Macroeconomics/Aggregate Expenditures

Introduction edit

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

 
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.

Select all correct statement(s).

 .
  if unplanned change in inventories is zero.
  if there is unplanned increase in inventories.
  if there is unplanned decrease in inventories.


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

Consumption edit

Consumption 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.

Select all correct statement(s).

Induced consumption is always greater than  .
 
 
 


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.

Select all correct statement(s).

When households have to pay less taxes,  .
When the government gives (US)  to each household,  .
When most firms are expected to adjust the wages downwards,  
 , i.e. incrase in price level and expected future income together implies decrease in consumption.


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.

Select all correct statement(s).

If firms have more optimistic view on their future profitability,  
Suppose now  .  .
Suppose unplanned change in inventory remains unchanged.  .


Government purchases edit

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

Net exports edit

Change 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.

Select all correct statement(s).

Suppose there is an expansion in the domestic economy such that the domestic price level and GDP increases much faster than foreign price level and GDP, then, domestic  .
Suppose the price level in US   by  , while the foreign price level   by  , then US  .
It is given that foreign products are suddenly very popular among the households, and thus they purchase much more foreign products than before. Then, domestic   ceteris paribus.
Suppose the     faster than   and  . Then, domestic  .


Macroeconomic equilibrium edit

AE function edit

We can plot the AE aginst GDP graph as follows:

 
The   line (with slope  ) is known as Keynesian cross, consisting every point at which  .   in the graph means  .

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 edit

In 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 edit

Recall 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 edit

In 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.

  1. 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  
  2. i.e. other currencies that can be exchanged by one dollar of domestic currency   ( )
  3. for each  ,   is above the level at which  , since it lies above the Keynesian cross
  4. for each  ,   is below the level at which  , since it lies below the Keynesian cross
  5. otherwise, there will not be sufficient inventories for future sale
  6. otherwise, there will be too much inventories
  7.  
  8.   if   in   is not solely caused by   in public saving ( , which should be true.
  9. 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