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An Analysis of Benefit Flows in New Zealand Using a Social Accounting Framework WP 13/01

Appendix E. The Matrix Multiplier

Consider the simple case of a two-state social accounting model, where the equilibrium stock, Equation. , is given by:

Equation.

and:

Equation.

The effect of a change in the vector of 'inflows' is thus straightforward, and is measured by the appropriate elements of the matrix Equation. : in matrix terms Equation. . Consider the effect on, for example, Equation. , of a change in the forward coefficient, Equation. , where:

Equation.

Differentiation gives:

Equation.

And using:

Equation.

Equation.

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:

Equation.

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