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How does Brouwer's fixed point theorem relate to Walrasian equilibrium?

I am trying to understand Walrasian equilibrium and its connection to fixed points, especially how we can apply Brouwer’s fixed point theorem to the notion of Walrasian equilibrium. I understand the ...
Robert Brown's user avatar
5 votes
1 answer
2k views

Convexity of Walrasian Demand

A follow up to my previous question: Given $u(x) = (x_1-b_1)^\alpha (x_2-b_2)^\beta(x_3-b_3)^\gamma$ and ending up with from a maximization that $$x(p,w) = (b_1, b_2, b_3) + (w - (b \cdot p))\...
Kitsune Cavalry's user avatar
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4 votes
1 answer
117 views

Walras Law in a production economy with fixed costs

Consider a price taking firm with fixed costs $fc \geq 0$: \begin{align*} \Pi &= \max_{n^D} \left\{ P_c F(n^D) - w\times n^D - fc \right\} \end{align*} A representative household owns this firm:...
Albert Zevelev's user avatar
4 votes
1 answer
233 views

Correct and complete characterisation of the Walrasian demand function

I would like to propose to you the following problem and my proposed solution. In particular, I am unsure in how to correctly characterize the Walrasian demand. Can you please have a look at it and ...
fennel's user avatar
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4 votes
2 answers
460 views

Solve for the Walrasian demand, Utility of three variables, and Convexity of Preferences?

I am given $U(x,y,z) = x^\frac{2}{3}y^\frac{1}{3} + z$. I am asked to solve the following: (i) Prove the convexity of these preferences (convex, strictly convex or neither?) (ii) Solve for the ...
Kinno's user avatar
  • 155
4 votes
1 answer
713 views

Show that $x(p,w)=w\cdot x(p,1)$ with homothetic preferences

Someone gave me a proof of this, but I am not sure if it is correct. Let $B(p,w) = \{x: p\cdot x \leq w\}$ (the budget set). Then: \begin{align} x(p,w) &= \arg \max_{x\in B(p,w)} u(x)\\ &=\...
möbius's user avatar
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4 votes
0 answers
61 views

How to find the General Equilibria allowing for infinitesimal prices?

I know there can’t exist a usual Walrasian Equilibrium when both agents have the same lexicographic preferences: If both agents had the preferences $(x,y) \succeq (x’,y’) \iff:$ $x > x’ \text{ or } ...
Nicolas Torres's user avatar
4 votes
0 answers
65 views

Walrasian Equilibrium in A Simple Assignment (Matching) Model

I am reading Acemoglu 1996 and the Walrasian allocation in section II makes me confused. The setting is following. The economy lasts for two periods and consists of two types of agents, firms and ...
Alalalalaki's user avatar
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3 votes
1 answer
1k views

Robinson Crusoe Economy Question

Question: Hypothetically, Robinson Crusoe is stuck on an island and can choose between working on gathering coconuts or leisure. The utility function is: $U(C,L)=C^{2/5}L^{3/5}$ where C is the num of ...
Lily B's user avatar
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3 votes
2 answers
233 views

What is the budget constraint when we assume a common utility function?

Let's consider an exchange economy with two identical consumers. The common utility function is: $$u^i (x_1, x_2) = x_1^α x_2^{1-α} \;\;\; \text{for} \;\;\; 0 < α < 1.$$ Society has 10 units of ...
aliosha karamazov's user avatar
3 votes
1 answer
162 views

Question about the relationship between Weak Axiom and Slutsky Matrix

We know that if a differentiable Walrasian demand function $x(p,w)$ satisfies Walras' law ($p^Tx=w$), homogeneity of degree zero ($x(\alpha p,\alpha w)=x(p,w)$), and the weak axiom of revealed ...
HXW's user avatar
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3 votes
1 answer
484 views

Blocking Coalitions

For $n$ individuals, can a blocking coalition only be formed by at least $n/2$ individuals? For example, if there are 6 individuals, can less than 3 individuals form blocking coalitions?
S.Rana's user avatar
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3 votes
1 answer
372 views

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 ...
quickhatch's user avatar
2 votes
2 answers
3k views

Finding Cobb-Douglas Hicksian Demand using Duality

I'm trying to follow the in-text examples from Mas-Colell. I can confirm I have the correct first order-conditions and hence the Marshallian demand functions for Example 3.D.1: $u(x_1,x_2) = x_1^\...
CorporateNationalism's user avatar
2 votes
1 answer
104 views

Effect of price on utility

A consumer has an endowment vector $w$; at prices $p$ his demand for the first good exceeds his endowment; $x_1^+(p; pw)>w_1$ then a small increase of $p_1$ will lower his utility. I was discussing ...
Ana Ellis's user avatar
2 votes
1 answer
151 views

How to find a Walrasian equilibrium for this production economy?

Consider a production economy with 2 consumers and 2 firms. There are 3 goods: leisure $l$ and two goods $x,y$. The utility function of both consumers are $u_i(l_i,x_i,y_i)=l_i(x_i)^2(y_i)^2$ and the ...
Ludwig Gershwin's user avatar
2 votes
1 answer
329 views

Walrasian demand with a twist of Leontief function

A consumer has the utility function $u(x_1; x_2) = \min(x_1; x_2) + 5 \max(x_1; x_2)$. Find its Walrasian demand $x^*(p; w)$. I've tried searching it up when we have two Leontief functions summed ...
Ana Ellis's user avatar
2 votes
1 answer
117 views

Exchange economy with trade

Consider an exchange economy with 2 goods $x,y$ and $n+1$ consumers $0,1,\cdots,n$. Consumer 0 has utility function $u_0(x,y)=x^\alpha y^{1-\alpha}$ and endowment $w_0=(1,0)$. Other consumers $1,\...
user avatar
2 votes
1 answer
272 views

Why is there a Walrasian Equilibrium if excess demand goes to infinity as price goes to 0?

In one exercise, we have to argue that a Walrasian Equilibrium exists and the solution says that if we can see that excess demand goes to infinity as price goes to 0, and as price goes to infinity, ...
aliosha karamazov's user avatar
2 votes
4 answers
621 views

Derive demand function $x(p,w)$ from utility function $u(x) = \min\{x_1, x_2\} + x_3$

I know how to solve the two-good case with $u(x) = \min\{x_1, x_2\}$, but the addition of $x_3$ confuses me. Problem Derive the demand function $x(p,w)$ from $u(x) = \min\{x_1, x_2\} + x_3$. What I ...
Anne1005's user avatar
2 votes
1 answer
370 views

Why does strictly Walrasian demand with quasi-concave utility function mean that the walrasian demand having only one single consumption bundle?

In the context of Walrasian demand: Suppose u is continuous, satisfies local nonsatiation, and is strictly quasi-concave, each $w(p, x)$ contains a single consumption bundle. The proof I got from a ...
Aqqqq's user avatar
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2 votes
1 answer
86 views

How can a "producer face a limited market as regards sales and yet a highly competitive market as regards price"?

Kaldor (1972): it is evident that the co-existence of increasing returns and competition—emphasised by Young and also by Marx, but wholly excluded by the axiomatic framework of Walrasian economics—is ...
user avatar
2 votes
0 answers
31 views

Becker Social Pricing and Lack of Smooth Demand

I was reading an article on Becker Social Pricing, where your demand for a good depends on other people's demand for the good. So basically this paper is meant to be a primer into explaining why some ...
Kitsune Cavalry's user avatar
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1 vote
1 answer
56 views

Prove the equation

Let $$ x^0=x^*(p^0,w)$$ then $$v(p,px^0)$$ is minimized at $$p=p^0$$ What theorem are we supposed to use in order to solve this, because I am a bit lost. Thanks in advance for all the suggestions/help....
Maybeline Lee's user avatar
1 vote
1 answer
147 views

How to show there is no a Walrasian equilibrium?

There are 3 agents, one seller and two buyers. There are two indivisible goods, apple(a) and banana(b), and one divisible good, money(m). The seller's endowment is $W_s=(1,1,0)$, and buyers's are $W_{...
user45481's user avatar
1 vote
1 answer
224 views

What does it mean "non-Walrasian" or "quasi-Walrasian"?

In a recent NBER workshop, Robert Hall started a discussion with his first slide summarizing the macro literature. I am not very familiar with either DMP model or New Keynesian model and have no idea ...
Alalalalaki's user avatar
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1 vote
1 answer
615 views

Doubt regarding Walrasian equilibrium with complements for both agents

There are two goods $1,2$ and two agents $ 1,2 $. Both have the utility function $ u_{i}=\min({x_{1i},x_{2i}}) $ for agent $i$ .The endowments are $(1,3)$ and $(3,1)$ for agent $1$ and $2$ ...
earthboy's user avatar
1 vote
3 answers
4k views

Perfect Complements - Walrasian Equilibrium

For a homework , I struggled to solve the following question but couldn't go further: endowment of person 1 = (30,0) endowment of person 2 = (0,20) utility ...
Ktp's user avatar
  • 13
1 vote
1 answer
2k views

General Equilibrium with Perfect Substitutes

I came across the following problem: The quantities of an economy’s only two goods are denoted by $X$ and $Y$; no production is possible. Ann’s and Ben’s preferences are described by the utility ...
Luca's user avatar
  • 25
1 vote
1 answer
231 views

Find equilibrium price using excess demand function

Consider an economy with two agents. There are two goods, x and y. Agents' preferences are Leontif ones as follows: $u_1(x,y)=\min(x,4y)$ and $u_2(x,y)=\min(x,y)$. Initial endowment for 1 is (2,4), ...
biden's user avatar
  • 111
1 vote
2 answers
219 views

What is the difference between solving for utility maximization separately or aggregately?

I derived the Walrasian equilibrium for an economy of two consumers with their respective utility functions (u1, u2) and initial endowments but later on I am asked to maximize (u1+u2). Does anyone ...
ceins's user avatar
  • 77
1 vote
1 answer
278 views

Whether the demand in the panels satisfies the weak axiom

The Walrasian demand function $x(p,w)$ satisfies the weak axiom of revealed preference if the following property holds for any two price wealth situations $(p,w)$ and $(p',w')$: If $p•x(p',w')\le w$ ...
Rowan's user avatar
  • 163
1 vote
2 answers
419 views

When is an allocation in the core of an economy but it's not a Walrasian equilibrium?

So I was given this problem where both agents have Cobb-Douglas utility functions and I'm asked to find an allocation that's in the core but not a Walrasian equilibrium. Isn't the core of an economy ...
Abraham Villalobos Jaimes's user avatar
1 vote
1 answer
63 views

How do I argue that "prices aggregate beliefs" in this Walrasian equilibrium?

Consider an $n$ person economy with only 1 good and 2 states of nature $r,s$. Consumer $i$'s utility function is $$u_i(x_{ir},x_{is})=\pi_i \ln x_{ir}+(1-\pi_i)\ln x_{is}$$ where $\pi_i\in (0,1)$ is ...
user avatar
1 vote
0 answers
57 views

How to find a Walrasian equilibrium for this economy?

Consider a production economy with 2 goods, 1 and 2, two consumers A and B, and 2 firms $\alpha,\beta$. The consumers' utility functions are \begin{align} u_A(x_1^A,x_2^A)=x_1^A+x_2^A-\frac{1}{24}(x_1^...
Ludwig Gershwin's user avatar
1 vote
0 answers
32 views

Endowment of an agent

I was going through the Shapley-Folkman-Starr Lemma (https://simons.berkeley.edu/sites/default/files/docs/3605/simons2.pdf) and I came across the term "endowment" of an agent. My assumption ...
user12632521's user avatar
1 vote
0 answers
320 views

How to prove that the walrasian demand function $x(p,w)$ is continuous in $p$ and $w$?

If the utility function $u$ is continuous and satisfies local nonsatiation, and walrasian demand function $x(p, w)$ is a function (i.e. always map to only single values), how to prove that $x(p,w)$ is ...
Aqqqq's user avatar
  • 392
1 vote
2 answers
437 views

What is the Walras law vs first welfare theorem

As far as I know, both of the first welfare theorem and the Walras law are closely tied to the invisible hand. what is the difference between them? thank you very much for your help
Fozoro's user avatar
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1 vote
0 answers
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Walras's law in a Fisher market

Walras's law is stated for an exchange market: each agent comes to the market with a certain endowment of each commodity, a price-vector is determined, and the demand of each agent is the best bundle ...
Erel Segal-Halevi's user avatar
1 vote
0 answers
137 views

MRS in Walrasian equilibrium price expression?

I have the utility function $U^A=\alpha ln(x^A)+\beta ln(y^A)$ for consumer $A$ and $U^B=\gamma ln(x^B)+\phi ln(y^B)$ for consumer $B$. They are endowed with $\omega^h_x$ and $\omega^h_y$ of $x$ and $...
GreenHumbollt's user avatar
1 vote
0 answers
141 views

General Equilibrium allocation holding fixed a consumer's utility

I'm having some issues with solving this general equilibrium exercise. The way I started off is by assuming that since consumer 2's utility is fixed, he will have a fixed utility function. Then ...
Tommy's user avatar
  • 11
0 votes
1 answer
2k views

Finding a walrasian demand function

If I have a sum utility function like this $U(y) = \sum_{j=1}^J u(y_j)$ where $u$ is convex. Is there then a way to find walrasian demand for such a function without using calculus or do you need to ...
user avatar
0 votes
1 answer
2k views

Pareto optimal and Walrasian equilibrium [closed]

There are 100 units of good 1 and good 2 in a economy. Consumer 1 and consumer 2 have 50 units of each good. Consumer 1 only wants good 1 whereas consumer 2 only wants good 2. Note: Neither of these ...
HARSHA VARDHAN's user avatar
0 votes
2 answers
7k views

What is excess demand/ excess supply?

I've been reading up on Walras' Law and thought I understood it pretty well. However one of my friends asked me point blank what is an example of excess demand or excess supply is and I had some ...
EconJohn's user avatar
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0 votes
1 answer
145 views

General Equilibrium Involving Production

I need a little conceptual clarification. For a standard $N*K*M$ general equilibrium model, would an allocation, say, $y^k$ be Pareto Optimal if it does not solve $max(py^k)$? I understand that the ...
S.Rana's user avatar
  • 401
0 votes
1 answer
123 views

How do I find the Walrasian equilibrium price?

Consider a production economy with 1 consumer and 2 firms. The consumer's utility is $u(x,y)=xy$. The first firm makes $x$ using capital and labor according to the production function $x=k^\alpha l^{1-...
Ludwig Gershwin's user avatar
0 votes
0 answers
88 views

How do I argue that the Walrasian equilibrium is unique?

Consider an economy with 1 consumer and 1 firm. There are only 2 goods, leisure $l$ and corn $c$ and the consumer is endowed with 24 hours of leisure. The consumer's utility function is $u(c,l)=cl$. ...
Ludwig Gershwin's user avatar
0 votes
0 answers
43 views

Pure exchange economy: What if a consumer can set prices?

I'm trying to solve the following problem: Consider an exchange economy with 2 consumers and 2 goods. First, consumer A sets the price, then consumer B maximizes utility according to the price. Then ...
Ludwig Gershwin's user avatar
0 votes
0 answers
107 views

Finding demand function in Walrasian equilibrium

Maybe the title doesn't reflect what I mean perfectly but basically, I wanna derive demand functions from those two utility functions: where $x_{11}$ is the consumption of good 1 by agent 1 and $x_{...
Tatanik501's user avatar
0 votes
0 answers
77 views

For each Pareto efficient allocation, suggest how we might change the endowments so that the Pareto efficient allocation is a walrasian equilibrium

I have a two-person exchange economy Each agent has the following utility $u_i(x_i,y_i)=v(x_i)+y_i$ for agent $i=\{A,B\}$ Assume that $v$ is strictly concave and increasing function that has a ...
ThePooh's user avatar
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