On preferences and taxation mechanisms in strategic bilateral exchange

Abstract

In this paper we investigate the effectiveness and the welfare implications of various fiscal policies in strategic bilateral trade. To this end, we reconsider the class of bilateral oligopoly models with taxation implemented in Gabszewicz and Grazzini (J Public Econ Theory 1999, 1:475–497, and in: Debreu G, Neuefeind W, Trockel W (eds) Economic essays, a Festschrift for Werner Hildenbrand, Springer, Berlin, 2001 109–126). Assuming the preferences of traders are represented by CES utility functions with non unitary shares on consumption, we determine the Cournot–Nash equilibria, and we study the effects of two fiscal policies with transfers. We show that both fiscal policies with transfers implement a first-best allocation only when commodities are perfect complements or perfect substitutes.

1 Introduction

Tax policies can be used to mitigate distortions in resource allocation caused by imperfectly competitive behavior (Guesnerie and Laffont 1978; Myles 1989) and also for redistributive purposes (Mirrless 1971). In this paper, we explore a model in which tax policies play this dual role. The main motivation is to study, in a simple framework, the extent to which the effects and effectiveness of redistributive tax policies depend on the preferences of individuals who behave strategically in trade. To this end, we reconsider the class of noncooperative strategic bilateral exchange models with taxation introduced by Gabszewicz and Grazzini (1999, 2001). To study the implementation and the effectiveness of fiscal policies with redistribution, Gabszewicz and Grazzini (2001) consider the case of an exchange economy in which two commodities are initially held by a finite number of inside agents, the traders, while one agent, the outside agent, owns nothing. The traders behave strategically and the outside agent does not participate in trade because s/he does not have any resources. The strategic behavior of traders is implemented by embedding the finite exchange economy within a non-cooperative game in which the players are the traders, the strategies are supplies of commodities they bring to the market, and the payoffs are the utility levels they achieve. Insofar as two commodities are exchanged between traders who behave strategically à la Cournot, and each type of trader is initially endowed with only one commodity, this strategic market game is akin to a bilateral oligopoly model.

The bilateral oligopoly model was introduced by Gabszewicz and Michel (1997), and explored in Bloch and Ghosal (1997); Bloch and Ferrer (2001a, 2001b), Dickson and Hartley (2012), Amir and Bloch (2009), and Busetto et al. (2020a, 2020b). This model of strategic bilateral exchange represents a two commodity version of the strategic market game models (Shapley and Shubik 1977; Dubey and Shubik 1978; Sahi and Yao 1989; Amir and Bloch 2009).¹ In bilateral oligopoly each trader has corner endowment but wants to consume both commodities. All participants to exchange, namely the traders, behave strategically using quantities as strategies. There is a trading post to which traders may offer a fraction of their endowment of the commodity to be exchanged for the other commodity. The trading post aggregates the strategic supplies of all traders and allocates the amounts traded to each trader in proportion of her/his supply. The Cournot–Nash equilibrium is the noncooperative equilibrium concept. The bilateral oligopoly model is a natural starting point for studying the distortions generated by strategic behaviors in interrelated markets and the public policies to be implemented to restore Pareto-optimality.

Within this framework of bilateral oligopoly, Gabszewicz and Grazzini (2001) study three kinds of fiscal policies to correct the market distortions caused by imperfectly competitive behaviors in markets and to collect resources for redistributive purpose. These fiscal policies consist of taxing trade and taxing endowments by subsidizing the outside agent; and taxing endowments and subsidizing the outside agent and the traders. By assuming that agents’ preferences are represented by the same Cobb–Douglas utility function, they show that the first two kinds of lump-sum taxes with transfers can only reach a second-best, whilst the third one leads to a first-best. Indeed, without transfers among insiders, such fiscal policies are not sufficiently powerful to neutralize the market power of strategic traders. A first-best analysis under endowment taxation with transfers for the linear, Cobb–Douglas, and CES utility functions is made in Gabszewicz and Grazzini (1999), who show that endowment taxation with transfers between the traders is Pareto optimal.²

In this paper, we propose to extend Gabszewicz and Grazzini’s contributions by considering two kinds of lump-sum tax and transfers mechanisms. The first kind of fiscal policy, namely ad valorem taxation, consists in levying an uniform tax on transactions. The second kind of fiscal policy, namely endowment taxation, consists in levying an uniform tax on the endowments of traders. In both cases, the product of the tax is transferred to the outside agent in such a way s/he reaches some preassigned fixed utility level.

The objectives of the paper are twofold. First, and rather than comparing the two fiscal policies with each other, we study the effect of taxes on strategic behavior when the individual decisions of traders depend on the degree of substitutability between commodities and also on the share of the consumption of these commodities in their utility. Second, we study the welfare implications of redistribution associated with these different taxation mechanisms in strategic bilateral exchange. To achieve these purposes, we consider that traders’ preferences are represented by generalized CES utility functions with different shares for the quantities of the two commodities consumed. This specification for traders’ utility function makes it possible to determine a broader set of strategic equilibria (with taxation) among which the existing equilibria constitute special cases. Thus, within this framework, our contribution to the literature is threefold.

First, the non unitary coefficients on consumption and the elasticity of substitution make it possible to compute the set of strategic equilibria of the CES bilateral oligopoly model without taxation, among which the Cournot–Nash equilibrium of Bloch and Ferrer (2001b) is a special case for unit consumption shares. Correlatively, we state one result which puts forward the effect of the elasticity of substitution on equilibrium supplies, and which will be useful to understand the effectiveness of taxation mechanisms in our framework. Above all, our CES specification offers a unified computation of the various symmetric strategic equilibria under the two kinds of taxation mechanisms with transfer. In this respect, our model extends the Cobb–Douglas bilateral oligopoly model of Gabszewicz and Grazzini (2001), and it allows to deal with ad valorem taxation in the three bilateral oligopoly examples with endowment taxation of Gabszewicz and Grazzini (1999).

Our second contribution concerns the effects of the two taxation mechanisms envisaged on the strategic supplies, by considering the possible values of the elasticity of substitution, and the role played by the non unitary shares on consumption. Indeed, we study the influence of the parameters of the CES utility functions on equilibrium strategies. More generally, our computations put forward the link between the effect of any fiscal policy and the local curvature of the indifference curves, through the possible values of the elasticity of substitution, in any Cournot–Nash equilibrium with taxation. We state one result which suggests that the reaction of strategic traders to variation in taxes depends on the value of the elasticity of substitution. This leads us to determine the extent to which the welfare effectiveness of tax policies with transfers depends on the elasticity of substitution.

Our third contribution is reminiscent of the second welfare theorem in general equilibrium analysis but with strategic trade. It focuses on the redistributive purpose of the tax as well as on the optimality of the corresponding fiscal policy. In particular, our model highlights the importance of traders’ preferences in the optimality of fiscal policy, notably through the elasticity of substitution between commodities. Indeed, we show that the two fiscal policies cannot reach a Pareto-optimal allocation for any strictly positive finite value of the elasticity of substitution, i.e. for any bilateral oligopoly model in which commodities are neither perfect complements nor perfect substitutes. But, we also show that any fiscal policy with transfer can reach a first-best allocation when commodities are either perfect complements or perfect substitutes. Our results put forward that the preferences of agents matter for the welfare effects of fiscal policies.

The paper is organized as follows. In Sect. 2 we describe the CES bilateral oligopoly model and study its properties in order to highlight the market distortions at work. Section 3 is devoted to the implementation of the taxation mechanisms with transfers and to the computation of the strategic equilibria with taxation. Section 4 deals with the effectiveness and welfare implications of the taxation mechanisms. Section 5 provides a comparison of the CES bilateral oligopoly model with taxation we use with the existing ones in the literature. In Sect. 6 we conclude.

2 The model

This section is devoted to the presentation of the CES bilateral oligopoly model we are considering. First, we describe the model, i.e. the exchange economy and the strategic market game associated with it. Second, we study the main properties of the Cournot–Nash equilibrium (henceforth CNE) of the game. In particular, the equilibrium analysis will focus on market inefficiencies at work as a prerequisite for the study of public policies that could be implemented.

2.1 The exchange economy

Consider an exchange economy, namely ${\mathcal {E}}$, with two divisible commodities X and Y. The corresponding unit prices are $p_{X}$ and $p_{Y}$, with commodity Y as the numeraire, so $p_{Y}=1$. The economy includes $2n+1$ agents, namely 2n traders who are endowed with some commodity; and one agent, namely $2n+1$, who is initially deprived from any commodity, and who does not participate to exchange.³ The 2n traders fall into two types, namely 1 and 2, with $n\geqslant 2$ traders of each type. Traders are indexed by i, and we let $T_{1}=\{1,\ldots ,n\}$ and $T_{2}=\{n+1,\ldots ,2n\}$.

Following Gabszewicz and Grazzini (2001), there are fixed initial endowments:

$$\begin{aligned} \omega _{i}=\left\{ \begin{array}{ccc} (1,0) &{} \text {for } &{} i\in T_{1} \\ &{} &{} \\ (0,1) &{} \text {for} &{} i\in T_{2} \\ &{} &{} \\ (0,0) &{} \text {for} &{} i=2n+1\text {.} \end{array} \right. \end{aligned}$$

(1)

The preferences of each agent are represented by a CES utility function. To reach the highest possible degree of generality, we consider the following generalized forms for CES functions:

$$\begin{aligned} u_{i}(x_{i},y_{i})=\left\{ \begin{array}{ccc} (\alpha x_{i}^{\rho }+\beta y_{i}^{\rho })^{\frac{\mu }{\rho }} &{} \text {for} &{} i\in T_{1} \\ &{} &{} \\ (\beta x_{i}^{\rho }+\alpha y_{i}^{\rho })^{\frac{\mu }{\rho }} &{} \text {for} &{} i\in T_{2} \\ &{} &{} \\ (x_{i}^{\rho }+y_{i}^{\rho })^{\frac{\mu }{\rho }} &{} \text {for} &{} \text { } i=2n+1\text {,} \end{array} \right. \end{aligned}$$

(2)

where $\alpha ,\beta >0$, with $\alpha +\beta =1$, and $\rho \leqslant 1$, with $\rho \ne 0$, and $\mu \in (0,\infty )$.⁴ It is common knowledged that $\rho \equiv \frac{\sigma -1}{\sigma }$, where $\sigma$ is the elasticity of substitution between commodities X and Y. As $\rho \in [-\infty ,1]$, we have $\sigma \in [0,\infty ]$. When $\sigma =1$, $\sigma \rightarrow \infty$, and $\sigma =0$, (2) correspond respectively to the Cobb–Douglas, linear, and Leontief utility functions. When $\sigma \in (0,\infty )$, commodities are neither perfect substitutes nor perfect complements. Let $MRS_{X/Y}^{i}(x_{i},y_{i})=(\frac{\partial u_{i}/\partial x_{i}}{\partial u_{i}/\partial y_{i}})_{\mid (x_{i},y_{i})}$, be agent i’s marginal rate of substitution at $(x_{i},y_{i})$, which we will denote hereafter by $MRS^{i}(x_{i},y_{i})$. As a reference, for $\sigma \in (0,\infty ]$, we have that $MRS^{i}(x_{i},y_{i})=\frac{\alpha }{\beta }( \frac{{{y}}_{i}}{{{x}}_{i}})^{\frac{1}{\sigma }}$, for each $i\in T_{1}$ , $MRS^{i}(x_{i},y_{i})=\frac{\beta }{\alpha }(\frac{y_{i}}{x_{i}})^{\frac{1 }{\sigma }}$, for each $i\in T_{2}$, and $MRS^{i}(x_{i},y_{i})=(\frac{{{y}} _{i}}{{{x}}_{i}})^{\frac{1}{\sigma }}$, for $i=2n+1$.

To highlight the essential role played by the parameter $\sigma$, let us consider, before studying the strategic equilibria of the game that will be associated with this exchange economy, the competitive equilibria of ${\mathcal {E}}$. The market clearing prices are the solutions to:⁵

$$\begin{aligned} p_{X}^{2\sigma -1}+\left( \frac{\beta }{\alpha }\right) ^{\sigma }(p_{X}^{\sigma }-p_{X}^{\sigma -1})-1=0. \end{aligned}$$

(3)

We will always refer to the following values for $\sigma$: $\sigma \in (0,\infty )$, $\sigma =0$, $\sigma =1$, and $\sigma \rightarrow \infty$. Indeed, when $\sigma =1$, there is a unique interior competitive equilibrium with relative price $p_{X}^{*}=1$ and allocations $(x_{i}^{*},y_{i}^{*})=(\frac{\alpha }{\alpha +\beta },\frac{\beta }{\alpha +\beta })$, $i\in T_{1}$, $(x_{i}^{*},y_{i}^{*})=(\frac{\beta }{\alpha +\beta },\frac{\alpha }{\alpha +\beta })$, $i\in T_{2}$, and $(x_{2n+1}^{*},y_{2n+1}^{*})=(0,0)$. When $\sigma =0$, there is a unique interior equilibrium, with relative price $p_{X}^{*}=1$ and allocations $(x_{i}^{*},y_{i}^{*})=(\frac{1}{2},\frac{1}{2})$, $i\in T_{1}\cup T_{2}$, and $(x_{2n+1}^{*},y_{2n+1}^{*})=(0,0)$. When $\sigma \rightarrow \infty$, with $\alpha =\beta$, there is a continuum of competitive equilibria, with relative price $p_{X}^{*}=1$ and allocations $(x_{i}^{*},y_{i}^{*})=(x,1-y)$, $i\in T_{1}$, and $(x_{i}^{*},y_{i}^{*})=(1-x,y)$, $i\in T_{2}$, with $x,y\in [0,1]$, and $(x_{2n+1}^{*},y_{2n+1}^{*})=(0,0)$. More generally, when $\sigma \in (0,\infty )$, with $\alpha \ne \beta$, there may be multiple competitive equilibria.⁶ Indeed, for instance, when $\sigma =\frac{1}{3}$, with $\frac{\beta }{\alpha }=\frac{1}{ 64}$, (3) may be written as $\frac{1}{4}p_{X}-p_{X}^{\frac{2}{3}}+p_{X}^{ \frac{1}{3}}-\frac{1}{4}=0$. The three competitive equilibria are given by $p_{X}^{*}=1$, $(x_{i}^{*},y_{i}^{*})=(\frac{4}{5},\frac{1}{5})$, $i\in T_{1}$, $(x_{i}^{*},y_{i}^{*})=(\frac{1}{5},\frac{4}{5})$, $i\in T_{2}$ and $(x_{2n+1}^{*},y_{2n+1}^{*})=(0,0)$; $p_{X}^{*}=( \frac{3+\sqrt{5}}{2})^{3}$, $(x_{i}^{*},y_{i}^{*})=(\frac{2(15+3 \sqrt{5})}{45},\frac{5+2\sqrt{5}}{15})$, $i\in T_{1}$, $(x_{i}^{*},y_{i}^{*})=(\frac{5-2\sqrt{5}}{15},\frac{2(5-\sqrt{5})}{15}),i\in T_{2}$, and $(x_{2n+1}^{*},y_{2n+1}^{*})=(0,0)$; and $p_{X}^{*}=(\frac{3-\sqrt{5}}{2})^{3}$, $(x_{i}^{*},y_{i}^{*})=(\frac{6(5- \sqrt{5})}{45},\frac{5-2\sqrt{5}}{15})$, $i\in T_{1}$, $(x_{i}^{*},y_{i}^{*})=(\frac{5+2\sqrt{5}}{15},\frac{2(5+\sqrt{5})}{15})$, $i\in T_{2}$, and $(x_{2n+1}^{*},y_{2n+1}^{*})=(0,0)$. Finally, from the First Welfare Theorem, it is possible to check, that, for all $\sigma \in [0,\infty ]$, each competitive equilibrium is Pareto-optimal.

2.2 The strategic market game without taxation

To ${\mathcal {E}}$, we associate the Dubey and Shubik (1978) strategic market game ${\Gamma }$ between the only traders who are henceforth oligopolists. Insofar as two commodities are exchanged between agents who behave strategically à la Cournot, and each type of agent is initially endowed with only one commodity, this strategic market game is akin to a bilateral oligopoly model (see Gabszewicz and Michel (1997)).

In this framework, the traders can offer only a fraction of the commodity they initially hold. Thus, by contracting her offer, each trader manipulates the relative price $p_{X}$. We denote by $q_{i}$ (resp. $b_{i}$ ) the supply of commodity X (resp. Y) by trader i of type 1 (resp. 2). It represents the amount of commodity X (resp. Y) that trader $i\in T_{1}$ (resp. $i\in T_{2}$) offers in exchange for commodity Y (resp. X ). Therefore, the strategy sets are given by:

$$\begin{aligned} Q_{i}= & {} \{q_{i}\in {\mathbb {R}} :0\leqslant q_{i}\leqslant 1\}\text {, }i\in T_{1}; \end{aligned}$$

(4)

$$\begin{aligned} {\mathcal {B}}_{i}= & {} \{b_{i}\in {\mathbb {R}} :0\leqslant b_{i}\leqslant 1\}\text {, }i\in T_{2}\text {.} \end{aligned}$$

(5)

Trader i of type 1 (resp. 2) consumes the amount $1-q_{i}$ of commodity X (resp. $1-b_{i}$ of commodity Y). A market price mechanism (see below) aggregates the strategic supplies of all traders and allocates the amounts traded to each trader. Indeed, trader $i\in T_{1}$ obtains in exchange for $q_{i}$ a quantity of commodity Y equal to $p_{X}q_{i}$ (recall $p_{Y}=1$), and ends up with the bundle of commodities $(x_{i},y_{i})=(1-q_{i}$ $,p_{X}q_{i})$. Therefore, her corresponding utility level is $u_{i}(1-q_{i},p_{X}q_{i})=(\alpha (1-q_{i}$ $)^{\frac{ \sigma -1}{\sigma }}+\beta \left( p_{X}q_{i}\text { }\right) ^{\frac{\sigma -1 }{\sigma }})^{\frac{\sigma }{\sigma -1}\mu }$. Likewise, $i\in T_{2}$ obtains in exchange for $b_{i}$ a quantity of commodity X equal to $\frac{1 }{p_{X}}b_{i}$, and ends up with the bundle of commodities $(x_{i},y_{i})=( \frac{1}{p_{X}}b_{i},1-b_{i})$, with corresponding utility level $u_{i}( \frac{1}{p_{X}}b_{i},1-b_{i})=(\alpha (\frac{1}{p_{X}}b_{i}$ $)^{\frac{ \sigma -1}{\sigma }}+\beta \left( 1-b_{i}\right) ^{\frac{\sigma -1}{\sigma } })^{^{\frac{\sigma }{\sigma -1}\mu }}$.

Then, given a 2n-tuple of supply strategies $({\mathbf {q}};{\mathbf {b}} )=(q_{1},\ldots ,q_{n};b_{n+1},\ldots ,b_{2n})$, with $({\mathbf {q}};{\mathbf {b}})\in \prod \nolimits _{i=1}^{n}Q_{i}\times \prod \nolimits _{i=n+1}^{2n}{\mathcal {B}} _{i}$, the market price $p_{X}$ must satisfy the market clearing condition $\sum \nolimits _{k\in T_{1}}q_{k}=\frac{1}{p_{X}}\sum \nolimits _{k\in T_{2}}b_{k}$, which may be written:

$$\begin{aligned} p_{X}=\frac{\sum \nolimits _{k\in T_{2}}b_{k}}{\sum \nolimits _{k\in T_{1}}q_{k}} \text {.} \end{aligned}$$

(6)

Then, by using (6), the final allocation that results from this market price mechanism may be written:

$$\begin{aligned} (x_{i},y_{i})&=\left( 1-q_{i}\text { },\frac{\sum \nolimits _{k\in T_{2}}b_{k}}{ \sum \nolimits _{k\in T_{1}}q_{k}}q_{i}\text { }\right) \text {, }i\in T_{1} \text {;} \end{aligned}$$

(7)

$$\begin{aligned} (x_{i},y_{i})&=\left( \frac{\sum \nolimits _{k\in T_{1}}q_{k}}{ \sum \nolimits _{k\in T_{2}}b_{k}}b_{i},1-b_{i}\right) \text {, }i\in T_{2} \text {.} \end{aligned}$$

(8)

Then, the indirect utility functions of traders may be written as payoffs at the vector of strategies $({\mathbf {q}};{\mathbf {b}})$. Formally, the payoffs in ${\Gamma }$ are defined by $\pi _{i}:\prod \nolimits _{i=1}^{n}Q_{i}\times \prod \nolimits _{i=n+1}^{2n} {\mathcal {B}}_{i}\rightarrow {\mathbb {R}} _{+}$, $\pi _{i}(q_{i},{\mathbf {q}}_{-i};{\mathbf {b}})=u_{i}(1-q_{i},p_{X}q_{i})$ , $i\in T_{1}$, and $\pi _{i}:\prod \nolimits _{i=1}^{n}Q_{i}\times \prod \nolimits _{i=n+1}^{2n}{\mathcal {B}}_{i}\rightarrow {\mathbb {R}} _{+}$, $\pi _{i}({\mathbf {q}};b_{i},{\mathbf {b}}_{-i})=u_{i}(\frac{1}{p_{X}} b_{i},1-b_{i})$, $i\in T_{2}$, and are given by:

$$\begin{aligned} \pi _{i}(q_{i},{\mathbf {q}}_{-i};{\mathbf {b}})&=\left( \alpha (1-q_{i}\text { })^{ \frac{\sigma -1}{\sigma }}+\beta \left( \frac{\sum \nolimits _{k\in T_{2}}b_{k} }{\sum \nolimits _{k\in T_{1}}q_{k}}q_{i}\text { }\right) ^{\frac{\sigma -1}{ \sigma }}\right) ^{\frac{\sigma }{\sigma -1}\mu }\text {, }i\in T_{1}\text {;} \end{aligned}$$

(9)

$$\begin{aligned} \pi _{i}({\mathbf {q}};b_{i},{\mathbf {b}}_{-i})&=\left( \beta \left( \frac{ \sum \nolimits _{k\in T_{1}}q_{k}}{\sum \nolimits _{k\in T_{2}}b_{k}} b_{i}\right) ^{\frac{\sigma -1}{\sigma }}+\alpha (1-b_{i})^{\frac{\sigma -1}{ \sigma }}\right) ^{\frac{\sigma }{\sigma -1}\mu }\text {, }i\in T_{2}\text {,} \end{aligned}$$

(10)

where ${\mathbf {q}}_{-i}$ (resp. ${\mathbf {b}}_{-i}$) is the $(n-1)$-tuple of supply strategies of all type 1 (resp. 2) traders but i.

A Cournot–Nash equilibrium (CNE thereafter) is given by a 2n-tuple of strategies $({\tilde{q}}_{1},\ldots ,{\tilde{q}}_{n};{\tilde{b}}_{n+1},\ldots ,{\tilde{b}} _{2n})\in \prod \nolimits _{i=1}^{n}Q_{i}\times \prod \nolimits _{i=n+1}^{2n} {\mathcal {B}}_{i}$, such that no trader has an advantage to deviate unilaterally from her choice. A CNE is said to be type-symmetric when ${\tilde{q}}_{i}={\tilde{q}}$, for each $i\in T_{1}$, and ${\tilde{b}}_{i}={\tilde{b}}$, for each $i\in T_{2}$.

Proposition 1

The Cournot–Nash equilibrium supplies of $\Gamma$ are type-symmetric and are given by:

$$\begin{aligned} {\tilde{q}}={\tilde{b}}=\frac{1}{1+\left( \frac{\alpha }{\beta }\frac{n}{n-1} \right) ^{\sigma }}\text {, with }\sigma \in [0,\infty )\text {.} \end{aligned}$$

(11)

Proof

See “Appendix A”.

First, consider the uniqueness and convergence properties of the CNE supplies. When $\sigma \in [0,\infty )$, the game $\Gamma$ has a unique CNE, which is interior and type-symmetric.⁷ For instance, when $\sigma =0$, the CNE is given by the 2n-tuple of strategies ${\tilde{q}}={\tilde{b}}=\frac{1}{2}$, and, when $\sigma =1$, the CNE is given by the 2n-tuple of strategies ${\tilde{q}}={\tilde{b}}=\frac{\beta (n-1)}{ (\alpha +\beta )n-\beta }$. In addition, from (11), we have that $\lim _{n\rightarrow \infty }{\tilde{q}}=\lim _{n\rightarrow \infty }{\tilde{b}}= \frac{\beta ^{\sigma }}{\alpha ^{\sigma }+\beta ^{\sigma }}$, which corresponds to the competitive equilibrium supply for the Cobb–Douglas utilities when $\sigma =1$ and to the competitive equilibrium supply for the Leontief utilities when $\sigma =0$. Otherwise, when $\sigma \rightarrow \infty$, with $\alpha \leqslant \beta$, there may be multiple Cournot–Nash equilibria: $\Gamma$ has the autarkic equilibrium (in which case ${\tilde{q}} =$ ${\tilde{b}}=0$) as a unique CNE when $\frac{\alpha }{\beta }>\frac{n-1}{n}$ (n finite); and has the autarkic equilibrium and the competitive equilibrium (in which case ${\tilde{q}}=$ ${\tilde{b}}=1$) as Cournot–Nash equilibria when $\frac{\alpha }{\beta }\leqslant \frac{n-1}{n}$, with $\alpha <\beta$.⁸

Second, consider the welfare property of the strategic equilibria. To this end, we compute the final allocations of traders:

$$\begin{aligned} ({\tilde{x}}_{i},{\tilde{y}}_{i})=\left( \frac{\left( \frac{\alpha }{\beta } \frac{n}{n-1}\right) ^{\sigma }}{1+\left( \frac{\alpha }{\beta }\frac{n}{n-1} \right) ^{\sigma }}\text { },\frac{1}{1+\left( \frac{\alpha }{\beta }\frac{n}{ n-1}\right) ^{\sigma }}\right) \text {, }i\in T_{1}\text {;} \end{aligned}$$

(12)

$$\begin{aligned} ({\tilde{x}}_{i},{\tilde{y}}_{i})=\left( \frac{1}{1+\left( \frac{\alpha }{\beta } \frac{n}{n-1}\right) ^{\sigma }}\text { },\frac{\left( \frac{\alpha }{\beta } \frac{n}{n-1}\right) ^{\sigma }}{1+\left( \frac{\alpha }{\beta }\frac{n}{n-1} \right) ^{\sigma }}\right) \text {, }i\in T_{2}\text {.} \end{aligned}$$

(13)

For finite values of n, the Cournot–Nash equilibria can be Pareto dominated by the competitive equilibria: when commodities are substitutable, the strategic traders manipulate the terms of trade to their own advantage by supplying less than the competitive supplies. To see this, consider trader i’s marginal rate of substitution at an interior CNE. When $\sigma \in (0,\infty )$, the traders use their market power as we have ${\tilde{p}} _{X}=\frac{n}{n-1}MRS^{i}({\tilde{x}}_{i},{\tilde{y}}_{i})$, for each $i\in T_{1}$, and $\frac{1}{{\tilde{p}}_{X}}=\frac{n}{n-1}MRS^{i}({\tilde{x}}_{i}, {\tilde{y}}_{i})$, for each $i\in T_{2}$.⁹ As ${\tilde{p}}_{X}=1$, the marginal rates of substitution differ across traders, i.e. $MRS^{i}({\tilde{x}}_{i},{\tilde{y}} _{i})=\frac{n-1}{n}$, for each $i\in T_{1}$, and $MRS^{i}({\tilde{x}}_{i}, {\tilde{y}}_{i})=\frac{n}{n-1}$, for each $i\in T_{2}$. Otherwise, when $\sigma \rightarrow \infty$, with $\frac{\alpha }{\beta }>\frac{n-1}{n}$, the autarkic CNE is not Pareto-optimal as there are still potential gains from trade which are not fully exploited (for each trader, the gains from manipulating prices, i.e. $\frac{1}{n}=\frac{\partial p_{X}}{\partial q} {\tilde{q}}=\frac{{\tilde{b}}}{\frac{\partial p_{X}}{\partial b}}$, are higher than the gains from participating to trade, i.e. $1-\frac{\alpha }{\beta }= {\tilde{p}}_{X}-\frac{\alpha }{\beta }$), while when $\frac{\alpha }{\beta } \leqslant \frac{n-1}{n}$, with $\alpha <\beta$, apart from the autarkic equilibrium allocation, the CNE allocation leads to a competitive allocation, which is thereby Pareto-optimal.

Finally, the next proposition, which puts forward the effect of the elasticity of substitution on equilibrium supplies, will be useful to understand the effectiveness of taxation mechanisms in our framework.

Proposition 2

When $\sigma \in (0,\infty )$, $\frac{ \partial {\tilde{q}}}{\partial \sigma }\geqslant 0$ if and only if $MRS^{i}({\tilde{x}}_{i},{\tilde{y}}_{i})\geqslant \frac{\alpha }{\beta }$ , for each $i\in T_{1}$. By symmetry, the same holds for $\frac{ \partial {\tilde{b}}}{\partial \sigma }$.

Proof

See “Appendix B”.

Paraphrasing the content of Proposition 2, the equilibrium supplies increase with the elasticity of substitution if and only if the marginal rate of substitution at a CNE is greater than the relative preference for the commodity being supplied. The parameters $\alpha$ and $\beta$ represent indifferently the shares of the consumption of the two goods in the utility or the preferences associated with each good. Thus, the traders are willing to participate in transactions to obtain gains from trade when their own exchange rate ($\frac{n-1}{n}$ for type 1 traders) exceeds their relative preference for the commodity initially held ($\frac{\alpha }{\beta }$ for type 1 traders), or equivalently, when their relative preference in favor of the good purchased ($\frac{\beta }{\alpha }$ for type 1 traders) exceeds the rate at which the other type traders are willing to trade ($\frac{n}{n-1}$ for type 2 traders). Two remarks can be made about the condition stated in Proposition 2. First, when the number of traders becomes arbitrarily large, the condition requires $\alpha >\beta$ for type 1 traders, and $\alpha <\beta$ for type 2 traders. Second, the condition holds at interior Cournot–Nash equilibria, and it satisfies the condition for the existence of Cournot–Nash equilibria with trade in Cordella and Gabszewicz (1998) which stipulates that type 1 traders are willing to trade if their marginal rates of substitution do not exceed $\frac{n-1}{n}$ at the point of initial endowment.

The non-optimality of the Cournot–Nash equilibria leads us to determine whether Pareto-optimality could be restored when some fiscal policies, through taxation and transfer mechanisms, are introduced.

3 Strategic equilibria with taxations

Let us now introduce two fiscal policies with transfers, namely, ad valorem and endowment taxations (Gabszewicz and Grazzini 1999, 2001).¹⁰ First, we specify the strategic market games with taxations. Second, we determine the strategic equilibria with taxations. Third, we perform one comparative statics exercise.

3.1 The strategic market game with taxations

Consider the two taxation mechanisms, namely ad valorem taxation and endowment taxation, for which the traders pay a tax and the total tax product is transferred to the outside agent $2n+1$. In each case, the fiscal policy generates a new strategic market game.

First, consider that, when exchange takes place, an uniform ad valorem tax $t\in (0,1)$ is charged on the supply of goods X and Y. Given an 2n -tuple of strategies $({\mathbf {q}};{\mathbf {b}})\in \prod \nolimits _{i=1}^{n}Q_{i}\times \prod \nolimits _{i=n+1}^{2n}{\mathcal {B}} _{i}$ and the uniform tax $t\in (0,1)$, a new market price $p_{X}$ is determined. Trader i’s revenue from sales obtains as $p_{X}(1-t)q_{i}= \frac{\sum \nolimits _{k\in T_{2}}b_{k}}{\sum \nolimits _{k\in T_{1}}q_{k}} (1-t)q_{i}$, for each $i\in T_{1}$, and as $\frac{1}{p_{X}}(1-t)b_{i}=\frac{ \sum \nolimits _{k\in T_{1}}q_{k}}{\sum \nolimits _{k\in T_{2}}b_{k}}(1-t)b_{i}$ , for each $i\in T_{2}$. The resulting post tax allocation is given by $(x_{i},y_{i})=(1-q_{i}$ $,p_{X}(1-t)q_{i})$, for each $i\in T_{1}$, and $(x_{i},y_{i})=(\frac{1}{p_{X}}(1-t)b_{i},1-b_{i})$, for each $i\in T_{2}$, and by $(x_{2n+1},y_{2n+1})=(t\sum \nolimits _{k\in T_{1}}q_{k},t\sum \nolimits _{k\in T_{2}}b_{k})$. Then, from (9) and (10), we can define the payoffs in ${\Gamma }^{t}$, i.e. $\pi _{i}:\prod \nolimits _{i=1}^{n}Q_{i}\times \prod \nolimits _{i=n+1}^{2n} {\mathcal {B}}_{i}\times (0,1)\rightarrow {\mathbb {R}} _{+}$, $\pi _{i}(q_{i},{\mathbf {q}}_{-i};{\mathbf {b}} ;t)=u_{i}(1-q_{i},p_{X}(1-t)q_{i})$, $i\in T_{1}$, and $\pi _{i}:\prod \nolimits _{i=1}^{n}Q_{i}\times \prod \nolimits _{i=n+1}^{2n} {\mathcal {B}}_{i}\times (0,1)\rightarrow {\mathbb {R}} _{+}$, $\pi _{i}({\mathbf {q}};b_{i},{\mathbf {b}}_{-i};t)=u_{i}(\frac{1}{p_{X}} (1-t)b_{i},1-b_{i})$, $i\in T_{2}$, which, from (2), may be written:

$$\begin{aligned} \pi _{i}(q_{i},{\mathbf {q}}_{-i};{\mathbf {b}};t)&=\left( \alpha (1-q_{i}\text { } )^{\frac{\sigma -1}{\sigma }}+\beta (\frac{\sum \nolimits _{k\in T_{2}}b_{k}}{\sum \nolimits _{k\in T_{1}}q_{k}}(1-t)q_{i})^{\frac{ \sigma -1}{\sigma }}\right) ^{\frac{\sigma }{\sigma -1}\mu }\text {, }i\in T_{1}\text {;}\nonumber \\ \end{aligned}$$

(14)

$$\begin{aligned} \pi _{i}({\mathbf {q}};b_{i},{\mathbf {b}}_{-i};t)&=\left( \beta (\frac{ \sum \nolimits _{k\in T_{1}}q_{k}}{\sum \nolimits _{k\in T_{2}}b_{k}}(1-t)b_{i} )^{\frac{\sigma -1}{\sigma }}+\alpha (1-b_{i})^{\frac{\sigma -1}{ \sigma }}\right) ^{\frac{\sigma }{\sigma -1}\mu }\text {, }i\in T_{2}\text {.}\nonumber \\ \end{aligned}$$

(15)

Second, consider that an uniform tax $\tau \in (0,1)$ is levied on endowments before exchange takes place. After trade has occurred the product of the tax is redistributed to the outside agent. The strategy sets are now given by $Q_{i}=\{q_{i}\in {\mathbb {R}} :$ $0\leqslant q_{i}\leqslant 1-\tau \}$, $i\in T_{1}$, and by ${\mathcal {B}} _{i}=\{b_{i}\in {\mathbb {R}} :0\leqslant b_{i}\leqslant 1-\tau \}$, $i\in T_{2}$. Given an n-tuple of strategies $({\mathbf {q}};{\mathbf {b}})\in \prod \nolimits _{i=1}^{n}Q_{i}\times \prod \nolimits _{i=n+1}^{2n}{\mathcal {B}}_{i}$, a new market price $p_{X}$ is determined, and the resulting post tax allocation is given by $(x_{i},y_{i})=(1-\tau -q_{i}$ $,p_{X}q_{i})$, for each $i\in T_{1}$, $(x_{i},y_{i})=(\frac{1}{p_{X}}b_{i},1-\tau -b_{i})$, for each $i\in T_{2}$, and by $(x_{2n+1},y_{2n+1})=(n\tau ,\ n\tau )$. Therefore, we can define the payoffs in ${\Gamma }^{\tau }$, i.e. $\pi _{i}:\prod \nolimits _{i=1}^{n}Q_{i}\times \prod \nolimits _{i=n+1}^{2n} {\mathcal {B}}_{i}\times (0,1)\rightarrow {\mathbb {R}} _{+}$, $\pi _{i}(q_{i},{\mathbf {q}}_{-i};{\mathbf {b}};\tau )=u_{i}(1-\tau -q_{i},p_{X}q_{i})$, $i\in T_{1}$, and $\pi _{i}:\prod \nolimits _{i=1}^{n}Q_{i}\times \prod \nolimits _{i=n+1}^{2n} {\mathcal {B}}_{i}\times (0,1)\rightarrow {\mathbb {R}} _{+}$, $\pi _{i}({\mathbf {q}};b_{i},{\mathbf {b}}_{-i};\tau )=u_{i}(\frac{1}{p_{X} }b_{i},1-\tau -b_{i})$, $i\in T_{2}$, which may be written:

$$\begin{aligned} \pi _{i}(q_{i},{\mathbf {q}}_{-i};\mathbf {b;}\tau )=\left( \alpha (1-\tau -q_{i} \text { })^{\frac{\sigma -1}{\sigma }}+\beta (\frac{ \sum \nolimits _{k\in T_{2}}b_{k}}{\sum \nolimits _{k\in T_{1}}q_{k}}q_{i}\text { })^{\frac{\sigma -1}{\sigma }}\right) ^{\frac{\sigma }{\sigma -1} \mu }\text {, }i\in T_{1}\text {;} \end{aligned}$$

(16)

$$\begin{aligned} \pi _{i}({\mathbf {q}};b_{i},{\mathbf {b}}_{-i};\tau )=\left( \beta ( \frac{\sum \nolimits _{k\in T_{1}}q_{k}}{\sum \nolimits _{k\in T_{2}}b_{k}}b_{i} )^{\frac{\sigma -1}{\sigma }}+\alpha (1-\tau -b_{i})^{\frac{\sigma -1}{\sigma }}\right) ^{\frac{\sigma }{\sigma -1}\mu }\text {, }i\in T_{2} \text {.} \end{aligned}$$

(17)

We now turn to the computation of the strategic equilibria in ${ \Gamma }^{\epsilon }$, $\epsilon =t,\tau$, where the notation $\Gamma ^{\epsilon }$ is used for the game $\Gamma$ with taxation $\epsilon$, $\epsilon =t,\tau$.

3.2 Derivation of strategic equilibria with taxation

Proposition 3

The Cournot–Nash equilibrium supplies of $\Gamma ^{t}$ and $\Gamma ^{\tau }$ are type-symmetric and are given respectively by:

$$\begin{aligned} {\tilde{q}}(t)&={\tilde{b}}(t)=\frac{1}{1+\left( \frac{\alpha }{\beta }\frac{n}{ n-1}\right) ^{\sigma }(1-t)^{1-\sigma }}\text {;} \end{aligned}$$

(18)

$$\begin{aligned} {\tilde{q}}(\tau )&={\tilde{b}}(\tau )=\frac{1-\tau }{1+\left( \frac{\alpha }{ \beta }\frac{n}{n-1}\right) ^{\sigma }}\text {, with }\sigma \in [0,\infty )\text {.} \end{aligned}$$

(19)

Proof

See “Appendix C”.

Proposition 3 shows that, when commodities are not perfectly substitutable, i.e. when $\sigma \in [0,\infty )$, for each taxation mechanism $\epsilon$, $\epsilon =t,\tau$, the game $\Gamma ^{\epsilon }$ has a unique CNE, which is interior and type-symmetric. When $\sigma \rightarrow +\infty$, with $\alpha <\beta$, then, for each taxation mechanism $\epsilon$, $\epsilon =t,\tau$, the game $\Gamma ^{\epsilon }$ has the autarkic equilibrium and the competitive equilibrium as CNE’s (see “Appendix C”).¹¹

It is worth noting that (18), (19) show that the equilibrium supplies depend linearly on the endowment tax, and nonlinearly on the ad valorem tax. Moreover, when $\alpha =\beta =\frac{1}{2}$ (with $\sigma =1$), (18) yields ${\tilde{q}}(t)={\tilde{b}}(t)=\frac{n-1}{2n-1}$, and (19) yields ${\tilde{q}} (\tau )={\tilde{b}}(\tau )=\frac{(1-\tau )(n-1)}{2n-1}$, each of these supplies coinciding with those that would be obtained with 2n traders in Gabszewicz and Grazzini (2001).

As a reference, we provide the final allocations and the payoffs that result from the two taxations mechanisms. At a CNE with ad valorem taxation, we have:

$$\begin{aligned} ({\tilde{x}}_{i}(t),{\tilde{y}}_{i}(t))&=\left( \frac{\chi }{1+\chi }\text { }, \frac{1-t}{1+\chi }\right) \text {, }i\in T_{1}\text {;} \end{aligned}$$

(20)

$$\begin{aligned} ({\tilde{x}}_{i}(t),{\tilde{y}}_{i}(t))&=\left( \frac{1-t}{1+\chi }\text { },\frac{ \chi }{1+\chi }\right) \text {, }i\in T_{2}\text {,} \end{aligned}$$

(21)

with corresponding payoffs:

$$\begin{aligned} {\tilde{\pi }}_{i}(t)= & {} \left( \alpha \left( \frac{\chi }{1+\chi }\right) ^{\frac{ \sigma -1}{\sigma }}+\beta \left( \frac{1-t}{1+\chi }\right) ^{\frac{\sigma -1}{\sigma }}\right) ^{\frac{\sigma }{\sigma -1}\mu }\text {, }i\in T_{1} \text {;} \end{aligned}$$

(22)

$$\begin{aligned} {\tilde{\pi }}_{i}(t)= & {} \left( \beta \left( \frac{1-t}{1+\chi }\right) ^{\frac{ \sigma -1}{\sigma }}+\alpha \left( \frac{\chi }{1+\chi }\right) ^{\frac{ \sigma -1}{\sigma }}\right) ^{\frac{\sigma }{\sigma -1}\mu }\text {, }i\in T_{2}\text {.} \end{aligned}$$

(23)

where $\chi \equiv \left( \frac{\alpha }{\beta }\frac{n}{n-1}\right) ^{\sigma }(1-t)^{1-\sigma }$.

Finally, in a CNE with endowment taxation, we have:

$$\begin{aligned} ({\tilde{x}}_{i}(\tau ),{\tilde{y}}_{i}(\tau ))= & {} \left( \frac{(1-\tau )\psi }{ 1+\psi }\text { },\frac{1-\tau }{1+\psi }\right) \text {, }i\in T_{1}\text {;} \end{aligned}$$

(24)

$$\begin{aligned} ({\tilde{x}}_{i}(\tau ),{\tilde{y}}_{i}(\tau ))= & {} \left( \frac{1-\tau }{1+\psi } \text { },\frac{(1-\tau )\psi }{1+\psi }\right) \text {, }i\in T_{2}\text {,} \end{aligned}$$

(25)

with corresponding payoffs:

$$\begin{aligned} {\tilde{\pi }}_{i}(\tau )= & {} \left( \alpha \left( \frac{(1-\tau )\psi }{1+\psi } \right) ^{\frac{\sigma -1}{\sigma }}+\beta \left( \frac{1-\tau }{1+\psi } \right) ^{\frac{\sigma -1}{\sigma }}\right) ^{\frac{\sigma }{\sigma -1}\mu } \text {, }i\in T_{1}\text {;}\end{aligned}$$

(26)

$$\begin{aligned} {\tilde{\pi }}_{i}(\tau )= & {} \left( \beta \left( \frac{1-\tau }{1+\psi }\right) ^{ \frac{\sigma -1}{\sigma }}+\alpha \left( \frac{(1-\tau )\psi }{1+\psi } \right) ^{\frac{\sigma -1}{\sigma }}\right) ^{\frac{\sigma }{\sigma -1}\mu } \text {, }i\in T_{2}\text {,} \end{aligned}$$

(27)

where $\psi \equiv \left( \frac{\alpha }{\beta }\frac{n}{n-1}\right) ^{\sigma }$.

3.3 Comparative statics on equilibrium supplies

Before investigating the welfare properties of the taxation mechanisms, we study the reaction of the equilibrium supplies with taxation when the tax rates are modified.

The next proposition shows that the effect of taxations has a contrasted impact on traders’ supply.

Proposition 4

The equilibrium supply of any trader increases with ad valorem taxation when $\sigma \in [0,1)$. In addition, the equilibrium supply of any trader always decreases with endowment taxation.

Proof

See “Appendix D”.

The positive effect of the ad valorem tax for low values of the elasticity of substitution may be explained as follows. Pick one $i\in T_{1}$. When the tax increases, from (20), trader i’s marginal rate of substitution at a CNE is now given by $MRS^{i}({\tilde{x}}_{i}(t),{\tilde{y}}_{i}(t))=\frac{n-1}{ n}(1-t)$. The tax increase reduces the rate at which the trader is willing to exchange good X for good Y. Then, the quantity of good Y obtained is lower, unless the supply of good X increases. However, Proposition 2 states that market power increases with substitutability when the purchased good (here Y) is more preferred than the sold good (here X). The condition for the equilibrium supply to increase with the degree of substitutability is now $\frac{\alpha }{\beta }<\frac{n-1}{n}(1-t)$: the amount of good Y bought is lower for the same amount of good X supplied. The effect of the tax increase on supply will therefore depend on the degree of substitutability between goods. Thus, the increase in supply is effective when substitutability is low, or equivalently, supply decreases with the increase in market power, i.e. when substitutability increases.

Besides, the negative effect of endowment taxation for all values of the elasticity of substitution stems from the fact that this tax contracts the set of admissible strategies of any trader. Indeed, the marginal rates of substitution do not depend on the endowment tax as, from (24), we have $MRS^{i}({\tilde{x}}_{i}(\tau ),{\tilde{y}}_{i}(\tau ))=\frac{n-1}{n}$, for each $i\in T_{1}$, and from (25), we have $MRS^{i}({\tilde{x}}_{i}(\tau ),{\tilde{y}} _{i}(\tau ))=\frac{n}{n-1}$, for each $i\in T_{2}$.

The preceding result suggests that the reaction of strategic traders to variation in taxes depends on the value of the elasticity of substitution. This leads us to determine the extent to which the welfare effectiveness of tax policies with transfers depends on the elasticity of substitution.

4 Welfare and taxation mechanisms

We now wonder whether some taxation mechanisms, i.e. tax and transfer schemes, could be sufficiently powerful to eliminate the distortions created by the strategic behaviors of traders when the number of market participants is finite. The main problem is to determine whether, in a market with a finite number of strategic traders, the ad valorem and endowment taxes can implement a Pareto-optimal allocation providing preassigned utility level to the outside agent. Indeed, can we find a tax and transfer scheme, such that a Pareto-optimal overall allocation of commodities among traders and the outside agent exists, that would simultaneously be a CNE outcome of the strategic market game played by the traders?

We are able to state the following proposition.

Proposition 5

If $\sigma \in (0,\infty )$, then there does not exist ad valorem tax ${\tilde{t}}$ and endowment tax ${\tilde{\tau }}$ such that, when the tax revenues are transferred to agent $2n+1$, (i) the overall-allocation resulting from these transfers and from the interior Cournot–Nash equilibria of the games with taxations $\Gamma ^{\epsilon }$, $\epsilon =t,\tau$, are Pareto efficient, and (ii) the utility of agent $2n+1$ is equal to $\bar{ u}_{2n+1}$. If $\sigma =\{0,\infty \}$, then there exist ad valorem tax ${\tilde{t}}$ and endowment tax ${\tilde{\tau }}$ such that, when the tax revenues are transferred to agent $2n+1$, (i’) the overall-allocation resulting from these transfers and from the corresponding Cournot–Nash equilibria of the games with taxations $\Gamma ^{\epsilon }$, $\epsilon =t,\tau ,$ are Pareto efficient, and (ii’) the utility of agent $2n+1$ is equal to ${\bar{u}}_{2n+1}$.

Proof

See “Appendix E”.

The first part of the result may be explained as follows. In any interior CNE, when commodities are imperfectly substitutable, no fiscal policy with transfers to the outside agent is sufficiently powerful to eliminate the market inefficiencies caused by the strategic interactions between traders.¹² In addition, by collecting resources for redistributive purpose, the overall-allocation reached with such taxation mechanisms does not lead to a first-best allocation, as the traders still behave strategically, but only to a second-best allocation.

Echoing Proposition 5, the following three numerical examples illustrate two things. First, they illustrate that the effectiveness of the two fiscal policies is limited when the elasticity of substitution is strictly positive but finite. Second, they show that the effects of the two fiscal policies differ according to the value of the elasticity of substitution. In all examples, we consider the following specification for the exchange economy ${\mathcal {E}}$: there are 9 agents, with $n=4$ traders of each type; (2) is such that $\alpha =\frac{1}{3}$, $\beta =\frac{2}{3}$, and $\mu =1$; and, we let ${\bar{u}}_{2n+1}=\frac{1}{4}$.¹³ To save notations, agent i’s marginal rate of substitution at $(x_{i},y_{i})$ is here denoted by $MRS^{i}$. Moreover, $({\tilde{q}},{\tilde{b}})$ will denote the CNE supplies of $\Gamma ^{\epsilon }$, $\epsilon =t,\tau$, at $\tilde{ \epsilon }={\tilde{t}},{\tilde{\tau }}$, in place of $({\tilde{q}}({\tilde{\in }}),\tilde{ b}({\tilde{\in }}))$, and $({\tilde{x}}_{i},{\tilde{y}}_{i})$ will denote the CNE post-tax allocation in place of $({\tilde{x}}_{i}({\tilde{\epsilon }}),{\tilde{y}} _{i}({\tilde{\epsilon }}))$, ${\tilde{\epsilon }}={\tilde{t}},{\tilde{\tau }}$: no confusion will arise. In examples 1 to 3 we consider successively $\sigma = \frac{1}{2},1,2$.¹⁴

Example 1

$\sigma =\frac{1}{2}$. With ad valorem taxation, (18) yields ${\tilde{q}}(t)={\tilde{b}}(t)=\frac{3}{3+\sqrt{6(1-t)}}$. From (20), (21), we have $({\tilde{x}}_{i}(t),{\tilde{y}}_{i}(t))=$ ${(}\frac{ \sqrt{6(1-t)}}{3+\sqrt{6(1-t)}},\frac{3(1-t)}{3+\sqrt{6(1-t)}}{)}$, for each $i\in T_{1}$, and $({\tilde{x}}_{i}(t),{\tilde{y}}_{i}(t))=$ ${(} \frac{3(1-t)}{3+\sqrt{6(1-t)}},\frac{\sqrt{6(1-t)}}{3+\sqrt{6(1-t)}}{) }$, for each $i\in T_{2}$, from which, by using (22), (23), we deduce $\tilde{ \pi }_{i}(t)=\frac{3}{3+\sqrt{6(1-t)}}\frac{3(1-t)\sqrt{6(1-t)}}{3(1-t)+2 \sqrt{6(1-t)}}$, for each $i\in T_{1}\cup T_{2}$. Then, (E1) may be written as $\underset{\{t\}}{\max }\frac{24}{3+\sqrt{6(1-t)}}\frac{3(1-t) \sqrt{6(1-t)}}{3(1-t)+2\sqrt{6(1-t)}}$ s.t. ${(}(\frac{12t}{3+\sqrt{ 6(1-t)}})^{-1}+(\frac{12t}{3+\sqrt{6(1-t)}})^{-1}{)}^{-1}=\frac{1}{4}$ . Some calculations show that the solution to (E1) is ${\tilde{t}}=\frac{23+ \sqrt{337}}{192}=0.215$. The corresponding equilibrium supplies are given by ${\tilde{q}}={\tilde{b}}=\frac{12}{12+\sqrt{\frac{169-\sqrt{337}}{2}}}=0.58$ . The relative price is ${\tilde{p}}_{X}=1$, and, by using (20), (21), the allocations are given by $({\tilde{x}}_{i},{\tilde{y}}_{i})={(}\frac{ \sqrt{\frac{169-\sqrt{337}}{2}}}{12+\sqrt{\frac{169-\sqrt{337}}{2}}},\frac{ 169-\sqrt{337}}{192+16\sqrt{\frac{169-\sqrt{337}}{2}}}{)} =(0.42,0.455)$, for each $i\in T_{1}$, $({\tilde{x}}_{i},{\tilde{y}}_{i})= {(}\frac{169-\sqrt{337}}{192+16\sqrt{\frac{169-\sqrt{337}}{2}}}$ $, \frac{\sqrt{\frac{169-\sqrt{337}}{2}}}{12+\sqrt{\frac{169-\sqrt{337}}{2}}} {)=}(0.455,0.42)$, for each $i\in T_{2}$, and $({\tilde{x}}_{2n+1}, {\tilde{y}}_{2n+1})=(\frac{23+\sqrt{337}}{48+4\sqrt{\frac{169-\sqrt{337}}{2}}}, \frac{23+\sqrt{337}}{48+4\sqrt{\frac{169-\sqrt{337}}{2}}})=(0.5,0.5)$. Then, the marginal rates of substitution differ among agents as $MRS^{i}= \frac{169-\sqrt{337}}{128}$, $i\in T_{1}$, $MRS^{i}=\frac{128}{169-\sqrt{337} }$, $i\in T_{2}$, and $MRS^{2n+1}=1$. With endowment taxation, (19) yields ${\tilde{q}}(\tau )={\tilde{b}}(\tau )=(3-\sqrt{6})(1-\tau )$. From (24), (25), we have $({\tilde{x}}_{i}(\tau ),{\tilde{y}}_{i}(\tau ))=$ $((\sqrt{6}-2)(1-\tau ),(3-\sqrt{6})(1-\tau ))$, for each $i\in T_{1}$, and $({\tilde{x}}_{i}(\tau ), {\tilde{y}}_{i}(\tau ))=$ $((3-\sqrt{6})(1-\tau ),(\sqrt{6}-2)(1-\tau ))$, for each $i\in T_{2}$, from which, by using (26), (27), we deduce ${\tilde{\pi }} _{i}(\tau )=\frac{3(18-7\sqrt{6})(1-\tau )}{5}$, for each $i\in T_{1}\cup T_{2}$. Then, (E2) may be written as $\underset{\{\tau \}}{\max }\frac{ 24(18-7\sqrt{6})(1-\tau )}{5}$ s.t. ${(}(4\tau )^{-1}+(4\tau )^{-1} {)}^{-1}=\frac{1}{4}$. The solution to (E2) is ${\tilde{\tau }}=\frac{1 }{8}$. Then, we deduce ${\tilde{q}}={\tilde{b}}=\frac{21-7\sqrt{6}}{8}$, ${\tilde{p}}_{X}=1$, and, by using (24), (25), $({\tilde{x}}_{i},{\tilde{y}}_{i})=( \frac{7\sqrt{6}-14}{8},\frac{21-7\sqrt{6}}{8})$, $i\in T_{1}$, $({\tilde{x}} _{i},{\tilde{y}}_{i})=(\frac{21-7\sqrt{6}}{8}$ $,\frac{7\sqrt{6}-14}{8})$, $i\in T_{2}$, and $({\tilde{x}}_{2n+1},{\tilde{y}}_{2n+1})=(\frac{1}{4},\frac{1}{4 })$. The marginal rates of substitution do not coincide as $MRS^{i}=\frac{3 }{4}$, $i\in T_{1}$, $MRS^{i}=\frac{4}{3}$, $i\in T_{2}$, and $MRS^{2n+1}=1$ .

Example 2

$\sigma =1$. With ad valorem taxation, (18) yields ${\tilde{q}}(t)={\tilde{b}}(t)=\frac{3}{5}$. From (20), (21), we have $({\tilde{x}} _{i}(t),{\tilde{y}}_{i}(t))=(\frac{2}{5},\frac{3}{5}(1-t))$, for each $i\in T_{1}$, and $({\tilde{x}}_{i}(t),{\tilde{y}}_{i}(t))=(\frac{3}{5}(1-t),\frac{2}{5 })$, for each $i\in T_{2}$, from which, by using (22), (23), we deduce ${\tilde{\pi }}_{i}(t)=(\frac{2}{5})^{\frac{1}{3}}(\frac{3}{5}(1-t))^{\frac{2}{3} }$, for each $i\in T_{1}\cup T_{2}$. Then, (E1) may be written as $\underset{ \{t\}}{\max }8(\frac{2}{5})^{\frac{1}{3}}(\frac{3}{5}(1-t))^{\frac{2}{3}}$ s.t. $(\frac{12}{5}t)(\frac{12}{5}t)=\frac{1}{4}$. The solution to (E1) is ${\tilde{t}}=\frac{5}{24}$. The equilibrium supplies are ${\tilde{q}}={\tilde{b}}= \frac{3}{5}$, the price is ${\tilde{p}}_{X}=1$, and, by using (20), (21), the allocations are $({\tilde{x}}_{i},{\tilde{y}}_{i})={(}\frac{2}{5},\frac{19 }{40}{)}$, $i\in T_{1}$, $({\tilde{x}}_{i},{\tilde{y}}_{i})={(} \frac{19}{40}$ $,\frac{2}{5}{)}$, $i\in T_{2}$, and $({\tilde{x}} _{2n+1},{\tilde{y}}_{2n+1})=(\frac{1}{4},\frac{1}{4})$. Then, the marginal rates of substitution differ among agents as $MRS^{i}=\frac{19}{32}$, $i\in T_{1}$, $MRS^{i}=\frac{32}{19}$, $i\in T_{2}$, and $MRS^{2n+1}=1$. With endowment taxation, (19) yields ${\tilde{q}}(\tau )={\tilde{b}}(\tau )=\frac{3}{5 }(1-\tau )$. From (24), (25), we have $({\tilde{x}}_{i}(\tau ),{\tilde{y}} _{i}(\tau ))=(\frac{2}{5}(1-\tau ),\frac{3}{5}(1-\tau ))$, for each $i\in T_{1}$, and $({\tilde{x}}_{i}(\tau ),{\tilde{y}}_{i}(\tau ))=(\frac{3}{5}(1-\tau ),\frac{2}{5}(1-\tau ))$, for each $i\in T_{2}$, from which, by using (26), (27), we deduce ${\tilde{\pi }}_{i}(\tau )=(\frac{2}{5}(1-\tau )^{\frac{1}{ 3}}(\frac{3}{5}(1-\tau ))^{\frac{2}{3}}$, for each $i\in T_{1}\cup T_{2}$. Then, (E2) may be written as $\underset{\{\tau \}}{\max }8(\frac{2}{5} (1-\tau )^{\frac{1}{3}}(\frac{3}{5}(1-\tau ))^{\frac{2}{3}}$ s.t. $(4\tau )(4\tau )=\frac{1}{4}$. The solution to (E2) is ${\tilde{\tau }}=\frac{1}{8}$. Then, we deduce ${\tilde{q}}={\tilde{b}}=\frac{21}{40}$, ${\tilde{p}}_{X}=1$, and, by using (24), (25), $({\tilde{x}}_{i},{\tilde{y}}_{i})=(\frac{7}{20},\frac{21}{40 })$, $i\in T_{1}$, $({\tilde{x}}_{i},{\tilde{y}}_{i})=(\frac{21}{40}$ $,\frac{7}{ 20})$, $i\in T_{2}$, and $({\tilde{x}}_{2n+1},{\tilde{y}}_{2n+1})=(\frac{1}{4}, \frac{1}{4})$. The marginal rates of substitution do not coincide as $MRS^{i}=\frac{3}{4}$, $i\in T_{1}$, $MRS^{i}=\frac{4}{3}$, $i\in T_{2}$, and $MRS^{2n+1}=1$.

Example 3

$\sigma =2$. With ad valorem taxation, (18) yields ${\tilde{q}}(t)={\tilde{b}}(t)=\frac{9(1-t)}{4+9(1-t)}$. From (20), (21), we have $({\tilde{x}}_{i}(t),{\tilde{y}}_{i}(t))=$ ${(}\frac{4}{4+9(1-t)},\frac{ 9(1-t)^{2}}{4+9(1-t)}{)}$, for each $i\in T_{1}$, and $({\tilde{x}} _{i}(t),{\tilde{y}}_{i}(t))=$ ${(}\frac{9(1-t)^{2}}{4+9(1-t)},\frac{4}{ 4+9(1-t)}{)}$, for each $i\in T_{2}$, from which, by using (22)-(23), we deduce ${\tilde{\pi }}_{i}(t)=\frac{4(4-3t)^{2}}{9(13-9t)}$, for each $i\in T_{1}\cup T_{2}$. Then, (E1) may be written as $\underset{\{t\}}{\max } \frac{32(4-3t)^{2}}{9(13-9t)}$ s.t. ${(}(\frac{36t(1-t)}{4+9(1-t)} )^{\frac{1}{2}}+(\frac{36t(1-t)}{4+9(1-t)})^{\frac{1}{2}}{)}^{2}= \frac{1}{4}$. By using (22), (23), some calculations show that the solution to (E1) is ${\tilde{t}}=\frac{585-\sqrt{312273}}{1152}=0.02$. Then, ${\tilde{q}}= {\tilde{b}}=\frac{567+3\sqrt{34697}}{1079+3\sqrt{34697}}=0.687$, ${\tilde{p}} _{X}=1$, and $({\tilde{x}}_{i},{\tilde{y}}_{i})={(}\frac{512}{1079+3\sqrt{ 34697}},\frac{(189+\sqrt{34697})^{2}}{128(1079+3\sqrt{34697})}{)} =(0.313,0.671)$, $i\in T_{1}$, $({\tilde{x}}_{i},{\tilde{y}}_{i})={(} \frac{(189+\sqrt{34697})^{2}}{128(1079+3\sqrt{34697})},\frac{512}{1079+3 \sqrt{34697}}{)}=(0.671,0.313)$, $i\in T_{2}$, and $({\tilde{x}}_{2n+1}, {\tilde{y}}_{2n+1})=(\frac{1}{16},\frac{1}{16})$. At this CNE, the marginal rates of substitution differ among agents as $MRS^{i}=\frac{189+\sqrt{34697} }{512}$, $i\in T_{1}$, $MRS^{i}=\frac{512}{189+\sqrt{34697}}$, $i\in T_{2}$, and $MRS^{2n+1}=1$. With endowment taxation, (19) yields ${\tilde{q}}(\tau )= {\tilde{b}}(\tau )=\frac{9}{13}(1-\tau )$. From (24), (25), we have $({\tilde{x}} _{i}(\tau ),{\tilde{y}}_{i}(\tau ))=(\frac{4}{13}(1-\tau ),\frac{9}{13}(1-\tau ))$, for each $i\in T_{1}$, and $({\tilde{x}}_{i}(\tau ),{\tilde{y}}_{i}(\tau ))=(\frac{9}{13}(1-\tau ),\frac{4}{13}(1-\tau ))$, for each $i\in T_{2}$, from which, by using (26)-(27), we deduce ${\tilde{\pi }}_{i}(\tau )=\frac{ 64(1-\tau )}{117}$, for each $i\in T_{1}\cup T_{2}$. Then, (E2) may be written as $\underset{\{\tau \}}{\max }\frac{512(1-\tau )}{117}$ s.t. ${(}(4\tau )^{\frac{1}{2}}+(4\tau )^{\frac{1}{2}}{)}^{2}=\frac{1 }{4}$. The solution to (E2) is ${\tilde{\tau }}=\frac{1}{64}$. Then, we deduce ${\tilde{q}}={\tilde{b}}=\frac{567}{832}$, ${\tilde{p}}_{X}=1$, and, by using (24), (25), $({\tilde{x}}_{i},{\tilde{y}}_{i})=(\frac{63}{208},\frac{567}{832})$, $i\in T_{1}$, $({\tilde{x}}_{i},{\tilde{y}}_{i})=(\frac{567}{832}$ $,\frac{63}{ 208})$, $i\in T_{2}$, and $({\tilde{x}}_{2n+1},{\tilde{y}}_{2n+1})=(\frac{1}{16}, \frac{1}{16})$. The marginal rates of substitution do not coincide as $MRS^{i}=\frac{3}{4}$, $i\in T_{1}$, $MRS^{i}=\frac{4}{3}$, $i\in T_{2}$, and $MRS^{2n+1}=1$.

The preceding numerical examples show that the higher the elasticity of substitution, the lower the equilibrium ad valorem tax, i.e. $\frac{585- \sqrt{312273}}{1152}<\frac{5}{24}<\frac{23+\sqrt{337}}{192}$. Moreover, the equilibrium endowment tax is the same for $\sigma =\frac{1}{2}$ and $\sigma =2$. In addition, from Proposition 5, under ad valorem taxation, the higher the elasticity of substitution, the higher the equilibrium supplies, i.e. $\frac{12}{12+\sqrt{\frac{169-\sqrt{337}}{2}}}<\frac{3}{5}<\frac{567+3 \sqrt{34697}}{1079+3\sqrt{34697}}$. It is worth noting, that, according to Proposition 5, we have $\frac{\partial {\tilde{q}}_{i}(t)}{\partial t} \gtreqqless 0$ when $\sigma \lesseqqgtr 1$ (as ${\tilde{q}}(t)=\frac{9(1-t)}{ 4+9(1-t)}$ when $\sigma =2$). Therefore, more (less) substitutability leads the traders to supply less (more) when the ad valorem tax increases. Moreover, under endowment taxation, the equilibrium supply of any trader increases with $\sigma$, i.e. $\frac{21-7\sqrt{6}}{8}<\frac{21}{40}<\frac{ 567}{832}$, while the equilibrium tax is the same when $\sigma =\frac{1}{2} ,1$, i.e. ${\tilde{\tau }}=\frac{1}{8}$. The reason stems from the fact that, according to Proposition 2, and by using (19), the equilibrium supply of any trader increases with $\sigma$, i.e. for all $\tau \in (0,1)$, $\frac{ \partial {\tilde{q}}(\tau )}{\partial \sigma }=\frac{(1-\tau )\ln (\frac{\beta }{\alpha }\frac{n-1}{n})(\frac{\alpha }{\beta }\frac{n}{n-1})^{\sigma }}{[1+( \frac{\alpha }{\beta }\frac{n}{n-1})^{\sigma }]^{2}}>0$ (the same holds with $\frac{\partial {\tilde{b}}(\tau )}{\partial \sigma }$).

When the two commodities are imperfectly substitutable, and regardless of the taxation mechanism implemented, traders are always able to use their market power by restricting their supply. Consequently, when considering the two taxation policies with transfers, we are led to wonder whether it exists some values of the elasticity of substitution for which the two taxation policies are likely to implement a Cournot–Nash equilibrium allocation with transfers that would be a Pareto optimum. The second part of Proposition 5 provides a positive answer to this question. When commodities are perfect substitutes (resp. complements), the traders can have no interest to exert (resp. can no longer exert) their market power by manipulating the market price through their supplies as the gains from participating to trade exceed the gains from manipulating the market price.

The following two examples, inspired by the exchange economy of the three previous examples, illustrate the second part of Proposition 6.

Example 4

$\sigma \rightarrow \infty$. (2) is $u_{i}(x_{i},y_{i})=\frac{1}{3}x_{i}+\frac{2}{3}y_{i}$, for each $i\in T_{1}$ , and $u_{i}(x_{i},y_{i})=\frac{2}{3}x_{i}+\frac{1}{3}y_{i}$, for each $i\in T_{2}$, and $u_{i}(x_{i},y_{i})=x_{i}+y_{i}$, for $i=2n+1$. With ad valorem taxation, from (20), (21), we have $({\tilde{x}}_{i}(t),{\tilde{y}} _{i}(t))=(0,1-t)$, for each $i\in T_{1}$, and $({\tilde{x}}_{i}(t),{\tilde{y}} _{i}(t))=(1-t,0)$, for each $i\in T_{2}$, from which, by using (22), (23), we deduce ${\tilde{\pi }}_{i}(t)=\frac{2}{3}(1-t)$, for each $i\in T_{1}\cup T_{2}$ . Then, (E13) may be written as $\underset{\{t\}}{\max }\frac{16}{3}(1-t)$ s.t. $4t+4t=\frac{1}{4}$. The solution (E14) to (E13) is ${\tilde{t}}=\frac{1}{ 32}$. With this tax, conditions (C6) and (C7) are satisfied for an interior CNE, i.e. $\frac{31}{32}>\frac{2}{3}$, so the equilibrium supplies are ${\tilde{q}}={\tilde{b}}=1$. The relative price is ${\tilde{p}}_{X}=1$, and, by using (20), (21), the allocations are $({\tilde{x}}_{i},{\tilde{y}}_{i})=(0,\frac{ 31}{32})$, for each $i\in T_{1}$, $({\tilde{x}}_{i},{\tilde{y}}_{i})=(\frac{31}{ 32},0)$, for each $i\in T_{2}$, and $({\tilde{x}}_{2n+1},{\tilde{y}}_{2n+1})=( \frac{1}{4},\frac{1}{4})$. Then, the post-tax allocation corresponds to a competitive equilibrium allocation with ad valorem taxation. With endowment taxation, from (24), (25), we have $({\tilde{x}}_{i}(\tau ),{\tilde{y}}_{i}(\tau ))=(0,1-\tau )$, for each $i\in T_{1}$, and $({\tilde{x}}_{i}(\tau ),{\tilde{y}} _{i}(\tau ))=(1-\tau ,0)$, for each $i\in T_{2}$, from which, by using (22), (23), we deduce ${\tilde{\pi }}_{i}(t)=\frac{2}{3}(1-\tau )$, for each $i\in T_{1}\cup T_{2}$. Then, (E15) may be written as $\underset{\{\tau \}}{ \max }\frac{16}{3}(1-\tau )$ s.t. $4\tau +4\tau =\frac{1}{4}$. The solution (E17) to (E15) is ${\tilde{\tau }}=\frac{1}{32}$. With this tax, conditions (C6) and (C7) are satisfied for an interior CNE, i.e. $\frac{31}{32}>\frac{2 }{3}$, so the equilibrium supplies are ${\tilde{q}}={\tilde{b}}=1-{\tilde{\tau }}$ . The relative price is ${\tilde{p}}_{X}=1$, and, by using (24), (25), the allocations are $({\tilde{x}}_{i},{\tilde{y}}_{i})=(0,\frac{31}{32})$, for each $i\in T_{1}$, $({\tilde{x}}_{i},{\tilde{y}}_{i})=(\frac{31}{32},0)$, for each $i\in T_{2}$, and $({\tilde{x}}_{2n+1},{\tilde{y}}_{2n+1})=(\frac{1}{4},\frac{1}{4 })$. Then, the post-tax allocation coincides with the competitive equilibrium allocation with endowment taxation.

Example 5

$\sigma =0$. (2) is $u_{i}(x_{i},y_{i})=\min (x_{i},y_{i})$, for each $i\in T_{1}\cup T_{2}\cup \{2n+1\}$. With ad valorem taxation, (18) yields ${\tilde{q}}(t)={\tilde{b}}(t)=\frac{1}{2-t}$. From (20)-(21), we have $({\tilde{x}}_{i}(t),{\tilde{y}}_{i}(t))=$ $(\frac{1-t}{ 2-t},\frac{1-t}{2-t})$, for each $i\in T_{1}\cup T_{2}$, from which, by using (22), (23), we deduce ${\tilde{\pi }}_{i}(t)=\frac{1-t}{2-t}$, for each $i\in T_{1}\cup T_{2}$. Then, (E22) may be written as $\underset{\{t\}}{\max }8\frac{1-t}{2-t}$ s.t.$\ \min (4\frac{t}{2-t},4\frac{t}{2-t})=\frac{1}{4}$ . The solution (E24) to (E22) is ${\tilde{t}}=\frac{2}{17}$. The equilibrium supplies are ${\tilde{q}}={\tilde{b}}=\frac{17}{32}$, the relative price is ${\tilde{p}}_{X}=1$, and, by using (20), (21), the allocations are $({\tilde{x}} _{i},{\tilde{y}}_{i})=(\frac{15}{32},\frac{15}{32})$, for each $i\in T_{1}$, $( {\tilde{x}}_{i},{\tilde{y}}_{i})=(\frac{15}{32},\frac{15}{32})$, for each $i\in T_{2}$, and $({\tilde{x}}_{2n+1},{\tilde{y}}_{2n+1})=(\frac{1}{4},\frac{1}{4})$ . Then, the post-tax allocation satisfies $\frac{{\tilde{y}}_{i}}{{\tilde{x}} _{i}}=1$, for each $i\in T_{1}\cup T_{2}\cup \{2n+1\}$, which coincides with the ratio in a competitive equilibrium allocation with ad valorem taxation. With endowment taxation, (19) yields ${\tilde{q}}(t)={\tilde{b}}(t)=\frac{1-\tau }{2}.$From (24), (25), we have $({\tilde{x}}_{i}(\tau ),{\tilde{y}}_{i}(\tau ))=$ $(\frac{1-\tau }{2},\frac{1-\tau }{2})$, for each $i\in T_{1}\cup T_{2}$, from which, by using (26)-(27), we deduce ${\tilde{\pi }}_{i}(\tau )=\frac{ 1-\tau }{2}$, for each $i\in T_{1}\cup T_{2}$. Then, (E26) may be written as $\underset{\{\tau \}}{\max }8\frac{1-\tau }{2}$ s.t. $\min (4\tau ,4\tau )=\frac{1}{4}$. The solution (E28) to (E26) is ${\tilde{\tau }}=\frac{1}{16}$ . The equilibrium supplies are ${\tilde{q}}={\tilde{b}}=\frac{15}{32}$, the relative price is ${\tilde{p}}_{X}=1$, and, by using (24), (25), the allocations are $({\tilde{x}}_{i},{\tilde{y}}_{i})=(\frac{15}{32},\frac{15}{32})$ , for each $i\in T_{1}\cup T_{2}$, and $({\tilde{x}}_{2n+1},{\tilde{y}}_{2n+1})=( \frac{1}{4},\frac{1}{4})$. Then, the post-tax allocation satisfies $\frac{ {\tilde{y}}_{i}}{{\tilde{x}}_{i}}=1$, for each $i\in T_{1}\cup T_{2}\cup \{2n+1\}$, which coincides with the competitive equilibrium allocation with ad valorem taxation.

5 Discussion of the literature

The effectiveness and welfare implications of ad valorem and endowment taxation mechanisms with transfer have been studied in Gabszewicz and Grazzini (2001) for a Cobb–Douglas bilateral oligopoly. These fiscal policies with transfers cannot reach a Pareto-optimal allocation as they are not sufficiently powerful to neutralize the market power of strategic traders. To circumvent this problem, Gabszewicz and Grazzini (2001) study a third kind of fiscal policy, namely endowment taxation with incentive transfers. As with the mechanism of taxation on endowments, this tax policy consists of levying a tax on traders’ endowments before exchange takes place, and redistributing after exchange has occured a share of the product of the tax among the traders so that they receive a quantity of the commodity they do not own initially in proportion to the quantity of the commodity they wish to sell, with the remaining share being transferred to the outside agent.¹⁵ This taxation mechanism leads to a Pareto-optimal allocation to the extent that the transfer received by each trader increases with her supply. In addition, through a class of bilateral oligopolies (linear, Cobb–Douglas, and CES), Gabszewicz and Grazzini (1999) show that endowment taxation with incentive transfers between the only traders leads to a Pareto-optimal allocation. The CES specification we use allows us to explore new features of fiscal policy in strategic bilateral exchange.

The introduction of the parameters of preferences $\alpha$ and $\beta$ in the CES utility functions of traders and the fact that the elasticity of substitution can take all values on $[0,\infty ]$ make it possible to determine a broader set of strategic equilibria among which the existing equilibria constitute special cases. Besides, it makes it possible to study the welfare implications of fiscal policies on markets with traders who behave strategically. In this regard, our model highlights the importance of agents’ preferences in the optimality of fiscal policy, notably through the parameters of preferences $\alpha$ and $\beta$ as well as the parameter of the elasticity of substitution $\sigma$. Thus, our generalized CES bilateral oligopoly model provides a unified framework that includes all the strategic equilibria without and with ad valorem and endowment taxation studied in the literature. Indeed, the CNE equilibrium supplies without taxation given in Proposition 1 generalize the equilibrium supplies provided in Bloch and Ferrer (2001b) and in Gabszewicz and Grazzini (1999) for the CES model as they coincide in the special for which $\alpha =\beta =\frac{1}{2}$, with $\sigma \in (0,\infty ]$. As a consequence, the CNE equilibrium supplies with ad valorem and endowment taxations given in Proposition 3 generalize the equilibrium supplies given in Gabszewicz and Grazzini (2001) as well as the equilibrium supplies with endowment taxation in Gabszewicz and Grazzini (1999). In particular, when $\sigma =1$, the equilibrium supplies with ad valorem and endowment taxations given in Proposition 3 coincide with those that would be obtained with 2n traders in Gabszewicz and Grazzini (2001) for $\alpha =\beta =\frac{1}{2}$. Finally, the case for which $\sigma =0$ has not been yet studied in the previous bilateral oligopoly models with taxation.

Beyond the unified computation of the various symmetric strategic equilibria with taxations, what matters here is that this general specification allows to capture some new features about the elasticity of substitution and the CNE supplies. On the one hand, we study the influence of the parameters of the CES utility functions on equilibrium strategies. Indeed, our computations put forward the link between the effect of any fiscal policy and the local curvature of the indifference curves, through the possible values of the elasticity of substitution, in any Cournot–Nash equilibrium with taxation. On the other hand, the parameters $\alpha$ and $\beta$ play a significant role. Indeed, as shown by Proposition 2, for $\sigma \in (0,\infty )$, the equilibrium supply of any trader increases with the elasticity of substitution if the relative share of the consumption of goods is lower than the marginal rate of substitution. In Gabszewicz and Grazzini (1999), who consider a CES utility function with unitary coefficients on consumption, for $\sigma \in (0,\infty )$, the equilibrium supply of any trader always decreases with the elasticity of substitution. Our Proposition 2 implies that, when $\alpha \ne \beta$, with ad valorem taxation, the equilibrium supply of any trader can increase with the elasticity of substitution. It illustrates the role played by preferences as, when commodities become more substitutable and traders have a relative preference in favor of the good purchased, the traders use their market power to obtain gains from trade. Examples 1 to 3 in Sect. 4 illustrate this. Obviously, this result cannot hold when $\alpha =\beta$ (or with unitary coefficients on consumption). Proposition 4 together with Proposition 2 also explain why strategic equilibria with ad valorem taxation differ from strategic equilibria with endowment taxation; for the latter, the marginal rates of substitution do not depend on the tax, so the effect of an increase of the tax is always negative.

More importantly, our study is reminiscent of the second welfare theorem in general equilibrium analysis. Indeed, it focuses on the redistributive aspects of the tax as well as on the optimality of the corresponding fiscal policy. The optimality of fiscal policies with transfers is based on the value of the elasticity of substitution. Hence, we show that any fiscal policy with transfer may reach a first-best allocation when commodities are either perfect complements or perfect substitutes. Thus, the first part of Proposition 5 reflects the fact that the Pareto inefficiency of the two fiscal policies prevails for any strictly positive finite value of the elasticity of substitution, i.e. for any bilateral oligopoly model in which commodities are neither perfect complements nor perfect substitutes. From this point of view, it includes as special cases the welfare implications of ad valorem and endowment taxations in the Cobb–Douglas bilateral oligopoly studied in Gabszewicz and Grazzini (2001). Moreover, by considering a CES bilateral oligopoly model, we allow the possibility of covering the perfect substitutability and the complementary cases. Thus, the second part of Proposition 5 shows that the two fiscal policies with transfer can reach a Pareto-optimal allocation when commodities are either perfect complements or perfect substitutes. Our results put forward that the preferences of agents matter for the welfare effects of fiscal policies. For instance, when commodities are perfect substitutes, the ad valorem taxation mechanism implements the competitive equilibrium when the gains from participating to trade, as measured by ${\tilde{p}}_{X}(1-t)-\frac{\alpha }{\beta }=1-t-\frac{ \alpha }{\beta }$, can exceed the gains from manipulating prices, as measured by $\frac{1-t}{n}=\frac{\partial p_{X}}{\partial q}{\tilde{q}}(1-t)$ . Otherwise, when commodities are complements, the traders cannot smoothly substitute the two commodities, and the taxation mechanism implements the competitive equilibrium as outcome. The redistributive purpose of taxation which holds when $\sigma =0$ partially echoes Theorem 1 in Busetto et al. (2020b), who show that, in a market with atoms and an atomless part, the Cournot–Nash allocation is a competitive allocation when the atoms have Leontief utility functions.

To circumvent the non Pareto-optimality of ad valorem and endowment taxations, Gabszewicz and Grazzini (2001), also consider another taxation mechanism based on endowment taxation with incentive transfers. Gabszewicz and Grazzini (1999), also study endowment taxation with transfers between the only traders within three kinds of bilateral oligopolies (linear, Cobb–Douglas, and CES). When the endowment taxation with transfers is such that a share of the resulting tax is redistributed among the traders in proportion of the amount of their supply, and the remaining part is transferred to the outside agent in such a way the transfer received by each trader increases, it leads to a Pareto-optimal allocation.

The next example shows that this result also holds in our framework: endowment taxation with transfer to the outside agent and incentive transfers between the sole traders could restore Pareto-optimality when $\sigma \in (0,\infty )$.

Example 6

$\sigma =\frac{1}{2}$. The competitive equilibrium with endowment taxation with transfer to the outside trader is given by the price $p_{X}^{*}=1$, and allocations $(x_{i}^{*},y_{i}^{*})=( \frac{15}{16}(\sqrt{2}-1),\frac{15}{16}(2-\sqrt{2}))$, for each $i\in T_{1}$ , $(x_{i}^{*},y_{i}^{*})=(\frac{15}{16}(2-\sqrt{2}),\frac{15}{16}( \sqrt{2}-1))$, for each $i\in T_{2}$, and $(x_{2n+1}^{*},y_{2n+1}^{*})=(\frac{1}{4},\frac{1}{4})$. Consider now the CNE with taxation. To this end, let a tax $\kappa =\tau +\frac{{\bar{u}}_{2n+1}}{4}$, is levied on each commodity. Assume that, after exchange has taken place, a share of the product of the tax is transferred to the only traders, with the remaining share $\frac{{\bar{u}}^{1/\mu }}{n}$ being transferred to the outside agent. In this example, the share of the total tax product of commodity X (resp. Y) transferred to each type 2 (resp. type 1) trader is $\frac{\tau }{\frac{ 15}{16}(2-\sqrt{2})-\tau }$, the denominator of which representing the solution $q_{i}(\tau )$ to $1-\tau -\frac{1}{16}-$ $q_{i}(\tau )=\frac{15}{ 16}(\sqrt{2}-1)$, while the remaining share of the tax product in each commodity corresponding to the utility level ${\bar{u}}_{2n+1}=\frac{1}{4}$, i.e. ${\bar{u}}_{2n+1}=\frac{1}{16}$, is redistributed to the outside agent. The payoffs of traders are now given by $\pi _{i}(\tau )={(}\frac{1}{3 }(1-\tau -\frac{1}{16}-q_{i})^{-1}+\frac{2}{3}(p_{X}q_{i}+\tau \frac{q_{i}}{ \frac{15}{16}(2-\sqrt{2})-\tau })^{-1}{)}^{-1}$, for each $i\in T_{1}$ , and $\pi _{i}(\tau )={(}\frac{2}{3}(\frac{1}{p_{X}}b_{i}+\tau \frac{b_{i}}{ \frac{15}{16}(2-\sqrt{2})-\tau })^{-1}+\frac{1}{3}(1-\tau -\frac{1}{16} -b_{i})^{-1}{)}^{-1}$, for each $i\in T_{2}$. The CNE with endowment taxation with incentive transfers is given by ${\tilde{q}}={\tilde{b}}=\frac{4}{5 }(2-\sqrt{2})$, with the tax on trader’s endowment given by ${\tilde{\tau }}= \frac{11}{80}(2-\sqrt{2})$. The corresponding equilibrium price is $\tilde{p }_{X}=1$, and, the allocations are $({\tilde{x}}_{i},{\tilde{y}}_{i})=(\frac{15}{ 16}(\sqrt{2}-1),\frac{15}{16}(2-\sqrt{2}))$, for each $i\in T_{1}$, $(\tilde{ x}_{i},{\tilde{y}}_{i})=(\frac{15}{16}(2-\sqrt{2}),\frac{15}{16}(\sqrt{2}-1))$ , for each $i\in T_{2}$, and $({\tilde{x}}_{2n+1},{\tilde{y}}_{2n+1})=(\frac{1}{4 },\frac{1}{4})$. The marginal rates of substitution coincide as $MRS^{i}= \frac{1}{2}(\frac{2-\sqrt{2}}{\sqrt{2}-1})^{2}=1$, for each $i\in T_{1}$, $MRS^{i}=2(\frac{\sqrt{2}-1}{2-\sqrt{2}})^{2}=1$, for each $i\in T_{2}$, and $MRS^{2n+1}=1$.

It should be noted that this redistribution mechanism, whether the transfers concern only the traders or the traders and the outside agent, only works with endowment taxation. In addition, such a taxation mechanism with transfers is difficult to implement in CES bilateral oligopoly with different shares on consumption. Indeed, the denominators in the shares $\frac{\tau }{y_{i}^{*}}=\frac{\tau }{_{1-\tau -\frac{{\bar{u}}^{1/\mu }}{n} -b_{i}(\tau )}}$ and $\frac{\tau }{x_{i}^{*}}=\frac{\tau }{_{1-\tau - \frac{{\bar{u}}^{1/\mu }}{n}-q_{i}(\tau )}}$ have to be computed at the competitive equilibrium without transfer between the traders. But, when $\alpha \ne \beta$, there is the possibility of multiple competitive equilibria (see notably Sect. 2.1). In the presence of multiple equilibria, comparative static exercises are more difficult to implement. In addition, from a redistributive viewpoint, there is the problem of selecting the competitive equilibrium for the implementation of this type of fiscal policy with transfers.

6 Conclusion

In this paper, we provided a model of a two-commodity exchange economy in which each trader is endowed with one commodity, while one agent was deprived from any commodity. Market distortions were caused by the strategic behavior of traders in a CES bilateral oligopoly model. To correct these distortions, two fiscal policies with redistribution in favor of the outside agent have been implemented.

It was shown that the effects and welfare implications of fiscal policies depend on the traders’ preferences. Thus, via the elasticity of substitution, and the parameters of preference, we were able to provide a sufficiently general model to compute a whole set of non trivial type-symmetric strategic equilibria. This allowed us to highlight the link between the effect of fiscal policy and the possible values of the elasticity of substitution, at any CNE with taxation. Finally, we showed that fiscal policies with redistribution implemented a first-best allocation when commodities are either perfectly substitutable or complements, and a second-best otherwise.

Modeling taxation in more complex environments, either by extending the model to markets with more than two commodities or to embody asymmetric strategic behavior within this setting is left for further research.

Appendix

1.1 Appendix A: Proof of Proposition 1

The CNE strategies for $\Gamma$ obtain as the simultaneous solution to the following 2n problems:

(A1)

The solution to these problems are given by the 2n first-order sufficient conditions $\frac{\partial \pi _{i}(.)}{\partial q_{i}}=0$, for $i\in T_{1}$ , and $\frac{\partial \pi _{i}(.)}{\partial b_{i}}=0$, for $i\in T_{2}$, which may be written:

(A2)

where $A\equiv \mu [a(1-q_{i})^{\frac{\sigma -1}{\sigma }}+\beta (p_{X}q_{i})^{\frac{\sigma -1}{\sigma }}]$ $^{\frac{\sigma }{\sigma -1}\mu -1}>0$ and $B\equiv \mu [\beta (\frac{1}{p_{X}}b_{i}$ $)^{\frac{ \sigma -1}{\sigma }}+\alpha (1-b_{i})^{\frac{\sigma -1}{\sigma }}]^{\frac{ \sigma }{\sigma -1}\mu -1}>0$. By adapting the approach of Gabszewicz and Grazzini (1999) to our framework, it is possible to show that all traders of the same type must adopt the same strategy at equilibrium. Indeed, we have ${\tilde{q}}_{i}={\tilde{q}}$, for each $i\in T_{1}$, and ${\tilde{b}}_{i}={\tilde{b}}$, for each $i\in T_{2}$. In addition, as for every point $(x,y)\in {\mathbb {R}} _{+}^{2}$ the utility functions of type 1 and type 2 traders satisfy $u_{1}(x,y)=u_{2}(y,x)$, it follows that ${\tilde{q}}={\tilde{b}}$, so the 2n first-order conditions in a type-symmetric interior CNE may be written:

(A3)

Then, after some algebra, we deduce the traders’ equilibrium strategies:

$$\begin{aligned} {\tilde{q}}={\tilde{b}}=\frac{1}{1+\left( \frac{\alpha }{\beta }\frac{n}{n-1} \right) ^{\sigma }}\text {.} \end{aligned}$$

(A4)

Therefore, (11) holds.$\blacksquare$

1.2 Appendix B: Proof of Proposition 2

Let $\sigma \in (0,\infty )$. Consider (11), and pick one trader of type 1. The effect of an increase of $\sigma$ on the equilibrium supply $\tilde{ q}$ is measured by the following partial derivative:

$$\begin{aligned} \frac{\partial {\tilde{q}}}{\partial \sigma }=-\ln \left( \frac{\alpha }{\beta }\frac{n}{n-1}\right) \left( \frac{\alpha }{\beta }\frac{n}{n-1}\right) ^{\sigma }{\tilde{q}}^{2}\text {.} \end{aligned}$$

(B1)

As $\left( \frac{\alpha }{\beta }\frac{n}{n-1}\right) ^{\sigma }>0$ and ${\tilde{q}}>0$ at a CNE when $\sigma \in (0,\infty )$, the sign of $\frac{ \partial {\tilde{q}}}{\partial \sigma }$ depends on the sign of the expression $\ln (\frac{\alpha }{\beta }\frac{n}{n-1})$. Then

$$\begin{aligned} \frac{\alpha }{\beta }\frac{n}{n-1}\leqslant 1\text { if and only if }\frac{ \partial {\tilde{q}}}{\partial \sigma }\geqslant 0\text {.} \end{aligned}$$

(B2)

In both implications the condition is $\frac{\alpha }{\beta }\leqslant \frac{ n-1}{n}$, where $MRS^{i}({\tilde{x}}_{i},{\tilde{y}}_{i})=\frac{n-1}{n}$, for each $i\in T_{1}$. Then, the conclusion follows. By symmetry, the same analysis holds for $\frac{\partial {\tilde{b}}}{\partial \sigma }$.$\blacksquare$

1.3 Appendix C: Proof of Proposition 3

In this Appendix we determine the CNE strategies by encompassing the two taxation mechanisms. To this end, for each $i\in T_{1}$, we some abuse of notations, let us set, by reference to (14) and (16), $\pi _{i}(.;t,\tau )$ for $\pi _{i}(q_{i},{\mathbf {q}}_{-i};{\mathbf {b}};t,\tau )$. Likewise, for each $i\in T_{2}$, let us set, by reference to (15) and (17), $\pi _{i}( \mathbf {.};t,\tau )$ for $\pi _{i}({\mathbf {q}};b_{i},{\mathbf {b}}_{-i};t,\tau )$. The CNE strategies for $\Gamma ^{\epsilon }$, $\epsilon =t,\tau$, obtain as the simultaneous solution to the following 2n problems:

(C1)

The 2n first-order sufficient conditions may be written:

(C2)

where $A\equiv \mu \{\alpha (1-\tau -q_{i}$ $)^{\frac{\sigma -1}{\sigma } }+\beta [p_{X}(1-t)q_{i}]^{\frac{\sigma -1}{\sigma }}\}$ $^{\frac{ \sigma }{\sigma -1}\mu -1}>0$ and $B\equiv \mu \{\beta [\frac{1}{p_{X} }(1-t)b_{i}$ $]^{\frac{\sigma -1}{\sigma }}+\alpha (1-\tau -b_{i})^{\frac{ \sigma -1}{\sigma }}\}$ $^{\frac{\sigma }{\sigma -1}\mu -1}>0$. By adapting once again the approach of Gabszewicz and Grazzini (1999) to our framework, it is possible to show that all traders of the same type must adopt the same strategy at equilibrium. Indeed, we have ${\tilde{q}}_{i}(t,\tau )={\tilde{q}} (t,\tau )$, $i\in T_{1}$, and ${\tilde{b}}_{i}(t,\tau )={\tilde{b}}(t,\tau )$, $i\in T_{2}$. Then, the 2n first-order conditions in a type-symmetric interior CNE with taxations may be written:

(C3)

Then, we deduce the traders’ equilibrium strategies:

$$\begin{aligned} {\tilde{q}}(t,\tau )={\tilde{b}}(t,\tau )=\frac{1-\tau }{1+(1-t)\left( \frac{ \alpha }{\beta }\frac{1}{1-t}\frac{n}{n-1}\right) ^{\sigma }}\text {.} \end{aligned}$$

(C4)

Therefore, (18) holds when $\tau =0$, and (19) holds when $t=0$.

Consider now the case $\sigma \rightarrow \infty$. Then, (2) may be written $u_{i}(x_{i},y_{i})=(\alpha x_{i}+\beta y_{i})^{\mu }$, for $i\in T_{1}$, and $u_{i}(x_{i},y_{i})=(\beta x_{i}+\alpha y_{i})^{\mu }$, for $i\in T_{2}$. Assume $\alpha <\beta$. The CNE strategies obtain as the simultaneous solution to the 2n problems:

(C5)

As $\frac{\partial \pi _{i}}{\partial q_{i}}=\mu (-\alpha +\beta \frac{n-1}{n }(1-t))(\pi _{i})^{\mu -1}$ and $\frac{\partial \pi _{i}}{\partial b_{i}}=$ $\mu (\beta \frac{n-1}{n}(1-t)-\alpha )(\pi _{i})^{\mu -1}$, with $\mu ,\pi _{i}>0$, the following 2n necessary and sufficient conditions

$$\begin{aligned} \frac{\partial \pi _{i}}{\partial q_{i}}_{\mid (\varvec{\tilde{q}};\varvec{\tilde{b}})}>0\text { \ }\Leftrightarrow \text { }1-t>\frac{\alpha }{\beta } \frac{n}{n-1}\text {, }i\in T_{1}\text {;} \end{aligned}$$

(C6)

and

$$\begin{aligned} \frac{\partial \pi _{i}}{\partial b_{i}}_{\mid (\varvec{\tilde{q}};\varvec{\tilde{b}})}>0\text { }\Leftrightarrow \text { }1-t>\frac{\alpha }{\beta } \frac{n}{n-1}\text {, }i\in T_{2}\text { } \end{aligned}$$

(C7)

show that there is a tax ${\tilde{t}}$, $0<t<1$, for which the vector of strategic supplies $(\varvec{\tilde{q}}(t);\varvec{\tilde{b}}(t))$ is a CNE of the game with taxation t. The function $f(t)=1-t$ which maps [0, 1] onto [0, 1] is strictly decreasing. In a interior equilibrium of the game without taxation, we have $\frac{\alpha }{\beta }\frac{n}{n-1}\leqslant 1$, and as $\frac{\alpha }{\beta }\frac{n}{n-1}>0$, there exists a value $\tilde{ t}$ of $t\in (0,1)$ such that (C6) and (C7) are satisfied. Then, ${\tilde{q}}= {\tilde{b}}=1$ is a CNE with ad valorem taxation. In addition, when $\alpha =\beta$, the two inequalites (C6) and (C7) cannot be satisfied: the following condition for an autarkic equilibrium, i.e. $\frac{\alpha }{\beta } \frac{n}{n-1}>1$, holds. The same holds with per unit taxation $\nu \in (0,1)$.$\blacksquare$

1.4 Appendix D: Proof of Proposition 4

We consider the effect of a tax variation on equilibrium supplies. In each case, pick one $i\in T_{1}$ (the same analysis may be handled for each $i\in T_{2}$).

First, let $t\in (0,1)$. By using (18), simple calculations lead to:

$$\begin{aligned} \frac{\partial {\tilde{q}}_{i}(t)}{\partial t}=(1-\sigma )\left( \frac{\alpha }{\beta }\frac{1}{1-t}\frac{n}{n-1}\right) ^{\sigma }[{\tilde{q}}_{i}(t)]^{2} \text {.} \end{aligned}$$

(D1)

Therefore, we have:

$$\begin{aligned} \frac{\partial {\tilde{q}}_{i}(t)}{\partial t}\gtreqqless 0\text { when }\sigma \lesseqqgtr 1\text {.} \end{aligned}$$

(D2)

Second, let $\tau \in (0,1)$. By using (19), some calculations lead to:

$$\begin{aligned} \frac{\partial {\tilde{q}}_{i}(\tau )}{\partial \tau }=-\frac{1}{1+\left( \frac{\alpha }{\beta }\frac{n}{n-1}\right) ^{\sigma }}<0\text {.} \end{aligned}$$

(D3)

$\blacksquare$

1.5 Appendix E: Proof of Proposition 5

There are two main cases to consider: first, $\sigma \in (0,\infty )$, and, second $\sigma =\{0,\infty \}$. Let $\epsilon =t,\tau$, and define the welfare function as $W(\epsilon ):=\sum \nolimits _{i\in T_{1}}{\tilde{\pi }} _{i}(\epsilon )+\sum \nolimits _{i\in T_{2}}{\tilde{\pi }}_{i}(\epsilon )$. In each case, we determine the optimal tax at a CNE, and, then, we check if it leads to a Pareto-optimal allocation.

1. Let $\sigma \in (0,\infty )$. Assume, contrary to the first part of Proposition 5 that there exist an ad-valorem tax t and an endowment tax $\tau$ for which requirements (i) and (ii) are satisfied.

With ad valorem taxation, from (22)–(23), we have that $W(t)=2n{\tilde{\pi }} _{i}(t)$. Therefore, any Pareto-optimal allocation, which would follow from ad valorem taxation and provide agent $2n+1$ with utility level ${\bar{u}} _{2n+1}$, with ${\bar{u}}_{2n+1}\in (0,\infty )$, must solve the problem:¹⁶

(E1)

With endowment taxation, from (26)-(27), we have that $W(\tau )=2n{\tilde{\pi }} _{i}(\tau )$. Therefore, any Pareto optimal allocation, which would follow from endowment taxation and provide agent $2n+1$ with utility level ${\bar{u}} _{2n+1}$, with ${\bar{u}}_{2n+1}\in (0,\infty )$, must solve the problem:

(E2)

As the welfare function $W(\epsilon )$ is a continuous function of $\epsilon$, and the constraint sets in $(P^{t})$ and $(P^{\tau })$ are (non linear) compact sets, then, for each taxation mechanism, $\epsilon =t,\tau$, there exists one solution to each of these problems, namely an optimal tax rate ${\tilde{\epsilon }}$, i.e. ${\tilde{t}}$ and ${\tilde{\tau }}$. The solution ${\tilde{t}}$ to problem $(P^{t})$ must satisfy the equation:

$$\begin{aligned} ((nt{\tilde{q}}(t))^{\frac{\sigma -1}{\sigma }}+(nt{\tilde{b}}(t))^{\frac{\sigma -1}{\sigma }})^{^{\frac{\sigma }{\sigma -1}\mu }}={\bar{u}}_{2n+1}\text {,} \end{aligned}$$

(E3)

and the solution ${\tilde{\tau }}$ to problem $(P^{\tau })$ must satisfy the equation:

$$\begin{aligned} ((n\tau )^{\frac{\sigma -1}{\sigma }}+(n\tau )^{\frac{\sigma -1}{\sigma } })^{^{\frac{\sigma }{\sigma -1}\mu }}={\bar{u}}_{2n+1}\text {.} \end{aligned}$$

(E4)

From Proposition 3, as $\sigma \in (0,\infty )$, we know that there exists an interior solution in $\Gamma ^{\epsilon }$, $\epsilon =t,\tau$, given by ${\tilde{q}}(\epsilon )$ and ${\tilde{b}}(\epsilon )$, for each $\epsilon$. In addition, as ${\tilde{q}}(\epsilon )={\tilde{b}}(\epsilon )$, for each $\epsilon$, (E3) and (E4) may be written respectively:

$$\begin{aligned} 2^{^{\frac{\sigma }{\sigma -1}\mu }}\left( nt\frac{1}{1+(\frac{\alpha }{ \beta }\frac{n}{n-1})^{\sigma }(1-t)^{1-\sigma }}\right) ^{\mu }-{\bar{u}} _{2n+1}=0\text {,} \end{aligned}$$

(E5)

by using (18), and

$$\begin{aligned} 2^{^{\frac{\sigma }{\sigma -1}\mu }}(n\tau )^{\mu }-{\bar{u}}_{2n+1}=0\text {.} \end{aligned}$$

(E6)

The solution to (E6) is given by:

$$\begin{aligned} {\tilde{\tau }}=\frac{{\bar{u}}_{2n+1}^{\frac{1}{\mu }}}{2^{^{\frac{\sigma }{ \sigma -1}}}n}\text {.} \end{aligned}$$

(E7)

In addition, as $\sigma \in (0,\infty )$ and $\mu >0$, there exists one solution to (E5). Finally, as $\sigma \ne 0$, the utility functions (2) are differentiable: the marginal rates of substitution at any overall allocation resulting from each CNE with taxation and from any kind of transfer to the outside agent, i.e. the allocation $({\tilde{x}}_{i}(\tilde{ \epsilon }),{\tilde{y}}_{i}({\tilde{\epsilon }}))$, ${\tilde{\epsilon }}={\tilde{t}}, {\tilde{\tau }}$, for each $i=1,\ldots ,2n+1$, are given by:

$$\begin{aligned} MRS^{i}({\tilde{x}}_{i}({\tilde{\epsilon }}),{\tilde{y}}_{i}({\tilde{\epsilon }}))= \frac{\alpha }{\beta }\left( \frac{{\tilde{y}}_{i}({\tilde{\epsilon }})}{{\tilde{x}} _{i}({\tilde{\epsilon }})}\right) ^{\frac{1}{\sigma }}\text {, }i\in T_{1}\text {; } \end{aligned}$$

(E8)

$$\begin{aligned} MRS^{i}({\tilde{x}}_{i}({\tilde{\epsilon }}),{\tilde{y}}_{i}({\tilde{\epsilon }}))= \frac{\beta }{\alpha }\left( \frac{{\tilde{y}}_{i}({\tilde{\epsilon }})}{{\tilde{x}} _{i}({\tilde{\epsilon }})}\right) ^{\frac{1}{\sigma }}\text {, }i\in T_{2}\text {; } \end{aligned}$$

(E9)

$$\begin{aligned} MRS^{i}({\tilde{x}}_{i}({\tilde{\epsilon }}),{\tilde{y}}_{i}({\tilde{\epsilon }} ))=\left( \frac{{\tilde{y}}_{i}({\tilde{\epsilon }})}{{\tilde{x}}_{i}(\tilde{ \epsilon })}\right) ^{\frac{1}{\sigma }}\text {, }i=2n+1\text {.} \end{aligned}$$

(E10)

where $-\left( \frac{dy_{i}}{dx_{i}}\right) _{\mid ({{\tilde{\mathbf {q}}}}( {\tilde{\epsilon }});{{\tilde{\mathbf {b}}}}({\tilde{\epsilon }}))}:=MRS^{i}({\tilde{x}} _{i}({\tilde{\epsilon }}),{\tilde{y}}_{i}({\tilde{\epsilon }}))$, is agent i’s marginal rate of substitution between commodities X and Y at a CNE with equilibrium tax ${\tilde{\epsilon }}={\tilde{t}},{\tilde{\tau }}$, $i\in T_{1}\cup T_{2}\cup \{2n+1\}$. Then, from (20)-(21), (E8)-(E10) are such that $MRS^{i}( {\tilde{x}}_{i}({\tilde{t}}),{\tilde{y}}_{i}({\tilde{t}}))=\frac{n-1}{n}(1-{\tilde{t}} )$, for each $i\in T_{1}$, $MRS^{i}({\tilde{x}}_{i}({\tilde{t}}),{\tilde{y}}_{i}( {\tilde{t}}))=\frac{n}{n-1}\frac{1}{1-{\tilde{t}}}$, for each $i\in T_{2}$, and $MRS^{2n+1}({\tilde{x}}_{i}({\tilde{t}}),{\tilde{y}}_{i}({\tilde{t}}))=1$. In addition, from (24)-(25), (E8)-(E10) lead to $MRS^{i}({\tilde{x}}_{i}(\tilde{ \tau }),{\tilde{y}}_{i}({\tilde{\tau }}))=\frac{n-1}{n}$, for each $i\in T_{1}$, $MRS^{i}({\tilde{x}}_{i}({\tilde{\tau }}),{\tilde{y}}_{i}({\tilde{\tau }}))=\frac{n}{n-1 }$, for each $i\in T_{2}$, and $MRS^{2n+1}({\tilde{x}}_{i}({\tilde{\tau }}), {\tilde{y}}_{i}({\tilde{\tau }}))=1$. In each case the marginal rates of substitution differ across agents, so that the resulting overall-allocation cannot be Pareto-optimal. A contradiction. Then, when $\sigma \in (0,\infty )$, there is no tax with transfer to the outside agent which implements a Pareto-optimal allocation.

2. Let $\sigma =\{0,\infty \}$. To avoid messy repetition, in each case, we encompass the two taxation mechanisms. For each taxation scheme, we have to show the allocation corresponding to the interior CNE of the corresponding game $\Gamma ^{\epsilon }$, $\epsilon =t,\tau$, and the associated transfer obtained by the agent $2n+1$ is optimal.

Consider $\sigma =\infty$. Then, (2) is $u_{i}(x_{i},y_{i})=(\alpha x_{i}+\beta y_{i})^{\mu }$, for each $i\in T_{1}$, and $u_{i}(x_{i},y_{i})=( \beta x_{i}+\alpha y_{i})^{\mu }$, for each $i\in T_{2}$, and $u_{i}(x_{i},y_{i})=(x_{i}+y_{i})^{\mu }$, for $i=2n+1$. From the payoffs given in (14)-(17), with $\rho =1$, the problem of any trader may be written as:

$$\begin{aligned} \max _{q_{i}\in Q_{i}}\text { }\left( \alpha (1-\tau -q_{i})+\beta \frac{ \sum \nolimits _{k\in T_{2}}b_{k}}{\sum \nolimits _{k\in T_{1}}q_{k}} (1-t)q_{i}\right) ^{\mu }\text {, }i\in T_{1}\text {;} \end{aligned}$$

(E11)

$$\begin{aligned} \max _{b_{i}\in {\mathcal {B}}_{i}}\text { }\left( \beta \frac{ \sum \nolimits _{k\in T_{1}}q_{k}}{\sum \nolimits _{k\in T_{2}}b_{k}} (1-t)b_{i}+\alpha (1-\tau -b_{i})\right) ^{\mu }\text {, }i\in T_{2}\text {.} \end{aligned}$$

(E12)

In “Appendix C”, it was shown that (C6) and (C7) were necessary and sufficient for the existence of a tax ${\tilde{\epsilon }}$, $\epsilon =t,\tau$, for which the vector of strategic supplies $({{\tilde{\mathbf {q}}}}(\epsilon ); {{\tilde{\mathbf {b}}}}(\epsilon ))$ is a CNE of the game with taxation $\epsilon$, $\epsilon =t,\tau$.

Consider the ad valorem tax. As ${\tilde{q}}(t)={\tilde{b}}(t)=1$, and $\tilde{p }_{X}=1$, the CNE allocation is given by $({\tilde{x}}_{i}(t),{\tilde{y}} _{i}(t))=(0,1-t)$, for each $i\in T_{1}$, $({\tilde{x}}_{i}(t),{\tilde{y}} _{i}(t))=(1-t,0)$, for each $i\in T_{2}$, and $({\tilde{x}}_{2n+1}(t),{\tilde{y}} _{2n+1}(t))=(tn,tn)$. The corresponding payoffs of traders are given by ${\tilde{\pi }}_{i}(t)=\left( \beta (1-t)\right) ^{\mu }$, for $i\in T_{1}$, and ${\tilde{\pi }}_{i}(t)=\left( \alpha (1-t)\right) ^{\mu }$, for $i\in T_{2}$. Therefore, any Pareto optimal allocation solves the following problem:

$$\begin{aligned} (P^{t})\text { }\left| \begin{array}{c} \underset{0\leqslant t\leqslant 1}{\max }\text { }n\left( \beta (1-t)\right) ^{\mu }+n\left( \alpha (1-t)\right) ^{\mu }\text { } \\ \text {s.t. }\left( tn+tn\right) ^{\mu }\geqslant {\bar{u}}_{2n+1}\text {.} \end{array} \right. \end{aligned}$$

(E13)

Then, the optimal ad-valorem tax, namely ${\tilde{t}}$, is given by:

$$\begin{aligned} {\tilde{t}}=\frac{{\bar{u}}_{2n+1}^{\frac{1}{\mu }}}{2n}\text {.} \end{aligned}$$

(E14)

The CNE post-ad valorem tax allocation obtains as $({\tilde{x}}_{i}({\tilde{t}}), {\tilde{y}}_{i}({\tilde{t}}))=(0,1-\frac{1}{2n}{\bar{u}}_{2n+1}^{1/\mu })$, $i\in T_{1}$, $({\tilde{x}}_{i}({\tilde{t}}),{\tilde{y}}_{i}({\tilde{t}}))=(1-\frac{1}{2n} {\bar{u}}_{2n+1}^{1/\mu },0)$, $i\in T_{2}$, and $({\tilde{x}}_{2n+1}({\tilde{t}}), {\tilde{y}}_{2n+1}({\tilde{t}}))=(\frac{1}{2}{\bar{u}}_{2n+1}^{1/\mu },\frac{1}{2} {\bar{u}}_{2n+1}^{1/\mu })$, which corresponds to a competitive equilibrium allocation with ad valorem taxation. As a consequence, the allocation of traders and the overall transfer obtained by agent $2n+1$ are such that the overall-allocation is Pareto optimal.

Next, consider the endowment tax. As ${\tilde{q}}(\tau )={\tilde{b}}(\tau )=1-\tau$, and ${\tilde{p}}_{X}=1$, the CNE allocation is given by $(\tilde{x }_{i}(\tau ),{\tilde{y}}_{i}(\tau ))=(0,1-\tau )$, for each $i\in T_{1}$, and $({\tilde{x}}_{i}(\tau ),{\tilde{y}}_{i}(\tau ))=(1-\tau ,0)$, for each $i\in T_{2}$, and $({\tilde{x}}_{2n+1}(\tau ),{\tilde{y}}_{2n+1}(\tau ))=(n\tau ,n\tau )$. The corresponding payoffs of traders are given by ${\tilde{\pi }}_{i}(\tau )=\left( \beta (1-\tau )\right) ^{\mu }$, for each $i\in T_{1}$, and $\tilde{ \pi }_{i}(\tau )=\left( \alpha (1-\tau )\right) ^{\mu }$, for each $i\in T_{2}$. Therefore, any Pareto optimal allocation solves the following problem:

$$\begin{aligned} (P^{\tau })\text { }\left| \begin{array}{c} \underset{0\leqslant \tau \leqslant 1}{\max }\text { }n\left( \beta (1-\tau )\right) ^{\mu }+n\left( \alpha (1-\tau )\right) ^{\mu }\text { } \\ \text {s.t. }\left( n\tau +n\tau \right) ^{\mu }\geqslant {\bar{u}}_{2n+1}\text { .} \end{array} \right. \end{aligned}$$

(E15)

Then, the optimal endowment tax, namely ${\tilde{\tau }}$, must satisfy the equation:

$$\begin{aligned} 2n{\tilde{\tau }}={\bar{u}}_{2n+1}^{\frac{1}{\mu }}\text {.} \end{aligned}$$

(E16)

Then, we deduce:

$$\begin{aligned} {\tilde{\tau }}=\frac{{\bar{u}}_{2n+1}^{\frac{1}{\mu }}}{2n}\text {.} \end{aligned}$$

(E17)

Then, as ${\tilde{q}}({\tilde{\tau }})={\tilde{b}}({\tilde{\tau }})=1-\frac{{\bar{u}} _{2n+1}^{1/\mu }}{2n}$, the CNE post-endowment tax allocation is given by $( {\tilde{x}}_{i}({\tilde{\tau }}),{\tilde{y}}_{i}({\tilde{\tau }}))=(0,1-\frac{{\bar{u}} _{2n+1}^{1/\mu }}{2n})$, for each $i\in T_{1}$, $({\tilde{x}}_{i}({\tilde{\tau }} ),{\tilde{y}}_{i}({\tilde{\tau }}))=(1-\frac{{\bar{u}}_{2n+1}^{1/\mu }}{2n},0)$, for each $i\in T_{2}$, and $({\tilde{x}}_{2n+1}({\tilde{\tau }}),{\tilde{y}}_{2n+1}( {\tilde{\tau }}))=(\frac{1}{2}{\bar{u}}_{2n+1}^{1/\mu },\frac{1}{2}{\bar{u}} _{2n+1}^{1/\mu })$, which corresponds to a competitive equilibrium allocation with endowment taxation. As a consequence, the allocation of traders and the overall transfer obtained by agent $2n+1$ are such that the overall-allocation is Pareto-optimal.

Consider $\sigma =0$. Then, (2) may be written as $u_{i}(x_{i},y_{i})= \left( \min (x_{i},y_{i})\right) ^{\mu }$, for $i\in T_{1}\cup T_{2}\cup \{2n+1\}$. The problem of any trader may be written as:

$$\begin{aligned} \max _{q_{i}\in Q_{i}}\text { }\min \left( 1-\tau -q_{i},\frac{ \sum \nolimits _{k\in T_{2}}b_{k}}{\sum \nolimits _{k\in T_{1}}q_{k}} (1-t)q_{i}\right) ^{\mu }\text {, }i\in T_{1}; \end{aligned}$$

(E18)

$$\begin{aligned} \max _{b_{i}\in {\mathcal {B}}_{i}}\min \left( \frac{\sum \nolimits _{k\in T_{1}}q_{k}}{\sum \nolimits _{k\in T_{2}}b_{k}}(1-t)b_{i},1-\tau -b_{i}\right) ^{\mu }\text {, }i\in T_{2}. \end{aligned}$$

(E19)

From Proposition 3, and (C4) in “Appendix C”, we deduce the unique interior symmetric CNE for each taxation scheme:

$$\begin{aligned} {\tilde{q}}(t)={\tilde{b}}(t)=\frac{1}{2-t}\text {, with ad valorem taxation;} \end{aligned}$$

(E20)

$$\begin{aligned} {\tilde{q}}(\tau )={\tilde{b}}(\tau )=\frac{1-\tau }{2}\text {, with endowment taxation.} \end{aligned}$$

(E21)

First, consider the ad valorem tax. From (E19) and as ${\tilde{p}}_{X}=1$, the CNE allocation is given by $({\tilde{x}}_{i}(t),{\tilde{y}}_{i}(t))=(\frac{ 1-t}{2-t},\frac{1-t}{2-t})$, for each $i\in T_{1}$, $({\tilde{x}}_{i}(t), {\tilde{y}}_{i}(t))=(\frac{1-t}{2-t},\frac{1-t}{2-t})$, for each $i\in T_{2}$, and $({\tilde{x}}_{2n+1}(t),{\tilde{y}}_{2n+1}(t))=(nt\frac{1}{2-t},nt\frac{1}{ 2-t})$. The corresponding payoffs of traders are given by ${\tilde{\pi }} _{i}(t)=(\frac{1-t}{2-t})^{\mu }$, for each $i\in T_{1}$, and ${\tilde{\pi }} _{i}(t)=(\frac{1-t}{2-t})^{\mu }$, for each $i\in T_{2}$. Any Pareto-optimal allocation solves the following problem:

$$\begin{aligned} (P^{t})\text { }\left| \begin{array}{c} \underset{0\leqslant t\leqslant 1}{\max }\text { }n\left( \frac{1-t}{2-t} \right) ^{\mu }+n\left( \frac{1-t}{2-t}\right) ^{\mu }\text { } \\ \text {s.t. }\left( \min {(}nt\frac{1}{2-t},nt\frac{1}{2-t}{)} \right) ^{\mu }\geqslant {\bar{u}}_{2n+1}\text {.} \end{array} \right. \end{aligned}$$

(E22)

As we must have that $x_{2n+1}=y_{2n+1}$, the optimal ad valorem tax, namely ${\tilde{t}}$, must satisfy the equation:

$$\begin{aligned} \frac{1}{2-t}tn={\bar{u}}_{2n+1}^{\frac{1}{\mu }}\text {,} \end{aligned}$$

(E23)

from which we deduce:

$$\begin{aligned} {\tilde{t}}=\frac{2{\bar{u}}_{2n+1}^{\frac{1}{\mu }}}{n+{\bar{u}}_{2n+1}^{\frac{1}{ \mu }}}\text {.} \end{aligned}$$

(E24)

Then, as from (E20) and (E24), we have that ${\tilde{q}}({\tilde{t}})={\tilde{b}}( {\tilde{t}})=\frac{n+{\bar{u}}_{2n+1}^{1/\mu }}{2n}$, the CNE post-ad valorem tax allocation is given by $({\tilde{x}}_{i}({\tilde{t}}),{\tilde{y}}_{i}({\tilde{t}} ))=(\frac{n-{\bar{u}}_{2n+1}^{1/\mu }}{2n},\frac{n-{\bar{u}}_{2n+1}^{1/\mu }}{2n} )$, for $i\in T_{1}$, $({\tilde{x}}_{i}({\tilde{t}}),{\tilde{y}}_{i}({\tilde{t}}))=( \frac{n-{\bar{u}}_{2n+1}^{1/\mu }}{2n},\frac{n-{\bar{u}}_{2n+1}^{1/\mu }}{2n})$, for $i\in T_{2}$, and $({\tilde{x}}_{2n+1}({\tilde{t}}),{\tilde{y}}_{2n+1}(\tilde{t }))=({\bar{u}}_{2n+1}^{1/\mu },{\bar{u}}_{2n+1}^{1/\mu })$. Then, the post-tax allocation satisfies:

$$\begin{aligned} {\tilde{y}}_{i}({\tilde{t}})={\tilde{x}}_{i}({\tilde{t}})\text {, }i\in T_{1}\cup T_{2}\cup \{2n+1\}\text {.} \end{aligned}$$

(E25)

which coincides with the competitive equilibrium allocation with ad valorem taxation.¹⁷

Second, consider the endowment tax. From (E21) and as ${\tilde{p}}_{X}=1$, the CNE allocation is given by $({\tilde{x}}_{i}(\tau ),{\tilde{y}}_{i}(\tau ))=(\frac{1-\tau }{2},\frac{1-\tau }{2})$, for each $i\in T_{1}$, $({\tilde{x}} _{i}(\tau ),{\tilde{y}}_{i}(\tau ))=(\frac{1-\tau }{2},\frac{1-\tau }{2})$, for each $i\in T_{2}$, and $({\tilde{x}}_{2n+1}(\tau ),{\tilde{y}}_{2n+1}(\tau ))=(n\tau ,n\tau )$. The corresponding payoffs of traders are given by ${\tilde{\pi }}_{i}(\tau )=(\frac{1-\tau }{2})^{\mu }$, for each $i\in T_{1}$, and ${\tilde{\pi }}_{i}(\tau )=(\frac{1-\tau }{2})^{\mu }$, for each $i\in T_{2}$. Any Pareto-optimal allocation solves the following problem:

$$\begin{aligned} (P^{\tau })\text { }\left| \begin{array}{c} \underset{0\leqslant \tau \leqslant 1}{\max }\text { }n\left( \frac{1-\tau }{2 }\right) ^{\mu }+n\left( \frac{1-\tau }{2}\right) ^{\mu }\text { } \\ \text {s.t. }\left( \min \left( n\tau ,n\tau \right) \right) ^{\mu }\geqslant {\bar{u}}_{2n+1}\text {.} \end{array} \right. \end{aligned}$$

(E26)

As we must have that $x_{2n+1}=y_{2n+1}$, the optimal endowment tax, namely ${\tilde{\tau }}$, must satisfy the equation:

$$\begin{aligned} n\tau ={\bar{u}}_{2n+1}^{\frac{1}{\mu }}.\ \end{aligned}$$

(E27)

Then, we deduce:

$$\begin{aligned} {\tilde{\tau }}=\frac{{\bar{u}}_{2n+1}^{\frac{1}{\mu }}}{n}\text {.} \end{aligned}$$

(E28)

Then, as from (E21) and (E28), we have that ${\tilde{q}}({\tilde{\tau }})=\tilde{b }({\tilde{\tau }})=\frac{n-{\bar{u}}_{2n+1}^{1/\mu }}{2n}$, the post-endowment tax allocation is given by $({\tilde{x}}_{i}({\tilde{\tau }}),{\tilde{y}}_{i}( {\tilde{\tau }}))=(\frac{n-{\bar{u}}_{2n+1}^{1/\mu }}{2n},\frac{n-{\bar{u}} _{2n+1}^{1/\mu }}{2n})$, for $i\in T_{1}$, $({\tilde{x}}_{i}({\tilde{\tau }}), {\tilde{y}}_{i}({\tilde{\tau }}))=(\frac{n-{\bar{u}}_{2n+1}^{1/\mu }}{2n},\frac{n- {\bar{u}}_{2n+1}^{1/\mu }}{2n})$, for $i\in T_{2}$, and $({\tilde{x}}_{2n+1}( {\tilde{\tau }}),{\tilde{y}}_{2n+1}({\tilde{\tau }}))=({\bar{u}}_{2n+1}^{1/\mu },\bar{u }_{2n+1}^{1/\mu })$. Then, $\frac{{\tilde{y}}_{i}({\tilde{\tau }})}{{\tilde{x}}_{i}( {\tilde{\tau }})}=1$, for $i\in T_{1}$, $\frac{{\tilde{y}}_{i}({\tilde{\tau }})}{ {\tilde{x}}_{i}({\tilde{\tau }})}=1$, for $i\in T_{2}$, and $\frac{{\tilde{y}} _{2n+1}({\tilde{\tau }})}{{\tilde{x}}_{2n+1}({\tilde{\tau }})}=1$. Then, the overall-allocation corresponding to the interior strategic equilibrium of this game corresponds to the competitive equilibrium allocation which results from the interactions between the only traders. As a consequence, the overall transfer obtained by the agent $2n+1$ is such that the overall-allocation is Pareto-optimal. $\blacksquare$