Finding Equilibrium Using Algebra | Macroeconomics

USING AN ALGEBRAIC APPROACH TO THE EXPENDITURE-OUTPUT MODEL

Imagine an economy defined by the following:

C = 140 + 0.9 (Yd).

This is the consumption function where 140 is autonomous consumption, 0.9 is the marginal propensity to consume, and Yd is disposable (i.e. after tax income).

Yd = Y- T, where Y is national income (or GDP) and T = Tax Revenues = 0.3Y; note that 0.3 is the average income tax rate.

I = Investment = 400

G = Government spending = 800

X = Exports = 600

M = Imports = 0.15Y

Step 1. Determine the aggregate expenditure function. Using the numbers from above, it is:

AE	=	C + I + G + X – M
AE	=	140 + 0.9(Y – T) + 400 + 800 + 600 – 0.15Y

Step 2. The equation for the 45-degree line is the set of points where GDP or national income on the horizontal axis is equal to aggregate expenditure on the vertical axis. Thus, the equation for the 45-degree line is: AE = Y.

Step 3. The next step is to solve these two equations for Y (or AE, since they will be equal to each other). Substitute Y for AE:

Y = AE = 140 + 0.9(Y – T) + 400 + 800 + 600 – 0.15Y

Step 4. Insert the term 0.3Y for the tax rate T. This produces an equation with only one variable, Y.

Step 5. Work through the algebra and solve for Y.

Y	=	140 + 0.9(Y – 0.3Y) + 400 + 800 + 600 – 0.15Y
Y	=	140 + 0.9Y –0.27Y + 1800 – 0.15Y
Y	=	1940 + 0.48Y
Y – 0.48Y	=	1940
0.52Y	=	1940
[latex]\displaystyle\frac{0.52\text{Y}}{0.52}[/latex]	=	[latex]\displaystyle\frac{1940}{0.52}[/latex]
Y	=	3730

This algebraic framework is flexible and useful in predicting how economic events and policy actions will affect real GDP.

Say, for example, that because of changes in the relative prices of domestic and foreign goods, the marginal propensity to import falls to 0.1. Calculate the equilibrium output when the marginal propensity to import is changed to 0.10.

Y	=	140 + 0.9(Y – 0.3Y) + 400 + 800 + 600 – 0.1Y
Y	=	1940 – 0.53Y
0.47Y	=	1940
Y	=	4127

Alternatively, suppose because of a surge of business confidence, investment rises to 500. Calculate the equilibrium output.

Y	=	140 + 0.9(Y – 0.3Y) + 500 + 800 + 600 – 0.15Y
Y	=	2040 + 0.48Y
Y – 0.48Y	=	2040
0.52Y	=	2040
Y	=	3923