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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?) 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

enter image description here

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.

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"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.

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  • $\begingroup$ Thank you for your answer! Could you please tell me in what part of Varian do they make this discussion? $\endgroup$ May 29, 2023 at 17:28
  • $\begingroup$ In the 8th edition: Chapter 9: Buying and Selling, subchapter: The Slutsky Equation Revisited $\endgroup$
    – Giskard
    May 29, 2023 at 18:34
  • $\begingroup$ Thank you! The 3 effect decomposition Varian makes is interesting to me. $\endgroup$ May 30, 2023 at 18:20

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