There are / individual consumers. Each consumer i's indirect utility function is given by v,(p, wy)= a, (p) + u. Consider the Bergon-Samuelson social welfare function W(u₁... u) = 1 Blog ₁. where 3, 20 for any i. Assume that the social welfare maximization problem has an interior solution. Find the optimal wealth distribution function w,(p, w).
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- A social welfare function is thought to be additive along the lines proposed by Bentham and Mill: W = U, + U, + .+ Un where U; is individual i's utility and there are n individuals. Suppose an economy is made up of two individuals, John and Jane. John's utility function is U John = 17X and Jane's utility function is U JJane = 2X. There is a fixed quantity of good X (40 units) evenly divided between John and Jane (20 units for each). Social welfare (W) is equal to: W = = utils. (Enter your response rounded to the nearest whole number.)The most general form of a social welfare function (SWF ) can be written as W = W(U1; ... ; UH). a. Explain the following properties that a SWF may satisfy: non paternalism, Pareto principle, anonymity (the names of the agents do not matter), and concavity (aversion to inequality). b. Consider two agents h = 1; 2 with utilities U1 and U2. Depict the social indifference curve of the utilitarian SWF in (U1; U2)-space. Which of the properties in part a does it satisfy? c. Depict the social indifference curves of the maxim in or Rawlins SWF. Contrast to the utilitarian SWF with respect to the aversion to inequality. Which properties does the Rawlins SWF satisfy? d. The Bernoulli-Nash social welfare function is given by the product of individual utilities. Discuss the distributional properties of the Bernoulli-Nash SWF.If the social welfare function produces concave social welfare indifference curves, social welfare can be increased by reallocating income between households if their marginal utilities of income differ. TRUE OR FALSE
- Give an example of monotonic preferenceSuppose X = R k + for some k ≥ 2, and we define x = (x1 , …, xk ) ≥ = (y1 , …, yk ) if x ≥ y; that is, if for each i = 1, …, k, xi ≥ yi . (This is known as the Pareto ordering on R k + ; it plays an important role in the context of social choice theory in Chapter 8.) (a) Show that ≥ is transitive but not complete. (b) Characterize ≥ Is asymmetric? Is≥ Negatively transitive? Prove your assertions. (c) Characterize ~ defined from ≥ in the usual fashion; that is, x ~ y if x ≥ y and y x. Is ~ reflexive? Symmetric? Transitive? Prove your assertions.We now focus on one country. Assume that the society is composed of two individuals A and B. The VIA total income available in the society is $550,000. The utility of individual A is given by UA= and individual B ones is given by UA = 0.5/IB. b) What should the income distribution be for the individuals to have the same utility level? c) Find the income distribution that maximizes social welfare in the case of an additive social welfare function. Now imagine that in reality, individual A has an income of $500,000 and individual B's income is $50,000. d) How far from the optimal income distribution defined in question c) are we? What redistri- bution should be chosen to reach the optimal social welfare calculated in c)?
- Suppose there are 90 consumers in the market with two goods x and y. There are two types of individuals A and B. 60 of the consumers are type A and the 30 of them are type B. Type A consumers have a utility function of u(x, y) = x¹/³y¹/³ and each of them has an income of mд. Type B consumers have a utility function of u(x, y) = x²/³y¹/³ and each of them has an income of mp. 3. Find the market demand for good x as a function of incomes and prices. 4. Find the market demand for good y as a function of incomes and prices. 5. Suppose price of good x is 2 and the price good y is 2. Further suppose mA = 60 and m³ = - 120. Find the market demand for good x and good y at the given prices and incomes.Individual that consumes two goods (X and Y) and has a CES Utility Function of the form: U = 100(X^(0.75) + Y^(0.75)). Income of 1000, the price of Good X is 10 and the Price of Good Y is 20 a) Find the Marginal Rate of Substitution as a function of the quantities consumed of Good X and Good Y. b) Write out the Lagrangian for this problem. c) Solve to find the demand for Good X, the demand for Good Y, and the highest level of utility for this individual. d) Now consider an increase in the price of Good X to 20. What is the demand for Good X and Good Y? What is the Utility of the consumer following the price change? e) Considering the change in demand for each good between parts c) and d), how much is due to the substitution effect and how much is due to the income effect? f) Show your answers on a graph.Show how to construct the reference dependent utility function for two friends Kate and Mary whose gains and losses are listed as follows: Kate's net worth is $4.5 million (decreased from $5.5 to $4.5 million) Mary's net worth $3.2 million (increased from $3 to $3.2 million) (First determine the reference point (use a parameter) and then derive reference utility function for each).
- Explain the difference between a utility function and correspondenceMarginal Utility MU, 0 $5,000 10,000 15,000 Income Singer Marginal Utility Multiple Choice MU, 0 $5,000 10,000 15,000 Income Catalano Refer to the above diagrams that show identical marginal utility from income curves for Singer and Catalano. If this initial distribution of $15,000 to Singer and $5,000 to Catalano is altered in favor of greater equality, it may be argued that: the combined total utility of the two consumers will decline because Catalano has a greater capacity to derive utility from income than does Singer incentives to produce will be weakened and total income will decrease.. incentives to produce will be enhanced and total income will increase. the combined total utility of the two consumers will decline because Singer has a greater capacity to derive utility from income than does Catalano.I need help with this homework problem. Suppose there are two consumers, A and B. The utility functions of each consumer are given by: UA(X,Y) = (X^1/2)*(Y^1/2) UB(X,Y) = X + Y The initial endowments are: A: X = 8; Y = 3 B: X = 4; Y = 5 What is the marginal rate of substitution for consumer A at the initial allocation? What is the marginal rate of substitution for consumer B at the initial allocation? Is the initial allocation Pareto Efficient?