150 Finding Equilibrium Using Algebra

Step 1. Calculate the amount of taxes for each level of national income (reminder: GDP = national income) for each level of national income using the following as an example:

[latex]\begin{array}{lr}\text{National Income (Y)}&\$300\\\text{Taxes = 0.2 or 20%}&\times0.2\\\text{Tax amount (T)}&\$60\end{array}[/latex]

Step 2. Calculate after-tax income by subtracting the tax amount from national income for each level of national income using the following as an example:

[latex]\begin{array}{lr}\text{National income minus taxes}&\$300\\&-\$60\\\text{After-tax income}&\$240\end{array}[/latex]

Step 3. Calculate consumption. The marginal propensity to save is given as 0.1. This means that the marginal propensity to consume is 0.9, since MPS + MPC = 1. Therefore, multiply 0.9 by the after-tax income amount using the following as an example:

[latex]\begin{array}{lr}\text{After-tax Income}&\$240\\\text{MPC}&\times0.9\\\text{Consumption}&\$216\end{array}[/latex]

Step 4. Consider why the table shows consumption of $236 in the first row. As mentioned earlier, the Keynesian model assumes that there is some level of consumption even without income. That amount is $236 – $216 = $20.

Step 5. There is now enough information to write the consumption function. The consumption function is found by figuring out the level of consumption that will happen when income is zero. Remember that:

[latex]\text{C}=\text{Consumption when national income is zero}+\text{MPC (after-tax income)}[/latex]

Let C represent the consumption function, Y represent national income, and T represent taxes.

[latex]\begin{array}{lcl}\text{C}&=&\$20+0.9\left(\text{Y}-\text{T}\right)\\&=&\$20+0.9\left(\$300-\$60\right)\\&=&\$236\end{array}[/latex]

Step 6. Use the consumption function to find consumption at each level of national income.

Step 7. Add investment (I), government spending (G), and exports (X). Remember that these do not change as national income changes:

Step 8. Find imports, which are 0.2 of after-tax income at each level of national income. For example:

[latex]\begin{array}{lr}\text{After-tax income}&\$240\\\text{Imports of 0.2 or 20% of Y}-\text{T}&\times0.2\\\text{Imports}&\$48\end{array}[/latex]

Step 9. Find aggregate expenditure by adding C + I + G + X – I for each level of national income. Your completed table should look like this:

Step 10. Answer the question: What is equilibrium? Equilibrium occurs where AE = Y. This table shows that equilibrium occurs where national income equals aggregate expenditure at $500.

Step 11. Find equilibrium mathematically, knowing that national income is equal to aggregate expenditure.Step 10. Answer the question: What is equilibrium? Equilibrium occurs where AE = Y. The table shows that equilibrium occurs where national income equals aggregate expenditure at $500.

[latex]\begin{array}{rcl}\text{Y}&=&\text{AE}\\&=&\text{C}+\text{I}+\text{G}+\text{X}-\text{M}\\&=&\$20+0.9\left(\text{Y}-\text{T}\right)+\$70+\$80+\$50-0.2\left(\text{Y}-\text{T}\right)\\&=&\$220+0.0\left(\text{Y}-\text{T}\right)-0.2\left(\text{Y}-\text{T}\right)\end{array}[/latex]

Since T is 0.2 of national income, substitute T with 0.2 Y so that:

[latex]\begin{array}{rcl}\text{Y}&=&\$220+0.9\left(\text{Y}-0.2\text{Y}\right)-0.2\left(\text{Y}-0.2\text{Y}\right)\\&=&\$220+0.9\text{Y}-0.18\text{Y}-0.2\text{Y}+0.04\text{Y}\\&=&\$220+0.56\text{Y}\end{array}[/latex]

Solve for Y.

[latex]\begin{array}{rcl}\text{Y}&=&\$220+0.56\text{Y}\\\text{Y}-0.56\text{Y}&=&\$220\\0.44\text{Y}&=&\$220\\\frac{0.44\text{Y}}{0.44}&=&\frac{\$200}{0.44}\\\text{Y}&=&\$500\end{array}[/latex]

Step 12. Answer this question: Why is a national income of $300 not an equilibrium? At national income of $300, aggregate expenditures are $388.

Step 13. Answer this question: How do expenditures and output compare at this point? Aggregate expenditures cannot exceed output (GDP) in the long run, since there would not be enough goods to be bought.