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7

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 $(\hat{x}_1,\hat{x}_2)$, but the marked point is not Pareto optimal.

6

Shell 1971 argues (in a ten page paper, so read it!) that the dynamic inefficiency stems from the double infinity of traders and goods, and not the dynamics. This allows us to do the Hilbert hotel switch. Therefore, even when all souls are able to transact business in the same Walrasian market, the absence of Pareto-optimality persists in the ...

6

(Note that this answer implicitly makes reference to the specific model in Lee and Saez.) Short answer: the increased taxes on high-skilled workers exactly offset the higher real wages they obtain from a decline in the minimum wage for low-skilled workers. Longer answer: Suppose that I'm the government, and I decide to lower the minimum wage $\bar{w}$. The ...

5

The reason is that at the same time the wage of the high-skilled increases. By reducing the minimum wage, the number of people working in the low-skilled sector increases (involuntary unemployment is reduced) which leads to an increase in the wage of the high-skilled. The corresponding proposition in the paper is Proposition 3, the argument of which is ...

5

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 from context which one is being used. The dictionary definition of Pareto Optimal is something like "An allocation from which any feasible change which makes any ...

5

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 of a preference ordering by a numerical function") states that a continuous utility function exists which represents the preferences. I will denote the utility ...

4

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 $L$ = Supply of $L$ Demand for $K$ = Supply of $K$ where these demands and supplies are either exogenously given or are derived by solving utility maximization ...

4

Let $\Omega$ be the set of all feasible allocations with an element $\omega \in \Omega$. Consider $I$ agents such that the utility of agent $i$ is described by $u_i(\omega)$. Definition 1: $\omega \in \Omega$ is weakly Pareto-optimal if $\nexists \omega' \in \Omega$ such that $\forall i$ $u_i(\omega') > u_i(\omega)$. Weak Pareto-optimality is basically ...

4

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 all $z \in [0,1]$ the distributions $$(x_A,y_A) = (z,1), \ (x_B,y_B) = (1-z,0).$$ are Pareto-efficient. This follows from $MRS_A(x_A,y_A) \geq MRS_B(x_B,y_B)... 4 To be a Pareto optimum, there must not exist another feasible allocation that makes every agent at least as well off and one or more agents strictly better off. So, let us consider the options here. Answer B - look at the individual allocations here and consider them relative to the economy's total endowment. Since the allocations given here do not ... 4 Well, when considering the minimal conditions necessary for an allocation to be Pareto optimal we must go to the primitives. First, we need the fact that all agents have rational preferences. What this means is that preferences are complete and transitive. Another thing we need is for preferences to be strictly convex. Let$\succcurlyeq$be the preference ... 4 There is an unpublished 1982 working paper by Donald Brown and John Geanakoplos, called “Understanding Overlapping Generations Economies as a Lack of Market Clearing at Infinity” (a scan used to be available at Brown's homepage). The authors show that there is a one-to-one correspondence between the equilibria of an OLG economy and almost-equilibria in a ... 4 Here $$\not\exists\;x_i^\star\; s.t.\; u_i(\alpha x_i)\geq u_i(\alpha x_i^\star)\;\forall i\;\text{and}\;u_i((1-\alpha)\hat{x}_i)\geq u_i((1-\alpha)x_i^\star)$$ $$\implies\not\exists\;x_i^\star\;s.t.\;u_i(\alpha x_i^\star+(1-\alpha)x_i^\star)\geq u_i(\alpha x_i+(1-\alpha)\hat{x}_i)$$ you assume that the function$u$is linear. Unfortunately the statement is ... 4 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 between the benefit of the reduced pollution and the loss to the pharmaceutical firm from the reduced production. But this last figure is small: at the competitive ... 4 The following plot has the answer to your question, observe it carefully: 4 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 increase his or her payoff by deviating to some alternative strategy. Let's consider$\{30,30,30\}$. In this 'putative' equilibrium, firm A's profit is$$\frac{D(... 4 By "potential Pareto improvement", what Fallis means is a "Kaldor-Hicks improvement" or more simply an overall increase in wealth. The argument by Kaldor (1939) and Hicks (1939) was that if there is an overall increase in wealth, then we could potentially compensate the losers, so that there is potentially a Pareto improvement. Therefore, a "potential ... 4 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 the union given$(w, L)$. What you are looking for are pairs$(w, L)$such that neither the firm nor the union can have a higher payoff without the other side ... 4 (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 the Pareto efficient alternative to (A,A) and (almost) never discuss (A,B) or (B,A). Keep in mind that Pareto efficiency always requires a starting point to be ... 3 It doesn't really make sense to ask what the "minimum" conditions are for a market equilibrium to be Pareto efficient. By minimum do you mean least restrictive? Do you mean simplest conditions? You have to specify what market we're looking to put conditions on, because otherwise I can make up a really trivial market, give it some conditions, and say that it ... 3 I don't think it is true in a standard pure exchange economy the question is referring to. Consider the following counterexample: Suppose$I = \{1,2\}$and$u_1(x_1, m_1) = \sqrt{x_1} + m_1$and$u_2(x_2, m_2) = \sqrt{x_2} + m_2$. and let the set of feasible allocations be$\{((x_1, m_1), (x_2, m_2))\in\mathbb{R}^2_+\times\mathbb{R}^2_+: x_1+x_2 = 2, ...

3

It seems to me that as long as every person likes wealth any allocation $W_t$ will be Pareto-optimal. Even if this was not the case, so supposing that $w_1, w_2,...$ are not real numbers representing wealth but vectors representing bundles of goods the distribution from one time to another can change drastically if you do not specify how growth occurs. At $... 3 Set of Pareto Efficient Allocations consists of feasible allocations$((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)$satisfying the following conditions: Budget Requirement:$p_xx_J+ y_J = 2p_x + 2$and$p_xx_D+ y_D = 4p_x + 1$Equilibrium Conditions:$\displaystyle\frac{2y_J}{x_J} ...

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Q1 Optimal usually means one of two things: Pareto optimal (there's no way to change the outcome such that everyone is at least as well-off, but someone is strictly better-off). Optimal in the utilitarian sense (we maximise the sum of everyone's profit/utility). If utility/profit is quasilinear then the two concepts coincide. This paper appears to be ...

3

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: $u_1(x_{11}, x_{12}) = x_{11}$ $u_2(x_{21}, x_{22}) = x_{22}$ Equilibrium price vector $(p_1, p_2=1)$ and allocation $((x_{11}, x_{12}), (x_{21}, x_{22}))$ satisfy the following: Optimality Conditions (...

3

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 made better off by another allocation (or, more weakly, if at least one of its members could be made better off without others being made worse off). And so, by ...

3

@HerrK. got it right in his comment (he should have deleted the somewhat confusing "yes" from the beginning and then posted it as an answer) It is possible that no pairwise improvements are possible but general Pareto-improvements are still possible. A simple counterexample for three actors and three goods is as follows. Let the utility functions be the same ...

3

This is essentially a variation on the answer of denesp that requires slightly fewer assumptions. Assume there are $l$ commodities and $m$ agents. An allocation is then a point in $\mathbb{R}^{lm}_+$. If the aggregate endowment is $e\in\mathbb{R^l}_+$, an allocation is a point in $\sum^{-1}(\{e\})$, where $\sum:\mathbb{R}^{lm}_+\to\mathbb{R}^l$ is the ...

3

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 possible to interpret this as an existence issue. The first welfare theorem is about all competitive equilibria and holds trivially if there are none. The second ...

3

The bottom right origin is actually not in the Pareto set. At that point, $(x_A,y_A)=(8,0)$, so $U_A(x_A,y_A)=0$. Similarly, $(x_B,y_B)=(0,4)$, so $U_B(x_B,y_B)=0$. As an example, $B$ could give one unit of $y$ to $A$ and thereby raise $A$'s utility to 8 without hurting $B$'s own utility (which would remain zero). As a matter of fact, any allocation ...

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