To define the composition effect it is convenient to work with x/y ratios. Let X = x/y denote the relative supply of X. Solving (2.9) and (2.10) for x and y and dividing yields
where к = K/L is the economy’s capital labor ratio. Note that X is increasing in к and p ; and therefore increasing in p and decreasing in t. 6 We will refer to any change in the economy that alters X (k,p ) as creating a composition effect. Using (2.15) and (2.16), we can now rewrite our expression for pollution (2.3) as:
where “A” denotes “percent change”, jy = y/(px+y) is the share of y in the value of gross output, Sa9 is the elasticity of a with respect to 9, and Z = 1a(9 )x/z is the ratio of abated pollution to actual pollution. The first term is the scale effect. Holding constant pollution abatement techniques and the mix of goods produced, an increase in the scale of economic activity will raise pollution. Next is the composition effect. Holding scale and techniques constant, a shift in the composition of production towards more pollution intensive goods will raise pollution. Finally, the technique effect: holding the scale and composition of economic activity constant, pollution levels will fall in response to an increase in the intensity of pollution abatement.
According to (2.18), the observed variation in our pollution data arises from variation in the scale, composition and techniques of economic activity across countries and over time. We will adopt a quantity index of output to proxy for scale in our empirical work. To relate the composition and technique effects to observable variables as well, we differentiate (2.16) and (2.6) to obtain expressions for X and 9 which we then substitute into (2.18). This yields
where S^ denotes the elasticity of i with respect to j, and at = t1[1-a(9)]/p . Since we do not observe policy directly in our data set, we must replace t in (2.19) with its determinants. From (2.13) and (2.14) we can write t as:
The pollution tax depends on population size, real per capita income, and consumer tastes. Now substitute (2.20) into (2.19), to obtain: