150 Finding Equilibrium Using Algebra
Learning Objectives
- Find the macro equilibrium using algebra
In the income-expenditure model, the equilibrium occurs at the level of GDP where aggregate expenditures equal national income (or GDP). We can identify this equilibrium using algebra as well as graphically. Given algebraic equations for the aggregate expenditure line and the income=expenditure line, the point where they cross can be readily calculated.
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 |
Exercise: Consumption in the Income-Expenditure Model
| National Income | Taxes | After-tax income | Consumption | I + G + X | Imports | Aggregate Expenditures |
|---|---|---|---|---|---|---|
| $300 | $236 | |||||
| $400 | ||||||
| $500 | ||||||
| $600 | |||