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

(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 ...


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

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 with Hart's 1975 paper wherein he shows that the opening of some new markets can make every agent worse off. Hart himself did not call this Pareto worsening, but ...


6

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 departures. For example, in the case of monopsony power in low-wage labor markets, legislated minimum wage increases can potentially move wages closer to efficient ...


5

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 ...


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

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 ...


5

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} ...


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

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 ...


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 ...


5

(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 ...


5

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"). By your assumptions $r(\theta_H) < \theta_H$ and $r(\theta_L)>\theta_L$ such that in PO, L shall stay at home and H shall work. However, you need to ...


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

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 ...


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

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

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 change where no one is better off and at least one person is worse of. Existing Antonyms to "improvement" As for what to call it, the Thersaurus does indeed ...


4

A short answer is that in case of provision of public goods in real-life we can never be completely sure if they are provided in optimal quantity due sheer difficulty of quantifying all costs and benefits, measurement problems when it comes to underlaying utility, due to uncertainties involved and many other factors. However, this does not mean that no ...


4

The Pareto distribution was derived based on the observation of Vilfredo Pareto that in Italy $80\%$ share of land belonged to $20\%$ of the country’s population, and he showed this held for many other countries as well (see Pareto Manual of political economy - the book is still being reprinted). Consequently, based on this empirical observation it was ...


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, ...


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