# Macroeconomics/Aggregate Expenditures
## Introduction

In 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 $AE=C+I^{\color {green}p}+G+NX$ in which
* $C$ is consumption
* $I^{\color {green}p}$ is planned investment
* $G$ is government purchases
* $NX$ is net export
Remark. For comparison, the real Gross Domestic Product $Y={\text{real }}GDP=C+I+G+NX$, in which $I$ 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 $I^{p}$ is planned spending by firms on capital goods (which includes, but not limited to planned inventory investment, causing planned change in inventories ($\Delta inv^{p}$) ), and by households on new homes. Equivalently, we can define $I^{p}$ by $I^{p}=I-{\text{unplanned change in inventories }}(\Delta inv^{unp})$ in which $I$ is actual investment.
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 $\downarrow ({\text{unexpectedly}})\;\&\;{\overline {I^{p}}}(\because {\text{ceteris paribus}})\Rightarrow I\downarrow$. There is an unplanned decrease in inventory.
Exercise.
We will examine $C,I^{p},G,NX$ in more details (more than their definitions) one by one in the following sections.

## Consumption

Consumption has two components, namely autonomous consumption (${\overline {C}}$) and induced consumption.
Definition. (Autonomous consumption) Autonomous consumption (${\overline {C}}$) is the amount of consumption that is not affected by the level of income ($Y$).
Remark.
* it may be interpreted as the expenditure on necessities, e.g. food., which is assumed to be remain unchanged no matter how $Y$ changes, and so ${\overline {C}}$ 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 ${\frac {\Delta C}{\Delta Y_{d}}}$ in which
* $\Delta$ is 'change in'
* $Y_{d}$ is disposable income ($=Y(={\text{income}})-{\text{taxes }}(T)$)
Remark.
* (assumption) suppose $\Delta T=0\Rightarrow \Delta (Y-Y_{d})=0$, so that $\Delta Y_{d}=\Delta Y$ .
* (assumption) suppose households do not borrow extra money, then it follows that $MPC\leq 1$, since households at most spend all income in consumption ($MPC=1$ 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 $MPC>0$
Another similar definition is marginal propensity to save (MPS).
Definition. (Marginal propensity to save) Marginal propensity to save (MPS) is ${\frac {\Delta S}{\Delta Y}}$.
Proposition. (Relationship between MPC and MPS) Then, $MPC+MPS=1.$
Proof. To determine the consumption level, each household allocates a portion of his wealth (i.e. asset minus liabilities) to $C$, allocates another portion to $T$, and allocates the remaining portion to $S$. Therefore, ${\text{wealth}}=C+S+T\Rightarrow \Delta {\text{wealth}}=\Delta C+\Delta S+\Delta T.$ ($\Rightarrow$ means 'implies')
When there is change in wealth (which equals $\Delta Y$), the portion of $\Delta {\text{wealth}}=\Delta Y$, allocated to $\Delta C$ is determined by MPC, by definition. Since $\Delta Y=\Delta C+\Delta S+\Delta T$, assuming $\Delta T=0$, we have $\Delta Y=\Delta C+\Delta S\Rightarrow {\frac {\Delta Y}{\Delta Y}}={\frac {\Delta C}{\Delta Y}}+{\frac {\Delta S}{\Delta Y}}\Rightarrow MPC+MPS=1.$ ($\Delta Y\neq 0$, since there is change in wealth)
$\Box$
Remark. In the following, we assume $\Delta T=0$, and so this equation is always true.
Then, we can use MPC to define induced consumption.
Definition. (Induced consumption) Induced consumption is $MPC\cdot Y_{d}$ in which $Y_{d}$ is the disposable income.
Then, we can express the consumption function as follows: $C={\overline {C}}+MPC\cdot Y_{d}=f(Y_{d}),$ which is a function in $Y_{d}$, and so the slope of consumption function is MPC (which is positive, and so $Y_{d}\uparrow \Rightarrow C\uparrow$), and $y$-intercept of consumption function is ${\overline {C}}$ (which is positive).
Example. (Consumption function) Given that a consumption function is $C=200+0.4Y_{d}$, we can see that $MPC=0.4$, and ${\overline {C}}=200$. It follows that $MPS=1-0.4=0.6$.
If $Y_{d}=1000$, the induced consumption is $0.4Y_{d}=400$, and $C=200+400=600.$
Exercise.
Then, we will discuss some important factors that affect $C$.
Proposition. (Factors affecting consumption) Ceteris paribus,
* (positive relationship) if $Y_{d}$, wealth or expected future income ($Y_{e}$) $\uparrow (\downarrow )$, then $C\uparrow (\downarrow )$
* (negative relationship) if price level ($P$) or real interest rate ($r$) $\uparrow (\downarrow )$, then $C\downarrow (\uparrow )$
Proof. Ceteris paribus,
* Current disposable income ($Y_{d}$): $Y_{d}\uparrow (\downarrow )(\&{\overline {MPC}})\Rightarrow C\uparrow (\downarrow )$, which follows from the consumption function
* Household wealth: ${\text{wealth}}\uparrow (\downarrow )(\Leftrightarrow \Delta Y>0(<0))\Rightarrow Y_{d}\uparrow (\downarrow )\Rightarrow C\uparrow (\downarrow )$
* Expected future income ($Y_{e}$): $Y_{e}\uparrow (\downarrow )\Rightarrow {\text{borrowing against future income}}\uparrow (\downarrow )\Rightarrow C\uparrow (\downarrow )$
* Price level ($P$): $P\uparrow (\downarrow )\Rightarrow {\text{real wealth}}:={\frac {\overline {\text{nominal wealth}}}{P^{\uparrow (\downarrow )}}}\downarrow (\uparrow )\Rightarrow Y_{d}\downarrow (\uparrow )\Rightarrow C\downarrow (\uparrow )$
* Real interest rate ($r$) ($S$ is saving): $r\uparrow (\downarrow )\Rightarrow {\text{saving earning}}\uparrow (\downarrow )\Rightarrow {\text{marginal benefit }}(MB){\text{ of saving }}\uparrow (\downarrow )(\&\;{\overline {{\text{its marginal cost}}(MC)}})\Rightarrow S\uparrow (\downarrow )(\&\;{\overline {Y}}\;\&\;{\overline {T}})\Rightarrow C\downarrow (\uparrow )$
   * since $\Delta Y=\Delta T=0$ (ceteris paribus), $\Delta Y=\Delta C+\Delta S+\Delta T\Rightarrow \Delta C=-\Delta S$, so $S\uparrow \Rightarrow C\downarrow$
$\Box$
Example. Suppose households expect their future income more pessimistically, then, since $Y_{e}\downarrow$, $C\downarrow$. (borrowing against future income $\downarrow$)
Exercise.

## Planned investment

$I^{p}$ is autonomous, which does not vary with $Y$. The following are some important factors affecting $I^{p}$ (which does not include $Y$).
Proposition. (Factors affecting planned investment) Ceteris paribus,
* (positive relationship) if expected future profitability ($\Pi _{e}$) or cash flow ($CF$) $\uparrow (\downarrow )$, then $I^{p}\uparrow (\downarrow )$
* (negative relationship) if $T$ or $r\uparrow (\downarrow )$, then $I^{p}\downarrow (\uparrow )$
Proof.
* $\Pi _{e}\uparrow (\downarrow )\Rightarrow MB{\text{ of firms' investment}}\uparrow (\downarrow )(\&\;{\overline {{\text{its }}MC}})\Rightarrow I^{p}\uparrow (\downarrow )$
* $CF\uparrow (\downarrow )\Rightarrow {\text{ability to finance investment}}\uparrow (\downarrow )\Rightarrow I^{p}\uparrow (\downarrow )$
* $T\uparrow (\downarrow )\Rightarrow \Pi _{e}\downarrow (\uparrow )\;\&\;CF\downarrow (\uparrow )\Rightarrow I^{p}\downarrow (\uparrow )$
* $r\uparrow (\downarrow )\Rightarrow MC{\text{ of borrowing}}\uparrow (\downarrow )(\&\;{\overline {{\text{its }}MB}})\Rightarrow I^{p}\downarrow (\uparrow )$
$\Box$
Example. Suppose the government decrease the profit tax on firms, then $I^{p}\uparrow$.
Exercise.

## Government purchases

Assume $G$ is solely determined by the government, and therefore $G$ is autonomous. So, its change depends on how the government changes $G$.

## Net exports

Change in $NX$ is mainly affected by the comparison between domestic country and foreign countries.
Proposition. (Factors affecting net exports)
* $\uparrow$ in domestic price level ($P_{\text{domestic}}$) $>(<)$ $\uparrow$ in foreign price level ($P_{\text{foreign}}$) $\Rightarrow NX\downarrow (\uparrow )$
* $\uparrow$ in domestic GDP ($Y_{\text{domestic}}$) $>(<)$ $\uparrow$ in foreign GDP ($Y_{\text{foreign}}$) $\Rightarrow NX\downarrow (\uparrow )$
* exchange rate of domestic currency to foreign currencies ($er$) $\uparrow$($\downarrow$) $\Rightarrow NX\downarrow (\uparrow )$
Proof.
* $\uparrow {\text{in }}P_{\text{domestic}}>(<)\uparrow {\text{in }}P_{\text{foreign}}\Rightarrow$ domestic goods become more (less) expensive relative to foreign goods $\Rightarrow X\downarrow (\uparrow )\;\&\;M\downarrow (\uparrow )\Rightarrow NX\downarrow (\uparrow )$
* $\uparrow {\text{in }}Y_{\text{domestic}}>(<)\uparrow {\text{in }}Y_{\text{foreign}}\Rightarrow$ $\uparrow$ in domestic demand for $M$ ($D_{M}$) $>(<)$ $\uparrow$ in foreign demand for domestic $X$ ($D_{X}$) $\Rightarrow X\downarrow (\uparrow )\;\&\;M\uparrow (\downarrow )\Rightarrow NX\downarrow (\uparrow )$
* $er\uparrow (\downarrow )\Rightarrow$ $M$ are less (more) expensive, and $X$ are more (less) expensive $\Rightarrow M\uparrow (\downarrow )\;\&\;X\downarrow (\uparrow )\Rightarrow NX\downarrow (\uparrow )$
$\Box$
Example. Assume one USD can be exchanged for 110 Japanese yen originally, and now one USD can be exchanged for 120 Japanese yen. $NX$ of US $\downarrow$, since $er\uparrow$ ceteris paribus.
Exercise.

## Macroeconomic equilibrium
### AE function

We can plot the AE aginst GDP graph as follows:
The blue line can be interpreted as the $AE$ curve with $I^{p}=G=NX=0$, i.e. the consumption function $C={\overline {C}}+(MPC)Y$.
Recall that $AE+C+I^{p}+G+NX$. Since $I^{p},G$ are autonomous, and $NX$ does not vary, ceteris paribus (the comparison between domestic country and foreign countries gives same results), we may denote them as ${\overline {I^{p}}},{\overline {G}},{\overline {NX}}$, to emphasize their invariance (they are constants which do not vary with $Y$).
Then, we can derive the $AE$ function (in $Y$) by adding back ${\overline {I^{p}}},{\overline {G}}$ and ${\overline {NX}}$ to the consumption function (which shifts the blue line upwards by ${\overline {I^{p}}}+{\overline {G}}+{\overline {NX}}$ parallelly, since the $y$-intercept changes from ${\overline {C}}$ to ${\overline {C}}+{\overline {I^{p}}}+{\overline {G}}+{\overline {NX}}$): $AE=\underbrace {{\overline {C}}+MPC\cdot Y} _{C}+{\overline {I^{p}}}+{\overline {G}}+{\overline {NX}}=MPC\cdot Y+{\text{constant}}=f(Y)$
We can observe that, at the region above the Keynesian cross, $Y<AE$ , and at the region below the Keynesian cross, $Y>AE$ .
Also, we can see from the $AE$ function that, its slope is $MPC$, which is the same as that of consumption function.

### Adjustment to macroeconomic equilibrium

Definition. (Macroeconomic equilibrium) Macroeconomic equilibrium is the point at which $Y=AE$, i.e. the intersection point between $AE$ curve and the Keynesian cross.
Sometimes, the economy is not at macroeconomic equilibrium, i.e. $AE>Y$ or $AE<Y$. Let's examine these two cases one by one.
Since $AE>Y\Rightarrow {\cancel {C+}}I^{p}{\cancel {+G+NX}}>{\cancel {C+}}I{\cancel {+G+NX}}\Rightarrow I^{p}>I\Rightarrow I-\Delta inv^{unp}>I\Rightarrow \Delta inv^{unp}<0$, there is unplanned decrease in inventories. In view of this, firms should refill the inventories by $\uparrow$ production $\Rightarrow I^{p}\uparrow \Rightarrow Y\uparrow$, until reaching $Y=AE$.
On the other hand, since $AE<Y\Rightarrow {\cancel {C+}}I^{p}{\cancel {+G+NX}}<{\cancel {C+}}I{\cancel {+G+NX}}\Rightarrow I^{p}<I\Rightarrow I-\Delta inv^{unp}<I\Rightarrow \Delta inv^{unp}<0$, there is unplanned increase in inventories. In view of this, firms should cut their production by $\downarrow$ production $\Rightarrow I^{p}\downarrow \Rightarrow Y\downarrow$, until reaching $Y=AE$.
After reaching the macroeconomic equilibrium, i.e. $Y=AE\Rightarrow {\cancel {C+}}I^{p}{\cancel {+G+NX}}={\cancel {C+}}I{\cancel {+G+NX}}\Rightarrow I^{p}=I\Rightarrow I-\Delta inv^{unp}=I\Rightarrow \Delta inv^{unp}=0,$ and thus there is no unplanned change in inventories, and so ${\overline {Y}}$ 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 ($Y^{p}$), but macroeconomic equilibrium may not be located at the point at which $AE=Y^{p}$. Macroeconomic equilibrium is at a point at which $AE=Y<(>)Y^{p}\Rightarrow {\text{unemployment rate }}(u)>(<){\text{ natural unemployment rate }}({\overline {u}})$.
Also, at macroeconomic equilibrium, $Y=AE\Rightarrow Y=MPC\cdot Y+{\overline {C}}+{\overline {I^{p}}}+{\overline {G}}+{\overline {NX}}\Rightarrow (1-MPC)Y={\overline {C}}+{\overline {I^{p}}}+{\overline {G}}+{\overline {NX}}\Rightarrow Y={\frac {{\overline {C}}+{\overline {I^{p}}}+{\overline {G}}+{\overline {NX}}}{1-MPC}}$

### The multiplier effect

In view of the above equation at macroeonomic equilibrium, when the autonomous expenditure (variables with a bar on top of it) changes by $1$, $Y$ changes by ${\frac {1}{1-MPC}}$ in the same direction. Since $0<MPC\leq 1$, this number is greater than one, and we give this number a name, namely multiplier:
Definition. (Multiplier) The multiplier (about $Y$) is ${\frac {\Delta Y}{\Delta {\text{ autonomous expenditure}}}}={\frac {\Delta Y}{\Delta ({\overline {C}}+{\overline {I^{p}}}+{\overline {G}}+{\overline {NX}})}}={\frac {1}{1-MPC}}$
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
* $MPC\uparrow (\downarrow )\Rightarrow {\frac {1}{(1-MPC^{\uparrow (\downarrow )})^{\downarrow }(\uparrow )}}\uparrow (\downarrow )\Rightarrow {\text{multiplier}}\uparrow (\downarrow )$

### The paradox of thrift

Recall that in closed economy in which $NX=0$, $S=I$. This implies $S$ is the key to long run (LR) growth (since $I$ is the key to LR growth). Thus, it has a positive effect on the economy.
However, in the short run (SR), $S\uparrow \Rightarrow {\text{private saving}}(S_{\text{private}})\uparrow \Rightarrow C\downarrow \Rightarrow Y\downarrow \Rightarrow C\downarrow \cdots$. 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 $S\uparrow \Rightarrow {\text{loanable fund supply}}\searrow (\&\;{\overline {\text{loanable fund demand}}})\Rightarrow r\downarrow \Rightarrow I\uparrow$, which may offset the $\downarrow$ in $C$.

## Aggregate demand (AD) curve

In the following, we will loosen the assumption that ${\overline {P}}$. After that, we can use the AE curve to derive aggregate demand (AD) curve.
$P$ affects AE as in the following proposition:
Proposition. (Relationship between $P$ and AE) $P\uparrow (\downarrow )\Rightarrow AE\searrow (\nwarrow )$.
Proof. We can prove this relationship in three ways.
* (wealth effect) $P\uparrow (\downarrow )\Rightarrow {\text{real wealth}}\downarrow (\uparrow )\Rightarrow C\downarrow (\uparrow )\Rightarrow AE\searrow (\nwarrow )$
* (interest rate effect) $P\uparrow (\downarrow )\Rightarrow {\text{need more money for transaction}}\Rightarrow {\text{money demand}}(m^{d})\nearrow (\swarrow )(\&\;{\overline {{\text{money supply}}(M^{s})}})\Rightarrow i\uparrow (\downarrow )\left((\&\;{\overline {{\text{inflation rate}}(\pi )}})\Rightarrow r\uparrow (\downarrow )\right)\Rightarrow I^{p}\downarrow (\uparrow )\Rightarrow AE\searrow (\nwarrow )$
* (net export effect) $\uparrow (\downarrow ){\text{ in }}P_{\text{domestic}}>(<)\uparrow (\downarrow ){\text{ in }}P_{\text{foreign}}\Rightarrow X\downarrow (\uparrow )\&\;M\uparrow (\downarrow )\Rightarrow NX\downarrow (\uparrow )\Rightarrow AE\searrow (\nwarrow )$
$\Box$
Remark.
* graphically, since ${\overline {MPC}}$ (ceteris paribus), the slope of AE curve does not change even if $P$ changes, so $P\uparrow (\downarrow )\Rightarrow AE\searrow (\nwarrow )$ parallelly
* the notations $\searrow ,\nwarrow$ mean the direction of shift
Since at macroecnomic equilibrium, $Y=AE$, $P\uparrow (\downarrow )\Rightarrow AE\searrow (\nwarrow )\Rightarrow Y\downarrow (\uparrow )$, and thus we have established the (inverse) relationship between $P$ and $Y$ 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 $P$ and $Y$ is reflected by AD curve.
Definition. (Aggregate demand (AD) curve) AD curve is a curve that shows the (inverse) relationship between $P$ and $AE(=Y{\text{ at macroecnomic equilibrium}}))$, 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 $P$ as $y$-axis and $Y$ as $x$-axis, then AD curve has a negative slope, i.e. it is downward sloping, because of the inverse relationship of $P$ and $Y$
* inverse relationship between $P$ and $Y$ means that when one of them changes, another one changes in opposite direction
* AD curve also consists of every possible point $(P,Q)$ (i.e. every possible pair of $P$ and $Q$)
Illustration of (a portion of) the downward sloping AD curve:
AD curve is essential in the AD-AS model, which will be discussed later.
* ↑ we can do this since $I$ includes both $I^{p}$ 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 $I$ gives $I^{p}$
* ↑ i.e. other currencies that can be exchanged by one dollar of domestic currency $\uparrow$ ($\downarrow$)
* ↑ for each $Y$, $AE$ is above the level at which $AE=Y$, since it lies above the Keynesian cross
* ↑ for each $AE$, $Y$ is below the level at which $Y=AE$, since it lies below the Keynesian cross
* ↑ otherwise, there will not be sufficient inventories for future sale
* ↑ otherwise, there will be too much inventories
* ↑ $Y=C+I+G{\cancel {+NX}}\Rightarrow \underbrace {Y-C-G} _{:=S\;({\text{public savings + private savings}})}=I$
* ↑ $S\uparrow \Rightarrow S_{\text{private}}$ if $\uparrow$ in $S$ is not solely caused by $\uparrow$ in public saving ($S_{\text{public}})$, which should be true.
* ↑ in particular, the point $(P,Q)$ should not be located at the $x$- and $y$-intercept, since it does not make sense for either one of them to be zero practically