# Effect of money supply on price level

Consider a macroeconomy defined by following equations:
$$M = kPy + L(r)$$ $$S(r) = I(r)$$ $$y = m$$
Where $$M$$ is money supply, $$P$$ is price level, $$y$$ is output, $$r$$ is interest rate, while $$k,m$$ are constants. $$S(r)$$ is saving function with $$S'(r) >0$$, $$I(r)$$ is investment function with $$I'(r) <0$$, and $$L(r)$$ is speculative money demand function with $$L'(r) <0$$.

Now how an increase in M affects P? Does P decrease or increase more than proportionately or less than proportionately or proportionately?

I am more interested in approach than in solution. I am not able to understand how to approach this problem.

• How are $L(r)$, $S(r)$, and $I(r)$ related? You could solve for the equlibrium $r$ from the second equation and then differentiate the first, maybe. – Weierstraß Ramirez Jun 21 '18 at 12:34

You can make $P$ the subject of the equation: $$P=\frac{M-L} {ky}$$ then find the derivative $$\frac{\mathrm d P} {\mathrm d M} = \frac 1{ky}>0$$ to show there is a positive relationship (when $M$ increases, $P$ increases). The elasticity answers your question about proportionality: \begin{align*} \frac{\mathrm d P} {\mathrm d M} \cdot \frac MP &= \frac 1{ky}\left( \frac {kPy} P + \frac {L \cdot ky} {M-L} \right) \\ &= 1 + \frac L{M-L} \end{align*} shows that $P$ increases more than proportionally (because the elasticity is greater than 1).