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Inverse demand and anti-giffen goods
European Economic Review 27 (1985) 397-404. North-Holland
INVERSE DEMAND AND ANTI-GIFFEN GOODS Ulrich K O H L I * Swiss National Bank, CH-8022 Zurich, Switzerland
Received August 1984, final version received February 1985 Consider the standard model of household theory and assume two goods, one of them a Giffen good. It is well known that the uncompensated direct demand for the Giffen good has a positive slope. However, contrary to common intuition, this note shows that the uncompensated inverse demand for the Giffen good has a negative slope, and it is the uncompensated inverse demand for the other good which is upward sloping. Positively sloped inverse demand curves tend to arise for vastly superior goods which can therefore be called anti-Giffen goods.
Consider 19th century Ireland, and assume two goods, potatoes and meat. Assume furthermore that potatoes are a Giffen good; meat, therefore, is a superior good. Assume an outbreak of foot and m o u t h disease resulting in a sharp decline in the aggregate supply of meat, but leaving the supply of potatoes untouched. H o w does this affect the price of meat? As it turns out, the increased scarcity of meat leads to a fall in its price.1 The question of the price effects of exogenous changes in quantities of goods can best be examined in the context of inverse demand analysis. This note draws heavily on recent contributions on this subject by Anderson (1980) and W e y m a r k (1980). Inverse demands express prices as functions of quantities. Assume two goods, and let the uncompensated direct (i.e., Marshallian) demand functions be given by
xi=fi(pl,p2,m),
i = 1,2,
(1)
where the x's denote quantities of goods, the p's are prices, and m is nominal
*I am grateful to P.E. Graves, W.E. Diewert, W.H. Kaempfer, E.R. Morey and the referees for useful comments and suggestions, but naturally I am solely responsible for any errors or omissions. ~Alternatively, if Ireland is viewed as an open economy, one would observe an increase in net exports of meat, and a reduction in net exports of potatoes.
0014-2921/85/$3.30 9 1985, Elsevier Science Publishers B.V. (North-Holland) E.E.R.-- F
398
U. Kohli, Inverse demand and anti-Giffen goods
income. The uncompensated inverse demand functions are defined as follows: 2
pi=gi(xl, x2, m),
i = 1,2.
(2)
This is obviously not to be confused with the inverse of the Marshallian demand functions, where only one price at a time is allowed to vary, p~ = hi(xi, p j, m),
i,j = 1, 2,
i ~j.
(3)
Most authors when discussing inverse demand do so in the one-good context, in which case the distinction between (2) and (3) vanishes? Note that for any good the slope of the inverted Marshallian demand function (3) is simply the inverse of the slope of the direct Marshallian demand function (1). This is generally not true for inverse demand functions; the calculation of the slope properties of (2) from knowledge of (1) requires the inversion of a Jacobian matrix. 4 Inverse demand functions such as (2) are useful to determine the set of prices which would induce a household with a given nominal income to consume a certain bundle of goods. If (1) is viewed as a system of market demand functions, then (2) can be used to determine the set of market equilibrium prices in a pure exchange economy for given endowments of goods and given nominal income. Indeed, Pearce (1964) refers to the p's as planning prices. It is well known from standard household theory that the uncompensated direct demand for a Giffen good has a positive slope, 5 while, in the two-good case, the demand for the other good is downward sloping. But what about the uncompensated inverse demands? Uncompensated inverse demand schedules do normally have negative slopes, 6 but one might expect the inverse demand for a Giffen good to be upward sloping. This note shows that this conjecture is incorrect: the uncompensated inverse demand for a Giffen good is downward sloping, and it is the uncompensated inverse demand for the other good which has a positive slope. Let e * = [ e * ] be the matrix of compensated price elasticities of direct demand, i.e., e* = 0 In d~(p, U)/O In p j,
i,j = 1, 2,
(4)
2See Henderson and Quandt (1971, p. 231), for instance. We use the terminology of Anderson (1980). Diewert (1980), and Weymark (1980). Other authors have used the terms reciprocal demand [Theil (1976)], and anti-demand I-Pearce(1964)]. 3See Varian (1978), and Henderson and Quandt (1971, p. 134) for instance. 4See Anderson (1980). One case where the slopes of (2) and (3) are identical is when all uncompensated cross price effectsare zero. SFor a history of the Giffen-goodparadox, see Stigler (1947). 6See Anderson (1980), for instance.
U. Kohli,Inoersedemandand anti-Giffengoods
399
where di(') is the compensated direct (i.e., Hicksian) demand function and U is utility; e* < 0 if the utility function is well behaved. Price homogeneity implies 7 ~e*=0, J
i,j= 1,2,
(5)
and it follows from symmetry that * -e12w1/w2, - * e21
>0, wi_
(6)
where w~ is the budget share of good i. Adding-up obviously implies Wl + w2 = 1.
(7)
Let r/=[rh] be the vector of income elasticities of direct demand. Engel aggregation requires rhw 1 + r/2w2 = 1.
(8)
We define e = l-e~j] as the matrix of uncompensated price elasticities of direct demand, i.e., eii= a In fi(p, m)/a In pp
(9)
eij can be obtained as follows (Slutsky equation in elasticity form): % = ~ - ~wj.
(10)
Taking (5)-(8) and (10) into account, e can be written as
1
-
e=lLE_e,i wi_wl(l_wlrll)S/(l_wl)
[e, Wl_(l_wl)(l_Wlrll)]/(l_wl) (11)
Let ~ = [~j] be the matrix of uncompensated quantity elasticities of inverse demand, i.e.,
~o=c~lng~(x,m)/c~lnxj,
i,j= 1,2.
7Relations (5)-(8) are well known; see Phlips (1974), for instance.
(12)
400
U. Kohli, Inverse demand and anti-Giffen goods
c5 can be obtained by inversion of e,s and it is therefore equal to 6
[ - - wl + (1 - w~)(1 - w, th)/e* l /
L-w~[l +(l-w~rh)le*~]
- ( 1 - wl) - ( 1 - w~)2rh/e*l]
(1-wx)(-l +wx~hle*O
I
_]" (13)
If we now assume that good 1 is a Giffen good, we have w~r/l<8*~ <0.
(14)
It follows that
~II =
- - W l + ( 1 - - Wl)(1 - - W11~1)/~1 < 0 ,
622 =(1 - w~)(- 1 + wx r/i/e*1) > 0,
(15) (16)
hence the uncompensated inverse demand for the Giffen good is downward sloping, while the uncompensated inverse demand for the other good has a positive slope. The slopes of the uncompensated direct and inverse demand curves can also be obtained graphically. Fig. 1 depicts two indifference curves in the (potatoes, meat) space. Initial relative prices are given by p0, and equilibrium obtains at X ~ An increase in the price of potatoes rotates the price line inwards, and equilibrium moves to X 1. The effect of the price change on the uncompensated direct demand for potatoes is given by the horizontal distance between X ~ and X 1. It is clear from fig. 1 that the rise in the price of potatoes results in an increase in the quantity demanded; this shows that potatoes are indeed a Giffen good. The reader can verify than an increase in the price of meat results in a fall in the demand for meat, thus revealing a downward sloping uncompensated direct demand schedule. We now examine the effects of changes in the quantities of goods available for consumption. A decrease in the quantity of potatoes, from X ~ to X 2, would lead to an increase in the relative price of potatoes, from p0 to in excess of P~; this indicates that the uncompensated inverse demand for a Giffen good is negatively sloped. Consider, on the other hand, a decrease in the consumption of meat, say from X ~ to X 3. Such a change requires an increase in the relative price of potatoes, from po to somewhat more than Pt, i.e., a fall in the relative price of meat. Moreover, the vertical intercept of the price line clearly moves upwards; hence, for given nominal income, the fall in the quantity of meat results in an absolute decrease in the price of meat. This
SSee Anderson (1980); this result follows from the definition of ~. We have indeed, d l n x = e dln p + rl dln m. Hence, dln p = e- l dln x - e- lrl dln m.
U. Kohli, Inverse demand and anti-Giffen goods
401
p0
..4.-I o~ tn~
I JX3
potatoes Fig. 1
reveals a positively sloped uncompensated inverse demand curve. An exogenous decrease in the aggregate supply of potatoes, as it occured in Ireland between 1845 and 1849, leads to an increase in the price of potatoes. 9 In this respect, Giffen goods behave no differently from most other goods. Meat is different, however, because the negative income effect resulting from a reduction in the available supply leads to a decline in demand sharp enough to generate an excess supply of meat and hence a fall in its relative price; actually, the effect is so strong as to cause an absolute fall in the price of meat. In spite of all the attention devoted to Giffen goods in the microeconomic literature, this property of the potatoes/meat model has gone unnoticed because attention is generally focused on potatoes, yet it is meat that behaves abnormally when it comes to inverse demand. It is tempting to use the term anti-Giffen to designate goods with an upward sloping uncompensated inverse demand. As suggested by our potatoes/meat model, anti-Giffen goods are not inferior goods; on the contrary, they are extremely superior goods. The above argument shows that a Giffen good is always accompanied by an anti-Giffen good in the twodimensional case. 9Dwyer and Lindsay (1984) incorrectly argue that a reduction in the supply of potatoes in a closed economy would lead to a drop in the price of potatoes under the Giffen-good hypothesis. Their mistake is to confuse (2) and (3). Nevertheless, their main conclusion, that potatoes were unlikely to be a Giffen good during the famine, seems to he supported by other evidence.
U. Kohli, Inverse demand and anti-Giffen goods
402
Hirshleifer (1980) uses the term ultra-superiorto designate, in the two-good case, the partner of an inferior good. Anti-Giffen goods are obviously ultrasuperior in the two-dimensional case, but the reverse is not true. A good that is ultra-superior without being anti-Giffen has a negatively sloped uncompensated inverse demand: a reduction in its supply indeed results in a fall in its relative price, but its absolute price nevertheless increases. The question of the slope of uncompensated inverse demand curves is closely related to the concept of the scale elasticity of inverse demand. Scale elasticities indicate how prices move when the quantities of all goods consumed are changed equiproportionately, i.e., when the consumption mix is held constant, but the level (or the scale) of consumption is altered, /1~, the scale elasticity of good i, is defined as follows:
i.ti=Olngi(kxx,kx2,m)/Olnk,
i= 1,2,
(17)
where k is a positive scalar. For given nominal income one would expect the prices of most goods to fall as a result of an increase in the scale of consumption, i.e., scale elasticities tend to be negative. Indeed, one can easily show that x~
~wi#i=-l,
i=1,2,
(18)
where w~ is again the budget share of good i. Naturally (18) does not rule out the possibility #~> 0 for some i. Let 6*= [6*] be the matrix of compensated quantity elasticities of inverse demand. 6* indicates the price effect of a change in consumption mix as one moves along an indifference curve. The compensated and the uncompensated inverse price elasticities are related in the following way: 1~ 6~i= 6 " + wlpi,
i , j = 1,2.
(19)
(19) is known as the Antonelli equation, and it is the dual of the Slutsky equation. The law of inverse demand implies that 6* < 0 (i= 1, 2), but it is visible from (19) that 6u<>0, although if #i<0, 6, is necessarily negative. AntiGiffen goods are goods for which 6 , > 0 ; this obviously requires a scale elasticity greater than zero. x2 1~ (1980). (18) is the dual of Engel aggregation (8) above. It is clear from (18) that the role of k in scaling the consumption vector is similar to the role of nominal income in scaling prices. 11Again, this is demonstrated in Anderson (1980). 12One can easily show that: p l = - 1 +(1--wl)(1-qi)/e*. The reader can verify that Pl < 0 and # z > 0 if (14) holds. One could use the term anti-inferior to designate goods with positive scale elasticities. An anti-Giffen good is necessarily anti-inferior, but the reverse is not true. Note also that an ultra-superior good is not necessarily anti-inferior, i.e., the scale elasticity of an ultrasuperior good can be of either sign. The concept of ultra-superiority, therefore, does not seem very useful for inverse demand analysis.
U. Kohli, Inverse demand and anti-Giffen goods
403
The meaning of the Antonelli equation in the presence of an anti-Giffen good can be illustrated graphically with the help of fig. 2 which reproduces the two indifference curves of fig. 1. Let the original consumption point be at X ~ and consider a reduction in the scale of consumption that shifts the consumption point from X ~ to X1; note that the consumption mix is kept unchanged. The reduction in the scale of consumption leads to an increase in the relative price of potatoes, from pO to p1. Furthermore it is visible from fig. 2 that the vertical intercept of the price line moves upwards, while the horizontal intercept moves to the left. Hence for given nominal income, the price of meat falls and the price of potatoes increases. This implies that the scale elasticity of the demand for meat is positive, while the scale elasticity of the demand for potatoes has the usual negative sign. Fig. 2 also shows that the total effect of a decrease in the supply of meat (that is the movement from po to pz) can be decomposed into a scale effect, from pO to P1, and into a pure substitution effect, from P* to pa. This is precisely the meaning of the Antonelli equation. It is apparent from fig. 2 that the scale effect dominates the substitution effect in the case of meat: meat is an anti-Giffen good. The discussion so far has been confined to the two-dimensional case. How much of it generalizes to higher dimensions? Definitions (1)-(4), (9), (12) and (17) can be extended for an arbitrary number of goods, and (18) and (19) still
p1
potatoes Fig. 2
404
U. Kohli, Inverse demand and anti-Giffen goods
hold in the general case. It is clear that some goods can have positive scale elasticities in a multi-dimensional setting, and that the scale effect can dominate the pure substitution effect in some cases: the possibility of antiGiffen goods arises even if the number of goods exceeds two. What does not hold in general, however, is the result that a Giffen good is necessarily accompanied by an anti-Giffen good (and vice-versa). This is always true when the number of goods is equal to two, but it is a result that is particular to the bi-dimensional case. 13 Indeed, a Giffen good must always be accompanied by at least one superior good, but it is easy to imagine cases - - when the number of goods exceeds two - - where a Giffen good is accompanied by several superior goods, without any one of them superior enough (or with a large enough budget share) to be anti-Giffen. Given the preponderance of the two-dimensional framework, it nevertheless seemed worthwhile pointing out this peculiarity of the Giffen-good model. More important, though, is the fact, which seems to have gone previously unnoticed, that some goods can have positively sloped uncompensated inverse demand schedules, and that, far from being inferior, they tend to be vastly superior goods. References Anderson, R.W., 1980, Some theory of inverse demand for applied demand analysis, European Economic Review 14, 281-290. Diewert, W.E., 1980, Duality approaches to microeconomic theory, in: K.F. Arrow and M.D. Intriligator, eds., Handbook of mathematical economics, vol. 2 (North-Holland, Anlsterdam). Dwyer, G,P., Jr. and C.M. Lindsay, 1984, Robert Giffen and the Irish potato, American Economic Review 74, 188-192. Henderson, J.M. and R.E. Quandt, 1971, Macroeconomic theory, 2nd ed. (McGraw-Hill, New York). Hirshleifer, J., 1980, Price theory and applications (Prentice-Hall, Englewood Cliffs, NJ). Pearce, I.F., 1964, A contribution to demand analysis (Oxford University Press, Oxford). Phlips, L., 1974, Applied consumption analysis (North-Holland, Amsterdam). Stigler, G.J., 1947, Notes on the history of the Giffen paradox, Journal of Political Economy 55, 152-156. Theft, H., 1976, Theory and measurement of consumer demand, vol. 2 (North-Holland, Amsterdam). Varian, H., 1978, Microeconomic analysis (Norton, New York). Weymark, J.A., 1980, Duality results in demand theory, European Economic Review 14, 377395.
a3One case, of course, where the two-dimensional framework is relevant even in the presence of many goods is when the utility function is weakly separable between two groups of goods.
INVERSE DEMAND AND ANTI-GIFFEN GOODS Ulrich K O H L I * Swiss National Bank, CH-8022 Zurich, Switzerland
Received August 1984, final version received February 1985 Consider the standard model of household theory and assume two goods, one of them a Giffen good. It is well known that the uncompensated direct demand for the Giffen good has a positive slope. However, contrary to common intuition, this note shows that the uncompensated inverse demand for the Giffen good has a negative slope, and it is the uncompensated inverse demand for the other good which is upward sloping. Positively sloped inverse demand curves tend to arise for vastly superior goods which can therefore be called anti-Giffen goods.
Consider 19th century Ireland, and assume two goods, potatoes and meat. Assume furthermore that potatoes are a Giffen good; meat, therefore, is a superior good. Assume an outbreak of foot and m o u t h disease resulting in a sharp decline in the aggregate supply of meat, but leaving the supply of potatoes untouched. H o w does this affect the price of meat? As it turns out, the increased scarcity of meat leads to a fall in its price.1 The question of the price effects of exogenous changes in quantities of goods can best be examined in the context of inverse demand analysis. This note draws heavily on recent contributions on this subject by Anderson (1980) and W e y m a r k (1980). Inverse demands express prices as functions of quantities. Assume two goods, and let the uncompensated direct (i.e., Marshallian) demand functions be given by
xi=fi(pl,p2,m),
i = 1,2,
(1)
where the x's denote quantities of goods, the p's are prices, and m is nominal
*I am grateful to P.E. Graves, W.E. Diewert, W.H. Kaempfer, E.R. Morey and the referees for useful comments and suggestions, but naturally I am solely responsible for any errors or omissions. ~Alternatively, if Ireland is viewed as an open economy, one would observe an increase in net exports of meat, and a reduction in net exports of potatoes.
0014-2921/85/$3.30 9 1985, Elsevier Science Publishers B.V. (North-Holland) E.E.R.-- F
398
U. Kohli, Inverse demand and anti-Giffen goods
income. The uncompensated inverse demand functions are defined as follows: 2
pi=gi(xl, x2, m),
i = 1,2.
(2)
This is obviously not to be confused with the inverse of the Marshallian demand functions, where only one price at a time is allowed to vary, p~ = hi(xi, p j, m),
i,j = 1, 2,
i ~j.
(3)
Most authors when discussing inverse demand do so in the one-good context, in which case the distinction between (2) and (3) vanishes? Note that for any good the slope of the inverted Marshallian demand function (3) is simply the inverse of the slope of the direct Marshallian demand function (1). This is generally not true for inverse demand functions; the calculation of the slope properties of (2) from knowledge of (1) requires the inversion of a Jacobian matrix. 4 Inverse demand functions such as (2) are useful to determine the set of prices which would induce a household with a given nominal income to consume a certain bundle of goods. If (1) is viewed as a system of market demand functions, then (2) can be used to determine the set of market equilibrium prices in a pure exchange economy for given endowments of goods and given nominal income. Indeed, Pearce (1964) refers to the p's as planning prices. It is well known from standard household theory that the uncompensated direct demand for a Giffen good has a positive slope, 5 while, in the two-good case, the demand for the other good is downward sloping. But what about the uncompensated inverse demands? Uncompensated inverse demand schedules do normally have negative slopes, 6 but one might expect the inverse demand for a Giffen good to be upward sloping. This note shows that this conjecture is incorrect: the uncompensated inverse demand for a Giffen good is downward sloping, and it is the uncompensated inverse demand for the other good which has a positive slope. Let e * = [ e * ] be the matrix of compensated price elasticities of direct demand, i.e., e* = 0 In d~(p, U)/O In p j,
i,j = 1, 2,
(4)
2See Henderson and Quandt (1971, p. 231), for instance. We use the terminology of Anderson (1980). Diewert (1980), and Weymark (1980). Other authors have used the terms reciprocal demand [Theil (1976)], and anti-demand I-Pearce(1964)]. 3See Varian (1978), and Henderson and Quandt (1971, p. 134) for instance. 4See Anderson (1980). One case where the slopes of (2) and (3) are identical is when all uncompensated cross price effectsare zero. SFor a history of the Giffen-goodparadox, see Stigler (1947). 6See Anderson (1980), for instance.
U. Kohli,Inoersedemandand anti-Giffengoods
399
where di(') is the compensated direct (i.e., Hicksian) demand function and U is utility; e* < 0 if the utility function is well behaved. Price homogeneity implies 7 ~e*=0, J
i,j= 1,2,
(5)
and it follows from symmetry that * -e12w1/w2, - * e21
>0, wi_
(6)
where w~ is the budget share of good i. Adding-up obviously implies Wl + w2 = 1.
(7)
Let r/=[rh] be the vector of income elasticities of direct demand. Engel aggregation requires rhw 1 + r/2w2 = 1.
(8)
We define e = l-e~j] as the matrix of uncompensated price elasticities of direct demand, i.e., eii= a In fi(p, m)/a In pp
(9)
eij can be obtained as follows (Slutsky equation in elasticity form): % = ~ - ~wj.
(10)
Taking (5)-(8) and (10) into account, e can be written as
1
-
e=lLE_e,i wi_wl(l_wlrll)S/(l_wl)
[e, Wl_(l_wl)(l_Wlrll)]/(l_wl) (11)
Let ~ = [~j] be the matrix of uncompensated quantity elasticities of inverse demand, i.e.,
~o=c~lng~(x,m)/c~lnxj,
i,j= 1,2.
7Relations (5)-(8) are well known; see Phlips (1974), for instance.
(12)
400
U. Kohli, Inverse demand and anti-Giffen goods
c5 can be obtained by inversion of e,s and it is therefore equal to 6
[ - - wl + (1 - w~)(1 - w, th)/e* l /
L-w~[l +(l-w~rh)le*~]
- ( 1 - wl) - ( 1 - w~)2rh/e*l]
(1-wx)(-l +wx~hle*O
I
_]" (13)
If we now assume that good 1 is a Giffen good, we have w~r/l<8*~ <0.
(14)
It follows that
~II =
- - W l + ( 1 - - Wl)(1 - - W11~1)/~1 < 0 ,
622 =(1 - w~)(- 1 + wx r/i/e*1) > 0,
(15) (16)
hence the uncompensated inverse demand for the Giffen good is downward sloping, while the uncompensated inverse demand for the other good has a positive slope. The slopes of the uncompensated direct and inverse demand curves can also be obtained graphically. Fig. 1 depicts two indifference curves in the (potatoes, meat) space. Initial relative prices are given by p0, and equilibrium obtains at X ~ An increase in the price of potatoes rotates the price line inwards, and equilibrium moves to X 1. The effect of the price change on the uncompensated direct demand for potatoes is given by the horizontal distance between X ~ and X 1. It is clear from fig. 1 that the rise in the price of potatoes results in an increase in the quantity demanded; this shows that potatoes are indeed a Giffen good. The reader can verify than an increase in the price of meat results in a fall in the demand for meat, thus revealing a downward sloping uncompensated direct demand schedule. We now examine the effects of changes in the quantities of goods available for consumption. A decrease in the quantity of potatoes, from X ~ to X 2, would lead to an increase in the relative price of potatoes, from p0 to in excess of P~; this indicates that the uncompensated inverse demand for a Giffen good is negatively sloped. Consider, on the other hand, a decrease in the consumption of meat, say from X ~ to X 3. Such a change requires an increase in the relative price of potatoes, from po to somewhat more than Pt, i.e., a fall in the relative price of meat. Moreover, the vertical intercept of the price line clearly moves upwards; hence, for given nominal income, the fall in the quantity of meat results in an absolute decrease in the price of meat. This
SSee Anderson (1980); this result follows from the definition of ~. We have indeed, d l n x = e dln p + rl dln m. Hence, dln p = e- l dln x - e- lrl dln m.
U. Kohli, Inverse demand and anti-Giffen goods
401
p0
..4.-I o~ tn~
I JX3
potatoes Fig. 1
reveals a positively sloped uncompensated inverse demand curve. An exogenous decrease in the aggregate supply of potatoes, as it occured in Ireland between 1845 and 1849, leads to an increase in the price of potatoes. 9 In this respect, Giffen goods behave no differently from most other goods. Meat is different, however, because the negative income effect resulting from a reduction in the available supply leads to a decline in demand sharp enough to generate an excess supply of meat and hence a fall in its relative price; actually, the effect is so strong as to cause an absolute fall in the price of meat. In spite of all the attention devoted to Giffen goods in the microeconomic literature, this property of the potatoes/meat model has gone unnoticed because attention is generally focused on potatoes, yet it is meat that behaves abnormally when it comes to inverse demand. It is tempting to use the term anti-Giffen to designate goods with an upward sloping uncompensated inverse demand. As suggested by our potatoes/meat model, anti-Giffen goods are not inferior goods; on the contrary, they are extremely superior goods. The above argument shows that a Giffen good is always accompanied by an anti-Giffen good in the twodimensional case. 9Dwyer and Lindsay (1984) incorrectly argue that a reduction in the supply of potatoes in a closed economy would lead to a drop in the price of potatoes under the Giffen-good hypothesis. Their mistake is to confuse (2) and (3). Nevertheless, their main conclusion, that potatoes were unlikely to be a Giffen good during the famine, seems to he supported by other evidence.
U. Kohli, Inverse demand and anti-Giffen goods
402
Hirshleifer (1980) uses the term ultra-superiorto designate, in the two-good case, the partner of an inferior good. Anti-Giffen goods are obviously ultrasuperior in the two-dimensional case, but the reverse is not true. A good that is ultra-superior without being anti-Giffen has a negatively sloped uncompensated inverse demand: a reduction in its supply indeed results in a fall in its relative price, but its absolute price nevertheless increases. The question of the slope of uncompensated inverse demand curves is closely related to the concept of the scale elasticity of inverse demand. Scale elasticities indicate how prices move when the quantities of all goods consumed are changed equiproportionately, i.e., when the consumption mix is held constant, but the level (or the scale) of consumption is altered, /1~, the scale elasticity of good i, is defined as follows:
i.ti=Olngi(kxx,kx2,m)/Olnk,
i= 1,2,
(17)
where k is a positive scalar. For given nominal income one would expect the prices of most goods to fall as a result of an increase in the scale of consumption, i.e., scale elasticities tend to be negative. Indeed, one can easily show that x~
~wi#i=-l,
i=1,2,
(18)
where w~ is again the budget share of good i. Naturally (18) does not rule out the possibility #~> 0 for some i. Let 6*= [6*] be the matrix of compensated quantity elasticities of inverse demand. 6* indicates the price effect of a change in consumption mix as one moves along an indifference curve. The compensated and the uncompensated inverse price elasticities are related in the following way: 1~ 6~i= 6 " + wlpi,
i , j = 1,2.
(19)
(19) is known as the Antonelli equation, and it is the dual of the Slutsky equation. The law of inverse demand implies that 6* < 0 (i= 1, 2), but it is visible from (19) that 6u<>0, although if #i<0, 6, is necessarily negative. AntiGiffen goods are goods for which 6 , > 0 ; this obviously requires a scale elasticity greater than zero. x2 1~ (1980). (18) is the dual of Engel aggregation (8) above. It is clear from (18) that the role of k in scaling the consumption vector is similar to the role of nominal income in scaling prices. 11Again, this is demonstrated in Anderson (1980). 12One can easily show that: p l = - 1 +(1--wl)(1-qi)/e*. The reader can verify that Pl < 0 and # z > 0 if (14) holds. One could use the term anti-inferior to designate goods with positive scale elasticities. An anti-Giffen good is necessarily anti-inferior, but the reverse is not true. Note also that an ultra-superior good is not necessarily anti-inferior, i.e., the scale elasticity of an ultrasuperior good can be of either sign. The concept of ultra-superiority, therefore, does not seem very useful for inverse demand analysis.
U. Kohli, Inverse demand and anti-Giffen goods
403
The meaning of the Antonelli equation in the presence of an anti-Giffen good can be illustrated graphically with the help of fig. 2 which reproduces the two indifference curves of fig. 1. Let the original consumption point be at X ~ and consider a reduction in the scale of consumption that shifts the consumption point from X ~ to X1; note that the consumption mix is kept unchanged. The reduction in the scale of consumption leads to an increase in the relative price of potatoes, from pO to p1. Furthermore it is visible from fig. 2 that the vertical intercept of the price line moves upwards, while the horizontal intercept moves to the left. Hence for given nominal income, the price of meat falls and the price of potatoes increases. This implies that the scale elasticity of the demand for meat is positive, while the scale elasticity of the demand for potatoes has the usual negative sign. Fig. 2 also shows that the total effect of a decrease in the supply of meat (that is the movement from po to pz) can be decomposed into a scale effect, from pO to P1, and into a pure substitution effect, from P* to pa. This is precisely the meaning of the Antonelli equation. It is apparent from fig. 2 that the scale effect dominates the substitution effect in the case of meat: meat is an anti-Giffen good. The discussion so far has been confined to the two-dimensional case. How much of it generalizes to higher dimensions? Definitions (1)-(4), (9), (12) and (17) can be extended for an arbitrary number of goods, and (18) and (19) still
p1
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hold in the general case. It is clear that some goods can have positive scale elasticities in a multi-dimensional setting, and that the scale effect can dominate the pure substitution effect in some cases: the possibility of antiGiffen goods arises even if the number of goods exceeds two. What does not hold in general, however, is the result that a Giffen good is necessarily accompanied by an anti-Giffen good (and vice-versa). This is always true when the number of goods is equal to two, but it is a result that is particular to the bi-dimensional case. 13 Indeed, a Giffen good must always be accompanied by at least one superior good, but it is easy to imagine cases - - when the number of goods exceeds two - - where a Giffen good is accompanied by several superior goods, without any one of them superior enough (or with a large enough budget share) to be anti-Giffen. Given the preponderance of the two-dimensional framework, it nevertheless seemed worthwhile pointing out this peculiarity of the Giffen-good model. More important, though, is the fact, which seems to have gone previously unnoticed, that some goods can have positively sloped uncompensated inverse demand schedules, and that, far from being inferior, they tend to be vastly superior goods. References Anderson, R.W., 1980, Some theory of inverse demand for applied demand analysis, European Economic Review 14, 281-290. Diewert, W.E., 1980, Duality approaches to microeconomic theory, in: K.F. Arrow and M.D. Intriligator, eds., Handbook of mathematical economics, vol. 2 (North-Holland, Anlsterdam). Dwyer, G,P., Jr. and C.M. Lindsay, 1984, Robert Giffen and the Irish potato, American Economic Review 74, 188-192. Henderson, J.M. and R.E. Quandt, 1971, Macroeconomic theory, 2nd ed. (McGraw-Hill, New York). Hirshleifer, J., 1980, Price theory and applications (Prentice-Hall, Englewood Cliffs, NJ). Pearce, I.F., 1964, A contribution to demand analysis (Oxford University Press, Oxford). Phlips, L., 1974, Applied consumption analysis (North-Holland, Amsterdam). Stigler, G.J., 1947, Notes on the history of the Giffen paradox, Journal of Political Economy 55, 152-156. Theft, H., 1976, Theory and measurement of consumer demand, vol. 2 (North-Holland, Amsterdam). Varian, H., 1978, Microeconomic analysis (Norton, New York). Weymark, J.A., 1980, Duality results in demand theory, European Economic Review 14, 377395.
a3One case, of course, where the two-dimensional framework is relevant even in the presence of many goods is when the utility function is weakly separable between two groups of goods.