/2. (a) Show that if x is an allocation in the y-core of an economy, then any two agents of the same type must receive the same bundles, provided the preferences of the agents are convex, strongly monotonic and continuous. (b) Suppose that an apple orchard is located next to a bee keeper. The orchard produces A apples and the bee keeper produces H units of honey. The cost function of the apple orchard is C(A) = A² and the cost function of the bee keeper is C,(H) = H² – A. Show that the socially efficient supply decisions involve apple orchard producing more apples than it would if it only considered its own costs. %3D

/2. (a) Show that if x is an allocation in the y-core of an economy, then any two agents of the same type must receive the same bundles, provided the preferences of the agents are convex, strongly monotonic and continuous. (b) Suppose that an apple orchard is located next to a bee keeper. The orchard produces A apples and the bee keeper produces H units of honey. The cost function of the apple orchard is C(A) = A² and the cost function of the bee keeper is C,(H) = H² – A. Show that the socially efficient supply decisions involve apple orchard producing more apples than it would if it only considered its own costs. %3D

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter13: General Equilibrium And Welfare

Section: Chapter Questions

Problem 13.1P

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why is there no inclusion of the parameter z in the solutions and the budget constraint?
Consider the production problem discussed in class in which 24 labor hours and 4 machinehours are to be allocated between food (x) production and wine (y) production. Suppose theproduction functions for x and y are as follows:Also, suppose the preferences of the agents are given by:Finally, suppose the MRT is given bywhere x and y denote, respectively, the aggregate quantities of food and wine produced.For each of the following allocations, determine whether they are (i) feasible and (ii) Paretoefficient. If they are inefficient, explain intuitively how it is possible to make someone betteroff without making the other person worse-off.a. ((14,2),(10,2),(5,10),(23,18))b. ((8,2),(16,2),(5,14),(11,18))c. ((14,2),(10,2),(14,21),(14,7))
Consider an economy inhabited by George and Harriet, whose utility functions are Ug : (ac)² (bc)2 Он тан + 2bн The total quantities of ale and bread that can be produced by the economy are a and b, and they are constrained by the production function b = 2(10 – a)/2 There are infinitely many Pareto optimal allocations. In one of them, Harriet's utility is 8. a) An allocation in this economy is described by a list of four variables. What are these variables? b) What four equations describe the Pareto optimal allocation in which Harriet's utility is 8? c) Find this Pareto optimal allocation.
Please refer to the following information to answer the question (in bold) below: You enjoy consuming apples (A) and oranges (O). Suppose that your utility function over both goods is given by Your marginal utility function for apples is . Your marginal utility function for oranges is U (A, O) = AO³ 2 apples, 28 oranges 3 apples, 26 oranges MUA 4 apples, 24 oranges 5 apples, 22 oranges 8 apples, 16 oranges 9 apples, 14 oranges MUO 3A0² . Currently, the price of apples is $10/peck, the price of oranges is $5/pound, and your income is $160. Assume that apples are your horizontal axis good and oranges are your vertical axis good. : 0³ - = Given your preferences and budget constraint, what bundle represents your consumer equilibrium for apples and oranges?
Dick, Jane, and their dog, Spot, share an apartment. Their utilities depend upon the cleanliness of the apartment and the hours spent doing housework. Dick and Jane are both averse to housework, though Dick finds it especially troublesome. Their utility functions are where c is an index of cleanliness and h represents hours of housework. The degree of cleanliness is determined by the “production function” a) Define the marginal rate of substitution MRSi as the increase in cleanliness that exactly compensates person i for one more hour of housework: and define the marginal rate of transformation as the number of hours of leisure that must must be given up to generate a one unit increase in cleanliness. Find the Samuelson condition for this economy. b) An allocation is a triplet (h D, hJ, c). What two conditions must a Pareto optimal allocation satisfy? Show that there are Pareto optimal allocations in which Dick, despite his aversion to housework, does more…
Ayana is pitching an idea for a startup company that makes and sells solar-powered phonechargers (C). Her market research has found that consumer demand for this product can beexpressed as a function of the price of the charger itself (PC), the price of phones (PF), andthe consmer’s income (I). Consumer demand can be described by the function C(PC, PF, I) =(i−10PC)/ (PF) Suppose her chargers come in all different capacities to meet any quantity demanded, so youdon’t need to worry about restricting C to whole numbers for this problem. (a) Does this product satisfy the law of demand?Explain. (b) Calculate consumers’ income elasticity as a function of the parameters. (c) Anaya is targeting a market whose income is expected to double over the next five years.Explain whether she should expect the demand for chargers to increase or decrease, andby how much, using the language of normal, inferior, luxury, and necessity goods. (d) Calculate the cross-price elasticity ofCwith respect toPF. (e)…
Ayana is pitching an idea for a startup company that makes and sells solar-powered phonechargers (C). Her market research has found that consumer demand for this product can beexpressed as a function of the price of the charger itself (PC), the price of phones (PF), andthe consmer’s income (I). Consumer demand can be described by the function C(PC, PF, I) =(i−10PC)/ (PF) Suppose her chargers come in all different capacities to meet any quantity demanded, so youdon’t need to worry about restricting C to whole numbers for this problem. (a) Does this product satisfy the law of demand?Explain.
Ginny likes to eat Bertie Bott’s Every Flavour Beans (b), and Chocolate Frogs (f). Herpreferences over these lollies are Cobb-Douglas, and can be presented by the utility functionu(b, f) = 1/5(b − 2)(f + 3). Each Every Flavour Bean costs 25 Knuts and each Chocolate Frogcosts 12 Knuts. Ginny has 314 Knuts to spend. Assume that beans and frogs are both perfectlydivisible goods.(a) Find Ginny’s optimal bundle.(b) Suppose that Ginny is mugged by Draco on her way to the lolly cart, so that by the timeshe is there, she is left only with 75 Knuts. What is her optimal bundle now?(c) Now, consider the general case, where Ginny’s income is m, the price of Every FlavourBeans is pb, and the price of Chocolate Frogs is given by pf . In other words, treat ChocolateFrogs as the numeraire. Find Ginny’s demand for each good in terms of m, pb and pf .(d) Find the good on which Ginny will always spend more than half of her income.(e) Find the own-price, cross-price, and income elasticities of good 2…
Ginny likes to eat Bertie Bott’s Every Flavour Beans (b), and Chocolate Frogs (f). Herpreferences over these lollies are Cobb-Douglas, and can be presented by the utility functionu(b, f) = 1/5(b − 2)(f + 3). Each Every Flavour Bean costs 25 Knuts and each Chocolate Frogcosts 12 Knuts. Ginny has 314 Knuts to spend. Assume that beans and frogs are both perfectlydivisible goods.(a) Find Ginny’s optimal bundle.(b) Suppose that Ginny is mugged by Draco on her way to the lolly cart, so that by the timeshe is there, she is left only with 75 Knuts. What is her optimal bundle now?(c) Now, consider the general case, where Ginny’s income is m, the price of Every FlavourBeans is pb, and the price of Chocolate Frogs is given by pf . In other words, treat ChocolateFrogs as the numeraire. Find Ginny’s demand for each good in terms of m, pb and pf .(d) Find the good on which Ginny will always spend more than half of her income.(e) Find the own-price, cross-price, and income elasticities of good 2.
You own a fast food restaurant and must decide on a pricing strategy for burgers and fries. The market you serve contains equal numbers of 3 types of consumers called "Average", "Burger Buffs", and "Fries Fiends". Each consumer will purchase at most 1 of each food type. Their valuations of the two goods are listed in the following table. Assume for this problem that the costs of production are zero. Consumer Valuations Consumer Types Average Burger Buffs Fries Fiends Burgers $5.33 $12 $3 Fries $8 $3 $11 a. What are the optimal (separate) prices for burgers and fries? b. What is the optimal (pure) bundle price for a meal? Did bundling increase your profits? Explain.
3) Oğuz has the utility function U(x1,X2) = x:*x2². (X: nuts, X2: berries). %3D a) If Oğuz has 25 units of nuts and 17 units of berries, the price of nuts is 2 and the price of berries is 1 liras, what would be the optimal consumption of Oğuz? b) Assume the prices change, so that nuts cost 1 and berries cost 3 liras. What is the new demand? In this change, What is the pure substitution effect? What is the income effect? c) In the change calculated in part (b), what is the pure substitution effect? d) In the change calculated in part (b), what is the income effect? In the change calculated in part (b), what is the endowment effect?