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A simple version of the new-Keynesian IS cuve is given as follows: $$ \ln Y_{t} =\ln Y_{t+1} -\frac{1}{\theta} r_t. $$ Here $r_t$ is the real interest rate and $1/\theta$ is the cross elasticity of consuming goods at different times. On the book I am reading (Advance Macroeconomics by David Romer, Chapter 6), the equation is derived based on consumption $C$ rather than income $Y$, by considering the marginal utility of comsuming right now vs saving and delaying consumption. Investment/net export is not a part of this model, so $C=Y.$

I have some doubt about this equation. The equation shows a direct relation between real interest rate $r_t$ and real GDP growth rate $\ln(Y_{t+1}/Y_t),$ which does not look quite right, as we often say that there is an inverse relation between GDP and real interest rate for an IS curve. Of course, it is the growth rate rather than the GDP $Y$ itself, but this direct relationship is still kind of hard to explain. Why is this the case?

If we keep $Y_{t+1}$ fixed, then there is an inverse relation between $r_t$ and $Y_t,$ as expected, but it seems unrealistic to hold the future $t+1$ as fixed and predetermined.

In summary, the additional term $\ln Y_{t+1}$ adds in some time-dependence, which makes this model difficult to understand in terms of an IS-LM diagram, which is a diagram at a fixed time.

How to explain the observations I make above about the interest rates?

Edit: It is easy to understand the above in terms of consumption $C,$ but it appears harder to believe when we replace $C$ by $Y,$ for the reasons above.

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As you already mentioned this equation can be derived from the households consumption rule.

the equation is derived based on consumption C rather than income Y, by considering the marginal utility of comsuming right now vs saving and delaying consumption. Investment/net export is not a part of this model, so C=Y

So lets take a look at the Euler condition for the household (where $\pi$ represents inflation, $\beta$ the discount factor and $R_t$ the noninal interest rate) and remember the market clearing condition $C=Y$:

$$\begin{align}C_t^\sigma = \beta^{-1} E \left[\frac{\pi_{t+1}}{R_t} C_{t+1}^\sigma \right]\end{align}$$

You can directely see, that if the household expects future consumption to increase, current consumption will also increase (assuming no changes in inflation and interest rates). This is called consumption smoothing. The households tries to balance consumption over different periods. A rise in nominal interest rate makes saving more attractive and thus households shifts some of the current consumption into future periods.

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  • $\begingroup$ So, since there are no investments, the increase in interest rate will not affect future output negatively? $\endgroup$
    – Ma Joad
    Feb 1 at 11:06
  • $\begingroup$ And can we draw an IS-LM diagram in 2D? The curve is now time-dependent. $\endgroup$
    – Ma Joad
    Feb 1 at 11:12
  • $\begingroup$ The increase in interest rate induces the households to save more i.e. consume less in t and more in t+n. A future increase of demand (consumption) will be met by an equal increase of Y (output). So an increase in interest will have a negative effect on current output and a positive on future output. $\endgroup$
    – Armenthus
    Feb 1 at 19:49

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