econ 28286...
Exercise 10 In a certain economy there are two types of goods: agricultural goods (X) and industrial goods (Y). The production possibilities set in this economy can be described by Y =9- X and the consumers' preferences by the utility function U = XY . The government imposed the exchange of 1 unit of industrial goods by 4 units of agricultural goods. a) What is the decision producers will make when facing this central restriction? b) Is this a general equilibrium? Why? c) If the government had let the market work by itself, would the price of the industrial good be higher or lower? Justify. Exercise 11 In order to achieve Pareto efficiency it is necessary that the technical rate of substitution between any two inputs be equal for all firms that use positive quantities of them, even if firms produce different goods. Comment. Exercise 12 The transformation curve of a certain economy is linear when all production functions have constant returns to scale. Comment. Exercise 13 If the marginal rate of transformation between any two goods is not equal to the marginal rate of substitution between them for all consumers, then at least one of the goods is not being efficiently produced. Comment. Exercise 14 A certain country has two inputs, K and L. in fixed quantities. These inputs are homogeneous and there are no barriers to intersectorial mobility. Capital and labor are used in two activities, X and Y, with similar CRS production functions. a) What is the shape of the contract curve in the space of the inputs? And the transformation curve? What is the inputs' equilibrium price? Justify. b) If there is only one consumer in this economy (consumer type A) whose preferences are homothetic and with MRS. 4X -, compute the ratio of the equilibrium quantities of X and Y. c) Explain the change in the equilibrium prices of the inputs (with respect to those in b)), if, instead 2X of consumer type A, the economy had only one consumer type B with MRS, YExercise 15 Two goods, X and Y, are produced in a certain country, with CRS production functions. There are 2 inputs in fixed quantitites (K =100 and Z = 9 ). All markets are competitive and with perfect intersectorial mobility of inputs. The inhabitants' preferences for goods X and Y can be represented by tha marginal rate of substitution: MRS =- 4X Y a) In a certain period of time the equilibrium prices in this economy are: PX = 1. W =50 and R P =5. How much is produced of X and Y? b) Having the information that the production function of Y is given by the expression: Y = 10V10KL" and that the productive efficiency implies ky > Ky , represent graphically the transformation curve and compute the point of efficient production when only Y is produced. c) Imagine that the production function of X changes to one represented by the same algebraic expression of Y and that the quantities of inputs remain the same. Compute the general equilibrium quantities for X and Y. Exercise 16 Consider an economy with only two firms, X and Y, which use two inputs, capital (K) and labor (L), in the following way: - The production function of X is described by X = Ly Ky . - The production of Y uses fixed coefficients of inputs, i.e., always uses 10 units of labor for each unit of capital. Assume that the producers of X have 30 units of labor and 6 units of capital, and those of Y have 20 and 4, respectivelly. a) Represent, in an Edgeworth box in the space of the inputs, the contract curve for this economy. How does the relative wage ( w/ r ) change along the curve when sector X expands? b) Show that, starting with the initial endowments and in order to achieve a competitive equilibrium, sector X will expand its units of capital and tranfer labor to sector Y and the relative wage ( w/r ) will equal 0,3, i.e., ten workers will cost three units of capital. c) Assume now that labor is so specific to each production that it cannot be transferred between them. Starting at the initial endowment, and given the immobility of labor, what are the Pareto optimal allocations? Identify those allocations that would be efficient if labor was flexible. (Make a graphical analysis and justify it carefully)Exercise 17 The Bedrock city, where Filinstons live, has no international economic relations. Two families live there: Fred's family (Fred, Wilma and the baby) and Barney's family (Barney, Betty and their daughter). Everything in this city is rocky. There are two kinds of rocks: the hard ones (H), used to build houses, and the soft ones (S), used as food. The utility of each family depends only on the consumption of these two types of rocks. While Fred's family wants to consume 4 Kilo of Soft Rocks (S) for each Kilo of Hard Rocks (H), Barney's family is more flexible in those proportions, having a utility function given by U, = He S, Both families can extract both types of rocks without any cost, but in different quantities. Their initial endowments are: HF= 20Kg SF= 50Kg He= 40Kg Sa= 100Kg a) Check if the initial endowment is a Pareto optimum for the Bedrock economy. Justify. b) Even though there are only two families, they agreed to exchange rocks in a competitive market. Find the equilibrium for this economy- c) Starting at the equilibrium in b), the government of the country which Bedrock belongs to, decided that each family must consume at least 75 Kilo of Soft Rocks (S). i. The government advocates that this measure implies a Pareto improvement. Comment. ii. For which conditions does this kind of measure imply an increase in social welfare? iii. If the measure implies an increase in social welfare, is it possible to find other measures that imply a bigger increase in social welfare? d) Assume that Barney's family can no longer extract Soft Rocks (S) costlessly. They must use 0.5 Kilo of Hard Rocks (H) to extract 1 Kilo of Soft Rocks (S). i. Explain why the equilibrium price for Hard Rocks must be 2. ii. Verify that the equilibrium consumption allocations for both families are: HE= 15Kg SF= 60Kg He = 20Kg SE= 40Kg Exercise 18 In a certain isolated island there are two families (1 and 2), which exchange, in a competitive way, food (A), collected from nature, and drinkable water (W) that each family gets from the closest river. a) If both families have utility functions given by U, = AW,". with / =1, 2 and 0 )= x +y . There are 10 units of x and 10 units of y to be divided among them. a) Represent their preferences and the contract curve in an Edgeworth box. b) Define the fair allocations. Exercise 4 Comment the following statements: a) According to Arrow's Impossibility Theorem, it is impossible to sort social preferences such that they are complete, reflexive and transitive. b) A certain allocation is fair if, when one person envies another, the latter does not envy the former. c) In a pure exchange economy, if an allocation is Pareto efficient, it is impossible to have two individuals that prefer the others' consumption bundle to his own. Exercise 5 a) If the Social Welfare function (SWF) is equal to the sum of individual utilities, when one divides a fixed amount of money, the SWF is maximized if there is an equal division. b) In which circumstances does an equal division correspond to the maximum social welfare? Exercise 6 One can express social preferences voting by ordered results. a) Verify that this social preferences relation is complete, reflexive and transitive. Justify. b) If all consumers prefer x to y, is & socially preferred to y? c) Does the social preferences relation respect the independence of irrelevant alternatives condition? Exercise 7 When the paintings were found at Foz Coa, there was a great discussion about building the planned dam. Different entities have different preferences with respect to the decision to make
