## Abstract

This paper investigates the efficiency properties of tax competition between sub-metropolitan jurisdictions when capital, residents and workers are mobile, and both households and firms compete for local land markets. We analyze two decentralized equilibria: (1) With a local tax on residents and two separate local taxes on capital and land inputs, efficiency is achieved and the existence of a *marginal fiscal cost* due to residents’ mobility is revealed; (2) Combination of the taxes on capital and land inputs into a single business property tax leads local authorities to charge inefficiently high taxation on capital. We show that capital mobility induces a reduction in the business land taxation and local public inputs are used to offset the distorting effects of the property tax, accounting for the distorting impact of workers’ mobility.

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## Notes

See Brülhart et al. (2015) for a comprehensive overview of this recent literature.

See, e.g., McKenzie (2013) for empirical evidence of the significance of county-to-county commuting in the USA.

While this paper is concerned with households’ perfect mobility, which may be relevant at the municipal level, other papers which investigate regional tax competition focus on imperfectly mobile households. Based on the model proposed by Mansoorian and Myers (1993), several authors, such as Burbidge and Myers (1994) and Wellisch (1994), study tax competition when individuals have different degrees of home attachment. Bucovetsky (2011) studies the situation where households face a fixed uniform mobility cost when moving from one jurisdiction to another.

Using Richter and Wellisch (1996)’s framework, Wellisch and Hulshorst (2000) analyze the distortions induced by the absence of either an undistortive tax on land or one of the direct taxes on households and firms. Wilson (1995) demonstrates that contrary to head taxes on households, labor taxes on individual labor supply induce under-provision of public goods.

Mieszkowski and Zodrow (1989) refer to this type of models as

*regional*models, whereas they qualify as*metropolitan*models, those that study tax competition within a metropolitan area.Gaigné et al. (2016) develop a urban economic model with asymmetric tax competition within metropolitan areas which also integrates both residential and labor mobility of households. Contrary to the present paper, in response to local policy changes residents are mobile within jurisdictions and not across them. Moreover, in Gaigné et al. (2016) public services are exogenous, while efficiency of endogenous local public service provision is central to our analysis. Thus, the results in Gaigné et al. (2016) should be regarded as complementary to those in this paper.

Business taxes are source-based since both local private inputs are partially owned by non-residents.

This corresponds to the first-best case in Wilson (1995). Similar outcomes are obtained by Wellisch and Hulshorst (2000) in Richter and Wellisch’s (1996) framework with several firms involving congestion but no capital. In their framework, due to firms’ congestion, the optimal poll business tax is a congestion fee.

Hettich and Winer (1988, 1999) characterize the optimal number of activities within a tax base, accounting for the administrative and political costs. Another rationale for a combined tax is that it might be difficult, in practice, to disentangle closely related tax bases. For instance, land improvements which constitute a type of capital are often integrated in the land tax base (Fisher 2015).

E.g., in the USA the legal restrictions in 40 out of the 50 states impose equal rates on real property (land and buildings) and personal property (equipment, machinery, inventories...). Source: the online database published by the Lincoln Institute of Land Policy (2014). In France, the

*taxe professionnelle*(1975–2010) was another example of an identical statutory tax rate on capital and land.When regional authorities are constrained to use a combined business property tax, Wilson (1995) shows that the distortions are balanced among all their tax instruments.

This provision rule echoes Matsumoto and Sugahara (2017). Their framework differs in several respects from the one proposed here, but the main difference is that we account for the existence of an untaxed mobile factor (labor).

Making the alternative assumption that \(F^i\) exhibits constant returns to scale in all factors including the public input would not affect significantly the analysis in this paper, since the number of firms is normalized to one (see Matsumoto 1998). The case of constant returns to scale in private factors is chosen for convenience and is usually considered as the empirically more relevant case (see footnote 56).

The usual assumption that factors are complements in production is reasonable given the aggregation of production.

For the sake of simplicity, we do not consider housing construction so that households and firms use land directly, and we assume that the individual demand for land is inelastic. In Hoyt (1991), Krelove (1993) and Wilson (1997), housing production uses mobile capital and fixed land, and the individual housing demand is elastic. Krelove shows that housing taxation entails usual distortions from optimal commodity tax theory. See Wilson (2003) for a survey of this literature.

Note that there may be differences in local land endowments \(\mathcal {L}_i\). This paper is in fact not restricted to the study of symmetric equilibria across jurisdictions.

The integration of commuting costs into the model is discussed in Sect. 5.

Commuting is a noticeable departure from the WRW framework. In this literature where commuting is not allowed, the benefits of local policies capitalize into the wage rate which is therefore an endogenous jurisdiction-specific variable even if jurisdictions are atomistic. This explains the central role of local labor markets in this literature.

For reasons of space, the framework does not consider local taxes on labor inputs. However, as discussed below, the results can easily be extended to labor taxes.

Note that the central planner does not need to use taxes in order to determine the Pareto-efficient allocation since its resources come from the direct control of firms.

The planner’s problem may well be solved by a so-called corner solution. That is, an allocation for which there is no production or no resident in some jurisdictions. We ignore this well-known problem in regional economics.

Recall that the central planner does not levy taxes on residents. This explains why from its viewpoint, residents only entail costs that it balances among jurisdictions. However, local governments tax residents, so that in the decentralized equilibria analyzed in Sect. 4, residents entail not only costs but also benefits from the individual jurisdictions’ viewpoint.

Notice that the symmetric role of capital and labor in this economy allows one to deduce that any uniform level of local labor tax would ensure an efficient location of workers across jurisdictions. This requirement is met since the absence of a labor tax can be considered a uniform zero-tax on labor.

It may be inferred from Result 1 that the location-based taxes on residents alone do not sustain efficiency. Jurisdictions must use the source-based taxes on capital and business land to finance the public services and perform the efficient interjurisdictional transfers of resources. Indeed, as shown in, for example, Myers (1990), Hercowitz and Pines (1991) and Krelove (1992), in a federation with household mobility, interjurisdictional transfers of resources are necessary to sustain efficiency.

This contrasts with the literature in which an undistortive tax—usually a source-based tax on (fixed) business land—clears the local budget constraints and achieves resource transfers [see, e.g., Proposition 2.3 in Wellisch (2006)].

A given increase (decrease) in the individual’s income will, ceteris paribus, increase (reduce) the satisfaction of the residents living in the community, and especially since the equilibrium amount of the local public good is higher (lower) than private consumption—assuming diminishing marginal rate of substitution.

Additive separability is a widespread hypothesis in tax competition models. However, utility need not be linear in consumption, e.g., Assumption 1 holds also for \(U(x,g,R)=\varPhi [x+v(g,R)]\), whenever \(\varPhi \) is a bijective function, which is guaranteed by the usual assumptions: \(\varPhi \) is continuous and \(\varPhi '>0\).

The result stated in Lemma 1 requires a separation of individuals’ decisions as consumers and landowners. It is ensured by the neutrality hypothesis (Assumption 1) introduced in this paper. Other approaches are also possible. For example, Wilson (1995) and Wellisch and Hulshorst (2000) assume an Arrow–Debreu separation, while Henderson (1985) assumes that the local policy is conducted by absentee landowners. Alternatively, Hoyt (1991) postulates that the policy instruments are controlled by some immobile landowners who aim at maximizing their net wealth. In all these frameworks, the individual income

*y*is treated as exogenous, while local authorities maximize the local land rent in order to maximize the income of landowners. Assumption 1 allows reconciliation of these two seamingly incompatible assumptions.To be consistent with this alternative argument, immobile landowners can be introduced in the model. It can be shown that this would not affect any of the results derived in this paper provided that mobile and immobile residents have the same marginal willingness to pay for the local public good. See Sect. 5 for further discussion.

The detailed derivation of the necessary conditions (30) is provided in Appendix C.

Derivation of the location responses requires that the local public good involves congestion, \(U^i_R<0\), as assumed in this paper (see footnote 87). This requirement must be met since all factors are variable; in the WRW model, deriving the location responses requires the presence of a fixed production factor.

The proof of this common result adapted to the present framework is available in an additional appendix, available from the author upon request.

See, e.g., Wilson (1995), Richter and Wellisch (1996) and Wellisch and Hulshorst (2000). Notice that Assumption 2 is a simplifying assumption which can be dropped without affecting any of the results of this paper. It would just require the interpretations to be adapted slightly, accounting for the fact that \(\tau ^L_i\) could be a subsidy.

The \(R_i/\mathcal {L}_i\) term in (35)—which can be written also as \(1-L_i/\mathcal {L}_i\)—simply recalls that from a budgetary perspective, broadening a tax base allows lowering of the related tax rate. However, this budgetary effect is of minor importance to this analysis.

Notice that \(\tau ^R_i\) and \(\tau ^L_i\) play a symmetric role from the local government’s viewpoint. Alternatively, \(\tau ^R_i\) can be used to clear the budget constraint and \(\tau ^L_i\) to internalize the net marginal fiscal cost of business land use \(\tau ^R_i-R_i|U_R^i|/U^i_x\). In this case, the level of \(\tau ^L_i\) is defined by (31) and the level of \(\tau ^R_i\) is obtained by inserting (31) into (35). This symmetry has important implications when the tax structure is constrained (Sect. 4.3).

Allowing workers to cause congestion would not change the results of this paper, provided that a local tax on labor is also introduced to allow local authorities to internalize this additional cost.

The conditions stated in Result 2 are essentially the same as the first-best results discussed in Wilson (1995) (section 3). However, Wilson’s results exclude two elements: (1) the marginal fiscal cost \(\tau ^L_i\) in (31); (2) condition (34) since his analysis focuses on local public goods. Result 2 can be seen also as an extension of the optimal behavioral rules in Wellisch and Hulshorst (2000) (section 2).

Household taxes also allow local governments to employ the business land tax so as to perform the efficient resource transfers (see footnote 32).

In the first-best setting (Sect. 4.2), only the allocation of local land between residents and firms was distorted by \(\tau ^L_i\). This distortion could be offset by the use of \(\tau ^R_i\). Here, \(\tau ^P_i\) also distorts the location of capital which leads to sub-efficiency. Indeed, \(\tau ^P_i\) does not enable local authorities to finance the provision of public services and achieve the resource transfers without distortion.

In this framework, a symmetric decentralized equilibrium is all the less likely since jurisdictions potentially have different initial land endowments \(\mathcal {L}_i\).

To our knowledge, the literature does not provide such a characterization of combined business property taxes. Most studies focus on the distortions caused by a tax structure incorporating a business property tax, without deriving its explicit level.

Since input prices ratios are not directly affected by changes in \(\tau ^R_i\), homogeneity of \(F^i\) implies that the relative demand for inputs \(K_i/L_i\) is unchanged.

In both cases \(\tau ^R_i=(C^i+L_i|U^i_R|/U^i_x)/(R_i+L_i)\).

See footnote 44.

In the WRW framework, each tax has its own role to play. Therefore, when the tax structure is constrained, each tax is used separately to alleviate directly or indirectly the lack of available instruments, while continuing to play its first-best role partially. See Wellisch (2006).

The literature on public inputs distinguishes two categories of public factors depending on the technology considered (Hillman 1978; McMillan 1979; Feehan 1989). This paper assumes “factor-augmenting” public inputs (i.e., \(F^i\) is CRS in private factors only), usually considered as the empirically more relevant case. It implies that public inputs only increase private factors’ productivity unlike “firm-augmenting” public inputs (i.e., \(F^i\) is CRS in all factors) which also increase the firms’ profit. Comparing the outcomes of these two specifications is beyond the scope of the present analysis. See Matsumoto (1998) for such a comparison when public inputs are financed by a capital tax.

Intuitively, condition (39) reads: The local public input is over(under)-provided if and only if it allows jurisdiction

*i*to attract capital(land)-intensive firms.Indeed, \(\tau ^P_i=0\) implies \(\varepsilon _i=0\) since \(\varepsilon _i=\frac{\tau ^P_i}{K_i+L_i}\frac{F^i_{WW}}{F^i_{WL}F^i_{KW}-F^i_{WW}F^i_{KL}}\) (see Appendix E).

See also Proposition 4 in Wellisch and Hulshorst (2000).

This result echoes the conclusions in Wellisch and Hulshorst (2000): When their tax instrument set is constrained, regional governments balance the distortions among their available instruments.

These considerations are not accounted for in Matsumoto and Sugahara (2017) since they do not consider an untaxed mobile factor, i.e., \(\left. (\partial W_i/\partial z_i)\right| _{(\bar{K}_i,\bar{L}_i)}=0\).

Mansoorian and Myers (1993) assume that residents have attachment to their jurisdiction so that they face a psychic cost when moving to another jurisdiction.

Since the model does not include a spatial dimension, introducing commuting costs dependent on the distance between locations would require changes in the framework much beyond the scope of the paper. See, e.g., Braid (2000) for a spatial tax competition model with commuting costs dependent on distance.

The online appendix provides a graphical representation of the labor market equilibrium for these three categories of jurisdictions.

The case of this category of jurisdictions is close to the WRW framework since residents work where they live. However, regional models still differ from the present case since they ignore residential land.

The online appendix formally develops this extension. An alternative approach to account for different degrees of household mobility is taken in Mansoorian and Myers (1993), which considers a continuum of individual home attachment degrees. However, introducing a Hotelling space of preferences as in Mansoorian and Myers (1993) might be complex in the present framework with many atomistic jurisdictions.

For instance, all residents have the same MWP if utility is additively separable, such that \(U(x,g,R)=x+v(g,R)\).

Note that in this framework, richer households’ MWP for the local public good is at least as high as that of poorer households due to diminishing marginal rates of substitution. This theoretical ordering of MWPs is also the most relevant case in practice.

See Sect. 1 for examples of countries where such reforms have been implemented.

Otherwise, such measures could be too unpopular to be implemented. For instance, all French local jurisdictions have been quasi-perfectly compensated by national grants after the removal of capital from the business property tax base in 2010.

See Fisher (2015) for a classification of intergovernmental grants and a presentation of their various economic effects.

It is straightforward to see from (40) that such a grant scheme provides local governments with the incentives to set a zero business property tax, which removes the distortions from (37) to (39). In words, this grant scheme allows local governments to use their household tax as a congestion fee while being ensured that their public service provision is financed without relying on the business property tax.

See the online appendix for further details about these simulations.

See the online appendix for a formal development of this extension.

Then, the location response pattern of Lemma 2 is altered by the presence of land use restriction policy.

The intuition is the following. Consider a jurisdiction where households pay a lower land rent than firms. Then, the replacement of a business land unit by a new resident entails a decrease in the land rent generated in the jurisdiction. This loss must be internalized by households through the resident tax.

See Stafford and DeBoer (2014) for a detailed discussion of such reforms in the USA.

See Keen and Konrad (2013) for a review of tax competition models with dynamic aspects.

The proof follows the methodological approach in Wellisch (2006) (Chapter 2).

In the following appendices, the jurisdiction index

*i*is dropped for notational convenience.Notice that from Schwarz’s theorem, \(\forall X,Y \in \lbrace W;K;L \rbrace , \ F_{XY}=F_{YX}\).

To solve (A.7) for \(W_t\), \(R_t\) and \(K_t\), the determinant |

*A*| must be nonzero which is the case since \(U_R < 0\) due to congestion, and \(D>0\) (see below).The derivations make extensive use of the following calculation rule for determinants: \( |c_1 \ldots c_j \ldots c_p|=\frac{1}{\alpha _j} |c_1 \ldots \underset{(column \ j)}{\sum _{k=1}^p \alpha _kc_k} \ldots c_p|\), where \(c_j\) is the \(j^{th}\) column vector. Since \(|^{t}{A}|=|A| \), the same rule applies to row operations.

To establish the signs of the location responses, recall that: \(U_x>0\), \(U_g>0\), \(U_R<0\), \(F_X>0\), \(F_{XX}<0\) and \(F_{XY}>0\) for all

*X*and*Y*in \(\lbrace W;K;L;z \rbrace \).

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## Acknowledgements

I thank the editor and an anonymous referee for helpful comments and suggestions. I am grateful to Sonia Paty, Florence Goffette-Nagot, Thierry Madiès, Stéphane Riou, Hubert Jayet, Etienne Farvaque, Albert Sollé-Ollé, Fransisco José Veiga, Stéphane Gauthier, Michael Suher for valuable comments. I also thank participants in the Annual Meeting of the North American Regional Science Council (Minneapolis), Public Economics at the Regional Level Workshop (Santiago de Compostela), Political Economy and Local Public Finance Workshop (Lille), GATE (Lyon), Journées de Microéconomie Appliquée (Besançon) and French Economic Association Meeting (Nancy) for their comments. Financial support from Région Auvergne-Rhne-Alpes (ARC 7 and Explora’Doc) is gratefully acknowledged.

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## Electronic supplementary material

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## Appendices

### Appendix A

The basic purpose of this appendix is to prove Result 1.^{Footnote 83} To this end, we assume that the economy is characterized by the resource constraints (1)–(3) and that the agents behave according to (4)–(11). Let us prove that a necessary and sufficient condition for implementing the efficient allocation—defined by the constraints (1)–(4) and (12), and the optimal conditions (13)–(17)—is that the instruments \(\{\tau ^R_i,\tau ^K_i,\tau _i^L,g_i,z_i\}\) satisfy conditions (18)–(21).^{Footnote 84}

First, observe that in a decentralized economy households migrate at no cost and all markets are first-order condition. Hence, the constraints (1)–(4) are satisfied. Besides, from the resident’s budget constraint (6) and the zero-profit condition (10), it follows that

where the last equality is obtained using the definition of the individual income (5) and the local government budget constraint (11). Therefore, the feasibility constraint of the economy (12) is satisfied. Also, noting that the wage rate *w* is the same in the whole federation, it follows from (7) that condition (15) is ensured in equilibrium.

Let us now turn to the tax system. Inserting (6) for \(x_i\) and (9) for \(\rho _i\) into both sides of the conditions for efficient allocation of residents (13) yields condition (18). This proves that conditions (13) and (18) are equivalent when allowing for private behaviors. And, inserting (8) for \(F_K^i\) into (14) proves that the conditions for efficient capital allocation (14) and (19) are equivalent.

Finally, noting that the public service provision rules (16) and (17) are, respectively, identical to (20) and (21) allows us to complete the proof.

### Appendix B

Consider the impact of a small local policy change on the representative resident’s utility. Inserting the budget constraint (6) into the utility function and accounting for the free mobility condition (23), total differentiation of \(U^i\) yields:

since \(\mathrm {d} \bar{u}=0\). That is, the only channel through which policy can increase utility is income variations; residents mobility compensates for any other effect. Using the income definition (5), the marginal utility change can be written as:

Thus, any utility gain resulting from the incremental policy is due to an increase in the return to domestic landowners.

### Appendix C

### 1.1 Necessary conditions

This paragraph derives the first-order conditions of the local government (30). Differentiating the local government’s objective (29) with respect to \(t \in \lbrace \tau ; \tau ^K ; g ; z \rbrace \) and the conditions for the optimal inputs demand (7)–(9), it follows that

where subscripts stand for derivatives.^{Footnote 85} Besides, differentiating the migration equilibrium (26), we have

Inserting (A.2) into (A.1), the first-order conditions (30) result.

### 1.2 Location responses

We now derive the location responses of residents, workers and capital to local policy changes. First, note that since *F* exhibits constant returns to scale in the private factors, Euler’s theorem requires that \(F=WF_W+KF_K+LF_L\). Differentiating this condition with respect to *W*, *K*, *L* and *z* yields

Differentiating the equilibrium conditions (24)–(26) with respect to \(t \in \lbrace \tau ,\tau ^K,g,z \rbrace \), it follows that

Let *A* denote the first matrix on the LHS of (A.7).^{Footnote 86} Performing the row operation

on the third row of |*A*|, and using (A.3)–(A.5) yields

where \(D= \begin{vmatrix} F_{WW}&\quad F_{WK} \\ F_{KW}&\quad F_{KK} \end{vmatrix} \).^{Footnote 87} It will prove useful in the sequel to notice that

The first (second) equality of (A.9) is obtained applying the column operation \(c_1(c_2) \leftarrow \frac{W}{L}c_1+\frac{K}{L}c_2\) to the first (second) column of *D*.^{Footnote 88}

Let us now derive the migration responses of residents. Applying Cramer’s rule to (A.7), it comes

Using the row operation (A.8) and equalities (A.3),(A.4) and (A.6), we obtain

where \(G_t \equiv \tau _t-\frac{K}{L}\tau ^K_t-\frac{U_g}{U_x}g_t+\frac{F_z}{L}z_t\). It follows that

which proves the signs of the responses \(R_t, \ t \in \lbrace \tau ; \tau ^K; g; z\rbrace \) in Lemma 2.^{Footnote 89}

Identically,

Inserting (A.9) into (A.12) yields

The location responses of workers are thus given by

which proves the signs of the responses \(W_t, \ t \in \lbrace \tau ; \tau ^K; g; z\rbrace \) in Lemma 2. Notice that the signs of the responses (A.13) are unambiguous, since \(D>0\) from (A.9).

Finally, the same calculations give

so that,

which proves the signs of the responses \(K_t, \ t \in \lbrace \tau ; \tau ^K; g; z\rbrace \) in Lemma 2.

### Appendix D

The basic purpose of this appendix is to prove Result 2. From the first-order condition (30),

Yet, using (A.11) and (A.14) yields

Then, equations (A.15) and (A.16) imply

which proves the optimal taxation rules (31) and (32), recalling that \(\tau =\tau ^R-\tau ^L\). Inserting (A.19) and (A.20) into (A.17) and (A.18), the Samuelson rules (33) and (34) follow.

Finally, using (A.19) and (A.20) to substitute \(\tau \) and \(\tau ^K\) into the local budget constraint (27) yields the optimal condition for \(\tau ^L\) (35).

### Appendix E

### 1.1 Locational system

The basic purpose of this appendix is to prove Result 3. Using (36) to substitute for \(\tau ^P\) into (25), the location system becomes

Differentiating (A.21)–(A.23) with respect to \(t \in \lbrace \tau ,g,z \rbrace \), it follows that

Let *B* denote the first matrix on the LHS of (A.24).

### 1.2 Household taxation rule

Let us start with the choice of \(\tau \) whose second-best value is determined by the necessary condition (30), where \(\tau ^P\) replaces \(\tau ^K\):

As in the first-best case, Cramer’s rule is used to solve for \(R_\tau \) and \(K_\tau \) from (A.24). Inserting these expressions into (A.25), the first-order condition becomes

Noting that the terms \(\frac{\tau ^P}{K+\mathcal {L}}\) and \(\frac{1}{K+\mathcal {L}}\left( \tau +R\frac{U_R}{U_x}\right) \) cancel each other and applying the row operation (A.8) to both determinants, it follows that

where \(\alpha =\frac{KR}{K+\mathcal {L}}+L\). Operating \(r_2 \leftarrow r_2+\frac{1}{\alpha }\frac{R}{K+\mathcal {L}}r_3\) on the right determinant, and developing the resulting determinants yields

Dividing (A.26) by *D* and using (A.9), it comes

which proves the second-best taxation rule (37).

Finally, using (A.27) to substitute \(\tau \) into the local budget constraint (36) yields the optimal condition for \(\tau ^P\) (40).

### 1.3 Public good provision rule

Consider now the choice of *g*. Replacing \(\tau +R\frac{U_R}{U_x}\) from the second-best taxation rule (A.27) into (30), it follows that

Using again Cramer’s rule and inverting the last two columns in the expression of \(R_g\), it comes

and operating \(c_2 \leftarrow -c_2+\frac{K}{L}c_3\) on the second column of |*B*|, we obtain

where \(\gamma ^W \equiv \frac{K}{L}F_{WK}+ F_{WL}\), \(\gamma ^K \equiv \frac{K}{L}F_{KK}+F_{KL}\) and \(\gamma ^L \equiv \frac{K}{L}F_{LK}+F_{LL}\). Multiplying (A.28) by \(-|B|\), introducing the explicit forms of \(R_g\), \(K_g\) and |*B*|, and adding determinants yields

where \(\varGamma \equiv R\frac{U_g}{U_x}-C_g\) has been introduced for convenience. Operating \(c_2 \leftarrow c_2+\frac{W}{L}c_1\), using Euler’s expressions (A.3)–(A.5) and condition (A.27) to simplify terms, it follows that

Performing \(r_2 \leftarrow Wr_1+(K+\mathcal {L})r_2-Rr_3\), simplifying from (A.3)–(A.5) and collecting terms, we obtain

It is straightforward to derive from (A.9) that the determinant in (A.30) equals \(\frac{K+L}{L}D \ne 0\). This proves the Samuelson rule (38).

### 1.4 Public input provision rule

Finally, let us prove the second-best public choice rule for *z*. This proof follows rigorously the same computation steps as in the derivation of the second-best public good provision rule. Therefore, we provide only the main steps of the proof. As, above, we start with replacing \(\tau +R\frac{U_R}{U_x}\) from (A.27) into (30), which yields

Then, applying Cramer’s rule to get explicit forms for \(R_z\) and \(K_z\), integrating them into (A.31) and operating \(c_2 \leftarrow c_2+\frac{W}{L}c_1\), it follows that

where \(\varLambda \equiv F_z-C_z\). Performing \(r_2 \leftarrow Wr_1+(K+\mathcal {L})r_2+Rr_3\), simplifying terms using (A.3)–(A.6) and collecting terms, we obtain

Since the first determinant in (A.30) equals \(\frac{K+L}{L}D \ne 0\) from (A.9). Straightforward manipulations yield

Notice that differentiating (A.21), given the equilibrium values \(\bar{K}\) and \(\bar{L}\), yields

Besides, the elasticity of the capital share in the overall business property with respect to property tax changes writes

where the second equality is obtained by recalling that \(L_{\tau ^P}=-R_{\tau ^P}\) from the land market first-order condition (3). And, replacing \(\tau ^K\) by \(\tau ^P\) and \(\tau \) by \(\tau ^R-\tau ^P\) into the location system (24)–(26) allows to derive by differentiation,

Integrating these expressions into (A.34) yields

Finally, inserting (A.33) and (A.35) into (A.32), the optimal second-best condition (39) follows.

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Ly, T. Sub-metropolitan tax competition with household and capital mobility.
*Int Tax Public Finance* **25**, 1129–1169 (2018). https://doi.org/10.1007/s10797-018-9490-7

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DOI: https://doi.org/10.1007/s10797-018-9490-7