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Chapter 25: Problem 5

Suppose there are two individuals in an economy. Utilities of those individuals under five possible social states are shown in the following table: $$\begin{array}{ccc} \text { State } & \text { Utility 1 } & \text { Utility 2 } \\ \hline \mathrm{A} & 50 & 50 \\ \mathrm{B} & 70 & 40 \\ \mathrm{C} & 45 & 54 \\ \mathrm{D} & 53 & 50.5 \\ \mathrm{E} & 30 & 84 \\ \hline \end{array}$$ Individuals do not know which number \((1 \text { or } 2)\) they will be assigned when the economy begins operating, hence they are uncertain about the actual utility they will receive under the alternative social states. Which social state will be preferred if an individual adopts the following strategies in his or her voting behavior to deal with this uncertainty? a. Choose that state which ensures the highest utility to the least well-off person. b. Assume there is a \(50-50\) chance of being either individual and choose that state with the highest expected utility. c. Assume that no matter what, the odds are always unfavorable such that there is a 60 percent chance of having the lower utility and a 40 percent chance of higher utility in any social state. Choose the state with the highest expected utility given these probabilities. d. Assume there is a \(50-50\) chance of being assigned either number and that each individual dislikes inequality, Fach will choose that state for which expected utility \(-\left|U_{1}-U_{2}\right|\) is as large as possible (where the \(|\ldots|\) notation denotes absolute value). e. What do you conclude from this problem about social choices under a "veil of ignorance" as to an individual's specific identity in society?

Short Answer

Expert verified
Answer: According to the Maximin rule strategy, State D is preferred, as it ensures the highest utility (50.5) to the least well-off person.

Step by step solution

01

(Strategy a: Maximin Rule)

The first strategy involves choosing the social state that ensures the highest utility to the least well-off person. To find this, we need to find the minimal utility for each state and pick the one that provides the highest minimal utility. Min. Utility - State A: 50 State B: 40 State C: 45 State D: 50.5 State E: 30 Here, State D ensures the highest utility to the least well-off person with 50.5.
02

(Strategy b: Expected Utility with A 50-50 Chance)

This strategy assumes there is a 50-50 chance of being either individual and chooses the state with the highest expected utility. The expected utility (EU) is given by the formula: EU = 0.5 × (Utility 1 + Utility 2). Expected Utilities - State A: 0.5*(50+50) = 50 State B: 0.5*(70+40) = 55 State C: 0.5*(45+54) = 49.5 State D: 0.5*(53+50.5) = 51.75 State E: 0.5*(30+84) = 57 Here, State E has the highest expected utility with 57.
03

(Strategy c: Expected Utility with Unfavorable Odds)

Assuming there is a 60% chance of having the lower utility and a 40% chance of the higher utility in any social state, we calculate the expected utility. Expected Utilities - State A: 0.6*(50)+0.4*(50) = 50 State B: 0.6*(40)+0.4*(70) = 52 State C: 0.6*(45)+0.4*(54) = 48.6 State D: 0.6*(50.5)+0.4*(53) = 51.7 State E: 0.6*(30)+0.4*(84) = 51.6 Under this strategy, the highest expected utility comes from State D with 51.7.
04

(Strategy d: Expected Utility considering Inequality Aversion)

This approach assumes there is a 50-50 chance of being assigned either number and that individuals dislike inequality. We need to find the state with the largest expected utility - |Utility 1 - Utility 2|. Inequality measure - State A: 50-50 = 0 State B: 70-40 = 30 State C: 45-54 = -9 State D: 53-50.5 = 2.5 State E: 30-84 = -54 The state with the largest expected utility without creating inequality is State A with an inequality measure of 0.
05

(Conclusion)

When individuals choose a social state under a "veil of ignorance" as to their specific identity in society, the preference over states depends on the strategy they adopt. This shows that different strategies lead to different outcomes, and careful evaluation is necessary to understand the trade-offs between preferences and aversions in social choices. In this example, different strategies lead to four different state preferences: A, B, D, and E.

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Most popular questions from this chapter

There are 200 pounds of food that must be allocated between two sailors marooned on an island. The utility function of the first sailor is given by \\[ \text { utility }=\sqrt{F_{1}} \\] where \(F_{1}\) is the quantity of food consumed by the first sailor. For the second sailor, utility (as a function of his food consumption) is given by \\[ \text { utility }=\frac{1}{2} \sqrt{F_{2}} \\] a. If the food is allocated equally between the sailors, how much utility will each receive? b. How should food be allocated between the sailors to ensure equality of utility? c. How should food be allocated so as to maximize the sum of the sailors' utilities? d. Suppose sailor 2 requires a utility level of at least 5 to remain alive. How should food be allocated so as to maximize the sum of utilities subject to the constraint that sailor 2 receive that minimum level of utility? e. Suppose both sailors agree on a social welfare function of the form \\[ W=U_{1}^{/ 2} U_{2}^{1 / 2} \\] How should food be allocated between the sailors so as to maximize social welfare?

How does the free rider problem arise in the decision of eligible voters to vote? How might voter participation decisions affect median voter results? How might it affect probabilistic voting models?

Suppose voters based their decisions on the ratio of utilities received from two candidates that is, Equation 25.6 would be \\[ \boldsymbol{\pi}_{i}=f_{i}\left[\left(U_{i}\left(\boldsymbol{\theta}_{1}\right) / U_{i}\left(\boldsymbol{\theta}_{2}\right)\right]\right. \\] Show that the results from a game involving net value platforms would in this case maximize the Nash Social Welfare function \\[ \boldsymbol{S W}=\prod_{i=1}^{n} \boldsymbol{U}_{i} \\]

Suppose an economy is characterized by a linear production possibility function for its two goods \((X \text { and } Y)\) of the form \\[ X+2 Y=180 \\] There are two individuals in this economy, each with an identical utility function for \(X\) and \(Y\) of the form \\[ \boldsymbol{U}(\boldsymbol{X}, \boldsymbol{Y})=\sqrt{\boldsymbol{X} \boldsymbol{Y}} \\] a. Suppose \(Y\) production is set at \(10 .\) What would the utility possibility frontier for this economy be? b. Suppose \(Y\) production is set at \(30 .\) What would the utility possibility frontier be? c. How should \(Y\) production be chosen so as to ensure the "best" utility possibility frontier? d. Under what conditions (contrary to those of this problem) might your answer to part (c) depend on the point on the utility possibility frontier being considered?

Suppose individuals face a probability of \(u\) that they will be unemployed next year. If they are unemployed they will receive unemployment benefits of \(b\), whereas if they are employed they receive \(w(1-t)\) where \(t\) is the tax used to finance unemployment benefits. Unemployment benefits are constrained by the government budget constraint \(u b=t w(1-u)\) a. Suppose the individual's utility function is given by \\[ U=\left(Y_{i}\right)^{8} / \delta \\] where \(1-\delta\) is the degree of constant relative risk aversion. What would be the utilitymaximizing choices for \(b\) and \(t\) b. How would the utility maximizing choices for \(b\) and \(t\) respond to changes in the probability of unemployment, \(u ?\) c. How would \(b\) and \(t\) change in response to changes in the risk aversion parameter \(\delta ?\)
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