<p>If <span class="math-container">$X$</span> is a Giffen good, then it must an inferior good.</p>
<p>Next, as the share-weighted sum of the income elasticities of all goods sum to 1 (Engel aggregation), and given that the income elasticity of <span class="math-container">$X$</span> is negative, the income elasticity of good <span class="math-container">$Y$</span> must be bigger than 1, which means that <span class="math-container">$Y$</span> is a luxury good.</p>
<p>To see that the latter, start from the budget constraint:
<span class="math-container">$$
p_X X(p,m) + p_Y Y(p,m) = m.
$$</span>
Take the derivative with respect to income, <span class="math-container">$m$</span>:
<span class="math-container">$$
p_X \frac{\partial X}{\partial m} + p_Y \frac{\partial Y}{\partial m} = 1.
$$</span>
This gives:
<span class="math-container">$$
\frac{p_X X}{m} \frac{\partial X}{\partial m} \frac{m}{X} + \frac{p_Y Y}{m} \frac{\partial Y}{\partial m}\frac{m}{Y} = 1.
$$</span>
Let <span class="math-container">$s_X = \frac{p_X X}{m}$</span> be the budget share of good <span class="math-container">$X$</span> and <span class="math-container">$s_Y = \frac{p_Y Y}{m}$</span> be the budget share of good <span class="math-container">$Y$</span> and denote by <span class="math-container">$\varepsilon^X_m$</span> and  <span class="math-container">$\varepsilon^Y_m$</span> the income elasticities of goods <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> respectively. Then:
<span class="math-container">$$
s_X \,\,\varepsilon^X_m + s_Y \,\,\varepsilon^Y_m = 1.
$$</span>
The first term is negative (as <span class="math-container">$X$</span> is inferior, <span class="math-container">$\varepsilon^X_m$</span> is negative). This means that <span class="math-container">$s_Y \varepsilon^Y_m \ge 1$</span>. As <span class="math-container">$s_Y \le 1$</span>, we obtain:
<span class="math-container">$$
\varepsilon^Y_m \ge \frac{1}{s_Y} \ge 1.
$$</span>
This shows that <span class="math-container">$Y$</span> is a luxury good.</p>

CC BY-SA 4.0
2024-02-28T14:15:31.460Z
I have another question. Could you help me check that? Thank you! https://economics.stackexchange.com/questions/57779/if-the-income-elasticity-of-walrasians-demand-for-all-property-is-constant-and
Jonathen
CC BY-SA 4.0
2024-02-28T14:43:55.527Z
Another question which is still relevant on income and price elasticity: But this time this have "no money illusion" condition, which confuses me a lot. I don't know how to use that condition. https://economics.stackexchange.com/questions/57778/true-or-false-if-there-is-no-money-illusion-once-all-price-elasticity-is-known
Jonathen
CC BY-SA 4.0
2024-02-28T14:44:24.007Z
This is a very good answer. I forgot to consider their mathematical connections. Thanks!
Jonathen
CC BY-SA 4.0
2024-02-28T06:13:14.107Z
2024-02-28T06:13:14.107Z
tdm
<p>In a two-good world, if one good is a Giffen good then the other is a good luxury.</p>
<p>This is a false statement. Reason: If X is a Giffen good then Y must be a normal good. But we cannot get any conclusion about luxury good.
Am I correct?</p>

<microeconomics>
In a two-good world, if one good is a Giffen good then the other is a good luxury
Jonathen
# In a two-good world, if one good is a Giffen good then the other is a good luxury

(Asked by: username_0 on 2024-02-27)

In a two-good world, if one good is a Giffen good then the other is a good luxury.

This is a false statement. Reason: If X is a Giffen good then Y must be a normal good. But we cannot get any conclusion about luxury good.
Am I correct?

**Tags:** microeconomics


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## 1 answers

### Answer by username_1 on 2024-02-28. Score: 3 | Accepted answer

If $X$ is a Giffen good, then it must an inferior good.

Next, as the share-weighted sum of the income elasticities of all goods sum to 1 (Engel aggregation), and given that the income elasticity of $X$ is negative, the income elasticity of good $Y$ must be bigger than 1, which means that $Y$ is a luxury good.

To see that the latter, start from the budget constraint:
$$
p\_X X(p,m) + p\_Y Y(p,m) = m.
$$
Take the derivative with respect to income, $m$:
$$
p\_X \frac{\partial X}{\partial m} + p\_Y \frac{\partial Y}{\partial m} = 1.
$$
This gives:
$$
\frac{p\_X X}{m} \frac{\partial X}{\partial m} \frac{m}{X} + \frac{p\_Y Y}{m} \frac{\partial Y}{\partial m}\frac{m}{Y} = 1.
$$
Let $s\_X = \frac{p\_X X}{m}$ be the budget share of good $X$ and $s\_Y = \frac{p\_Y Y}{m}$ be the budget share of good $Y$ and denote by $\varepsilon^X\_m$ and $\varepsilon^Y\_m$ the income elasticities of goods $X$ and $Y$ respectively. Then:
$$
s\_X \,\,\varepsilon^X\_m + s\_Y \,\,\varepsilon^Y\_m = 1.
$$
The first term is negative (as $X$ is inferior, $\varepsilon^X\_m$ is negative). This means that $s\_Y \varepsilon^Y\_m \ge 1$. As $s\_Y \le 1$, we obtain:
$$
\varepsilon^Y\_m \ge \frac{1}{s\_Y} \ge 1.
$$
This shows that $Y$ is a luxury good.

> **Comments:**
> **username_0**: I have another question. Could you help me check that? Thank you! https://economics.stackexchange.com/questions/57779/if-the-income-elasticity-of-walrasians-demand-for-all-property-is-constant-and  
> **username_0**: Another question which is still relevant on income and price elasticity: But this time this have "no money illusion" condition, which confuses me a lot. I don't know how to use that condition. https://economics.stackexchange.com/questions/57778/true-or-false-if-there-is-no-money-illusion-once-all-price-elasticity-is-known  
> **username_0**: This is a very good answer. I forgot to consider their mathematical connections. Thanks!  

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