Efficient Taxation of 'Diamond' Goods

(Ng, AER, 1987)

Consider a consumer who derives utility from three consumption goods:

x and y and z, which prices are given, respectively, by x y z p , p and p . We

let z denote the numeraire good and normalize its prices to unity. The

consumer possesses an initial endowment of Z>0 units of z.

The utility function of the consumer is given by:

U (x, y, z)= v(x) + u( y . p?_y) + z

Thus, the consumer is deriving utility from the amount she spends on y

and not from the number of units consumed.

The government is seeking to raise funds in the total amount of R>0

(units of z). For this purpose it considers the following two alternative

options: (i) levying a lump-sum tax; (ii) imposing an ad-valorem tax on

consumption good y.

(1) Provide a theoretical explanation for the special form of the

utility function we employ.

(2) What would be the optimal choice of the government?

My professor just give more information on this homework. hope this give you more ideas to solve the problem. Thank you!

The question is based on an article published in the American Economic Review in 1987 (the article is attached, for those interested in reading it). The question illustrates a scenario in which a tax levied on a consumption good is more efficient than a lump-sum tax. Prima facie, this sounds paradoxical given what we have been arguing in class. The reason for the surprising prediction (which you need to prove) stems from the special features of the utility function. Notice that the consumer derives utility from the amount spent on y rather than (as is typically the case) from the quantity consumed. In the first part, you should provide an interpretation for the utility specification (and the resulting consumption patterns). In the second part, you need to show that taxing y would be superior to a lump-sum. We know that lump-sum tax entails only an income effect (which implies a decrease in the utility). To prove the claim, you need to show that a tax on y entails, in fact, no effect on the utility of the consumer. Put differently - the optimal choices of the consumer with and w/o a tax on y will derive the same level of utility! This is clearly superior to a lump-sum tax which indeed entails no substitution effect but does have a negative income effect which implies a reduction in the utility.

Diamonds Are a Government's Best Friend: Burden-Free Taxes on Goods Valued for Their Values Author(s): Yew-Kwang Ng Source: The American Economic Review , Mar., 1987, Vol. 77, No. 1 (Mar., 1987), pp. 186191 Published by: American Economic Association Stable URL: http://www.jstor.com/stable/1806737 JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms is collaborating with JSTOR to digitize, preserve and extend access to The American Economic Review This content downloaded from 132.72.138.1 on Thu, 09 Jul 2020 07:23:39 UTC All use subject to https://about.jstor.org/terms Diamonds are a Government's Best Friend: Burden-Free Taxes on Goods Valued for their Values By YEW-KWANG NG* plies to most other precious stones and metals, including gold. In various lesser degrees, this "diamond effect" also applies to other items of conspicuous consumption such as expensive fur coats and luxurious cars. With increasing affluence, "diamond goods" will To most economists, it seems almost axiomatic that taxes (except corrective taxes) impose not just a burden equal to the amount of the tax collected, but also an excess burden by distorting individual choices, not to mention administrative, compliance, and policing costs (loosely called transaction costs below). Lump sum taxes with no excess burden exist only in theory. Yet there exists at least one class of goods that can be taxed and the tax will not only not produce an excess burden, but it will not be a burden at all (ignoring transaction costs)! This sounds like a miracle but it is really quite simple once it is recognized that some goods are valued for their values, not for their intrinsic consumption effects. Taxes on these goods increase their prices. But consumers can reduce quantities consumed without changing the values of these goods, suffering no loss in utility. Thus, no burden is imposed, not to mention excess burden. For example, after a doubling in the price of diamond, a $1000 gift of diamond is still valued at $1000, though the size of the stone is smaller. A carat of diamond can be worth thousands of dollars, but costume jewelry that looks similar may cost only a few dollars. Imitation diamonds look virtually the same as real diamonds and it takes experts with fine instruments to tell the difference. Surely, diamonds are valued not for their intrinsic consumption effects but because they are costly. Consumers of diamonds either derive utility by showing off their wealth (Veblen's conspicuous consumption), by using it as a store of value, or by giving it as a gift of value. In virtually all cases, it is the value, not the diamond itself, that counts. This ap- become more important. The diamond effect (valuing something for its value rather than the consumption effect) must be distinguished from other similar or related but different phenomena. First, " the habit of judging quality by price" (Tibor Scitovsky, 1945) is the belief that higherpriced brands give higher intrinsic consumption utilities. Second, while Thorstein Veblen's (1899) conspicuous consumption may partly contribute to the diamond effect, they are conceptually distinct. A person's desire to go on a world trip may be partly to show off to his (her) admiring friends who cannot afford to go. But as long as he can and they cannot, it has value for conspicuous consumption. Increases in the price of the trip may add little to the value and half a world trip is not as good, even if the price of the world trip has doubled. On the other hand, a man's gift of a $1000 diamond ring to his wife is worth that much irrespective of the size of the stone, and they may never show the ring to anybody. Also, some people show off their heavy gold bracelets (conspicuous consumption with a diamond effect) while others hide their gold bullion (a diamond effect without conspicuous consumption). Third, there are goods whose intrinsic consumption effects depend on whether other people also consume them (for example, telephones, unusual clothing, fashions). To some extent, this consideration may also affect the diamond effect. However, to concentrate on the pure diamond effect, I will abstract these complications away. As in the distinction between private and public goods where different degrees of pub- *Department of Economics, Monash University, Clayton, Victoria 3168 Australia, and University of Maryland-College Park. 186 This content downloaded from 132.72.138.1 on Thu, 09 Jul 2020 07:23:39 UTC All use subject to https://about.jstor.org/terms VOL. 77 NO. 1 NG: BURDEN-FREE licness are involved, most goods are valued partly for their intrinsic consumption effects (approaching 100 percent for ordinary items like bread) and their values (approaching 100 percent for precious jewels). However, for analytical simplicity, I consider only two polar cases and adopt an atemporal model commonly used in welfare analysis. The complicated questions of the dynamics of transition and some other practical complications are touched on in the concluding section, but are not formally analyzed. The basic result is that a change in the price of a diamond good leaves its own value unchanged and the amount of all other goods consumed, and hence the utility levels of consumers unchanged, and that it is optimal to place arbitrarily high taxes on diamond goods which impose no burden and no excess burden. A corollary is that the demand curve for a diamond good is a rectangular hyperbola with unit price elasticity throughout. Such an obvious phenomenon as the diamond effect has not of course completely escaped the economist's attention. For example, Pigou touched on the "desire to possess what other people do not possess" (1932, p. 226) and used diamonds as an example. But there is a curious lack of for- TAXES 187 I. A Simple Analysis For simplicity, consider the case with only one diamond good (say, the first) and ignore all other complications (externalities, etc.). Generalization to impure diamond goods is straightforward, but impure or mixed goods bring some complications not considered in this paper. The utility function of an individual may thus be written as (1) U(p1X11pn x2 Xn) where xi is the amount of the i th good consumed, pi its price, and the last good is being used as a numeraire. In the long run, the consumer is unlikely to suffer from significant money illusion. Thus, instead of the money value of the diamond good p1x1, we should replace the money price p1 by real or relative price pl/pn 2 For impure diamond goods, both plXl/pn and xi enter the utility function. Taking prices as given, the consumer maximizes (1) subject to (2) >'pixi= M, where the summation is over all the n goods mal analysis,1 and almost complete disregard and M is the given amount of income. This in the public finance texts (for example, Richard Musgrave and Peggy Musgrave, 1980) and actual policy debate on taxation issues (for example, the great Australian tax reform in 1985). This paper provides a modest attempt at a formal analysis which may attract, hopefully, more attention both by theorists and those concerned with policy decision. problem is homogeneous of degree zero in all prices and money income; no money illusion is involved. Assuming an interior solution for notational simplicity, the first-order conditions for optimality are (3a) U1 = Xpn (3b) Ui= Xpi (i = 2,... , n) where Ui is the partial derivative (margin utility) of the ith element in the utility func 'Peter Kalman (1968) provides a rigorous analysis of tion (1), and X is the Lagrangian multiplier consumer behavior when prices enter the utility function. This is a very general analysis which may be said to include "judging quality by price," conspicuous consumption, and the diamond effect. However, partly because it is too general and partly because of its exclusive concern with the positive theory of consumer behavior, it reaches none of the results of this paper. 2Altematively, we may replace pn in (1) by P, a price index of all nondiamond goods, with the same result except that pn below is replaced by P. This content downloaded from 132.72.138.1 on Thu, 09 Jul 2020 07:23:39 UTC All use subject to https://about.jstor.org/terms 188 THE AMERICAN ECONOMIC REVIEW MARCH 1987 associated with (2) or the marginal utility of amounts of all other goods consumed, and income. hence the utility level of the consumer un- The price of a diamond, pi, does not appear in the system of equations (3) describing the optimal solution. It is tempting but wrong to infer from this that the optimal x's are independent of p1. This is wrong because p1 appears in (2) which is included in the set of equations, together with (3), defining the optimal solution. However, it is affected. COROLLARY 1: The demand curve for a diamond good is a rectangular hyperbola with unit elasticity throughout the whole range where it remains a pure diamond good. This is true not only for an individual true that plxl/pn and x2,..., Xn and hence demand curve, but also for a market demand U are independent ofp', as shown below. The maximization problem above may be written, with no change of any substance, as the maximization of (4) U(Yl y 2 Yn) curve as long as the good is viewed by all consumers as a diamond good (assumed here for simplicity) because the horizontal summation of rectangular hyperbolas is also a rectangular hyperbola. For simplicity, assume a horizontal supply curve. A 100 percent tax on the diamond good then doubles its price and a 200 percent tax triples it, etc. The higher the tax rate, the larger the tax revenue, while con(5) qyi= M sumers remain no worse off. The tax revenue where collected thus represents pure gain, imposing not only no excess burden, but also no burden at all! yI p1x1/pn, y (i= 2,...,n), There is an upper limit beyond which the qI _ pn qi =_ pi (E=2.,n). tax revenue cannot exceed. This supremum (the maximum does not exist) is the pre-tax This rewritten problem is identical in its ( = post-tax) value of the good. The amount of tax revenue that can be raised without mathematical form to the traditional conburden is limited by the amount of expendisumer optimization problem with no diamond effect, and with the following familiar ture on diamond goods (which may be exfirst-order conditions for an interior solupected to increase relatively and absolutely tion, with increasing affluence). The net gain is the amount of resources saved due to a smaller output after the imposition of the diamond (6) Ui = Xq (i =1, ...,In), subject to tax. which, with constraint equation (5), define the optimal y's. With the rewritten problem, if we work in terms of y's instead of x's (the only dif- ference is to take pxll/pn as an integral variable instead of breaking it up into its constituent parts), it is clear that p1 appears neither in the constraint (5) nor in the firstorder condition (6). The optimal set of y's and hence the maximized utility level are thus independent of p1. This result may be expressed as PROPOSITION 1: A change in the price of a diamond good leaves its value and the II. A Model of Optimal Taxation The above analysis may be regarded as somewhat partial and/or intuitive. Here, I present a standard model of optimal taxa- tion, except that I allow for pure diamond goods. Since no changes in the relative price between private goods is considered, I lump them into a composite good y. Similarly, I lump all diamond goods into another composite good d. As in the standard optimal taxation literature, I concentrate on the tax side by assuming a constant government revenue requirement and assume that the con- This content downloaded from 132.72.138.1 on Thu, 09 Jul 2020 07:23:39 UTC All use subject to https://about.jstor.org/terms VOL. 77 NO. 1 NG: BURDEN-FREE sumer side of the economy can be represented by one consumer or a community utility function, (7) U(D, y), where D (q + t)d/(Q + T) is the (relative) value of the diamond good, q and Q are the fixed producer prices, and t and T the per unit taxes on diamond and the private good, respectively. The assumption of a representative consumer does assume away distributional issues, but may be justified by the predominant concern on efficiency issues and the argument on separating equity and efficiency issues even in the presence of second-best factors and other complications (see my 1984 paper). The consumer maximizes (7) with respect to d and y, subject to (8) (q + t)d + (Q + T)y = M, TAXES 189 it can immediately be seen that the con- sumer price (Q + T) of the private good serves as the price for both the private good y and for the relative value of diamond D, and equation (9) is thus obvious. The consumer allocates his (her) income M between two goods D and y that have the same price, so MRS = 1 for an interior maximum. As discussed in Section I, the consumer's optimal choice between D and y is independent of the consumer diamond price, q + t, which entered neither (8') nor (9). We thus have (10) AD/dt = 0 = ay/dt. From the first inequality in (10) and the definition of D, we have (11) nqd t/(q + t), where money income M is taken as given. where qdt =(d/d t)d/t is the elasticity of While this may seem to abstract away work- d with respect to t. The government maximizes (7) with respect to t and T, subject to the consumer's choice described above and to the fixed revenue constraint leisure choice, we may alternatively interpret M as full income and include leisure in the composite good y, with the result that leisure is regarded as taxable. If I can establish the result on the optimality of imposing a high tax on diamond even in a model where leisure is taxable, the desirability of doing so where leisure is not taxable seems to apply a fortiori. The first-order condition for the consumer maximization is (12) dt+yT=R. The first-order conditions for an interior solution are (13) d + q+t Ad ay (9) UD/UY=1, (13) Q+T +Q+Tdt Da Ad ay where UD is the marginal utility of the value of diamond (relative to the price of private good) consumer and U the marginal utility of the private good. In other words, PROPOSITION 2: In equilibrium, the mar- = 9d+ a+ Tat (14) { q+t dd (q+t)d +TaT (Q?) 2J UD ginal rate of substitution between the (rela- tive) value of diamond and the private good equals unity. This may appear too simple to be true. But if I write the budget constraint (8) as (8') (Q+T)D+(Q+T)y=M, ay / dy Ad 4aTaUy=0 ~ aT aT'y+T-+ where 0 is the Lagrangian multiplier associated with (12). Eliminate 0 between (13) and (14) and rewrite expressions in elasticity This content downloaded from 132.72.138.1 on Thu, 09 Jul 2020 07:23:39 UTC All use subject to https://about.jstor.org/terms 190 THE AMERICAN ECONOMIC REVIEW MARCH 1987 Taking account of this, a very high tax rather form, that is, qxy = (dx/dy)y/x, we have than an infinite tax is optimal. (15) III. Concluding Remarks d t q+t Yd t4Q+T Q+T 1U+