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Using corrective taxes to remedy consumer misperceptions
Journal
of Public
USING
Economics
28 (1985) 85-94.
North-Holland
CORRECTIVE TAXES TO REMEDY MISPERCEPTIONS
CONSUMER
Amihai GLAZER* School
ofSocialSciences,
University of California, Irvine CA 92717, U.S.A.
Received
1983, revised version
January
received
April 1985
When consumers are misinformed about the amount of characteristics obtained from some good, government may attempt to correct the misperceptions by imposing taxes or subsidies on these goods. Such a policy, however, faces difliculties: only with separable utility functions does a single tax suffke, and this tax should not induce consumers to purchase the same bundle they would under perfect information.
1. Introduction
Consumers, alas, are not always perfectly informed about the characteristics of goods they purchase. A buyer may overestimate the reliability of an appliance; he may underestimate the effectiveness of airbags in automobiles; misleading advertising may convince him that some cereal is more nutritious than it actually is. Government may therefore be called upon to improve consumers’ consumption choices. Two such policies come to mind: government can educate consumers, providing them with more accurate information about the goods; it can levy taxes, thereby inducing consumers to make the proper consumption choices in spite of their misperceptions. This paper focuses on analyzing such corrective taxes. The inspiration for the present work comes from Atkinson (1973) and Harris (1980) who examine the problem of consumer misconceptions concerning the hazards of cigarette smoking. In his seminal paper, Atkinson points out a major difficulty in analyzing such problems: it is unclear whether social welfare should be evaluated on the basis of ex ante or of ex post consumer utility. In this paper I follow the example of Harris and consider only ex post utility - the utility a consumer obtains after having consumed a particular bundle of goods. Using this assumption, Harris considers a model in which each consumer decides how many cigarettes to smoke and also chooses among *I am grateful to Duran Bell, Jossi Berechman, Ivor Pearce, and an anonymous their comments and suggestions. All remaining errors are, of course, my own.
0047-2727/85/!§3.30
0
1985, Elsevier Science Publishers
B.V. (North-Holland)
referee for
86
A. Glazer, Using corrective
taxes
brands of cigarettes distinguished by different tar and nicotine contents. If not all consumers are identical, and if consumers must choose among two variables - quantity of cigarettes and their tar and nicotine content - then it is clear that the imposition of a single tax cannot lead all consumers to make consumption choices identical to those they would make with perfect information. In a different context, Green and Sheshinski (1976) reach similar conclusions about the limitations of corrective taxes. Studying the problem of consumption externalities, they find that if not all consumers are identical, the optimal corrective tax policy may require that taxes be imposed on several goods, and not only on the good that causes the externality. My analysis, which yields similar conclusions, relies on the Lancasterian approach to consumer demand [see Auld (1972), Colantoni et al. (1976), Lancaster (1966)]. I extend previous work by considering more than two goods and more than two characteristics, and by introducing into the model issues of taxation. Although my results bear some resemblance to the Theory of the Second Best [see Lipsey and Lancaster (1956)], that theory cannot be directly applied to the problem discussed here. The literature on the Second Best considers price distortions, and not informational problems. In addition, the analysis here applies to a single distortion - a consumer’s misperception concerning the quantity of one characteristic contained in one good - and not to multiple distortions. The next section presents a graphical exposition of the problem. Section 3 presents a more general analysis. Section 4 considers corrective policy which taxes only the one good subject to misperceptions. 2. A two-goods model A good understanding of the problem is gained by considering the simple case in which there are only two goods, each of which yields only one characteristic. Suppose all consumers are identical, and let a consumer’s budget line for the two goods be line QzQ, in fig. 1. Let each unit of good 1 yield one unit of characteristic one, and let each unit of good 2 yield one unit of characteristic 2; thus the line QzQl also represents the consumer’s budget line in terms of characteristics. The consumer obtains utility from characteristics, and not directly from goods; he chooses to purchase the bundle of goods that yields a utility-maximizing bundle of characteristics. Given budget line Q,Q,, the consumer will purchase b, units of good 1 and b, units of good 2, obtaining the characteristics bundle shown by point B in fig. 1. Suppose next that the consumer overestimates the quantity of characteristic 2 obtainable from good 2, so that he believes his consumption possibility frontier (in terms of characteristics) to be line Q;Q1, The
A. Glazer, Using corrective taxes
Fig. 1. Misperceptions
cause underconsumption
87
of good.
consumer now wishes to consume characteristics bundle F’, and he therefore purchases b; units of good 1 and b; units of good 2. He thus in fact obtains characteristics bundle B’, which lies on a lower indifference curve than does bundle B. The consumer could be induced to purchase bundle B by the imposition of a subsidy on good 2 coupled with a lump-sum reduction of his income.’ This is shown in fig. 2. Let the tax shift the consumer’s budget line in terms of goods from line Qz QI to line T2TI . With this budget line and this overestimate of the amount of characteristic 2 contained in good 2, the ‘Equivalently, a tax could be imposed on good income. I assume that both characteristics are normal
1, coupled ones.
with
a lump-sum
increase
in
A. Glazer, Using
correctivetaxes
Fig. 2. Corrective
taxation.
consumer perceives his budget line in terms of characteristics to be line T; Tr.’ The consumer wishes to consume characteristics bundle G’, and he therefore purchases goods bundle B, the same bundle he would have purchased had he faced no taxes and suffered from no misperceptions. To be more concrete, suppose the consumer overestimates the amount of vitamins contained in cereals and that he has some target level of vitamin intake. This overestimate may lead him to consume less cereal than he would under perfect information. On the other hand, the overestimate leads the consumer to view the effective price of vitamins to be quite low; if his demand for vitamins is sufficiently elastic, his misperception would cause him ‘1 have drawn fig. 2 such that the ratio of the segments OQ;/OQ, in fig. 1; that is, the extent of consumer misperceptions
OTz/OT, is equal to the ratio is identical in the two cases.
A. Glazer, Using corrective
taxes
89
to consume too much, rather than too little cereal. This means that the corrective tax policy may require that the relative price of the good subject to misperceptions be either decreased or increased. 3. Corrective taxes in a general model This section extends the model presented above by considering a world with more than two goods and characteristics. Such a generalization introduces a major new element into the discussion. Can a corrective tax policy be designed that requires taxing only the good subject to misperceptions, or must taxes instead be imposed on other goods as well? In the two goods case, of course, such an issue cannot arise. A tax on both goods is equivalent to a tax on only one good and a lump-sum income transfer, so that it makes no sense to ask how many goods must be taxed or subsidized. Moreover, in the two goods case, any policy that reduces the consumer’s overconsumption of some good will necessarily increase his welfare. That may not be so if the consumer buys more than two goods. In the most general Lancasterian model, the number of goods on the market bears no necessary relation to the number of characteristics entering a utility function; which goods are actually purchased depends on the prices in effect and on the particular technology relating goods to characteristics. It greatly simplifies matters, however, to focus on the quantities of goods purchased rather than on which goods are purchased. Analysis is also simplified if the number of goods is at least as great as the number of characteristics. I shall therefore henceforth assume that there are n goods, purchased in quantities xi,. . . ,x,, and n characteristics, consumed in quantities ci, . . . , c,. Let the price of good i be pi. For brevity, denote these lists by the (1 x n) vectors X, c, and p. The consumer’s income is denoted by M. The true technological relation between goods and characteristics is given by the (n x n) matrix B*,each of whose elements, b& gives the amount of characteristic j obtainable from good i. The matrix fi represents the consumer’s erroneous estimates of the technological relations: he believes that by purchasing the goods bundle x he obtains the characteristics bundle S=xB. The consumer’s problem is then to: maximize u(xB) s.t.
(1) xp’ 5 M.
where a prime denotes transposition.
Substitute x=&l
in expression (1) to
90
A. Glazer, Using corrective
taxes
write the budget constraint in the more useful form of &l~‘sM. The vector s^- &- ‘p’ can thus be thought of as the shadow prices of the characteristics. Note that a single informational distortion entails, in general, multiple distortions in the shadow prices of the characteristics. An example may prove useful. Let n = 3, let p = (1, 1, l), and let the true technology relating goods to characteristics be represented by the matrix:
B*=
The shadow prices of the characteristics are then s* =(0.063, 0.063, 0.063). If the consumer overestimates the value of b, 1, so that
B=
(
11
1
5
5
10
1
1
5
10
1 ,
the shadow prices of the characteristics are 0=(0.057, 0.065, 0.062). A change in the value of only one element in B* causes a change in all the shadow prices. The solution to the consumer’s maximization problem can be expressed as a function of these shadow prices. His demand functions for characteristics are c=c(s, M). From the relation f =&-I, one finds demand functions for the goods:
a=c(i,M)P.
(2)
But under conditions of perfect information, the consumer would purchase not the bundle 4, but the bundle x* = c(s*, M) B*-l,which is the solution to the problem: maximize u(xB*) s.t.
(3) xp’ 5 M.
The consumer can be induced to purchase this bundle x* by imposing a set of taxes (or subsidies), t, on the goods, coupled with a lump-sum transfer of t’x*. The demand functions can be written as functions of these taxes and transfers, so that f = c(B- ’ . (p + t)‘, M+ t’x*)& l. Set f = x* in this equation and solve for t to obtain the set of corrective taxes.
A. Glazer, Using corrective taxes
91
Note that the corrective tax policy cannot simply levy taxes that would make the consumer’s perceived shadow prices of the characteristics coincide with the true shadow prices. To see this, let the true shadow prices be s*~B*-~p’ and let the taxes be such that ?=B-’ (p+t)‘=s*. Now if s^=s*, the consumer would wish to purchase a bundle of characteristics which is identical to the bundle c* purchased by a perfectly informed consumer facing no corrective taxes. But in attempting to obtain the bundle c*, the consumer suffering from misperceptions will in fact obtain the bundle c*&‘B*, which is not identical to bundle c*. 4. A single tax In practice, imposing a single corrective tax - on the good subject to misperceptions - is preferable to imposing a tax on several goods, including those not subject to misperceptions. This section therefore considers the properties of such a policy. A single tax will suffice to attain optimality if the consumer’s utility function is separable. It will not, in general, suffice otherwise. Nevertheless, a single tax might still improve consumer welfare; this tax should not induce consumers to purchase the same bundle that they would under perfect information. I shall first consider the question of whether a single tax can yield optimality. Let the consumer’s misperceptions be limited to mis-estimating the quantity of characteristic 1 contained in good 1. That is, 6,, #bT, and gij =l$ for all i, j, where i #j # 1. Let a tax on good 1 and a lump-sum transfer induce the consumer to purchase x: units of good 1, leaving him with M --pIx: dollars to spend on goods 2,. . .,n. The question is whether he will purchase the correct quantities of these latter goods. Define a utility function u(x2,. . . ,xn;bllxl,. . ., b,,xJ such that That is to say, the function u( .) repreu(X2,...,Xn;bllxl, . . . . blnxl)=u(xB). sents the consumer’s perceived utility from consuming goods 2,. . .,n in quantities x2,. . . , x,, given that he believes he is obtaining some quantities of characteristics 1,...,n from good 1. For any specified values of expenditure, of p2,. . .,p,,, and of x1, the consumer maximizes his utility by maximizing u(.) subject to his budget constraint. The first-order condition for such maximization is that
Sij(bllxl,...,bl*Xl)~
axi aotx z,...,x,;bllx1
,-..,blnXl)
=E
pj
for i,j#l.
axj (4) The question here is whether a consumer who suffers from misperceptions,
92
A. Glazer, Using corrective taxes
who is faced with a tax on good one coupled with a lump-sum transfer, who is thereby induced to purchase x7 units of good 1, and who can spend M-p,x: on goods 2,..., n, will in fact purchase quantities xt , . . . , x,* of these goods. The answer is in the affirmative if and only if sij(b, r x7,. . . ,6,&) = Sij(b:,X:, . . .) br,,x:) for all i, j # 1. This represents the necessary and sufficient condition for a single tax and a lump-sum transfer to yield optimal consumption choices. To obtain further insight about this condition, suppose good 1 is specialized in producing characteristic 1, and no other good yields characteristic 1. A tax on good 1 (coupled with an income transfer) can lead the consumer to purchase x7 units of good 1. But the consumer believes he obtains 6, 1~: units of characteristic 1, instead of bfl xf units of characteristic 1 that he in fact obtains. This misperception affects the consumer’s marginal rate of substitution between goods i and j (where i# 1 and j# 1). The consumer’s purchases of goods 2,. . . , n therefore differ from the optimal quantities xr , . . . , x,*. If, however, the utility function u(ci,. . . , CJ is weakly separable between characteristics 1 and all other goods, there is no such difficulty. With separability, a single tax can completely correct the consumer’s misperceptions, and the optimal policy makes the consumer purchase the bundle of goods he would purchase under perfect information. When utility is not separable, a consumer’s misperception may cause him both to consume the wrong amount of good 1, and for any level of expenditures on goods 2,. . . , n to consume them in the wrong proportions. A single tax then cannot correct both distortions, though it can improve consumer welfare. This reasoning also means that the optimal tax need not have the consumer purchase the same quantity of x1 that he would under perfect information. An extreme example will clarify the assertion. Suppose that u(cl,cZ,cJ = min (cl,cZ,cs) for c,<2, and that u(c1,c2,c3)=min($c,,c,) for c,>2. One might think of an antacid that is effective after consumption of small pizzas, and useless after the eating of larger ones. Let bZ = 1 and bc=O for if j. The price of each good is 1 and’ the consumer’s income is 3 dollars. Under perfect information, the consumer’s utility would be min(1, 1,1) = 1. Now suppose 6,, equals 2.1, not 1, and that the corrective tax makes the consumer choose xi = 1. Since 2, =2.1, which is greater than 2, the consumer perceives his utility function to be min ()c,, c,), so that he lets x2 = 0; his ex post utility, however, is min (l,O, 2) = 0. A better policy would be to make the consumer purchase 3/5.2 units of x1 ; he would choose (2.1)(3)/5.2 units of x2 and x3, have a perceived utility of (2.1)(3)/5.2, and obtain an actual utility of 3/5.2. In this example, the optimal corrective tax makes the consumer’s perceived consumption of a characteristic greater than what his consumption would be under perfect information; his purchase of the good subject to misperceptions is not the same as under perfect information.
A. Glazer, Using corrective
93
taxes
The effect can be shown graphically as in fig. 3. Let there be only three goods so that (given the consumer’s income and the prices of the goods) a consumption bundle is specified by the quantities of x1 and x2 consumed. The indifference contour in fig. 3 shows quantities of x1 and x2 among which (given his budget constraint) the consumer is indifferent; such an indifference contour is derived by slicing a three-dimensional indifference curve by a three-dimensional budget plane, and then projecting this cut on the x1 -x2 plane.
_-I-------
Q c\ \\
lB -$
l I I
Fig. 3. Consumption
X1
of good is suboptimal.
Let point A in fig. 3 be the optimal consumption choice under conditions of perfect information, so that all points lying within the indifference contour are preferred to points lying on or outside the contour. Suppose that the consumer consumes at point C, consuming too much of good 2 and too little of good 1 compared to the choices he would make under perfect information. By subsidizing good 1, or taxing good 2, government can induce the consumer to consume at II, with the same quantity of x2 as at point A. The consumer’s utility, however, is lower at point B that at point C, and his utility was decreased after he was induced to purchase the correct amount of the good subject to misperceptions. 5. Conclusion Many governmental programs are designed to deal with problems of imperfect information. Thus, government may require that drinks labelled ‘orange juice’ meet certain minimum standards, or that manufacturers of
J.PE
D
94
A. Glazer, Using corrective taxes
cigarettes and saccharin print notices concerning the hazards inherent in using the products. Similarly, government may tax certain goods not only to collect revenue, but also in the hope of correcting market failures caused by consumer ignorance. The purpose of this paper is to offer a cautionary note. It is not straightforward to correct consumers’ misperceptions. Welfare improvements may require that consumers increase rather than decrease purchases of a good they overvalue; full correction of even one misperception may require taxing several goods; a single tax will suffice to attain optimality only if the consumer’s utility function is separable; the policymaker must rely on careful investigation of consumer preferences to determine whether the change in consumer behavior should be towards the purchase of the goods bundle that obtained under perfect information, or instead towards the purchase of the perceived characteristics bundle obtained under perfect information.
References Atkinson, Anthony B., 1973, Smoking and the economics of government intervention, in: Mark Perlman, ed., The economics of health and medical care, New York, 1973, pp. 42841. Auld, Douglas A.L., 1972, Imperfect knowledge and the new theory of demand, Journal of Political Economy, 80, 1287-1294. Colantoni, Claude S., Otto A. Davis and Malatai Swaminuthan, 1976, Imperfect consumers and welfare comparisons of policies concerning information and regulation, Bell Journal of Economics 7,602-615. Green, Jerry and Eytan Sheshinski, 1976, Direct versus indirect remedies for externalities. Journal of Political Economy 84, no. 4, 797-808. Harris, Jeffrey E., 1980, Taxing tar and nicotine, American Economic Review 70, 300-311. Lancaster, K., 1966, A new approach to consumer theory, Journal of Political Economy 74, 132157. Lipsey, R.G. and K. Lancaster, 1956, The general theory of second best, Review of Economic Studies 24, 1 l-32.
of Public
USING
Economics
28 (1985) 85-94.
North-Holland
CORRECTIVE TAXES TO REMEDY MISPERCEPTIONS
CONSUMER
Amihai GLAZER* School
ofSocialSciences,
University of California, Irvine CA 92717, U.S.A.
Received
1983, revised version
January
received
April 1985
When consumers are misinformed about the amount of characteristics obtained from some good, government may attempt to correct the misperceptions by imposing taxes or subsidies on these goods. Such a policy, however, faces difliculties: only with separable utility functions does a single tax suffke, and this tax should not induce consumers to purchase the same bundle they would under perfect information.
1. Introduction
Consumers, alas, are not always perfectly informed about the characteristics of goods they purchase. A buyer may overestimate the reliability of an appliance; he may underestimate the effectiveness of airbags in automobiles; misleading advertising may convince him that some cereal is more nutritious than it actually is. Government may therefore be called upon to improve consumers’ consumption choices. Two such policies come to mind: government can educate consumers, providing them with more accurate information about the goods; it can levy taxes, thereby inducing consumers to make the proper consumption choices in spite of their misperceptions. This paper focuses on analyzing such corrective taxes. The inspiration for the present work comes from Atkinson (1973) and Harris (1980) who examine the problem of consumer misconceptions concerning the hazards of cigarette smoking. In his seminal paper, Atkinson points out a major difficulty in analyzing such problems: it is unclear whether social welfare should be evaluated on the basis of ex ante or of ex post consumer utility. In this paper I follow the example of Harris and consider only ex post utility - the utility a consumer obtains after having consumed a particular bundle of goods. Using this assumption, Harris considers a model in which each consumer decides how many cigarettes to smoke and also chooses among *I am grateful to Duran Bell, Jossi Berechman, Ivor Pearce, and an anonymous their comments and suggestions. All remaining errors are, of course, my own.
0047-2727/85/!§3.30
0
1985, Elsevier Science Publishers
B.V. (North-Holland)
referee for
86
A. Glazer, Using corrective
taxes
brands of cigarettes distinguished by different tar and nicotine contents. If not all consumers are identical, and if consumers must choose among two variables - quantity of cigarettes and their tar and nicotine content - then it is clear that the imposition of a single tax cannot lead all consumers to make consumption choices identical to those they would make with perfect information. In a different context, Green and Sheshinski (1976) reach similar conclusions about the limitations of corrective taxes. Studying the problem of consumption externalities, they find that if not all consumers are identical, the optimal corrective tax policy may require that taxes be imposed on several goods, and not only on the good that causes the externality. My analysis, which yields similar conclusions, relies on the Lancasterian approach to consumer demand [see Auld (1972), Colantoni et al. (1976), Lancaster (1966)]. I extend previous work by considering more than two goods and more than two characteristics, and by introducing into the model issues of taxation. Although my results bear some resemblance to the Theory of the Second Best [see Lipsey and Lancaster (1956)], that theory cannot be directly applied to the problem discussed here. The literature on the Second Best considers price distortions, and not informational problems. In addition, the analysis here applies to a single distortion - a consumer’s misperception concerning the quantity of one characteristic contained in one good - and not to multiple distortions. The next section presents a graphical exposition of the problem. Section 3 presents a more general analysis. Section 4 considers corrective policy which taxes only the one good subject to misperceptions. 2. A two-goods model A good understanding of the problem is gained by considering the simple case in which there are only two goods, each of which yields only one characteristic. Suppose all consumers are identical, and let a consumer’s budget line for the two goods be line QzQ, in fig. 1. Let each unit of good 1 yield one unit of characteristic one, and let each unit of good 2 yield one unit of characteristic 2; thus the line QzQl also represents the consumer’s budget line in terms of characteristics. The consumer obtains utility from characteristics, and not directly from goods; he chooses to purchase the bundle of goods that yields a utility-maximizing bundle of characteristics. Given budget line Q,Q,, the consumer will purchase b, units of good 1 and b, units of good 2, obtaining the characteristics bundle shown by point B in fig. 1. Suppose next that the consumer overestimates the quantity of characteristic 2 obtainable from good 2, so that he believes his consumption possibility frontier (in terms of characteristics) to be line Q;Q1, The
A. Glazer, Using corrective taxes
Fig. 1. Misperceptions
cause underconsumption
87
of good.
consumer now wishes to consume characteristics bundle F’, and he therefore purchases b; units of good 1 and b; units of good 2. He thus in fact obtains characteristics bundle B’, which lies on a lower indifference curve than does bundle B. The consumer could be induced to purchase bundle B by the imposition of a subsidy on good 2 coupled with a lump-sum reduction of his income.’ This is shown in fig. 2. Let the tax shift the consumer’s budget line in terms of goods from line Qz QI to line T2TI . With this budget line and this overestimate of the amount of characteristic 2 contained in good 2, the ‘Equivalently, a tax could be imposed on good income. I assume that both characteristics are normal
1, coupled ones.
with
a lump-sum
increase
in
A. Glazer, Using
correctivetaxes
Fig. 2. Corrective
taxation.
consumer perceives his budget line in terms of characteristics to be line T; Tr.’ The consumer wishes to consume characteristics bundle G’, and he therefore purchases goods bundle B, the same bundle he would have purchased had he faced no taxes and suffered from no misperceptions. To be more concrete, suppose the consumer overestimates the amount of vitamins contained in cereals and that he has some target level of vitamin intake. This overestimate may lead him to consume less cereal than he would under perfect information. On the other hand, the overestimate leads the consumer to view the effective price of vitamins to be quite low; if his demand for vitamins is sufficiently elastic, his misperception would cause him ‘1 have drawn fig. 2 such that the ratio of the segments OQ;/OQ, in fig. 1; that is, the extent of consumer misperceptions
OTz/OT, is equal to the ratio is identical in the two cases.
A. Glazer, Using corrective
taxes
89
to consume too much, rather than too little cereal. This means that the corrective tax policy may require that the relative price of the good subject to misperceptions be either decreased or increased. 3. Corrective taxes in a general model This section extends the model presented above by considering a world with more than two goods and characteristics. Such a generalization introduces a major new element into the discussion. Can a corrective tax policy be designed that requires taxing only the good subject to misperceptions, or must taxes instead be imposed on other goods as well? In the two goods case, of course, such an issue cannot arise. A tax on both goods is equivalent to a tax on only one good and a lump-sum income transfer, so that it makes no sense to ask how many goods must be taxed or subsidized. Moreover, in the two goods case, any policy that reduces the consumer’s overconsumption of some good will necessarily increase his welfare. That may not be so if the consumer buys more than two goods. In the most general Lancasterian model, the number of goods on the market bears no necessary relation to the number of characteristics entering a utility function; which goods are actually purchased depends on the prices in effect and on the particular technology relating goods to characteristics. It greatly simplifies matters, however, to focus on the quantities of goods purchased rather than on which goods are purchased. Analysis is also simplified if the number of goods is at least as great as the number of characteristics. I shall therefore henceforth assume that there are n goods, purchased in quantities xi,. . . ,x,, and n characteristics, consumed in quantities ci, . . . , c,. Let the price of good i be pi. For brevity, denote these lists by the (1 x n) vectors X, c, and p. The consumer’s income is denoted by M. The true technological relation between goods and characteristics is given by the (n x n) matrix B*,each of whose elements, b& gives the amount of characteristic j obtainable from good i. The matrix fi represents the consumer’s erroneous estimates of the technological relations: he believes that by purchasing the goods bundle x he obtains the characteristics bundle S=xB. The consumer’s problem is then to: maximize u(xB) s.t.
(1) xp’ 5 M.
where a prime denotes transposition.
Substitute x=&l
in expression (1) to
90
A. Glazer, Using corrective
taxes
write the budget constraint in the more useful form of &l~‘sM. The vector s^- &- ‘p’ can thus be thought of as the shadow prices of the characteristics. Note that a single informational distortion entails, in general, multiple distortions in the shadow prices of the characteristics. An example may prove useful. Let n = 3, let p = (1, 1, l), and let the true technology relating goods to characteristics be represented by the matrix:
B*=
The shadow prices of the characteristics are then s* =(0.063, 0.063, 0.063). If the consumer overestimates the value of b, 1, so that
B=
(
11
1
5
5
10
1
1
5
10
1 ,
the shadow prices of the characteristics are 0=(0.057, 0.065, 0.062). A change in the value of only one element in B* causes a change in all the shadow prices. The solution to the consumer’s maximization problem can be expressed as a function of these shadow prices. His demand functions for characteristics are c=c(s, M). From the relation f =&-I, one finds demand functions for the goods:
a=c(i,M)P.
(2)
But under conditions of perfect information, the consumer would purchase not the bundle 4, but the bundle x* = c(s*, M) B*-l,which is the solution to the problem: maximize u(xB*) s.t.
(3) xp’ 5 M.
The consumer can be induced to purchase this bundle x* by imposing a set of taxes (or subsidies), t, on the goods, coupled with a lump-sum transfer of t’x*. The demand functions can be written as functions of these taxes and transfers, so that f = c(B- ’ . (p + t)‘, M+ t’x*)& l. Set f = x* in this equation and solve for t to obtain the set of corrective taxes.
A. Glazer, Using corrective taxes
91
Note that the corrective tax policy cannot simply levy taxes that would make the consumer’s perceived shadow prices of the characteristics coincide with the true shadow prices. To see this, let the true shadow prices be s*~B*-~p’ and let the taxes be such that ?=B-’ (p+t)‘=s*. Now if s^=s*, the consumer would wish to purchase a bundle of characteristics which is identical to the bundle c* purchased by a perfectly informed consumer facing no corrective taxes. But in attempting to obtain the bundle c*, the consumer suffering from misperceptions will in fact obtain the bundle c*&‘B*, which is not identical to bundle c*. 4. A single tax In practice, imposing a single corrective tax - on the good subject to misperceptions - is preferable to imposing a tax on several goods, including those not subject to misperceptions. This section therefore considers the properties of such a policy. A single tax will suffice to attain optimality if the consumer’s utility function is separable. It will not, in general, suffice otherwise. Nevertheless, a single tax might still improve consumer welfare; this tax should not induce consumers to purchase the same bundle that they would under perfect information. I shall first consider the question of whether a single tax can yield optimality. Let the consumer’s misperceptions be limited to mis-estimating the quantity of characteristic 1 contained in good 1. That is, 6,, #bT, and gij =l$ for all i, j, where i #j # 1. Let a tax on good 1 and a lump-sum transfer induce the consumer to purchase x: units of good 1, leaving him with M --pIx: dollars to spend on goods 2,. . .,n. The question is whether he will purchase the correct quantities of these latter goods. Define a utility function u(x2,. . . ,xn;bllxl,. . ., b,,xJ such that That is to say, the function u( .) repreu(X2,...,Xn;bllxl, . . . . blnxl)=u(xB). sents the consumer’s perceived utility from consuming goods 2,. . .,n in quantities x2,. . . , x,, given that he believes he is obtaining some quantities of characteristics 1,...,n from good 1. For any specified values of expenditure, of p2,. . .,p,,, and of x1, the consumer maximizes his utility by maximizing u(.) subject to his budget constraint. The first-order condition for such maximization is that
Sij(bllxl,...,bl*Xl)~
axi aotx z,...,x,;bllx1
,-..,blnXl)
=E
pj
for i,j#l.
axj (4) The question here is whether a consumer who suffers from misperceptions,
92
A. Glazer, Using corrective taxes
who is faced with a tax on good one coupled with a lump-sum transfer, who is thereby induced to purchase x7 units of good 1, and who can spend M-p,x: on goods 2,..., n, will in fact purchase quantities xt , . . . , x,* of these goods. The answer is in the affirmative if and only if sij(b, r x7,. . . ,6,&) = Sij(b:,X:, . . .) br,,x:) for all i, j # 1. This represents the necessary and sufficient condition for a single tax and a lump-sum transfer to yield optimal consumption choices. To obtain further insight about this condition, suppose good 1 is specialized in producing characteristic 1, and no other good yields characteristic 1. A tax on good 1 (coupled with an income transfer) can lead the consumer to purchase x7 units of good 1. But the consumer believes he obtains 6, 1~: units of characteristic 1, instead of bfl xf units of characteristic 1 that he in fact obtains. This misperception affects the consumer’s marginal rate of substitution between goods i and j (where i# 1 and j# 1). The consumer’s purchases of goods 2,. . . , n therefore differ from the optimal quantities xr , . . . , x,*. If, however, the utility function u(ci,. . . , CJ is weakly separable between characteristics 1 and all other goods, there is no such difficulty. With separability, a single tax can completely correct the consumer’s misperceptions, and the optimal policy makes the consumer purchase the bundle of goods he would purchase under perfect information. When utility is not separable, a consumer’s misperception may cause him both to consume the wrong amount of good 1, and for any level of expenditures on goods 2,. . . , n to consume them in the wrong proportions. A single tax then cannot correct both distortions, though it can improve consumer welfare. This reasoning also means that the optimal tax need not have the consumer purchase the same quantity of x1 that he would under perfect information. An extreme example will clarify the assertion. Suppose that u(cl,cZ,cJ = min (cl,cZ,cs) for c,<2, and that u(c1,c2,c3)=min($c,,c,) for c,>2. One might think of an antacid that is effective after consumption of small pizzas, and useless after the eating of larger ones. Let bZ = 1 and bc=O for if j. The price of each good is 1 and’ the consumer’s income is 3 dollars. Under perfect information, the consumer’s utility would be min(1, 1,1) = 1. Now suppose 6,, equals 2.1, not 1, and that the corrective tax makes the consumer choose xi = 1. Since 2, =2.1, which is greater than 2, the consumer perceives his utility function to be min ()c,, c,), so that he lets x2 = 0; his ex post utility, however, is min (l,O, 2) = 0. A better policy would be to make the consumer purchase 3/5.2 units of x1 ; he would choose (2.1)(3)/5.2 units of x2 and x3, have a perceived utility of (2.1)(3)/5.2, and obtain an actual utility of 3/5.2. In this example, the optimal corrective tax makes the consumer’s perceived consumption of a characteristic greater than what his consumption would be under perfect information; his purchase of the good subject to misperceptions is not the same as under perfect information.
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The effect can be shown graphically as in fig. 3. Let there be only three goods so that (given the consumer’s income and the prices of the goods) a consumption bundle is specified by the quantities of x1 and x2 consumed. The indifference contour in fig. 3 shows quantities of x1 and x2 among which (given his budget constraint) the consumer is indifferent; such an indifference contour is derived by slicing a three-dimensional indifference curve by a three-dimensional budget plane, and then projecting this cut on the x1 -x2 plane.
_-I-------
Q c\ \\
lB -$
l I I
Fig. 3. Consumption
X1
of good is suboptimal.
Let point A in fig. 3 be the optimal consumption choice under conditions of perfect information, so that all points lying within the indifference contour are preferred to points lying on or outside the contour. Suppose that the consumer consumes at point C, consuming too much of good 2 and too little of good 1 compared to the choices he would make under perfect information. By subsidizing good 1, or taxing good 2, government can induce the consumer to consume at II, with the same quantity of x2 as at point A. The consumer’s utility, however, is lower at point B that at point C, and his utility was decreased after he was induced to purchase the correct amount of the good subject to misperceptions. 5. Conclusion Many governmental programs are designed to deal with problems of imperfect information. Thus, government may require that drinks labelled ‘orange juice’ meet certain minimum standards, or that manufacturers of
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cigarettes and saccharin print notices concerning the hazards inherent in using the products. Similarly, government may tax certain goods not only to collect revenue, but also in the hope of correcting market failures caused by consumer ignorance. The purpose of this paper is to offer a cautionary note. It is not straightforward to correct consumers’ misperceptions. Welfare improvements may require that consumers increase rather than decrease purchases of a good they overvalue; full correction of even one misperception may require taxing several goods; a single tax will suffice to attain optimality only if the consumer’s utility function is separable; the policymaker must rely on careful investigation of consumer preferences to determine whether the change in consumer behavior should be towards the purchase of the goods bundle that obtained under perfect information, or instead towards the purchase of the perceived characteristics bundle obtained under perfect information.
References Atkinson, Anthony B., 1973, Smoking and the economics of government intervention, in: Mark Perlman, ed., The economics of health and medical care, New York, 1973, pp. 42841. Auld, Douglas A.L., 1972, Imperfect knowledge and the new theory of demand, Journal of Political Economy, 80, 1287-1294. Colantoni, Claude S., Otto A. Davis and Malatai Swaminuthan, 1976, Imperfect consumers and welfare comparisons of policies concerning information and regulation, Bell Journal of Economics 7,602-615. Green, Jerry and Eytan Sheshinski, 1976, Direct versus indirect remedies for externalities. Journal of Political Economy 84, no. 4, 797-808. Harris, Jeffrey E., 1980, Taxing tar and nicotine, American Economic Review 70, 300-311. Lancaster, K., 1966, A new approach to consumer theory, Journal of Political Economy 74, 132157. Lipsey, R.G. and K. Lancaster, 1956, The general theory of second best, Review of Economic Studies 24, 1 l-32.