Intuition behind the positive slope of LM curve : 'Income must increase'?

Question

Source: by John Floyd PhD (in Economics; U Chicago)

The intuition behind the positive slope of LM is as follows:
An increase in the interest rate reduces the demand for money
and an increase in income increases it [the demand for money].
To [...] [equalize] the demand for money equal to a constant money supply
as the interest rate rises and we move along the LM curve,
the level of income must increase.

Would someone please explain the bolded? It sounds too imperious and presumptuous:
how MUST a rise in interest rate increase consumers' level of income?
Eg, employers can still refuse to increase salary. Then income would NOT increase?

Answer 1

The actors of an economy have a liquidity demand function $L(Y,i)$ where $Y$ is the real income in the economy and $i$ is the nominal interest rate. This function shows how much money the actors keep. (Not wealth, but wealth that is not tied down assets.) It is assumed in the IS-LM model that $L(Y,i)$ is increasing in $Y$ because if you have more income you also spend more and to do this you need more money. It is also assumed that $L(Y,i)$ is decreasing in $i$. If the interest rates are high you have an incentive to deposit more of your wealth and keep less of it liquid/instantly accessible.

Now the points of the LM (Liquidity - Money Supply) curve are those $(Y,i)$ pairs for which the liquidity demand equals the money supply. Suppose that given some money supply $M$ and price level $P$ you have $$ L(Y,i) = \frac{M}{P}, $$ so the market for money (liquidity) is in equilibrium. Then this point (Y,i) is on the LM curve. While we keep the exogenous variables $M$ and $P$ unchanged we would like to see if there is another pair (Y',i') such that $$ L(Y',i') = \frac{M}{P}. $$ Suppose that $i'>i$. Because the function $L$ is decreasing in $i$ this means $$ L(Y,i') < L(Y,i) = \frac{M}{P} = L(Y',i'). $$ Since $L(Y,i') < L(Y',i')$ and $L$ is increasing in $Y$ we have $Y < Y'$.

Thus if there are two points on the same LM curve and the nominal interest rate is higher in one point, income must the higher in that point as well. I believe this is what Floyd meant.

If you dislike dealing with general function forms you may assume $$ L(Y,i) = \frac{Y}{i} \mbox { or } \frac{Y}{1+i}. $$

Answer 2

The intertemporal consumption equation is :

$$Y_{1}+\frac{Y_{2}}{(1+r)} = W$$

where $W$ is total wealth and $Y$ is revenue.

When interest rate rise investiment drops and savings rise, so lets say someone choses to save all his revenu from period 1 (this is just to calculate his total wealth), then in period 2 his total wealth will be :

$$Y_{1}(1+r)+Y_{2} = W(1+r)$$

So as you can see his total wealth in 2nd period grows when interest rate grows

Also we can isolate $Y_{1}$ :

$$Y_{1}= W-\frac{Y_{2}}{(1+r)}$$

And you see that when interest rate rises your revenu in period one rises.

I am not sure if this is a valid answer but it seems correct