Macroeconomics/Aggregate Expenditures

Introduction

In this chapter, we will discuss Template:Colored em. Its definition is as follows: Template:Colored definition To be more precise, AE means the following: Template:Colored definition Template:Colored remark We will define Template:Colored em in the following. For other expenditures, we have defined them in the chapter about GDP, and they have the same definitions here. Template:Colored definition Template:Colored example Template:Colored 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 Template:Colored em consumption ($\bar{C}$) and Template:Colored em consumption. Template:Colored definition Template:Colored remark Before defining induced consumption, let us define a term which will be used in its definition. Template:Colored definition Template:Colored remark Another similar definition is Template:Colored em (MPS). Template:Colored definition Template:Colored proposition

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, $$\text{wealth}=C+S+T\Rightarrow \mathrm{\Delta }\text{wealth}=\mathrm{\Delta }C+\mathrm{\Delta }S+\mathrm{\Delta }T.$$ ($\Rightarrow$ means 'implies')

When there is change in wealth (which equals $\mathrm{\Delta }Y$), the portion of $\mathrm{\Delta }\text{wealth}=\mathrm{\Delta }Y$, allocated to $\mathrm{\Delta }C$ is determined by MPC, by definition. Since $\mathrm{\Delta }Y=\mathrm{\Delta }C+\mathrm{\Delta }S+\mathrm{\Delta }T$, Template:Colored em, we have $$\mathrm{\Delta }Y=\mathrm{\Delta }C+\mathrm{\Delta }S\Rightarrow \frac{\mathrm{\Delta }Y}{\mathrm{\Delta }Y}=\frac{\mathrm{\Delta }C}{\mathrm{\Delta }Y}+\frac{\mathrm{\Delta }S}{\mathrm{\Delta }Y}\Rightarrow MPC+MPS=1.$$ ($\mathrm{\Delta }Y\ne 0$, since Template:Colored em change in wealth)

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Template:Colored remark Then, we can use MPC to define induced consumption. Template:Colored definition Then, we can express the consumption function as follows: $$C=\bar{C}+MPC\cdot Y_d=f(Y_d),$$ which is a function in $Y_d$, and so the Template:Colored em of consumption function is MPC (which is positive, and so $Y_d\uparrow \Rightarrow C\uparrow$), and Template:Colored em of consumption function is $\bar{C}$ (which is positive). Template:Colored example Template:Colored exercise Then, we will discuss some important factors that affect $C$. Template:Colored proposition

Proof. Ceteris paribus,

• Current disposable income ($Y_d$): $Y_d\uparrow (\downarrow )(\mathrm{\&}\overline{MPC})\Rightarrow C\uparrow (\downarrow )$, which follows from the consumption function
• Household wealth: $\text{wealth}\uparrow (\downarrow )(\iff \mathrm{\Delta }Y>0(<0))\Rightarrow Y_d\uparrow (\downarrow )\Rightarrow C\uparrow (\downarrow )$
• 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): $r\uparrow (\downarrow )\Rightarrow \text{saving earning}\uparrow (\downarrow )\Rightarrow \text{marginal benefit }(MB)\text{ of saving }\uparrow (\downarrow )(\mathrm{\&}\overline{\text{its marginal cost}(MC)})\Rightarrow S\uparrow (\downarrow )(\mathrm{\&}\bar{Y}\mathrm{\&}\bar{T})\Rightarrow C\downarrow (\uparrow )$

• since $\mathrm{\Delta }Y=\mathrm{\Delta }T=0$ (ceteris paribus), $\mathrm{\Delta }Y=\mathrm{\Delta }C+\mathrm{\Delta }S+\mathrm{\Delta }T\Rightarrow \mathrm{\Delta }C=-\mathrm{\Delta }S$, so $S\uparrow \Rightarrow C\downarrow$

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Planned investment

$I^p$ is Template:Colored em, which does not vary with $Y$. The following are some important factors affecting $I^p$ (which does not include $Y$). Template:Colored proposition

Proof.

• ${\mathrm{\Pi }}_e\uparrow (\downarrow )\Rightarrow MB\text{ of firms' investment}\uparrow (\downarrow )(\mathrm{\&}\overline{\text{its }MC})\Rightarrow I^p\uparrow (\downarrow )$
• $CF\uparrow (\downarrow )\Rightarrow \text{ability to finance investment}\uparrow (\downarrow )\Rightarrow I^p\uparrow (\downarrow )$
• $T\uparrow (\downarrow )\Rightarrow {\mathrm{\Pi }}_e\downarrow (\uparrow )\mathrm{\&}CF\downarrow (\uparrow )\Rightarrow I^p\downarrow (\uparrow )$
• $r\uparrow (\downarrow )\Rightarrow MC\text{ of borrowing}\uparrow (\downarrow )(\mathrm{\&}\overline{\text{its }MB})\Rightarrow I^p\downarrow (\uparrow )$

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Government purchases

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

Net exports

Change in $NX$ is mainly affected by the Template:Colored em between domestic country and foreign countries. Template:Colored proposition

Proof.

• $\uparrow \text{in }{P}_{\text{domestic}}>(<)\uparrow \text{in }{P}_{\text{foreign}}\Rightarrow$ domestic goods become more (less) Template:Colored em relative to foreign goods $\Rightarrow X\downarrow (\uparrow )\mathrm{\&}M\downarrow (\uparrow )\Rightarrow NX\downarrow (\uparrow )$
• $\uparrow \text{in }{Y}_{\text{domestic}}>(<)\uparrow \text{in }{Y}_{\text{foreign}}\Rightarrow$ $\uparrow$ in domestic demand for $M$ ($D_M$) $>(<)$ $\uparrow$ in foreign demand for domestic $X$ ($D_X$) $\Rightarrow X\downarrow (\uparrow )\mathrm{\&}M\uparrow (\downarrow )\Rightarrow NX\downarrow (\uparrow )$
• $er\uparrow (\downarrow )\Rightarrow$ $M$ are less (more) expensive, and $X$ are more (less) expensive $\Rightarrow M\uparrow (\downarrow )\mathrm{\&}X\downarrow (\uparrow )\Rightarrow NX\downarrow (\uparrow )$

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Macroeconomic equilibrium

AE function

We can plot the AE aginst GDP graph as follows:

The $45^{\circ }$ line (with slope $1$) is known as Template:Colored em, consisting every point at which $Y(\text{real GDP})=AE$. $AD$ in the graph means $AE$.

The blue line can be interpreted as the $AE$ curve with $I^p=G=NX=0$, i.e. the consumption function $C=\bar{C}+(MPC)Y$.

Recall that $AE+C+I^p+G+NX$. Since $I^p,G$ are Template:Colored em, and $NX$ does not vary, ceteris paribus (the comparison between domestic country and foreign countries gives same results), we may denote them as $\overline{I^p},\bar{G},\overline{NX}$, 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 $\overline{I^p},\bar{G}$ and $\overline{NX}$ to the consumption function (which shifts the blue line Template:Colored em by $\overline{I^p}+\bar{G}+\overline{NX}$ parallelly, since the $y$-intercept changes from $\bar{C}$ to $\bar{C}+\overline{I^p}+\bar{G}+\overline{NX}$): $$AE=\underset{C}{\underbrace{\bar{C}+MPC\cdot Y}}+\overline{I^p}+\bar{G}+\overline{NX}=MPC\cdot Y+\text{constant}=f(Y)$$

We can observe that, at the region Template:Colored em the Keynesian cross, $Y<AE$ , and at the region Template:Colored em 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

Sometimes, the economy is Template:Colored em at macroeconomic equilibrium, i.e. $AE>Y$ or $AE<Y$. Let's examine these two cases one by one.

Since $$AE>Y\Rightarrow \cancel{C+}{I}^p\cancel{+G+NX}>\cancel{C+}I\cancel{+G+NX}\Rightarrow I^p>I\Rightarrow I-\mathrm{\Delta }in{v}^{unp}>I\Rightarrow \mathrm{\Delta }in{v}^{unp}<0$$, there is unplanned Template:Colored em 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 $$AE<Y\Rightarrow \cancel{C+}{I}^p\cancel{+G+NX}<\cancel{C+}I\cancel{+G+NX}\Rightarrow I^p<I\Rightarrow I-\mathrm{\Delta }in{v}^{unp}<I\Rightarrow \mathrm{\Delta }in{v}^{unp}<0$$, there is unplanned Template:Colored em 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. $$Y=AE\Rightarrow \cancel{C+}{I}^p\cancel{+G+NX}=\cancel{C+}I\cancel{+G+NX}\Rightarrow I^p=I\Rightarrow I-\mathrm{\Delta }in{v}^{unp}=I\Rightarrow \mathrm{\Delta }in{v}^{unp}=0,$$ and thus there is no unplanned change in inventories, and so $\bar{Y}$ 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 $AE=Y<(>)Y^p\Rightarrow \text{unemployment rate }(u)>(<)\text{ natural unemployment rate }(\bar{u})$.

Also, at macroeconomic equilibrium, $$Y=AE\Rightarrow Y=MPC\cdot Y+\bar{C}+\overline{I^p}+\bar{G}+\overline{NX}\Rightarrow (1-MPC)Y=\bar{C}+\overline{I^p}+\bar{G}+\overline{NX}\Rightarrow Y=\frac{\bar{C}+\overline{I^p}+\bar{G}+\overline{NX}}{1-MPC}$$

The multiplier effect

In view of the above equation at macroeonomic equilibrium, when the Template:Colored em expenditure (variables with a bar on top of it) changes by $1$, $Y$ changes by $\frac{1}{1-MPC}$ in the same direction. Since $0<MPC\le 1$, this number is greater than one, and we give this number a name, namely Template:Colored em: Template:Colored definition Template:Colored remark

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 Template:Colored em, 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 $S\uparrow \Rightarrow \text{loanable fund supply}\searrow (\mathrm{\&}\overline{\text{loanable fund demand}})\Rightarrow r\downarrow \Rightarrow I\uparrow$, which Template:Colored em offset the $\downarrow$ in $C$.

Aggregate demand (AD) curve

In the following, we will loosen the assumption that $\bar{P}$. After that, we can use the AE curve to derive aggregate demand (AD) curve.

$P$ affects AE as in the following proposition: Template:Colored proposition

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) $P\uparrow (\downarrow )\Rightarrow \text{need more money for transaction}\Rightarrow \text{money demand}(m^d)\nearrow (\swarrow )(\mathrm{\&}\overline{\text{money supply}(M^s)})\Rightarrow i\uparrow (\downarrow )((\mathrm{\&}\overline{\text{inflation rate}(\pi )})\Rightarrow r\uparrow (\downarrow ))\Rightarrow I^p\downarrow (\uparrow )\Rightarrow AE\searrow (\nwarrow )$
• (net export effect) $\uparrow (\downarrow )\text{ in }{P}_{\text{domestic}}>(<)\uparrow (\downarrow )\text{ in }{P}_{\text{foreign}}\Rightarrow X\downarrow (\uparrow )\mathrm{\&}M\uparrow (\downarrow )\Rightarrow NX\downarrow (\uparrow )\Rightarrow AE\searrow (\nwarrow )$

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Template:Colored remark 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. Template:Colored definition Template:Colored remark Illustration of (a portion of) the downward sloping AD curve:

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

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