The AE Model

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This is an article from Macroeconomics. To get back to the list of all topics click here.

[1]The AE (Aggregate Expenditure) model is an attempt to explain contractionary and expansionary gaps in the economy. It is based on the ideas by economist John Maynard Keynes, who argued that there is room for governments to intervene in the economy in order to reduce output gaps.

[2]Since the analysis that leads to the theory is quite cumbersome, it is left to later in this article, but the basic outline of the theory is that:

  1. From microeconomics, we know that firms will respond to changes in demand for their products by changing prices
  2. However changing prices on a regular basis (i.e. in the short run) introduces menu costs, which may be larger than the costs of reduced revenues from reduced demand
  3. As a consequence, firms would rather change their levels of output as a response to changes in demand (or aggregate spending)
  4. This change in output affects GDP (output) and (from Okun's Law) also unemployment rates
  5. This explains fluctuation in GDP and hence output gaps
  6. Hence, in order to reduce output gaps in the short run, there is incentive for governments to intervene and affect the level of demand (aggregate spending) in the economy to reduce output gaps
  7. In the long run however firms will eventually respond to changes in demand by changing their prices so that on the whole output gaps are small

The following article explains the full analysis and conclusions of the model.

Contents

Textbook Readings

Bernanke, Olekalns and Frank, Principles of Macroeconomics, (3rd ed, Sydney, McGraw Hill, 2011), pp. 122 - 124 and 130 - 160 (End of Chapter 4 and all of Chapter 5).

Key Assumption

[3]The key assumption of the AE model is that in the short run firms do not change their prices to meet demand for their products. Rather, firms change their level of output to meet the demand. This is because changing prices whenever demand changes involves menu costs. An important note is that menu costs will not prevent firms from changing their price indefinitely, only in the short run.

Aggregate Expenditure

[4]In order to understand what happens when firms change their output as opposed to price (on an aggregate level), we look at the difference between actual and planned output. From the definition of GDP, we know that the national accounting identity is:

Y = AE = C + I + G + NX

And since this is our method of calculating GDP by the expenditure method, we can say that this is a measure of aggregate expenditure (AE) or actual GDP. Notice that in this method, inventories that are produced but not sold are included under investment, as if firms have "bought" these inventories. Under this calculation method, expenditure in the economy is equal to production. However this may not be the case: if demand falls and overall expenditure falls, then sales of inventory falls without it affecting our value of GDP (since any unsold products are sill used under investment). In order to see the difference between actual and planned aggregate expenditure, we define planned aggregate expenditure (PAE) as:

PAE = C + IP + G + NX

Where:

IP = Planned Investment

And the difference between AE and PAE is the change in inventories (that is the inventories that were not sold but are held by firms as "investment").

Now we move on to see what determines this difference.

Consumption and Expenditure

Consumption as a Function of Disposable Income

[5]A large part of aggregate expenditure (nearly two thirds) is spending by households (C). We therefore begin the analysis of the difference between PAE and AE by analysing how much households will consume.

We assume that households consume at a rate relative to their disposable income, that is their total income after tax. On an aggregate level, we say that actual GDP (Y) corresponds to total income so that the aggregate disposable income is given by (Y - T) and that as (Y- T) increases, so does spending. However, consumption is also dependant on other factors which are independent of disposable income. These factors are known as exogenous and are denoted as ExogenousConsumption1.png. Hence, the consumption function is given by:

C = ExogenousConsumption1.png + c(Y - T)

Where:

c = marginal propensity to consume- the amount by which consumption will increase if disposable income were to increase by one dollar. Normally, 0 < c < 1.

[6]Notice that consumption is dependent on the exogenous term (which is defined outside of this model but assumed constant here) and GDP (Y), which is known as induced expenditure. Now if PAE depends on consumption, but consumption is dependent on output, then we can say that PAE is dependent on output. That is, overall spending in the economy is dependent on output.

Two Sector Model

Investment, Consumption and PAE as functions of Output

[7]To put the relationship between PAE and consumption in a mathematical way, we begin with the assumption that the economy only consists of two sectors: households and businesses. Then:

PAE = C + IP

But, C = ExogenousConsumption1.png + cY (since no government to pay tax to) so by substitution:

PAE = ExogenousConsumption1.png + cY + IP

By rearranging:

PAE = [ ExogenousConsumption1.png + IP] + cY

Now, in order to find the equilibrium output, we simply say that PAE = Y yielding:

Y - cY = [ ExogenousConsumption1.png + IP]

and therefore:

Ye = [1/(1-c)][ ExogenousConsumption1.png + IP]

Graphically speaking, this can be explained in the following way: Investments are assumed to be constant so that it is just a flat line while consumption is an upward sloping function of output. Since PAE is an addition of the two, it is seen as upward sloping function of output as well. The equilibrium point is wherever PAE = Y, or alternatively, wherever a 45° line intersects the PAE line.

Withdrawals and Injections

Savings and Investment as Functions of Output

[8]Another way of looking at the equilibrium output is by using the understanding of injections to and withdrawals from the economy. Under the two sector model, the amount being injected into circulation in the economy is equal to the amount invested. Conversely, the amount withdrawn from circulation is the amount that is saved. Noting that savings are given by income minus consumption yields:

PAE = C + IP

But under equilibrium, PAE = Y. By rearranging:

Y - C = IP

But Y- C = Savings. Therefore under equilibrium:

S = IP

Or in other words, WD = INJ (withdrawals equal injections).

Now the Savings function can be derived by noting that S = Y - C. But C = ExogenousConsumption1.png + cY yielding:

S = Y -ExogenousConsumption1.png - cY
S = -ExogenousConsumption1.png + (1-c)Y

This is interpreted as an upward sloping line, but because ExogenousConsumption1.png is negative, the savings function begins below the 0 line. Now since equilibrium occurs when S = I (or when INJ = WD), graphically equilibrium is attained at the intersection of the Savings and Investment curves.

Explaining Output Gaps

[9]From this analysis, it can be seen that expenditure is related to output. Now there can exist times when PAE does not equal Y, so that there is diesquilibrium. Under this model we can say that if:

PAE > Y then firms have too many products left over, and since they cannot change prices, they will attempt to meat the lowered aggregate consumption by reducing production which in turn will reduce GDP, moving towards the equilibrium point
PAE < Y then firms have not met the increased demand for their products. As a result, firms will increase production which in turn increases GDP, moving towards the equilibrium point

Hence the model explains why fluctuations in GDP and output gap occur (in the short run).

Four Sector Model

[10]We now move on to try expanding the analysis to the full economy. That is, we reintroduce the government (G) and International (NX) sectors. That is:

PAE = C + IP + G + NX
PAE = C + IP + G + X - M

We have already shown that

C = ExogenousConsumption1.png + c(Y - T)

and we can do the same analysis for T and for M (imports). We say that:

T = Ť + tY and
M = m(Y - T)

Where:

Ť is the exogenous value of tax
t = marginal tax rate (0 < t < 1)
m = marginal propensity to import (0 < m < 1)

Essentially saying that both tax (or transfer payments) and imports are functions of output. Now by subbing these values into PAE equation and rearranging, we can say that:

PAE = [ ExogenousConsumption1.png - cŤ + IP + G + X + mŤ] + (c - m)(1 - t)Y

Noting that in equilibrium PAE = Y gives:

FourSectorEquilibrium.png

The Multiplier

[11]Notice that graphically, PAE looks the same- it is still an increasing function of Y but quite flat. This flatness results in a heavy increase in GDP (Y) for very small increases in any of the exogenous variables (ExogenousConsumption1.png, Ť, IP, G or X), or PAE. This is due to the multiplier, defined as:

Multiplier = Multiplier.png

Where all c, m and t are defined as being between zero and one. Hence the term on the denominator (1-(c-m)(1-t)) of the multiplier becomes less than one. Now 1 divided by a number less than one yields a result that is greater than one. Hence any small changes in the exogenous factors will be multiplied to give larger values for Ye.

Explaining GDP Fluctuations

As can be seen by the equation for equilibrium GDP for the four sector model, deviation from the equilibrium point can be due to changes in:

  1. Exogenous variable (ExogenousConsumption1.png, Ť, IP, G or X) or
  2. Parameters (c, t and m)

The conclusion is that in order to reduce output gaps, governments can intervene as they can control government spending (G) or the transfer payments (T) as well as the parameter t. Government policies on intervention in the economy are the next topic.

References

"Textbook" refers to Bernanke, Olekalns and Frank, Principles of Macroeconomics, (3rd ed, Sydney, McGraw Hill, 2011).

  1. Textbook p. 130
  2. Textbook p. 122
  3. Textbook p. 131
  4. Textbook p.133
  5. Textbook p. 134
  6. Textbook p.136
  7. Textbook p.142
  8. Textbook p. 138
  9. Textbook p. 146
  10. Textbook p.149
  11. Textbook p. 157
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