# Chapter 7 The IS curve: the interest rate and the goods market equilibrium

In the previous chapter, we examined the demand-side equilibrium of the economy. In doing so, we have already established that economic policy can influence aggregate demand and that this has important consequences for employment in an economy. In this context, we have discussed in particular the important role of fiscal policy. However, we have also already introduced potentially interest-elastic investment into the model. This gives economic policy another instrument to which we will now turn: the central bank’s interest rate policy.

To analyse the effects of interest rate policy in a compact way, we make use of the $$IS$$ curve, which represents the goods market equilibrium as a function of the real interest rate. The real interest rate is determined in the monetary sphere of the economy, e.g. by the central bank.

We will see here that the $$IS$$ curve in itself does not really hold any new information for us. Rather, it is based on the elements introduced in the chapters on aggregate demand and goods market equilibrium as we presented them in chapter 6. The $$IS$$ curve allows us to represent several equilibrium situations in the goods market simultaneously. The different situations differ only with respect to the change of exactly one determinant of the equilibrium, the real interest rate. This is precisely the benefit of the $$IS$$ curve.

It allows us to see how our model economy would react to an economic policy change in the interest rate. The interest rate in our model is still chosen completely freely by the central bank. We will see later that the central bank can set its interest rate policy according to certain rules. In this chapter, however, we first assume that the central bank does not follow any particular rule.

Thus, in this chapter we already have two economic policy instruments at our disposal to influence demand. Through fiscal policy, we determine government spending, and through interest rate policy, we determine the interest rate.

Why “IS”?

The abbreviation “$$IS$$” stands for investment = saving. The reason for this is that the equality of aggregate investment and aggregate saving always holds along the IS curve, i.e., in the goods market equilibrium. We can illustrate this by understanding how the goods market equilibrium is related to the investment, $$I$$, and saving, $$S$$, of the various agents in an economy.

To do this, let us use the simplest model of the demand side of a closed economy with no taxes and no government spending. Demand, output and income of this economy, $$Y$$, are given in goods market equilibrium by, $$Y = C + I$$. At the same time, households in this economy can either consume, $$C$$, or save, $$S$$, the income generated in the production process, i.e., $$Y = C + S$$ also holds. By equating these two equations, we obtain:

$C + I = C + S$

so:

$I = S$

In the goods market equilibrium, aggregate saving thus coincides exactly with the level of aggregate investment. The output not demanded by household saving is demanded by business investment. Since we are always in a goods market equilibrium along the $$IS$$ curve, equality of $$I$$ and $$S$$ also always holds. Hence the name “$$IS$$” curve.

IS curve with government spending:

Let us now again assume our closed economy with government spending, in which no taxes are levied. Thus, government spending here is deficit financed.

$Y = C + I + G$

Households can either use their income for consumption or save it:

$Y = C + S$

We thus obtain:

$C + I + G = C + S$

It holds:

$S = I + G$

In equilibrium, the output not demanded by household saving is demanded by business investment and government consumption.

## 7.1 The interest rate elastic IS curve

As we discussed in chapter 4, the interest rate can be an important determinant of investment and hence aggregate demand. Interest rate decisions by the central bank should normally therefore have an impact on aggregate demand. Even the simple income-expenditure model from chapter 6, extended by an interest rate-sensitive investment function, leads to a an interest-elastic $$IS$$ curve and thus opens up the possibility of active central bank influence on aggregate demand.30

The $$IS$$ curve here represents the goods market equilibrium for different real interest rates under otherwise identical conditions. Since the real interest rate has a dampening effect on investment demand in this model, the relationship represented by the $$IS$$ curve is negative: a higher real interest rate leads to lower equilibrium output, all other things being equal, via falling investment demand and the multiplier process. The shape of the $$IS$$ curve is best understood in conjunction with the Keynesian cross of the goods market. For the graphical representation here, we again use the numerical example of the model from chapter 6. The upper part of the figure 7.1 shows two different equilibrium situations of the goods market in the income-expenditure model. The first equilibrium is determined by the intersection of the blue dotted demand curve and the 45-degree line. Exactly this equilibrium situation is also shown in the lower part of the figure, where the current interest rate of 20% is indicated on the horizontal axis.

Let us now assume that the central bank lowers the interest rate from 20% to 0.5%. Business investment demand will now increase due to the lower interest cost. As we saw in chapter 6, such an increase in demand independent of income is accompanied by an upward shift in the demand function. The new demand curve is given by the blue solid line and its intersection with the 45 degree line determines the new equilibrium. This equilibrium is established by the multiplier process triggered by excess demand. Thus, the interest rate cut to 0.5% has led to an increase in equilibrium GDP to 118. This equilibrium is also shown in the lower part of the figure 7.1.

Figure 7.1: Interest-dependent goods market equilibrium: the IS curve.

The line on which the two equilibrium points are located in the lower part of the figure is the $$IS$$ curve of this model. We have moved from the first to the second equilibrium point on the $$IS$$ curve as a result of the interest rate cut. If we did not lower the interest rate quite so much, say only to 15%, then the demand curve would not shift up quite so much in the Keynesian cross, and the equilibrium on the $$IS$$ curve would be about halfway between the points shown in the figure. The $$IS$$ curve thus represents all possible equilibrium situations that would arise with different interest rates under otherwise identical conditions.

For us, the $$IS$$ curve thus provides a simple way to analyze the impact of interest rate changes on the GDP of our economy without having to shift the demand curve in the income-expenditure model each time. The $$IS$$ curve tells us what interest rate we would need to set in order to achieve a given level of output. Suppose that our economy is hit by a negative demand shock. As a result, the $$IS$$ curve shifts to the left and output and employment fall below their previous levels. We can now respond to this situation with both interest rate policy and fiscal policy. In the interactive app below, negative or positive demand shocks can now be simulated.

## 7.2 The IS curve and the multiplier

In the sections and interactive figures above, we have already seen that the $$IS$$ curve shifts horizontally in the case of a non-interest rate related demand shock. If instead the change in demand is the result of a change in the interest rate, the $$IS$$ curve stays in place and we move along the $$IS$$ curve to a new equilibrium point. But what determines the strength of this movement? How far do we move along the $$IS$$ curve when the interest rate changes? Clearly, the interest rate responsiveness of investment determines the strength of the equilibrium output response to a change in the interest rate. However, since the $$IS$$ curve depicts equilibrium situations, the adjustment process to equilibrium and thus the multiplier also play an important role here. This can best be deduced from the formal representation of the $$IS$$ curve. The $$IS$$ curve is formally determined by the equation of equilibrium GDP. For the simple model of the examples above (equation (6.18)):

$Y^* = \underbrace{\frac{1}{1 - c_Y}}_{\text{multiplier}} \quad \underbrace{(c_a + a_a - a_r r + G)}_{\text{autonomous demand}}$

Since the $$IS$$ curve represents the relationship between output and interest rate, it makes sense to divide the equation into an interest-independent and an interest-dependent term by simply rearranging it:

$Y^* = \underbrace{\frac{1}{1 - c_Y} (c_a + a_a + G)}_{\text{interest rate independent}} - \underbrace{\frac{ a_r }{ 1 - c_Y } r}_{\text{interest rate dependent}}$

Where $$c_Y$$ is again the marginal propensity to consume, which determines the multiplier, and $$a_r$$ is the interest rate sensitivity of investment. We can label the multiplier contained in the equations with $$\mu=\frac{1}{1 - c_Y}$$ to write down the equation a bit more clearly:

$$$Y^* = \mu (c_a + a_a +G ) - \mu a_r r \tag{7.1}$$$

The constants $$c_a$$, $$a_a$$ and $$G$$ are the autonomous components (both from income and the interest rate) of consumption, of investment and government demand. If we collect these autonomous expenditures components multiplied by the multiplier, $$A = \mu(c_a + a_a + G)$$, and denote the product of the multiplier and the interest rate responsiveness of investment as $$\alpha = \mu a_r$$, we can also write down the equation again in simplified form:

$$$Y^* = A - \alpha r \tag{7.2}$$$

where:

$Y^* = \underbrace{A}_{\text{interest rate independent}} - \underbrace{\alpha r}_{\text{interest rate dependent}}$

In this way, we can easily see the connection with the graphical representation of the $$IS$$ curve: The equation plots equilibrium GDP as a function of the central bank’s real interest rate, i.e. exactly the combination of equilibrium points whose connection yields the $$IS$$ curve. The strength of this negative correlation is determined by the parameter $$\alpha = \mu a_r$$. This parameter determines the negative slope of the $$IS$$ curve. The greater the multiplier and/or interest rate responsiveness of investment, the more responsive equilibrium GDP is to a change in the interest rate (i.e. the more it falls when the interest rate rises). Our movement on the $$IS$$ curve will therefore be greater the greater the product of the multiplier and interest rate sensitivity.

The graphical representation of the IS curve

In the above figures we have $$Y^*$$ on the horizontal axis, while the interest rate is plotted on the vertical axis. For this typical representation of the $$IS$$ curve, we simply need to rearrange the equation for equilibrium GDP to $$r$$:

$Y^* = A - \alpha r$

$Y^* - A = - \alpha r$

$\frac{Y^* - A}{- \alpha} = r$

$$$r = \frac{1}{\alpha}A - \frac{1}{\alpha}Y^* \tag{7.3}$$$

Because of the necessary conversion to $$r$$, the $$IS$$ curve with the interest rate on the vertical (y) axis is sometimes called the inverse $$IS$$ curve. The slope of the inverse $$IS$$ curve is thus given here by $$- \frac{1}{\alpha}$$. Rearranging the equation to $$r$$ is a purely formal step for graphical representation. The causality underlying the equation is still the same: the real interest rate is a determinant of equilibrium GDP. The strength of the relationship between equilibrium GDP and the real interest rate is thus still determined by the product of the multiplier and the interest rate responsiveness of investment $$\alpha = \mu a_r$$.

Figure 7.2: Shift of the real interest IS curve after a positive or negative demand shock.

If one or more components of autonomous expenditure in $$A$$ change instead of the real interest rate, the $$IS$$ curve shifts. With the same real interest rate, the result is a different equilibrium GDP. A negative demand shock, e.g. due to lower government final demand, would imply lower equilibrium output for each level of the real interest rate, causing a leftward shift of the $$IS$$ curve. Again, the size of the multiplier determines how far the $$IS$$ curve shifts in response to a demand shock: the larger the multiplier, the further the $$IS$$ curve shifts (equation (7.1)). The changes in the position and slope of the $$IS$$ curve are best illustrated by an interactive example. In the interactive model presented at the end of this chapter, all determinants of the $$IS$$ curve can be freely selected, resulting in a different position of the (inverse) $$IS$$ curve in each case.

## 7.3 When interest rate policy is ineffective - the vertical IS curve

Especially in a severe economic crisis, such as the Great Depression of 1929, the Great Recession of 2007 - 2009 or the current global economic crisis triggered by the Corona pandemic, the steering of aggregate demand by interest rate policy is significantly limited. This phenomenon can be explained, among other things, by the theoretical concept of the “investment trap”. Strongly pessimistic expectations regarding future economic conditions, aggregate demand and the profitability of investment mean that even a significant reduction in the interest rate has little, if any, incentive effect on investment.

We can illustrate a situation in which interest rate policy is completely ineffective for demand management with the simplest income-expenditure model from chapter 6. In this model, the real interest rate is irrelevant for the equilibrium of the goods market. A change in the interest rate therefore does not lead to a change in the goods market equilibrium represented by the $$IS$$ curve; the $$IS$$ curve in this case can also be described as interest rate inelastic. We will return to the economic policy implications of such an $$IS$$ curve below. But first we will look at the graphical representation.

For the simplest income-expenditure model, in the last chapter we used the equilibrium GDP in equation (6.5) as:

$$$Y^* = \frac{1}{\left(1 - c_Y \right)} \left( c_0 + \overline{I} + \overline{G} \right) \tag{6.5}$$$

The first expression in (6.5) is the multiplier and the second expression is the sum of non-income expenditure. Obviously, there is no interest rate in this equation. This means that even if the interest rate were to change in this model economy, the equilibrium output would remain at the value given by the equation (6.5). We can represent this by a fully interest rate inelastic $$IS$$ curve, as figure 7.3 shows.

Figure 7.3: The interest rate inelastic IS curve: the real interest rate has no effect on demand.

In figure 7.3, the interest rate is shown on the vertical axis. The horizontal axis shows the equilibrium output. Since equilibrium output is formed independently of the interest rate, the $$IS$$ curve shown in the figure is vertical. There is therefore no relationship between the interest rate and GDP. The equilibrium is given solely by the influencing factors in equation (6.5). Here we have again used our numerical example from chapter 7 for the numerical values. There, the equilibrium GDP had taken the value 120. As can be seen in the figure, a change in the real interest rate would not change this value.

What does such an $$IS$$ curve now mean for our economic policy instruments? Quite simply, interest rate policy fails as an instrument in the case of an interest rate inelastic $$IS$$ curve. The interest rate policy of the central bank cannot influence the overall economic development by changing the interest rate. In such a situation, fiscal policy is left to its own devices when it comes to coping with a negative demand shock, for example. Let us assume that autonomous household consumption suddenly falls. The equilibrium GDP of the economy falls as a result, which is also accompanied by a decline in employment. How can we now return employment to its old level? Since interest rate policy is ineffective with a vertical $$IS$$ curve, only an economic policy increase in government demand can achieve this goal in this model. This is illustrated in the following interactive app.

The case of a completely interest rate inelastic $$IS$$ curve presented in this section is an exceptional situation. In our example, it results from the fact that none of the demand components reacts to a change in the interest rate. Only the simplest income-expenditure model from the previous chapter excludes any influence of the interest rate. In the further models we will now integrate an effect of interest rate policy on aggregate demand via the investment function.

## 7.4 Delayed effect of interest rate policy

So far, we have assumed that a change in the real interest rate by the central bank has an immediate effect on investment demand and thus simultaneously on the macroeconomic equilibrium. In the simple model with interest rate-dependent investment, the central bank can thus effectively react immediately to a decline in demand as long as investment is interest-elastic. In reality, however, such an immediate effect is hardly to be expected. Rather, it will take some time for a change in interest rates to affect the real economy. We then speak of a delayed interest rate reaction of investment demand (cf. section 4.2). In the investment function we can express this by a so-called “lag effect” of the interest rate. For example, we assume that investments react to a change in the interest rate with a lag of one period (equation (4.7)):

$I = a_a - a_r r_{-1}$

The term $$-1$$ in the index of the real interest rate, $$r_{-1}$$, indicates that this real interest rate is the value of the previous period. The values without time index, $$I, a_a$$ and $$a_r$$, refer to the current period. This investment function thus indicates that the current value of investment demand depends on the interest rate of the previous period.

What are: leads, lags and time series?

In macroeconomics we often deal with dynamic models. Dynamic in this context primarily means that a model has a time dimension. The variables of the model are linked to each other over time. Let’s take the values of real GDP as an example. In each time period, our model will lead to a certain equilibrium. This equilibrium will only change if a determinant of the equilibrium GDP changes. For example, the government might decide to reduce government spending in the next period. Accordingly, GDP in the next period will be lower than before because of the spending cuts, provided that no other determinant of GDP changes.

The value of an economic variable $$x$$ at time $$t$$ is often indicated by an index: $$x_t$$. So the value of output in period $$t$$ (e.g. $$t = 2020$$) would be denoted as $$Y_t = Y_{2020}$$. The same can be applied to a parameter if it changes over time. However, since we already have so many indices for each variable and parameter in our models, we simplify the notation and simply write $$x$$ or $$Y$$ for the value in the current period (e.g. $$Y_{2020} = Y$$). If we want to talk about a past or future value of the same variable from the perspective of the current period, we mark that value with an index $$-p$$ or $$+p$$, where $$p$$ is the exact number of periods we are looking into the past or future. So if the current year is 2021, then we mark the past value of GDP in 2019 as $$Y_{-1}$$ and the future value of GDP in 2021 as $$Y_{+1}$$. The past value of an economic variable, in this case GDP, $$Y_{-1}$$, is called the lag and the future value, $$Y_{+1}$$, is called the lead. The sequence of values for several consecutive periods (here: $$Y_{-1},Y,Y_{+1}$$ or $$Y_{2019},Y_{2020},Y_{2021}$$) is called a “time series”.

Time notation in figures

Above we have clarified time notation for the theoretical representation of the model. For the time notation in the illustrations, however, we follow a slightly different rule. In order to clearly show the time lapse in the figures, we have numerically indexed the different diagram elements (curves, equilibrium values, etc.) (e.g. $$\pi_0$$, $$\pi_1$$, $$\pi_2$$, …), where the subscripted numbers correspond to the different rounds of the model simulations. For a one-time curve shift (e.g. after a price-setting shock), we use the labels “old” and “new” (e.g. $$PS_{old}$$, $$PS_{new}$$). Time notation has been omitted where not strictly necessary.

If we incorporate this new investment function into our previous model, this also leads to a lagged effect of interest rate policy on equilibrium GDP. Here, the lag arises from the reaction of investment, as described above. Once investment changes, the multiplier process leading to the new equilibrium then plays out within one period. While this is somewhat unrealistic, it reduces the dynamic complexity of our model so that we can continue to discuss it using relatively simple methods. The central bank must therefore take into account in its interest rate decisions that it cannot influence GDP in the current period. For example, if the economy is subject to a negative demand shock, the central bank can trigger an expansionary impulse by lowering the interest rate, but its expansionary effect will only take effect in the next period. The following interactive application illustrates this situation.

## 7.5 The IS curve with a nominal interest rate

So far, we have tacitly assumed that the central bank can directly control the real interest rate. In reality, however, the central bank does not control the real interest rate, but the nominal interest rate. This makes little difference to the model world we have discussed so far. Why doesn’t it? To understand this, we can refer to our definition of the real interest rate from section 4.2. There we saw that the relationship between nominal and real interest rates can be approximated by the difference between the nominal interest rate, $$i$$, and the inflation rate, $$\pi$$:

$$$r = i - \pi \tag{7.4}$$$

In our simple goods market equilibria so far, however, we have assumed that firms react to a change in demand with quantity adjustments rather than price changes. This assumption is quite realistic in the short run in goods markets characterised by imperfect competition. With this assumption, the price level is thus constant in the very short run and the inflation rate assumes the value $$\pi = 0$$. This means that in the very short run there is no difference between real and nominal interest rates. So if the central bank changes the nominal interest rate, the real interest rate changes in the same way.

However, since we have introduced a lag of the interest rate in our investment function, it is problematic to maintain the very short run assumption. If it takes some time for a change in the interest rate to affect the real economy, a more or less pronounced change in the price level could occur in the meantime (we will see how this can happen in chapter 9). The inflation rate would then no longer be zero and there would indeed be a difference between real and nominal interest rates. It is therefore appropriate to take this difference into account in interest rate policy.

We have shown the previous $$IS$$ curves for the real interest rate. If there is a difference between $$r$$ and $$i$$ and the central bank can only control $$i$$ directly, then we need a second plot of the $$IS$$ curve with the nominal interest rate to analyse our interest rate policy. The difference between the two $$IS$$ curves is the inflation rate, by whose value we need to shift the $$IS$$ curve for the real interest rates in parallel in order to derive the $$IS$$ curve for the nominal interest rate. Graphically, we can thus illustrate this quite simply as in figure 7.4.

Figure 7.4: IS curve with real interest rate vs. IS curve with nominal interest rate.

The lower $$IS$$ curve shows equilibrium values of real GDP in relation to the real interest rate. The upper $$IS$$ curve runs parallel to the $$IS$$ curve for the real interest rate, with the distance between the two curves given by the inflation rate. We have assumed an inflation rate of 2% as an example. If the central bank now wants to influence the economy in a certain way, it can see directly from the upper $$IS$$ curve what the effect of a change in the nominal interest rate would be if the inflation rate remains at 2%. In most cases, we will show only one of the two $$IS$$ curves for simplicity, but the existence of one curve always implies the existence of the other, and it is therefore important to know the difference.

Formally, we can easily understand this difference. To do this, we can simply insert the approximation equation for the real interest rate, $$r = i-\pi$$, into the $$IS$$ curve of the real interest rate from equation (7.2) to obtain the $$IS$$ curve of the nominal interest rate:

$Y^* = A - \alpha r$ $$$Y^* = A - \alpha (i - \pi) \tag{7.5}$$$

The graphical representation of the IS curve for the nominal interest rate

That the difference between the real interest rate and nominal interest rate $$IS$$ curves consists only in a parallel shift around the inflation rate, $$\pi$$, can be seen from the inverse $$IS$$ curve. We obtain this for the real interest rate by rearranging the equation for the equilibrium output according to $$r$$:

$r = \frac{A}{\alpha} - \frac{Y^*}{\alpha}$

If we now set the $$i-\pi$$ for $$r$$ and convert to $$i$$, we get the inverse nominal interest rate $$IS$$ curve:

$i - \pi = \frac{1}{\alpha}A - \frac{1}{\alpha}Y^*$

$$$i = \frac{1}{\alpha}A - \frac{1}{\alpha}Y^* +\pi \tag{7.6}$$$

The difference between the two curves is the inflation rate, $$\pi$$, as illustrated in the interactive app here below.

Finally, the following app illustrates how fiscal and monetary policy can respond to a shock in aggregate demand.

## Further reading on chapter 7

Textbooks:

### Literature

Carlin, W., and D. W. Soskice. 2015. Macroeconomics: Institutions, Instability, and the Financial System. Oxford University Press.
Heine, M., and H. Herr. 2013. Volkswirtschaftslehre: Paradigmenorientierte Einführung in Die Mikro- Und Makroökonomie. 4. Aufl. München: Oldenbourg.

1. One could, of course, think of other demand components that could be influenced by the interest rate, e.g., credit-financed consumption.↩︎