Sub-metropolitan tax competition with household and capital mobility

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.

Appendices

Appendix A

The basic purpose of this appendix is to prove Result 1.⁸³ 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).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

$$\begin{aligned} \sum _{i=1}^n[F^i-R_ix_i-C^i]= & {} \sum _{i=1}^n\left[ wW_i+\left( r+\tau ^K_i\right) K_i+\left( \rho _i+\tau _i^L\right) L_i \right. \\&\left. -R_i\left( y-\rho _i-\tau ^R_i\right) -C^i\right] = 0 \end{aligned}$$

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:

$$\begin{aligned} \mathrm {d}U^i= U^i_x \mathrm {d}y+\mathrm {d}U(\bar{y}-\rho _i-\tau ^R_i,g_i,R_i)= U^i_x \mathrm {d} y+\mathrm {d} \bar{u}=U^i_x \mathrm {d} y, \end{aligned}$$

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:

$$\begin{aligned} \mathrm {d} U^i= U^i_x\frac{\mathcal {L}_i}{\mathcal {P}}\mathrm {d} \rho _i. \end{aligned}$$

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

$$\begin{aligned} \frac{\hbox {d}\varOmega }{\hbox {d}t} = \tau R_t+\tau ^KK_t-C_gg_t+(F_z-C_z)z_t+R(F_{Lt}+\tau _t) = 0 \end{aligned}$$

(A.1)

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

$$\begin{aligned} F_{Lt}+\tau _t=\frac{U_R}{U_x}R_t+\frac{U_g}{U_x}g_t \end{aligned}$$

(A.2)

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

$$\begin{aligned}&WF_{WW}+KF_{WK}+LF_{WL}=0 \end{aligned}$$

(A.3)

$$\begin{aligned}&WF_{KW}+KF_{KK}+LF_{KL}=0 \end{aligned}$$

(A.4)

$$\begin{aligned}&WF_{LW}+KF_{LK}+LF_{LL}=0 \end{aligned}$$

(A.5)

$$\begin{aligned}&WF_{zW}+KF_{zK}+LF_{zL}=F_z \end{aligned}$$

(A.6)

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

(A.7)

Let A denote the first matrix on the LHS of (A.7).⁸⁶ Performing the row operation

$$\begin{aligned} r_3 \leftarrow Wr_1+Kr_2+Lr_3 \end{aligned}$$

(A.8)

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

$$\begin{aligned} |A| =\frac{U_R}{U_x}D \end{aligned}$$

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

$$\begin{aligned} D= -\frac{L}{W}\begin{vmatrix} F_{WL}&\quad F_{WK} \\ F_{KL}&\quad F_{KK} \end{vmatrix} = -\frac{L}{K}\begin{vmatrix} F_{WW}&\quad F_{WL} \\ F_{KW}&\quad F_{KL} \end{vmatrix} >0 \end{aligned}$$

(A.9)

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.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

$$\begin{aligned} R_t = G_t\frac{U_x}{U_R} \end{aligned}$$

(A.10)

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

$$\begin{aligned} R_\tau =\frac{U_x}{U_R} \text {, }&R_{\tau ^K}=-\frac{K}{L}\frac{U_x}{U_R} \text {, }&R_g = -\frac{U_g}{U_R} \text {, }&R_z = \frac{U_x}{U_R}\frac{F_z}{L} \text {. } \end{aligned}$$

(A.11)

which proves the signs of the responses $R_t, \ t \in \lbrace \tau ; \tau ^K; g; z\rbrace$ in Lemma 2.⁸⁹

Identically,

(A.12)

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

The location responses of workers are thus given by

$$\begin{aligned} \begin{aligned}&W_\tau =-\frac{W}{L}\frac{U_x}{U_R} \text {, } \quad W_{\tau ^K}=\frac{KW}{L^2}\frac{U_x}{U_R}-\frac{F_{WK}}{D} \text {, } \\&W_g = \frac{W}{L}\frac{U_g}{U_R} \text {, } \qquad W_z = -\frac{W}{L^2}F_z\frac{U_x}{U_R}+\frac{F_{WK}F_{Kz}-F_{KK}F_{Wz}}{D} \text {. } \end{aligned} \end{aligned}$$

(A.13)

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,

$$\begin{aligned} \begin{aligned}&K_\tau =-\frac{K}{L}\frac{U_x}{U_R} \text {, } \quad K_{\tau ^K}=\left( \frac{K}{L}\right) ^2\frac{U_x}{U_R}+\frac{F_{WW}}{D} \text {, } \\&K_g = \frac{K}{L}\frac{U_g}{U_R} \text {, } \qquad K_z = -\frac{K}{L^2}F_z\frac{U_x}{U_R}+\frac{F_{KW}F_{Wz}-F_{WW}F_{Kz}}{D} \text {. } \end{aligned} \end{aligned}$$

(A.14)

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),

$$\begin{aligned}&\left( \tau +R\frac{U_R}{U_x}\right) R_\tau +\tau ^KK_\tau =0 \end{aligned}$$

(A.15)

$$\begin{aligned}&\left( \tau +R\frac{U_R}{U_x}\right) R_{\tau ^K}+\tau ^KK_{\tau ^K}=0 \end{aligned}$$

(A.16)

$$\begin{aligned}&\left( \tau +R\frac{U_R}{U_x}\right) R_g+\tau ^KK_g+R\frac{U_g}{U_x}-C_g=0 \end{aligned}$$

(A.17)

$$\begin{aligned}&\left( \tau +R\frac{U_R}{U_x}\right) R_z+\tau ^KK_z+F_z-C_z=0 \end{aligned}$$

(A.18)

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

$$\begin{aligned} \begin{vmatrix} R_\tau&\quad K_\tau \\ R_{\tau ^K}&\quad K_{\tau ^K} \end{vmatrix} =\frac{U_x}{U_R}\frac{F_{WW}}{D} \ne 0 \end{aligned}$$

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

$$\begin{aligned} \tau +R\frac{U_R}{U_x}&=0 \end{aligned}$$

(A.19)

$$\begin{aligned} \tau ^K&=0 \end{aligned}$$

(A.20)

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

$$\begin{aligned}&F_W(W,K,\mathcal {L}-R,z)-w=0 \end{aligned}$$

(A.21)

$$\begin{aligned}&F_K(W,K,\mathcal {L}-R,z)+\frac{\tau R-C}{K+\mathcal {L}}-r=0 \end{aligned}$$

(A.22)

$$\begin{aligned}&U[\bar{y}-F_L(W,K,\mathcal {L}-R,z)-\tau ,g,R]-\bar{u}=0 \end{aligned}$$

(A.23)

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

(A.24)

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 $\tau ^K$:

$$\begin{aligned} \left( \tau +R\frac{U_R}{U_x}\right) R_\tau +\tau ^PK_\tau =0 \end{aligned}$$

(A.25)

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

$$\begin{aligned}&\left( \tau +R\frac{U_R}{U_x}\right) \begin{vmatrix} F_{WW}&\quad 0&\quad F_{WK} \\ F_{KW}&\quad -\frac{R}{K+\mathcal {L}}&\quad F_{KK} \\ 0&\quad \alpha&\quad 0 \end{vmatrix} \\&\quad +\,\tau ^P \begin{vmatrix} F_{WW}&\quad -F_{WL}&\quad 0 \\ F_{KW}&\quad -F_{KL}-\frac{R}{K+\mathcal {L}}\frac{U_R}{U_x}&\quad -\frac{R}{K+\mathcal {L}} \\ 0&\quad \alpha \frac{U_R}{U_x}&\quad \alpha \end{vmatrix} =0 \end{aligned}$$

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

$$\begin{aligned} \left( \tau +R\frac{U_R}{U_x}\right) D +\tau ^P \begin{vmatrix} F_{WW}&\quad F_{WL} \\ F_{KW}&\quad F_{KL} \end{vmatrix} =0 \end{aligned}$$

(A.26)

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

$$\begin{aligned} \tau +R\frac{U_R}{U_x} -\frac{K}{L}\tau ^P =0 \end{aligned}$$

(A.27)

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

$$\begin{aligned} \tau ^P\left( \frac{K}{L}R_g+K_g\right) +R\frac{U_g}{U_x}-C_g=0 \end{aligned}$$

(A.28)

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

(A.29)

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

(A.30)

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

$$\begin{aligned} \tau ^P\left( \frac{K}{L}R_z+K_z\right) +F_z-C_z=0 \end{aligned}$$

(A.31)

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

$$\begin{aligned} \frac{F_z-C_z}{K}=\frac{1}{K}\tau ^P\frac{L}{K+L}\frac{F_{WW}}{D}\left[ F_{Kz}-F_{Lz}-\frac{F_{Wz}}{F_{WW}}(F_{KW}-F_{LW}) \right] \end{aligned}$$

(A.32)

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

$$\begin{aligned} \left. \frac{\partial W}{\partial z} \right| _{(\bar{K},\bar{L})}=-\frac{F_{Wz}}{F_{WW}}>0 \end{aligned}$$

(A.33)

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

$$\begin{aligned} \varepsilon =\tau ^P\frac{K+L}{K}\left( \frac{K}{K+L}\right) _{\tau ^P}=\frac{1}{K}\tau ^P\frac{L}{K+L}\left( K_{\tau ^P}+\frac{K}{L}R_{\tau ^P}\right) \end{aligned}$$

(A.34)

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,

$$\begin{aligned}&R_{\tau ^P}=-\frac{K+L}{L}\frac{U_x}{U_R}&\text {and}&K_{\tau ^P}=\frac{F_{WW}}{D}+\frac{K(K+L)}{L^2}\frac{U_x}{U_R} \end{aligned}$$

Integrating these expressions into (A.34) yields

$$\begin{aligned} \varepsilon =\frac{1}{K}\tau ^P\frac{L}{K+L}\frac{F_{WW}}{D} <0 \end{aligned}$$

(A.35)

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