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.

  • $\begingroup$ 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. $\endgroup$ – 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).

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