Marshallian Static Stability

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Let $P_d=D^{-1}(q)$¹ and $P_S=S^{-1}(q)$.²

We defined excess demand price as the difference between $P_d$ and $P_S$

$$F(q)=P_d-P_S=D^{-1}(q)-S^{-1}(q)$$

Marshall assumes that if excess demand price is positive, then the seller will rise the quantity, $q\uparrow$ . Similarly, if excess demand price is negative then the seller will reduce the quantity.

Then the excess demand price function is an increasing function of its derivative.³

$$\begin{array}{rlrl}\frac{dF(q)}{dq}& =D^{-1}(q{)}^{\prime }-S^{-1}(q{)}^{\prime }& <& 0\\ & =(D^{-1}(q){)}^{\prime }& <& S^{-1}(q{)}^{\prime }\\ & =\frac{1}{(D(q){)}^{\prime }}& <& \frac{1}{S(q{)}^{\prime }}\\ & =\frac{1}{\text{slope of }D}& <& \frac{1}{\text{slope of }S}\end{array}$$

Hence, there will be a static stability according to the Marshallian assumption, if the reciprocal of the slope of the demand function is less than the reciprocal of the slope of the demand function.⁴

Thus it is true for normal goods and so, for both the approaches (Walrasian and Marshallian ) there will be static stability in the market for a normal good.

For a non-normal good, if there is stability according to one assumption, there may not be stability according to another assumption, since one implies the opposite of the other.

Footnotes

1. Inverse Demand Function ↩

2. Inverse Supply Function ↩

3. The behavior of price level when there is a difference between the demand price and the supply price.

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   ↩

4. We will never graph the inverse demand function for this purpose. ↩