# Competitive equilibrium in a two-person economy with substitutes and complements

Recently came across this question on a microeconomics test and there was something that did not sit quite right with me.

In an economy with two agents, A and B, and two goods, milk and honey, the preferences of each consumer are the following:

$$U_A=\min(2m_A,h_A)$$ $$U_B=m_B+h_B$$ There is 100 of each good, no production in the economy, and the initial endowment is:

$$E_A(m_A,h_A)=(60,40)$$ $$E_B(m_B,h_B)=(40,60)$$

What is the competitive equilibrium allocation with this endowment?

After drawing the Edgeworth box, it is easy to see that the contract curve is defined by consumer A because of his preferences. Following that, we can see one trivial solution in point A, where the price ratio is 1. My problem with this is that I cannot comprehend that this would be the only solution in such an economy. For any other price ratio $${{p_M}\over{p_H}}\in (0,1)$$, couldn't the equilibrium be in point B drawn on the graph? If not, why not? Coming from an assumption of rationality and utility maximisation of the two players, although B cannot reach its highest indifference curve, $$I_3$$, he can reach $$I_2$$, which clearly provides more utility to him than $$I_1$$. Please help me understand whether I have overlooked something, or if my thought process is wrong.

• Have you considered solving this competitive equilibrium exercise with demand functions? Apr 5 at 3:40
• Yes, but I am not sure how would I derive these demand functions considering the nature of the two goods. Apr 5 at 10:16
• Hint: For any price ratio, consumer B would always demand 100 of the cheaper good and 0 of the more expensive one. So it seems like the market could never clear. But there is a single exception... Apr 5 at 11:07
• If you’re talking about the exception being when the prices are the same I already mentioned that as one solution. If you’re not getting at that, I’m not sure what you’re getting at. I understand that with any other price ratio other than 1, B would demand only one good, but does that mean that in this model, any price ratio other than one produces no equilibria? As I’ve mentioned, this to me, does not seem as rational behaviour from the consumers. Apr 5 at 12:46

Given the economy with two consumers: $$u_1=\min(2m_1,h_1)$$ and $$u_2=m_2+h_2$$ with endowment allocation $$E=((60,40),(40,60))$$, we observe the case where the price ratio is $$\frac{p_M}{p_H}\in (0,1)$$. In such a case, as depicted in the figure, point $$B$$ is the optimal choice of consumer $$1$$, and point $$C$$ is the optimal choice of consumer $$2$$ and it will offer higher satisfaction to consumer $$2$$ in comparison to point $$B$$ and is also affordable. Therefore, such a price ratio cannot be competitive equilibrium as there is excess demand for commodity $$M$$.
I will denote the price ratio $$p_M/p_H$$ by $$p$$.
The $$m$$ demand of $$A$$ is: $$m_A = \frac{60p + 40}{p + 2}$$ The $$m$$ demand of $$B$$ is: \begin{align*} m_B = \left\{ \begin{array}{ll} 0 & \text{ if } p > 1 \\ [0,100] & \text{ if } p = 1 \\ \frac{40p+60}{p} & \text{ if } p < 1 \end{array} \right. \end{align*} Middle part represents that any $$m_B$$ consumption in the $$[0,100]$$ interval can be optimal. Investigating the three cases ($$p>1$$, $$p=1$$, $$p<1$$) separately, you will find there is only one price where demand equals supply $$m_A + m_B = 60 + 40.$$