets. The hoie between using omensated or unomensated demand urves in eonomi analysis is largely a matter of onveniene. In most emirial work unomensated urves (whih are sometimes alled Marshallian demand urves ) are used beause the data on ries and nominal inomes needed to estimate them are readily available. In the Etensions to Chater 10 we will desribe some of these estimates and show how they might be emloyed for ratial oliy uroses. For some theoretial uroses, however, omensated demand urves are a more aroriate onet, beause the ability to hold utility onstant offers some advantages. Our disussion of onsumer surlus in the final setion of this hater offers one illustration of these advantages. EXAMPLE 5.3 Comensated Demand Funtions In Eamle 3.1, we assumed that the utility funtion for hamburgers (y) and soft drinks () was given by utility = U(, y) =.5 y.5, (5.15) and in Eamle 4.1, we showed that we an alulate the Marshallian demand funtions for suh utility funtions as y I = ` I = a = y I = 2 (5.16) Also, in Eamle 4.3, we alulated the indiret utility funtion by ombining Equations 5.15 and 5.16 as I 2 y. utility = V ( I, I, y ) = y (5.17) To obtain the omensated demand funtions for and y, we simly use Equation 5.17 to solve for I and then substitute this eression involving V into Equations This ermits us to interhange inome and utility so we may hold the latter onstant, as is required for the omensated demand onet. Making these substitutions yields V =. V. 5 y =.. 5 y (5.18) These are the omensated demand funtions for and y. Notie that now demand deends on utility (V) rather than on inome. Holding utility onstant, it is lear that inreases in redue the demand for and this now reflets only the substitution effet (see Eamle 5.4 also).. 5 y 5

15 3076-C5.df 3/12/04 10:56 AM Page 135 Chater 5 Inome and Substitution Effets 135 Although y did not enter into the unomensated demand funtion for good, it does enter into the omensated funtion inreases in y shift the omensated demand urve for outward. The two demand onets agree at the assumed initial oint = 1, y = 4, I = 8, and V = 2 Equations 5.16 redit = 4, y = 1 at this oint as do Equations For > 1 or < 1, the demands differ under the two onets, however. If, say, = 4, the unomensated funtions (Equations 5.16) redit = 1, y = 1 whereas the omensated funtions (Equations 5.18) redit = 2, y = 2. The redution in resulting from the rise in its rie is smaller with the omensated demand funtion than it is with the unomensated funtion beause the former onet adjusts for the negative effet on urhasing ower that omes about from the rie rise. This eamle makes lear the different eteris aribus assumtions inherent in the two demand onets. With unomensated demand, eenditures are held onstant at I = 2 so the rise in from 1 to 4 results in a loss of utility in this ase, utility falls from 2 to 1. In the omensated demand ase, utility is held onstant at V = 2. To kee utility onstant, eenditures must rise to E = 1(2) + 1(2) = 4 in order to offset the effets of the rie rise (see Equation 5.17). Query: Are the omensated demand funtions given in Equations 5.18 homogeneous of degree zero in and y if utility is held onstant? Would you eet that to be true for all omensated demand funtions? A mathematial develoment of resonse to rie hanges U to this oint we have largely relied on grahial devies to desribe how individuals resond to rie hanges. Additional insights are rovided by a more mathematial aroah. Our basi goal is to eamine the artial derivative / ; that is, how a eteris aribus hange in the rie of a good affets its urhase. In the net hater, we take u the question of how hanges in the rie of one ommodity affet urhases of another ommodity. Diret aroah Our goal is to use the utility-maimization model to learn something about how the demand for good hanges when hanges; that is, we wish to alulate /. The diret aroah to this roblem makes use of the first-order onditions for utility maimization (Equations 4.8). Differentiation of these n + 1 equations yields a new system of n + 1 equations, whih eventually an be solved for the derivative we seek. 3 Unfortunately, obtaining this solution is quite umbersome and the stes required yield little in the way of eonomi insights. Hene, we will instead adot an indiret aroah that relies on the onet of duality. In the end, both aroahes yield the same onlusion, but the indiret aroah is muh riher in terms of the eonomis it ontains. Indiret aroah To begin our indiret aroah 4 we will assume (as before) there are only two goods ( and y) and fous on the omensated demand funtion, (, y, U), introdued in Equation We now wish to illustrate the onnetion between this demand funtion and the ordinary demand 3 See, for eamle, Paul A. Samuelson, Foundations of Eonomi Analysis (Cambridge, MA: Harvard University Press, 1947), The following roof is adated from Philli J. Cook, A One Line Proof of the Slutsky Equation, Amerian Eonomi Review 62 (Marh 1972): 139.

16 3076-C5.df 3/12/04 10:56 AM Page Part 2 Choie and Demand funtion, (, y, I). In Chater 4 we introdued the eenditure funtion, whih reords the minimal eenditure neessary to attain a given utility level. If we denote this funtion by minimum eenditure = E(, y, U), (5.19) then by definition, (, y, U) = [, y, E(, y, U)]. (5.20) This onlusion was already introdued in onnetion with Figure 5.7, whih showed that the quantity demanded is idential for the omensated and unomensated demand funtions when inome is eatly what is needed to attain the required utility level. Equation 5.20 is obtained by inserting that eenditure level into the demand funtion, (, y, I). Now we an roeed by artially differentiating Equation 5.20 with reset to and reognizing that this variable enters into the ordinary demand funtion in two laes. Hene, and rearranging terms, ì ì ì ì = ì ì = ì ì + ì ì E, ìe ì - ì ì E. ìe ì (5.21) (5.22) The substitution effet Consequently, the derivative we seek has two terms. Interretation of the first term is straightforward: It is the sloe of the omensated demand urve. But that sloe reresents movement along a single indifferene urve it is in fat what we alled the substitution effet earlier. The first term on the right of Equation 5.22 is a mathematial reresentation of that effet. The inome effet The seond term in Equation 5.22 reflets the way in whih hanges in affet the demand for through hanges in neessary eenditure levels (that is, hanges in urhasing ower). This term therefore reflets the inome effet. The negative sign in Equation 5.22 shows the diretion of the effet. For eamle, an inrease in inreases the eenditure level that would have been needed to kee utility onstant (mathematially, E/ > 0). But beause nominal inome is in fat held onstant in Marshallian demand, these etra eenditures are not available. Hene (and y) must be redued to meet this shortfall. The etent of the redution in is given by / E. On the other hand, if falls, the eenditure level required to attain a given utility falls too. The deline in that would normally aomany suh a fall in eenditures is reisely the amount that must be added bak through the inome effet. Notie that in this ase the inome effet works to inrease. The Slutsky equation The relationshis embodied in Equation 5.22 were first disovered by the Russian eonomist Eugen Slutsky in the late nineteenth entury. A slight hange in notation is required to state the result the way Slutsky did. First, we write the substitution effet as substitution effet = ì = ì U = onstant (5.23) ì ì to indiate movement along a single indifferene urve. For the inome effet we have inome effet =- ì ì E =- ì (5.24) ì ì ì ì E, E I ì beause hanges in inome or eenditures amount to the same thing in the funtion (, y, I).

17 3076-C5.df 3/12/04 10:56 AM Page 137 Chater 5 Inome and Substitution Effets It is a relatively easy matter to show that ìe =. (5.25) ì Intuitively, a $1 inrease in raises neessary eenditures by dollars, beause $1 etra must be aid for eah unit of urhased. A formal roof of this assertion, whih relies on the enveloe theorem (see Chater 2), will be relegated to a footnote By ombining Equations we an arrive at the following: OPTIMIZATION PRINCIPLE Slutsky equation. The utility-maimization hyothesis shows that the substitution and inome effets arising from a rie hange an be reresented by ì = substitution effet + inome effet, (5.26) ì or ì = ì ì = - U onstant. (5.27) ì ì ìi The Slutsky equation allows a more definitive treatment of the diretion and size of substitution and inome effets than was ossible with only a grahi analysis. First, the substitution effet ( / U = onstant) is always negative as long as the MRS is diminishing. A fall (rise) in redues (inreases) / y, and utility maimization requires that the MRS fall (rise) too. But this an only our along an indifferene urve if inreases (or, in the ase of a rise in, dereases). Hene, insofar as the substitution effet is onerned, rie and quantity always move in oosite diretions. Equivalently, the sloe of the omensated demand urve must be negative. 6 We will show this result in a somewhat different way in the final setion to this hater. The sign of the inome effet ( / I) deends on the sign of / I. If is a normal good, / I is ositive and the entire inome effet, like the substitution effet, is negative. Thus for normal goods, rie and quantity always move in oosite diretions. For eamle, a fall in raises real inome, and beause is a normal good, urhases of rise. Similarly, a rise in redues real inome and urhases of fall. Overall then, as we desribed reviously using a grahi analysis, substitution and inome effets work in the same diretion to yield a negatively sloed demand urve. In the ase of an inferior good, / I < 0 and the two terms in Equation 5.27 would have different signs. It is at least theoretially ossible that in this ase the seond term ould dominate the first, leading to Giffen s arado ( d / > 0). 5 Remember that the individual s dual roblem is to minimize E = + y y, subjet to U = U(, y). The Lagrangian eression for this rolem is L = + y y + k[u U(,y)], and the enveloe theorem alied to onstrained minimization roblems states that at the otimal oint, ìe = ì L =. ì ì This is the result in Equation The result, and similar ones that we will enounter in the theory of firms osts, is sometimes alled Shehard s lemma. Its imortane in emirial work is that the demand funtion for good an be found diretly from the eenditure funtion by simle artial differentiation. The demand funtions generated in this way will deend on U, so they should be interreted as omensated demand funtions. In Eamle 4.4 we found that the eenditure funtion was E = 2 V.5.5 y. Partial differentiation of this eression with reset to yields the omensated demand funtion in Equations For a further disussion, see the Etensions to this hater. 6 It is ossible that substitution effets would be 0 if indifferene urves have an L-shae (imlying that and y are used in fied roortions). Some eamles are rovided in the Chater 5 roblems.

18 3076-C5.df 3/12/04 10:56 AM Page Part 2 Choie and Demand EXAMPLE 5.4 A Slutsky Deomosition The deomosition of a rie effet that was first disovered by Slutsky an be niely illustrated with the Cobb-Douglas eamle studied reviously. In Eamle 5.3 we found that the Marshallian demand funtion for good was (,, I) y. I = 05, (5.28) and the Hiksian (omensated) demand funtion was V (, y y, V ) 05. =. 05. (5.29) The overall effet of a rie hange on the demand for good an be found by differentiating the Marshallian demand funtion: ì ì = 0. 5I 2. (5.30) Now we wish to show that this effet is the sum of the two effets that Slutsky identified. As before, the substitution effet is found by differentiating the omensated demand funtion: substitution effet = ì 0. 5V y = 05.. (5.31) ì 15. We an eliminate indiret utility, V, by substitution from Equation 5.17: (. I 025 substitution effet = ) 05. y y. I = (5.32) Calulation of the inome effet in this eamle is onsiderably easier. Alying the results from Equation 5.27 we have inome effet =- ì 05. I I =- =- (5.33) ìi. 2 A omarison of Equations 5.30 with Equations 5.32 and 5.33 shows that we have indeed deomosed the rie derivative of this demand funtion into substitution and inome omonents. Interestingly, the substitution and inome effets are of reisely the same size. This, as we will see in later eamles, is one of the reasons that the Cobb- Douglas is a very seial ase. The well-worn numerial eamle we have been using also demonstrates this deomosition. When the rie of rises from $1 to $4, the (unomensated) demand for falls from = 4 to = 1. But the omensated demand for falls only from = 4 to = 2. That deline of 50 erent is the substitution effet. The further 50 erent fall from = 2 to = 1 reresents reations to the deline in urhasing ower inororated in Marshallian demand funtion. This inome effet does not our when the omensated demand notion is used. Query: In this eamle the individual sends half his or her inome on good and half on good y. How would the relative sizes of the substitution and inome effets be altered if the eonents of the Cobb-Douglas utility funtion were not equal?

19 3076-C5.df 3/12/04 10:56 AM Page 139 Demand elastiities Chater 5 Inome and Substitution Effets So far in this hater we have been eamining how individuals resond to hanges in ries and inome by looking at the derivatives of the demand funtion. For many analytial questions this is a good way to roeed beause alulus methods an be diretly alied. However, as we ointed out in Chater 2, fousing on derivatives has one major disadvantage for emirial work the size of derivatives deends diretly on how variables are measured. That an make omarisons among goods or aross ountries and time eriods very diffiult. For this reason, most emirial work in miroeonomis uses some form of elastiity measure. In this setion we introdue the three most ommon tyes of demand elastiities and elore some of the methematial relations among them. Again, for simliity, we will look at a situation where the individual hooses between only two goods, though these ideas an be easily generalized. Marshallian demand elastiities Most of the ommonly used demand elastiities are derived from the Marshallian demand funtion (, y, I). Seifially, the following definitions are used: 139 DEFINITION 1. Prie elastiity of demand (e, ): This measures the roortionate hange in quantity demanded in resonse to a roortionate hange in a good s own rie. Mathematially, / e, = D = D = ì. (5.34) D/ D ì 2. Inome elastiity of demand (e,i ): This measures the roortionate hange in quantity demanded in resonse to a roortionate hange in inome. In mathematial terms, / I I e, I = D = D. (5.35) DI/ I D I = ì ì I 3. Cross-rie elastiity of demand (e,y ): This measures the roortionate hange in the quantity of demanded in resonse to a roortionate hange in the rie of some other good (y): / y y e, y = D = D = ì. (5.36) Dy/ y D y ì y Notie that all of these definitions use artial derivatives thereby signifying that all other determinants of demand are to be held onstant when eamining the imat of a seifi variable. In the remainder of this setion we will elore the own rie elastiity definition in some detail. Eamining the ross-rie elastiity of demand is the rimary toi of Chater 6. Prie elastiity of demand The (own) rie elastiity of demand is robably the most imortant elastiity onet in all of miroeonomis. Not only does it rovide a onvenient way of summarizing how eole resond to rie hanges for a wide variety of eonomi goods, but it is also a entral onet in the theory of how firms reat to the demand urves faing them. As you robably already learned in earlier eonomis ourses, a distintion is usually made between ases of elasti demand (where rie affets quantity signifiantly) and inelasti demand (where the effet of rie is small). One mathematial omliation in making these ideas reise is that the rie elastiity of demand itself is negative 7 beause, eet in the 7 Sometimes eonomists use the absolute value of the rie elastiity of demand in their disussions. Although this is mathematially inorret, suh usage is quite ommon. For eamle, a study that finds that e, = 1.2 may sometimes reort the rie elastiity of demand as 1.2.

20 3076-C5.df 3/12/04 10:56 AM Page Part 2 Choie and Demand DEFINITION unlikely ase of Giffen s arado, / is negative. The dividing line between large and small resonses is generally set at 1. If e, = 1, hanges in and are of the same roortionate size. That is, a 1 erent inrease in rie leads to a fall of 1 erent in quantity demanded. In this ase demand is said to be unit-elasi. Alternatively, if e, < 1 quantity hanges are roortionately larger than rie hanges and we say that demand is elasti. For eamle, if e, = 3, eah 1 erent rise in rie leads to a fall of 3 erent in quantity demanded. Finally, if e, > 1, demand is inelasti quantity hanges are roortionately smaller than rie hanges. A value of e, = 0.3, for eamle, means that a 1 erent inrease in rie leads to a fall in quantity demanded of 0.3 erent. In Chaters 10 and 11 we will see how aggregate data are used to estimate the tyial individual s rie elastiity of demand for a good and how suh estimates are used in a variety of questions in alied miroeonomis. Prie elastiity and total sending The rie elastiity of demand determines how a hange in rie, eteris aribus, affets total sending on a good. The onnetion is most easily shown with alulus: ì ( ) = ì + = [ e, + 1]. (5.37) ì ì So, the sign of this derivative deends on whether e, is larger or smaller than 1. If demand is inelasti (0 > e, > 1), the derivative is ositive and rie and total sending move in the same diretion. Intuitively, if rie does not affet quantity demanded very muh, quantity stays relatively onstant as rie hanges and total sending reflets mainly those rie movements. This is the ase, for eamle, for the demand for most agriultural roduts. Weather-indued hanges in rie for seifi ros usually ause total sending on those ros to move in the same diretion. On the other hand, if demand is elasti (e, < 1) reation to a rie hange are so large that the effet on total sending is reversed a rise in rie auses total sending to fall (beause quantity falls a lot) and a fall in rie auses total sending to rise (quantity) inreases signifiantly). For the unit-elasti ase (e, = 1) total sending is onstant no matter how rie hanges. Comensated rie elastiities Beause some miroeonomi analyses fous on the omensated demand funtion, it is also useful to define elastiities based on that onet. Suh definitions follow diretly from their Marshallian ounterarts: If the omensated demand funtion is given by (, y, U) we define: 1. Comensated own rie elastiity of demand (e, ): This elastiity measures the roortionate omensated hange in quantity demanded in resonse to a roortionate hange in a good s own rie: / e, = D = D = ì. (5.38) D / D ì 2. Comensated ross-rie elastiity of demand (e, ): This measures the roortionate omensated hange in quantity demanded in resonse to a roortionate hange in the rie of another good: y / y e, = D = D = ì. y (5.39) Dy/ y D y ì y Whether these rie elastiities differ very muh from their Marshallian ounterarts deends on the imortane of inome omensation in the overall demand for good. The

21 3076-C5.df 3/12/04 10:56 AM Page 141 Chater 5 Inome and Substitution Effets reise onnetion between the two an be shown by multilying the Slutsky result from Equation 5.27 by the fator /: ì ì = e = - ì, = e, - s e, I, (5.40) ì ì ìi where s is the share of total inome devoted to the urhase of good. = I Equation 5.40 therefore shows that omensated and unomensated own rie elastiities of demand will be similar if either of two onditions hold: (1) The share of inome devoted to good (s ) is small; or (2) The inome elastiity of demand for good (e,i ) is small. Either of these onditions serves to redue the imortane of the inome omensation emloyed in the onstrution of the omensated demand funtion. If good is unimortant in a erson s budget, the amount of inome omensation required to offset a rie hange will be small. Even if a good is imortant in the budget, if a erson does not reat very strongly to omensating hanges in inome, results of either demand onet will be similar. Hene, there will be many irumstanes where one an use the two rie elastiity onets more or less interhangeably. Put another way, there are many eonomi irumstanes in whih substitution effets onstitute the most imortant omonent of rie resonses. 141 Relationshis among demand elastiities There are a number of relationshis among the elastiity onets that have been develoed in this setion. All of these are derived from the underlying model of utility maimization. Here we look at three suh relationshis that rovide further insight on the nature of individual demand. Homogeneity. The homogeneity of demand funtions an also be eressed in elastiity terms. Beause any roortional inrease in all ries and inome leaves quantity demanded unhanged, the net sum of all rie elastiities together with the inome elastiity for a artiular good must sum to zero. A formal roof of this roerty relies on Euler s theorem (see Chater 2). Alying that theorem to the demand funtion (, y, I) and remembering that this funtion is homogeneous of degree zero yields 0 = ì + ì + ì y I. ì (5.41) ì y ìi If we simly divide Equation 5.41 by we get 0 = e, + e, y + e, I (5.42) as intuition suggests. This result shows that the elastiities of demand for any good an not follow a omletely fleible attern. They must ehibit a sort of internal onsisteny that reflets the basi utility-maimizing aroah on whih the theory of demand is based. Engel aggregation In the Etensions to Chater 4 we disussed the emirial analysis of market shares and took seial note of Engel s law that the share of inome devoted to food delines as inome inreases. From an elastiity ersetive, Engel s law is a statement of the emirial regularity that the inome elastiity of demand for food is generally found to be onsiderably less than one. Beause of this, it must be the ase that the inome elastiity of all nonfood items must be greater than one. If an individual eerienes an inrease in his or her inome we would eet food eenditures to inrease by a smaller roortional amount, but the inome must be sent somewhere. In the aggregate, these other eenditures must inrease roortionally faster than inome.

22 3076-C5.df 3/12/04 10:56 AM Page Part 2 Choie and Demand A more formal statement of this roerty of inome elastiities an be derived by differentiating the individual s budget onstraint (I = + y y) with reset to inome while treating the ries as onstants: 1 = ì + ì y ìi ìi y. A bit of algebrai maniulation of this eression yields (5.43) 1 = ì (5.44) ì I + ì y ì yi y = se, I + se y y, I, I I I yi where, as before, s i reresents the share of inome sent on good i. Equation 5.44 shows that the weighted average on inome elastiities for all goods that a erson buys must be one. If we knew, say, that a erson sent one-fourth of inome on food and the inome elastiity of demand for food were 0.5, the inome elastiity of demand for everything else must be aroimately 1.17 [= ( )/0.75]. Beause food is an imortant neessity, everything else is in some sense a luury. Cournot aggregation. The eighteenth-entury Frenh eonomist Antoine Cournot rovided one of the first mathematial analyses of rie hanges using alulus. His most imortant disovery was the onet of marginal revenue a onet entral to the rofit-maimization hyothesis for firms. Cournot was also onerned with how the hange in a single rie might affet the demand for all goods. Our final relationshi shows that there is indeed a onnetion among all of the reations to the hange in a single rie. We begin by differentiating the budget onstraint again, this time with reset to, say, : Multiliation of this equation by /I yields so the final Cournot result is ìi ì = = ì ìy y. ì ì 0 = ì ì + y y + y ì I I ì I y 0 = se + s + se, y y,, (5.45) s e, + s y e y, = s. (5.46) This equation shows that the size of the ross-rie effet of a hange in the rie of on the quantity of y onsumed is restrited beause of the budget onstraint. Diret, own rie effets annot be totally overwhelmed by ross-rie effets. This is the first of many onnetions among the demands for goods that we will study more intensively in the net hater. Generalizations. Although we have shown these aggregation results only for the ase of two goods, they are in fat easily generalized to the ase of many goods. You are asked to do just that in Problem 5.9. A more diffiult issue is whether these results should be eeted to hold in tyial eonomi data in whih the demands of many eole are ombined. Often eonomists treat aggregate demand relationshis as desribing the behavior of a tyial erson, and these relationshis should in fat hold for suh a erson. But the situation may not be quite that simle, as we will show when we disuss aggregation later in this book.