on the existence of the optimum commodity tax system: a heuristic proof

Journal of Public Economics 43 (1990) 339-352. North-Holland

ON THE EXISTENCE OF THE OPTIMUM COMMODITY TAX SYSTEM

A Heuristic Proof

Masatoshi YAMADA*

Nagoya City University, Mizuho, Nagoya 467. Jupan

Received September 1988, revised version received April 1990

This paper presents a basic proof of the existence of the optimum commodity tax system in a typical economy where a social preference is defined. The proof in this paper utilizes only the fundamental apparatus familiar in the proof of the existence of competitive equilibrium [as in Debreu (1959)] and the maxima-minima theorem for continuous functions. The purpose of the paper is to facilitate the understanding of how the optimum tax is assured, especially for policy- oriented researchers. The paper also intends to show when the existence of the optimum commodity tax is not guaranteed.

1. Introduction

While much literature on optimal taxation since the seminal work by Diamond and Mirrlees (1971) has been directed towards clarifying the optimal tax rules, little has been done on the most basic problem of the existence of the optimum tax system. An exception is the work by Mantel (1975). His argument, i.e., his existence proof, however, is mathematical, and it is not so easy to understand how the existence is secured, at least for policy-oriented researchers. It is therefore of some interest to present another proof of existence of the optimum tax system following a very basic and well- known procedure used to prove the existence of a competitive equilibrium. The purpose of this paper is thus to give an existence proof of the optimum commodity tax system making use of the fundamental Arrow-Debreu apparatus to prove the existence of competitive equilibrium and the maxima- minima theorem for continuous functions.’ In addition, this paper intends to show when the existence of the optimum tax system is not guaranteed, which is another interesting point to which Mantel paid little attention.

*I am indebted to Professor K. Kuga for alerting me to this problem, and to Professor D. Kestenbaum for polishing the English. I also gratefully appreciate the thoughtful comments by an anonymous referee which helped correct some errors in an earlier version.

‘We should, however, note the model below loses generality compared to Mantel’s in that taxes have only one rate, dismissing the progressive case, and it assumes preference represented by a (continuous) function.

0047-2727/90/$03,50 0 199bElsevier Science Publishers B.V. (North-Holland)

340 M. Yamada, Optimum commodity tax system

The basic strategy in the discussion below is to follow an elementary process in the proof of existence. First, we analyze whether an equilibrium with a commodity tax system to finance a given government budget exists, following the traditional method of the proof of competitive equilibrium.2 Then we proceed to ask if it is possible to choose a commodity tax system that maximizes social welfare, i.e., attains the highest social welfare, among commodity taxes which secure a competitive equilibrium and finance the government’s budget. By following this procedure we can also find where the optimal tax system is not assured.

We shall differentiate two situations depending on whether the government raises positive revenue or not because the proving arguments below differ in respective cases. The first is the situation where the government needs no positive revenue. Then all the commodity taxes work for a redistributive purpose, i.e., consumption of some commodities is subsidized while that of others is taxed, so as a whole they bring no revenue to the government. Here the existence of competitive equilibrium with commodity taxes is provided by direct application of the Arrow-Debreu existence theorem. Existence of the optimal commodity tax system in that case is then proved by applying the maxima-minima theorem for continuous functions.

The other case is the usual and more natural situation where the government intends to raise positive revenue. However, no optimum tax system may exist, and thus we need an assumption to assure its existence. We shall present a proof of existence under this assumption following the same procedure as in the first case. Non-existence of the optimum tax system will be easily understood if we imagine that the amount of government revenue is too high to be assured in the economy considered.

In addition to proving the existence of the optimum taxes, we shall pay further attention to finding a structural feature of the optimum commodity tax in the case where government needs positive revenue and there exists a positive production profit. Special attention to this case comes from the intuition that a proportional tax on all goods is possible, and it is then questioned whether the proportional tax is optimal, since it works as if it were an income or profit tax and is thought of to be non-distortive. We shall find that the optimal tax in this case can be considered to be composed of two kinds of taxes, one proportional and the other purely redistributive, which further suggests that the optimum commodity tax has a differing feature depending on whether or not production profit exists.

*This part of the argument is close to that of Shoven (1974). The arguments differ, however, on the following points. First, the amount, and therefore the detail, of the government expenditure is not specified ex ante, but determined in the model in Shoven’s analysis. Second, this treatment of the government ensures that equilibrium always exists in his analysis, while non-existence of equilibrium cannot be excluded in our model. See section 6 below. I am indebted to Professor 0. lchioka for pointing out this reference.

M. Yamada, Optimum commodity tas system 341

Lastly, we shall examine the case where no optimum commodity taxes exist, making use of examples, since we need an assumption to prove the existence of the optimum tax in the second situation above. This exemplitica- tion may bring also another insight into the structure of the optimum commodity tax system.

The remainder of the paper is organized as follows. Section 2 presents a model economy appropriate for our purpose. Section 3 analyzes the case where the government needs no revenue and thus commodity taxes are purely redistributive. Section 4 considers the case where the goverment aims to raise a positive amount of revenue for financing a specified amount of its demand. Section 5 considers a property of the optimum commodity tax system in a situation where government raises positive revenue and production profit exists. More concretely, we shall examine whether the conjecture is true that the optimum tax system composes of a proportionate tax and a redistributive one as in section 3. For this purpose, we prove that a proportionate tax, financing the government’s demand, attains a competitive equilibrium, and show that the optimum tax is decomposed into a proportionate or income (or profit) tax, tinancing just the government’s demand, and a purely distributive commodity tax. The final section presents examples and explains the case where the optimal tax system is not assured to exist.

2. The model

The model economy to be considered is a standardized one in the optimal tax literature but specified with more details than usual. We consider an economy composed of N housholds, K firms and the government, where there are J private goods. (Public goods are assumed away.) Firm k is held by households with the share {(Ik1,Qk2,. . .,HkN) (ekizO, ~~ZI1ki= 1 for any i and/or k).

The production possibility set for firm k is denoted by Yk (with element yk), which is supposed to be closed, convex, containing the non-positive orthant R< (implying free disposability), irreversible ( Yk n (- Yk) = {0}) and implying no Cockaigne ( Yk n R!+ = (0)). Define Y =cf= r Yk, and denote firm k’s profit by rck and rr E cf= 1 nk.

Consumption set of household i, Xi, is assumed to be closed and contained in R$. Household i’s preference is given by a continuous, strictly quasi-concave (except in section 5) and strictly increasing function ui defined on Xi. It also has an initial holding of w’( E int Xi).3

Third, the government can impose any consumption tax, by which it finances a given public expenditure g = (gr , , gJ). This government expendi-

‘int S denotes the set of the interior points of S. In addition, the vector inequality is defined as: For a,hERJ,a>hoa,>h,; a>hoa,zh,,j= I,..., J; azhsaeh and a#h.


ture is assumed not to directly affect the households’ utility as usual. It also has a continuous social welfare function W(u’, . , uN).

Lastly, taxes on commodity consumption are denoted by t =(t,, . . , tJ). Then firms or producers face price p=(pr,. . ,pJ), and households or consumers face price p + t = q = (ql,. . , qJ).

3. The case of redistributive commodity taxes

We consider the case where g=O and the government intends to raise no positive amount of tax revenue, and therefore commodity taxes work just redistributively. As seen below, the existence of the optimal tax is ascertained here without any more assumptions, i.e., roughly under the same conditions that assure the existence of a competitive equilibrium without any tax.

Let us start by imagining that the government imposes commodity tax t and then an equilibrium of all markets is attained with producer price p( 20) and consumer price q =p+ t( 20). This implies the following two conditions are met, denoting household i’s net demand by xi and firm k’s supply by yk:

yk maximizes profit pyk among all elements in Yk, and xi maximizes utility ui among any element in the budget set

K x’Iqx’5 c ~ki7Ck,nk=pl.k,Xi+WiEXi (1)

k=l

and

N x

c xi= c yk (market equilibrium). i=l k=l

We note that when (1) and (2) hold the government’s budget also balances. That is, since we consider now redistributive taxes, the following holds

t i: xi=o.4 (3) i=l

“This is easily ascertained as:

N K

=En”-pCyk=C(pyk-py”)=O,

where the second equality uses the market balance condition cp=, xi=Ct=, yx, and the third utilizes the households’ budget balance condition.

M. Yamada, Optimum commodity tax system 343

Thus, (1) and (2) are considered to represent a competitive equilibrium under commodity tax system t.

Because the optimal tax system also must be one achieving a competitive equilibrium, let us define a set E” of prices under which a competitive

equilibrium exists. This is defined as

E”={(p,q)zOl(p,q) satisfies (1) and (2)).

Then a tax system securing a competitive equilibrium is given by t=

q-P,(P,4)EE”. Since it is well known that a competitive equilibrium without tax exists under the assumptions specified in the preceding section, and the situation without any tax corresponds just to the case of r = 0, we know that there exists a p such that (P,JI)EE” and E” is not empty.

We note a feature of the price system (p,q) E El’: let us suppose a set of equilibrium prices (p, q), and a scalar c(( >O). Then (ccp, clq) is also an equilibrium price system, bringing the same allocation of goods as under (p, q). This property comes from the zeroth-order homogeneity of households’ demand and firms’ supply so that they do not change with any proportionate change of prices. This implies that there is no loss of generality in restricting E” to a (25 - I)-dimensional simplex.

Noting this property, we can restrict the range of producer and consumer prices to be considered to

(p,q)ZOl(p,q) satisfies (1) (2) and i (pJ+qJ)=l .I=1 1

E’ is bounded by its definition and is known to be closed from the continuity of the ui’s and the closedness of Yk’s (see the proof in the appendix).

Third, suppose (p, q) E E’, and denote the utility attainable under (p,q) by u’(p,q)=u’(x’(p, q)). Then the problem of optimum taxation or of government optimization is expressed as: to maximize W(u’(p, q), . . . , uN(p, q)) with respect to (p, q) E E ‘. Then, the optimal tax rate is defined by q-p.

When the optimum tax problem is viewed as above, its existence is easily ascertained simply by the maxima-minima theorem for continuous functions. That is, first note that W is continuous with respect to ui,ui with regard to xi, and x’(p, q) is continuous as to (p,q) which is shown in the proof of the closedness of E’ (see the appendix). Then since El is known to be compact, there exists at least a (p*, q*) maximizing W in E’. By definition this t*=q*-p* must be an optimum tax system.


4. The (general) case where the government needs positive revenue

(i) We now consider the more general case where the government needs a positive amount of revenue to finance its specified amount of expenditure, g. Then, the existence of the optimum tax system is not assured in general. The reason for this is rather simple, for suppose that the goverment demand, g, is larger than what the economy considered can produce. Then it is simply impossible for the government to finance g even by taxes with infinite rates.

Thus, at the beginning of the discussion we need an assumption to ensure that there exists at least a tax system achieving a competitive equilibrium and financing the government demand, g. Therefore we begin by stating this assumption. For this purpose, let us first note that the market equilibrium condition is now given as

N K

c x’+g= c yk, i=l k=l

(2’)

where {xl,..., x”} and {y’,..., y”> are households’ demands and firms’ supplies, respectively. Note further that if there exist p and 4 satisfying (1) and (2’) then the government budget also is fulfilled. This is easily seen as follows. Noting that a tax vector, t, is by definition given by q-p, and making use of households’ budgets and demand-supply balances, the relation sought is obtained as

t ; x’=(q-p) 2 i=l i=l

xi=4j11 Xl-Ffk$ +g)

Eq. (3’) apparently corresponds to (3) in the case where no government revenue is needed.

Thus, we know, recalling (3’) that (1) and (2’) express a situation where there exists a commodity tax system financing the public demand, g, as well as attaining a competitive equilibrium. This is the condition we need in subsequent discussions. We give it as an assumption:

Assumption 1. For a given g, there exists a set of producer and consumer prices (p, q) satisfying (1) and (2’).

(ii) From the above discussion we know that when searching for an optimum tax system we can restrict our concern only to taxes satisfying (1)


and (2’). Then following the argument in the preceding section, let us define the set

E2 = (p, 4) 2 0 1 (p, q) satisfies (l), (2’) and i (pJ + qJ) = 1 .I=1

This definition of E2 reflects that we can restrict p and q in a (25- l)- dimensional simplex by the zeroth-order homogeneity of demand and supply.

We can further find first that E2 is not empty by Assumption 1. Second, E* is bounded by definition. Lastly, we find E2 is closed by the same reasoning as with E’ above (see the appendix). E2 is therefore known to be non-empty and compact.

The problem for an optimum tax system is clearly to find a (p*,q*) in E2 which maximizes W(u’(p, q), . . . , #(p, q)), where u’(p, q) = u’(x’(p, q)) and (p, q)

is a set of prices leading to a competitive equilibrium. Because we have ascertained that E2 is non-empty and compact, there exists a (p*,q*) in E2 maximizing the continuous function W5 by the maxima-minima theorem

which, by definition, gives the optimum commodity tax system t* = q* -p*.

5. A feature of the optimum commodity tax system with production profit

This section explains a specific feature of the optimum commodity tax in the case where production profit exists and the government raises positive revenue. We first prove that a price system with a uniform commodity tax, i.e. (p,( 1 + fl)p), will belong to E2. Noting the intuition that a uniform commodity tax works as if it were an income or profit tax, then this feature would suggest further that, first, a uniform commodity tax is optimal if no redistribution of income is required, and second, the optimum commodity tax in general is composed of a proportionate commodity tax to finance the government demand, and a redistributive one as argued in section 3 above. We shall show this conjecture is correct by proving that the optimum commodity tax is decomposed into a proportionate commodity (or income) tax and a purely redistributive commodity tax.

(a) As noted above, it is conjectured that in an economy with production profit, a proportionate tax on all goods ensures a competitive equilibrium allocation. We confirm this is true, assuming further that the ui’s are strictly concave (rather than quasi-concave) and smooth.

In the remainder of this section we follow Foley’s (1967) argument proving

%ee the discussion in section 3


the existence of a public competitive equilibrium in a public goods economy. So we shall give only a rough sketch of the proof and not enter into detail.

First, define the set D of consumption allocation attainable through the present production technology:

x1 ,..., xN(xi+w%Xi, i=l,..., N, F .w’+g~ Y .

i=l

D is assumed non-empty. Also, D is bounded and closed under the assumptions in section 2 [see Debreu (1959)].

Second, consider a weight vector C.X =(E’, . . ,a”) satisfying ~20 and crZ 1 cti = 1, and define the following function based on this:

N

W(cc) = c &‘(x’+ wi). i=l

Here, think of an Z? maximizing W(E) among allocations in D. Because W’(X) is continuous and D is compact, there is such an ,?, which is denoted by j?(c()=(x’(a),..., xN(~)) and is unique by strict concavity of the z?s. That is to say, the correspondence cr-+_C(cc) is single-valued and, furthermore, it is known to be continuous.6

Third, define a set of commodity allocations which can attain a utility for every household higher than x(a):

for any i=l,...,N ,

D* is strictly convex by strict concavity of the ui’s. Fourth, we find the following properties pertaining to D*(a). (i) Two sets, Y-g and D*(a), share no point, and Y-g and the closure of

D*(a) share only xx’(a). Furthermore, since both sets are convex and Y-g is closed, there is a hyperplane H separating them as well as supporting

6W(~)=~~‘, r’u’(x’+w’) is strictly concave if ~20. Furthermore, if a’=0 for some i, then x’(a) must be -w‘. These two points mean that x(a) is uniquely defined depending on a. See also Foley.


Y-g. Then, we call p( 20) as normal to H, denote by z the point of tangency between H and Y-g (the point H and Y-g share), and define rc=p.(z+g).

(ii) Then in fact z=~x’(cx). Since cx’(cc) is unique with regard to Z, and thus z, p and H are uniquely determined depending on c( by strict convexity of D*(a). Furthermore, since it is ascertained that the correspondences a-+p or rt are upper semi-continuous [see Foley], they are really continuous.

(iii) Because z~‘(a)+ge Y, there exists a productive configuration

(Y’ , . . . , y”) corresponding to the consumption allocation xxi(~) +g=xyk and, furthermore, a hyperplane Hk, parallel to H, supports Yk at yk. Then if we define rrk=pyk, nk corresponds to cx continuously by the same reasoning as with n.

(iv) Note further that, from the definitions of D* and H, each x’(ct) must be on a household i’s indifference curve which is tangent to (or supported by) the hyperplane Hi= {x 1 px=px’(sr)}. This means that xi(~) is household i’s optimal bundle of consumption if the (consumer) price is p, and household i has an income of I’=px’(a), supposing px’(cr)20. Similarly, ~xi(a)+g=~yk, which is supposed to be positive, maximizes profit for producers as a whole facing price p. However, it does not hold that the each firm’s profit (7L’,..., nK) obtainable by producing (y’,. .,yK) will necessarily yield the income I’ defined above, and thus the allocation ,?(LY) cannot be achived as a competitive equilibrium in general.

Fifth, we now show that an a, and the corresponding allocations (xl(~)),. . ,~~(a)), (y’, . , yK), price p and profit (rc’, . . , nK), defined as above depending on c(, form a competitive equilibrium. To prove this, consider the following adjustment equation of CC

c(it _ ma! [O, d + u{( 1 - z)(CF, 1 Nki nk) -px’}] C~=“=,max[O,cci+a{(l-~)(C,K_I~ki~k)-ppXi}]’ i=l”‘.‘N’

where r =pg/n, z = I:=, nk, a is a positive scalar, and p, fi = (nl,. . . , nK) and (x’, . .,x”) are the ones defined depending on c( as above. Since p, fi and 1 are unique with respect to c(, so is LX’. It is simple to check that this denominator is never zero and cc’=(cc”,.. .,aN’) belongs to (N-l)- dimensional unit simplex. It is understood also that this function works as follows. First, define the ratio of the value of the government purchase to the total profit, r. Then the weight c? is increased (or decreased) if the value of households i’s net consumption x’(a) evaluated by p, px’(z), is larger (or smaller) than its profit income after tax with a rate of r, (1 -z)C,“=, Okink.

Sixth, it is easily understood from the continuity of p, fi and i that the correspondence c(-+cx’ is continuous [see earlier discussion and Foley]. Then, by Brouwer’s fixed point theorem, there exists a fixed point c(*. The definition of tl’ above means that the following holds at cc*:


k=l

i=l ,..., N,

where the variables with an asterisk denote those defined corresponding to c[*.

Lastly, recalling the fourth and sixth points, it is known that the allocation ~‘(a*) corresponding to a * is chosen by household i under the income tax with rate r* and consumer price p*, i= 1,. ., N, and also that the government demand, g, is financed. It is easy to check that the allocation

(x1 (cc*), . . . , ~~(a*)) is attainable also under the proportional commodity tax with a rate of r*/( 1 -r*). This completes the discussion.

(b) The result above, together with that in section 3 (the purely redistributive tax), suggests that the optimum tax is a mix of a uniform tax and a redistributive one. We shall show that this conjecture is correct by demon- strating that the optimum commodity tax system here is decomposed into a proportionate tax and a purely redistributive one, the former just financing the government demand. This will imply, further, that the proportionate tax is optimal if the marginal social welfare is equal for all households and no income redistribution is required.7

To prove this, suppose {x’ *, . . , xN*; y'*, . . . , yK*;p*, q*} denote the optimum allocation and prices, and note that the budget equation (p* + t*)x’* = xeikp*yk* is rearranged as follows:

This shows that, if the commodity tax, (t*-vp*)/l +v, and the profit tax, v/l +v, with v=p*g/p*(xyk*-g), are taxed, the optimum allocation, x*, is attained. Moreover, if the profit tax with rate v/l +v is taxed, then the optimum commodity tax rate must be (t* -vp*)/l + v, because otherwise t* cannot be optimum by following the above reasoning in reverse order. Then, we know also that, since the profit tax with rate v/l +v just finances the government demand, the commodity tax, (t*-vp*)/l +v, works purely redistributively as shown in section 3.

Then, if there is no need for income redistribution, no role for a redistributive tax is left, and the commodity tax, (t*-vp*)/l +v, defined above must not work. Thus, the optimum tax system must be the income tax with rate v/l + v or, equivalently, the proportionate commodity tax, t* = VP*.

‘See Yamada (1988)


good 2 b

3b

4

n, II ’ Y-g

-b

good 1 >

Fig. 1. An example without optimum commodity tax.

6. Opposing examples

In section 4, to show the necessity of the assumption that there exists a commodity tax system which assures an equilibrium, i.e., belongs to E2, we referred to an absurd case where the government’s demand g is out of YtCw’. We now present a more plausible example with no optimum commodity tax, so as to ascertain that the assumption is indispensable, and to show how it works.

(i) To show this, let us assume the following economy: there exist a representative household, a firm and two kinds of goods. The household initially holds h units of goods 1 only. Its preference is given by the utility function u=(x1)u~(x2)‘*. The firm’s production set, Y, is given as in fig. 1. Lastly, the government intends to finance the public demand of g = (0, b/2).

To find the optimum allocation, it will suffice to use the following procedure. First, move the production set by -g so that the government demand is secured. Then find a point where the household’s indifference curve, depicted with its origin at (- b,O), is tangent to the frontier of the production set after the move above, Y-g. This procedure allows us to find an allocation that attains the maximum utility for the household after


securing the government demand. Thus, this leads us to an optimum equilibrium allocation if the point found through this procedure is chosen by the household with the corresponding income that the firm earns.

To see that the last property is not always secured, we further restrict ourselves to the situation with a, =2/3 and uz= l/3. Then Y-g and the highest indifference curve shares the point (-h/2, h/4). [The utility attained there is u =( 1/4)“3h.] The gradient of the indifference curve at ( -b/2, b/4) is 1, which is less than that of the production frontier between (0,O) and (-b/2, 3b/4), 312. Now let us show that an allocation given by x=(-b/2, b/4) and y=x+g=( - b/2,3b/4) is not competitively achieved. As fig. 1 shows, the household must face the consumer price q1/q2= 1 and have an income of -b/4 for the point (-b/2, b/4) to be chosen as optimum. That is to say, for the point to be a subjective optimum for the household, the househod lump- sum income must be negative. However, any commodity tax cannot reduce the household income to negative, and thus we know that the point (-b/2, b/4) is not attainable through any commodity tax system in a competitive economy.

Fourth, let us ascertain that no competitive equilibrium to finance the government demand, g, exists under any commodity tax system. This is confirmed by considering whether or not there exists an intersection between the household’s offer curve and the production frontier, Y-g. The former is now given as x,= -a,b+a,n/q, (n-firm’s profit), which is larger than -azb. This implies no intersection between the two above if Y is specified as in fig. 1 and a2 5 l/3.

Lastly, it may be noted that the existence of the optimum taxes is not assured despite the fact that the maximum profit of 3b/4 is bigger than the government’s demand, g [see fig. 11.

(ii) We now present an example with no optimum tax system in the case without production profit. This is done by rearranging the above example as follows. The household initially holds (b, b), and the production frontier is given by the straight line through the origin and the point (-b/2,3b/4). Let us denote this production set by Y’. Other settings remain the same. Then the household’s offer curve is given by (x1 +u,b)(xz+a,b)=u,u,b2, which is known to share no point with Y’-g. This means there is no (optimum) commodity tax system achieving a competitive equilibrium.

Appendix: Proof of the closedness of E’ and E2 ’

To prove the closedness of E’ or E2 it suffices to show that for any -- --

converging sequence {p”,q”j EE’ or E2 with the limit (p,q), (p,q) also belongs

‘The author is indebted to Professor H. Sasaki for revising the proof below.

M. Yamoda, Optimum commodity tax system 351

to E’ or E2. This is shown below. Subsequently, we proceed primarily

referring to the set E2. First note that from the definition of E2, for (p”,q”) E E2 there exist a

demand P=(xrn,..., x”‘) and a supply y=(yln,. , yKn) satisfying the following:

p”yk” 2 p”#, for anyykEYk, k=l,..., K,

u’(x’“+ w’) 2 u’(x’ + w’), for any

i=l,...,N,

and

where nkn = p”yk”.

Since the attainable set of the economy is compact [see Debreu], we may assume f” and j converge to i and y’, respectively. To conclude the closedness of E2, or (P,$E E’, it is sufficient to show that (x’;i) satisfies the conditions of a competitive equilibrium (1) and (2’).9 We shall show this in what follows.

To prove this, first it is simple to see that the following hold:

42’5 f @kink (5” =pyk),

k=l

Second, let us show that jk is a maximizer of profit in Yk under j?. Note that (i) jk~ Yk since Yk is closed. Also, (ii) from the definition of ykn,

p”yk” 1 pnz, for any ZE Yk,

‘Let us note this implies that (x’(p, q),. .,.?(p,q)) and (y’(p), , yK(p)) are upper semi- continuous with respect to (p.q), and furthermore, recalling the strict concavity of the 12’s and therefore the uniqueness of the former, that (x’(p, q). , xN(p, q)) is continuous with regard to

(P. 4).


which implies, by the continuity of the inner product,

jjk 2 pz, for any z E Yk.

Third, let us show that Xi maximizes household i’s utility under the budget there. To prove this, note that the following holds from the definition of xi’?

ui(xin + w’) 2 ui(xi + w’), for any xi E B’”

= XiIqnXi~~eki7rk”,Xi+WiEXi i=l,...,N, k

and also that B’” is continuous with respect to (p”,q”; 7~“) for q">O. Note further that we may confine our attention to q >O and therefore q can be restricted so that q>E for a sufficiently small E, since ui is monotone and wi E int Xi. Noting these properties, the following is simply true by continuity of the utility function and of B’” (in a restricted region):

ui(Xi+wi)~~i(xi+wi), for any x’EB’

= Xi~cjXi~~Bkifik,xi+wiEXi i=l,..., N, k

Xi must be in B’ and Xi is a maximizer of ui among any allocations belonging to the budget set B’.

The second and third points prove that i and i satisfy conditions (1) and (2) or (2’) for prices (p,q).

References

Debreu, G., 1959, Theory of value (Wiley, New York). Diamond, P. and J. Mirrlees, 1971, Optimal taxation and public production, I, 11, American

Economic Review 61, 8-27, 261-278. Foley, D., 1967, Resource allocation and the public sector, Yale Economic Essays 7, 45-98. Mantel, R., 1975, General equilibrium and optimum taxes, Journal of Mathematical Economics

2, 1877200. Munk, K., 1977, Optimum taxation and pure profit, Scandinavian Journal of Economics 80,

l-19. Rockafellar, R., 1972, Convex analysis (Princeton University Press, Princeton). Shoven, J., 1974, A proof of the existence of a general equilibrium with ad valorem commodity

taxes, Journal of Economic Theory 8, l-25. Yamada, M., 1988, Generalized optimal tax rule for an economy with pure profit, Discussion

paper no. 93 (Nagoya City University).

on the existence of the optimum commodity tax system: a heuristic proof

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