7
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Is the Convex Combination of Two Pareto Optimal Allocations Also Pareto Optimal?
To complement densep's answer, here is a schematic Edgeworth box illustration of what can go wrong. The points on the dashed line are convex combinations of the Pareto optimal points $(x_1,x_2)$ and $(...
- 16.9k
6
votes
Accepted
What's the opposite of a Pareto improvement called?
I usually use the phrase "Pareto worsening". It is not really widespread, in fact I am not sure I have ever heard anyone else use it. However now I googled it and people seem to use it in connection ...
- 28.7k
6
votes
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Can the theory of the second best mathematically justify labor unions in some scenarios?
Quoting your quote, emphasis altered by me:
The “theory of the second best” clearly argues that once markets depart at all from perfect competition, efficiency may well be increased by further ...
- 28.7k
6
votes
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Proving Pareto-efficiency with MRS
One way to check that this allocation is Pareto efficient is using the first welfare theorem. Consider the exchange economy with utility functions
\begin{eqnarray*} u_i(x_i, y_i) = \sqrt{x_iy_i}\end{...
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5
votes
Does a General Equilibrium here require Pareto Optimality?
Competitive equilibrium is the price vector $(p_x, p_y, w =1, r)$ such that it solves the following system of equations:
Demand for $X$ = Supply of $X$
Demand for $Y$ = Supply of $Y$
Demand for $L$ =...
- 8,306
5
votes
Two quasilinear utility functions
The statement is not true.
Let $x_A + x_B = 1$, $y_A + y_B = 1$.
Let $U_A(x_A,y_A) = x_A + \ln(y_A)$, $U_B(x_B,y_B) = x_B + \ln(y_B + 1)$.
$f \neq g$ are both strictly increasing & concave. For ...
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5
votes
Perfect complement preferences in an exchange economy
Set of Pareto Efficient Allocations consists of feasible allocations $((x_J, y_J), (x_D, y_D))$ satisfying $y_J=\displaystyle\frac{x_J}{2}$.
Competitive Equilibrium is the price $(p_x, p_y=1)$ ...
- 8,306
5
votes
Accepted
Pareto optimality with externalities
I feel like I do not understand the exact meaning behind the notion of
the Pareto optimality.
It's not you. There are different senses of the phrase "Pareto Optimal," and you have to figure out ...
- 231
5
votes
Pareto Optimality and Core
An allocation is defined as being a part of the "core" of an economy if there's no coalition of people that blocks the allocation. A coalition will block an allocation if all of its members could be ...
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5
votes
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Does every allocation have a maximal Pareto-improvement?
I think there is a short proof if you also assume that the number of agents $n$ is finite and that the preferences are continuous.
Given the second assumption Debreu's theorem (1954, "Representation ...
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5
votes
Why is Pareto efficiency not always equitable or equilibrium?
You seem to be confusing concepts from partial and general equilibrium.
Partial equilibrum
Supply and demand functions are usually mapped in such a way that quantity is on the $x$-axis and price is ...
- 28.7k
5
votes
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Definition of Pareto efficiency and prisoner's dilemma
(A, B) and (B, A) are in fact Pareto efficient.
I believe that your confusion may be because when discussing the Pareto inefficiency of the Prisoner's dilemma equilibrium, we always discuss (B,B) as ...
- 6,128
5
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Adverse Selection: Positive Selection of Worker Types (Mas-Collel)
In PO, you want all types with opportunity cost $r(\theta)\leq \theta$ to trade, because the firm gets more productivity than the worker has to give up on home productivity ("opportunity cost&...
- 5,270
5
votes
Accepted
Finding the Pareto efficient allocations
Set of feasible allocations in this economy is:
$\mathcal{F}=\{(x_1,x_2,y)\in\mathbb{R}^3_+|x_1+x_2+y^2=30\}$
This set can also be represented in graph in the following way:
To determine (interior) ...
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5
votes
Question for general equilibrium
To find efficient allocations in this economy, we can first determine the production possibility frontier (PPF) which is given by the line segment $\dfrac{x}{2}+y=100$ where $x\in[0,200]$. Pareto ...
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5
votes
Accepted
How to find the contract curve when both agents have linear utilities?
I rewrite the problem of maximization you wrote (I omit the endowments):
$\max x_A + y_A \;\;\qquad (1)$
subject to
$s x_B + y_B = \overline{U}\qquad (2)$.
This problem can be seen as a problem of ...
- 3,322
5
votes
Pareto efficient allocations for non-monotonic, quasi-linear utility function
Given the economy:
Utility functions: $u_A(x_A,y_A) = x_A - |y_A-\alpha_A|$, $u_B(x_B,y_B) = x_B - |y_B-\alpha_B|$, where $\alpha_A \geq 0$, and $\alpha_B\geq 0$ are given.
$\omega = (\omega_X, \...
- 8,306
4
votes
Accepted
In modern economics, what are the least restrictive conditions shown to theoretically achieve Pareto Efficiency?
Well, when considering the minimal conditions necessary for an allocation to be Pareto optimal we must go to the primitives. First, we need the fact that all agents have rational preferences. What ...
- 1,558
4
votes
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Is the Convex Combination of Two Pareto Optimal Allocations Also Pareto Optimal?
Here
$$\not\exists\;x_i^\star\; s.t.\; u_i(\alpha x_i)\geq u_i(\alpha x_i^\star)\;\forall i\;\text{and}\;u_i((1-\alpha)\hat{x}_i)\geq u_i((1-\alpha)x_i^\star)$$
$$\implies\not\exists\;x_i^\star\;s.t.\...
- 28.7k
4
votes
Accepted
Finding Pareto Optimality
To be a Pareto optimum, there must not exist another feasible allocation that makes every agent at least as well off and one or more agents strictly better off.
So, let us consider the options here. ...
- 2,911
4
votes
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Why do externalities lead to a Pareto-inefficient outcome?
If you allow side payments then the issue you identify goes away in a Coase sense.
The citizens being polluted could pay for production to be reduced by one unit. This amount would have to be ...
- 4,755
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_{...
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4
votes
How do we find pareto optimal points in a 2 goods simple exhange economy?
The following plot has the answer to your question, observe it carefully:
- 8,306
4
votes
Accepted
Is the convexity of production sets necessary for the welfare theorems?
Convexity of the production set is indeed not needed for the proof of the first welfare theorem but for the proof of the second welfare theorem. It is not a necessary condition though.
It is ...
- 11.7k
4
votes
Bertrand-equilibrium with discrete price set
The key is the definition of the Nash equilibrium solution concept that you are applying to solve your game. In non-formal terms, a Nash equilibrium is a set of strategies such that no player can ...
- 16.9k
4
votes
How to calculate Pareto Optimal outcome in a game with a Nash Equilibrium
For Pareto optimality, you can ignore the timing. Also, anything you know about Nash equilibria in this game is irrelevant.
Let $v_F(w, L)$ be the payoff of the firm and $v_U(w, L)$ be the payoff of ...
- 11.7k
4
votes
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Walrasian Equilibrium intuition given prices and some initial allocation
The "trick" of this question is that the fact that agents do not want to trade at the given prices does not mean the allocation is Pareto. The only thing you know is that if there is an allocation ...
- 4,188
4
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What's the opposite of a Pareto improvement called?
The strict logic opposite of course is simply any change where at least one person is worse off but not sure we need a name for that. The conceptual opposite to Pareto improvement can as you say be a ...
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