<p>&quot;Giffenness&quot; may follow only from the preferences, you have to filter out endowment income effects. (Substitute a fixed monetary income <span class="math-container">$m$</span>, for details, see e.g.; Varian's Intermediate Microeconomics.) So it plays no role that this is a pure exchange economy.</p>
<p>Given the identical preferences we can drop the index <span class="math-container">$i$</span>. Given prices <span class="math-container">$p_x,p_y$</span>, in all interior points
<span class="math-container">$$
\begin{equation*}
|MRS(x,y)| = \frac{p_x}{p_y} \\
e^{y-x} = \frac{p_x}{p_y} \\
y-x = \ln p_x - \ln p_y \\
\end{equation*}
$$</span>
thus all goods consumption is decreasing in their own price for either consumer, thus also for the market as a whole.</p>

CC BY-SA 4.0
2023-05-29T17:28:20.547Z
Thank you for your answer! Could you please tell me in what part of Varian do they make this discussion?
Nicolas Torres
CC BY-SA 4.0
2023-05-29T18:34:55.073Z
In the 8th edition: Chapter 9: Buying and Selling, subchapter: The Slutsky Equation Revisited
Giskard
CC BY-SA 4.0
2023-05-30T18:20:15.777Z
Thank you! The 3 effect decomposition Varian makes is interesting to me.
Nicolas Torres
CC BY-SA 4.0
2023-05-29T08:28:47.460Z
2023-05-29T18:35:38.390Z
2023-05-29T18:35:38.390Z
Giskard
<p>I was checking some questions I recently answered here on General Equilibrium, and a result from this one (<a href="https://economics.stackexchange.com/questions/55483/exchange-economy-with-two-agents-whats-the-competitive-equilibrium/55488#55488">Exchange economy with two agents, what&#39;s the competitive equilibrium?</a>) drew my attention:</p>
<p>Both agents <span class="math-container">$i = A,B$</span> have the following utility function:</p>
<p><span class="math-container">$U_i(x_i,y_i) = - e^{-x_i} - e^{-y_i}$</span></p>
<p>and the endowments are <span class="math-container">$((w_{x_A},w_{y_A}), (w_{x_B},w_{y_B})) = ((1,5),(3,3))$</span>.</p>
<p>For this problem I got the following demands for good <span class="math-container">$x$</span></p>
<p><span class="math-container">${x_A}^\star = \frac{p_x - \ln(p_x) + 5}{p_x + 1}$</span></p>
<p><span class="math-container">${x_B}^\star = \frac{3 p_x - \ln(p_x) + 3}{p_x + 1}$</span></p>
<p>Note I took good <span class="math-container">$p_y = 1$</span> as numeraire, so the demands for good <span class="math-container">$x$</span> are functions of its own price.</p>
<p>Adding them both up, I get the following &quot;aggregate demand function&quot; for good <span class="math-container">$x$</span></p>
<p><span class="math-container">$x^\star = \frac{4 p_x - 2 \ln(p_x) + 8}{p_x + 1}$</span></p>
<p>I plotted this function on Geogebra and it seemed to be decreasing up to a certain price, from which, it would become increasing in its own price, analogous to a Giffen good.</p>
<p>I then proceded to take its derivative to check what the graph showed. I got the following expression, which I'd then <span class="math-container">$\overset{set}{=} 0$</span> to find extrema:</p>
<p><span class="math-container">$\frac{dx^\star}{dp_x} = \frac{2p_x \ln(p_x) - 6p_x - 2}{p_x (p_x + 1)^2} \overset{set}{=} 0$</span></p>
<p>GeoGebra and WolframAlpha yield a numerical solution near <span class="math-container">$p_x = 21.06$</span></p>
<p>Here I show graphs of both the aggregate <span class="math-container">$x$</span>-demand function and its derivative, respectively:</p>
<p>Note: My Geogebra is in Spanish, the words &quot;extremo&quot; and &quot;raíz&quot; mean &quot;extremum&quot; and &quot;root&quot;, respectively.</p>
<p><a href="https://i.sstatic.net/aAl1H.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/aAl1H.jpg" alt="enter image description here" /></a></p>
<p><a href="https://i.sstatic.net/vgmXQ.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/vgmXQ.jpg" alt="enter image description here" /></a></p>
<p>If there is an upward sloping portion of the aggregate demand function, in particular, at least one agent has a demand function with an upward sloping portion.</p>
<p>The fact that Walrasian demands with upward sloping portions are mathematically possible made me think about the following question:</p>
<p>Can we talk about Giffen goods in pure exchange economies (the usual Edgeworth box)? If so, then there must be a way to define income and substitution effects, and an interpretation of the Giffen good scenario.</p>

<microeconomics><mathematical-economics><general-equilibrium><pure-exchange-economy><giffen-goods>
Is there/can we define a notion of Giffen goods in pure exchange economies?
Nicolas Torres
# Is there/can we define a notion of Giffen goods in pure exchange economies?

(Asked by: username_0 on 2023-05-29)

I was checking some questions I recently answered here on General Equilibrium, and a result from this one ([Exchange economy with two agents, what's the competitive equilibrium?](https://economics.stackexchange.com/questions/55483/exchange-economy-with-two-agents-whats-the-competitive-equilibrium/55488#55488)) drew my attention:

Both agents $i = A,B$ have the following utility function:

$U\_i(x\_i,y\_i) = - e^{-x\_i} - e^{-y\_i}$

and the endowments are $((w\_{x\_A},w\_{y\_A}), (w\_{x\_B},w\_{y\_B})) = ((1,5),(3,3))$.

For this problem I got the following demands for good $x$

${x\_A}^\star = \frac{p\_x - \ln(p\_x) + 5}{p\_x + 1}$

${x\_B}^\star = \frac{3 p\_x - \ln(p\_x) + 3}{p\_x + 1}$

Note I took good $p\_y = 1$ as numeraire, so the demands for good $x$ are functions of its own price.

Adding them both up, I get the following "aggregate demand function" for good $x$

$x^\star = \frac{4 p\_x - 2 \ln(p\_x) + 8}{p\_x + 1}$

I plotted this function on Geogebra and it seemed to be decreasing up to a certain price, from which, it would become increasing in its own price, analogous to a Giffen good.

I then proceded to take its derivative to check what the graph showed. I got the following expression, which I'd then $\overset{set}{=} 0$ to find extrema:

$\frac{dx^\star}{dp\_x} = \frac{2p\_x \ln(p\_x) - 6p\_x - 2}{p\_x (p\_x + 1)^2} \overset{set}{=} 0$

GeoGebra and WolframAlpha yield a numerical solution near $p\_x = 21.06$

Here I show graphs of both the aggregate $x$-demand function and its derivative, respectively:

Note: My Geogebra is in Spanish, the words "extremo" and "raíz" mean "extremum" and "root", respectively.

[![enter image description here](https://i.sstatic.net/aAl1H.jpg)](https://i.sstatic.net/aAl1H.jpg)

[![enter image description here](https://i.sstatic.net/vgmXQ.jpg)](https://i.sstatic.net/vgmXQ.jpg)

If there is an upward sloping portion of the aggregate demand function, in particular, at least one agent has a demand function with an upward sloping portion.

The fact that Walrasian demands with upward sloping portions are mathematically possible made me think about the following question:

Can we talk about Giffen goods in pure exchange economies (the usual Edgeworth box)? If so, then there must be a way to define income and substitution effects, and an interpretation of the Giffen good scenario.

**Tags:** microeconomics, mathematical-economics, general-equilibrium, pure-exchange-economy, giffen-goods

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## 1 answers

### Answer by username_1 on 2023-05-29. Score: 2 | Accepted answer

"Giffenness" may follow only from the preferences, you have to filter out endowment income effects. (Substitute a fixed monetary income $m$, for details, see e.g.; Varian's Intermediate Microeconomics.) So it plays no role that this is a pure exchange economy.

Given the identical preferences we can drop the index $i$. Given prices $p\_x,p\_y$, in all interior points
$$
\begin{equation\*}
|MRS(x,y)| = \frac{p\_x}{p\_y} \\
e^{y-x} = \frac{p\_x}{p\_y} \\
y-x = \ln p\_x - \ln p\_y \\
\end{equation\*}
$$
thus all goods consumption is decreasing in their own price for either consumer, thus also for the market as a whole.

> **Comments:**
> **username_0**: Thank you for your answer! Could you please tell me in what part of Varian do they make this discussion?  
> **username_1**: In the 8th edition: Chapter 9: Buying and Selling, subchapter: The Slutsky Equation Revisited  
> **username_0**: Thank you! The 3 effect decomposition Varian makes is interesting to me.  

---

