The Walrasian General Equilibrium Theory

Essay, 2017

28 Pages, Grade: 2,0


Table of contents


Part I. General Equilibrium in a Pure Exchange Economy
1.1. Positive Analysis
1.1.1 Pure Exchange Economy
1.1.2. Walras’ Law and the Walrasian Equilibrium: Definitions
1.1.3 Existence of Walrasian Equilibrium
1.1.4 Uniqueness of Walrasian Equilibrium
1.1.4 Stability of Walrasian Equilibrium
1.2. Normative Analysis: Welfare Economics
1.2.1 The First Fundamental Theorem of Welfare Economics and Its Implication
1.2.2 The Second Fundamental Theorem of Welfare Economics and Its Implication

Part II : General Equilibrium with Production
2.1. Positive Analysis : Walrasian Equilibrium with Production
2.2. Welfare Economics


List of references

List of internet sources


The question about how the different economic units such as consumers, governments, and enterprises operate in a competitive markets und make the decisions under conditions of limited amounts of resources is a main subject of studies of the economic science. The lack of resources has turned into a vexed problem in recent years, while people’s desires keep increasing. In order to deal with such a conflict, these three basic questions should be answered by economic systems:

(1) What services and goods should be produced and in what quantities?
(2) How should these services and goods be produced?
(3) Who would be the end customers of such services und goods and how should these should be distributed?

Microeconomic theory defines and models an economic activity as an interaction of individuals (economic agents) pursuing their private interests.[1]

Considering different microeconomic processes for better understanding the principles of the whole economy we are interested in how the goods are distributed among consumers within different markets. What condition markets’ outcomes, i.e. prices of goods and services and quantities traded, are agents’ behavioral characteristics and the market mechanisms, described of law of supply and demand. It is common to divide two large and groups of agents - households and firms, each of them plays its own role in the market.

The general equilibrium analysis, which will be studied precisely in this essay, takes into account all the relationships that arise between markets’ participants and elements while describing it. This model is based on the analysis of supply and demand.

There are two approaches to carry out the market equilibrium analysis, depending on the number of considered markets - partial and general. By the analysis of partial equilibrium only the price of the particular good is normally taken into account, disregarding the prices of other goods, that influence people’s preferences pretty much in reality. Since partial equilibrium considers only one market at a time and so ignores the interaction between markets, and general equilibrium theory considers equilibrium in many markets simultaneously, we will investigate only the general equilibrium markets’.

General equilibrium theory examines the market, where all prices are variable and requires that all markets clear, considering all effects (in terms of prices and quantities) of a potential change in an economy.

In following pages we want to examine the neoclassical model of general equilibrium of Swiss economist, one of the progenitors of Lausanne School, Leon Walras.

The aim of this essay is to examine competitive market economies from a general equilibrium perspective, especially in terms of how Leon Walras had interpreted it. At such markets all prices are variable and equilibrium requires that all markets clear, considering both positive and normative properties of general equilibrium as well. We are also about to investigate the relationship between the competitive equilibrium and Pareto efficiency and to study fundamental theorems of welfare economics and its meaning for the economics in context of general equilibrium analysis.

Part I. General Equilibrium in a Pure Exchange Economy

1.1. Positive Analysis

1.1.1 Pare Exchange Economy

From a positive point of view, the general equilibrium theory is a theory of the determination of both equilibrium prices and quantities in a system of perfectly competitive markets. It is often called the Walrasian theory of market equilibrium because it was first introduced in Leon Walras’ Elements of Pure Economics in 1874.

At first we examine the case of a pure exchange, which is the special case of the general equilibrium analysis, where all of the economic agents are individuals and no production takes place. This case examines the situation, where consumers have fixed endowments of goods and interact in order to trade these goods among themselves in all possible ways, taking into account the preferences of both sides.

In this general equilibrium model, goods are identical, the market is concentrated in a single point of space and the exchange occurs instantaneously. Moreover, individuals have full information about the goods that are the subject of exchange, and the terms of the transaction are known to both parties. As a result, the exchange does not require any other efforts, except for the use of an appropriate amount of cash. Thus, prices become an important allocation tool, allowing each exchange participant to obtain maximum utility.[2]

In order to simplify the exposition of the case of the pure exchange economy, a graphical tool, which can be used to analyze the exchange of two goods between two individuals, known as the Edgeworth box.

Figure 1

illustration not visible in this excerpt

Source: Varian, H.R. (2010), p. 584

Both width and height of the box measure the entire amounts of goods 1 and 2 respectively. Consumption bundles of the person в are represented in the upper right-hand corner of the box, while A’s are represented in the lower-left comer. This concept could be represented in the following way.

Consider an exchange economy with two goods 1 and 2 and two individuals, A and B, who are involved in the exchange. We can define A ’s consumption bundle byXi = (xaj, X4[2]), where X4J represents A ’s consumption of good 1 and X.I[2] represents A ’s consumption of good 2. Then В ’s consumption bundle is defined by Хв = (xb[1], Xu[2])■ A pair of consumption bundles, Xa and Xb, is called an allocation. The total endowment is 11’ 11’ / + w2. A particular feasible allocation that is of interest is the initial endowment allocation: (WA[1], и 7/) and (wr, W2B).

Thus, the point w in the Edgeworth Box, can be used to represent the initial endowments of two persons, (see fig.l) Advantage of the Edgeworth Box is that it gives all the possible (balanced) trading points. That is, an allocation is a feasible allocation if the total amount of each good consumed is equal to the total amount available:

illustration not visible in this excerpt

The Edgeworth box contains all possible consumption bundles for individuals A und в (feasible allocations), as well as their preferences. In general, the points in this graphical tool demonstrate all feasible allocations in the simple economy.[3]

1.1.2. Walras’ Law and the Walrasian Equilibrium : Definitions

Now we suppose that there is a third party who is acting as an “auctioneer” for the two market agents A and B. The auctioneer chooses a price for good 1 and 2 and presents these prices to the agents A and B. Each agent can understand now how much his endowment is worth at the prices (pi, P2) and decides how much of each good he would want to buy at those prices. Next figure illustrates the two demand consumption bundles of the both agents.

Figure 2

Gross and net demands of the persons A and в

illustration not visible in this excerpt

As it could be seen on the figure, there are two concepts of demand to be distinguished: gross demand (total amount of good person wants to buy at the given prices) and net demand (amount of good person wants to purchase). Considering the term of net demand in the context of general equilibrium study, it presents, in fact, the difference between the total demand and the initial endowment of the good, that agent possesses.

In other words it can be described as excess demand, so A’s excess demand for a good 1 takes following form:

illustration not visible in this excerpt

For the arbitrary prices (pi, pi) no guarantees exist that supply will equal demand; in the example above the individuals can’t proceed with the transaction they would prefer because the markets are not clear.

While this condition dissatisfy the terms of general equilibrium, it describes the case, when the market is in a disequilibrium.

To achieve the market equilibrium, the exchange process between two individuals should continue until demand for each of the goods equals supply. This happens, if individual A wants to buy the same amount of a good 1, that individual в wants to sell. In this case each individual chooses the most affordable consumption bundle for him, such as their choices are compatible and demand equals supply in every market. Such Equilibrium is illustrated on the following page.

Figure 3

Equilibrium in the Edgeworth box

illustration not visible in this excerpt

Source: Varian, H.R. (2010), p.591

Mathematically equilibrium could be described by following equations with a set of prices

illustration not visible in this excerpt

Where Xf] (pi*, pi*’), xŕ (pi*, P2*) -A’s demand function for goods 1 and 2; X1B(p!*, P2*), X2B (pi*, P2*) -B’s demand function for goods 1 and 2 respectively.

As we stated earlier, the excess demand is the difference between the individuals’ total demand and its initial endowments (3). If we readjust the equations above in terms of excess demand, we will get another set of equations, which shows, that the sum of net demands for each individual should equal zero.[4]

illustration not visible in this excerpt

Such equilibrium definition comes from a concept of aggregate excess demand function. If we summate individual A’s and B’s excess demands for good 1 and then do the same for the good 2 respectively, we get aggregate excess demand functions for both goods:

illustration not visible in this excerpt

General equilibrium is such a state of the economic system, where demand equals supply or, in other words, where aggregate excess demand as the excess of demand over supply is zero in each of the markets. We can describe an equilibrium (pi*, P2), saying that the aggregate excess demand for each good is zero:

illustration not visible in this excerpt

The equilibrium economy is governed by Walras' law, which states that the value of aggregated excess demand equals to zero.

illustration not visible in this excerpt

It is not difficult to establish its fairness. In the conditions of the optimum for each, j-th, individual, the budget constraint is fulfilled in the form of equality:

illustration not visible in this excerpt

The sum of such equations for all individuals from 1 toj gives zero aggregate net demand in value terms, what Walras’ law states:

illustration not visible in this excerpt

Traditionally, the general equilibrium is defined as equality to zero or negativity, i.e. non­positivity of excess demand in all markets in the economy:

Zi < 0

by non-negative prices (pi >0)

with the condition : Pi Zi = 0, 7= {1, ...,k}

The last condition means that either the price or excess demand on each market is zero.

This is important because a situation could be possible, when at a non-negative price the equality of supply and demand is not achieved. Then we can talk about equilibrium at zero price and negative excess demand (in this situation there are no reasons for changing the existing situation, so it is equilibrium). This is a situation of a free good, eliminating of which does not require resources.

In Walrasian model of general equilibrium, a closed market economy is considered without taking into account the role of the state in the conditions of perfect competition.

The Walras’ law holds for any price vectors and not just for equilibrium prices. If non- satiation.condition is fulfilled for all households (which means that they completely spend their income by optimal choice), the expression in the Walras’ law will be fulfilled with equality and then value of excess demand will be exactly zero.

Two important consequences follow from Walras' law. First consequence of Walras' law is that in economy consisting of / markets, at some price vector p = {phfh=1, Ph>0, existing excess demand is zero at 1-1 markets, it turns out zero on the last /-market as well. This can be proved by an example of two markets. Let Zi(p1, P2) = 0, then from Walras' law follows, becauseP2>0, Zi(p1, P2) = 0. It means that if existing demand is zero in the first market, it will be zero in the second market too.

Due to induction that statement extends to an economy consisting of 1 markets. Let excess demand be zero on 1-1 markets: z! = 0, Z1-1 = 0. Because pi > 0, from Walras' law follows,

that Zi = 0. Thus, according to the law of Walras, if all the markets, excluding given one, are in the state of equilibrium, excess demand on the given one will also be zero.

Second consequence of Walras' law is that situation, when the price of at least one good is different from zero, the excess demand values cannot be the same simultaneously.

That could be proved by an example of economy consisting of two markets. In that case P!Z1 = -P2Z2, what follows Walras' law. In case, when both prices of economic goods are positive (pi>0,1 = 1,2), then Z1>0, Z2<0 or Z1<0, Z2>0. When the first good is a free good (pi = 0), then its excess demand cannot be positive Z1<0. The same works for the second good also. If it is a free good (p2 = 0), then its excess demand cannot be positive Z2<0. In that case, since pi > 0, then due to P!Z1 = ~P2Z2, Zi = 0. Thus, the second consequence of the Walras law is proved.

Due to natural balance constraints for each of the markets, Z1<0, Z2<0. Therefore, the sum of the excess demand in value terms across all markets in the economy should be non-positive: P!Z1 + ... + pi-izi-1 + p!Zi = Σ/ι=1Ρ/ι zh <0· But, since in the state of equilibrium either the price of a good or its excess demand must be zero, the sum of the corresponding multiplication will be zero, which was established above by the Walras' law.

Thus, the aggregate of budget constraints of consumers and natural balance correlations, relating to the volume of consumed and available products for all markets, contains the same information. Consequently, one of these conditions in a state of equilibrium is linearly dependent on the others.

The required values in the model are the volumes of consumption of all (1) goods by each of the m consumers, as well as the equilibrium price vector consisting of 1 components. Hence, in the model (l+l)m of the unknowns and l+lm -1 is the independent condition. Consequently, it could be possible to determine not the absolute level of prices, but their relative values, when the price of one of the goods or, equivalently, the sum of all their prices will serve as the unit of account[3] [4].

1.1.3 Existence of Walrasian Equilibrium

Though Leon Walras was the first to describe an equilibrium as the solution of a system of equations, reflecting how goods are allocated between agents for a specific set of prices in a set of markets, he didn’t give any formal proof of existence of the solution for this system (and existence of a competitive equilibrium). Moreover, he presupposed this solution to be unique.

Later studies by Wald (1934; 1935; 1936) and Arrow & Debreu (1956) have examined specific conditions under which this market system of equations could have a solution.[5]

In case of the pure exchange economy proving the existence of a general equilibrium is equivalent to proving that there is at least one set of prices in the non-negative price set, that makes all commodity excess demands equal to zero.

The two main theorems that were instrumental to Arrow and Debreu in answering the existence question of Walrasian general equilibrium theory are Brouwer’s (1911:12) and Kakutani’s (1941) fixed point theorems, the latter being an extension of the former studies.

In the context of a simple exchange economy, we should look for at least one specific instance determined by a set of prices and quantities, a ‘point’ where everybody agrees.

The big question is does such an equilibrium necessarily exist for all economies?

At first we will show that non-satiation and desirability of goods lead to demand equaling supply at equilibrium. A key aspect to ensure the existence of equilibrium is that the aggregate excess demand function in each market must be continuous. This occurs when:

a) Each agent demand function is continuous (sufficient condition). That is to say that preferences are convex: small variations in prices lead to small variations in quantities
b) Even though, each agent demand functions were non-continuous, the aggregate demand function is continuous (necessary and sufficient condition).

The basic theorem of the general economic equilibrium is the Arrow-Debreu theorem, which establishes sufficient conditions for its existence.

It is supposed that the assumptions about the continuity and strict quasi concavity of utility functions, as well as the continuity and strict concavity of production functions are satisfied.

Then, for any initial distribution of the stock of goods between consumers, there is a price vector in which general competitive equilibrium is achieved.

To prove the general equilibrium existence theorem, the example of an economy consisting of two markets for economic goods (pi>0, I = 1,2) is considered. Figure 4 shows the spatial curve of excess demand Zi(p1) = (z1(p1), Z2(pi)) and its projection onto the plane Z1ÜZ2. Here p! = —— is the relative price of the first good. It is possible not to include P2 = 1 - pi in the P1+P! number of arguments of functions of excess demand. According to the second consequence of Walras's law, in the absence of an equilibrium, the economy will be in some state A in the second quadrant or in some state в in the fourth quadrant. All states in the first and third quadrants are unavailable because there the values of excess demand Zi and Z2 have the same sign: they are either positive or negative.

For real-valued functions z: p —> z(p), the Cauchy-Bolzano theorem is known, which states that if z(p) e c°[a,b], then zip) takes on every interval ία,β) c [a,b] all the intermediate values between z(a) and ζ(β). From the Cauchy-Bolzano theorem it follows that a function from a class c°[a,b], that has values different in sign at the points a and Д where ία,β) c [a,b], necessarily takes a zero value in some intermediate point between the points a and β. That consequence could be used to prove the general equilibrium existence.

In the state A: Z1<0, Z2>0. In the state B: zf>0, Z2<0. According to the Cauchy-Bolzano theorem, since each coordinate of the excess demand Zi(p1) = (z!(p1), 22(pi)) is continuous, for the transition of the economy from state A to state B, each coordinate must take a zero value. Let in the state El Zi = 0. According to the first consequence of Walras's law, excess demand Ζ2 must also be zero. So, in the state El, the excess demand Σ1 and Ζ2 values are both zero at positive prices, i.e. the general economic equilibrium is achieved. Thus, the existence of a general equilibrium in an economy with two markets with positive prices (pi>0, I = 1,2) has been proved. State E2 reflects the situation when the second good is free (p2 = o, Z2 <0), and the first - economic (pi = 0,21 = 0)\ and E3 - when, on the contrary, the first - free (pi = 0, 21 <0), and the second - the economic (p2 = o, Z2 = 0)[6]

Figure 4

Possible equilibrium situations

a) Spatial illustration b) Projection on the plane of excess demand

illustration not visible in this excerpt

Source:Verenikin, A. p.6

1.1.4 Uniqueness of Walrasian Equilibrium

Existence of a competitive equilibrium can be proved under quite general conditions. Equilibria are unique only under very strong restrictions. There are several approaches with its own restrictions to prove that the equilibrium is unique. Two assumptions that play an important role in discussions of uniqueness of equilibrium since Wald (1936) are gross substitutability and weak axiom of reveal preferences.[7]

For a pure exchange economy, uniqueness of equilibrium can be assured if aggregate excess demand functions satisfy the so-called gross substitutability property. We bring in some restrictions on the excess demand z, that would assure the uniqueness of the equilibrium. While the case of free goods is not particularly interesting, in order to prove the uniqueness we will use the desirability assumption, i.e. that every equilibrium price of each good must by strictly positive (p* > 0) and the aggregate excess demand function must be continuous differential. In another case, if indifference curves have kinks in them, it could be found the whole dimension of prices that are all market equilibria.

illustration not visible in this excerpt

Two goods i and j are gross substitutes at a price vector p if TU > 0 for i f j. If in a ״Pi pure exchange economy demand function of each agent satisfies the condition of gross substitutability property, then the aggregated demand function is satisfied too. The property of gross substitutability is sufficient. If the demand function fulfils this property, the Walrasian equilibrium is unique in this economy.[8]

We take the following proposition and prove its uniqueness:

Theorem: Suppose all goods are desirable and gross substitutes at all price > 0 for /' f j. If p* is competitive equilibrium price vector and Walras’ Law holds, then it is the unique equilibrium price vector. Proof. Since all goods are desirable, thenp* > 0. Suppose p’ is some other equilibrium ปี . price vector. We can define m = max■^ Ψ 0 . Let the good at which m is obtained be denoted Pi by /, i.e. m = ^7 (at least one / satisfies this, but there can be more). In this setup, good / n becomes the most relatively expensive good at pricesp. By homogeneity and the fact that/?* is an equilibrium, we know that zip*) = z (mp*) = O.Then we lower each price mp* other than Pi successively to p'i . Since the price of each good other than / goes down in the movement from mp* toP׳, we must have the demand for good / also going down. Thus Zi (p) < 0 which implies thatP׳ can not be an equilibrium.[9]

If demand functions do not satisfy the condition of gross substitutability, it is necessary to use the Weak Axiom of Reveal Preferences (Kehoe, 1992). This Axiom states that consumer decisions are stable in the sense that when they face the same price they behave in the same manner, i.e. demand is exactly the same. If the aggregate demand function satisfies the Weak Axiom of Revealed Preference and Walras’ Law holds, then the competitive equilibrium is unique. The Weak Axiom of Revealed Preference (WARP of the aggregate excess demand function):

illustration not visible in this excerpt

The idea of this definition is the same as that for a single consumer: if the excess demand zip') was affordable under priceP but were not chosen, then z(p) is revealed preferred to zip'). Therefore, under priceP׳, since zip') was chosen, the excess demand zip) must not be affordable at P׳. If zip') could have been bought at P where zip) was bought (that z(p) > zip') since z(p) is the optimal choice, then at priceP׳ z (p) is outside of budget constraint (otherwise it contradicts that z (p) is an optimal choice).

Lemma 1. Under the assumptions of strong Walras’ Law and WARP: p*z(p) >0 for all p f lp*.(p*is a competitive equilibrium).

We can prove the statement above in the following way: if /?*is a competitive equilibrium, then z(p*) < 0. According to Walras’ Law, z(p) = 0. Because of the fact, that p ER+, pz(p*) <0. So, we have p z(p) >pz(p*). Hence, by WARP, p*z(p) > p*z(p*) = 0, which implies that p* z(p) > 0 for all pf l p*.

Theorem: Under the assumptions of strong Walras’ Law and WARP of the aggregate excess demand the competitive equilibrium is unique.


[1] Tian, G. Lecture Notes. Microeconomic Theory. 2004. Department of Economics of Texas A&M University, January 2004 pp. 8,9 (

[2] North, D.c. 1997. Institutions, Institutional Change and Economic Performance. Cambridge University Press [Translation into Russian by Nesterenko A.N.] Moscow, Center for Evolutionary Economics. 1997 (http://cee- p. 49 ISBN 5-88581-006-0

[3] Verenikin, A.o. Lecture Notes: Micro Economy 3. Chapter 7 : Walras Law and Existence of Generai Economical Equilibrium / Moscow State University, pp. 4,

[4] ( %B3°/0D 1 %80%D0%B0%D 1 %84%207_1 .pdf)

[5] Cárdente, M.A., Guerra, A-I. and Sancho F. 2012 Applied Generai Equilibrium, An Introduction. Springer, pp. 11,12 ( setKey=AS%3 A324000914313216%401454259245756)

[6] Verenikin, A.o. Lecture Notes: Micro Economy 3. Chapter 7 : Walras Law and Existence of Generai Economical Equilibrium / Moscow State University, pp. 5,

6 ( %B3°/0D 1 %80%D0%B0%D 1 %84%207_1 .pdf).

[7] Kehoe T. J. Elements of Genera! Equilibrium Analysis. Uniqueness and Stability / Basil Blackwell, 1998, pp.46,46 (

[8] Smooha, N. 2014. Positive Theoiy of Equilibrium: Existence, Uniqueness, and Stability, Chapter 7. p.22, 23 ( Uniqueness-and-Stability.pdf)

[9] Elori, K. 2004. Economic Theoiy and Applications I, Microeconomics. General Equilibrium /Birkbeck College, University of London, 2004. p. 9 (

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The Walrasian General Equilibrium Theory
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