6 votes
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Effect of price on utility

That was tricky. The idea is as follows: First, under standard assumptions, demand is continuous. If you change prices a little bit, demand will not change a lot. In particular, if your excess demand ...
Michael Greinecker's user avatar
6 votes

Finding Cobb-Douglas Hicksian Demand using Duality

You have found the value function $V(p,I)$ which in your case is given as $$V(p,I) = \alpha_1^{\alpha_1}\alpha_2^{\alpha_2}\frac{I}{p_1^{\alpha_1}p_2^{\alpha_2}},$$ where $\alpha_2 = 1- \alpha_1$ and $...
Jesper Hybel's user avatar
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6 votes
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Prove the equation

You don't need to use any fancy theorem, the trick is to disentangle the definitions. Everything follows directly from the definitions. $x^0=x^*(p^0,w)$ means that $p^0x^0\leq w$ and that $u(x)\leq u(...
Michael Greinecker's user avatar
6 votes
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Robinson Crusoe Economy Question

If you want to find Pareto efficient allocation in this economy, then you can determine that by solving the following system for $(C, L, H)$: $L+H=1$ $C = 8\sqrt{H}$ $\text{MRS} = \dfrac{3C}{2L} = \...
Amit's user avatar
  • 8,396
5 votes

Finding a walrasian demand function

Given that the utility maximization problem is: \begin{eqnarray*} \max_{\{y_j \geq 0 : 1\leq j\leq J\}} && \sum_{j=1}^{J} u(y_j) \\ \text{s.t.} && \sum_{j=1}^{J} p_jy_j \leq m\end{...
Amit's user avatar
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5 votes
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Correct and complete characterisation of the Walrasian demand function

To solve $$\max_{x \geq 0} \ (x_1+1)^\alpha(x_2 + 1)^\beta$$ $$s.t. \ \ I \geq p_1x_1 + p_2x_2,$$ I would define $y_1 = x_1+1$ and $y_2 = x_2 + 1$ to get the problem $$\max_{y \geq 0} \ y_1^\alpha y_2^...
Jesper Hybel's user avatar
  • 3,296
5 votes
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Walras Law in a production economy with fixed costs

Partial answer: for simplicity let $P_c =1$. The budget constraint: $c= wn + \Pi$ Simplify (plug in $\Pi$): $c= F(n)- fc$ Goods clearing: $c = F(n)$ The household's budget constraint is inconsistent w/...
Albert Zevelev's user avatar
4 votes
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Walrasian demand with a twist of Leontief function

Why don't you just plug in some values for $x_1$ and $x_2$? Start with something like $x_1=1, x_2=1$ and find utility $u(1,1)=1+5*1$. Then increase $x_1$ or $x_2$ and let $x_1=2, x_2=1$ and find ...
Bayesian's user avatar
  • 5,280
4 votes

Pareto optimal and Walrasian equilibrium

We have two Consumers 1 and 2, and two goods 1 and 2 pure exchange economy. The following utility functions can be used to represent their preferences: $u_1(x_{11}, x_{12}) = x_{11}$ $u_2(x_{21}, x_{...
Amit's user avatar
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4 votes
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How to find a Walrasian equilibrium for this production economy?

Solving the profit maximisation problem of firm producing $x$, $\max_{x\geq 0, l_X\geq 0} \ p_Xx - l_X$ subject to $x = 2\sqrt{l_X}$ gives the demand for labor by firm producing $x$ and supply of $x$ ...
Amit's user avatar
  • 8,396
3 votes
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What does it mean "non-Walrasian" or "quasi-Walrasian"?

Well non-Walrasian is an model that is not Walrasian so to explain that we should explain what a Walrasian model is. Generally speaking in macro context when we talk about a Walrasian model it is a ...
1muflon1's user avatar
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3 votes

Doubt regarding Walrasian equilibrium with complements for both agents

Since demand equals supply holds for every price $p$, this simply means that every $p$ is an equilibrium price. However, the equilibrium allocation that $p$ supports varies with $p$. To be precise, ...
Amit's user avatar
  • 8,396
3 votes
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Why is there a Walrasian Equilibrium if excess demand goes to infinity as price goes to 0?

Let $z_j(p)$ be the excess demand function for good $j$, where $p := \frac{p_2}{p_1}$ is the relative price between the two goods. Note it is possible to express the excess demand functions as single ...
Nicolas Torres's user avatar
3 votes

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

Common utility function means: Here utility functions of the two consumers are of the same form: $u_i (x_i, y_i) = x_i^\alpha y_i^{1-\alpha}$  where $0 < \alpha< 1$ and $i\in\{1,2\}$ Society has ...
Amit's user avatar
  • 8,396
3 votes
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Exchange economy with trade

First, let us observe the autarky equilibrium: Consumer $0$'s utility maximisation problem is $\displaystyle\max_{x_0\geq 0,y_0\geq 0}x_0^\alpha y_0^{1-\alpha}$ subject to $px_0+y_0\leq p$ Solving it ...
Amit's user avatar
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3 votes
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How to show there is no a Walrasian equilibrium?

Given the exchange economy, Three Goods: X (Apple), Y (Banana), M (Money) Three Agents: Seller (S), Buyer 1 (A), Buyer 2 (B) Endowment of the seller: $W_S=(1,1,0)$; Endowment of the buyers: $W_A=W_B=(...
Amit's user avatar
  • 8,396
2 votes

Perfect Complements - Walrasian Equilibrium

If the prices $p_1$ and $p_2$ are positive than as you pointed out the equations \begin{eqnarray*} \frac{30p_1}{p_1+p_2} + \frac{20p_2}{p_1+4p_2} & = & 30 \\ \\ \frac{30p_1}{p_1+p_2} + ...
Giskard's user avatar
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2 votes

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

An indirect proof. Suppose $$ x(p,w) = w\cdot x(p,1) $$ does not hold. This is equivalent with stating $$ U(x(p,w)) \neq U(w\cdot x(p,1)). $$ (To be precise: $x(p,w)$ and $x(p,1)$ may be set valued. ...
Giskard's user avatar
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2 votes

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

In standard proofs of the existence of Walrasian equilibrium, after one applies the appropriate fixed point theorem (typically Kakutani's, but sometimes Brouwer's), one then shows that the given fixed ...
Theoretical Economist's user avatar
2 votes
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Whether the demand in the panels satisfies the weak axiom

So WARP says that if bundle x' you choose under budget B' is affordable at budget B'' (and you don't choose x' - i.e x'$\not=$x''), then x'' cannot be affordable under budget B'. The picture you ...
VCG's user avatar
  • 766
2 votes

Find equilibrium price using excess demand function

Let $m_i$ be the market value of the endowment of player $i\in\{1,2\}$. In optimum player $i$ spends everything such that $p_x x_i + p_y y_i =m_i$. And because of the preferences in optimum also $x_1 =...
Bayesian's user avatar
  • 5,280
2 votes

Question about the relationship between Weak Axiom and Slutsky Matrix

It is almost true. There are examples of demand that have a negative definite Slutsky matrix but fails the Weak Axiom. However, if we ask that $$v \cdot S(p,w) v <0 $$ whenever $v \not = \alpha ...
Walrasian Auctioneer's user avatar
2 votes
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Solve for the Walrasian demand, Utility of three variables, and Convexity of Preferences?

For part (i), in complete rigor, you should also check the determinants of all the leading principal minors of the bordered Hessian and make sure that they have alternating signs. Your final ...
Herr K.'s user avatar
  • 15.4k
2 votes

General Equilibrium with Perfect Substitutes

To solve for competitive equilibrium, we can first find the demand : Demand for commodity $X$ by A is $x_A = \frac{5}{p_x}$ if $p_x < 1$, $x_A \in [0,5]$ if $p_x = 1$, $x_A = 0$ otherwise. Demand ...
Amit's user avatar
  • 8,396
2 votes
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Blocking Coalitions

Coalition formation is not the same as majority voting. Say, in a single good economy, individual 1 has the entire endowment, so $\mathbf e=(1,0,0,0,0,0)$. Then individual 1 forms a blocking coalition ...
Herr K.'s user avatar
  • 15.4k
1 vote

Finding Cobb-Douglas Hicksian Demand using Duality

I prefer another approach. I use duality to find the cost function and then use Shepard's Lemma to derive the Hicksian demands. I will demonstrate below for the simple 2 good case, rather than ...
Brennan's user avatar
  • 620
1 vote

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

Under what constraints is $u_1+u_2$ maximized? The exercise may be about the First Welfare Theorem: a competitive equilibrium is Pareto efficient. In this case, maximizing the sum may be designed to ...
Sander Heinsalu's user avatar
1 vote
Accepted

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

For case 1, you can argue $z$ will be unique by contradiction: Suppose ad absurdum there is another $z'$ that is feasible (i.e. $p^Tz' =w$) optimal and $z'\neq z$. Then you can consider a convex ...
GabMac's user avatar
  • 354

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