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# Annex 2: Deriving model equation

In Section 6, we describe the modelling of the expenditure on public services. Here we provide further detail on the derivation of this equation.

First we assume that nominal expenditure (Et) on a given output equals the quantity of inputs (It) multiplied by the nominal price of a unit of input (Wt). The subscript t denotes the time period - which is a fiscal year in the model. Assuming a homogenous input, we can write this as:

EtWt X It

Now assume that there is a simple production function, in which the quantity of outputs (Qt) is the product of the quantity of inputs and their productivity (At). Then:

QtAt X It

Thus nominal expenditure is a function of nominal input prices, the quantity of outputs supplied and the productivity of the production process:

Converting the form of this equation into growth rates, we can see that:

Now assume that the growth in quantity demanded for outputs is a function of a demographic growth factor (dt) and non-demographic demand factors (pt):

Assume that nominal input price growth equals inflation (πt), plus a factor for real growth (wt) such as for wages:

We define public sector productivity growth rate as follows:

Putting this together we can see that:

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