Answer

Solution

1. 1

   The Slutsky equation gives

2. 2
   $$\frac{\partial x}{\partial p_x}= \underbrace{\frac{\partial x^{h}}{\partial p_x}}_{\text{substitution}\,\le 0}- x\cdot\frac{\partial x}{\partial m}.$$
3. 3

   Since the substitution effect is non-positive, the Marshallian demand $x$ rises when $p_x$ falls unless the income effect is sufficiently negative and dominates. The given observation $\partial x/\partial p_x>0$ (a price fall reduces consumption) implies

4. 4
   $$-x\cdot\frac{\partial x}{\partial m}>-\frac{\partial x^{h}}{\partial p_x}\ge 0\ \Longrightarrow\ \frac{\partial x}{\partial m}<0.$$
5. 5

   That is, $X$ is a Giffen good and hence inferior. When income rises (prices held fixed), consumption of an inferior good falls.

Answer structure / marking notes

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