The proof that environmental disruption is an increasing function of foreign debt will proceed as follows. We will represent the optimal values of emissions and consumption as functions of the shadow price A. Then the canonical equations, (8.2) and (8.4), will be linearized in the equilibrium and the initial results will be used to eliminate e and c from this system, such that there are only two differential equations in two variables. The shape of the saddle path will be examined. Then environmental pollution, z, will be expressed as a function of A and from this we can infer the desired relationship between z and v.
Equation (8.25) can be rewritten in matrix notation:
λϑ = su′ a (8.A1)
where ϑ is a column vector whose ith component is, a is a vector of the same dimension containing the ais as its components. Moreover u′ is used to represent un+1 for notational brevity and convenience. Total differentiation yields
ϑdλ + λΘde = -(sa)u″ A de.
e is the vector of emission rates, ei. Θ is matrix with elements equalling the second derivatives of gf with respect to e. They are all negative and the matrix is negative definite since all fis are strictly concave in the eis and the externalities, represented by the gi functions, are negative. A is a matrix whose ith-row jth-column element is aiaj. It is symmetric and its rank is 1. Finally, u″. has been used to replace un+1,n+1The equation can be solved for de:
de = -(Θ + s2u″ A)-1 ϑdλ. (8.A2)
The matrix in the right-hand side of this equation is symmetric. It consists of negative elements and its diagonal elements are dominant (i.e. their absolute values exceed the absolute values of the off-diagonal elements in the same row or column). Thus this matrix is negative definite.
In the same fashion, we can derive
dc = U-1pdλ, (8.A3)
where U is the matrix of second derivatives of the utility function, which is negative definite, and p is the vector of goods prices.
These results can be used to linearize the state equation (8.2) in the equilibrium and the dynamics of the optimal path are characterized by, (8.A4)
where the superscript T denotes the transpose of a matrix or a vector. Due to the definiteness properties of the matrices, the first-row second-column element of the matrix on the right-hand side of this equation is positive. The system is stable in the saddle- point sense and saddle path has a negative slope in the (ν,λ) phase space. This is shown in Figure 8A.1.