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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 ? whether P will 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.

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  • $\begingroup$ 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
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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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