### 5.1 The IS curve with expectations

In the following section, the model will be extended to include the time dimension. To simplify matters, the assumption is made that there are only two periods. One is the current period, which means this present year, the other is the future. The IS curve for the current period is modified in such a way that expectations of future developments of consumption and investment are included. Consumption and investment decisions C (Y-T) + I (Y, r)are combined in the total private demand A(Y, T, $r$) Now, the IS curve can be written as follows: Y = A(Y, T, $r$) + G. Please note, we use the real interest rate $r$ instead of the nominal interest rate $r$.
Only now the equation of the IS curve is extended with expectations. We insert the dependence of private demand on the values expected in the future. Y = A (Y, T, $r$, Y’<sup>e</sup>, T’<sup>e</sup>, $r$’<sup>e</sup>) + G describes the equilibrium on the goods market and thus the IS curve with expectations. The dash indicates future values, the superscript e means "expected".
The effects of a tax increase or reduction in government expenditure are the same as for the IS curve without expectations. Private demand A increases with income Y, because both consumption and investment increase with this variable. Private demand decreases with rising taxes because consumption demand decreases and finally private demand decreases with increasing real interest rate r because investments decrease in this case.
If the current or expected income increases, private demand A increases.
If current or expected taxes or the current or expected real interest rate r increases, A decreases.
The following conditions apply to the newly derived IS curve: The IS curve with expectations is much steeper than the IS curve without expectations, which is indicated in the chart in light pink, i.e. with given expectations, a reduction in real interest rates leads only to a small increase in output. In the model without expectations, a change of the interest rate means a change of the entire interest rate curve, that is, a shift in the current and future interest rates. This obviously has greater effects than a change in the current interest rate alone - as in the case of the model with expectations. The graph above illustrates this relationship. If the slider is changed, both the current real interest rate $r$ and the expected real interest rate $r$’<sup>e</sup> are changed, i.e. the entire interest rate curve is changed. This should produce the same effect as for the IS curve without expectations. The change in the current interest rate is shown by the shift of the interest marker on the y-axis. The change in the expected future interest rate causes a shift of the IS curve. The change of the equilibrium point to the different interest rate levels leaves a trace that corresponds exactly to an IS curve without expectations.
In general, changes to the current values have a relatively small effect if they cannot change expectations.
The decline in real interest rates has two consequences: First, a decline in the current real interest rate will not have a strong impact on demand if expectations regarding the future real interest rate remain unchanged, because the present values of the real interest rate hardly change. Companies will not immediately revise their investment plans if the real interest rate falls in the current year. They cannot expect the real interest rate to remain so low in the future. Also, the multiplier effect is probably small, because its magnitude depends on how strongly a change in current income affects demand. If only the current income changes, but expectations about future income remain constant, the effect on demand cannot be very large. Temporary changes in income have little impact on consumer and investment demand.
For example, if consumers expect their income to increase only in the current year, they will increase their consumption proportionately less than the increase in income. If companies fear that their sales figures will only increase in the current year, they will hardly revise their investment plans. A sharp decline in the real interest rate will therefore result in only a small increase in income.

(c) by Christian Bauer
Prof. Dr. Christian Bauer
Chair of monetary economics
Trier University
D-54296 Trier
Tel.: +49 (0)651/201-2743
E-mail: Bauer@uni-trier.de
URL: https://www.cbauer.de