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This was asked in my online midterm test, and I proved it using the fact that a Giffen good and an inferior good behave in the same manner with respect to increase or decrease in purchasing power. Although I know that there exist inferior goods that are also giffen, I'm not able to find a rigorous proof.

Thank you for your help.

Edit 1 :As pointed out in the comments, I feel its necessary to clarify some aspects of the question Firstly, the proof can be mathematical or verbal with the support of a graph. Secondly, this is the original question, it was not given the other way round. The question was preceded by the statement "A Giffen Good is a highly inferior good" and then the question was given.

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    $\begingroup$ Are you sure it is not the other way around? A Giffen good must be an inferior good, but not the other way around. $\endgroup$ Nov 24 '20 at 23:04
  • $\begingroup$ You may like to change the title as pointed out by @MichaelGreinecker. An inferior goods is that which has negative income effect. When the income effect is negative enough to overshadow substitution effect (in magnitude) it becomes a giffen good. So every giffen good is inferior but the opposite is not necessarily true. $\endgroup$
    – Dayne
    Nov 25 '20 at 2:39
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As pointed out in one of the comments, a descriptive answer to the question is available here. However, here OP has asked for a rigorous proof. It is not true that if a good is inferior, it is also Giffen. On the contrary, if a good is Giffen, it is inferior. Here's a proof for this. Note that I am using some of the properties directly as including their proofs will make the answer undesirably long.

First some definitions:

Let $h_i(\mathbf p,u)$ and $x_i(\mathbf p, m)$ be the compensated (or Hicksian) demand function and Marshallian demand function for good $i$, respectively. Here, $\mathbf p$ is the price vector, $u: \mathbb R^n \to \mathbb R$ be the utility function meeting all the regularity conditions (quasi-concavity, continuity, etc.), and $m$ is the income.

Formally,

\begin{align} \mathbf h(\mathbf p,u) = \arg \min\limits_{\mathbf x} \, \mathbf {p \cdot x} \\ \text{such that } u(\mathbf x) \geq u \end{align}

Further, let $e(\mathbf p, u)$ be the expenditure minimizing function for a given minimum utility level $u$. So, $e(\mathbf p, u) = \mathbf{p\cdot h}(\mathbf p,u)$.

For Marshallian demand function:

\begin{align} \mathbf x(\mathbf p,m) = \arg \max\limits_{\mathbf x} \, u(\mathbf x) \\ \text{such that } \mathbf {p \cdot x} \leq m \end{align}

Now, from duality we have:

\begin{align} h_i(\mathbf p,u) \equiv x_i(\mathbf p,e(\mathbf p, u)) \\ \end{align}

Differentiating this equation w.r.t $p_i$:

\begin{align} \frac{\partial h_i(\mathbf p,u)}{\partial p_i} & = \frac{\partial x_i(\mathbf p,e(\mathbf p, u))}{\partial p_i} + \frac{\partial x_i(\mathbf p,e(\mathbf p, u))}{\partial e}\frac{\partial e(\mathbf p, u)}{\partial p_i} \\ \end{align}

In the terms, $ \frac{\partial x_i(\mathbf p,e(\mathbf p, u))}{\partial p_i}, \frac{x_i(\mathbf p,e(\mathbf p, u))}{\partial e}$, $e$ is not really being treated as a function, so we replace with $m$, giving:

\begin{align} \frac{\partial h_i(\mathbf p,u)}{\partial p_i} & = \frac{\partial x_i(\mathbf p,m)}{\partial p_i} + \frac{\partial x_i(\mathbf p,m)}{\partial m}\frac{\partial e(\mathbf p, u)}{\partial p_i} \\ \end{align}

Rearranging gives:

\begin{align} \frac{\partial x_i(\mathbf p,m)}{\partial p_i} & = \frac{\partial h_i(\mathbf p,u)}{\partial p_i} - \frac{\partial x_i(\mathbf p,m)}{\partial m}\frac{\partial e(\mathbf p, u)}{\partial p_i} \tag1 \end{align}

Now, note the following properties:

(1) From convexity of preferences (or quasi-concavity of utility function), $h_i(\mathbf p, u)$ is non-increasing in $\mathbf p$. Therefore, we have that:

$$\frac{\partial h_i(\mathbf p,u)}{\partial p_i} \leq 0$$

(2) $e(\mathbf p, u)$ is non-decreasing in $\mathbf p$. So we have:

$$\frac{\partial e(\mathbf p, u)}{\partial p_i} \geq 0$$

Using these in $(1)$, we can say:

$$\frac{\partial x_i(\mathbf p,m)}{\partial p_i} > 0 \text{ only if}\,\, \,\, \frac{\partial x_i(\mathbf p,m)}{\partial m} < 0$$


A side note: it seems that at some places (in some comments and the descriptive answer sited above) that a Giffen good is the one that does not follow Law of demand. This is not strictly speaking true. The law of demand pertains to Hicksian demand not the Marshallian demand. As it can be seen in the proof, $h_i(\mathbf p, u)$ is always non-increasing in $p_i$, but $x_i(\mathbf p, m)$ may not necessarily be. So the law of demand always hold under the regularity conditions on preferences.

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    $\begingroup$ Interestingly, this answer has had 3 downvotes and 2 upvotes. But none of the downvoters have left a comment citing any reason, something which is usually a practice on other stack exchange sites. $\endgroup$
    – Dayne
    Nov 27 '20 at 16:03

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