# How to solve a general equilibrium problem with lexicographic preferences?

I have been unable to find a good example of this type of GE problem in our textbooks, and our professor has indicated that something like this may appear on our exam. So, here is a hypothetical construction:

Let Consumer A have lexicographic preferences, e.g., suppose $$X=\rm I\!R_+^2$$ and that $$x \succeq y$$ if and only if either $$x_1>y_1$$ or $$x_1=y_1$$ and $$x_2≥y_2$$.

Let Consumer B have quasilinear preferences: $$u(x_1,x_2)=x_1+\varphi(x_2)$$, where $$\varphi$$(•) is invertible, non-decreasing in $$x_1$$ and $$x_2$$, and continuously differentiable.

Each consumer is endowed with bundles $$\omega^i=\omega_1^i+\omega_2^i$$, $$i\in\{A,B\}$$.

Given this information, how would one solve the following questions?

(a) Solve each consumer's utility maximization problem in terms of some market clearing price vector $$\textbf{p}$$, normalized such that $$p_1=p$$ and $$p_2=1$$.

(b) Do the first and second welfare theorems hold? Does Walras' law hold?

(c) Does a competitive equilibrium exist? If so, find all equilibria.

(d) Find and characterize the contract curve and core of the Walrasian equilibrium.

I have tried to leave things as general as possible, but if it is useful to substitute specific numbers and functions, feel free to do so. And if anything is underspecified please let me know and I'll try to elaborate.

• Hi! Have you tried to figure these out yourself? Some of these seem pretty doable. Since A's preferences are not continuous, there is little to do but use logic. Commented Dec 23, 2021 at 16:10
• Also, some details seem unclear. I am guessing $\varphi$ is also increasing and convex? And what do you mean by "Solve each consumer's utility maximization problem"? Assuming a price vector $\textbf{p}$, or in what context? Commented Dec 23, 2021 at 16:11
• Good point about phi, I suppose that would have to be stipulated. I've added some details regarding the price vector as well. With regard to your first comment, I think that's precisely my confusion – I'm not sure how to establish general equilibria solely through logical argument instead of just solving with straightforward maths. Commented Dec 24, 2021 at 11:55
• Do you really want $\varphi$ to be convex? That will give you nonconvex preferences. Commented Dec 24, 2021 at 15:35
• Solving the general equilibrium should be similar whether the preferences are lexicographic or not. And you basically have a step-by-step guide to do it. Solving consumer's utility maximization problem when the preferences are lexicographic should be quite trivial, the solution almost directly follows from the definition of lexicographic preferences. Commented Dec 27, 2021 at 16:23

Let me answer this question for the following exchange economy:

Consumer $$A$$ has Lexicographic preferences. Lexicographic preference relation $$\succsim$$ on $$\mathbb{R}^2_+$$ is defined as follows:

for any $$(x_1,y_1), (x_2,y_2) \in \mathbb{R}^2_+$$, $$(x_1,y_1)\succsim (x_2,y_2)$$ if and only if  either $$x_1 > x_2$$ or $$(x_1 = x_2 \text{ and } y_1\geq y_2)$$

Suppose consumer $$B$$'s preferences are represented by a Quasi-linear utility function $$u_B(x_B,y_B)=x_B+2\sqrt{y_B}$$

Let $$\omega_X > 0$$ and $$\omega_Y > 0$$ denotes the total endowments of goods X and Y respectively. Set of feasible allocations is $$\mathcal{F}=\{((x_A, y_A), (x_B, y_B))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_A+x_B=\omega_X \ \wedge \ y_A+y_B=\omega_Y\}$$

Q. What is the set of Pareto efficient allocations?

Any feasible allocation $$((x_A, y_A), (x_B, y_B))$$ satisfying $$y_A>0$$ and $$x_B >0$$ is not Pareto efficient. This is because we can find $$\epsilon > 0$$ small enough such that $$((x_A+\epsilon, 0), (x_B-\epsilon, \omega_Y))$$ is feasible and Pareto superior to $$((x_A, y_A), (x_B, y_B))$$. Therefore, we are left with the following subset of feasible allocations:

$$\{((x_A, y_A), (x_B, y_B)) | y_A = 0 \ \text{or} \ x_B = 0 \}$$

Check that all these allocations are Pareto efficient.

Q. What is the competitive equilibrium?

Competitive equilibrium may or may not exist depending on our choice of endowment allocation. For example if endowment allocation is that $$A$$ has all of X i.e. $$(\omega_X,0)$$ and $$B$$ has all of Y i.e. $$(0,\omega_Y)$$ then any $$(p_X, p_Y=1)$$ satisfying $$\sqrt{\omega_Y}\leq p_X$$ will support $$((\omega_X,0),(0,\omega_Y))$$ in equilibrium.

Q. Do the first welfare theorem hold?

Yes, it holds because preferences of both individuals are monotonic.

Q. Do the second welfare theorem hold?

No, because consider an efficient allocation where $$A$$ consumes all of X and some positive amount of $$Y$$ i.e. an allocation $$((\omega_X, \theta\omega_Y), (0, (1-\theta)\omega_Y))$$ for some $$\theta\in (0,1]$$. Such an allocation is Pareto efficient but it cannot be supported as a competitive equilibrium allocation no matter how we re-distribute the endowment.