<p>As pointed out in one of the comments, a descriptive answer to the question is available <a href="https://economics.stackexchange.com/questions/25323/difference-between-giffen-and-inferior-goods-why-arent-all-inferior-goods-giff">here</a>. 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.</p>
<p>First some definitions:</p>
<p>Let <span class="math-container">$h_i(\mathbf p,u)$</span> and <span class="math-container">$x_i(\mathbf p, m)$</span> be the compensated (or Hicksian) demand function and Marshallian demand function for good <span class="math-container">$i$</span>, respectively. Here, <span class="math-container">$\mathbf p$</span> is the price vector, <span class="math-container">$u: \mathbb R^n \to \mathbb R$</span> be the utility function meeting all the regularity conditions (quasi-concavity, continuity, etc.), and <span class="math-container">$m$</span> is the income.</p>
<p>Formally,</p>
<p><span class="math-container">\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}</span></p>
<p>Further, let <span class="math-container">$e(\mathbf p, u)$</span> be the expenditure minimizing function for a given minimum utility level <span class="math-container">$u$</span>. So, <span class="math-container">$e(\mathbf p, u) = \mathbf{p\cdot h}(\mathbf p,u)$</span>.</p>
<p>For Marshallian demand function:</p>
<p><span class="math-container">\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}</span></p>
<p>Now, from duality we have:</p>
<p><span class="math-container">\begin{align}
h_i(\mathbf p,u) \equiv x_i(\mathbf p,e(\mathbf p, u))  \\
\end{align}</span></p>
<p>Differentiating this equation w.r.t <span class="math-container">$p_i$</span>:</p>
<p><span class="math-container">\begin{align}
\frac{\partial h_i(\mathbf p,u)}{\partial p_i} &amp; = \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}</span></p>
<p>In the terms, <span class="math-container">$ \frac{\partial x_i(\mathbf p,e(\mathbf p, u))}{\partial p_i}, \frac{x_i(\mathbf p,e(\mathbf p, u))}{\partial e}$</span>, <span class="math-container">$e$</span> is not really being treated as a function, so we replace with <span class="math-container">$m$</span>, giving:</p>
<p><span class="math-container">\begin{align}
\frac{\partial h_i(\mathbf p,u)}{\partial p_i} &amp; = \frac{\partial x_i(\mathbf p,m)}{\partial p_i} + \frac{\partial x_i(\mathbf p,m)}{\partial m}\frac{\partial e(\mathbf p, u)}{\partial p_i}  \\
\end{align}</span></p>
<p>Rearranging gives:</p>
<p><span class="math-container">\begin{align}
\frac{\partial x_i(\mathbf p,m)}{\partial p_i} &amp; =  \frac{\partial h_i(\mathbf p,u)}{\partial p_i} - \frac{\partial x_i(\mathbf p,m)}{\partial m}\frac{\partial e(\mathbf p, u)}{\partial p_i}  \tag1
\end{align}</span></p>
<p>Now, note the following properties:</p>
<p>(1) From convexity of preferences (or quasi-concavity of utility function), <span class="math-container">$h_i(\mathbf p, u)$</span> is non-increasing in <span class="math-container">$\mathbf p$</span>. Therefore, we have that:</p>
<p><span class="math-container">$$\frac{\partial h_i(\mathbf p,u)}{\partial p_i} \leq 0$$</span></p>
<p>(2) <span class="math-container">$e(\mathbf p, u)$</span> is non-decreasing in <span class="math-container">$\mathbf p$</span>. So we have:</p>
<p><span class="math-container">$$\frac{\partial e(\mathbf p, u)}{\partial p_i} \geq 0$$</span></p>
<p>Using these in <span class="math-container">$(1)$</span>, we can say:</p>
<p><span class="math-container">$$\frac{\partial x_i(\mathbf p,m)}{\partial p_i} &gt; 0 \text{     only if}\,\, \,\, \frac{\partial x_i(\mathbf p,m)}{\partial m} &lt; 0$$</span></p>
<hr />
<p>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 <strong>Law of demand</strong>. This is not strictly speaking true. The law of demand pertains to <em>Hicksian</em> demand not the Marshallian demand. As it can be seen in the proof, <span class="math-container">$h_i(\mathbf p, u)$</span> is <strong>always</strong> non-increasing in <span class="math-container">$p_i$</span>, but <span class="math-container">$x_i(\mathbf p, m)$</span> may not necessarily be. So the law of demand always hold under the regularity conditions on preferences.</p>

CC BY-SA 4.0
2020-11-27T16:03:34.427Z
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.
Dayne
CC BY-SA 4.0
2020-11-25T06:01:32.067Z
2020-11-25T06:19:22.207Z
2020-11-25T06:19:22.207Z
Dayne
<p>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.</p>
<p>Thank you for your help.</p>
<p>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 &quot;A Giffen Good is a highly inferior good&quot; and then the question was given.</p>

CC BY-SA 4.0
2020-11-24T23:04:17.087Z
Are you sure it is not the other way around? A Giffen good must be an inferior good, but not the other way around.
Michael Greinecker
CC BY-SA 4.0
2020-11-25T02:39:45.173Z
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.
Dayne
<microeconomics>
Prove that if a good is inferior, it is also a Giffen good
Roshan Kandukuri
# Prove that if a good is inferior, it is also a Giffen good

(Asked by: username_0 on 2020-11-24)

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.

**Tags:** microeconomics
> **Comments:**
> **username_1**: Are you sure it is not the other way around? A Giffen good must be an inferior good, but not the other way around.  
> **username_2**: 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.  

---

## 1 answers

### Answer by username_2 on 2020-11-25. Score: 6

As pointed out in one of the comments, a descriptive answer to the question is available [here](https://economics.stackexchange.com/questions/25323/difference-between-giffen-and-inferior-goods-why-arent-all-inferior-goods-giff). 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.

> **Comments:**
> **username_2**: 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.  

---

