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The equilibrium output equation, $Y = k(c_0 + I)$, expresses the relationship where equilibrium output (Y) is the product of total autonomous demand $(c_0 + I)$ and the multiplier factor $(k)$. In the simplified model with only consumption and investment, the multiplier is defined as $k = \frac{1}{1-c_1}$. While the specific formula for $k$ varies in more complex models, it consistently represents the factor by which output changes in response to a change in autonomous demand.

Equilibrium Output Equation using the Multiplier (k)

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.

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: $\Delta Y = k \Delta I$.

Derivation of the Change in Output from an Investment Shock

In a simplified economic model, the equilibrium output (Y) is determined by the equation Y = k(c₀ + I), where 'k' is a multiplier with a value greater than 1, 'c₀' represents spending that does not depend on income, and 'I' represents investment spending. If businesses suddenly become more pessimistic about the future and reduce their investment spending (I), while 'k' and 'c₀' remain unchanged, what will be the resulting effect on the equilibrium output (Y)?

An economy is described by the equilibrium output equation Y = k(c₀ + I), where Y is the equilibrium output, k is the spending multiplier, c₀ is spending that does not depend on income, and I is investment spending. Using the data provided in the case study, calculate the value of the equilibrium output (Y).

Calculating Equilibrium Output

In the equation for equilibrium output, $Y = k(c_0 + I)$, analyze the distinct roles of the two components on the right-hand side: total autonomous demand $(c_0 + I)$ and the multiplier $(k)$. How does each component contribute to determining the final level of output $(Y)$?

Deconstructing the Equilibrium Output Equation

In the macroeconomic model where equilibrium output is represented by the equation `Y = k(c₀ + I)`, match each component of the equation to its correct economic description.

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