We consider the following simple model:

We have one country and the rest of the world. In each country there is a representative consumer with the following utility function:

where:

c_p is the consumption of good p;

c_c is the consumption of good c;

e is the emission level;

q is the parameter who indicate the elasticity of preferences for pollution intensive good, with .

We suppose that p is the pollution intensive good and its production is associated with emission. c is the "clean" good with no level of emission.

e = x_p

where x_p is output of good p.

The production possibility frontier in any given country is linear, and is given by:

x_p + x_c = P

where P is the total capital stock (including human capital) of the country. For simplicity we assume the frontier have the same slope of 1 in each country. (In this case we neutralise all comparative advantages considerations.)

We assume too that in each country there exists a system of taxes and transfers which allows the government to implement the first-best allocation.

First, we consider the solution in absence of trade. In this case we have c_p = x_p and c_c = x_c .

The solution is given by the following problem:

The first order condition is given by:

Its easy to see that the solution of this problem there exists.

It is clear from the previous equation that:

That means: emission levels monotonically increase with output.

Let us now consider what happens when that country is able to trade with the rest of the world. Let p_p the world price for the pollution intensive good and p_c = 1 the price of "clean" good, and R the revenue in the country.

Then each individual country maximises U subject to some restrictions.

We solve this problem in two steps.

Conditional on R, the problem solutions are:

c_c = (1 - q ) R

Substituting in (2) we have the problem:

Solving this problem we have the following solution:

a) If P ³ (p_p –1) / b then x_p = 0. The country fully specialises in "clean" good production and imports the pollution intensive good.

b) If . The country produce both goods. We observe that , so the income rises, the emission level fall. This is because in that range it pays to give up part of extra income to produce less pollution intensive goods.

c) If then x_p = P . The country fully specialises in p good. It will produce more of it as it gets richer, so that > 0.