DERIVATION OF THE INVESTMENT MULTIPLIER

KEYNESIAN THEORY AND POLICY AT A GLANCE DERIVATION OF THE INVESTMENT MULTIPLIER

The notion of an investment multiplier is most relevant when (1) the economy is functioning somewhere below its full-employment level and (2) market forces, which normally impinge on prices, wages and the interest rate, are--for some reason--not working. In these circumstances, a (Keynesian) macroeconomic equilibrium--one involving a substantial amount of economywide unemployment--is achieved through inventory-induced adjustments in the levels of employment, income, and spending. When the level of investment increases by some amount, DI, the equilibrium level of income will increase by some greater amount, DY. The ratio of DY to DI is called the investment multiplier. It can be derived, as follows, from the equilibrium condition (Y = C + I + G) together with the consumption equation (C = a + bY). Study the 10-step derivation below and then mimic the procedure to derive the government-spending multiplier.

	I N S T R U C T I O N S	E Q U A T I O N S
1.	Write the equilibrium condition letting it describe the initial equilibrium.	Y = C + I + G, where C = a + bY
2.	Replace the C in this equation with its algebraic equivalent, a + bY.	Y = a + bY + I + G
3.	Rewrite equation 2 substituting Y + DY for Y and I +DI for I. (This equation describes the new equilibrium, once the economy has adjusted to the increase in investment.	XXXXY + DY = a + b(Y + DY) + I + DI + G
4.	Remove the parentheses algebraically.	XXXXY + DY = a + bY + bDY + I + DI + G
5.	Rewrite equation 2 aligning the corresponding terms.	XXXXY + DY = a + bY + bDY + I + DI + G XXXX______________________________
6.	Subtract equation 5 from equation 4.	XXXXY + DY = a + bY + bDY + I + DI + G
7.	Transpose bDY to the left side of the equation.	DY - bDY = DI
8.	Factor out the DY.	(1 - b)DY = DI
9.	Now divide both sides of the equation by (1 - b). This equation tells us that if we know the level of investment has increased by DI, we can multiply by 1/(1 - b) to determine the corresponding increase (DY) in the level of income.	DY = [1/(1 - b)] DI
10.	Alternatively, divide both sides of this equation by DI to get the definitional statement of the investment multiplier. Note that the investment multiplier is simply the reciprocal of the marginal propensity to save.	DY/DI = 1/(1 - b)