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Unit 18 Partial and General Equilibrium Approaches

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Unit 18 Partial and General Equilibrium Approaches UNIT 18 PARTIAL AND GENERAL EQUILIBRIUM APPROACHES: PURE EXCHANGE MODEL Structure Objectives Introduction A Pure Exchange Economy 18.2.1 Description of the Economy Walrasian Equilibrium 18.3.1 Excess Demand Functions 18.3.2 Existence of Equilibrium Prices Brouwer's Fixed Point Theorem Mechanism for Attaining Walrasian Equilibrium 18.5.1 Stabilityanduniqueness Competitive Equilibrium and Pareto Efficiency 18.6.1 Edgeworth Box 18.6.2 Pareto Optimal Allocations 18.6.3 Process of Reaching Equilibrium Through Trade 18.6.4 Pareto Efficiency of the Market Allocation Let Us Sum Up Key Words Some Useful Books Answer or Hints to Check Your Progress Exercises 18.0 OBJECTIVES After going through this unit, you will be able to: understand the usefulness of general equilibrium for an economy; appreciate the Walrasian formulation and solution of general equilibrium; and evaluate the efficiency of competitive equilibrium and welfare implications. 18.1 INTRODUCTION Addressing the consumer and producer's objectives in the preceding analysis, we have considered only so-called partial equil ibri um. Note that the attribute "partial7' refers to looking at an equilibrium result in one market for a particular good only. It is just like a scenario you have worked out where the increased demand for agricultural products due to an increased income only and nothing has happened to other activities in the agricultural sector. That is to say, the impact of changed demand on markets such as inputs and employment has not come into effect. See that these other markets will also experience the change, which has been overlooked by us. A model that includes the interdependencies of all the markets in the economy can account for the fact that if the equilibrium price in one market changes, the equilibrium prices and hence quantities in other markets are also affected. General Equilibrium To understand such dimensions, we need a model that can accommodate the interactions of all market simultaneously and determine the properties of equilibria in all the markets. We have to develop a general equilibrium model, in contrast to the .partial equilibrium models used thus far. 18.2 A PURE EXCHANGE ECONOMY 18.2.1 Description of the Economy Let us consider a pure exchange model where no production takes place. Consumers have initial bundles of goods, initial endowments. They exchange with each other these goods according to their preferences. For example, you have quantities of apples and 1 have oranges. We enter into an exchange, your apple and my oranges. Note that the exchange to take, you must be willing to consume my oranges and 1 am willing to consume your apples. 4 We can think of n consumers and k commodities in the economy. Each consumer has initial endowments and preferences. Whereas endowment refers to the commodity held by a consumer, her preferences are represented by a utility function y = u, (x,' ,xI2,x,', ...., x:) where x, = ('x, ,x, ,xj ,...., x:) is the ith individual's consumption bundle. We introduce a price system P such that P = (PI, Pz, ...., Pk). Note that the economy you are presented with, does required payment terms of money as people trade one good for another (exchange in barter system). But the price we intend to use is for the exchange rates. For example, the price of one unit of good X is one unit good Y. Such a price, therefore, can be called relative price of good X. If the price of Potato is Rs. 5 per kg and the price of apple is Rs. 2.5 per kg, then the relative price of Potato in terms of apple is 2 (i.e., each unit of Potato is worth 2 units of apple). Similarly, the relative price of apples 1 in terms of Potato is 0.5 (i.e., one unit of apple is worth half unit of Potato). Remember that we will use relative price in the following analysis. Imagine that the consumer i purchases x,' units of good j at price PJ. Then PJ.X,gives the amount of expenditure incurred by her and to that extent her income stands reduced. On the other hand, when she delivers goods of equal quantity, the income t',x,' is added to her income. To arrive at the equilibrium of the model, let us start with consumer's utility maximisation. See that the ith consumer maximises Ui(Xi) subject to her budget constraint PXi = PWi. Remember that solution to this problem yields the demand functions Xi = X,(P, PWi), i =1,2,...n and demand for each commodity depends on all prices and the initial endowment. 18.3 WALRASION EOUILIBRIUM A solution to consumer's utility maximisation problem when we take prices as given yields the demand function for a consumer. In the equilibrium the aggregate demand cannot exceed the endowment. The Walrasian Equilibrium refers to a pair of price and consumption bundle (P*,x*) such that xi*= X, (P*, P'Y ) and Partial and General If we assume that all markets are perfectly competitive and consumers EquilibriumAvproaches participating there are price takers, then Walrasian equilibrium is called a competitive equilibrium. 18.3.1 Excess Demand Functions Define the excess demand function ED, (P) = Di(PFS. That is, ED(P) = X, (P,Pay) - x 4 where Xi's are demand functions for the ith individual. Making use of this notation, the equilibrium conditions can be written as ED~(P*)= D,(P*FS~= 0. This condition states that at the equilibrium prices, excess demand is to be equal to zero in all markets. Two interesting results immediately follow: 1) If there is equilibrium in the market for n-l ..... then in Walrasian equilibrium, the remaining market will also be in equilibrium. 2) One can only solve for the relative prices in the model. Attempts to solve for absolute prices require adoption of normalisation. This would involve making additional assumption of one of the prices is equal to 1. Another form of commonly employed normalisation is to assume that el=1 with el=7. E: xt Several interesting features emerge from such a formulation: 1) The aggregate excess demand functions (and demand functions) is homogenous of degree zero in all prices. That is to say, if all prices were to double, the quantity demanded of every good would remain unchanged. 2) Demand functions are continuous. If prices were to change by only a small amount, quantities demanded would change by only small amount. 3) n excess demand functions are not independent of one another and the equations are related by the formula, This formulation is called Walrus' Law. It states that the total value of excess demand is zero at any set of prices. There can be neither excess demand for all goods together nor excess supply. To prove this, take General Equilibrium = 0 by budget constraint of ith individual. 4) If for some price system P, we get all prices to be strictly positive and (k-1) markets clear, then the kthmarket also clears. By Walras' law we have [Pi EDI(P) + P2 ED2(P) + .. .. + PR-1EDR-I(P)]+PK EDR(P) = 0. If the first k-1 markets clear, than ED,(P) = ED2(P)= ... = EDR-I(P)= 0. Hence, PK EDK(P)= 0. If PpO, then EDR(P)= 0, i.e., the kth market also clears. i 5) If a commodity is in excess supply in Walrasian equilibrium, then its price must be zero in equilibrium. That is, if (P*, x*)is a WE and ED,(P*)<o, then Pi= 0. Proof: Since (P*, x*)is a WE, ED(P*)L 0 and P* t 0, where 0 denotes the vector (0,0,0.. ..). \ Consequently, for P,*ED,(P*)<o for all j. However, by Walras law, P*ED(P*)= 0. lf ED,(P*)<o and P,*>o, then P;ED,(P*)<o. We will have P*ED(P*)<Oas all other terms P;ED,(P*)<o. This Contradicts the Walras law. Example 1: , Suppose there are two people (A and B), two goods (F and S), with initial endowment (o: ,w:),(oA ,&). A feasible allocation X = [(F,s,)F,, s,] . [f prices are PF and Ps, write the budget equations in terms of relative price. i The budgets are: A: Agent 1 P,. FA + P,SA = P,@; + p~@i Since there is no money, we need to use relative prices. Let us express the prices of goods in terms of the second commodity S. Dividing both the budgets by P,, we get 8.. Agent 1 -FAq,. +S, =-w:+w: 9s 4 Agent 2 -F,+,S,4'. =-LoRp, +aRs 9s 9s The price of F in terms of S is p = -4 and the price of S in terms of S is I. P, Pr Using p = -, we write ps Agent 1 PFA+SA=Pw;+w: Agent 2 PF, + S, = PW; + wi Partial and General Example 2: 'Equilibrium Approach In addition to information given in Example 1, you are told that the initial endowment is (u:u:) = (6,2);(u; ui ) = (2,4) and the preferences are represented by the following utility functions: .A (FAsA) = (F,.s*) u, (FUSU ) = (4SH ). Find the general equilibrium in this economy. Solution: For solving the problem, follow the steps: 1) find the demand for each consumer; 2) equate demand = supply in one of the markets to find the equilibrium prices; 3) plug the price in each consumer's demand, to find the equilibrium allocation Step 1: Check that the demand curves are a result of Cobb-Douglas preferences. Note that exponents of the utility functions are equal. So, the consumer spends half of her income in each good.
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