Uncertainty and optimally uniform commodity taxes.1. Introduction Starting with Atkinson Atkinson may refer to: Places In Canada:
pertaining to logarithm. logarithmic relationship when the logs of two variables plotted against each other create a straight line. ), a linear income tax is a sufficient tool for optimal tax policy (see Atkinson [1977], Deaton [1979], and Deaton and Stern [1986]). Recently, Cremer and Gahvari (1995a) showed that these results do not carry to an economy in which individuals face wage uncertainty. In models with uncertainty, taxation is useful for the insurance it provides. If consumption takes place both before and after the resolution of uncertainty, commodity taxes will have different insurance properties depending on when consumption occurs. This property makes differential tax treatment of these two types of goods optimal. Cremer and Gahvari (1995a) divided commodities into two categories: the consumption levels in one category are committed to before the resolution of uncertainty and those in the other after.(1) Their study pinpoints a fundamental difference between these categories of goods that calls for their tax treatment to be different. However, their analysis leaves open the question of the uniformity of tax rates within each category. The main aim of the current paper is to address this question. We shall examine the type of preferences for which the tax rates within each category have to be uniform. In this way, we will build a bridge between the results of Cremer and Gahvari (1995a) and the literature on uniform taxation in the presence of a linear income tax. There are three types of tax uniformity results in the literature. The first is due to Sandmo (1974). He proved that if preferences are separable sep·a·ra·ble adj. Possible to separate: separable sheets of paper. sep in labor supply and goods,(2) with the sub-utility for goods being homothetic, optimal commodity taxes are proportionately pro·por·tion·ate adj. Being in due proportion; proportional. tr.v. pro·por·tion·at·ed, pro·por·tion·at·ing, pro·por·tion·ates To make proportionate. uniform. Sandmo's result is derived within the context of the traditional one-consumer Ramsey problem The Ramsey problem or Ramsey-Boiteux pricing, is a policy rule concerning what price a monopolist should set, in order to maximize social welfare, subject to a constraint on profit. A closely related problem arises in relation to optimal taxation of commodities. . As such, the result embodies only efficiency considerations. Redistributive goals do not come into play. The second and most influential uniformity result is due to Atkinson and Stiglitz (1976). This classic paper on the design of tax structures was particularly concerned with the usefulness of commodity taxes in the presence of income taxes in many-consumer economies.(3) It proved that if the government can levy a general income tax, and if preferences are weakly weak·ly adj. weak·li·er, weak·li·est Delicate in constitution; frail or sickly. adv. 1. With little physical strength or force. 2. With little strength of character. separable in labor supply and goods, then commodity taxes are not needed as instruments of optimal tax policy. Atkinson and Stiglitz assumed that individuals have identical preferences and differ only in their earning abilities. Mirrlees (1976) generalized gen·er·al·ized adj. 1. Involving an entire organ, as when an epileptic seizure involves all parts of the brain. 2. Not specifically adapted to a particular environment or function; not specialized. 3. the result by examining the type of preferences that make commodity taxes redundant while allowing for taste differentiation. The third type of uniformity results also has its origin in Atkinson and Stiglitz (1976). They showed that, for a particular example of preferences, even a linear income tax calls for proportionately uniform commodity taxes. Atkinson (1977) showed that the result is more general and holds if preferences over labor supply and goods correspond to the linear expenditure system. (Atkinson and Stiglitz [1976] had in fact made this claim.) Deaton (1979) generalized this result and proved that it holds for all preferences that are weakly separable between labor supply and goods, provided that all consumers have linear Engel curves for goods in terms of income. In Deaton's setup See BIOS setup and install program. all individuals have identical tastes, differing only in earning abilities. Deaton and Stern (1986) further generalized the result by allowing a bit of taste differentiation. Engel curves can now have different intercepts while the government is enabled to make differential lump-sum grants conditioned on observable ob·serv·a·ble adj. 1. Possible to observe: observable phenomena; an observable change in demeanor. See Synonyms at noticeable. 2. household characteristics.(4) In the context of an economy with wage uncertainty, when consumption takes place before as well as after uncertainty, Cremer and Gahvari (1995b, 1999) reexamine re·ex·am·ine also re-ex·am·ine tr.v. re·ex·am·ined, re·ex·am·in·ing, re·ex·am·ines 1. To examine again or anew; review. 2. Law To question (a witness) again after cross-examination. the uniformity question in the presence of a general income tax. They show that the fundamental difference between the two categories of good remains. The ability to levy a general income tax does not allow uniform tax treatment of the two categories even if preferences are separable in labor supply and goods. Nevertheless, Cremer and Gahvari (1995b, 1999) are able to prove that with a general income tax the noncommitted goods must be taxed uniformly. Furthermore, they prove that if the subutilities of noncommitted and precommitted goods are strongly separable, then the precommitted goods must also be taxed uniformly. It is apparent that in the absence of a general income tax, the subutilities associated with the two categories of goods must be severely restricted in order for the tax rates within each category to be uniform. We will show that if the subutility of noncommitted goods is logarithmic, then noncommitted goods must be taxed uniformly. This result requires no further restriction on the subutility for precommitted goods except for its strong separability sep·a·ra·ble adj. Possible to separate: separable sheets of paper. sep from noncommitted goods. Furthermore, we show that if the subutility for precommitted goods is also logarithmic, then these goods should also be taxed uniformly, albeit, as proved by Cremer and Gahvari (1995a), at a lower rate. The intuition intuition, in philosophy, way of knowing directly; immediate apprehension. The Greeks understood intuition to be the grasp of universal principles by the intelligence (nous), as distinguished from the fleeting impressions of the senses. for our uniformity result can best be seen in the light of the intuition behind the optimality of the differential tax treatment of the two categories. That the precommitted goods must be taxed at a lower rate than the noncommitted goods may be explained as follows. A risk-averse Risk-averse Describes an investor who, when faced with two investments with the same expected return but different risks, prefers the one with the lower risk. individual being concerned about a low realization of the wage tends, in the absence of insurance markets, to "under-consume" the precommitted goods (as compared to a situation where full insurance is available). Taxing these goods at a lower rate relative to noncommitted goods counters this. It induces individuals to consume a higher amount of precommitted goods. In this way, it reduces the gap between the total after-tax expenditures in cases of bad and good states of nature, thus providing individuals with some degree of insurance. It also allows them to achieve this reduction by changing their consumption of different goods differently. The above discussion leads one to understand the reason behind our uniformity result. Logarithmic preferences imply that whereas aggregate expenditures on noncommitted and precommitted goods respond to commodity taxes, within each category, expenditure share of every commodity is fixed. Now, in our setup, commodity taxes are useful as an insurance mechanism by determining how much resources one has after the resolution of uncertainty as compared to before. Consequently, commodity taxation can help only through differentiating between the categories of goods and not within them. The intuition for uniformity of tax rates within a category thus follows that of the uniformity result in general (i.e., in the absence of uncertainty). When expenditure shares are fixed, there are no commodities that are consumed con·sume v. con·sumed, con·sum·ing, con·sumes v.tr. 1. To take in as food; eat or drink up. See Synonyms at eat. 2. a. disproportionately dis·pro·por·tion·ate adj. Out of proportion, as in size, shape, or amount. dis  pro·por more by the rich or by
the poor. It is this property that makes differential commodity taxes
sterile sterile /ster·ile/ (ster´il)1. unable to produce offspring. 2. aseptic. ster·ile adj. 1. Not producing or incapable of producing offspring. 2. as far as redistribution re·dis·tri·bu·tion n. 1. The act or process of redistributing. 2. An economic theory or policy that advocates reducing inequalities in the distribution of wealth. is concerned. It is true that the result also holds with preferences that give rise to linear Engel curves (with positive intercepts) - namely with goods that are classified as luxuries and necessities. However, this is simply a matter of "chance." Efficiency calls for taxation of necessities, whereas redistribution calls for taxing luxuries. The two effects cancel given the types of preferences that are stipulated. The paper also shows that for the preferences discussed, the tax rates on both categories must be nonzero non·ze·ro adj. Not equal to zero. nonzero Not equal to zero. . Earlier, Cremer and Gahvari (1995a) had been able to prove only that the noncommitted goods must be taxed (relative to the wage and at a positive rate). The importance of this extension arises because one may a priori a priori In epistemology, knowledge that is independent of all particular experiences, as opposed to a posteriori (or empirical) knowledge, which derives from experience. suspect that, relative to the wage, the precommitted goods should not be taxed at all. After all, their consumption levels are, as with the labor supply, chosen before the resolution of uncertainty and thus provide no insurance.(5) We will show why this intuition is wrong. 2. The Model The model is based on Cremer and Gahvari (1995a). A large number of identical individuals face uncertainty about their wage when deciding how much to work and to consume. The uncertain wage, w, is continuously distributed over some interval. Goods are classified according to according to prep. 1. As stated or indicated by; on the authority of: according to historians. 2. In keeping with: according to instructions. 3. whether they are determined before or after the resolution of uncertainty. There are two goods in each category.(6) The production technology is linear so that the producer prices are fixed. The vector of precommitted goods is denoted by [Mathematical Expression A group of characters or symbols representing a quantity or an operation. See arithmetic expression. Omitted] and noncommitted goods by [Mathematical Expression Omitted]. Labor supply, L, is also committed to prior to the resolution of uncertainty. The time endowment A transfer, generally as a gift, of money or property to an institution for a particular purpose. The bestowal of money as a permanent fund, the income of which is to be used for the benefit of a charity, college, or other institution. is normalized to one. Preferences are separable in [Mathematical Expression Omitted], [Mathematical Expression Omitted], and L; they are represented by [Mathematical Expression Omitted], (1) where U is quasi-concave, twice continuously differentiable dif·fer·en·tia·ble adj. 1. That can be differentiated: differentiable species. 2. Mathematics Possessing a derivative. and strictly increasing in [Mathematical Expression Omitted], [Mathematical Expression Omitted], and 1 - L. To ensure risk aversion risk aversion The tendency of investors to avoid risky investments. Thus, if two investments offer the same expected yield but have different risk characteristics, investors will choose the one with the lowest variability in returns. , also assume that u is strictly concave Concave Property that a curve is below a straight line connecting two end points. If the curve falls above the straight line, it is called convex. . Denote de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. the nonwage income by G, the price of [x.sub.k] by [p.sub.k] and of [y.sub.i] by [q.sub.i] (k and i run from 1 to 2). The individual's budget constraint A Budget Constraint represents the combinations of goods and services that a consumer can purchase given current prices and his income. Consumer theory uses the concepts of a budget constraint and a preference ordering to analyze consumer choices. , which must hold for every w, is given by [summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) of] [p.sub.k][x.sub.k](w) where k = 1 to 2 + [summation of] [q.sub.i][y.sub.i] where i = 1 to 2 = wL + G. (2) The individual's decision problem may be decomposed de·com·pose v. de·com·posed, de·com·pos·ing, de·com·pos·es v.tr. 1. To separate into components or basic elements. 2. To cause to rot. v.intr. 1. into two parts: he chooses L and [Mathematical Expression Omitted] in the first stage and [Mathematical Expression Omitted] in the second stage. Of course, in the first stage, the individual will take into account his (optimal) second-stage decisions regarding [Mathematical Expression Omitted]. Throughout the paper, we assume that the subutility in [Mathematical Expression Omitted] is logarithmic. That is, [Mathematical Expression Omitted]. (3) For the time being, however, we impose no further restrictions on the subutility for [Mathematical Expression Omitted]. Formally, the individual's optimization problem In computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. More formally, an optimization problem  is a quadruple can be stated as follows. In the
second stage, after the resolution of the uncertainty, the individual
has
I [equivalent to] wL + G - [summation of] [q.sub.i][y.sub.i] where i = 1 to 2 (4) to spend on [x.sub.k]'s. He solves a standard consumer problem under certainty and chooses [Mathematical Expression Omitted] to maximize [Mathematical Expression Omitted] subject to having an income equal to I. This optimization problem yields the demand functions [Mathematical Expression Omitted]. (5) Note that [Mathematical Expression Omitted] is the standard (ordinary) demand function possessing all the traditional properties associated with logarithmic preferences. Moreover, substituting for [Mathematical Expression Omitted] from Equation 5 into Equation 3 yields the second-stage indirect utility function In economics, a consumer's indirect utility function  gives the consumer's maximal utility when faced with a price level
[Mathematical Expression Omitted]. Denoting the marginal utility marginal utility In economics, the additional satisfaction or benefit (utility) that a consumer derives from buying an additional unit of a commodity or service. The law of diminishing utility implies that utility or benefit is inversely related to the number of units of income in the second stage by [Alpha] [equivalent to] [Delta] [Upsilon up·si·lon or yp·si·lon n. Symbol  The 20th letter of the Greek alphabet. ]/[Delta]I,
we will have, from Roy's identity Roy's identity (named for French economist Rene Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma relates the ordinary demand function to the derivatives of the indirect utility function. , [Mathematical Expression Omitted]. In the first stage, w and thus I are random variables. The problem of the individual is to choose [Mathematical Expression Omitted] and L to maximize [Mathematical Expression Omitted], (6) where E denotes the expectation operator and I is defined by Equation 4. Assuming an interior solution, the first-order conditions after a bit of algebraic 1. (language) ALGEBRAIC - An early system on MIT's Whirlwind. [CACM 2(5):16 (May 1959)]. 2. (theory) algebraic - In domain theory, a complete partial order is algebraic if every element is the least upper bound of some chain of compact elements. manipulation can be written as [Mathematical Expression Omitted], (7a) [Mathematical Expression Omitted], (7b) where [f.sub.i] denotes the partial derivative partial derivative In differential calculus, the derivative of a function of several variables with respect to change in just one of its variables. Partial derivatives are useful in analyzing surfaces for maximum and minimum points and give rise to partial differential of f with respect to [y.sub.i]. Equations 7a and 7b, along with Equation 4, determine the supply of L and the demand for [y.sub.i] as functions of [Mathematical Expression Omitted] and [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. (Of course, the functional forms for L and [y.sub.i] also depend on the probability distribution Probability distribution A function that describes all the values a random variable can take and the probability associated with each. Also called a probability function. probability distribution of w.) Next, substituting for L and [y.sub.i] in Equation 4 yields [Mathematical Expression Omitted]. It is important to note that L, [y.sub.i] (i = 1, 2) and I are all independent of [Mathematical Expression Omitted]. This property is due to the logarithmic specification of preferences for [Mathematical Expression Omitted]. Substituting for [Mathematical Expression Omitted] in Equation 5 then gives the demand function for [Mathematical Expression Omitted]. Formally, we have [Mathematical Expression Omitted]. (8) This demand function, unlike those for [Mathematical Expression Omitted] and the supply function for L, does depend on [Mathematical Expression Omitted]. Note that before the resolution of the uncertainty, [x.sub.k]'s are random variables whereas L and [y.sub.i]'s are not. Note also that whereas [Mathematical Expression Omitted] is a standard logarithmic demand, [x.sub.k] is not. Finally, following Cremer and Gahvari (1995a), one can define the compensated demand functions for L, [y.sub.i], and [x.sub.k] (denoted by [L.sup.c], [Mathematical Expression Omitted] and [Mathematical Expression Omitted]) and show that the following "Slutsky type" decompositions apply, for all i, j, k, s = 1, 2, [Mathematical Expression Omitted], (9a) [Mathematical Expression Omitted], (9b) with similar expressions for [Mathematical Expression Omitted] and [L.sub.c]. Given these decompositions, and using Identity 8, straightforward calculations result in the relationships reported in Table 1, where coy coy adj. coy·er, coy·est 1. Tending to avoid people and social situations; reserved. 2. Affectedly and usually flirtatiously shy or modest. See Synonyms at shy1. 3. (., .) denotes covariance Covariance A measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covariance means returns vary inversely. . [TABULAR tab·u·lar adj. 1. Having a plane surface; flat. 2. Organized as a table or list. 3. Calculated by means of a table. tabular resembling a table. DATA FOR TABLE 1 OMITTED] 3. Optimal Taxation of Noncommitted Goods The government's tax instruments consist of commodity taxes and a linear income tax with a lump-sum element. Denote the tax rate on [y.sub.i] by [[Tau].sub.i] and on [x.sub.k] by [t.sub.k]. Normalize normalize to convert a set of data by, for example, converting them to logarithms or reciprocals so that their previous non-normal distribution is converted to a normal one. all producer prices at one so that [q.sub.i] = 1 + [[Tau].sub.i] and [p.sub.k] = 1 + [t.sub.k]. Regarding the income tax parameters, use G to denote the lump-sum element of the linear tax system. Given that a full set of commodity taxes is being used, the tax rate on the wage is superfluous su·per·flu·ous adj. Being beyond what is required or sufficient. [Middle English, from Old French superflueux, from Latin superfluus, from superfluere, to overflow : and can be set at zero without imposing any restrictions.(7) The per-capita tax revenue requirement is [R.sub.0], where [R.sub.0] [less than or equal to] 0. The government's problem is to set [[Tau].sub.i]'s, [t.sub.k]'s, and G to maximize the (indirect) utility of a representative individual, subject to the tax revenue constraint Constraint A restriction on the natural degrees of freedom of a system. If n and m are the numbers of the natural and actual degrees of freedom, the difference n - m is the number of constraints. . The indirect utility is found by substituting [Mathematical Expression Omitted], [Mathematical Expression Omitted], and [Mathematical Expression Omitted] derived in section 2 in Equation 6. This yields the maximum value of V, denoted by [V.sup.*], as a function of [Mathematical Expression Omitted], [Mathematical Expression Omitted], and G. One then maximizes [V.sup.*] subject to the following revenue constraint [Mathematical Expression Omitted] (10) where the population is assumed large enough so that the realized means of the random variables will be equal to their expected values Expected value The weighted average of a probability distribution. Also known as the mean value. . We are now in a position to state our first main result. PROPOSITION 1. Assume the [Mathematical Expression Omitted] subutility is logarithmic. Then, the optimal tax rates on [x.sub.k]'s are uniform. PROOF. From Proposition 1 in Cremer and Gahvari (1995a) we have that optimal tax rates satisfy [Mathematical Expression Omitted] (11) Substituting the expressions derived in Table 1 into these two equations, dividing the first equation (s = 1) by [a.sub.1]/[p.sub.1] and the second equation (s = 2) by [a.sub.2]/[p.sub.2], subtracting one equation from the other, and simplifying results in [Mathematical Expression Omitted] (12) Given [p.sub.i] = 1 + [t.sub.i], Equation 12 implies [t.sub.1] = [t.sub.2]. This is an interesting result. It tells us that if the subutility in noncommitted goods is logarithmic, these goods must be taxed uniformly regardless of the nature of preferences for precommitted goods. We have learned from Cremer and Gahvari (1995a) that differential taxation of precommitted and noncommitted goods is useful because this corrects the inefficiencies due to lack of insurance. Whether or not the goods within each category should also be taxed differentially thus depends on whether or not such a differential treatment can bring about a higher degree of insurance. The degree of insurance depends on the relative importance of certain income G to uncertain income wL, in the total expenditures on all goods. It is reflected in the relative size of the total expenditures on [Mathematical Expression Omitted] versus total expenditures on [Mathematical Expression Omitted]. With logarithmic preferences, the total expenditures on [Mathematical Expression Omitted] cannot be affected by the taxes imposed on these goods. Consequently, differential taxation of these goods is not useful. Note, on the other hand, that differential taxation of [y.sub.i]'s is useful because it affects the size of expenditures on [y.sub.i]'s. 4. Optimal Taxation of Precommitted Goods The logarithmic specification for the x subutility was sufficient to yield a uniformity result for taxing [x.sub.k]'s. On the other hand, without restricting the [Mathematical Expression Omitted] subutility, the tax rates on [y.sub.i]'s will continue to be nonuniform. It is nevertheless the case that if in addition the subutility for [Mathematical Expression Omitted] is logarithmic, their taxes must also be uniform. To prove this, we will first derive certain additional properties concerning the demand functions for [y.sub.i]'s and the supply function for L. They will facilitate the derivation derivation, in grammar: see inflection. of our result. Assume, here and in the rest of the paper, that f([y.sub.1], [y.sub.2]) = [summation of] [b.sub.i]ln [y.sub.i] where i = 1 to 2. so that preferences (Eqn. 1) are now represented by [Mathematical Expression Omitted]. (13) Given these preferences, the first-order conditions in Equation 7b will become [Mathematical Expression Omitted], i = 1, 2. It is important to point out that despite the logarithmic specification for the [Mathematical Expression Omitted] subutility, in addition to the [Mathematical Expression Omitted] subutility, the resulting demand functions for [y.sub.i]'s will not be logarithmic,(8) This must be obvious from the procedure for deriving these functions that constrains y and L to be determined independently of w. These are Equations 7a and 14. The above observations notwithstanding, the following lemma lemma (lĕm`ə): see theorem. (logic) lemma - A result already proved, which is needed in the proof of some further result. proves that the demand functions for L and y have very simple structures. LEMMA 1. (i) L is a function of G only. (ii) The demand for [y.sub.i] has the following specific structure [y.sub.i] = [b.sub.i] / ([b.sub.1] + [b.sub.2])[q.sub.i] [Phi] (G), i = 1, 2, where [Phi](G) is some function of G. PROOF. Let y [equivalent to] [q.sub.1][y.sub.1] + [q.sub.2][y.sub.2]; summing Equation 14 over i = 1, 2 then results in [b.sub.1] + [b.sub.2] ([a.sub.1] + [a.sub.2])E[(wL + G - y).sup.-1] = y. (15) Equations 7a and 15 determine L and y as functions of G only. This proves (i). Denote the solution for y as [Phi](G) so that we have [q.sub.1][y.sub.1] + [q.sub.2][y.sub.2] = [Phi](G). (16) In addition, from Equation 14, one has [q.sub.1][y.sub.1]/[q.sub.2][y.sub.2] = [b.sub.1]/[b.sub.2]. (17) Solving Equations 16 and 17 for [y.sub.1] and [y.sub.2] proves (ii). Note that Lemma 1 rests on logarithmic specifications for [Mathematical Expression Omitted] and [Mathematical Expression Omitted] subutilities. Next we restate re·state tr.v. re·stat·ed, re·stat·ing, re·states To state again or in a new form. See Synonyms at repeat. re·state Lemma 2 from Cremer and Gahvari (1995a). This lemma, unlike Lemma 1, holds for the general specification of [Mathematical Expression Omitted] and [Mathematical Expression Omitted] subutilities. LEMMA 2. (a) The following relationship holds for all values of [Mathematical Expression Omitted] and [Mathematical Expression Omitted]: [Mathematical Expression Omitted] (18) (b) The following relationship holds at the optimal values of the tax rates: [Mathematical Expression Omitted] (19) where from Lemma 1 in Cremer and Gahvari (1995a), [Delta][L.sup.c]/[Delta][q.sub.j] [not equal to] 0. We are now in a position to prove our second major result. PROPOSITION 2. Assume [Mathematical Expression Omitted] and [Mathematical Expression Omitted] subutilities are logarithmic. Then, the optimal tax rates on precommitted goods are uniform. PROOF. Set [t.sub.1] = [t.sub.2] = t, multiply mul·ti·ply v. 1. To increase the amount, number, or degree of. 2. To breed or propagate. Equation 19 by 1 + t, and subtract A relational DBMS operation that generates a third file from all the records in one file that are not in a second file. the resulting expression from Equation 18. We will have [Mathematical Expression Omitted]. (20) Dividing Equation 20 by [y.sub.j], writing out the equations for j = 1, 2 and subtracting from one another yields [Mathematical Expression Omitted]. (21) Substituting from the Slutsky-type decompositions (Eqn. 9a) into Equation 21, while noting from Letorna 1 that for i [not equal to] j, [Delta][y.sub.i]/[Delta][q.sub.j] = [Delta]L/[Delta][q.sub.j] = 0, we obtain [Mathematical Expression Omitted] (22) Moreover, given the expression for [q.sub.i][y.sub.i] in part (ii) of Lemma 1, we can easily show by differentiation that [Mathematical Expression Omitted] (23) Substituting from Equation 23 in Equation 22 and simplifying yields [[Tau].sub.1] = [[Tau].sub.2].(9) This means that although commodity taxes are necessary, they have a simple structure. It is sufficient to levy one tax rate per each category of goods. No tax differentials are needed within categories. The intuition for this result is simple. As argued earlier, logarithmic preferences imply that while aggregate expenditures on [Mathematical Expression Omitted] and [Mathematical Expression Omitted] categories respond to commodity taxes, within each category, expenditure share of every commodity is fixed. (From Equation [Mathematical Expression Omitted], and from Equation 14, [q.sub.i][y.sub.1]/[q.sub.2][y.sub.2] = [b.sub.1][b.sub.2].) Now, in our setup, commodity taxes are useful as an insurance mechanism by determining how much resources one has after the resolution of uncertainty as compared to before. Consequently, commodity taxation can help only through differentiating between the categories of goods and not within them. We next turn to our third result and show that for preferences represented by Equation 13, [[Tau].sub.1] = [[Tau].sub.2] [not equal to] 0. To facilitate the derivation of this result, we begin by stating and proving another lemma. LEMMA 3. The compensated demand functions [Mathematical Expression Omitted] in general satisfy: [Mathematical Expression Omitted], j = 1, 2. PROOF. From Equation 2, and the definition of I, we have [Mathematical Expression Omitted]. With [Mathematical Expression Omitted], from (i) and (ii) in Lemma 1 it follows that E(I) is a function of G only. Consequently, [[Sigma SIGMA - A scientific visual programming environment from NASA. http://fi-www.arc.nasa.gov/fia/projects/sigma/. ].sub.k][p.sub.k]E([x.sub.k]) will also be a function of G only. Differentiating it partially with respect to G and [q.sub.j] then (in general) implies [Mathematical Expression Omitted], (24a) [Mathematical Expression Omitted]. (24b) Now from the Slutsky-type decomposition decomposition /de·com·po·si·tion/ (de-kom?pah-zish´un) the separation of compound bodies into their constituent principles. de·com·po·si·tion n. 1. (Eqn. 9a) for [x.sub.k], one has [Mathematical Expression Omitted]. Substituting from Equations 24a and 24b in above proves the lemma. Note that Lemma 3 is based on preference structure stipulated by Equation 13. We are now in a position to state our fourth result. This is summarized as: PROPOSITION 3. If [Mathematical Expression Omitted] and [Mathematical Expression Omitted] subutilities are logarithmic, the tax rate on [y.sub.i]'s must be nonzero. That is, [[Tau].sub.1] = [[Tau].sub.2] = 0 cannot be optimal. PROOF. The proof is by contradiction CONTRADICTION. The incompatibility, contrariety, and evident opposition of two ideas, which are the subject of one and the same proposition. 2. In general, when a party accused of a crime contradicts himself, it is presumed he does so because he is guilty for . From Proposition 1 in Cremer and Gahvari (1995a) we have that the optimal tax rates satisfy [Mathematical Expression Omitted] (25) It then follows from Equation 25 and Proposition 2 that when [t.sub.1] = [t.sub.2] = t is set optimally, if [[Tau].sub.1] = [[Tau].sub.2] = 0: [Mathematical Expression Omitted] (26) Now from Equation 5, we have [Mathematical Expression Omitted], implying that [Mathematical Expression Omitted]. Differentiating this relationship with respect to [q.sub.1] yields [Mathematical Expression Omitted] (27) With [p.sub.i] = 1 + [t.sub.i] and [t.sub.2] = [t.sub.2] [not equal to] 0 (a result from Cremer and Gahvari [1995a]), it follows from Equations 26 and 27 that [Mathematical Expression Omitted]. but this contradicts Lemma 3. The intuition is this. Although taxation of [y.sub.i]'s does not provide any insurance (in that their consumption will be unaffected by the particular realization of the wage), it does not follow that these goods should not be taxed. What is required for optimality is that the net marginal cost Marginal cost The increase or decrease in a firm's total cost of production as a result of changing production by one unit. marginal cost The additional cost needed to produce or purchase one more unit of a good or service. of the tax on [y.sub.i] equals its marginal benefit. Now, having a zero marginal benefit for [[Tau].sub.i] only implies that, at the optimum, the net marginal cost of [[Tau].sub.i] should also be zero. Given that there are other taxes in the system, this does not mean that [[Tau].sub.i] = 0. Indeed, it must be the case here that the net marginal cost of [[Tau].sub.i] is zero when [[Tau].sub.i] [not equal to] 0. As a final point, it is important to emphasize the relevance of logarithmic preferences for [Mathematical Expression Omitted] and [Mathematical Expression Omitted] for Cremer and Gahvari's (1995a) result concerning the relative size of the tax rates on noncommitted versus precommitted goods. They had proved that if the goods within each category are to be taxed uniformly, the tax rate on goods that are precommitted must be smaller than that on noncommitted goods. Their result critically rests on the assumption that the goods within each category will be taxed uniformly. The logarithmic preferences thus provide an important avenue for the application of their result. We state this result as COROLLARY corollary: see theorem. 1. Assume [Mathematical Expression Omitted] and [Mathematical Expression Omitted] subutilities are logarithmic. Then, the optimal tax rate on [y.sub.i]'s must be smaller than-the optimal tax rate on [x.sub.s]'s. In the absence of insurance markets, risk-averse persons tend to "under-consume" the precommitted goods (as compared to a situation where full insurance is available). Taxing these goods at a lower rate relative to noncommitted goods counters this. In this way, it reduces the gap between the total after tax expenditures in cases of bad and good states of nature, thus providing individuals with some measure of insurance. 5. Concluding Remarks Differential commodity taxation is a useful instrument of optimal tax policy in the presence of uncertainty. There exists a fundamental difference between goods that consumers commit to their consumption prior to the resolution of uncertainty and to those that are determined after. The difference makes their differential tax treatment optimal. This paper has shown that optimal commodity taxes have nevertheless a very simple structure if preferences are separable between labor and goods and logarithmic in the latter. Although taxes differ according to whether consumers commit to the consumption of goods before the resolution of uncertainty or after, the tax treatment of each type of good must be uniform. All that is needed is for the government to levy two sets of tax rates. As a prescription for applied tax policy, this is very useful because of its sheer simplicity. The paper has also shown that the uniformity result for the two "sectors" can be separated. It is possible to have uniform taxes in one but nonuniform taxes in the other. However, the two categories are not symmetric No difference in opposing modes. It typically refers to speed. For example, in symmetric operations, it takes the same time to compress and encrypt data as it does to decompress and decrypt it. Contrast with asymmetric. (mathematics) symmetric - 1. . In particular, to have uniform taxes for noncommitted goods, all that matters is preferences for these goods. On the other hand, our result suggests that uniform taxation of precommitted goods is based on the structure of preferences for both types of goods. This is also a useful policy result. We conclude by observing that any attempts at devising the "right" mix of commodity taxes must rest on a far better empirical understanding of the structure of preferences. This has not been the subject of the current study. However, this is where future research must concentrate. We thank Jonathan Hamilton Hamilton, city, Bermuda Hamilton, city (1990 est. pop. 3,100), capital of Bermuda, on Bermuda Island. It is a port at the head of Great Sound, a huge lagoon and deepwater harbor protected by coral reefs. and an anonymous referee A judicial officer who presides over civil hearings but usually does not have the authority or power to render judgment. Referees are usually appointed by a judge in the district in which the judge presides. for useful suggestions. 1 This allows them to short-circuit the timing question, which is a crucial aspect of insurance, and to recast re·cast tr.v. re·cast, re·cast·ing, re·casts 1. To mold again: recast a bell. 2. their problem in a framework similar to Atkinson and Stiglitz's (1976) original piece. Varian (1980), Diamond, Helms, and Mirrlees (1980), and Hamilton (1987), on the other hand, have explicit two-period models in which first-period consumption takes place before, and the second-period consumption after, the resolution of uncertainty. 2 By goods we mean nonleisure goods. This terminology is adopted for brevity Brevity Adonis’ garden of short life. [Br. Lit.: I Henry IV] bubbles symbolic of transitoriness of life. [Art: Hall, 54] cherry fair cherry orchards where fruit was briefly sold; symbolic of transience. . 3 The ineffectiveness in·ef·fec·tive adj. 1. Not producing an intended effect; ineffectual: an ineffective plea. 2. Inadequate; incompetent: an ineffective teacher. of commodity taxes and their proportionately uniform tax treatment boil down boil 1 v. boiled, boil·ing, boils v.intr. 1. a. To change from a liquid to a vapor by the application of heat: to the same thing. In the absence of exogenous Exogenous Describes facts outside the control of the firm. Converse of endogenous. incomes, the government will have an extra degree of freedom in setting its income and commodity tax instruments. This reflects the fact that demand function will then be homogeneous The same. Contrast with heterogeneous. homogeneous - (Or "homogenous") Of uniform nature, similar in kind. 1. In the context of distributed systems, middleware makes heterogeneous systems appear as a homogeneous entity. For example see: interoperable network. of degree zero in consumer prices, including the price of leisure, and lump-sum grants, if any. In consequence, the government can, without any loss of generality gen·er·al·i·ty n. pl. gen·er·al·i·ties 1. The state or quality of being general. 2. An observation or principle having general application; a generalization. 3. , set one of the commodity taxes at zero (i.e., set one of the commodity prices at one). Under this normalization In relational database management, a process that breaks down data into record groups for efficient processing. There are six stages. By the third stage (third normal form), data are identified only by the key field in their record. , uniform rates imply absence of commodity taxes. 4 Deaton (1981) reexamines the uniformity issue on the basis of quasi- or implicit separability. Besley and Jewitt (1995) also discuss the uniformity issue with particular emphasis on the circumstances CIRCUMSTANCES, evidence. The particulars which accompany a fact. 2. The facts proved are either possible or impossible, ordinary and probable, or extraordinary and improbable, recent or ancient; they may have happened near us, or afar off; they are public or under which first-order conditions are sufficient for the problem. 5 Recall that insurance is achieved by reducing the relative importance of uncertain wages to certain income (tax rebates tax rebate n → devolución f de impuestos; reembolso fiscal tax rebate n → ristourne f d'impôt tax rebate ) in determining expenditures that are uncertain, thus reducing the variability in their consumption. 6 This assumption is made to simplify the exposition exposition or exhibition, term frequently applied to an organized public fair or display of industrial and artistic productions, designed usually to promote trade and to reflect cultural progress. ; none of our results depends on it. 7 This implies that by "a commodity tax" we mean a tax on the price of that commodity relative to the wage. 8 Nor will the demands for [x.sub.k]'s be logarithmic. (These will not change even if [Phi][[center dot]] was logarithmic.) On the other hand, as was shown earlier, [Mathematical Expression Omitted] is a standard logarithmic demand. 9 It is important to emphasize that Equation 23 is not the familiar result that the cross-price elasticity of ordinary demand curve is - 1 when preferences are logarithmic. As we pointed out earlier, unlike the second stage, the first-stage first-stage said of larva; the first of several larval stages. demand curves are not the standard demand curves. One cannot thus simply assume that they necessarily possess the traditional properties of demand functions. References Atkinson, A. B. 1977. Optimal taxation and the direct versus indirect tax controversy. Canadian Canadian (kənā`dēən), river, 906 mi (1,458 km) long, rising in NE New Mexico. and flowing E across N Texas and central Oklahoma into the Arkansas River in E Oklahoma. Journal of Economics 10:590-606. Atkinson, A. B., and J. E. Stiglitz. 1976. The design of tax structure: Direct versus indirect taxation. Journal of Public Economics 6:55-75. Besley, T., and I. Jewitt. 1995. Uniform taxation and consumer preferences. Journal of Public' Economics 58:73-84. Cremer, H., and E Gahvari. 1995a. Uncertainty and optimal taxation: In defense of commodity taxes. Journal of Public Economics 56:291-310. Cremer, H., and F. Gahvari. 1995b. Uncertainty, optimal taxation and the direct versus indirect tax controversy. Economic Journal 105:1165-79. Cremer, H., and E Gahvari. 1999. Uncertainty, commitment and optimal taxation. Journal of Public Economic Theory 1:51-70. Deaton, A. 1979. Optimally uniform commodity taxes. Economics Letters Economics Letters is a scholarly peer-reviewed journal of economics that publishes concise communications (letters) that provide a means of rapid and efficient dissemination of new results, models and methods in all fields of economic research. Published by Elsevier. 2:357-61. Deaton, A. 1981. Optimal taxes and the structure of preferences. Econometrica 49:1245-60. Deaton, A., and N. Stern. 1986. Optimally uniform commodity taxes, taste differences and lump-sum grants. Economics Letters 20:263-6. Diamond, P. A., L. J. Helms, and J. A. Mirrlees. 1980. Optimal taxation in a stochastic By guesswork; by chance; using or containing random values. stochastic - probabilistic economy: A Cobb-Douglas example. Journal of Public Economics 14:1-29. Hamilton, J. 1987. Optimal wage and income taxation with wage uncertainty. International Economic Review 28:373-88. Mirrlees, J. A. 1976. Optimal tax theory: A synthesis. Journal of Public Economics 6:327-58. Sandmo, A. 1974. A note on the structure of optimal taxation. American Economic Review 64:701-6. Varian, H. R. 1980. Redistributive taxation as social insurance. Journal of Public Economics 14:49-68. |
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