A theory of interregional tax competition*1

JOURNAL OF URBAN ECONOMICS 19,2%-315 (1986) A Theory of Interregional Tax Competition JOHN D. WILSON* Indiana University Received January 27,1984; revised September 14,1984 A general equilibrium model is constructed to study tax competition, where local governments compete for capital by holding down property tax rates and public expenditure levels. An exact definition of tax competition is provided, and both the existence and nonexistence of tax competition are shown to be theoretically possible. It is argued, however, that tax competition must occur under empirically reasonable conditions. Inefficiency in public production is also explicitly modeled. The amount of capital used to produce a given level of public service output is shown to be greater than that which is required to minimize costs evaluated at the prices facing private firms. 0 1986 Academic Press. Inc. 1. INTRODUCTION Tax competition between local governments can take two forms. First, each region’s local government attempts to encourage particular types of investment by taxing some types of property at relatively low rates (e.g., business property is taxed less than residential property). Second, the local government attempts to increase the total level of capital investment in the region by lowering the average rate at which all property is taxed. Elements of both types of tax competition are contained in Oates’s [13, p. 1431 claim that “the result of tax competition may well be a tendency toward less than efficient levels of output of local public services. In an attempt to keep tax rates low to attract business investment, local officials may hold spending below levels for which marginal benefits equal marginal costs, particularly for programs that do not offer direct benefits to local business.” In a previous paper [15], I examined the first type of tax competition and conclude that, under likely conditions, each local government has an incentive to tax the property used to produce goods which are traded between regions at a lower rate than property used to produce nontraded goods (e.g., housing). It is theoretically possible for the incentive to be reversed, but not empirically likely. The present paper is concerned with the second type of tax competition. I present a general equilibrium model in which a region’s local public *I am grateful to Jack Mintz, Jon Sonstelie, and an anonymous referee for helpful comments and suggestions. Research support was provided by the Council for Research in the Social Sciences at Columbia University. 296 0094-1190/86 Copyright All rights $3.00 0 19X6 by Academic Press. Inc. of reproduction in any form reserved. INTERREGIONAL TAX COMPETITION 297 expenditures are financed by a uniform tax on all of the region’s property. Tax competition is defined as a situation where public service outputs and tax rates are “too low” in the sense that a federal government could raise the nation’s welfare by requiring each region to increase its public service output. In my model, this forced rise in public service outputs is equivalent to requiring that tax rates be raised. Surprisingly, even though I consider only public services which do not directly benefit local business, I am able to identify cases where the opposite restriction is desirable: a forced reduction in each region’s public service output raises welfare. My analysis suggests, however, that such cases are not plausible. Theoretical models used to study efficiency of property taxation simplify the structure of production and demand in unsatisfactory ways. Some models assume that all private goods are traded between regions (no housing); some allow for nontraded goods but assume that all or some goods are produced by a linear technology (constant marginal costs and no substitutability between inputs); some assume that public services consist of public purchases of private goods; and some specify the functional forms of production and utility functions (e.g., Cobb-Douglas).’ I construct a model which includes none of these assumptions and demonstrate that it can be usefully employed to analyze the property tax. Rather than obscuring the basic arguments, this added generality is utilized to explain tax competition. Furthermore, I am able explicitly to model not only the inefficiencies in the chosen outputs of public services, but also inefficiencies in techniques used to produce these outputs. In particular, I demonstrate that each local government has an incentive to produce its chosen public service output using more capital than minimizes costs at prices faced by private firms. The plan of this paper is as follows. The next section describes the model, and Section 3 presents an expression for the shadow factor prices used by local governments to design their public production plans. This expression is subsequently used in my investigation of tax competition. Section 4 defines tax competition and presents an important preliminary result about its existence. In Section 5, sufficient conditions for existence are presented, and the basic arguments behind them are provided. These arguments are kept verbal so as to prevent their economic content from becoming concealed by complex algebra. Section 6 provides a formal statement of two necessary and sufficient conditions for tax competition which imply the sufficient conditions in Section 5. The Appendix contains their proofs. Concluding remarks are made in Section 7. ‘See, for example, Beck [3], Boadway [4], Hamilton [7], Zodrow and Mieszkowski [16], and Starrett (141. Hamilton [8] reviews the efficiency issues associated with the property tax. 298 JOHN D. WILSON 2. THE MODEL The economy consists of a large number of regions. Two primary factors are supplied to each region. One factor, capital, is perfectly mobile between regions, while the other factor is immobile. A region’s fixed supply of the latter factor is owned entirely by the region’s residents. To concentrate on capital mobility, commuting and migration between regions are ignored.* I call the immobile factor labor. Both capital and labor are perfectly mobile between private firms within any region, but they are fixed in supply for the nation as a whole. There are two competitive industries within each region, the national good industry (N) and the local good industry (C); and each region’s government provides a public service (E).3 All three goods are produced from capital and labor by a constant returns to scale production technology. Equilibrium in a region’s factor markets requires that K,+K,+K,=K (1) and L,+ L,+ L,= L, (2) where K, and Li are industry i’s capital and labor demands, and K and L are the total quantities of capital and labor supplied to the region. The national good is traded between regions in exchange for capital, while the local good and public service are nontraded goods, consumed only where they are produced. The national good serves as the numhaire, with its price set equal to one in each region. Capital moves between regions until all regions face the same after-tax return to capital, p. I assume that each region is small in the sense that p can be treated exogenous from its viewpoint. Each region’s public service output is distributed uniformly across its residents. Expenditures on the public service are financed by a property tax, consisting of a single specific tax rate, t, on the region’s total capital stock, K. Because capital is mobile, the tax base K depends on the region’s tax and expenditure policy. The government budget constraint for a region is tK = rK, + wL,, (3) ’ Mintz and Tulkens [ll] provide an insightful analysis of “commodity tax competition” in a model which allows for commuting between regions. However, they do not consider taxes on capital. 3The public service may be interpreted as either a publicly provided private good or a Samuelson public good, where each unit produced is consumed jointly by all residents. INTERREGIONAL TAX COMPETITION 299 where r is the before-tax return to capital (r = p + t), and w is the return to labor. Since the government is not constrained to minimize public production costs evaluated at market prices, my analysis would in no way change if only private capital, K, + Kc, served as the tax base. In this case, revenue would equal t(KN + K,), and expenditures would equal pK, + WL,. Since r = p + t, however, (3) is equivalent to the constraint, ?(KN + K,) = pK, + WL,. To isolate efficiency considerations from equity considerations, I assume that all individuals possess identical factor endowments and preferences. I also assume that all regions possess identical production technologies and contain identical numbers of residents.4 For notational convenience, the population of each region is normalized to equal one. The representative individual’s utility is a function of his consumption of the national good, the local good, and the public service. He chooses his consumption of private goods to maximize this function, subject to the budget constraint, D,+pD,=pK+wE, (4 where Di is the individual’s (and region’s) demand for good i, p is the price of the local good, and K and z are the individual’s endowments of capital and labor. (z = L always, but K + K if inter-regional trade occurs.) To simplify the analysis, two assumptions are placed on the form of the utility function. First, utility is assumed to be weakly separable between private goods and the public service: U = U(V( D,, DC), DE), where V( D,, DC) is called “private utility.” Second, V is homogeneous of degree one. Given the high level of aggregation in the model, these two assumptions appear to be reasonable. The equilibrium for a single region is described by (l)-(4) and the following additional conditions, where Xi is the region’s output of good i: D, = Xc, (5) D, = X,, (6) rK, + wL, = X,, (7) rK, + wL, = pX,. (8) Equations (5) and (6) hold because local good and public service outputs are 41f the public service is a publicly provided private good, then the size of regions is irrelevant, since all production functions exhibit constant returns to scale. In this case, my results hold when population sizes differ. 300 JOHN D. WILSON consumed only by residents; and (7) and (8) are the zero profit conditions for private production. Each region’s government chooses its property tax rate and demands for capital and labor (the “public factor demands”) to maximize the common utility of its residents. The chosen public policy is referred to as the region’s “optimal” public policy. Since all regions are effectively identical, they all possess the same optimal public policies when the economy is in equilibrium.5 Thus, there is no trade between regions in equilibrium: K = x and D, = Xv. However, a region’s government can induce trade by pursuing a policy which differs from that followed by other local governments. This potential for trade is recognized by each local government when it chooses its optimal policy. 3. PUBLIC SHADOW PRICES Consider a single region with an optimal public policy. The public shadow an additional unit of i in public service production, as measured from the region’s viewpoint. The optimality of the initial public policy implies that sK/sL equals the marginal rate of substitution between capital and labor (MRS,,) in public production. To achieve economic efficiency from the nation’s viewpoint, sK/sL must also equal the rental-wage ratio, r/w, since private firms minimize costs by equating r/w with their MRS,,. But what is optimal for a single region need not be optimal for the nation as a whole. In fact, I shall argue that each local government chooses a production plan where sK/.sL falls short of the ratio of after-tax factor prices, p/w, which is, of course, less than r/w. In this sense, public production is too capital intensive: it is technologically possible to raise utility in every region by reallocating labor and capital so that the capital-labor ratio falls in each region’s public production sector. I next present the basic argument behind this result. For a more detailed argument, I then derive an explicit expression for sK/sL. The subsequent sections utilize this expression to investigate the existence of tax competition, which concerns inefficiencies in the outputs of public services. Intuition suggests that sK/sL is less than r/w, because r includes tax payments to the local government, which should not represent a social cost for the region. This intuition is correct, but it does not imply that sK/sL equals the after-tax price ratio, p/w. A local government can certainly rent capital at the after-tax price p, and it can employ labor at the wage rate w. But when it lowers the capital-labor ratio in public service production, while price of factor i, si, is the social cost of employing 5There may exist equilibria where identical regions choose different public policies. I ignore this possibility, however, by assuming that the economy is always at the equilibrium where all chosen public policies are identical. INTERREGIONAL TAX COMPETITION 301 keeping output fixed, less labor remains for private production. As a result, the marginal product of capital falls in private production, and private firms respond by reducing their employment of capital, thereby causing the region’s property tax revenue to fall. This decline in tax revenue is an additional social cost associated with a rise in the labor intensity of public production. And the presence of this cost should lower sK/sL below p/w. But what “market imperfection” causes the local government to employ a production technique which is inefficient from the viewpoint of the nation? The answer is that when the government increases the labor intensity of public production, it creates a positive externality by causing an outflow of capital from the region to the rest of the nation. The rest of the nation benefits from the resulting rise in its tax base. By ignoring this benefit, the local government chooses a public production technique which is too capital intensive from the nation’s viewpoint. Of course, the assumed “smallness” of each region implies that any change in a region’s public production plan affects utility elsewhere by only a negligible amount. But this does not imply that the externality problem is unimportant, just as the “smallness” of automobile drivers does not imply that congestion externalities are unimportant. Although each region is “small,” the number of regions is “large.” Thus, the total social cost of inefficient government behavior is significant. I now prove that where k, is the capital-labor ratio in national good production. Equation (9) confirms my argument that sK/sL < p/w. Starting from the optimal policy for a region, consider a small change in the region’s public factor demands (&,, dL,) which does not move the government budget out of balance at the prevailing property tax rate. Since this policy change does not involve a change in t, it cannot change the before-tax return to capital (recall that a single region has a negligible influence over p). Then the wage rate must remain constant to maintain zero profits in the national good industry. With no changes in factor prices, product prices must also remain unchanged. The optimality of the initial public policy implies that the small policy change does not alter utility (to a first-order approximation). With product and factor prices unchanged, this means that the policy change does not alter public service output. But both X, and utility can remain unchanged only if the social cost of providing the public service does not change: sKdK, + sLdL, = 0. 00) 302 JOHN I next show that (&,, D. WILSON dL,) also satisfies pdK, + (w + tk,)dL, = 0. (11) Equation (9) follows from (10) and (11). With all prices unchanged following the small policy change, the residents’ demands for private goods stay fixed, as do the capital-labor ratios in national and local good production. Thus, the total quantities of capital and labor used in local good production do not change: dK, = dL, = 0. Labor market equilibrium then requires that dL, = -dL,, from which it follows that equilibrium in the region’s capital market is achieved by a change in the region’s capital stock satisfying dK = dK, - kN(dLE). Since prices are unchanged, the government budget [Eq. (3)] implies that tdK = rdK, + wdL,. These last two equalities yield (11). The proof of (9) is now complete. Although sK/sL differs from r/w, (9) does imply that both ratios decline as p increases with t held fixed.6 Furthermore, both ratios also decline as t increases with p held fixed.7 These comparative statics results will be used in my study of tax competition, to which I now turn. 4. TAX COMPETITION: A PRELIMINARY RESULT As discussed in Section 1, I define tax competition as a situation where a federal government could raise utility in each region by requiring all regions to increase their public service outputs (or tax rates) by identical small amounts. Thus, tax competition is defined in terms of inefficiencies in the outputs of the public service. The previous section has shown that there is another type of inefficiency in the model: the ratio of public shadow prices, sK/sL, differs from the rental-wage ratio, r/w, in each region. Since public production is inefficient in this way, the federal government cannot completely restore efficiency by restricting only the public service outputs which local governments are allowed to choose. Public production techniques must also be controlled. If, however, restrictions on production techniques entail high “administrative costs” relative to output restrictions, then it may be desirable to impose only output restrictions. In any case, examining the welfare effects of output restrictions appears to be a useful way of characterizing inefficiencies in public service outputs, as distinct from inefficiencies in their production. ‘Proof. Given t, r increases with national good industry. The rise in rises. ‘Proof. A rise in t increases r/w, national good industry implies that approximation. Then (9) implies that p, and w must decline to maintain zero profits in the T/W lowers k,, thereby insuring [by (9)] that sK/sr, also Q.E.D. thereby causing k, to decline. Profit maximization in the the change in w satisfies dw + k,dr = 0, to a first-order Q.E.D. sK/sI. rises. INTERREGIONAL TAX 303 COMPETITION Note that I have restricted the analysis to an examination of local welfare improvements. If welfare is a concave function of the X, which the federal government requires each region to produce, then my results allow me to compare the optimal restriction on X, with the equilibrium X, in the absence of restrictions. However, this concavity assumption need not hold. My analysis will be based on the crucial observation (proved below) that tax competition exists if and only if a rise in a single region’s public service output causes capital to flow out of the region. To state this condition in symbols, consider a region which is forced by the federal government to supply a particular public service output, X,, and assume that this region faces an after-tax return to capital equal to p. I define K(p, X,) as the region’s equilibrium capital stock. In other words, K(p, X,) gives each region’s capital stock as a function of p and X,, assuming that K,, L,, and t are set at their utility-maximizing values under the given p and X,. Using this notation, my claim can be stated as follows. Claim: Tax competition exists if and only if aK(p, X,)/G’X, < 0. Before presenting a formal proof of this claim, I first provide a simple explanation. Each local government views the property tax as distortionary: by raising the unit cost of capital above the after-tax return, the property tax encourages private firms to use too little capital. Thus, the requirement that public expenditures be financed by property taxation induces each local government to alter its policies so as to stimulate additional investment in the region. In particular, if a capital inflow accompanies a reduction in a single region’s X,, then the region’s government treats this inflow as a benefit associated with a lower X,. Since the nation’s total capital stock is fixed, however, this capital inflow implies a capital outflow of identical magnitude from the rest of the nation. The capital outflow represents a negative externality, in that it imposes a cost on the other regions by lowering their property tax base. The local government creating this externality ignores it, since its objective is to maximize the utility of only its residents. By so doing, the local government sets its X, at a level which is “too low,” as defined by my concept of tax competition. Since regions are small, a single local government cannot significantly alter utility elsewhere by changing its X,. But the economy’s equilibrium utility level is significantly reduced when all local governments set their XE’s too low. Turning to the formal proof of the claim, recall that a single region has a negligible influence over the after-tax return to capital, p. Each local government therefore chooses the public policy which provides its residents with the highest feasible utility under the prevailing p. Let u(p) denote this maximum utility level. When the federal government forces all regions to raise their public service levels by a given amount, utility changes because p must adjust to maintain equilibrium in the nation’s capital market. Let 304 JOHN D. WILSON dp/dX, represent the marginal change in p from a forced rise in every region’s X,. The resulting change in utility is (du/dp)( dp/dX,). Tax competition exists, as defined above, if and only if this utility change is positive. Intuition suggests that u(p) should decline as p rises: for a single region, the property tax distorts firm behavior by discouraging the use of capital, and a rise in p aggravates this distortion by encouraging firms to use even less capital. In the Appendix, I confirm this intuition by showing that du/dp must be negative (Lemma l), but the proof is far from intuitive. It can then be said that tax competition exists if and only if dp/dX, < 0. Now the forced rise in every region’s X, cannot alter any single region’s capital stock because all regions are identical and the nation’s total capital stock is fixed. Thus, p must change to keep K(p, X,) fixed as X, rises: aK dp aK --=o. ap dx, + ax, (12) A rise in p raises both r/w and sK/sL (see Sect. 3), and these price changes lower the capital-labor ratios used to produce the public service and the private goods. This observation suggests, but does not prove, that aK/ap is negative. In the Appendix, I show that aK/ap must be negative (Lemma 2). Then (12) implies that aK/aX, and dp/dX, possess the same signs. I can therefore conclude that tax competition exists if and only if aK/aX, is negative. My claim is proved. 5. TAX COMPETITION: THE BASIC ARGUMENT Armed with the results of the previous two sections, I can now discuss the conditions under which tax competition occurs. This section considers only sufficient conditions, since the basic arguments behind them can be presented in a relatively simple way. The next section gives two necessary and sufficient conditions which imply the conditions discussed.here. All formal proofs are in the Appendix. My first conclusion is that tax competition exists whenever public production is labor intensive relative to private production. In symbols, tax competition exists if (but not only if) X,, > XKE, where Ai, is the public production sector’s share of a region’s total supply of factor i (A KE = KJK and h,, = L,/L). The basic argument behind this result relies on my conclusion that tax competition exists if and only if a rise in a region’s public service output lowers its capital stock: aK(p, X,)/ax, < 0. I shall sign this derivative by decomposing it into two effects: an “output effect” and a “factor substitution effect.” For the output effect, I hold fixed the tax rate and all prices INTERREGIONAL TAX COMPETITION 305 faced by private firms, and I consider a “compensated” increase in X,, where the resident’s lump sum income is reduced to keep utility fixed. The factor substitution effect is obtained by returning the income to residents and then letting the tax rate and all prices adjust to their equilibrium levels under the higher X,. To sign the output effect, note first that the constancy of factor prices implies that the capital-labor ratios in national and local good production do not change. Then the ratio of public shadow prices, sJs,., stays fixed (see Sect. 3) implying that the capital-labor ratio in public production does not change. And the constancy of product prices implies that the ratio of national to local good consumption in the region remains unchanged.’ But X, can rise only if labor is transferred from private to public production. If public production is labor (capital) intensive, then a fall (rise) in the region’s capital stock is needed to complete this transfer without altering the capital-labor ratios or private consumption pattern. In symbols, the output effect is negative (K falls) if X,, > XKE, and it is positive if X,, < h,,. Turning to the factor substitution effect, observe that the rise in X, must be accompanied by a rise in the tax rate to keep the government budget balanced. As a result, both T/W and sK/sL rise (see Sect. 3). Thus, there is a drop in the capital-labor ratios used to produce the two private goods and the public service. And the relative price of the more capital intensive private good rises, causing consumers to switch their consumption pattern toward the more labor intensive private good. With both production and private consumption becoming more labor intensive, the region’s total capital stock must fall to clear its capital market. Thus, the factor substitution effect is necessarily negative: K falls. The assumption that public production is labor intensive insures that the output and factor substitution effects are both negative. Thus, this assumption implies that an increase in X, lowers K, in which case regions engage in tax competition. However, the two effects have opposite signs when public production is capital intensive. Examples can therefore be constructed where regions do not engage in tax competition. In other words, there exists cases where a forced reduction in every region’s public service output causes utility to rise. There do not appear to be any strong arguments for signing X,, - X,, one way or the other.’ It is still possible, however, to provide a condition for sThis result requires the restrictions which I impose in Section 2 on the form of the utility function. ‘After reviewing the empirical evidence, Beck [3] assumes for his numerical calculations that the public service which enters utility functions has the same factor intensity as his traded private good, while his nontraded private good, housing, uses only capital. Under these assumptions, my Proposition 1 supports his finding that the level of public services chosen by local governments is inefficiently low, although our comparisons differ somewhat. 306 JOHN D. WILSON tax competition which appears to be empirically reasonable. Tax competition occurs if the negative factor substitution effect exceeds the output effect in absolute value. The magnitude of the factor substitution effect obviously depends positively on the substitutability between labor and capital in private and public production. The elasticity of substitution between labor and capital, as normally defined, serves as the appropriate measure of factor substitutability in public service production. For private production, a simple measure can be obtained by using my assumption that the utility function has the form U( V( D,, D,), DE), where the function I/ is homogeneous of degree one. This assumption allows me to treat private utility, V, as a good which is “produced” from “intermediate inputs” iV and C by means of a constant returns to scale production technology. I can then define uy as the elasticity of substitution between labor and capital in the production of private utility. The following remarkable result can then be proved: the factor substitution effect outweighs the output effect if uy exceeds a lower bound which is always less than one. I derive this lower bound in the following section and use it to argue that tax competition exists under empirically reasonable parameter values. 6. NECESSARY AND SUFFICIENT CONDITIONS This section presents two necessary and sufficient conditions for tax competition. (See the Appendix for proofs.) To state these conditions, I use the notation introduced in Section 2 and the notational conventions of Jones [9]: X ;, is industry j’s share of a region’s total supply of factor i, e,, is factor i’s income share in industry j (e.g., eKC = (rK,)/( pX,)), and uE and uy are the substitution elasticities between labor and capital in public and private production ( uy is discussed in detail below). The subscript “F ” is used to denote the value of a variable for the entire private sector (e.g., f%, = r(K, + &)/VN + z-W); and “I ” denotes the total after-tax income of a region’s residents. Finally, I define p = [(sJsK) - (w/ p)]/[sJsK], which equals (tkN)/(w + tkN) by Eq. (9). The following propositions give the necessary and sufficient conditions. PROPOSITION 1. if and only if In particular, Regions engage in tax competition tax competition occurs if A,, > A,, (and 8K/aX, (or A KP > A LF). < 0) INTERREGIONAL PROPOSITION 2. TAX COMPETITION 307 Regions engage in tax competition if and only if [(1-$2) - o,jhLFAKFT [l-q<o, -XKEXLEPaEa” - XLJKF where 1 > r 7. In particular, 04) regions engage in tax competition if KF (I”> 1- PK eKN I eKF --. (15) The first term in condition (13) corresponds to the factor substitution effect discussed in the previous section, and the second term corresponds to the output effect. As previously argued, both effects are negative if A,, > x KE' The validity of condition (15) can be assessed by relating uy to the elasticities of substitution between capital and labor in N and C production, uN and a,, and the elasticity of substitution between N and C in consumption, E. If aij is the amount of factor i used to produce a unit of good j, then a y is defined mathematically as aV= d(WL,) r/w d(r/w) Kv/L, 06) ’ where and Kv=aKNDN+aK~DC L, = a,,D, + a,,Dc. (17) The changes in the ajj’s from a rise in r/w depend on uN and a, (see the Appendix), while the changes in D, and DC depend on E. Using these dependencies, the following expression is easily derived: a” = teLNxgN +teKC- + eKNh:N)aN eKN)(xk- + teLCAk '?C)", + eKCA~CbC (18) where Xyj is industry j’s (j = N, C) share of the total amount of factor i 308 JOHN D. WILSON used to produce private utility. The coefficients of uN, uc, and E are all positive and sum to one. The term involving E accounts for the change in relative product prices caused by a rise in T/W. Specifically, if N is capital (labor) intensive relative to C, then a rise in T/W lowers (raises) p, which lowers (raises) D,/D,. In other words, a rise in T/W always shifts consumption toward the labor intensive good, implying that u y is positively related to the substitutability between N and C. Empirical evidence on production elasticities suggests values of uN and uc near one. If the local good and national good are interpreted as housing and “other private goods,” respectively, then empirical studies on housing demand suggest values of E around .7 or .8 (see Mayo [lo]). It then appears reasonable to assume that uy lies close enough to one for the sufficient condition for tax competition given by (15) to hold under reasonable parameter values. In any case, it appears that u,, would have to be unrealistically low for (14) to be violated. But because the range of estimated production elasticities is quite wide (see Nerlove [12]), the possibility that tax competition does not occur cannot be completely dismissed. 7. CONCLUDING REMARKS I have demonstrated that, when the federal government forces each region to raise its public service output, the resulting change in the nation’s welfare may be either positive or negative. Atkinson and Stiglitz [2, p. 4941 obtain a similar ambiguity in their analysis of optimal commodity taxation and public goods: distorting commodity taxes have an ambiguous effect on the optimal level of public good provision. lo In their model, commodity taxation distorts household consumption decisions. In my model, the constraint that local public expenditures be financed by property taxation distorts local government decision making. In fact, it is easily shown that local governments would behave efficiently if they were allowed to raise revenue by imposing lump sum taxes on their residents. With lump sum taxation, local governments would no longer believe that private production is not sufficiently capital intensive, and they would therefore no longer attempt to stimulate capital investment by pursuing inefficient tax and public expenditure policies. Since all local governments choose the same policies in my model, no region is able to increase its share of the nation’s capital stock above that of any other region. Although Atkinson and Stiglitz are unable to obtain global results, they are at least able to obtain the local result that the optimal public goods supply declines when the availability of lump sum taxation is reduced by a small amount from the first-best optimum. Given my particular choice of ‘“See, also, Atkinson and Stem [l]. INTERREGIONAL TAX COMPETITION 309 comparisons, I am unable to obtain even local results without imposing further restrictions. I have considered some restrictions which appear to be empirically reasonable, and I have shown that they insure the existence of tax competition, where the federal government can raise welfare by making each region increase its public service output. To satisfy these restrictions, the substitutability between capital and the immobile input, labor, must be sufficiently high. The intuition here is that this substitutability is closely related to the responsiveness of investment decisions to property taxation, and local governments can be expected to hold down tax rates if capital is sufficiently “ mobile.” By disaggregating the private production sector into two goods, I have been able to explicitly analyze two distinct ways in which labor is substituted for capital when property tax rates are raised: (1) firms use more labor intensive production techniques; and (2) the prices of capital intensive goods rise relative to labor intensive goods, causing demands and supplies to switch toward labor intensive goods. Models which ignore the second source of substitutability may omit an important response of investment to property taxation. My model explicitly recognizes that the public production techniques chosen by local governments differ from those which minimize costs at private producer prices. In particular, I have shown that the chosen capital-labor ratios in public production are inefficiently high. Production inefficiency in government activities is a relatively unexplored area of research. Courant and Rubinfeld [5, p. 2921 observe that “with the exception of Fiorina and No11 [6], there has been essentially no scholarly literature on the mechanisms by which such inefficiency might arise, although its existence is clearly on the minds of voters according to some survey results we have obtained in other work.” Fiorina and Noll concentrate on the inefficiencies created by the need to use government “bureaucracy” as an input in the production of public goods and services. In contrast, I have ignored problems of bureaucracy and concentrated on inefficiencies resulting from decentralized decision making by local governments. A model which integrates both approaches would be useful. APPENDIX LEMMA 1. 4P)/dP < 0. Proof. Since the region’s public policy is initially optimal, conditional on p, the envelope theorem implies that the marginal impact of p on utility is independent of the particular change in the region’s public policy which accompanies the rise in p. Thus, I shall prove the lemma by showing that a small rise in p reduces the maximum X, which can be produced when k, is held fixed and the tax rate is adjusted to keep the value of private goods 310 consumption JOHN D. WILSON fixed, dD,+pdDc= 0. (A-1) Given (A.l), utility must decline if dX, < 0. For zero profits, the (first-order) price changes accompanying the rise in p must satisfy X,dp = K,dr + L,dw, (A4 where K, and L, are the total quantities of capital and labor in private production, and K,=K,+Kc L,= L,+ L,. 64.3) These price changes must also satisfy the resident’s budget constraint: dD, + pdDc + Dcdp = Kdp + Ldw. 64.4) Since all regions are identical and choose the same public policies, there is no trade at the initial equilibrium. Consequently, D, = Xc, Equations (A.2)-(A.5) D, = X,, andK= K. (A-5) and the equality, dr = dp + dt, imply that dD, + pdDc = K,dr + L,dw - Kdt. 64.6) Then (A.l) and the government budget constraint [Eq. (8)] give 0 = rdK, + wdL, - tdK. 64.7) Suppose, contrary to the lemma, that dX, 2 0. With k, held fixed, (A.7) implies that dK> 0. 64.8) And the fixity of the region’s total labor supply gives dL, I 0. (A.9) I shall use (A.8) and (A.9) to derive a contradiction. As mentioned in the text (Sect. 5) “private utility” can be viewed as a final good which is produced from intermediate inputs N and C by means INTERREGIONAL TAX 311 COMPETITION of a constant returns to scale production technology. Letting aji denote the amount of factor i used to produce a unit of good i, I can then use the equilibrium conditions in Section 2 to write L, = aLNXN + a,,&; LF=aLNDN+ aLcW- (ILN txN - DN); L, = a,,V(D,, 0,) + a,,(X, - D,); L, = a&D,, 4) + pa,dK - K). (A.lO) Since K = K initially, (A&-(A.lO) imply that da,, I 0. In other words, the capital-labor ratio in private utility production does not fall. But this implies that T/W does not rise. Thus, for zero profits in national good production, dr I 0 and dw 2 0. Since dp = dr - dt > 0 by assumption, I can conclude that But (A.ll) -dt and dw 2 0 > -dr (A.ll) 2 0. Q.E.D. contradicts (A.l) and (A.6). LEMMA 2. aK(p, X,)/ap < 0 at the equilibrium (p, X,). Prooj With X, held fixed, consider a small rise in p which is accompanied by the utility-maximizing adjustments in the public factor demands and tax rate. Assume, contrary to the lemma, that the resulting change in the region’s capital stock is non-negative: (A.12) dK20. I now derive a contradiction. There are two cases to consider. First, assume that the rise in p increases r/w. Then dk, < 0. Using the equality, sK/.sL = p/(w + tkN), it is easily seen that d(sK/sL) > 0. l1 With X, fixed, this implies that dK, -c 0 Equation and (A.13) dL, > 0. (A.6) and the government budget constraint [Eq. (8)] imply that - (dD, By the optimality + pdD,) + wdL, (A.14) - tdK. of the initial public policy, sKdK, “Whatever change X,w. See footnote I. = rdK, in r accompanies + sLdL, (A.15) = 0. the rise in p only affects sK/sI. through its impact on 312 JOHN D. WILSON Since T/W > sK/sL, however, (A.13) and (A.15) imply that rdK, + wdL, < 0. (~.16) Inequalities (A.12) and (A.16) imply that the right side of (A.14) is negative. But, with X, held fixed, Lemma 1 implies that private utility drops, from which it follows that the left side of (A.14) is positive. This is a contradiction. Finally, suppose that d(r/w) I 0. Using dp > 0, it follows that -dr > -dr 2 0 and dw 2 0. Then the right side of (A.6) is positive, whereas the left side is negative. This is a contradiction. Q.E.D. Proof of Proposition 1. With the economy initially in equilibrium, consider a rise in a single region’s X,, with p held fixed. I first consider separately the resulting first-order changes in the region’s K,, L,, K,, and L,. The notation introduced in Sec$ons 2 and 5 is used here, and a “hat” denotes a percentage change (e.g., K, = dKF/KF). Recall the expression for L, given by (A.lO). By similar arguments, K, = qJ’/(& DC) + pa,,(K - E). (A.17) Since the initial equilibrium is characterized by an absence of trade, K = K, X, = D,, K, = K,, and L, = L,. Thus, differentiating (A-17) shows that the percentage change in K, from the rise in X, satisfies (~.18) It is easily seen that F. = B,,a,(~ UK” - i); (A.19) and, by the zero profit constraint for national good production, f&J + eKNiG = 0, (A.20) -i/e,,. (A.21) or, G-F= It follows from (A.18)-(A.21) that kF = -ay(eLV/eLN)i + C + $e,,R. F (A.22) INTERREGIONAL TAX 313 COMPETITION By similar reasoning, i, = a,(e,,/e,,)i + P+ $s,,B. (A.23) F The fixity of the region’s total labor supply gives i, (A.24) = - (L,/L,)i,. Turning to R,, observe first that the definition elasticity, uE, gives RE - i, = a,(& of the substitution - sIK). (A.25) Recall that the public shadow prices satisfy SK/SL. = PAW+ fkv). (9) Equations (A.20) and (13) imply that the changes in r and w affect sK/sL only through their effects on k,. In particular, it is easily shown that A SK - s^L = uJ3(i - q, tw/d = w (~.26) where p = bLhK) - hk sL/sK Equations . (A.27) N (A.21) (A.26) and (A.27) give R, - i, = -uEu”/3(?/eLN). Combining (~.28) (A.28) with (A.24) and (A.25) yields R,=-(L,/L,) +P+ge,,z1 I(e,,/e,,)u,i F -%Gw/4N). (A.29) The change in private utility, V,_can be obtained by using the resident’s budget constraint, D, + pD, = pK + WL = I. Let p v be the gross value of the capital and labor embodied in a unit of private utility. Then the budget constraint can be written, p ,,V = & + wL, which gives P= (wL/l)k-3”. (A.30) 314 JOHN D. WILSON The zero profit conditions for N and C production imply that eKyi + e,,ci, = a,; (A.31) [f&Y- v4dLv)~LVli =a,. (A.32) or, using (A.20) Substituting (A.32) into (A.30) and using (A.20) yields (A.33) Equations (A.22), (A.29), and (A.33) give expressions for Z?,, Z?r, and p, respectively. The change in the region’s capital stock is R = A& (A.34) + h,,R,. After several mepulations, these four equations can be shown to yield an expression for K which has the same sign as 3 multiplied by the expression in Proposition 1 [condition (13)].12 But i is positive because p is fixed and the tax rate must rise to finance the given increase in X,. Thus, Z? has the same sign as the expression in Proposition 1. The proof is complete. Q.E.D. Proof of Proposition 2. The proof consists of showing that the condition (14) follows from (13). Turning to (13) first observe that [eKN(wLE/z) + &FeLN] =eK~(WL/z) = eKNeKF Thus, the term 1 - (e,,/e,,)(pK/Z) claimed. 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