### The model

Assume that two types of countries exist. One of the countries is an advanced industrialized or developed country which emits CO_{2}. The other country is a developing country that does not emit CO_{2}. There are *n* pairs of a developed and a developing country (*d*_{
j
}, *u*_{
j
}) between which the emission right is traded, where *d*_{
j
} and *u*_{
j
} are the *j* th developed and developing country respectively, comprising the *j* th block of emission trading.

Each country has the same utility function {U}_{j}^{i}:

\begin{array}{l}{U}_{j}^{i}\equiv {c}_{j}^{i}-\mathrm{\Psi}\left({e}_{1}^{d},\cdot \cdot \cdot ,{e}_{j}^{d},\cdot \cdot \cdot ,{e}_{n}^{d},{e}_{1}^{u},\cdot \cdot \cdot ,{e}_{j}^{u},\cdot \cdot \cdot ,{e}_{n}^{u}\right)\\ \phantom{\rule{1.5em}{0ex}}\equiv {c}_{j}^{i}-\mathrm{\Psi}\left(\overrightarrow{{e}^{d}},\overrightarrow{{e}^{u}}\right),\end{array}

(1)

where {c}_{j}^{i} and {e}_{j}^{i} denote the consumption and CO_{2} emission level of the *j* th country that belongs to type *i* (*i = d, u*), respectively. Ψ represents the disutility from the CO_{2} emission via the production process of a consumption good. We assume that Ψ is linear homogenous^{b}, quasi concave and *symmetric* in the following sense. That is,

\mathrm{\Psi}\left(\cdot \cdot ,{e}_{l}^{k},\cdot \cdot \cdot ,{e}_{l\text{'}}^{k\text{'}}\cdot \cdot \right)\equiv \mathrm{\Psi}\left(\cdot \cdot ,{e}_{l\text{'}}^{k\text{'}},\cdot \cdot \cdot ,{e}_{l}^{k}\cdot \cdot \right),\forall k,k\text{'},l,l\text{'}.

(2)

This symmetric assumption implies that the disutility derived from the emission does not depend on where it is emitted. It is a plausible assumption when we consider the diffusion speed of CO_{2} in the atmosphere.

The semi-reduced form production function, which represents the relationship between the consumption *c* and the adjoined emission *e*, is *c* = *α F (e)*, *α* is some positive constant.

F\text{'}>0,F\text{'}\text{'}<0,\left(e<\overline{e}\right).\phantom{\rule{0.5em}{0ex}}F\text{'}=F\text{'}\text{'}=0,\left(e\ge \overline{e}\right).

(3)

That is, *F*(∙) means that if a factory emits CO_{2} by weight *e*, it produces *c* units of goods. \overline{e} is the maximal effective CO_{2} emission in the sense that the emission exceeding this limit bears no additional consumption goods.

We assume that only developed countries possess the production technology. No developing country can access such an opportunity until the bilateral emission trading is settled or some type of proportional carbon tax is levied.

### The complete *laissez-faire* situation

Assume that, in the *laissez-faire* situation, the decisions of the production sectors within a country are separated from the consumer sector; they do not consider the nuisance incurred by their emission. Hence, they produce goods amounting to the maximum level F\left(\overline{e}\right). In such a case, the unit price of carbon is zero.

As noted by Lovins and Cohen (2011), ch.8 and evidenced by history, this fact implies that without government intervention, the carbon market will surely collapse and the unit price of carbon will fall and become negligible.

### Unilateral proportional carbon tax: implementation of the Nash equilibrium

#### Formatting a unilateral proportional carbon tax

Moving from the complete *laissez-faire* situation, each country is separately incentivized to levy a carbon tax to suppress its emissions for its own wellbeing, and thus, a unilateral proportional carbon tax is formatted by the solution of the following optimization problem:

\begin{array}{l}{max}_{{e}_{l}^{k}}\left[\mathit{\alpha F}\left({e}_{l}^{k}\right)-\mathrm{\Psi}\left({e}_{l}^{k},\overrightarrow{{e}_{l}^{k*}}\right)\right],\\ \phantom{\rule{3.5em}{0ex}}\overrightarrow{{e}_{l}^{k*}}\equiv \left({e}_{l}^{k\text{'}*},{\overrightarrow{e}}_{2}^{*},\cdot \cdot \cdot ,{\overrightarrow{e}}_{l-1}^{*},{\overrightarrow{e}}_{l+1}^{*},\cdot \cdot \cdot ,{\overrightarrow{e}}_{\mathit{n}}^{*}\right),\\ \phantom{\rule{3.5em}{0ex}}{\overrightarrow{e}}_{\mathit{j}}^{*}\equiv \left({e}_{j}^{d*},{e}_{j}^{u*}\right).\end{array}

(4)

Since the solution of this problem satisfies

F\text{'}\left({e}_{l}^{k*}\right)=\frac{1}{\alpha}\cdot \frac{\partial \psi}{\partial {c}_{1}^{d}}{|}_{{\u03f5}_{l}^{k}=\frac{1}{n}},

(5)

we find that the allocation implemented by a unilateral carbon tax \frac{\partial \psi}{\partial {c}_{1}^{l}}{|}_{{\u03f5}_{j}^{l*}=\frac{1}{n}} is identical to that implemented by the Nash equilibrium of an international game of CO_{2} emission. In this sense, a unilateral proportional carbon tax belongs to a kind of international *laissez-faire* scheme and does not require any cooperative behavior between countries. However, it is evident that the resource allocation is improved compared with the complete *laissez-faire* situation, because the equilibrium emissions are determined by weaving the social disutility from excess emissions.

### Bilateral emissions trading

#### Formatting the bilateral emission trading scheme

Bilateral emissions trading between the developed and the developing country in the *j* th pair is defined by the following optimization problem. That is,

\begin{array}{ll}\hfill {max}_{{e}_{j}^{d},{e}_{j}^{u},P}\left\{\phantom{\rule{0.25em}{0ex}}\left[\mathit{\alpha F}\left({e}_{j}^{d}\right)\right.\right.& +\mathit{\alpha F}\left({e}_{j}^{u}\right)-P\\ -\mathrm{\Psi}\left(\right)close="]">\left({e}_{j}^{d},{e}_{j}^{u},{e}_{1}^{d*},{e}_{1}^{u*}\cdot \cdot \cdot {e}_{n}^{d*},{e}_{n}^{u*}\right)\end{array}+\lambda \left(\right)close="\}">\left[P-\mathrm{\Psi}\left({e}_{j}^{u},{e}_{j}^{d},{e}_{1}^{d*},{e}_{1}^{u*}\cdot \cdot \cdot {e}_{n}^{d*},{e}_{n}^{u*}\right)-\overline{U}\right]\phantom{\rule{0.25em}{0ex}}& ,\n

(6)

where *P* is the total payment for the carbon emissions. * denotes the optimal contract emissions of the other pairs. *λ* is the Lagrangean multiplier of this problem. \overline{U} denotes the reservation utility of the corresponding developing country. The above constrained maximization problem means that an advanced country maximizes its economic welfare guaranteeing the minimal utility level of the counterpart developing country at the level of \overline{U}. The first term of (6) is the developed-country’s utility derived from this emission trading, and the term within the square brackets of the second term is the net welfare gain of the developing country.

Let us denote

{\u03f5}_{j}^{l}\phantom{\rule{0.2em}{0ex}}\equiv \frac{{e}_{j}^{l}}{{\displaystyle \sum _{l,k=1}^{n}{e}_{k}^{l}}}\equiv \frac{{e}_{j}^{l}}{E}.

(7)

By using {\u03f5}_{j}^{l} and *E*, Ψ can be transformed into

\mathrm{\Psi}=\psi \left(\overrightarrow{{\u03f5}^{d},}\overrightarrow{{\u03f5}^{u}}\right)E\equiv \psi \left(\overrightarrow{\u03f5}\right)E.

(8)

Then by the symmetry of Ψ, the optimality condition for the above contract problem under perfect information and symmetric equilibrium can be represented as

\begin{array}{l}F\text{'}\left({e}_{l}^{k}*\right)=\frac{2}{\alpha}\cdot \frac{\partial \psi}{\partial {c}_{1}^{l}}{|}_{{\u03f5}_{j}^{l*}=\frac{1}{n}}\equiv \frac{\theta \left(2\right)}{\alpha}.\forall j,l.\end{array}

(9)

\begin{array}{l}{P}^{*}=\overline{U}+\Psi \left(\overrightarrow{{e}^{*}}\right)=\overline{U}+\theta \left(2\right)\cdot n{e}^{*}\\ \phantom{\rule{8.5em}{0ex}}=\overline{U}+\theta \left(2\right)\cdot {e}^{*}+\left[n-1\right]\theta \left(2\right)\cdot {e}^{*}\equiv \overline{U}\\ \phantom{\rule{9.5em}{0ex}}+T\left(2\right)+\left[n-1\right]\theta \left(2\right)\cdot {e}^{*}.\end{array}

(10)

\theta \left(2\right)\equiv 2\frac{\partial \psi}{\partial {c}_{1}^{l}}{|}_{{\u03f5}_{j}^{l*}=\frac{1}{n}}

is the unit carbon price in this trading scheme.

This scheme can be regarded as the ideal offset-trade scheme. The carbon price should be determined as the sum of the marginal disutility from the global warming of two countries with partnership. The transfer from a developed country is, aside from the direct payment for carbon emissions *T*(2), the sum of the reservation utility of the developing country in concern \overline{U} and the pecuniary nuisance from emissions of other developed countries as a whole [*n* − 1]*θ*(2)⋅ *e**.

We must note that such an ideal system does not require setting the baseline of the project, which is unavoidable in the current system. Since the procedures for such setting is very much complicated and lapses much time, it seems to be desirable to transform the existing offset trading scheme to that based on the formulae (9) and (10).

### Universal proportional carbon tax

#### Formatting a universal proportional carbon tax scheme

A universal proportional carbon tax is formatted by the following optimization problem:

{max}_{\overrightarrow{\u03f5}}\left[\mathit{\alpha F}\left({e}_{1}^{d}\right)-\mathrm{\Psi}\left(\overrightarrow{\u03f5}\right)E+{\displaystyle \sum _{\left(l,k\right)\ne \left(1,d\right)}{\lambda}_{l}^{k}}\left[\mathit{\alpha F}\left({e}_{l}^{k}\right)-\mathrm{\Psi}\left(\overrightarrow{\u03f5}\right)E-\overline{{U}^{m}}\right]\right].

(11)

The above problem means that an advanced country maximizes its economic utility presuming it provides the rest of the world with the minimal utility \overline{U}. It is clear that the attained allocation under such a scheme is Pareto efficient by definition. By the symmetry of the problem, it is also clear that every Lagrangean multiplier under optimal planning takes the value unity^{c}. Thus, we obtain the following formula concerning the optimal emission:

F\text{'}\left({e}^{*}\right)=\frac{n}{\alpha}\frac{\partial \psi}{\partial {c}_{1}^{d}}{|}_{{\u03f5}_{j}^{l*}=\frac{1}{n}}.

(12)

This is the modified Samuelson (1954) rule concerning the optimal public good (bad) provision: The marginal benefit accrued from the country’s emissions *αF* ' (*e**) should be equalized to the sum of the marginal disutility diffused all over the world, n\frac{\partial \psi}{\partial {c}_{1}^{d}}{|}_{{\u03f5}_{j}^{l*}=\frac{1}{n}}. The right-hand side of (12) is the optimal tax rate common to all constituents.

### Welfare ordering for various emission suppressing measures

Although it is clear that a proportional carbon tax, with the rate expressed by (12), is the first-best policy, it is important to understand the manner in which the other two measures are ordered in terms of Pareto efficiency. We can deal with this problem by using the symmetry and linear homogeneity of Ψ.

The utility of each country {U}_{l}^{k} can be written as

\begin{array}{l}{U}_{l}^{k}\left(j\right)=\mathit{\alpha F}\left({e}^{*}\left(j\right)\right)-\frac{\partial \psi}{\partial {c}_{1}^{l}}{|}_{{\u03f5}_{l}^{k*}={\frac{1}{n}}^{*}}\cdot {E}^{*}\left(j\right)\\ \phantom{\rule{3em}{0ex}}=\mathit{\alpha F}\left({e}^{*}\left(j\right)\right)-n\frac{\partial \psi}{\partial {c}_{1}^{l}}{|}_{{\u03f5}_{l}^{k*}={\frac{1}{n}}^{*}}\cdot {e}^{*}\left(j\right).\end{array}

(13)

Employing the envelop theorem,

\begin{array}{l}\frac{d{U}_{l}^{k}\left(j\right)}{\mathit{dj}}=\frac{\partial \psi}{\partial {c}_{1}^{l}}{|}_{{\u03f5}_{l}^{k}=\frac{1}{n}}\cdot {e}^{*}\left(j\right)>0,\left(1\le j<n\right)\\ \frac{d{U}_{l}^{k}\left(j\right)}{\mathit{dj}}=0,\left(j=n\right)\\ \frac{d{U}_{l}^{k}\left(j\right)}{\mathit{dj}}=\frac{\partial \psi}{\partial {c}_{1}^{l}}{|}_{{\u03f5}_{l}^{k}=\frac{1}{n}}\cdot {e}^{*}\left(j\right)<0,\left(n<j\right)\end{array}

(14)

holds. Thus, it is clear from the above equations that the bilateral emission trading scheme Pareto-dominates a unilateral carbon tax scheme, whereby a country can set a proportional carbon tax rate at its discretion.

This fact implies that although a proportional carbon tax possibly attains the first-best allocation, emission trading is the second-best measure, unless all countries concur about the seriousness of global warming, in which case a much higher carbon tax rate than that in the unilateral case can be adopted. In addition, since *n* is likely to far exceed two, the suppression effect of emission trading is estimated to be rather restrictive from the view point of the first-best allocation.

### Welfare analysis of emission-saving technological progress

Consider the effect of emission-saving technological progress to the world economy as a whole. This progress is expressed by an increase in *α* in this model. Before proceeding to the general equilibrium analysis, we must note that every trading pair increases emissions in conjunction with technological progress. Although it seems to be counterintuitive, if we note the fact that technological progress makes the imputed price of CO_{2} cheaper as shown by the right-hand sides of (5), (9), and (12), it is natural that emission-saving technological progress conversely heightens the accumulation of CO_{2}.

With this precaution in mind, we shall proceed with the general equilibrium analysis, into which the mutual negative externalities between trading pairs are woven. Then, from (13) and the envelop theorem, we obtain

\begin{array}{l}\frac{d{U}_{j}^{d}}{\mathit{d\alpha}}=F\left({e}^{*}\right)-\left[n-j\right]\frac{\partial \psi}{\partial {e}_{1}^{d}}{|}_{{\u03f5}_{l}^{k}=\frac{1}{n}}\frac{d{e}^{*}}{\mathit{d\alpha}}>\frac{1}{\alpha}\left[\mathit{\alpha F}\left({e}^{*}\right)-\mathit{\eta \Psi}\right],\\ \phantom{\rule{3.5em}{0ex}}\eta \equiv \frac{\raisebox{1ex}{$d{e}^{*}$}\!\left/ \!\raisebox{-1ex}{${e}^{*}$}\right.}{\raisebox{1ex}{$\mathit{\text{d\alpha}}$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.},\forall j,\phantom{\rule{0.5em}{0ex}}\frac{d{U}_{n}^{d}}{\mathit{d\alpha}}>0,\end{array}

(15)

where *η* is the elasticity of the emission volume to the unit of the technological progress. Since *αF*(*e**) − *Ψ* > 0, if *η* is small enough and the increase in the emission generated by the technological progress is not so serious^{d}, the advance in the emission-saving technology improves worldwide utility, although this advance also increases the total amount of CO_{2} emissions. This fact is underscored in that when we extend the scope of analysis to dynamic and intergenerational emission allocation (e.g., Otaki 2013), we may have to modify the obtained result, because the acceleration in emissions diffuses the negative externality to future generations.

In other words, although the emission-saving technological progress lowers the imputed price of CO_{2} and stimulates the current generation’s consumption, such current prosperity may conversely worsen the descendants’ utility via the resulting massive emissions. However, such a dynamic prospect is beyond of the scope of this article, and itself requires solving the simultaneous optimization concerning intertemporal and international emission problems.

### On income distribution between countries: the possibility of nonlinear pricing

Thus far, this article has assumed that a developed country directly invests in the corresponding developing country and that it receives revenues after deducting the carbon tax. Thus,

\mathit{\alpha F}\left({e}^{*}\left(j\right)\right)-j\cdot \frac{\partial \psi}{\partial {e}_{1}^{l}}{|}_{{\u03f5}_{l}^{k}=\frac{1}{n}}\cdot {e}^{*}\left(j\right)\equiv \mathit{\alpha F}\left({e}^{*}\right)-\tau \left(j\right)\cdot {e}^{*}\left(j\right),

(16)

where *τ*_{
j
} is the carbon tax rate, which is identical to the unit carbon price in emission trading. Hence, the developing country obtains tax revenues*R* (*j*), which amount to

R\left(j\right)\equiv \tau \left(j\right)\cdot {e}^{*}\left(j\right),

(17)

from the investing developed country.

Since every constraint concerning the joint utility from such a trading scheme binds whenever planning is optimal (see Uzawa 1958), the net surplus from the trading in terms of consumption becomes \overline{{U}^{m}}^{e}. Although we have not yet analyzed the possibility of additional lump-sum transfer from the investing developed country to its counterpart developing country (or the transfer inverted direction, which is possible if the tax payment is too heavy for the developed country), one cannot envisage a universal proportional carbon tax without some fair division of the surplus earned by direct investment through this lump-sum transfer, specifically because the standard of living of the remitting /recipient country depends decisively on its share of this surplus (Uzawa 2003).

Hereafter this article analyzes both directions of the transfer and clarifies how the direction affects the *effective* tax rate. First, consider the transfer from the developed country to the developing country. This is an application of nonlinear pricing, which appears in basic microeconomics (e.g., see Tirole 1988). Let the sum of the transfer be *s*. Then, the total payment of a developed country to her counterpart *T* becomes

T\equiv \tau \left(j\right)e\left(j\right)+s=\left[\tau \left(j\right)+\frac{s}{e\left(j\right)}\right]\cdot e\left(j\right).

(18)

The term within the square brackets is the *effective* tax rate, which is illustrated by Figure 1. Thus, the effective tax rate is digressive although such a transfer enriches the developing country. This is owing to the economy of scale from the de-facto massive purchase of the right of emission.

Figure 2 illustrates the locus of the *effective* tax rate for the inverted transfer from the developing country to the developed country (*s* < 0). It is apparent that the *effective* tax rate becomes progressive despite impoverishing the developing country. The progressive *effective* tax rate owes its existence entirely to the diseconomy of scale concerning emissions. However, we must still note that whether the *effective* tax rate is progressive or otherwise does not affect the efficacy of emission allocation. Moreover, this discussion exemplifies that the progressive tax rate is not necessarily advantageous to the developing country.