Assignment 5 Advanced Microeconomics

Transcription

1 LONDON SCHOOL OF ECONOMICS Department of Economics Leonardo Felli S.478; x7525 Assignment 5 Advanced Microeconomics 1. Consider a two consumers exchange economy where the two people (A and B) act as price takers. There are two goods, food F and clothing C sold at prices p F and p C respectively. A and B have endowments ( F A, C A ) and ( F B, C B ), respectively. To make things easy, A and B have the same utility function. Unfortunately, that function is (omitting subscripts): U(F, C) = 7 F α C β + 44 F 2α C 2β + α ln(f ) + β ln(c) + (F α C β ) e F 3α C 3β where α > 0 and β > 0. (i) Identify the Walrasian Equilibrium of this economy in terms of endowments and the parameters of the utility function. [Hint: is there some principle of consumer theory and of general equilibrium theory that can simplify things a little?] 2. In a two-persons (Ms. A and Mr. B), two-commodities (apples and oranges), pure exchange economy, Ms. A likes only apples and does not care how many oranges she has. On the other hand, Mr. B likes only oranges and does not care how many apples he has. Both people behave as price-takers. (i) Suppose that A owns all the apples and B all the oranges. Is there a Walrasian equilibrium? If so, what is (are) the equilibrium price and allocation(s)? (ii) What are the Pareto optimal allocations in such an economy? 1

2 (iii) Suppose now that the initial endowments are such that Ms. A owns some oranges and Mr. B some apples. Is there a Walrasian equilibrium? If so, describe any equilibrium price and allocation. 3. In a two-persons (Ms. A and Mr. B), two-goods (good X 1 and good X 2 ), pure exchange economy, Mr. A likes only good X 1 and does not care how many units of good X 2 he has. In other words, Mr. A s preferences are represented by the utility function: U A (X1 A, X2 A ) = X1 A. On the other hand, Ms. B likes only good X 2 she does not care how many units of good X 1 she consumes, but she suffers from a negative externality imposed on her by Mr. A s consumption of good X 1. In other words, Ms. B s preferences are represented by the following utility function: U B (X1 B, X2 B, X1 A ) = X2 B X1 A. Finally, assume that Mr. A owns 5 units of good X 1 and 15 units of good X 2, while Ms. B owns 15 units of good X 1 and 5 units of good X 2. (i) Is there any Walrasian equilibrium in this economy? (ii) If so, what is (are) the Walrasian equilibrium price(s) and allocation(s)? (iii) Find the Pareto efficient allocation(s) in such an economy. (vi) Is the Walrasian equilibrium allocation Pareto efficient? Explain your answer. 4. Draw an Edgeworth box diagram for a pure exchange economy in which no Walrasian Equilibrium exists but in which there are many (as many as you can find) Pareto Optima. 5. Consider the following economy with production, two consumption goods, labelled 1 and 2 and two inputs labour and capital, labelled L and K. A large number of firms have access to a technology which comprises two activities: ( ak1 a L1 ) ( ) ( ) ( ) 3 ak2 2 = = a L2

3 where a Kh, a Lh are the inputs of capital and labour required to produce 1 unit of commodity h, (h = 1, 2). Capitalists own the total stock K = 22, and workers supply total labour L = 40, that are sold to firms at prices r, respectively w. Assume that the capitalists and workers have preferences respectively represented by the utility functions: u K (x 1, x 2 ) = 8 ln x ln x 2 u L (x 1, x 2 ) = 2 ln x ln x 2 There are three equilibria: Equilibrium A: there is surplus labour; Equilibrium B: there is surplus capital; Equilibrium C: there is no surplus labour or capital. In each equilibrium let the price of the consumption commodities 1 and 2 be p and (1 p) respectively. (i) Find the equilibrium values of p, w and r. (ii) Find the equilibrium allocations of commodities x 1 and x Does Walras Law hold when the output market is monopolistic (i.e. producers have monopoly power)? 7. In a general equilibrium economy consumers owns firms. Firms pay all profits out to their owners. Suppose, however, that consumers and firms have their own (and not necessarily equal) expectations as to what prices will be (where expectations are here point expectations). (i) What becomes of Walras Law in such an economy? [You may assume that there exists one firm and one consumer] 3

4 8. A perfectly competitive industry consists of N identical firms. Each firm has a cost function c(w, y) = f(w) y 2 where y is the amount of output produced and w is the price of the sole input. The aggregate demand in this industry is D(p), where dd(p) dp < 0. Calculate the elasticity of the equilibrium price in the market for the output y with respect to w and determine the sign. Explain your answer. 9. The supply side of a perfectly competitive market is represented by 100 identical producers each one endowed with a technology that uses two inputs, x 1 and x 2, and produces one output y described by the following cost function: c(w 1, w 2, y) = min {w α 1, w α 2 } y where y denotes the output, w 1 and w 2 the input prices and we assume α > 0. (i) What is the value of α compatible with the properties of cost functions? Explain your answer. (ii) What are the returns to scale of each producer s technology? Explain your answer. (iii) Identify the production function associated with each producer s technology. (iv) Identify and plot in a graph the aggregate supply function of commodity y in this market. The demand side of this market is represented by 20 identical consumers. These consumers have an exogenous income of size 1 and consume two commodities y and z. Their preferences are represented by the strictly monotonic utility function u(y, z) = yz (v) Identify the Marshallian demand of each consumer for commodity y. 4

5 (vi) Identify and plot in a graph the consumers aggregate demand in the market for y. (vii) What is the equilibrium price and quantity in this market? Explain your answer. (viii) Can you identify the quantity supplied by each individual producer in this market and his profits? Explain your answer. (ix) Can you identify the quantity of commodity y demanded by each individual consumer at the market equilibrium price? Explain your answer. 5