Economics 102H
Principles of Macroeconomics (Honors Section)
Spring 2004

Lecture 11

The Expenditure Model (Chapter 10)

Introduction

This chapter can be viewed from two perspectives:

Suppose that the SAS curve is perfectly elastic. That is, suppose that the SAS curve is perfectly flat or horizontal at the price level, P₀. How will the equilibrium level of real GDP be determined? It will be determined by planned expenditures of the household, business, government, and foreign sectors of the economy.

What determines the aggregate quantity demanded at any given price level, say P₀?

You should understand that in both cases we are really asking the same question.

Introduction to the Aggregate Expenditure Model

Let's assume that the price level is given and equal to P₀ and that firms are willing to produce and sell whatever quantity goods market buyers would like to purchase at that price level. That is, assume that the aggregate supply curve is perfectly elastic at P₀. What determines the aggregate quantity demanded at P₀?

Following the textbook, we refer to the aggregate quantity demanded at a given price level as planned expenditure and we'll refer to our theory of planned expenditures as the expenditure model.

The four components of aggregate planned expenditure are:

planned consumption expenditure
planned investment
planned government purchases
planned net exports ( = planned exports - planned imports)

It will be useful to relate planned expenditures to national income, or, simply, GDP.

We assume that planned investment, planned government purchases, and planned exports are not related to GDP. As a result, we say that they are autonomous expenditures. We assume that planned consumption and planned imports are partly determined by GDP and partly determined by other factors. The components of consumption and imports that are determined independently of GDP are referred to as the autonomous components of consumption and import expenditures. The components of consumption and imports that are determined by GDP are referred to as induced expenditure components.

Thus, we will end up arguing that we can express aggregate planned expenditure (AE) by an equation of the form

AE = a + bY

where a = autonomous expenditure and bY = induced expenditure. The coefficient b measures how planned expenditure varies with Y.

In equilibrium, planned aggregate expenditure (AE) must be equal to actual aggregate expenditure (Y) and so in equilibrium

Y = a + bY

i.e.,

Y = a/(1-b)

which is the equilibrium level of real GDP implied by the expenditure model.

Our next tasks are to think a bit more about how a and b are determined and, therefore, how this equilibrium level of GDP is determined.


Derivation of Aggregate Expenditure

Planned Consumption - The Consumption Function

Recall (from Chapter 8) that desired consumption depends upon:

real interest rates (negatively)
current disposable income (positively)
real net wealth (positively)
expected future income (positively)

The consumption function is the relationship between planned consumption expenditure and disposable income, holding all other factors constant. The simplest form of the consumption function is the linear consumption function:

C = " + $(Y-T)

where Y-T = disposable income, 0 < $ < 1, and " > 0. That is, planned consumption is equal to some amount " plus a component that varies directly with disposable income. The coefficient $ measures )C/)(Y-T) and is called the marginal propensity to consume (out of disposable income). The assumption that 0 < $ < 1 means that part of each additional unit of disposable income is consumed and part of each additional unit of disposable income is saved. Rough estimates of $ for the U.S. economy suggest that $ is on the order of about .8.


Changes in the other factors that affect consumption are assumed to be reflected in this consumption function through changes in the parameter ". Thus, e.g., an increase in real interest rates, all else equal, are assumed to decrease the magnitude of " (i.e., shift the consumption function's graph to the right).

For our main current purpose, it is useful to rewrite the consumption function in terms of an autonomous and induced component as discussed previously. There are two ways we will approach this.

The simplest approach is to assume that net tax collections are independent of GDP in which case we can rewrite the consumption function as

C = ("-$T) + $Y.

Then " - $T is the autonomous component of consumption and $Y is the induced component of consumption.

A more realistic approach is to assume that net tax collections vary directly with real GDP. For example, consider a version of the flat tax such that

T = t₀ + t₁Y

where t₀ > 0 and 0 < t₁ < 1.

In this case, we can rewrite the consumption function as

C = " + $Y - $(t₀ + t₁Y)

= ("-$t₀) + $(1-t₁)Y.

Then " - $t₀ is the autonomous component of consumption and $(1-t₁)Y is the induced component. Notice that the marginal propensity to consume out of GDP in this case is equal to $(1-t₁), which is less than $ since 0 < t₁ < 1.

Planned Imports - The Import Function

U.S. imports are determined primarily by three factors:

U.S. real GDP
prices of foreign-made goods relative to the prices of similar U.S.-made goods
foreign exchange rates

All else equal, we assume that planned import expenditures: vary directly with U.S. real GDP, vary inversely with the ratio of the foreign price level to the domestic price level, and vary directly with the value of the dollar. We can formalize this set of assumptions via the import function:

IM = ( + *Y ( > 0, 0 < * < 1

The parameter ( accounts for the autonomous component of import expenditure and will vary as the foreign price level and foreign exchange rate vary. * measures the marginal propensity to import out of GDP. It lies between 0 and 1 to reflect the assumption that as U.S. income increases, all else equal, some of that additional income is used to increase import expenditures. In 1970 the marginal propensity to import was about 0.07; it rose to about 0.15 by 1994.

Planned Investment, Government Purchases, and Exports

We assume that planned investment, government purchases, and exports are independent of the level of real GDP, i.e., they are completely autonomous. Thus,

I = I₀ , G = G₀, and EX = EX₀

where I₀, G₀, and EX₀ are the particular levels of planned investment, government, export expenditures, respectively.

Planned Aggregate Expenditure

Based on the preceding discussion, we derive the planned aggregate expenditure function as follows:

AE = C + I + G + EX - IM

= ("-$T₀) + $Y + I₀ + G₀ + EX₀ - ( - *Y

= a + bY

where a = " - $T₀ + I₀ + G₀ + EX₀ - ( , b = $ - *, and T₀ is the amount of "lump-sum" taxes we assume the government will collect.

{If we assume instead that T = t₀ + t₁Y , then
a = " -$t₀+I₀+G₀+EX₀-( and b = $(1-t₁) - *.}


Equilibrium Expenditure

In equilibrium, it is necessary that actual aggregate expenditure, Y, is equal to planned aggregate expenditure, AE. That is,

Y = a + bY

i.e.,

Y = a/(1-b).

Notice that so long as the marginal propensity to import out of real GDP is smaller than the marginal propensity to consume out of real GDP, then 0 < 1-b < 1.

Recall from the introduction to these notes that we can interpret a/(1-b) as the equilibrium level of real GDP when aggregate supply is perfectly elastic at P₀ . Equivalently, we can interpret a/(1-b) as the aggregate quantity demanded at P = P₀ , i.e., as the quantity on the AD curve corresponding to the price level P₀.

We assume that adjustments to equilibrium occur (i.e., discrepancies between planned and actual aggregate expenditures are resolved) by unplanned inventory adjustments (i.e., through I > I₀ or I < I₀).

Multiplier Analysis

Multiplier analysis is concerned with how equilibrium expenditure changes in response to changes in autonomous expenditure.

Note that if Y = a/(1-b) then )Y = )a/(1-b). That is, a change in autonomous expenditure in the amount )a will change the equilibrium level of real GDP by )a multiplied by 1/(1-b). Thus, we call 1/(1-b) the autonomous expenditure multiplier. Notice that the autonomous expenditure multiplier is greater than one, which means that if autonomous expenditure increases (decreases) by an amount X then the equilibrium level of real GDP will increase (decrease) by more than X. The basic reason why this occurs is because of induced expenditure: each increase in aggregate expenditure lead to an increase in income and, hence, another increase in aggregate expenditure. Notice that the size of the multiplier will vary directly with the magnitude of the coefficient b.


Applications and Implications

Business Cycles

Changes in business sector forecasts of future profits cause changes in planned current investment which translate into amplified changes in equilibrium GDP.

Paradox of Thrift

If the household sector increases (decreases) its propensity to save out of disposable income, aggregate saving could decrease (increase).

Proof:
As the MPS increases, the MPC must decrease. That is
the coefficient b decreases which flattens the planned aggregate expenditure line and reduced GDP.
If the MPS increase but GDP fall, saving could increase or decrease. (S = -" + (1-$)DI.)

Fiscal Policy (G,T, and G-T)

Assume that taxes are lump sum taxes.

An increase in G by the amount X, holding T fixed, will increase Y by the amount Y/(1-b). The government spending multiplier is 1/(1-b).

A decrease in T by the amount X, holding G fixed, will increase Y by the amount $/(1-b), since this decrease in T will increase autonomous expenditures by $. Note that the net tax multiplier is less than the government spending multiplier, so that if G increases by X and T increases by X, Y will increase. It will increase by X/(1-b) - X$/(1-b) = X(1-$)/(1-b). (1-$)/(1-b) is called the balanced budget multiplier.

Why is the net tax multiplier less than the government spending multiplier? A change in G directly affects Y through Y = C + I + G + NX. A change in T indirectly affects Y through its affect on C, which is less than the change in T itself.

{Suppose that taxes depend on GDP according to T = t₀ + t₁Y

In this case, the magnitude of the autonomous expenditure multiplier (and, therefore, the government spending multiplier) decreases. That is, the autonomous expenditure multiplier varies inversely with t₁. The reason is that increases in GDP have smaller effects on DI the larger t₁ is.

This means that the impact of private sector autonomous expenditure shocks (I,EX,",() will be muffled or dampened by the stabilizing response of net tax collections. These stabilizing effects occur automatically and so we say that income taxes and income or employment dependent transfer payments are automatic stabilizers.

The downside is that these automatic stabilizers also muffle the effects of changes in G and therefore they reduce the effectiveness of such changes in countering the effects of adverse private sector shocks.}

This model suggests that the government can use its fiscal policy to help control the behavior of GDP. Keep in mind, however, that we are assuming at this point that i) the price level is fixed and ii) firms are willing and able to supply whatever amount is desired at that price level. These assumptions may be approximately correct when the economy is operating with substantial excess capacity (i.e., during recession/depressions) or to the extent that there is widespread short-run price stickiness in the economy.

Implications of the Expenditure Model for the AD Curve

The question that we have now answered is: given P = P₀, how much final output would the business sector have to supply in order to exactly accommodate planned purchases of final output? Let Y₀ denote that output level.

It follows definitionally that the price-output pair (P₀,Y₀) is a point on the aggregate demand curve, given all of the other factors that determine planned expenditures.

To determine the complete AD curve we must answer the following question: all else equal, how does the equilibrium GDP level implied by the AE model vary with the price level?

Assume that autonomous expenditures vary inversely with P (due to the intertemporal substitution effect, the international substitution effect, and the real money balance effect). It follows that the AD curve will be downward sloping.

What causes AD to shift? The AD curve will shift if holding P fixed at any particular level, the aggregate quantity demanded increases (AD shifts rightward; AD increases) or the aggregate quantity demanded decreases (AD shifts leftward; AD decreases). It follows from our preceding discussion that AD will increase whenever autonomous expenditure increases for any reason other than a decrease in P: a decrease in the real interest rate, an increase in G, a decrease in T, an increase in foreign income,... AD will decrease whenever autonomous expenditure decreases for any reason other than an increase in P.

Note that a change in autonomous expenditure in the amount X will cause AD to shift by an amount greater than X because the autonomous expenditure multiplier is greater than 1.