Appendix E. The Matrix Multiplier
Consider the simple case of a two-state social accounting model, where the equilibrium stock, 
, is given by:
and:
The effect of a change in the vector of 'inflows' is thus straightforward, and is measured by the appropriate elements of the matrix 
: in matrix terms 
. Consider the effect on, for example, 
, of a change in the forward coefficient, 
, where:
Differentiation gives:
And using:
Given the definition, , the proportional change in the equilibrium stock of individuals in state 1 arising from the combination of a change in inflows and a change in the proportion remaining in the state from one period to the next is given by:
