# Macroeconomics/Aggregate Expenditures

## Introduction

editIn this chapter, we will discuss *aggregate expenditure model*. Its definition is as follows:

**Definition.**
(Aggregate expenditure model)
Aggregate expenditure model is a macroeconomics model that focuses on the *short-run* relationship between *aggregate expenditure* (AE)
and *real GDP*, assuming the *price level* is *constant*.

To be more precise, AE means the following:

**Definition.**
(Aggregate expenditure)
Aggregate expenditure (AE) is the total spending in the economy, which is
in which

- is consumption
- is
*planned*investment - is government purchases
- is net export

**Remark.**
For comparison, the real Gross Domestic Product , in which is the *actual* investment, and other same notations have the same meanings. We should be careful that *planned* investment is slightly different from the actual investment.

We will define *planned investment* in the following. For other expenditures, we have defined them in the chapter about GDP, and they have the same definitions here.

**Definition.**
(Planned investment)
*Planned* investment is *planned* spending by firms on capital goods (which includes, but not limited to
*planned* inventory investment, causing *planned* change in inventories ( ) ), and by households on new homes.
Equivalently, we can define by
in which is actual investment ^{[1]}.

**Example.**
Ceteris paribus (assume this from now on, unless otherwise specified),
assume households purchase much more products sold by firms than firms expected. It causes the inventory .
There is an unplanned *decrease* in inventory.

**Exercise.**

We will examine in more details (more than their definitions) one by one in the following sections.

## Consumption

editConsumption has two components, namely *autonomous* consumption ( ) and *induced* consumption.

**Definition.**
(Autonomous consumption)
Autonomous consumption ( ) is the amount of consumption that is *not affected* by the level of *income* ( ).

**Remark.**

- it may be interpreted as the expenditure on
*necessities*, e.g. food., which is assumed to be remain unchanged no matter how changes, and so should be positive - putting a bar at the top of a variable means that it remains unchanged

Before defining induced consumption, let us define a term which will be used in its definition.

**Definition.**
(Marginal propensity to consume)
Marginal propensity to consume (MPC) is
in which

- is 'change in'
- is disposable income ( )

**Remark.**

- (assumption) suppose , so that .
- (assumption) suppose households do not borrow extra money, then it follows that , since households at most spend all income in consumption ( in this case), without borrowing.
- (assumption) we would expect that, ceteris paribus, when income increases, most households will consume more, so consumption should increase, and thus

Another similar definition is *marginal propensity to save* (MPS).

**Definition.**
(Marginal propensity to save)
Marginal propensity to save (MPS) is .

**Proposition.**
(Relationship between MPC and MPS)
Then,

**Proof.**
To determine the consumption level, each household allocates a portion of his wealth (i.e. asset minus liabilities) to ,
allocates another portion to , and allocates the remaining portion to . Therefore, ( means 'implies')

When there is change in wealth (which equals ), the portion of , allocated to is determined by MPC, by definition.
Since , *assuming *,
we have
( , since *there is* change in wealth)

**Remark.**
In the following, we assume , and so this equation is always true.

Then, we can use MPC to define induced consumption.

**Definition.**
(Induced consumption)
Induced consumption is
in which is the disposable income.

Then, we can express the consumption function as follows:
which is a function in ,
and so the *slope* of consumption function is MPC (which is positive, and so ), and * -intercept* of consumption function is (which is positive).

**Example.**
(Consumption function)
Given that a consumption function is , we can see that , and .
It follows that .

If , the induced consumption is , and

**Exercise.**

Then, we will discuss some important factors that affect .

**Proposition.**
(Factors affecting consumption)
Ceteris paribus,

- (positive relationship) if , wealth or expected future income ( ) , then
- (negative relationship) if price level ( ) or real interest rate ( ) , then

**Proof.**
Ceteris paribus,

- Current disposable income ( ): , which follows from the consumption function
- Household wealth:
- Expected future income ( ):
- Price level ( ):
- Real interest rate ( ) ( is saving):

- since (ceteris paribus), , so

**Example.**
Suppose households expect their future income more *pessimistically*, then, since , .
(borrowing against future income )

**Exercise.**

## Planned investment

edit is *autonomous*, which does not vary with .
The following are some important factors affecting (which does not include ).

**Proposition.**
(Factors affecting planned investment)
Ceteris paribus,

- (positive relationship) if expected future profitability ( ) or cash flow ( ) , then
- (negative relationship) if or , then

**Proof.**

**Example.**
Suppose the government decrease the profit tax on firms, then .

**Exercise.**

## Government purchases

editAssume is solely determined by the government, and therefore is *autonomous*.
So, its change depends on how the government changes .

## Net exports

editChange in is mainly affected by the *comparison* between domestic country and foreign countries.

**Proposition.**
(Factors affecting net exports)

- in domestic price level ( ) in foreign price level ( )
- in domestic GDP ( ) in foreign GDP ( )
- exchange rate of domestic currency to foreign currencies ( ) ( )
^{[2]}

**Proof.**

- domestic goods become more (less)
*expensive*relative to foreign goods - in domestic demand for ( ) in foreign demand for domestic ( )
- are less (more) expensive, and are more (less) expensive

**Example.**
Assume one USD can be exchanged for 110 Japanese yen originally, and now one USD can be exchanged for 120 Japanese yen.
of *US* , since ceteris paribus.

**Exercise.**

## Macroeconomic equilibrium

edit### AE function

editWe can plot the AE aginst GDP graph as follows:

The blue line can be interpreted as the curve with , i.e. the consumption function .

Recall that .
Since are *autonomous*, and does not vary, ceteris paribus (the comparison between domestic country and foreign countries gives same results),
we may denote them as , to emphasize their invariance (they are constants which do not vary with ).

Then, we can derive the function (in ) by adding back and to the consumption function (which shifts the blue line *upwards* by parallelly,
since the -intercept changes from to ):

We can observe that, at the region *above* the Keynesian cross, ^{[3]}
, and at the region *below* the Keynesian cross,
^{[4]}.

Also, we can see from the function that, its slope is , which is the same as that of consumption function.

### Adjustment to macroeconomic equilibrium

edit **Definition.**
(Macroeconomic equilibrium)
Macroeconomic equilibrium is the point at which , i.e. the intersection point between curve and the Keynesian cross.

Sometimes, the economy is *not* at macroeconomic equilibrium, i.e. or .
Let's examine these two cases one by one.

Since ,
there is unplanned *decrease* in inventories.
In view of this, firms should refill the inventories ^{[5]} by production , until reaching .

On the other hand,
since ,
there is unplanned *increase* in inventories.
In view of this, firms should cut their production ^{[6]} by production , until reaching .

After reaching the macroeconomic equilibrium, i.e. and thus there is no unplanned change in inventories, and so ceteris paribus.

Therefore, eventually, we will reach macroeconomic equilibrium, and macroeconomic equilibrium can occur at arbitrary point at the Keynesian cross.

Recall the economy has a level of potential GDP ( ), but macroeconomic equilibrium may not be located at the point at which . Macroeconomic equilibrium is at a point at which .

Also, at macroeconomic equilibrium,

### The multiplier effect

editIn view of the above equation at macroeonomic equilibrium, when the *autonomous* expenditure (variables with a bar on top of it) changes by ,
changes by in the same direction.
Since , this number is greater than one, and we give this number a name, namely *multiplier*:

**Definition.**
(Multiplier)
The *multiplier* (about ) is

**Remark.**

- it reflects the magnitude of change (in the same direction) when autonomous expenditure changes: the larger (smaller) the multiplier, the larger (smaller) the magnitude of the change

### The paradox of thrift

editRecall that in closed economy in which , ^{[7]}.
This implies is the key to long run (LR) growth (since is the key to LR growth). Thus, it has a positive effect on the economy.

However, in the short run (SR), ^{[8]}.
This can push the economy into *recession*, and thus have a negative effect on the economy.

Here is the paradox, since what appears to be favourable in LR may be unfavourable in SR.

However, the existence of this paradox is questionable, since it is argued that , which *may* offset the in .

## Aggregate demand (AD) curve

editIn the following, we will loosen the assumption that . After that, we can use the AE curve to derive aggregate demand (AD) curve.

affects AE as in the following proposition:

**Proposition.**
(Relationship between and AE)
.

**Proof.**
We can prove this relationship in three ways.

- (wealth effect)
- (interest rate effect)
- (net export effect)

**Remark.**

- graphically, since (ceteris paribus), the slope of AE curve does not change even if changes, so
*parallelly* - the notations mean the direction of
*shift*

Since at macroecnomic equilibrium, , , and thus we have established the (inverse) relationship between and at macroeconomic equilibrium. We can assume that the economy is at macroecnomic equilibrium unless otherwise specified, since it is likely that the economy is at macroeconomic equilibrium, considering that the economy will adjust to the macroeconomic equilibrium eventually.

This inverse relationship between and is reflected by AD curve.

**Definition.**
(Aggregate demand (AD) curve)
AD curve is a curve that shows the (inverse) relationship between and ,
ceteris paribus (paricularly, holding constant all factors that affect AE other than P).

**Remark.**

- for simplicity, we assume the AD curve is linear in this book, but we should notice that it can be a curve

- after assuming this, we can see that, if we plot the AD curve in a graph with as -axis and as -axis, then AD curve has a negative slope, i.e. it is downward sloping, because of the inverse relationship of and

- inverse relationship between and means that when one of them changes, another one changes in
*opposite*direction - AD curve also consists of every possible point (i.e. every possible pair of and )

Illustration of (a portion of) the downward sloping AD curve: ^{[9]}

P | | | \ | \ AD | \ |--------- Y

AD curve is essential in the AD-AS model, which will be discussed later.

- ↑ we can do this since includes both and unplanned investment, which only includes
*unplanned*inventory investment causing*unplanned*change in inventories, since other investment categories are planned. So, subtracting*unplanned*change in inventories from gives - ↑ i.e. other currencies that can be exchanged by one dollar of domestic currency ( )
- ↑ for each , is above the level at which , since it lies above the Keynesian cross
- ↑ for each , is below the level at which , since it lies below the Keynesian cross
- ↑ otherwise, there will not be sufficient inventories for future sale
- ↑ otherwise, there will be too much inventories
- ↑
- ↑ if in is not solely caused by in public saving ( , which should be true.
- ↑ in particular, the point should not be located at the - and -intercept, since it does not make sense for either one of them to be zero practically