# G Derivation of the multiplier

Back to chapter 6.1.2: “The adjustment process to equilibrium and the multiplier”

We can now derive the multiplier from the simple income-expenditure model seen in chapter 6.1.2 by assuming an initial demand surplus, as shown in figure 6.7.

The initial demand surplus, \(Y^N - Y\), leads to another increase in demand equal to the product of the initial demand surplus (= initial income adjustment) and the marginal propensity to consume:

\[\Delta_1 Y^N = c_Y (Y^N - Y)\]

Since in the second round supply and income increases by this amount, in the second round demand increases by:

\[\Delta_2 Y^N = c_Y \left(c_Y \left(Y^N - Y \right) \right) = c_Y^2 \left(Y^N - Y \right)\]

In each round, the increase in demand is given by the product of the initial demand surplus and the marginal propensity to consume, which is raised to the power of the number of the current round. In the \(n^{th}\) round, the increase in demand becomes:

\[\Delta_n Y^N = c_Y^n \left(Y^N - Y \right)\]

The total increase in demand reached at the end of the adjustment to equilibrium is given by the sum of the initial demand surplus (\(Y^N - Y\)) and all increases in each round of adjustment:

\[\Delta Y^N = \left(Y^N - Y\right) + c_Y \left(Y^N - Y \right) + c_Y^2 \left(Y^N - Y \right) + ... = \left(Y^N - Y\right) + \sum_{n = 1}^{\infty} c_Y^n \left(Y^N - Y \right)\]

By factoring out the initial demand surplus, this equation can be transformed into:

\[\begin{equation} \Delta Y^N = \left(1 + c_Y+ c_Y^2 + \dots \right) \left(Y^N - Y\right) = \left(1 + \sum_{n = 1}^{\infty} c_Y^n\right) \left(Y^N - Y\right) \tag{G.1} \end{equation}\]

For simplicity, we denote the term \(\left(1 + c_Y + c_Y^2 + ... \right)\) or \(\left( 1 + \sum_{n = 1}^{\infty} c_Y^n \right)\) as \(\mu\). We can write equation (G.1) as:

\[\Delta Y^N = \mu \left(Y^N - Y\right)\]

We see that \(\mu\) is exactly the factor by which the initial demand surplus was increased so that the equilibrium condition is fulfilled again,

\(\mu\) is therefore our multiplier!

The term \(\mu = \left(1 + c_Y + c_Y^2 + ... \right) = \left( 1 + \sum_{n = 1}^{\infty} c_Y^n \right)\) is the geometric series. The simplified representation of this geometric series looks like this:

\[\mu = \frac{1}{1 - c_Y}\]