A change in investment, denoted as ΔI, is a common example used to illustrate a shock to autonomous demand. This initial change in spending initiates the multiplier process, leading to a larger, magnified change in the equilibrium level of output.

Change in Investment (ΔI) as an Autonomous Demand Shock

The change in equilibrium output (ΔY) resulting from a change in autonomous spending, such as a change in investment (ΔI), is calculated by multiplying the change in spending by the multiplier (k). The formula is expressed as: $ΔY = k \times ΔI$.

Formula for Change in Output from an Autonomous Spending Shock

A change in autonomous demand, which consists of autonomous consumption $(c_0)$ and investment $(I)$, leads to a multiplied effect on the equilibrium output $(Y)$. Specifically, any change in autonomous demand will cause the equilibrium output to change by an amount equal to the multiplier $(k)$ times the initial change.

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The equilibrium output equation can be written compactly as $Y = k(c_0 + I)$, where $k$ is the general symbol for the multiplier. In the simplified model that includes only consumption and investment, the multiplier is calculated with the specific formula $k = \frac{1}{1-c_1}$. However, in more complex models that include government and net exports, the formula for $k$ changes, though it always represents the total effect on output from an exogenous demand shock.

Equilibrium Output Equation using the Multiplier (k)

To cite the ebook, use The CORE Econ Team 2023 The Economy 2.0: Macroeconomics Open access e-text https://core-econ.org/the-economy/.

To cite a unit, use The CORE Econ Team 2023 The Economy 2.0: Macroeconomics Open access e-text https://core-econ.org/the-economy/ Unit [unit number] (url).

CORE Econ - The Economy 2.0: Macroeconomics

Multiplier Effect of Autonomous Demand on Equilibrium Output

A numerical example can illustrate how to calculate the value of the multiplier (k). Given a marginal propensity to consume ($c_1$) of 0.6, the multiplier is calculated using the formula $k = \frac{1}{1-c_1}$. The calculation is as follows: $k = \frac{1}{1-0.6} = \frac{1}{0.4} = 2.5$.

Example Calculation of the Multiplier (k)

The change in equilibrium output ($\Delta Y$) resulting from a change in investment ($\Delta I$) is derived by calculating the difference between the new and original equilibrium output levels. Starting with the equilibrium equation $Y = k(c_0 + I)$, the change is expressed as $\Delta Y = k(c_0 + I + \Delta I) - k(c_0 + I)$. By simplifying this expression, the terms $k(c_0+I)$ cancel out, leaving the final relationship: $\Delta Y = k \Delta I$.

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