Unit 3 Aggregate demand and the multiplier model
3.8 The multiplier model: Including the government and net exports
In this section, we add the government and the rest of the world to the model, and provide a more detailed model of investment. As before, we assume that firms are willing to supply any amount of goods demanded, so that in equilibrium:
\[\begin{align} \text{output} &= \text{aggregate demand} \\ Y &= \text{AD} \end{align}\]When we include the government and interactions with the rest of the world through exports and imports, aggregate demand can be split into these components:
\[\begin{align} \text{aggregate demand} &= \text{consumption} \\ &+ \text{planned investment} \\ &+ \text{government spending} \\ &+ \text{net exports} \end{align}\]To understand the aggregate demand function, it is useful to go through each component in turn:
Consumption
To simplify the multiplier model, we ignore the role of transfers such as unemployment benefits. To include them, you can interpret the tax rate as net of income-related transfers, and add an extra term for non income-related transfers to disposable income.
Our model of household consumption is essentially as set out previously in Section 3.6, but now depends on post-tax income. The government charges a tax, \(t\), which we assume is proportional to income. The income left after the payment of tax, \((1 − t)Y\), is called disposable income. The marginal propensity to consume, \(c_1\), is now the fraction of disposable income (not pre-tax income) consumed. This means that in the aggregate consumption function:
- Spending on consumption is written as: \(C\) = \(c_0 + c_1 (1 − t)Y\).
- All of the influences on consumption, apart from current disposable income, are again included in autonomous consumption, \(c_0\), and will therefore shift the consumption function in the multiplier diagram. We explore these other determinants in more detail in Section 3.9.
Investment
- interest rate, rate of interest
- The price of bringing buying power forward in time, by borrowing; it is the additional amount that the borrower promises to repay; the rate of interest is the amount of interest to be repaid per period, as a proportion of the loan. See also: nominal interest rate, real interest rate.
The determinants of investment are explored further in Sections 3.11 and 3.12, and, in more depth, in Unit 5. At this stage, we introduce just one additional factor: namely a role for the interest rate. We therefore assume that aggregate investment is given by
\[I = a_0 − a_1r\]Since the price level in the economy changes over time, we need to distinguish between real and nominal GDP. Similarly, inflation affects the cost of borrowing and in Section 5.2 we explain the distinction between real and nominal interest rates. When we refer to the interest rate, \(r\), in this unit, we mean the real interest rate.
- autonomous investment
- In a model of investment demand, autonomous investment is planned investment expenditure that does not depend on other variables in the model (such as income, or the interest rate).
- where \(a_0\) is autonomous investment and \(a_1\) reflects how sensitive investment is to the interest rate.
- Ceteris paribus, we assume that a higher interest rate reduces investment spending. At this stage, we simply focus on the basic intuition that a higher cost of borrowing for firms will make them less willing to borrow the funds they need to undertake investment. We examine this process in more detail in Unit 5.
- All of the influences on investment apart from the interest rate are included in the autonomous investment, \(a_0\). For example, higher expected profits will increase investment spending, which we capture by an increase in \(a_0\). Government investment is included in \(a_0\).
Government spending
Much of government spending (excluding transfers) is on general public services, health, and education. Government spending does not change in a systematic way with changes in income. It is referred to as exogenous to the model.
An increase in government spending shifts the aggregate demand curve up in the multiplier diagram.
Net exports
- marginal propensity to import
- The change in total imports when aggregate income changes by one unit.
The home economy sells goods and services abroad, which are its exports. The amount of foreign goods the home economy demands (its imports) will depend on domestic incomes. In the multiplier model, it is assumed that imports only depend on the level of income. The fraction of each additional unit of income that is spent on imports is termed the marginal propensity to import \((m)\), which is between 0 and 1. So we have:
\[\begin{align} \text{net exports} &= X − M \\ &= X − mY \end{align}\]We also ignore the fact that imports can be driven by exports, which is the case for export-oriented countries that buy their components from suppliers abroad. An increase in exports would then be accompanied by an increase in imports.
If a country’s costs of production fall so that it can sell its goods at a lower price on world markets compared to the prices of other countries, the demand for its exports will increase, and domestic demand for imports will fall. The exchange rate affects the prices of a country’s goods on world markets. Growth in world markets also increases exports. However, for now, we will ignore these effects and assume that imports depend only on income and that exports, like government spending, are exogenous (not explained by the model).
The aggregate demand curve
Putting together each of the components of aggregate demand, we have:
\[\text{AD} = \underbrace{c_0 + c_1(1−t)Y}_{\text{consumption}} + \underbrace{a_0 − a_1r}_{\text{investment}} + G \ + \underbrace{X − mY}_{\text{net exports}}\]As in the previous section, aggregate demand depends on income, \(Y\), and we can draw it in a diagram with \(Y\) on the horizontal axis. We call this graph ‘the aggregate demand function’ or the ‘aggregate demand curve’, although in our model it is a straight line.
Changes in autonomous consumption, \(c_0\), or government spending, \(G\), lead to a parallel shift in the aggregate demand curve as we analysed before. Likewise, an increase in exports, \(X\), or autonomous investment, \(a_0\), would shift the AD curve upward.
Now, the aggregate demand also depends on the interest rate, \(r\). When we draw the diagram of the AD curve, the level of the interest rate determines how high or low AD is at each level of income. We assume that an increase in the interest rate, \(r\), will reduce investment, and hence would cause a parallel downward shift of the AD curve; a decrease in \(r\) would shift it upward.
The multiplier
Previously, the slope of the aggregate demand curve was \(c_1\), the marginal propensity to consume. Now the slope also depends on the tax rate, \(t\), and the marginal propensity to import, \(m\)—so they change the multiplier. Exports and government spending, in contrast, are now additional components of autonomous demand.
Both taxes and imports reduce the size of the multiplier. Recall that the multiplier tells us the amount by which an increase in spending (such as a rise in autonomous consumption, investment, government spending, or exports) raises GDP in the economy. When we include taxation and imports in the model, the indirect multiplier effect of a given rise in spending on GDP is smaller. This is because some household income goes straight to the government as taxation, and some is used to buy goods and services produced abroad. But in the model, we assume that the government does not increase its spending when taxes go up, and foreign buyers do not import more of our goods when we buy more of theirs. So some of the initial increase in income resulting from a demand shock does not lead to further indirect income increases in the domestic economy. The result is to reduce the indirect effects on aggregate demand, output, and employment.
Follow the steps in Figure 3.16 to understand how different components of the aggregate demand equation affect the AD curve.
Figure 3.16 Changes in the AD curve.
Goods market equilibrium
A rise in G
A rise in r
A fall in t
A rise in m
To summarize:
- A higher marginal propensity to import reduces the size of the multiplier: This makes the aggregate demand curve flatter.
- An increase in the tax rate reduces the size of the multiplier: This also makes the aggregate demand curve flatter.
- An increase in exports or government spending shifts the aggregate demand curve up in the multiplier diagram.
Calculating the multiplier in an economy with a government and foreign trade
To calculate the multiplier in the full model, we can again use the fact that there is equilibrium in the goods market when output is equal to aggregate demand. (Equilibrium is where the aggregate demand curve crosses the 45-degree line in the multiplier diagram.) The aggregate demand equation can be rearranged to solve for output, and consequently, the multiplier:
\[\begin{align} \text{output} &= \text{consumption} \\ &+ \text{planned investment} \\ &+ \text{government spending} \\ &+ \text{net exports} \end{align}\]Therefore:
\[\begin{align} \text{AD} &= C + I(r) + G + X − M \\ &= c_0 + c_1(1-t)Y + a_0 − a_1 r + G + X − mY \end{align}\]Here we have written \(I(r)\) as a reminder that investment is a function of \(r\). Rearranging this equation to find \(Y\), we get:
\[\begin{align} Y &= c_0 + c_1(1−t)Y + a_0 - a_1 r + G + X − mY \\ Y(1−c_1(1−t) + m) &= c_0 + a_0 − a_1 r + G + X \\ Y &= \underbrace{\frac{1}{(1 − c_1(1-t) + m)}}_\text{multiplier} \times \underbrace{(c_0 + a_0 − a_1 r + G + X)}_\text{demand that doesn't depend on income} \\ &= k \times (c_0 + a_0 − a_1 r + G + X) \end{align}\]So, in the full model, the multiplier is equal to:
\[k=\frac{1}{1−c_1(1−t)+m}\]The multiplier is smaller when we introduce the government and foreign trade:
\[\begin{align} \frac{1}{(1−c_1(1−t) + m)} < \frac{1}{(1−c_1)} \end{align}\]The reason is that the denominator on the left-hand side is larger than that on the right:
\[\begin{align} 1−c_1(1-t) + m > (1−c_1) \end{align}\]To illustrate, suppose as before that \(c_1=0.6\), but now the tax rate is 20% (0.2) and the marginal propensity to import is 0.1. Substituting these numbers in the formula for the multiplier, we find that \(k = 1.6\), compared to 2.5 without including taxation and imports.
In Unit 5, we investigate how economists have estimated the size of the multiplier from data, why their estimates differ, and why it matters.
Exercise 3.5 The multiplier model
Consider the multiplier model discussed in this section.
- Compare two economies, which differ only in their share of credit constrained households but are identical otherwise. In which economy is the multiplier larger? Illustrate your answer using a diagram.
- On the basis of your comparison of the two economies, would you expect the multiplier in an economy to vary over its business cycle?
Question 3.10 Choose the correct answer(s)
The aggregate demand of an open economy is given by the after-tax domestic consumption, C, the investment, I (which depends on the interest rate r), the government spending, G, and net exports, \((X − M)\):
\[\begin{align} \text{AD} &= C + I(r) + G + X - M \\ &= c_0 + c_1(1-t)Y + I(r) + G + X - mY \end{align}\]\(c_0\) is autonomous consumption, \(c_1\) is the marginal propensity to consume, and m is the marginal propensity to import. In the economy’s equilibrium, this equals its output: AD = Y.
Use the fact that AD = Y in equilibrium to rearrange this equation, with Y on the left-hand side and all other terms on the right-hand side. Given this rearranged equation, which of the following increases the multiplier?
- G affects the level of AD, but not the multiplier.
- r affects I(r), which in turn affects the level of AD but not the multiplier. Investment is not a function of the level of output.
- A rise in t reduces the multiplier.
- A fall in m increases the multiplier. This is because it reduces spending on foreign-produced goods and services.
