# Macroeconomics/Aggregate Supply
## Introduction

Aggregate supply (AS) curve shows the relationship between $P$ of the economy and $Y$ supplied of the economy. SRAS is the AS curve in SR, and LRAS is the AS curve in LR.
Definition. (SRAS curve) SRAS curve describes the relationship between $P$ and $Y$ firms are willing to supply, holding constant all other variables that affect the willingness of firms to supply goods and services.
The following proposition states the shape of SRAS and LRAS curves.
Proposition. (Shape of SRAS and LRAS curves) SRAS curve is upward sloping (i.e. with positive slope), and LRAS curve is vertical.
Proof.
* SRAS is upward sloping
   * main: sticky-price and sticky-wage theory: prices (e.g. input cost) and nominal wages ($w^{n}$) do not respond quickly to $\Delta P$
      * e.g. $w^{n}$ may be sticky because of contract in which the $w^{n}$ may be fixed for a certain period. Therefore, when output price increases, while the production cost, which includes the $w^{n}$ paid, is sticky and $\uparrow$ more slowly than output price, the profitability $\uparrow$ . Therefore, firms are willing to supply more goods and services, which causes $\uparrow$ in $Y$ supplied.
   * slow $w^{n}$ adjustment
      * e.g. salary may be reviewed annually only, and firms may dislike cutting $w^{n}$
   * menu costs making some prices sticky: it costs firms money to change prices stated at the menu, so slight change in price may not be worthy, causing the price to be sticky, then having the effects from sticky price theory
      * e.g. printing cost
   * misperception theory: Some firms may fail to accurately predict $\Delta P$ . Therefore, they may be slow to $\uparrow$ its output prices when $P\uparrow$. Consequently, customers find that their goods and services are cheaper, and buy more. Then, firms' sale $\uparrow$, and therefore they $\uparrow$ their production because of this signal
* LRAS is vertical
   * $Y$ in LR, which is known as potential GDP ($Y^{p}$) (or full employment GDP), is determined by the number of workers ($N$), capital stocks ($K$) and the available technology. These factors are not affected by $P$. Therefore, $Y=Y^{p}$ for each $P$, and thus LRAS curve is vertical.
$\Box$
We can see from above that LRAS curve is vertically located at $Y^{p}$, and $Y$ supplied is not affected by $\Delta P$.
Illustration of plotted SRAS and LRAS curves:

## Movements of SRAS curve

A change in $P$ not caused by factors that would otherwise affect SRAS results in movement along a stationary SRAS curve. However, some factors shift SRAS curve.
Illustration of $\nearrow$ movement along SRAS curve:
Illustration of $\searrow$ shift of SRAS curve from $SRAS_{0}$ to $SRAS_{1}$:

### Shift of SRAS curve

Proposition. (Factors shifting SRAS curve)
* (positively related) $N,K$ or available technology $\uparrow (\downarrow )\Rightarrow SRAS\searrow (\nwarrow )$
* (negatively related) expected future price level ($P^{e}$) $\uparrow (\downarrow )\Rightarrow SRAS\nwarrow (\searrow )$
* positive (negative) supply shock (i.e. unexpected event that causes SRAS to increase (decrease), typically through $\uparrow (\downarrow )$ production cost) $\Rightarrow SRAS\searrow (\nwarrow )$
* adjustments to errors in past workers' and firms' $P^{e}\Rightarrow SRAS\searrow {\text{ OR }}\nwarrow )$
Proof.
* $N,K\uparrow (\downarrow )\Rightarrow Y{\text{ supplied}}\uparrow (\downarrow ){\text{ for each }}P\Rightarrow SRAS\searrow (\nwarrow )$
* available technology $\uparrow (\downarrow ({\text{unlikely}}))\Rightarrow$ productivity $\uparrow (\downarrow )\Rightarrow {\text{production cost}}\downarrow (\uparrow )\Rightarrow SRAS\searrow (\nwarrow )$
* $P^{e}\uparrow (\downarrow )\Rightarrow {\text{workers' demand for }}w^{n}\uparrow (\downarrow )\Rightarrow {\text{production cost}}\uparrow (\downarrow )\Rightarrow SRAS\nwarrow (\searrow )$
* positive (negative) supply shock $\Rightarrow {\text{production cost}}\uparrow (\downarrow )\Rightarrow SRAS\searrow (\nwarrow )$
* adjustments to errors in past $P^{e}$:
   * $P\uparrow$ unexpectedly: workers and firms realize that they underestimate $P\Rightarrow P^{e}\uparrow ({\text{adjustment}})\Rightarrow SRAS\nwarrow (\searrow )$
   * $P\downarrow$ unexpectedly: workers and firms realize that they overestimate $P\Rightarrow P^{e}\downarrow ({\text{adjustment}})\Rightarrow SRAS\searrow (\nwarrow )$
$\Box$

## Movements of LRAS curve

Since $Y=Y^{p}$ for each $P$, changing $P$ does not have any effect on LRAS curve. Therefore, arbitrary $\Delta P$ would only cause movement along a stationary LRAS curve ($\Delta P$ may be caused by various reasons). However, some factors shift LRAS curve by changing $Y^{p}$ .
Illustration of $\uparrow$ movement along LRAS curve:
Illustration of $\rightarrow$ shift of LRAS curve from $LRAS_{0}$ to $LRAS_{1}$:

### Shift of LRAS curve

Proposition. (Factors shifting LRAS curve)
* (positively related) $N,K$ or available technology $\uparrow (\downarrow )\Rightarrow LRAS\rightarrow (\leftarrow )$
Proof. $N,K$ or available technology $\uparrow (\downarrow )\Rightarrow Y^{p}\uparrow (\downarrow )\Rightarrow LRAS\rightarrow (\leftarrow )$
$\Box$

## Functions of aggregate supply (optional)

There are different functions of short run aggregate supply based on different models. We will give one example in the following. In the short run, a function which describes the aggregate supply which is known as Lucas aggregate supply function is
    $Y=f(P-P^{e})=Y^{p}+\alpha (P-P^{e})$
in which coefficient $\alpha$ is a positive real number. This is a function of $P-P^{e}$ which is known as 'price surprise'.
In the long run, the level of real GDP simply equals its natural level, i.e.
    $Y=Y^{p},$
and therefore LRAS curve is vertical.