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.

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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.


1 Answer 1


"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.

  • $\begingroup$ 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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