<p>Let me answer this question for the following exchange economy:</p>
<p>Consumer <span class="math-container">$A$</span> has Lexicographic preferences. Lexicographic preference relation <span class="math-container">$\succsim$</span> on <span class="math-container">$\mathbb{R}^2_+$</span> is defined as follows:</p>
<p>for any <span class="math-container">$(x_1,y_1), (x_2,y_2) \in \mathbb{R}^2_+$</span>,
<span class="math-container">$(x_1,y_1)\succsim (x_2,y_2)$</span> if and only if 
either <span class="math-container">$x_1 &gt; x_2$</span> or <span class="math-container">$(x_1 = x_2 \text{ and } y_1\geq y_2)$</span></p>
<p>Suppose consumer <span class="math-container">$B$</span>'s preferences are represented by a Quasi-linear utility function <span class="math-container">$u_B(x_B,y_B)=x_B+2\sqrt{y_B}$</span></p>
<p>Let <span class="math-container">$\omega_X &gt; 0$</span> and <span class="math-container">$\omega_Y &gt; 0$</span> denotes the total endowments of goods X and Y respectively. Set of feasible allocations is
<span class="math-container">$\mathcal{F}=\{((x_A, y_A), (x_B, y_B))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_A+x_B=\omega_X \ \wedge \ y_A+y_B=\omega_Y\}$</span></p>
<p>Q. What is the set of Pareto efficient allocations?</p>
<p>Any feasible allocation <span class="math-container">$((x_A, y_A), (x_B, y_B))$</span> satisfying <span class="math-container">$y_A&gt;0$</span> and <span class="math-container">$x_B &gt;0$</span> is not Pareto efficient. This is because we can find <span class="math-container">$\epsilon &gt; 0$</span> small enough such that <span class="math-container">$((x_A+\epsilon, 0), (x_B-\epsilon, \omega_Y))$</span> is feasible and Pareto superior to <span class="math-container">$((x_A, y_A), (x_B, y_B))$</span>. Therefore, we are left with the following subset of feasible allocations:</p>
<p><span class="math-container">$\{((x_A, y_A), (x_B, y_B)) | y_A = 0 \ \text{or} \ x_B = 0 \}$</span></p>
<p>Check that all these allocations are Pareto efficient.</p>
<p>Q. What is the competitive equilibrium?</p>
<p>Competitive equilibrium may or may not exist depending on our choice of endowment allocation. For example if endowment allocation is that <span class="math-container">$A$</span> has all of X i.e. <span class="math-container">$(\omega_X,0)$</span> and <span class="math-container">$B$</span> has all of Y i.e. <span class="math-container">$(0,\omega_Y)$</span> then any <span class="math-container">$(p_X, p_Y=1)$</span> satisfying <span class="math-container">$\sqrt{\omega_Y}\leq p_X$</span> will support <span class="math-container">$((\omega_X,0),(0,\omega_Y))$</span> in equilibrium.</p>
<p>Q. Do the first welfare theorem hold?</p>
<p>Yes, it holds because preferences of both individuals are monotonic.</p>
<p>Q. Do the second welfare theorem hold?</p>
<p>No, because consider an efficient allocation where <span class="math-container">$A$</span> consumes all of X and some positive amount of <span class="math-container">$Y$</span> i.e. an allocation <span class="math-container">$((\omega_X, \theta\omega_Y), (0, (1-\theta)\omega_Y))$</span> for some <span class="math-container">$\theta\in (0,1]$</span>. Such an allocation is Pareto efficient but it cannot be supported as a competitive equilibrium allocation no matter how we re-distribute the endowment.</p>

CC BY-SA 4.0
2022-10-07T09:45:40.220Z
2023-10-03T02:33:55.543Z
2023-10-03T02:33:55.543Z
Amit
<p>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:</p>
<p>Let Consumer A have lexicographic preferences, e.g., suppose <span class="math-container">$X=\rm I\!R_+^2$</span> and that <span class="math-container">$x \succeq y$</span> if and only if either <span class="math-container">$x_1&gt;y_1$</span> or <span class="math-container">$x_1=y_1$</span> and <span class="math-container">$x_2≥y_2$</span>.</p>
<p>Let Consumer B have quasilinear preferences: <span class="math-container">$u(x_1,x_2)=x_1+\varphi(x_2)$</span>, where <span class="math-container">$\varphi$</span>(•) is invertible, non-decreasing in <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>, and continuously differentiable.</p>
<p>Each consumer is endowed with bundles <span class="math-container">$\omega^i=\omega_1^i+\omega_2^i$</span>, <span class="math-container">$i\in\{A,B\}$</span>.</p>
<p>Given this information, how would one solve the following questions?</p>
<p>(a) Solve each consumer's utility maximization problem in terms of some market clearing price vector <span class="math-container">$\textbf{p}$</span>, normalized such that <span class="math-container">$p_1=p$</span> and <span class="math-container">$p_2=1$</span>.</p>
<p>(b) Do the first and second welfare theorems hold? Does Walras' law hold?</p>
<p>(c) Does a competitive equilibrium exist? If so, find all equilibria.</p>
<p>(d) Find and characterize the contract curve and core of the Walrasian equilibrium.</p>
<p>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.</p>

CC BY-SA 4.0
2021-12-23T16:10:15.817Z
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.
Giskard
CC BY-SA 4.0
2021-12-23T16:11:14.180Z
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?
Giskard
CC BY-SA 4.0
2021-12-24T11:55:49.177Z
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.
quickhatch
CC BY-SA 4.0
2021-12-24T14:46:06.313Z
? You are correct, you need to use straightforward math. Does a price $p$ (as defined in the question) exist for which the aggregate demand equals aggregate supply? Map individual demands as a function of $p$, and proceed from there?
Giskard
CC BY-SA 4.0
2021-12-24T15:35:07.253Z
Do you really want $\varphi$ to be convex? That will give you nonconvex preferences.
Michael Greinecker
CC BY-SA 4.0
2021-12-27T14:33:14.747Z
@Giskard yes, the vector p would correspond to the market clearing prices. Sorry if I misunderstood your point about the math vs. logic in this case. Even if someone could point me to a textbook example of this it would be helpful, we've never covered this type of problem in class so I really don't know how to tackle it.
quickhatch
CC BY-SA 4.0
2021-12-27T16:23:08.377Z
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.
LudwigNagasena
<utility><preferences><general-equilibrium><walrasian>
How to solve a general equilibrium problem with lexicographic preferences?
quickhatch
# How to solve a general equilibrium problem with lexicographic preferences?

(Asked by: username_0 on 2021-12-23)

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.

**Tags:** utility, preferences, general-equilibrium, walrasian
> **Comments:**
> **username_1**: 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.  
> **username_1**: 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?  
> **username_0**: 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.  
> **username_1**: ? You are correct, you need to use straightforward math. Does a price $p$ (as defined in the question) exist for which the aggregate demand equals aggregate supply? Map individual demands as a function of $p$, and proceed from there?  
> **username_2**: Do you really want $\varphi$ to be convex? That will give you nonconvex preferences.  
> **username_0**: @Giskard yes, the vector p would correspond to the market clearing prices. Sorry if I misunderstood your point about the math vs. logic in this case. Even if someone could point me to a textbook example of this it would be helpful, we've never covered this type of problem in class so I really don't know how to tackle it.  
> **username_3**: 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.  

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## 1 answers

### Answer by username_4 on 2022-10-07. Score: 1

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.


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