6
votes
Accepted
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 ...
- 9,405
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 $...
- 3,247
6
votes
Accepted
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(...
- 9,405
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{...
- 4,417
5
votes
Accepted
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
$$\max_{y \geq 0} \ y_1^\alpha y_2^...
- 3,247
5
votes
Accepted
Walras Law in a production economy with fixed costs
Partial answer:
for simplicity let $P_c =1$.
The budget constraint: $c= wn + \Pi$
Simplify (plug in $\Pi$): $c= F(n)- fc$
Goods clearing: $c = F(n)$
The household's budget constraint is inconsistent w/...
- 300
3
votes
Accepted
Walrasian demand with a twist of Leontief function
Why don't you just plug in some values for $x_1$ and $x_2$?
Start with something like
$x_1=1, x_2=1$ and find utility $u(1,1)=1+5*1$.
Then increase $x_1$ or $x_2$ and let
$x_1=2, x_2=1$ and find ...
- 5,090
3
votes
Accepted
What does it mean "non-Walrasian" or "quasi-Walrasian"?
Well non-Walrasian is an model that is not Walrasian so to explain that we should explain what a Walrasian model is. Generally speaking in macro context when we talk about a Walrasian model it is a ...
- 41.5k
3
votes
Pareto optimal and Walrasian equilibrium
We have two Consumers 1 and 2, and two goods 1 and 2 pure exchange economy. The following utility functions can be used to represent their preferences:
$u_1(x_{11}, x_{12}) = x_{11}$
$u_2(x_{21}, x_{...
- 4,417
3
votes
Doubt regarding Walrasian equilibrium with complements for both agents
Since demand equals supply holds for every price $p$, this simply means that every $p$ is an equilibrium price. However, the equilibrium allocation that $p$ supports varies with $p$. To be precise, ...
- 4,417
2
votes
Show that $x(p,w)=w\cdot x(p,1)$ with homothetic preferences
An indirect proof. Suppose
$$
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)).
$$
(To be precise: $x(p,w)$ and $x(p,1)$ may be set valued. ...
- 26k
2
votes
Accepted
Convexity of Walrasian Demand
I am not sure whether the question asks about the function being convex, or simply the demand correspondence/function being a convex set. The latter is often asked and if that is the case here is an ...
- 5,394
2
votes
How does Brouwer's fixed point theorem relate to Walrasian equilibrium?
In standard proofs of the existence of Walrasian equilibrium, after one applies the appropriate fixed point theorem (typically Kakutani's, but sometimes Brouwer's), one then shows that the given fixed ...
- 2,097
2
votes
Accepted
Whether the demand in the panels satisfies the weak axiom
So WARP says that if bundle x' you choose under budget B' is affordable at budget B'' (and you don't choose x' - i.e x'$\not=$x''), then x'' cannot be affordable under budget B'.
The picture you ...
- 756
2
votes
Find equilibrium price using excess demand function
Let $m_i$ be the market value of the endowment of player $i\in\{1,2\}$.
In optimum player $i$ spends everything such that $p_x x_i + p_y y_i =m_i$.
And because of the preferences in optimum also $x_1 =...
- 5,090
2
votes
Question about the relationship between Weak Axiom and Slutsky Matrix
It is almost true.
There are examples of demand that have a negative definite Slutsky matrix but fails the Weak Axiom.
However, if we ask that
$$v \cdot S(p,w) v <0 $$
whenever $v \not = \alpha ...
- 1,829
2
votes
Accepted
Solve for the Walrasian demand, Utility of three variables, and Convexity of Preferences?
For part (i), in complete rigor, you should also check the determinants of all the leading principal minors of the bordered Hessian and make sure that they have alternating signs. Your final ...
- 14.3k
2
votes
General Equilibrium with Perfect Substitutes
To solve for competitive equilibrium, we can first find the demand :
Demand for commodity $X$ by A is $x_A = \frac{5}{p_x}$ if $p_x < 1$, $x_A \in [0,5]$ if $p_x = 1$, $x_A = 0$ otherwise.
Demand ...
- 4,417
2
votes
Accepted
Blocking Coalitions
Coalition formation is not the same as majority voting. Say, in a single good economy, individual 1 has the entire endowment, so $\mathbf e=(1,0,0,0,0,0)$. Then individual 1 forms a blocking coalition ...
- 14.3k
1
vote
Finding Cobb-Douglas Hicksian Demand using Duality
I prefer another approach. I use duality to find the cost function and then use Shepard's Lemma to derive the Hicksian demands. I will demonstrate below for the simple 2 good case, rather than ...
- 620
1
vote
What is the difference between solving for utility maximization separately or aggregately?
Under what constraints is $u_1+u_2$ maximized? The exercise may be about the First Welfare Theorem: a competitive equilibrium is Pareto efficient. In this case, maximizing the sum may be designed to ...
- 928
1
vote
Accepted
Why does strictly Walrasian demand with quasi-concave utility function mean that the walrasian demand having only one single consumption bundle?
For case 1, you can argue $z$ will be unique by contradiction:
Suppose ad absurdum there is another $z'$ that is feasible (i.e. $p^Tz' =w$) optimal and $z'\neq z$. Then you can consider a convex ...
- 342
1
vote
Derive demand function $x(p,w)$ from utility function $u(x) = \min\{x_1, x_2\} + x_3$
for this problem you must conciser two possible branches of the utility function:
$$u(\text{x})=x_1+x_3\ \ \text{if} \ \ x_1<x_2$$
$$u(\text{x})=x_2+x_3\ \ \text{if} \ \ x_1>x_2$$
The demands of ...
- 7,685
1
vote
Derive demand function $x(p,w)$ from utility function $u(x) = \min\{x_1, x_2\} + x_3$
Hint: Imagine that there are two coffee bars, $A$ and $B$. There is only one type of coffee in the world. My preferences are such that I always want 1 unit of sugar with 1 unit of coffee; if I consume ...
- 743
1
vote
What is the Walras law vs first welfare theorem
Walras' law describes market equilibria conditions, which states roughly speaking that if there exists within an exchange economy market equilibria for $n-1$ good markets, then the last good market $n$...
1
vote
General Equilibrium Involving Production
You seem to be looking for the Second Welfare Theorem.
The Second Theorem states that out of all possible Pareto optimal outcomes one can achieve any particular one by enacting a lump-sum wealth ...
- 26k
1
vote
What is excess demand/ excess supply?
Excess demand is the function describing the amount of quantity demanded above quantity supplied at each price level. Mathematically, it is allowed to be negative (so then we have excess quantity ...
- 31.8k
1
vote
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} + ...
- 26k
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