What happens to the LM curve when the interest rate is fixed?

Question

Suppose the interest rate $i$ is fixed at $\bar{i}$. How does the LM curve look like in this case? I suspect that it looks like a vertical line on the $Y$ (x axis) v/s $i$ (y axis) graph. I think so because $\bar{i} = \frac{1}{\theta_1}(\theta_2 Y - \frac{M}{P})$ for some $\theta_1, \theta_2 \in (0,1)$. Please verify this and also tell me why the output gets fixed when the interest rate is fixed.

Answer 1

The LM curve would not be a vertical line, it would be the same as always, unless we have some deeper knowledge of why the interest rate i is fixed, i.e. it is fixed partially because for some reason the interest rate and money demand have become unrelated.

I believe you may be confusing points and curves, which is very common for students in this area.

The LM curve shows the combinations of interest rates and levels of real income for which the money market is in equilibrium. It shows where money demand equals money supply.

When the interest rate is fixed by some kind of market intervention, that just means we are enforcing a point on the cure. But in the absence of the intervention, those other points still hold, because they still represent an equilibrium. So the curve is the same, because the curve is just the collection of all possible points, which are still valid in general.

The LM curve would be vertical when changes in the interest rate have no effect on the demand for money. But that is not necessarily the case, just because the interest rate is being fixed, e.g. by the Central Bank, because the latent demand for money still has the same relationships and determinants.

In other words, just because we have policies to keep an interest rate fixed somehow (e.g. by shifting IS-LM curves), does not mean that the fundamental relationship between interest rates and money demand have changed. As the curve depicts these fundamental relationships, the curve would be unaffected by a fixed interest rate.