metadata: authors: 0: author: Kakwani, Nanak
1: author: Ray, Ranjan
docna: 0: docna: Optimal commodity taxes under rationing
docty: Policy Research Working Paper
subtopic: Tax Administration,Macro-Fiscal Policy,Taxation & Subsidies,Economic Adjustment and Lending,Tax Law,Disability,Public Sector Management and Reform,Public Sector Economics,Tax Policy,Services & Transfers to Poor,International Trade and Trade Rules,Economic Assistance,Access of Poor to Social Services,Public Finance Decentralization and
            Poverty Reduction
teratopic: Public Sector Development
abstracts: cdata!: How useful and relevant are the results
            of standard optimal commodity tax models when one or more
            commodities are rationed? This paper investigates the
            implications of optimal commodity taxation under rationing.
            In a single person economy, optimal policy dictates that the
            rationed commodity bears the entire tax. The implication for
            developing countries is that if the government has a fixed
            budget to subsidize certain commodities, optimal policy will
            be to subsidize only the rationed commodities. In a multi
            person economy, optimal policy will tax all nonrationed
            commodities at an infinite rate if the rule is that taxes on
            all commodities are proportional to prices. The more a
            society is concerned about inequality, the greater the tax
            should be on nonrationed commodities. The alternative model
            presented here overcomes some of the restrictive features of
            the previous rationing model.
repnme: repnme: Optimal commodity taxes under rationing
display_title: Optimal commodity taxes under rationing
listing_relative_url: /research/1989/06/700509/optimal-commodity-taxes-under-rationing
topicv3: Human Resources m1326304 1211,Trade m1326924 698,Social Assistance m1327201 1466,Taxation m1327615 1121,Tax Policy m1327616 983
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Public Sector Development
origu: Policy Plan & Prog Review Dept
no_of_pages: 28
model_extractions: **Plc,Planning, and Research**

# **Welfare and Human Resourc**

How useful and relcvant are the rcsults of standard optimal commodity tax models when one or more commodities are rationed? Kakwani and Ray investigatcd the implications of optimal commodity taxation under ratior ng and reached these conclusions:

- In a single-person economy, optimal policy dictates that the rationed commodity bears the entire tax. The implication for developing countries using ihis model is that if the government has a fixed budget to subsidize certain commodities, optimal policy will be to subsidize only the iationed commodities, such as food.

- In a many-person economy (which reflects reality), optimal policy will tax all nonrationed

commoo.es at an infinite rate if the rule is that taxes on all commoditit. are proportional to prices.

* The widely used linear expenditure system cannot be used to find a sensible optimal commodity tax structure under rationing.

* The more a society is concemed out inequality, the greater the tax should be on nonrationed commodities.

Kakwani and Ray present an alternative (more realistic) model of rationing that overcomes some of the restrictive features of the previous rationing model.

This paper is a product of the Welfare and Human Resources Division, Population and Human Resources Department. Copies are a"lailable free from the World Bank, 1818 H Street NW, Washington DC 20433. Please contact Maria Felix, room No. S9- 114, extension 33724 (23 pages).

Thc PPR Working Paper Scrics disscminates thc findings of work under way in thc Bank's Policy, Planning, and Rcscarci Complex. An objectivc of thc series is to get thesc findings out quickly, even if prcsentations are lcss than fully polished. The findings, interpretations, and conclusions in thcsc papers do not necessarily represent official policy of the Bank.

Produccd at thc PPR Dissemination Ccnter
Optimal Commodity Taxes in the Presence of Rationing by Nanak Kakwani* and Ranjan Ray**

# Table of Contents

|1.|Introduction|1|
|---|---|---|
|2.|Single Person Case|5|
|3.|Many Person Case under General Welfare Function|8|
|4.|A Specific Social Welfare Function|11|
|5.|Linear ExpenditureSystem|14|
|6.|An Alternative Method of Rationing|16|
|7.|Conclusion|19|
||References|22|

*Welfare and Human Resources Division, the World Bank and Department of Econometrics, The University of South Wales, Australia.

**Department of Econometrics, University of Manchester, England.
**OPTIMAL** COMMODITY **TAXES IN THE PRZSENCE** OF **RATIONING**

# 1. **INTRODUCTION**

Commodity taxes play an important role in resource mobilisation in many developing countries (see Tanzi (1987)]. The predominance of indirect taxes in the overall tax effort in such economies reflects, partly, the severe political constraints that exist on direct taxes. The authorities in many less developed economies have traditionally, and more so in recent years, also relied heavily on indirect taxes as the principal tool for securing redistribution Esee, for example, the recent volume edited by Newberj and Stern (1987)]. The commodity tax rate is, hence, a paraneter of considerable policy importance in a developing country.

The literature on optimal commodity taxes dates back to Ramsey's (1927) seminal contribution which sought to answer the question (p.47): 'if a given revenue is to be raised by proportionate taxes on some or all uses of income... being possibly at different rates, how should these rates be adjusted in order that the decrement of utility may be a minimum? Ramsey's original treatment was devoted exclusively to the efficiency issue and ignored the equity/redistributive aspects of commodity taxation. Diamon4 and Mirrlees (1971) introduced optimal taxation in a many person economy and used a social welfare function approach which allowed examination of the equity aspect of indirect taxation.

The 'efficiency' aspect of commodity taxation essentially involves the government raising a pre-specified amount of aggregate revenue in the 'least cost' way - i.e. the authorities choose a set of commodity tax rates that maximise social welfare, expressed as a function of individual welfaze/utility. It is clear that the solution, namely, the optimal tax rates would depend on: (a) society's perception of individual.welfare, and
2 (b) the assumed form for consumer preferences, namely, the adopted utility/demand functional forms. The importance of assumptions about consumer preferences for optimal commodity taxes is now widely accepted in the literature (see, for example, Atkinson and Stiglitz (1980), Ch. 12, 14)J. As Ray (1986) has, recently, shown using numerical demand estimates for India, optimal commodity taxes are extremely sensitive to departure from the linearity/separability assumptions of the Linear Expenditure System (LES) demand functional form that is widely used in tax studies (Atkinson and Stiglitz (1972), Harris and Mackinnon (1978), Deaton (1975), Ahmad and Stern (1984), Deaton and Stern (1986)].

Even setting aside the question of the wisdom of using LES demand estimates in view of the well known incompatibility of the linearity/separability assumptions with data (Deaton (1974), Ray (1985a)], there is the issue of the relevance of optimal commodity tax theory to developing countries because of its dependence on the common assumption underlying empirical dernand analysis, namely, that given his/her budget and commodity prices, the consumer has chosen freely the quantities that he/she is observed to have purchased. It is a matter of common observation that the assumption of free choice is unlikely to be valid in reality. There are various reasons why the quantities consumed may not be directly under the control of those who consume them. The most obvious example is that of necessities like Food, which are scarce in relation to damand in developing countries, and are subject to rationing. Since optimal commodity tax estimates are crucially dependent on the calculated price/expenditure elasticities, and since the latter are likely to be sensitive to the introduction of quantity constraint, the formulation of the standard Ramsey-Diamond-Mirrlees optimal tax problem in the presence of rationing is an important issue, especially in the case of LDCs. It is worth noting that while the optimal tax literature, especially in recent years, has
3 oncentrated on the link between optimal tax tbeory and consumer preferences, no attention bas been paid to the question of the robustness of optimal commodity taxes to the presence of quantity constraint on demand viz. rationing. That is the central motivation of this paper.

Not surprisingly, some of the earliest studies on consumer demand under rationing took place during and soon after the Warl (e.g. Rothbarth (1941), Tobin and Houthakker (1951)]. After a period of relative neglect in the sixties and early seventies, there has recently been renewed - interest in the subject e.g. Pollak (1969], Howard 11977], Weitzaan (19771, Neary and Roberts [1980], Sab 11987]. The methodology of generating rationed demand equations rests on the concept of 'virtual prices' introduced by Rothbarth [1941] and developed recently in the elegant analysis of Neary and Roberts (19801 using duality and expenditure functions. The 'virtual price' of the rationed coumodity is defined as that price which, in conjunction with actual prices of the non-rationed commodities and a 'virtual income' i.e. with the consumer compensated for the price change, will induce him to choose the rationed level as the result of (i.e. as if as) free choice. in their important contribution, Neary and Roberts (19801 bave shown that virtual prices must exist, that the support or virtual prices of unrationed goods coincide with actual prices, and that, via the use of the expenditure function, one can generate a matching system of rationed demand equations from an unrationed demand systeo. It is, however, interesting to note that in most of the recent exercises on rationed demand systems (e.g. Neary and Roberts (19801] Sah [19871), the LES has been chosen as an illustrativ **e** xample. This is of course, not without reason, since LES is one of the few demand systeas that 1See Tobin (1952) for a survey of these early studies.
4 generate an explicit expression for virtual price and, hence, the rationed LA,S is simple to write down and easy to estimate. A principal finding of this study, however, is that the rationed LES, in the context of optimal commodity tax theory, has some strong (and absurd) implications that should rule it out as a candidate for welfare applications.

In the context of the present exercise, only one commodity is rationed and the ramaining are freely chosen items. It is important to note that, because of the 'complete systems' framework, quantity constraint on one item will have an impact on the demand levels and price/expenditure elasticities of the unrationed items, and hence on their optimal commodity taxes. Section 2 investigates the structure of optimal comsodity taxes in a single person economy under rationing. The validity of the result, derived therein, in a many person economy is investigated in Section 3, using the most general Bergson-Samuelson form of social welfare function. In order to get some stronger results, an additive separable social welfare function of Atkinson's (1970) type is assumed in Section 4 with implications for optimal taxation under rationing outlined. The theory developed in the latter section is applied in Section 5 to the familiar case of the rationed LES, to derive some restrictive (indeed absurd) optimal tax recults that cast serious doubt on the rationing framework as far as considered in this paper. Section 6 presents an alternative (and a more realistic) model of rationing where the rationed commodity can be bought up to the quantity of ration in the ration shops at a subsidised price, and beyond in the open market at the market price. As this section demonstrates, this more realistic rationing model has significant implications for optimal tax theory and helps to overcome some of the restrictive features of the previous rationing model. The paper ends on the concluding note of section 7.
s

# 2. **SiNCLE** PnRSoN CASK

Suppose there are (nWl) commodities in the economy and one of them is rationed. Let rationed commodity *e q and denote the fixed quantity to be consumed by 40 and a price p is ch'rged for this commodity. Further, let q (q , q **_,...,q_** ) be the vector of n freely chosen goods and p = (p , p _...,p ) be the vector of prices. Then, the indirect utility function may be denoted by where x is the lump-sum income of the household.

The optimal tax problem under rationing involves maximizing u subject to the government revehue constraint ti being the tax rates. (p -t ) for i = 0,1,2,...,n, are the producer prices which are assumed to be fixed in face of changes in the pattern of demand. Setting up the Lagrangean and differentiating L with respect to po and pi and equating deriiatives to zero, the first order conditions may be written as where j varies from 1 to a.
6 Using Roy's theorem, it can be shown that where A - au is the marginal utility of income.

Let us write the Slutsky equations and where s*i is tIe compensated change in q with respect to p.; i and j vary **g1 1** 1 from 1 to n.

Since s** must be symmetric, (2.4) will imply s - 0 for all i which means that there is zero substitution possibility into or away from the rationed commodity. Then differentiating (2.1) with respect to p0 and p 0 yields and respectively. Utilizing the fact that sio **=** 0 for all i and sij = sji, (2.6) and (2.7) can be written as
7 (2.8) and (2.9) respectively, where ° pPi 1 tbeing the marginal propensity to spend, e* i and

-i 'iT° (2.11)

7ii being the compensated price elasticities for non-rationed commodities.

Substituting (2.3), (2.8) and (2.9) into (2.2) leads to . = 0 for all , so that for j **a** 1, 2,...,n. The only solution to (2.12) occurs when °j a 0 for all i 1,...,n; i.e. commodity tax rates on non-rationed goods are all equal to zero, so that rationed good bears the entire tax.

In the context of developing countries, this result has an important policy implication. If the government has a fixed budget to subsidize certain commodities, the optimai policy will be to subsidize only the rationed commodities (non-rationed commodities should receive zero subsidy).
8 It should be noted however, that this result is such less startling than it appears. The conventional wisdom in the optimal tax structure is that lump **sum** taxes are preferable to distortionary coamodity taxes. In the present context, tax on the rationed items is like a lump sum tax, since there is zero substitution possibility into or away from the rationed item. It is hardly surprising, therefore, that the rationed item bears the entire tax. Does this result generalize to the many person case? The answer is provided in the next section.

# 3. NM PERSON CASE UNDER GENERAL WELFARE FUNCTION

In a many person economy, the government is concerned with the maximization of a soc.al welfare function. We assume that the social welfare function is of the Bergson-Samuelson form which is a function of the utilities of the H individuals. The optimal taxation problem involves choosing ti (i **=** O.l...,n) to maximize V subject to the revenue constraint where S i qh being the aggregate demand of the ith commodity. **h I** Using the Lagrangean framework, the first order optimization conditions may be
9

**where** J varies from 1 to A.

Using Roy's identity where eV O u is the social marginal utility of income of the hth consumer or the welfare weight.

Differentiating (3.1) with respect to po and pi yields and which on substituting into (3.2) and using (3.3) and (3.4) give the optimization conditions as

[ ° hal ] **IQ + i'.g iPi tj!]** L U, **0** tJ [Rio + **i' oi i** 35

# Let us assume that taxes on all non-rationed commodities are

proportional to prices, i.e. a = a for all j = 1,2,...,n, then (3.5) simplifies to
10

# [q b l **°[Qj + &i'.Pi** ti]tlS U **#'i][Rio** + s oit] 36

The budget constraint for all consumers **may** be written as

**n** X S P. Q. + UP q

H where X = x xh is the total income of all consumsrs. Differentiating this ht equation with respect to p0 and p yields and which on substituting into (3.6) leads to when O are all not equal, (3.7) will hold true only if o * 1, which means that all non-rationed comodities must be taxed at infinite rate., In the case of a single person economy, the optimus tax policy implied zero taxes on all non-rationed co modities. Thus, the single person result does not generalize to the many person case.

2 This follows from the fact that °i = _ti -pi being the producer price so Pi+ti that _ei_ = 1.0 will hold true only if ti approaches infinity (in view of the fact that pi # 0 for all i).
11 The above conclusions may change, however if we do not impose the restriction that taxes on all non-rationed commodities are proportional to prices. in order to get stronger results it would be necessary to make some assumptions about the social welfare function. This is considered in the next section.

# 4. A SPECIFIC SOCIAL VELFARE FUNCTION

Ve assume that all consumers have identical tastes and differ only in income.3 The preferences are given by the indirect utility function u(q , pO , p, x). The government maximizes the social welfare function where a is the minimum income, f(x) the probability density function describing the distribution of total expenditure and c is a measure of the government's aversion to inequality. If c is zero, the social welfare is given by the mean utility level. As c approaches infinity, social welfare is sensitive only to the utility of the poorest consumer in the economy. Note that Atkinson's (1970) welfare function is defined over money incomes whereas here we use the utility function (see Deaton 1977).

The government's revenue constraint as given in (3.1) can also be written as 3The differences due to size and composition of households to which consumers belong can be taken into account under this framework by means of consumer unit scales (see for instance Deaton and Muellbauer (1980), Muellbauer (1974, 1980), Kakwani (1977, 1980) and more recently Ray (1985).
12 where E(qi) is the mathematical expectation of qi<sup>4</sup> .

Using Roy's theorem, it can be shown that and where X = au is the marginal utility of income as defined in (2.3). ax Utilizing Slutsky's equation it can be shown that and where 0 and 6j are defined in (2.10) and (2.11), respectively.

Substituting (4.3), (4.4), (4.5) and (4.6) into the first-order conditions (3.2) yields where It can be seen that when aj = a for all j = 1, 2,...,n, (4.7) will hold true only if aj = 1 for all j, which confirms our earlier result that 4The expectation is evaluated over the probability distribution of the total consumer expenditure. This formulation uses the assumption that the demand function is the same for all consumers.
13 all non-rationed goods in the many person case must be taxed at infinite rates.

Let us write where E(qjkj) is the average compensated change in the demand for good j as a result of imposing taxes on goods other than j. (4.8), thus gives us a measure of the substitutability of good j. Substituting (4.8) into (4.7) gives Since X is the marginal utility of income, it must be a decreasing function of income and e > 0 implies that u E decreases with x, so that E(u x) and qj will be negatively correlated which would mean which implies that T. < 1 for all j. Also as c increases, the magnitude of -e correlation between u X and qj increases, so that (1-T) will be an increasing function of E * Since the first term in the righthand side of (4.9) is positive, ai will be an increasing function of e<sup>,</sup> i.e. the more a society is concerned about inequality, the greater will be the tax on the non-rationed commodity.

Further, if commodity j is an absolutely inelastic commodity in the sense that its consumption is completely insensitive to changes in income,
14 then T. = 1 and E(q.Q) = E(q.)E(Q) which imply that the first two terms in i **1** ) the righthand side of (4.9) will be zero. This means that the tax on commodity j will depend on the third term. If in addition commodity j is of limited substitutability, it will attract zero tax rate.

The substitution term E(q.k.) is likely to be positive for necessities and negative for luxuries and, therefore, this term will dictate that necessities should attract higher tax rates. But this effect can be offset by the first term which implies lower tax rates on necessities. If the social welfare function is highly egalitarian, i.e. f is high, the first term in (4.9) will dominate the third term. The policy implication of this is that the government will tax luxury goods more heavily than necessities. Thus, the major conclusion emerging from this section is that tax rates on non-rationed commodities should not be uniformly proportional to prices. If, however, the restriction of uniformity is imposed, then the optimum policy will be to tax all non-rationed commodities at infinite rate which in our view is not a feasible solution.

# 5. LINEAR EXPENDITURE SYSTEM

The demand functions under rationing for the linear expenditure system is given by (Neary and Roberts 1980): where i is the subsistence consumption of the ith commodity and _5_ is its marginal budget share so that
15 (5.2) n £ Bi '.

Using the Slutsky equation, immediately yields the compensated price elasticities as <!-- Start of picture text --> = 0.(q.-Y.)<br>8.(qI-y) (q.-Y..j<br><!-- End of picture text --> which gives 6. = _ii (4-a.)_ (5.3) ~~~~~~~~~~.3 n o where $ = E M. i=i (1-B 0<sup>)</sup>

> When the Engel curves are linear, * in (2.10) will be non-stochastic which simplifies (4.7) to

which in view of (5.3) becomes Substituting the demand function (5.1) into (5.4) leads to where uL = E(z), the mean total expenditure. This equation demonstrates
16 that * is same for all j, i.e. the taxes on non-rationed commodities must be uniformly proportional. Thus, it follows from the<sup>previous sections</sup> that the non-rationed comodities must be taxed at infinite rate. This is an important finding. It demonstrates that the linear expenditure systea cannot be used to find a sensible optimal commodity<sup>tax structure under</sup> rationing.

# 6. AN ALTERNATIVE MODEL OF RATIONING

We now consider an alternative model of rationing in<sup>which the</sup> government provides a fixed asount of one commodity,<sup>say, food at a</sup> subs. ized price. Suppose there are n commodities q , q ...,q in the2' **n** economy and q is the amount of the first commodity<sup>provided at a</sup> subsidized price **p.** If p = (p , p , ,p ) is the vector of market **I 2 n** prices, the consumer maximizes his or her utility function subject to the budget constraint Note that if q > q , the consumer satisfies his or her excess demand of the first commodity over q by buying it at the prevailing market price. Thus, there is a dual market for the first commodity.

It can be seen that the above model is equivalent to the usual consumer demand model when the consumer has been given a cash subsidy of
17 (p -p )q 5 Thus, the demand equation of the ith commodity will be given by Note that if p is fixed, this demand equation is not a homogenous function of degree zero in prices p and incose x.

Substituting (6.3) into (6.1) yields the indirect utility function of the consumer with income x as which on differentiating with respect to p and p. and using the Roy's identity and respectively.

Note that 0 0 if q q which means that the consumers who do not buy the rationed commodity at the market price are not adversely affected by the increase in the market price of that good.

The optimal problem now involves determining tax rates t , t ... ,t so that consumer's indirect utility function (6.4) is maximized subject to the government's revenue constraint sIt is assumed that the ration quota is always less than the actual requirement of that commodity.
18 which is based on the assumption that the government buys the rationed commodity at the producer's price and sells it to the consumer at a lower price of pi. 6 Using the Slursky equation and the demand equation (6.3), it can be shown that and where 0. and 6. are defined in (2.10) and (2.11), respectively.

Substituting (6.5), (6.6), (6.8) and (6.9) into the first order optimization condition for a single person we obtain Suppose that all texes are uniformly proportional, i.e. vi = a for all i which will imply Si and 8. equal to zero for all j = 2,...,n. Under these - conditions (6.10) will be satisfied only if q, 0, i.e. when no commodity is subsidized. Thus, when a fixed quantity of any commodity is provided at a subsidized price, the optimum tax policy dictates'that we do not have all uniformly proportional taxes.

6Note that the government collects revenue on the first commodity only on the amount bought in the open market, i.e. tqj-ql).
19 Equation (6.10) can also be written as where kj is the compensated proportional change in the demand for the jth good as a result of imposing taxes on goods other than j.<sup>k. is likely to be</sup> positive for necessities and negative luxury goods. It can be seen<sup>from</sup> (6.11) that a. will be higher if the ith commodity is a necessity. Further, the magnitude of e. is inversely proportional to the compensated own price elasticity. However, the most interesting finding is that 1. is a decreasing function of kl, an implication of which is that if the rationed cemmodity is a necessity, the taxes on non-rationed item is<sup>generally a</sup> necessity and therefore, should attract greater tax than<sup>the non-rationed</sup> item.

# 7. CONCLUSIONS

In this paper we have investigated the implications of the optimal commodity taxation in the presence of rationing. The<sup>analytical results</sup> presented in the paper raise the fundamental issue of the relevance and usefulness of the standard optimal commodity tax results<sup>in evaluating</sup> existing indirect taxes and providing policy advice. Following are<sup>some of</sup> the implications for tax policies that emerge from this analysis.

1. In the presence of a single person economy the optimal policy dictates that the rationed commodity bears the entire tax. In the<sup>context of</sup> developing countries, this result implies that if the government<sup>has a</sup> fixed budget to subsidize certain commodities, the optimal policy will
20

# be to subsidize only the rationed commodities (non-rationed commodities should receive zero subsidy).

2. The single person result of zero tax on non-rationed commodities does not generalize to the many person case. In the many person case if we impose the restriction that taxes on all non-rationed comsodities are proportional to prices, the optimal policy will tax all non-rationed commodities at infinite rate.

3. The linear expenditure system always results in uniformly proportional taxes on non-rationed comaodities. It follows that the non-rationed commoiities must always be taxed at infinite rate. Thus, the LES cannot be used to find a sensible optimal commodity tax structure under rationing. The linearity/separability assumptions of LES are already known to have some restrictive implications for optimal tax theory in the standard formulation. We have extended them to the rationed case.

4. The tax on a non-rationed commodity should be an increasing function of the inequality aversion parameter, i.e. the more a society is concerned about inequality, the greater will be the tax on the non-rationed commodity.

5. When a fixed quantity of any commodity is provided at a subsidized price, the optimal tax policy dictates that we do not have **all** uniformly proportional taxes. This result differs from the standard optimal commodity tax result which states that all taxes should be uniformly proportional for the single consumer economy.
21 A useful extension of this study and one that we hope to carry out in further work would be the empirical calculation of optimal commodity taxes in the presence of rationing and within the framework of a nonlinear (and, hence, more realistic) demand model. It is worth noting that while a good deal can be said a priori about optimal commodity taxes if preferences are separable, very little can be asserted if they are not. As Ray (1986)'s results have shown, optimal commodity taxes under realistic demand systems bear very little resemblance to LES-based tax rates, especially from the viewpoint of an equity conscious planner. As we have shown in this paper, the situation gets more comrlUcated even for the restrictive rationed LES. Empirical calculation of optimal commodity taxes under rationed nonlinear demand systems, while requiring complex and expensive estimation proceedings, would be a valuable contribution to the optimal tax literature, especially in developing countries.