Indirect tax harmonization and global public goods

Abstract

This paper identifies conditions under which, starting from any tax-distorting equilibrium, destination- and origin-based indirect tax-harmonizing reforms are potentially Pareto improving in the presence of global public goods. The first condition (unrequited transfers between governments) requires that transfers are designed in such a way that the marginal valuations of the global public goods are equalized, whereas the second (conditional revenue changes) requires that the change in global tax revenues, as a consequence of tax harmonization, is consistent with the under/over-provision of global public goods relative to the (modified) Samuelson rule. Under these conditions, tax harmonization results in redistributing the gains from a reduction in global deadweight loss and any changes in global tax revenues according to the Pareto principle. And this is the case independently of the tax principle in place (destination or origin).

Appendices

Appendix A

Proof of the statement that the reform in (12) and (13) implies that dP=dX=dX ^∗=0

Re-write, for convenience, the market clearing condition in (3) and the first-order conditions in (9) and (10) given by, respectively,

(A.1)

(A.2)

(A.3)

Equations (A.1)–(A.3) define the equilibrium of output and the world producer price of the tradeable good. Notice that sufficiency for the choice of X and X ^∗ requires, respectively, that

$$ \varPi_{XX}\equiv\alpha_{d}=2P^{\prime}+XP^{\prime\prime}-C^{\prime \prime}<0 , $$

(A.4)

and

$$ \varPi_{X^{\ast}X^{\ast}}^{\ast}\equiv\alpha_{d}^{\ast}=2P^{\prime }+X^{\ast}P^{\prime\prime}-C^{\ast\,\prime\prime}<0 . $$

(A.5)

It is also assumed that

(A.6)

(A.7)

and so the firms’ best response functions are downward sloping in quantity space. Stability of equilibrium (in the Cournot stage) requires that

$$ \varDelta _{d}=\alpha_{d}\alpha_{d}^{\ast}- \beta_{d}\beta_{d}^{\ast }>0 . $$

(A.8)

Perturbation (abusing notation somewhat) of (A.1)–(A.3)—after using the fact that, following from the demand functions, dD=D′(dP+dt _d ) ($dD^{\ast}=D^{\ast\prime}(dP+dt_{d}^{\ast})$), but also that P′=(D′+D ^∗′)⁻¹—gives in matrix form

$$ \left [ \begin{array}{c@{\quad }c@{\quad }c} 1 & -P^{\prime} & -P^{\prime} \\ 1 & \alpha_{d}-P^{\prime} & \beta_{d}-P^{\prime} \\ 1 & \beta_{d}^{\ast}-P^{\prime} & \alpha_{d}^{\ast}-P^{\prime}\end{array} \right ] \left [ \begin{array}{c} dP \\ dX \\ dX^{\ast}\end{array} \right ] =\left [ \begin{array}{c} -P^{\prime}D^{\prime}dt_{d}-P^{\prime}D^{\ast\prime}dt_{d}^{\ast} \\ 0 \\ 0\end{array} \right ] . $$

(A.9)

It can be easily verified that the determinant of the left-hand side matrix is given by (A.8). As is typically the case, without further restrictions on the structure of the model the comparative statics are indeterminate. This, in the present context, is not problematic: All that is required here is that the comparative statics are ‘well defined’ in the sense that the coefficients of the components of $D^{\prime }dt_{d}+D^{\ast \prime}dt_{d}^{\ast}$, are non-zero. It is assumed this to be the case. Solving the system of equations in (A.9) for dP,dX, and dX ^∗, one obtains

(A.10)

(A.11)

(A.12)

Close inspection of (A.10) reveals that if $D^{\prime }dt_{d}+D^{\ast \prime}dt_{d}^{\ast}=0$, then, dP=dX=dX ^∗=0. □

Appendix B

Proof of the statement that the reform in ( 27 ) and ( 28 ) implies that dQ=0

Re-write, for convenience, the market clearing condition in (3) and the first-order conditions in (25)

(B.1)

(B.2)

(B.3)

Equations (B.1)–(B.3) define the equilibrium of output and the world consumer price of the tradeable good. Notice that sufficiency for the choice of X and X ^∗ requires, respectively, that

$$ \varPi_{XX}\equiv\alpha_{o}=2Q^{\prime}+XQ^{\prime\prime}-C^{\prime \prime}<0 , $$

(B.4)

and

$$ \varPi_{X^{\ast}X^{\ast}}^{\ast}\equiv\alpha_{o}^{\ast}=2Q^{\prime }+X^{\ast}Q^{\prime\prime}-C^{\ast\prime\prime}<0 . $$

(B.5)

It is also assumed that

(B.6)

(B.7)

and so the firms’ best response function are downward sloping in quantity space. Stability of equilibrium (in the Cournot stage) requires that

$$ \varDelta _{o}=\alpha_{o}\alpha_{o}^{\ast}- \beta_{o}\beta_{o}^{\ast }>0 . $$

(B.8)

Perturbing now (B.1)–(B.3) gives (again abusing notation somewhat) in matrix form

$$ \left [ \begin{array}{c@{\quad }c@{\quad }c} 1 & -Q^{\prime} & -Q^{\prime} \\ 1 & \alpha_{o}-Q^{\prime} & \beta_{o}-Q^{\prime} \\ 1 & \beta_{o}^{\ast}-Q^{\prime} & \alpha_{o}^{\ast}-Q^{\prime}\end{array} \right ] \left [ \begin{array}{c} dQ \\ dX \\ dX^{\ast}\end{array} \right ] =\left [ \begin{array}{c} 0 \\ dt_{o} \\ dt_{o}^{\ast}\end{array} \right ] . $$

(B.9)

Solving the system of equations in (B.9) for dQ, dX, and dX ^∗, one obtains

(B.10)

(B.11)

(B.12)

Since

$$ \alpha^{\ast}_o-\beta^{\ast}_o=Q^{\prime}-C^{\ast\,\prime\prime } \equiv A^{\ast} , $$

(B.13)

and

$$ \alpha_o -\beta_o =Q^{\prime}-C^{\prime\prime} \equiv A , $$

(B.14)

it is the case that, following from (31), the origin-based tax-harmonizing reforms imply that dQ=0. □

Appendix C

Destination principle: numerical example based on Proposition 2 To simplify matters, suppose that both demands and costs functions are linear and given, respectively, by

(C.1)

(C.2)

and so

$$ P^{\prime}=- \bigl( \beta+\beta^{\ast} \bigr)^{-1}, $$

(C.3)

and the utility the consumer derives from global public goods in the home (foreign) country is Γ(G,G ^∗) (Γ ^∗(G ^∗,G)). It can be easily shown, in this case, that (A.10)–(A.12) reduce to, respectively,

(C.4)

(C.5)

and that, by making use of,

$$ dD=-\beta ( dP+dt_{d} ) ,\qquad dD^{\ast}=- \beta^{\ast } \bigl( dP+dt_{d}^{\ast} \bigr) , $$

(C.6)

perturbation of the home country utility function in (11) gives

(C.7)

(An analogous condition applies to the foreign country). Non-cooperative taxes (denoted by the subscript N) are given by setting the derivative of (C.7) with respect to t _d equal to zero, that is

(C.8)

(a similar expression holds for $t_{d}^{\ast}$).

Suppose now that the demand parameters are a=10, a ^∗=9, β=1.2, β ^∗=1.3, the marginal utilities of the public goods are Γ _G =1.9, $\varGamma_{G^{\ast}}=1.2$, $\varGamma_{G}^{\ast }=1.1$, $\varGamma_{G^{\ast}}^{\ast}=1.8$—implying that $\varGamma_{G}+\varGamma_{G}^{\ast}=3=\varGamma_{G^{\ast}}+\varGamma_{G^{\ast }}^{\ast}>1$ (and so global public goods are under-provided with respect to the Samuelson rule)—the costs are c=5>c ^∗=4, and the reform parameters are given by ψ=1=ψ ^∗ and δ=1.

It is the case that (computation performed with MAPLE v12—and all numbers have been rounded to two decimal points) D=2.25, D ^∗=1.63, P=5.28, X=0.69, X ^∗=3.19, and $t_{d}^{N}=1.18$, $t_{d}^{\ast N}=0.39$ (and so it is the home country that is the high tax country). Adding (C.7) and its foreign counterpart gives dV+dV ^∗=0.56>0 and so tax harmonization is welfare improving. It is easy also to verify that (following (21)) d(G+G ^∗)=0.09 (with $( Q/e+t_{d} ) - ( Q^{\ast }/e^{\ast}+t_{d}^{\ast} ) =0.17>0$).

Origin principle: Numerical example based on Proposition 4 Following the same steps as above, it is the case that

(C.9)

(C.10)

(C.11)

Perturbing (11) the non-cooperative origin-based tax in the home country is given by

(C.12)

(an analogous condition holds for the foreign country).

Suppose now that demand parameters are a=20, a ^∗=14, β=1.8, β ^∗=1.9, the marginal utilities of the public goods are Γ _G =1.7, $\varGamma_{G^{\ast}}=1.6$, $\varGamma_{G}^{\ast }=1.8$, $\varGamma_{G^{\ast}}^{\ast}=1.9$, whereas the costs are c=2>c ^∗=1, and the reform parameters ψ=1, ψ ^∗=1, δ=1. Then it is the case that D=12.11, D ^∗=5.67, Q=4.38, X=8.28, X ^∗=9.50, and $t_{o}^{\ast N}=0.81>t_{o}^{N}=0.14$ (so the foreign country is the high tax one). Then, in this case, dV+dV ^∗=2.29>0 (with dG+dG ^∗=0.42, and $( t_{o}+AX ) - ( t_{o}^{\ast}+A^{\ast}X^{\ast} ) =-0.34<0$). □