This is “The Algebra of Equilibrium”, section 2.1 (from appendix 2) from the book Macroeconomics Principles (v. 2.0).

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Suppose an economy can be represented by the following equations:

Equation 22.1

$$C={C}_{\text{a}}+b{Y}_{\text{d}}$$Equation 22.2

$$T={T}_{\text{a}}+\mathit{tY}$$Equation 22.3

$${I}_{\text{p}}={I}_{\text{a}}$$Equation 22.4

$$G={G}_{\text{a}}$$Equation 22.5

$${X}_{\text{n}}={X}_{{\text{n}}_{\text{a}}}$$As in our specific example in the chapter, the consumption function given in Equation 22.1 has an autonomous component (*C*_{a}) and an induced component (*bY*_{d}), where *b* is the marginal propensity to consume (*MPC*). In the example in the chapter, *C*_{a} was $300 billion and the *MPC*, or *b*, was 0.8. Equation 22.2 shows that total taxes, *T*, include an autonomous component *T*_{a} (for example, property taxes, licenses, fees, and any other taxes that do not vary with the level of income) and an induced component that is a fraction of real GDP, *Y*. That fraction is the tax rate, *t*. Disposable personal income is just the difference between real GDP and total taxes:

Equation 22.6

$${Y}_{\text{a}}=Y-T$$In Equation 22.3, Equation 22.4, and Equation 22.5, *I*_{a}, *G*_{a}, and
${X}_{{\text{n}}_{\text{a}}}$
are specific values for the other components of aggregate expenditures: investment (*I*_{p}), government purchases (*G*), and net exports (*X*_{n}). In this model, planned investments, government purchases, and net exports are all assumed to be autonomous. For this reason, we add the subscript “a” to each of them.

We use the equations that describe each of the components as aggregate expenditures to solve for the equilibrium level of real GDP. The equilibrium condition in the aggregate expenditures model requires that aggregate expenditures for a period equal real GDP in the period. We specify that condition algebraically:

Equation 22.7

$$Y=\mathit{AE}$$Aggregate expenditures *AE* consist of consumption plus planned investment plus government purchases plus net exports. We thus replace the right-hand side of Equation 22.7 with those terms to get

Equation 22.8

$$Y=C+{I}_{\text{p}}+G+{X}_{\text{n}}$$Consumption is given by Equation 22.1 and the other components of aggregate expenditures by Equation 22.3, Equation 22.4, and Equation 22.5. Inserting these equations into Equation 22.8, we have

Equation 22.9

$$Y={C}_{\text{a}}+b{Y}_{\text{d}}+{I}_{\text{a}}+{G}_{\text{a}}+{X}_{{\text{n}}_{\text{a}}}$$We have one equation with two unknowns, *Y* and *Y*_{d}. We therefore need to express *Y*_{d} in terms of *Y*. From Equation 22.2 and Equation 22.6, we can write

And remove the parentheses to obtain

Equation 22.10

$${Y}_{\text{d}}=Y-{T}_{\text{a}}-tY$$We then factor out the *Y* term on the right-hand side to get

Equation 22.11

$${Y}_{\text{d}}=(1-t)Y-{T}_{\text{a}}$$We now substitute this expression for *Y*_{d} into Equation 22.9 to get

Equation 22.12

$$Y={C}_{\text{a}}-b{T}_{\text{a}}+b(1-t)Y+{I}_{\text{a}}+{G}_{\text{a}}+{X}_{{\text{n}}_{\text{a}}}$$The first two terms (*C*_{a} − *bT*_{a}) show that the autonomous portion of consumption is reduced by the marginal propensity to consume times autonomous taxes. For example, suppose *T*_{a} is $10 billion. If the marginal propensity to consume is 0.8, then consumption is $8 billion less than it would have been if *T*_{a} were zero.

Combining the autonomous terms in Equation 22.12 in brackets, we have

Equation 22.13

$$Y=[{C}_{\text{a}}-b\left({T}_{\text{a}}\right)+{I}_{\text{a}}+{G}_{\text{a}}+{X}_{{n}_{\text{a}}}]+b(1-t)(Y)$$Letting $\overline{A}$ stand for all the terms in brackets, we can simplify Equation 22.13:

Equation 22.14

$$Y=\overline{A}+b(1-t)Y$$The coefficient of real GDP (*Y*) on the right-hand side of Equation 22.14, *b*(1 − *t*), gives the fraction of an additional dollar of real GDP that will be spent for consumption: it is the slope of the aggregate expenditures function for this representation of the economy. The aggregate expenditures function for the simplified economy that we presented in the chapter has a slope that was simply the marginal propensity to consume; there were no taxes in that model, and disposable personal income and real GDP were assumed to be the same. Notice that in using this more realistic aggregate expenditures function, the slope is less by a factor of (1 − *t*).

We solve Equation 22.14 for Y:

$$Y-b(1-t)(Y)=\overline{A}$$$$Y[1-b(1-t)]=\overline{A}$$

Equation 22.15

$$Y=\frac{1}{1-b(1-t)}(\overline{A})$$In Equation 22.15, 1/[1 − *b*(1 − *t*)] is the multiplier. Equilibrium real GDP is achieved at a level of income equal to the multiplier times the amount of autonomous spending. Notice that because the slope of the aggregate expenditures function is less than it would be in an economy without induced taxes, the value of the multiplier is also less, all other things the same. In this representation of the economy, the value of the multiplier depends on the marginal propensity to consume and on the tax rate. The higher the tax rate, the lower the multiplier; the lower the tax rate, the greater the multiplier.

For example, suppose the marginal propensity to consume is 0.8. If the tax rate were 0, then the multiplier would be 5. If the tax rate were 0.25, then the multiplier would be 2.5.