Charles University in Prague Faculty of Social Sciences Institute of Economic Studies

BACHELOR THESIS

Differential equations and stability of competitive economy

ˇ Author: Marek Sabata Supervisor: RNDr. Tom´ aˇ s B´ arta, Ph.D. Academic Year: 2013/2014

Declaration of Authorship The author hereby declares that he compiled this thesis independently, using only the listed resources and literature. The author grants to Charles University permission to reproduce and to distribute copies of this thesis document in whole or in part.

Prague, May 15, 2014 Signature

Acknowledgments I wish to thank to RNDr. Tom´aˇs B´arta, Ph.D., who not only introduced me to the topic of stability of competitive economies, but also supported me with helpful comments and suggestions throughout the writing of the thesis.

Bibliographic record ˇ SABATA, Marek. Differential equations and stability of competitive economy. Prague, 2014. 43 pages. Bachelor thesis (Bc.), Charles University, Faculty of social sciences, Institute of economic studies. Supervisor of the bachelor thesis RNDr. Tom´aˇs B´arta, Ph.D.

Bibliografick´ a evidence ˇ SABATA, Marek. Differential equations and stability of competitive economy. Praha, 2014. 43 str´anek. Bakal´aˇrsk´a pr´ace (Bc.), Univerzita Karlova, Fakulta soci´aln´ıch vˇed, Institut ekonomick´ ych studi´ı. Vedouc´ı bakal´aˇrsk´e pr´ace RNDr. Tom´aˇs B´arta, Ph.D.

Abstract In the thesis, the author will analyse the theory of differential equations and its applications in economic model of price adjustment processes in competitive markets. First of all, the economic model sufficient to study stability of the market is introduced. Next microeconomic theory of competitive markets is presented and theory of differential equations is laid out, including the stability theory. Differences between the general model and the pure exchange model are discussed. Under certain microeconomic assumptions such as weak axiom of revealed preferences, gross substitutability, Walras’s law and other properties of competitive markets, local and global stability of the market is proved. JEL Classification Keywords

D50, D41, C62 differential equations, stability, competitive equilibrium, tatonnement process

Author’s e-mail Supervisor’s e-mail

[email protected] [email protected]

Abstrakt Pr´ace se zab´ yv´a teori´ı diferenci´aln´ıch rovnic a jej´ı vyuˇzit´ı v ekonomick´em modelu stability cen na dokonale konkurenˇcn´ıch trz´ıch. Nejprve je uveden ekonomick´ y model vhodn´ y ke studiu stability trˇzn´ıch ekonomik. D´ale je pˇredstavena mikroekonomick´a teorie trˇzn´ıch ekonomik a vyloˇzena teorie diferenci´aln´ıch rovnic, vˇcetnˇe teorie stability. Jsou diskutov´any rozd´ıly mezi obecn´ ym modelem a modelem ˇcist´e smˇeny. Za jist´ ych mikroekonomick´ ych pˇredpoklad˚ u jako je slab´ y axiom odhalen´ ych preferenc´ı, hrub´a substitutabilita, Walras˚ uv z´akon a dalˇs´ıch vlastnost´ı trˇzn´ıch ekonomik, je dok´az´ana lok´aln´ı a glob´aln´ı stabilitu trhu. Klasifikace JEL Kl´ıˇ cov´ a slova

D50, D41, C62 diferenci´aln´ı rovnice, stabilita, rovnov´aha, proces t´ap´an´ı

E-mail autora [email protected] E-mail vedouc´ıho pr´ ace [email protected]

trˇzn´ı

Contents Thesis Proposal

viii

1 Introduction

1

2 Basic notions and derivation of the model 2.1 Introduction to the model and basic thoughts . . . 2.2 General and pure exchange models of tatonnement 2.2.1 Origination of differential equations . . . . . 2.2.2 Model’s premises . . . . . . . . . . . . . . . 2.3 Walras’ price adjustment process . . . . . . . . . . 2.3.1 Walras’ process of tatonnement . . . . . . . 3 Theoretical background 3.1 Economic background . . . . . . . . . . 3.2 Mathematical background . . . . . . . . 3.2.1 Theory of differential equations . 3.2.2 Linear systems and their stability 3.2.3 Stability of a general equation . . 3.2.4 Lyapunov’s concept of stability .

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4 Stability of competitive equilibrium 4.1 History - literature review . . . . . . . . . . 4.1.1 In the beginning . . . . . . . . . . . 4.1.2 First results . . . . . . . . . . . . . . 4.2 Stability of the general model . . . . . . . . 4.2.1 Framework of the analysis . . . . . . 4.2.2 Local stability of the general model . 4.2.3 Global stability of the general model 5 Conclusion

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Contents

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Bibliography

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Bachelor Thesis Proposal Author Supervisor Proposed topic

ˇ Marek Sabata RNDr. Tom´aˇs B´arta, Ph.D. Differential equations and stability of competitive economy

Topic characteristics To study selected parts of the theory of differential equations (mainly stability results), to get acquainted with mathematical models of price adjustment for markets with n commodities and known results about stability of price adjustment process. Finally, to present these results in the frame of general theory. Outline 1. Introduction 2. Basic notions and the derivation of the model 3. Theoretical background 4. The stability of competitive equilibrium 5. Conclusion In the first part of the thesis, we are going to make an introduction to the topic of stability of competitive economy. In the second part, the problem of stability of competitive economy will be outlined and the underlying market models introduced. The third chapter will go through the theoretical background, namely the theory of differential equations, microeconomic theory and their relations. The fourth chapter will contain brief literature review and then stability of the market equilibrium will be studied. In conclusion, we will sum up the thesis.

Bachelor Thesis Proposal

ix

Core bibliography 1. Takayama, A. (1985): “Mathematical economics.” 2nd edition, Cambridge university press, 1985 : pp. 295–358. 2. Vrabie, I.I.(2004): “Differential equations: an introduction to basic concepts, results, and applications.” New Jersey, World Scientific 3. Negishi, T. (1962): “The stability of a competitive economy: a survey.” Econometrica 30 : pp. 635–669. 4. Mukherji, A. (2008): “ Stability of a competitive economy: A reconsideration.” International Journal of Economic Theory, The International Society for Economic Theory 4(2): pp. 317–336.

Author

Supervisor

Chapter 1 Introduction The theory of competitive economies has for a long time belonged among the major economic topics studied by both microeconomists and macroeconomists all around the world. Almost all economic policies nowadays stem from the theory of competitive economies and follow its principles, since they have proven themselves to be a very good models of the real life economy. It is essential for the economists to study the models of competitive economies even nowadays, since they give us a very good preview of the things to come not only after positive or negative economic shocks, but also after imposing market policies in times of crisis and other economic tools that should help guide the market to the equilibrium. Questions regarding competitive economies are generally concerned with the competitive market equilibrium, under which conditions could the equilibrium be reached and if it is optimal in some way. Many famous economists such as Samuelson, Arrow and Hicks have studied these questions in the past and much has been written about this topic. However the focus has mostly been set on the comparative static of the market equilibrium and the dynamic process of attaining the equilibrium and its stability were by far not studied so thoroughly. The question regarding the stability of equilibriums is how the price will behave after deflection from the market equilibrium, or saying it in Adam Smith’s way, how the price will be guided by the invisible hand of the market. This question is exactly the one we are going to study in this thesis. We are going to introduce the models of competitive markets, which are suitable for modelling stability of competitive market equilibrium. Next we will go through the question of what generates the process lying behind the price adjustment and we will take a look at the differences between the pure

1. Introduction

2

exchange and the general model for tatonnement process. The proofs of global asymptotic stability and local asymptotic stability of a market satisfying certain economic assumptions will be presented. When modelling variables which evolve dynamically over time, such as prices, it naturally arises to use rich mathematical apparatus of differential equations. As we will see in the thesis, apart from modelling the price adjustment process itself, we can also utilize the well developed theory of stability of differential equations. Together with few microeconomic assumptions about the competitive economy, the stability theory of differential equations is going to help us capture the questions concerning stability of the underlying price adjustment processes. Not much has been written about this topic. Among the main contributors we highlight Takashi Negishi, who wrote an excellent article, which gathered all the findings from the late 1950’s and early 1960’s, when the topic was introduced, and made suggestions for further studies. Another great source is the book of Akira Takayama (1986), who devoted whole chapter to the stability of competitive equilibrium, in which he summarized the known results and few open questions concerning the topic. In recent years Anjan Mukherji has been the leading contributor to the topic. In his work Mukherji (2003) he solved the problem of stability of competitive equilibrium almost completely. The thesis is divided as follows. The second chapter gives introduction to the studied problematic and defines the general model and the pure exchange model of competitive economy under which we can study the stability of underlying price adjustment processes. In the third chapter, we state all necessary economic assumptions, which are used in the model and we provide rigorous treatment of theory of differential equations and the theory of stability. The fourth chapter begins with brief literature review, which contains all major findings in the topic up to today. Then we get to the analysis of local and global stability of the market equilibrium, we propose all necessary conditions for the equilibrium to be stable and then give the proofs. Last chapter gives a short conclusion and summarises the thesis. We assume that the reader has a basic knowledge of the microeconomic theory, mathematical analysis and linear algebra.

Chapter 2 Basic notions and derivation of the model In this chapter, we first give a fundamental example of competitive economy model and price adjustment process to show which problems and difficulties can arise in such models. We proceed further and state the general model and the pure exchange model of competitive economy, which enables the study of its stability. We turn next to explanation of the Walras’ price adjustment process of tatonnement, which was one of the first processes enabling the study of stability of competitive markets and we also look at its drawbacks.

2.1

Introduction to the model and basic thoughts

As we mentioned in the introduction, competitive markets play very important role in current economic theory. Naturally their complex modelling can be quite tough, so let’s start just informally with simple model of one commodity economy to obtain basic insight to stability of the competitive economy issues. Assume that supply and demand functions of the commodity are determined only by the price p of the commodity. Denote demand and supply functions D(p) and S(p) respectively. Further assume that there exists price p∗ , such that D(p∗ ) = S(p∗ ), meaning that the market is in equilibrium under p∗ . Now suppose that the economy is hit by positive or negative shock, that will cause the price to change from the equilibrium price p∗ to pˆ, which is different from the initial price and the market is not in the equilibrium under such price. Our question is, what the market forces will do to the new price pˆ. Few questions arise. Is the price going to converge to its initial equilibrium

2. Basic notions and derivation of the model

4

price p∗ ? Or is it going to converge to some other equilibrium price? Is it possible, that the price will not converge to any equilibrium at all, meaning it will be diverging? These questions can be all viewed as questions of stability of the underlying dynamic price adjustment process and are exactly those which we are going to study thoroughly. A question in a quite different manner is what exactly generates the price adjustment process or the motion of invisible hand, which governs in the market. As was pointed out by Samuelson (1946), we cannot study stability of a competitive economy without analysing its underlying dynamic process. When analysing stability of competitive market, it is convenient to adopt some basic economic assumptions, on which we will base our mathematical analysis of the models. One of the basic and most natural assumptions concerns what happens to demand and supply when the economy is in disequilibrium. It is quite intuitive to assume that when the demand for a commodity is greater than its supply, that is D(p) > S(p), the price will go up and vice versa. Under this assumption we can get the general idea how the price will behave when facing disequilibrium. However even in the simplest model of the market, this assumption is neither sufficient nor necessary condition for the economy to be stable. We now present a short example, in which the demand and supply functions satisfy the above condition, however the market is unstable. As it is common in economic graphs, we denote the independent variable p on the vertical axis and the dependent variable q on the horizontal axis and p∗ denotes the equilibrium price. Suppose that the market prevailing price is p0 so economy’s output is q 0 . Under such price, the supply is greater than the demand, so D(p0 ) − S(p0 ) < 0 and the market price adjustment mechanism lowers the price to p00 . However, under p00 the suppliers supply only the quantity q 00 , so there is now greater demand than the supply in the market, so D(p00 ) − S(p00 ) > 0 and the price will jump to p000 . It should be clear from the picture, that the price is going to diverge away, never reaching the equilibrium price p∗ .

2. Basic notions and derivation of the model

5

Figure 2.1: Simple example of instability

As we can see from the outcome of our example, we need to introduce more sophisticated mathematical tools and economic premises, which would give us sufficient conditions for the equilibrium to be stable. Without further specification, the analysis would be very tedious. This will be the content of the few following paragraphs.

2.2

General and pure exchange models of tatonnement

After a short introduction to the topic studied, we are now going to give a formal definitions of two key models of competitive economy with all its essentials. The models are freely paraphrased from the book Takayama (1986). We consider the model of competitive economy with n commodities and we assume that the economy can be described by n so called equilibrium equations, which represent the relations between explanatory variables and market governing mechanism, for now let aside what the mechanism exactly is. Mathematically formulated we will be working with an open connected set Ω ⊆ Rn in positive orthant of Rn , representing all possible prices p of the commodities and functions fi (p) : Ω → R, i = 1, ..., n, standing for some form

2. Basic notions and derivation of the model

6

of excess demand on the i-th market, which will be specified later depending on the model and further we write f (p) = (f1 (p), ..., fn (p)). We suppose that the functions fi (p) are continuously differentiable functions of p for all i. These are going to be standing assumptions throughout the thesis. To be mathematically correct, any type of convergence we are going to consider, is in n P the Eucleidian norm of the space Rn , so kpk2 = k(p1 , ..., pn )k2 = p2i . i=1

Now we get to the question of stability of the market. Suppose there exists a vector p∗ = (p∗1 , ..., p∗n ) ∈ Ω satisfying fi (p∗1 , ..., p∗n ) = 0 for all i = 1, ..., n, or equivalently f (p∗ ) = 0. This simply means, that the price vector p∗ is the market equilibrium price. Generally speaking, we are interested in what happens to prices, when they deviate from the equilibrium price p∗ . Our aim is to observe behaviour of the system depending on the variable p. Initially we have observed value p0 such that f (p0 ) 6= 0 (p0 is not an equilibrium) and we suppose that time adjustment process is generated, guiding the value p from p0 to some equilibrium. This means that the value p will change over time, being a time dependent variable in fact. We will denote the time path of p ≡ p(t), where t is a single variable representing time and we consider t to be from the interval [0, ∞), so p0 = p(0). As we already mentioned, the price in the competitive market is changing in time according to the excess demand function, so we can think of it in a following manner - the greater the excess demand is, the greater will be the change of price and if the excess demand is positive, the price will grow and if it is negative, the price will fall. This concept of modelling naturally leads to systems of differential equations. Having this in mind, we now present the price adjustment process in the market and subsequently the two models of competitive economy suitable for stability analysis. The system of differential equations describing the price adjustment process may be written as follows: p0i = hi (fi (p)),

i = 1, ..., n,

where functions fi represents some form of excess demand function on the i-th market, depending on the model and hi is a sign-preserving differentiable function, so the price is modelled according to the value of excess demand. We can simplify this price adjustment model and replace the functions hi by constants ki > 0 representing speed of adjustment on the i-th market, so we arrive at equation p0i = ki fi (p). Because ki ’s are fixed constants, they do not have any impact on stability or instability of the model, so finally we can assume, that

2. Basic notions and derivation of the model

7

the equation for the price adjustment process states as p0i = fi (p), i = 1, ..., n. So even thought the function representing the price adjustment process is generally different than the excess demand function, which models the equilibrium in the economy, we may suppose without much loss of generality that they are the same. Let’s move on to the definitions of the models. The equation in both models will be following: p0 = f (p)

⇔

p0i = fi (p), i = 1, ..., n.

In the general model, the excess demand function is in the form: fi (p) = Di (p) − Si (p),

i = 1, ..., n

⇔ f (p) = D(p) − S(p), where Di (p) represents the market demand function for the i-th commodity, Si (p) represents the market supply function for the i-th commodity, D(p) = (D1 (p), ..., Dn (p)) and S(p) = (S1 (p), ..., Sn (p)). In the pure exchange model, we suppose that there are m agents in the economy, xik (p) denotes k-th’s agent demand for the i-th commodity under price p and xˆik (p), denotes the holdings of the commodity i by the k-th agent under m P price p. Thus xi (p) = xik (p) is the total demand for the i-th commodity k=1 m P

in the market and xˆi =

xˆik (p) is the total holding of the i-th commodity,

k=1

which is a constant. The excess demand functions are in the following form: fi (p) = xi (p) − xˆi =

m X

xik (p) −

k=1

m X

xˆik (p),

i = 1, ..., n

k=1

⇔ f (p) = x(p) − xˆ, where x(p) = (x1 (p), ..., xn (p)) is the market demand function and xˆ = (ˆ x1 , ..., xˆn ). Both models are formed in a way that the price p behaves according to the value of excess demand function. We can immediately see, that the pure exchange model is a special case of the general model, since the supply in the pure exchange model is a given constant for all prices, whereas the supply function in the general model varies with price. Thanks to this, all the results about

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the general model apply even for the pure exchange model. In the following subsection, we will clarify the origination of modelling the dynamic system using differential equations.

2.2.1

Origination of differential equations

We now give the basic idea which lies behind the origination of differential equations. Suppose we have some unknown time dependent variable p, for instance price of a commodity, which is a function of time, and we observe the value p(t). We are interested in a change p(t+h) impending in a small time unit h - that is what happens to the prices in a small time horizon. Further assume that the change in a short time can be expressed by a function f (invisible hand of the competitive market), which depends on the value of variable p. This leads us to approximate the value p(t + h) by a following equation: p(t + h) = p(t) + hf (p(t)) meaning that the value of a variable in time t + h can be approximated by its value at time t plus h times the direction of change in time t. By rearranging the equation we get: p(t + h) − p(t) = f (p(t)), h

h 6= 0.

Of course the larger h is, the larger is the error in our prediction, so we would like h to be as small as possible. This leads us to making a limit passage as h → 0: p(t + h) − p(t) = f (p(t)). lim h→0 h We can easily see now, that we obtain the derivative of function p in time t (of course assuming it does exist) on the left side of the equation, resulting in a differential equation p0 (t) = f (p(t)). Hence the adjustment process governing our variable in time, can be modelled by a system of differential equations p0 (t) = f (p(t)) satisfying the initial condition p(0) = p0 , which represents the initial price vector. As you can notice, we used the same function f for modelling the equilibrium relation in the economy, as well as for expressing the direction in which the variable p moves in time, or equivalently for the dynamic process. This

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helps us a lot in the analysis of stability of the price adjustment process, since we can use freely all the properties of function f derived from economic assumptions. Whether this attitude is just, is a very complex question, regarding the underlying price adjustment process and we will get to the answer later. It requires a very good economic intuition to determine how much the process can be simplified, so that it will still describe the market forces well and yet will be closely linked with the excess demand functions. We already sketched the idea how is the price adjustment process related to the excess demand function in this thesis in the definition of the model, so we will proceed in this way.

2.2.2

Model’s premises

There are two main premises of the model according to Negishi (1962). The first one is that the economy is competitive. This means that neither demanders nor suppliers can affect the price that prevails in the market (we do impeach monopolistic or oligopolistic markets). This assumption is very important, as it guarantees that the price cannot be distort in favour of any player in the market and our price adjustment process is driven purely by the economy’s invisible hand. Furthermore, based on the theory of competitive economies we know how the price is going to behave, when there is disequilibrium in the market. When the excess demand is positive, the price will go up, because demand is greater than supply, so producers can raise the price and still sell all of their commodities. On the other hand when the excess demand is negative, the price goes down. The second premise is that the price is the only time adjusting parameter of the market. At each instant of time, demand and supply adjust their quantities demanded/supplied according to information about price at given time and the adjustment is assumed to be immediate. This assumption is important because apart from price adjustment in the market, we can also consider quantity adjustment mechanism with demand D(q) and supply S(q) for quantities, telling us the price that buyers/sellers are willing to pay/charge when the quantity is q. The quantity adjusting concept leads to the concept of Marshallian stability, which is not the subject of this work. In this thesis we will only be concerned with Walrasian stability, that is stability of price adjustment processes. Just note that these two concepts do not coincide. There are situations that the market can be Marshallian stable and Walrasian unstable and vice versa. Put simply, we can roughly say that the Marshallian concept of stability is more

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10

suitable for the theory of production (where quantities play a dominant role), whereas the Walrasian concept is more suitable for the theory of exchange (where prices play the dominant role).

2.3

Walras’ price adjustment process

We now get to the intricate question, what exactly generates the dynamic price adjustment process and if there is more than one, which one is the most suitable for the real world modelling. In our terms, we would like to find a mechanism which generates stable adjustment processes and satisfies some natural economic assumptions. Even thought there are two major concepts of price adjustment mechanism - the process of tatonnmenet and the nontatonnement process - in this thesis we will deal only with the tatonnement process, so we omit the explanation of non-tatonnement processes. If the reader is interested in the non-tatonnement processes, we may recommend the book Takayama (1986), pages 341-345.

2.3.1

Walras’ process of tatonnement

Walras’ tatonnement process is a market clearing process, in which no exchange takes place before some equilibrium is met. In this process, there exists a central coordinator of the market, which announces market price p to all demanders and suppliers in the market. In return, the market actors tell the coordinator their planned supplies and demands under this price. Thanks to this information, the coordinator obtains the excess demand and thereafter adjusts the price, depending on whether the excess demand is positive or negative. When the excess demand is positive, he raises the price and when it is negative, he lowers the price. This is consistent with the assumption of competitive market, that prices behave according to the value of excess demand. He then announces the price again and the process goes on and on, till the excess demand is zero for all commodities, meaning the price p converges to equilibrium price p∗ . We consider that the process is carried out simultaneously for all commodities. Since Walras considered time as a continuous variable in the process, there can occur infinitely many iterations in arbitrarily small time interval, so it makes no difficulties to consider our model continuous and not discrete, as one may think given the form of the process. We shall remember two important assumptions about the Walras’ process - first that the price in

2. Basic notions and derivation of the model

11

each market adjusts in the direction of excess demand and second that trades occur only at equilibrium prices. Should the reader want a deeper knowledge about the market exchange, we highly recommend chapter 31 in Varian (2010), pages 583 - 608, which is devoted to all aspects of the market exchange theory. Is Walras’ process a good way to model the price adjustment process? If we cannot be sure that the underlying price adjustment process guarantees that function f can be used in modelling both the economy equilibrium and the time path of the process it would be inadequate to proceed in this way. The real adjustment process and our modelling function describing the equilibrium relation in the economy would not have much in common, henceforth the outcomes of our analysis would probably be incorrect. Considering the Walras’ simultaneous process of tatonnement, we can say that the equation p0 = f (p) represents the adjustment process governed by the market coordinator, who behaves according to the function of excess demand. Because no transactions takes place before the equilibrium price is announced, there is no redistribution of initial wealth in the economy, hence the supply and demand functions do not alter throughout the process. It is therefore possible, to model the price adjustment process using the excess demand function. If the wealth could be redistributed before the price announcement, the demand functions and the supply functions of all agents would change, resulting in a different excess demand function. One may ask, if such price adjustment mechanism has some real life foundations. For example stock-exchange brokers at financial markets or markets with scarce minerals are highly organized, so every available data are reflected in the price very quickly and the price adjustment process can therefore be considered as the Walras’ process. However, we can immediately see a few drawbacks of this process. One of the most important is that in reality there hardly ever exists any coordinator of the market, who sets the price. The other problem arising in such process is the one we already mentioned, that potential coordinator does not have any concrete rule, under which he announces the price p - the same reason, why this process is called tatonnement. The last shortcoming of this approach is that we cannot possibly have enough time to wait till an equilibrium occurs, until the exchange takes place. Even thought this model has a few shortcomings, it is a very good stepping stone in explaining what happens behind the price adjustment process, according to Charles Plott from Caltech university. In article Mukherji (2008), the author is referencing to the experiments conducted by Ch. Plott and his team,

2. Basic notions and derivation of the model

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who showed that models based on the tatonnement processes have a remarkable explanatory power even when applied to the non-tatonnement processes. So it is not irrelevant to study such processes even though we are not sure whose behaviour we are analysing as we already mentioned before. It is going to be a good predictor of what to expect. As of now, we expounded all the models we are going to study and we can now proceed to the theoretical background of the thesis.

Chapter 3 Theoretical background In this chapter, we are going to state precisely all the economic and mathematical assumptions needed for rigorous study of stability of competitive economy. We will start with the more familiar economic background of competitive economy and then we will go more deeply through the mathematical aspects of stability of competitive market.

3.1

Economic background

In this section, we will focus on economic concepts occurring in the modelling of price adjustment processes. These should be well known from the microeconomic theory, so we will go through them just briefly, in order to make the reader familiar with the ideas. One of the main aspects when it comes to the analysis of stability of the price adjustment process is the condition of gross substitutability. This condition was proposed by Hicks, when he first proved stability of competitive market with two commodities and it turned out that it really is one of the most important assumptions. In short, when commodity i is a gross substitute of commodity j, then if the price of the j-th commodity rises, the demand for the i-th also rises and vice versa. We state the gross substitutability assumption for the excess demand function. Denote fi (p) the excess demand function for the i-th commodity, where p = (p1 , ..., pn ) is a vector of prices. Then the > 0. For detailed information good i is a gross substitute of the good j, if ∂f∂pi (p) j about gross substitutes, the reader can take a look into Varian (2010), page 113. Another key assumption is Walras’ law. Walras’ law states that the sum of

3. Theoretical background

14

excess demands is identically zero, whether or not the market is in the equilibn P rium. Mathematically written, the equation pi f (p1 , ..., pn ) = 0 holds for all i=1

prices. The reader can again take a closer look on what lies behind the Walras’ law in Varian (2010) on page 592. The third assumption is that demand and supply functions are homogenous of degree zero in prices. It has a nice economic interpretation - the agents in the market do not sustain money illusion. Money illusion means that proportionate change in prices of all goods in the economy and in wealth of all agents, does not alter the preferences of agents. We can express it as follows: S(αp) = α0 S(p) = S(p) and D(αp) = α0 D(p) = D(p) for any α > 0, where p of course represents prices and D(p) and S(p) demand and supply. The fourth assumption - boundary condition - is quite intuitive, even though the mathematical formula can look quite misty. We will suppose, that if the price of one of the goods in the market converges to zero, then the excess demand function will tend to infinity. This comes from a common sense, that the cheaper a good is, the greater is the demand, so when the good is for free, the demand will be infinite. We can mathematically state this as follows s if we have a sequence of prices {ps }∞ s=1 , such that kp k ≥ δ > 0 (prices are bounded away from zero) for some δ ∈ (0, ∞) and psk → 0 as s → ∞ for some n P k ∈ {1, ..., n}, then fj (ps ) → ∞. The last sum means that there is at least j=1

one commodity in the market, whose excess demand will tend to infinity. We just mention, that this condition is relatively new in the theory of stability of competitive economy, in comparison with the former three. The fifth assumption we will need is the well-known weak axiom of revealed preference or the WARP. The WARP is stated in Varian (2010), page 124, as follows: If bundle A is directly revealed preferred to bundle B, and the two bundles are not the same, then it cannot happen that bundle B is directly revealed preferred to bundle A. When we consider the equilibrium price p∗ and the excess demand function f (p) = D(p)−S(p), we can formulate the WARP mathematin P cally in this way: we say that the WARP holds, if (p∗ −p)T f (p) = p∗i fi (p) > 0 i=1

for all p (the second term is zero thanks to Walras’ law). We further state the generalized version of WARP. We say that generalized WARP holds, if there exists positive definite matrix B, such that (p∗ − p)T Bf (p) > 0. It can be easily seen, that when we take B = I in the generalized WARP, we get the normal version of WARP. The idea behind generalization of the WARP is that we can assign different weights to prices and excess demand functions.

3. Theoretical background

15

Among the last, but equally important assumptions in the analysis of stability of competitive equilibrium, are existence of at least one equilibrium price vector and positivity of all prices. The existence of at least one equilibrium is a quite natural assumption and the theory about existence of the market equilibrium is well explained in most microeconomic text books, for example in Varian (2010), page 595. As most of the commodities in economy are not for free, the assumption about strictly positive prices is also quite native for the model of competitive economy. Now when we have in mind all the economic assumptions, we carry forward to the mathematical background of the thesis.

3.2

Mathematical background

We are now going to lay out rigorous theory of differential equations needed for our analysis. Our main reference is the book Vrabie (2004), which treats the theory of differential equations thoroughly, so the reader can take a look into it for elaborate explanation.

3.2.1

Theory of differential equations

We will begin this chapter by introducing the framework we will be working with. Because in this thesis we deal with the autonomous systems, all definitions and propositions will concern only them. In the rest of this chapter we assume that Ω ⊆ Rn is open connected set, f : Ω → Rn is a continuously differentiable function, where f = (f1 , ..., fn ) and fi = fi (p1 , ..., pn ), ∀i = 1, ..., n. We will be working with a system of autonomous differential equations of first order in explicit form p0 = f (p). For easier description we refer to the system described as (DE). First of all, we need to define the solution of (DE) precisely, so we can work with it in a proper manner. Definition 3.1 (Solution of ODE of first order, initial condition). We say that a function p : I → Rn , where I ⊆ R is an open interval, is a solution to (DE) if: i) ∀t ∈ I, p(t) ∈ Ω, ii) ∀t ∈ I, the derivative exists and satisfies iii) ∀t ∈ I: p0 (t) = f (p(t)) (e.g. p0 (t) is finite) Further on, let (p0 , t0 ) ∈ Ω × I be given. We say that the solution of (DE) satisfies the initial condition with (p0 , t0 ) if p(t0 ) = p0 .

3. Theoretical background

16

We remark, that if we restrict a solution to a smaller interval, we get again a solution. Therefore we speak about a maximal solution if it is not a restriction of another solution. Let us explain the idea behind initial condition. From the economic point of view, the initial condition can be a result of some measurements - we know that the dynamical system passes through this point in a given time, or it could represent the initial prices in the competitive economy model. The mathematical idea behind initial condition has a more direct interpretation. Without initial condition, the solution to the system (DE) cannot be uniquely determined. So specifying the initial condition is a necessary condition for the system to have unique a solution. Just note that it is not a sufficient condition. A sufficient conditions will be stated later. We would like to know that there exists a solution to our problem (DE) and that it is uniquely determined by the initial condition. This issue solves the following theorem. Theorem 3.1. (Picard’s theorem) If f is a continuously differentiable function, then there exists a unique maximal solution to (DE) for each initial condition p(t0 ) = p(0) with (t0 , p0 ) ∈ Ω. Picard’s theorem entitles us to define a fundamental term of theory of differential equations - solving operator. A solving operator assigns to every initial condition (p0 , t0 ) the maximal solution passing through the point p0 in time t0 . A formal definition follows. Definition 3.2 (Solving operator). The function φ : G ⊂ Rn+2 → Rn defined as φ(t, t0 , p0 ) := p(t), where p(t) is the maximal solution to (DE) satisfying p(t0 ) = p0 , is called the solving operator/function of (DE). Here G is the maximal set, where the defined equality makes sense. Without loss of generality, we can assume that t0 = 0 and we will write φ(t, 0, p0 ) = φ(t, p0 ). We have defined and stated all fundamental notions, concerning the existence of a solution, its uniqueness and maximality. This a posteriori assures us, that under very general assumptions (particularly differentiability of function f ), there indeed exists a solution to the equation specified in the models, which we considered previously, without justification of its existence. We’ll now proceed to the theory of stability of the system. Throughout the thesis, we were and we are going to talk a lot about equilibrium points. Even though it is very intuitive, what an equilibrium point is, let us state its definition.

3. Theoretical background

17

Definition 3.3 (Equilibrium point). We say that a point p∗ is an equilibrium point of the system (DE), if f (p∗ ) = 0. We have already made a few heuristic approaches to what does stability of equilibrium means, so simply put, we are interested whether the solving operator φ(t, p0 ) → p∗ , for t → ∞, where p∗ is an equilibrium point. Finally, proper definitions of stability follows. Definition 3.4 (Stability). Let p∗ be an equilibrium point of system (DE), and φ the solving operator. Then we say that the equilibrium p∗ is: i) locally stable, if ∀ > 0, ∃δ > 0 so that kp0 − p∗ k < δ ⇒ |φ(t, p0 ) − p∗ | < , ∀t > 0, ii) unstable, if it is not locally stable, iii) local attractor, if ∃δ > 0 such that if kp0 −p∗ k < δ ⇒ limt→+∞ φ(t, p0 ) = p∗ , iv) locally asymptotically stable, if it is locally stable and local attractor v) globally asymptotically stable, if limt→+∞ φ(t, p0 ) = p∗ , ∀(p0 , t0 ) ∈ Ω × R initial conditions. In the fourth chapter, we will clarify what the definitions could represent in the economic sense. We now provide a key theorem and two definitions, that will help us to prove the global asymptotic stability of the equilibrium in the fourth chapter. Proposition 3.1. Let p(t, p0 ) be a solution to the (DE). If limt→+∞ p(t, p0 ) = pˆ for some finite pˆ ∈ Ω then pˆ is an equilibrium of (DE). By the virtue of this theorem, we can state the following two definitions. Definition 3.5 (Globally stable system). We say that the system (DE) is globally stable, if for all p0 ∈ Ω, the solution p(t, p0 ) converges. Definition 3.6 (Quasi globally stable system). Consider that the solution p(t, p0 ) of the system (DE) with initial condition (p0 , 0) is contained in some compact subset K ⊂ Ω for all t > 0. We say that pˆ ∈ K is a limit point of the solution p(t, p0 ), if there exists sequence {ts }∞ ˆ. s=1 such that lims→+∞ p(ts , p0 ) = p We say that the system (DE) is quasi-globally stable, if for all pˆ ∈ Ω, the limit points of the solution p(t, p0 ) are equilibrium points of the system (DE). In the next subsection, we finally start to uncover the conditions for the equilibrium points to be stable.

3. Theoretical background

3.2.2

18

Linear systems and their stability

We begin with a definition of the linear system with constant coefficients. Definition 3.7 (Linear system with constant coefficients). We say that the system (DE) is a linear system with constant coefficients, if f (p) = Ap, where A ∈ Rn×n is a constant matrix. We now further develop the theory of systems of linear differential equations with constant coefficients, because they are of major importance in stability theory. Recall that by linear system with constant coefficients we have in mind differential equation in the form p0 = Ap and p(t0 ) = p0 is a given initial condition. We will begin with a fundamental definition of matrix exponential, which plays key role in the theory of differential equations with constant coefficients. Further we state theorem showing why the matrix exponential is so important for the theory. Definition 3.8 (Matrix exponential). Let A ∈ Cn×n be an arbitrary matrix. We ∞ P 1 (tA)k . define the matrix exponential etA := k=0

k!

One can ask whether the definition is correct, that is whether the sum converges regardless of the matrix A. This indeed is true, because the norm of the sum is equal or less than the sum of the norms, which equals etkAk and that is a finite number for all t and all matrices. Here the norm of the matrix is defined as kAk = max{x:kxk=1} kAxk. Proposition 3.2 (Solution to p0 = Ap). Consider an arbitrary constant matrix A ∈ Rn×n and the equation p0 = Ap. Then the solution satisfying the initial condition p(t0 ) = p0 is p(t) = eA(t−t0 ) p0 . At present, we are going to show under what assumptions the zero solution to linear system with constant coefficients is (asymptotically) stable. We just recall that λ is an eigenvalue of A if there exists a vector p 6= 0 such that Ap = λp. We denote the set of all eigenvalues of A by σ(A). In the following few paragraphs, we will try to show the idea, which lies behind the stability of linear equations with constant coefficients. For simplicity, we will consider a symmetric matrix A ∈ Cn×n and denote its eigenvalues λi , i = 1, ..., n. For the matrices, that are not symmetric, the derivation is more technical, but the idea is the same. It is well-known from linear algebra that

3. Theoretical background

19

for symmetric matrix A ∈ Cn×n , there exists that  λ1 0 . . .   0 ... ...  A=P. .  .. .. ...  0 ... 0

a regular matrix P ∈ Cn×n such  0 ..  .  −1 P 0  λn

From here we can also easily see, that when computing powers of matrix, that is Ak , it suffices to compute only the products of the diagonalized matrix. Write ˆ −1 )k = (P AP ˆ −1 P AP ˆ −1 · · · P AP ˆ −1 P AP ˆ −1 ) where A ˆ denotes Ak = (P AP the diagonalized matrix belonging to matrix A. Truly we can see, that the products P −1 P = I, where I denotes the identity matrix, and what remains ˆk P −1 . Knowing this result, we can rewrite the formula in the product is: P A from Proposition 3.1 as follows p0 = Ap with initial condition (t0 , p0 ) as follows: ∞ ∞ X X 1 1 k k A (t−t0 ) ˆ −1 )k )p0 = (t − t0 ) A )p0 = ( (t − t0 )k (P AP e p0 = ( k! k! k=0 k=0

= P(

∞ X 1 ˆ −1 ˆk )P −1 p0 = P e(t−t0 )A (t − t0 )k A P p0 . k! k=0

ˆ We will take a closer look at the term e(t−t0 )A . Without major difficulties, we deduce that the matrix exponential of a diagonal matrix is in fact a diagonal matrix of exponentials. Take an arbitrary diagonal matrix A and let’s see what we get when multiplying it. Since 

λ1

 0 ˆ = A .  ..  0 k

0 ... .. .. . . .. .. . . ...

0

 k  0 λk1 0 . . . 0   ..   0 . . . . . . ...  .     = . .  ..   . . . . 0 0  . k λn 0 . . . 0 λn

and consequently from the definition of the matrix exponential we get that   ˆ   (t−t0 )A e =  

e(t−t0 )λ1 0 .. . 0

0 ... ... .. . .. .. . . ...

0

0 .. . 0 e

(t−t0 )λn

      

3. Theoretical background

20

At this time, we are just a step away from deriving the sufficient conditions for stability of the solution of system p0 = Ap with initial condition (p0 , t0 ). As we derived earlier, the general solution to such a system is in form: ˆ p(t) = P e(t−t0 )A P −1 p0 . Since P , P −1 are constant and regular matrices and p0 is a constant vector, ˆ they do not have any influence on stability, so the term e(t−t0 )A is of major concern. With the knowledge that an exponential function converges to zero, when its exponent converges to minus infinity, we can see that the zero solution will be asymptotically stable if and only if all the eigenvalues have negative real parts. When all the eigenvalues have negative real parts, all the elements on the diagonal will converge to zero for t → +∞, so the whole matrix will converge to zero matrix very quickly, regardless of the initial condition. Thus the solution will converge to the equilibrium point zero, confirming our statement, that zero is asymptotically stable. As was mentioned above, the situation is more complicated if A is not diagonalizable, but following theorem holds. Theorem 3.2 (Stability of a linear system of differential equations with constant coefficients). Let A ∈ Rn×n be an arbitrary matrix. Then the equilibrium point zero of p0 = Ap is asymptotically stable if and only if Reλ < 0 ∀λ ∈ σ(A) Remark that the same theorem holds for the system p0 = A(p − p∗ ) and its equilibrium p∗ . We propose one more theorem concerning stability of a linear equation, which is sometimes known as the Lyapunov’s theorem. Its proof can be found for example in Vrabie (2004). It will have a great use in the fourth chapter. Theorem 3.3 (Lyapunov’s theorem). Let A ∈ Rn×n . Then the following is equivalent: i) Reλ < 0 for all λ ∈ σ(A), ii) there exists positive definite matrix B such that BA + AT B = −I, where I is the identity matrix.

3.2.3

Stability of a general equation

In the previous subsection, we have derived under what assumptions the linear system is stable. Now we are going to show how and under what assumptions this information helps us with stability of the general system (DE). The idea for analysing stability of a general equation is to expand the function f into the Taylor series about its equilibrium point p∗ . Of course, the function f

3. Theoretical background

21

has to be at least once differentiable for the expansion to be possible. Since we assume the function f to be continuously differentiable, it causes no difficulties to us. Expanding the function f leads us to the following equation: p0 = f (p∗ ) + ∇f (p∗ )T (p − p∗ ) + o(kp − p∗ k) kw(p)k where the symbol o(kp−p∗ k) denotes the class of functions ω satisfying lim∗ kp−p ∗k = p→p

∗

∗

∗

0. Denote A := ∇f (p ). Since f (p ) = 0 (p is an equilibrium point), we can rewrite the equations as follows: p0 = A(p − p∗ ) + o(kp − p∗ k) This suggests that stability of the system (DE) is determined by the linearised part of the system, since the term o(kp − p∗ k) is very small in a small neighbourhood of p∗ . A rigorous statement follows. Theorem 3.4 (About linearised stability). Let p∗ be an equilibrium of (DE). Denote A = ∇f (p∗ ) and further suppose that Reλ < 0 ∀λ ∈ σ(A). Then p∗ is locally asymptotically stable. We add one more theorem about stability and our brief exposition to the theory of linearised stability will be almost complete. Theorem 3.5 (About linearised instability). Let p∗ be an equilibrium of (DE). Denote A = ∇f (p∗ ). If ∃λ ∈ σ(A) such that Re λ > 0, then p∗ is unstable. One may see the above theorems about stability a bit misty, concerning the economic interpretation, so let’s give a brief explanation what happens there. For simplicity, consider the excess demand function in a one dimensional situation f (p) = D(p) − S(p), whether in the general model or in the pure exchange model, and p∗ is an equilibrium point. Taking linear approximation of the excess demand function f (p) about the equilibrium, we arrive at equation p0 = f 0 (p∗ )(p − p∗ ). As is well known, the first derivative determines the direction of the graph - whether it is increasing or decreasing. In this simple case, f 0 (p∗ ) = A ∈ R1×1 and obviously A is the only eigenvalue of A. First assume that f 0 (p∗ ) is positive. Then, we can see from the differential equation that when the price p is greater than the equilibrium price p∗ , then p0 is positive, meaning that the price will grow even more. When the price is smaller than p∗ , the derivative is negative so the price will continue to decrease. From this explanation, one can directly see that the price can never converge to

3. Theoretical background

22

the equilibrium price p∗ , unless the excess demand is zero from the beginning. A different situation occurs when the value f 0 (p∗ ) is negative. Then if the price p is greater than the equilibrium, the derivative is negative and the price will be decreasing towards the equilibrium. Also when the price is smaller than the equilibrium, the derivative is positive, so the price will increase towards the equilibrium price p∗ . It follows that the price will converge to the equilibrium, making the market stable. The last case is the one where f 0 (p∗ ) = 0, this is equivalent to the case when the eigenvalue equals zero. As you may notice, the theorems about linearised stability/instability does not concern this case. It is given by a simple reason when the term f 0 (p∗ ) = 0, the linear approximation term is also zero. Then a small perturbation (the higher order terms in the Taylor expansion) can change the behaviour to stability or instability and different methods are needed to obtain stability in such cases. We propose one of them in the following section. Beforehand we propose one more proposition about the global stability, that we will use in the fourth chapter. Proposition 3.3. If the equilibrium points of the system (DE) are isolated, so for all equilibriums p∗ , there exists δ > 0 such that f (p) 6= 0 for all p ∈ U (p∗ , δ)\{p∗ }, then quasi global stability of the system implies global asymptotic stability.

3.2.4

Lyapunov’s concept of stability

The method of linear approximation of the system and subsequent study of the linearised part to get information about stability of the whole system, which we presented in the last subsection, can be viewed as an indirect approach to studying stability. We now introduce another, more direct approach for investigating stability, namely the method of Lyapunov’s function. This concept is quite different than the previous one, yet very useful, since it is going to help us prove the global asymptotic stability of the general model. We start again with a definition. Definition 3.9 (Lyapunov’s function). Consider the system (DE). A function V (p) : Ω → [0, ∞) is called Lyapunov’s function for (DE) in Ω, if: i) V is continuous on Ω, ii) V (p(t, p0 )) converges as t → +∞ for all p0 ∈ Ω, where p(t, p0 ) is solution to (DE) satisfying p(0) = p0 ,

3. Theoretical background

23

iii) dtd V (p(t)) = V˙ (p(t, p0 )) exists and is zero if and only if p(t, p0 ) is an equilibrium point of (DE). The method of Lyapunov’s function is a powerful tool in the stability theory. The main result of Lyapunov’s theory states the following. Theorem 3.6 (Lyapunov’s about stability). Consider the system (DE) and suppose there exists Lyapunov’s function for this system. Then if the solution to (DE) is bounded for any initial condition p0 ∈ Ω, then (DE) is quasi globally stable. If the reader wants a further explanation and possibly more results concerning the Lyapunov’s method, we highly recommend the book by Vrabie (2004) and in connection with its use in economic models, the book Takayama (1986), pages 347-357 and monograph Mukherji (2003). Now we can proceed to the analysis of stability of the competitive equilibrium model.

Chapter 4 Stability of competitive equilibrium We now have a theoretical background deep enough to analyse stability of competitive market equilibria. Before we proceed to the analysis of local and global stability of the general model, we give a brief literature review, to make the reader familiar with the history of the topic and known results.

4.1

History - literature review

In this sub chapter, we are going the give a short survey of the main results relating to the area of stability of competitive economy, which were achieved in the early years of studying this topic. Almost all of the major results in the theory were very well summarized in the book Takayama (1986) and in one of the very first articles regarding this topic Negishi (1962), so we will draw the results from these two sources. Both the article by Negishi and the book by Takayama contain extensive enumeration of all literature sources concerning this topic. With regard to the more recent results that were achieved after the year 1986, we use mainly articles from Mukherji, who is the leading contributor to the subject of stability of competitive economy throughout the last thirty years. In his monograph Mukherji (2003), he very well summarized and solved lot of questions concerning the price adjustment processes.

4.1.1

In the beginning

The idea about stability of competitive economy was first brought by Walras, who developed the static theory of general equilibrium and proposed the idea for the two commodity market. It was also Walras who came up with the theory of tatonnement process. Hicks (1946) extended the analysis to the

4. Stability of competitive equilibrium

25

multi commodity market. Even though Walras solved the problem for the two commodity economy, his theoretical background was not rigorous enough for a comprehensive dynamical analysis of the multi commodity case. The first systematic investigation of stability within proper framework of general equilibrium analysis of the multi commodity market was done by Arrow & Hurwicz (1958). Scarf (1960) offered an example of instability in the model of competitive economy, therefore showed that the tatonnement process can be stable only under certain conditions. The three key economic assumptions - Walras’ law, gross substitutability and zero homogeneity - come from this era. For an exhausting historical survey of the topic, we highly recommend the article from Negishi (1962), pages 642-644.

4.1.2

First results

Although the concept of modelling the market equilibrium stability by tatonnement process was first proposed by Walras, it was not Walras who showed stability of the process in general. Walras proved stability in the two commodity market under the assumption that the market behaves according to the sign of excess demand, that is - as we already mentioned a few times when the excess demand is positive, the prices goes up and vice versa. As was pointed out, the first generalisation was proposed by Hicks for the multi commodity market. Hicks defined the condition under which the market is stable in the static sense. The sufficient condition proposed by Hicks for the multi commodity case to be perfectly stable (that is stable in Hicksian sense) was the following one - the sign of the derivative of excess demand of a commodity with respect to its own price must be negative, even when arbitrary subset of the other prices are kept unchanged, while the remaining ones are adjusted so as to maintain equilibrium in the respective markets. This condition is exactly the famous Hicksian condition for the market to be stable. For further details, we would refer the reader to Takayama (1986), page 314. This condition is however static, since it does not explore the nature of the dynamics of the market adjustment process. The first person who observed that stability of equilibrium cannot be studied without specifying the dynamic adjustment process was Samuelson (1946). He was also the first who gave a rigorous statement of the problem and gave a definition of stability of the equilibrium. Samuelson considered stability as stability of the system of differential equations, which we specified in the gen-

4. Stability of competitive equilibrium

26

eral model. He proved that the system is stable if all eigenvalues of the linear approximation matrix have negative real parts. This condition is generally different from the Hicks condition. Takayama (1986) mentions four main results, which were achieved by Samuelson, Lange, Metzler and Morishima, about the relation between the Samuelson’s approach and the Hicks’ approach. For now, denote matrix A = ∗ i (p ) n ( ∂f∂p )i,j=1 the matrix of partial derivatives of excess demand of the i-th comj modity with respect to the price of the j-th commodity. Then the following propositions holds: 1) If A is symmetrical, the Hicksian condition and the Samuelson’s condition coincide with each other (Samuelson, Lange). 2) If A is quasi-negative-definite, both Hicksian and the Samuelson’s conditions are satisfied (Samuelson). 3) The Hicksian condition is necessary for the dynamical process to be stable (Metzler). 4) If the matrix A has all off-diagonal elements positive (all goods are gross substitutes), the Hicksian condition and the Samuelsons condition coincide (Metzler). Although Samuelson made a great deal in the analysis, in most cases he examined stability in a small neighbourhood of the equilibrium - local stability - by the methods of linear approximation. Also Samuelson did not consider the asymptotic stability, which is our concern. The first ones who explored the competitive economy with its relation to stability of the price adjustment process were Hahn (1958), Arrow & Hurwicz (1958), and Negishi (1958). In 1958 they proved under the assumptions of Walras’s law, homogeneity of demand functions with respect to all prices and the gross substitutability, that not only Hicksian and Samuelson’s conditions coincide (as was stated above), but also the dynamic stability holds. This means that the linear approximation of the model about the equilibrium point is also stable, implying local stability of the initial system of differential equations. Finally, in 1959, Kenneth J. Arrow (1959) proved that the original system is globally stable, if the gross substitutability assumption holds for all commodities. In the subsequent years, some other areas of the topic were explored. For example expectations were introduced into the model and some attempts were made to reduce the gross substitutability assumption. Since then, the question is to what extent can we relax these assumptions and replace it with less strict

4. Stability of competitive equilibrium

27

ones, such that equilibrium will still be stable. In the early 1960’s the nontatonnement pure exchange models were introduced, to solve the shortcomings of the tatonnement processes. From the recent past, we highlight the results achieved by Mukherji, who deals with the topic of stability of competitive equilibrium throughout his whole academic career. Mukherji’s major contribution to the topic is his monograph Mukherji (2003), where he analyses tatonnement and non-tatonnement processes and in great detail solves many questions regarding their stability. With respect to other Mukherji’s work, we recommend the articles Mukherji (2008), in which Mukherji introduces wealth endowments into the model and Mukherji (2010), where he analyses stability in economy, where there are diverse economic agents. We now state rigorously the results, that we mentioned in this section. Among the first major result in the theory was the proof of global stability in the three commodity case market, which was done by Walras. Proposition 4.1 (Global stability for the three commodity case). Let fi (p) where p = (p1 , p2 , p3 ), denote the excess demand function of the i-th commodity, where i = 1, 2, 3. Further assume that the following holds: gross substitutability, homogeneity of degree zero of the supply and demand in prices, Walras’ law, prices are strictly positive, that is pi > 0 for i = 1, 2, 3 and suppose there exists at least one equilibrium price pˆ so fi (ˆ p) = 0, i = 1, 2, 3. Then the system p0i = fi (p1 , ..., p3 ), i = 1, 2, 3 is globally stable. One of the major propositions concerning the pure exchange model follows. It is the theorem about stability of the pure exchange model from Arrow, Block and Hurwicz, which we mentioned in the end of our history tour. Proposition 4.2 (Global stability of the market). Consider the system p0i (t) = xi (p1 (t), ..., pn (t)) − xˆi , where xi are continuously differentiable. Denote pˆ the equilibrium point of the system. Further assume that Walras’ law, homogeneity of degree zero of the prices and the gross substitutability assumptions hold. Then the system is globally stable.

4.2

Stability of the general model

In this part of the thesis, we will specify under what assumptions is the general model of competitive market locally and globally stable and we will

4. Stability of competitive equilibrium

28

prove corresponding theorems. First let us discuss the framework of the analysis. It arises that reasonable analysis of stability can only be done when we consider one good as a numeraire. Next we will link the definitions of stability made in the third chapter with the economic theory. Finally, we examine conditions for local asymptotic stability and then we step up to the proof of global asymptotic stability of the model.

4.2.1

Framework of the analysis

As we already pointed out, we have to make a certain adjustments in the model, if we want to arrive at some sufficient results. We explain what exactly causes the trouble in the analysis. From the zero homogeneity in prices, it comes that if a price p∗ is equilibrium then also all prices in form αp∗ , α > 0 are equilibrium prices, because f (αp∗ ) = f (p∗ ) = 0, ∀α > 0. So in fact we have a half line of equilibrium points. We can see that the system can never be locally asymptotically stable, because in every neighbourhood of the price p∗ there is another price pˆ, such that f (ˆ p) = 0, hence the solution to p0 = f (p) with initial condition p(0) = pˆ can never reach the equilibrium price p∗ , because it is constant. It follows, that the equilibrium can only be stable, but not asymptotically and if we are interested in the asymptotic stability, we need to modify its definition in an appropriate way. The approach we are going to take is that we will think of the n-th good as a numeraire, meaning that the price of the n-th good equals one. We will consider such price adjustment process, under which the price of the n-th acts as a numeraire, so pn = 1. When we considered the model without numeraire, we had a subset {αp∗ , α > 0} ⊂ Rn of equilibrium prices all lying on one half line. Now with pn = 1, we fixed one coordinate in the n-dimensional plane, so we reduced the half line in Rn to a single point in Rn . By making this adjustment, we no longer have the problem described in previous paragraph, while not making any restrictions to the model. Having in mind the idea about numeraire, we can restate the n-dimensional system of differential equations as an n − 1 dimensional system with additional assumption for the n-th commodity, that pn = 1. As we already discussed in the second chapter, the function describing the adjustment process does not have to be exactly the excess demand function, however based on the discussion in the second chapter, we will proceed in this way. The model with numeraire is the following. We have functions fi : (0, +∞)(n−1) × {1} → R, i = 1, ..., n,

4. Stability of competitive equilibrium

29

that are continuously differentiable functions of p ∈ Ω = (0, +∞)n−1 × {1} for all i and they represent the excess demand function on the i-th market. We suppose that there exists at least one vector p∗ = (p∗1 , ..., p∗n−1 , 1) ∈ Ω such that f (p∗ ) = 0, meaning that p∗ is a market equilibrium. Note that functions fi , i = 1, ..., n are in fact functions of n − 1 variables, when the n-th good is numeraire. We will denote f = (f1 , ..., fn ) and f−n = (f1 , ..., fn−1 ). The model then states as follows: p0i = fi (p1 , ..., pn−1 , 1),

i = 1, ..., n − 1,

and pn = 1.

We will now work with the system of differential equations in this n − 1 dimensional subspace, where the equilibrium can be unique and the definitions of stability and all the theorems about stability from the third chapter are well defined and have a good meaning. We want to avoid unnecessary technicalities in labelling, so we still write p = (p1 , ..., pn−1 , 1) ∈ Ω for prices, even though they are different than in the initial system and refer to the system of n − 1 differential equations as p0 = f−n (p). We now state the economic assumptions made in the third chapter for the case when we consider the n-th good to be numeraire. They are essentially the same, but we feel that it would be good to provide them in a proper manner. The assumptions we will need are following: n−1 P (WL) (Walras’ law) pT f (p) = pi fi (p) + fn (p) = 0 for all p ∈ Ω, i=1

s (BC) (Boundary condition) if there is sequence {ps }∞ s=1 ⊂ Ω such that kp k ≥ δ > 0 for some δ > 0 and ∀s ∈ N and psk → 0 as s → ∞ for some k ∈ n P {1, ..., n − 1} then fi (ps ) → ∞ as s → ∞, i=1

(GS) (Gross substitutability) ∂f∂pi (p) > 0 for all i = 1, ..., n, j = 1, ..., n − 1 and j i 6= j, ˆ ∈ Rn×n (GWARP) (Generalized WARP) there exists positive definite matrix B ˆ (p) > 0 for all p ∈ Ω \ {p∗ }. such that (p∗ − p)T Bf In the following paragraphs, we will go through the definitions of stability given in the third chapter and we try to explain what do they mean in the economic sense. Then we proceed to the analysis of stability. The definitions in the third chapter probably look a bit opaque, so we’ll clarify them now and explain why they are important in the economic modelling of competitive economy, before we proceed to the analysis of stability. Suppose we have an equilibrium point p∗ , meaning that the constant vector p∗

4. Stability of competitive equilibrium

30

solves the system (DE). By saying that the solution is locally stable, we mean that when we take any initial condition close to the point p∗ , then the maximal solution satisfying this initial condition will not get much far away from the point p∗ for all future times. Note that under local stability they do not have to reach the point p∗ . So in the economic concept, when the price does not deviate much from the equilibrium price p∗ , then the price vector will still be near the equilibrium price. When the solution p∗ is local attractor, it means that, when our initial condition is close to the point p∗ , the solution will converge to the point p∗ in large enough time. Putting these two together, we get asymptotically stable solution. When the solution p∗ is asymptotically stable, taking initial condition close to it will guarantee that the solution passing through this point will never get far away from the point p∗ (note that if it is only local attractor, the solution can get as far away as it wants from the point p∗ ) and further it will converge to the point p∗ . Again taken from economic perspective, when the initial price p is near the equilibrium price p∗ , we know that it will converge to the equilibrium. Global stability simply means that regardless of the initial condition, all the solutions will converge to the point p∗ . When the shocks in the economy are so profound, that the price of commodities plummets, it is the case of instability - regardless how close the initial price starts to the equilibrium price, it will diverge from it. It is also interesting to ask, whether it is more suitable to focus on the local stability or the global stability of the market. As one may expect, there are pros and cons for both of the concepts. When the economy undergoes a shock, positive or negative, the change in prices is usually not so large. In other words the new vector of prices will be in a close neighbourhood of the initial equilibrium. As we know from our experience, the prices of almost all goods fluctuate in time horizon, some more, some less, but generally there are very few goods, whose price does not change over a long period of time. It is therefore consistent with our idea of local stability. Hence it seems quite intuitive, that the concept of local stability will be more efficient and studying the global stability would be too restrictive. On the other side, even local stability has its drawbacks. Generally, we are more interested whether the market is globally stable and not just locally. Furthermore there can be more equilibrium points and some of them may be locally unstable, even though the economy is globally stable. So the price can

4. Stability of competitive equilibrium

31

actually diverge from its initial equilibrium, being unstable, but it does not cause a major problem, since it converges to a new equilibrium point. Another situation, which may not be so clear at first sight is that after the disturbance of equilibria, the underlying adjustment process governing the prices, does not have to converge to a single point, but it can actually create a periodic solution, representing an economic cycle. This concept leads to study of dynamical systems, but it is behind the scope of this work.

4.2.2

Local stability of the general model

In this section, we will build up a theory showing under what assumptions is the equilibrium in the general model locally stable under the tatonnement process. The aim is not only to derive under what conditions is the equilibrium locally stable, but also to get the idea what the necessary assumptions for global stability are, since local stability is necessary for global stability to hold. Let us recall that we are interested in stability of an equilibrium point ∗ p ∈ Ω of the system p0i = fi (p), for i = 1, ..., n − 1. We suppose that such equilibirum exists and if there is more than one, we consider one arbitrarily chosen but fixed. Throughout the whole chapter, we will use the notation ∗ i (p ) n−1 )i,j=1 , so A is the Jacobian matrix of the excess demand function A = ( ∂f∂p j evaluated at the equilibrium. Then the linearised system reads p0 = A(p − p∗ ). As we mentioned in the third chapter, if all the eigenvalues of A have negative real parts, then the system p0 = f−n (p) is locally asymptotically stable. When this condition is satisfied, we will say that the system is linear approximation stable. Moreover by the Lyapunov’s theorem, this occurs if and only if there exists positive definite matrix B such that AB + AT B = −I. We now show in a series of theorems a few consequences of Lyapunov’s theorem in relation to the excess demand function. Proposition 4.3. If eigenvalues of A have negative real parts, then there exists ˆ (p) > 0 for a positive definite matrix B ∈ R(n−1)×(n−1) , such that (p∗ − p)T Bf all p ∈ U (p∗ , δ) \ {p∗ } = {p ∈ Rn : pn = 1, kp − p∗ k < δ} \ {p∗ }, for some δ > 0, where ! B 0 ˆ= B 0T 1 Proof. We use the Lyapunov’s theorem, which gives us the existence of a T positive definite matrix B = (bij )n−1 i,j=1 such that BA+A B is negative definite. ˆ (p) attains a strict local We now show that the function g(p) = (p∗ − p)T Bf

4. Stability of competitive equilibrium

32

minimum at p∗ , hence there is a δ > 0 such that g(p) > g(p∗ ) = 0 for all p ∈ U (p∗ , δ) \ {p∗ }. Take notice, that the function g(p) is a function of n − 1 variables. It can easily be seen, that g(p∗ ) = 0 from the definition. Further, note that because pn = 1 = p∗n , then (p∗n −pn )fn (p) = 0 so g(p) does not contain the term fn (p). Now for k = 1, ..., n − 1 write n−1 n−1 X ∂g(p) ∂fj (p) X ∗ = (pi − pi )bij − bkj fj (p) ∂pk ∂p k i,j=1 j=1

for the first derivative and the second derivative: n−1 n−1 n−1 X ∂ 2 g(p) ∂ 2 fj (p) X ∂fj (p) X ∂fj (p) ∗ = − − . (p − pi )bij blj bkj ∂pk ∂pl i,j=1 i ∂pk ∂pl ∂pk ∂pl j=1 j=1 ∗

) By evaluating the first partial derivative at p∗ , we get that ∂g(p = 0 for all ∂pk ∗ ∗ k because the excess demand vanishes at p , so p is a critical point of the function g. Further for k, l 6= n we have that n−1

n−1

n−1

n−1

∂ 2 g(p∗ ) X ∂fj (p∗ ) X ∂fj (p∗ ) X ∂fj (p∗ ) X ∂fj (p∗ ) = blj − bkj = blj − bjk ∂pk ∂pl ∂pk ∂pl ∂pk ∂pl j=1 j=1 j=1 j=1 thanks to the symmetry of matrix B. Henceforth ∂ 2 g(p∗ ) = −(BA + AT B), ∂pk ∂pl so the Hessian matrix of the function g(p) evaluated at p∗ is positive definite implying the function g truly has a strict local minimum at p∗ . As we stated in the beginning of the proof, it follows that there exists δ > 0 such that ˆ (p) = g(p) > g(p∗ ) = 0 for all p ∈ U (p∗ , δ) \ {p∗ }. (p∗ − p)T Bf This theorem tells us, that if the system p0 = Ap is locally asymptotically stable then the function g(p) attains a strict local minimum at p∗ or ˆ (p) > 0 for all p 6= p∗ . Now we link the theorem equivalently (p∗ − p)T Bf to the weak axiom of revealed preference. As an easy consequence, we know that if the system is linear approximation stable, then the local version of the generalized weak axiom of revealed preference holds. Specially, if the previous theorem holds with B = I, then the weak axiom of revealed preference holds, because (p∗ − p)T f (p) = (p∗ )T f (p) > 0 (the second term is zero thanks to Walras’ law), for all p ∈ U (p∗ , δ) \ {p∗ }, which is the definition of the WARP.

4. Stability of competitive equilibrium

33

We would like to prove the converse to the previous theorem. However we need one additional assumption, for the converse to be true. ˆ such that (p∗ − Proposition 4.4. Suppose there is a positive definite matrix B ˆ (p) > 0 for all p ∈ U (p∗ , δ) \ {p∗ } for some δ > 0, where p)T Bf ˆ= B

! B b bT 1

and the matrix BA + AT B has rank (n − 1), then all eigenvalues of A have negative real parts. ˆ (p). From the assumptions, we Proof. We again define g(p) = (p∗ − p)T Bf know that g has a strict local minimum at p∗ with respect to p ∈ U (p∗ , δ)\{p∗ }, so the Hessian matrix, which equals −(BA + AT B) as was shown in the previous implication, has to be positive semi-definite, implying that BA+AT B is negative semi-definite. Since the rank of the matrix is n − 1, the matrix is negative definite. It now follows from the Lyapunov’s theorem, that all eigenvalues of A have negative real parts. We showed that if the generalized WARP is satisfied and moreover the rank condition holds, then the system is linear approximation stable. If the rank condition is not satisfied, we cannot be sure that the matrix will be positive definite, hence we cannot make any conclusions about stability based on the linearised part of the equation. Fortunately, the asymptotic stability can be proved directly, even without the rank condition. Proposition 4.5. Suppose there is a positive definite matrix B ∈ R(n−1)×(n−1) ˆ (p) > 0 for all p ∈ U (p∗ , δ) \ {p∗ } for some δ > 0, where such that (p∗ − p)T Bf ˆ= B

B b bT a

!

then p∗ is a locally asymptotically stable equilibrium point. Proof. We already sketched out, that the linearisation is insufficient in this case, so the method of Lyapunov’s function might be of use. Define a function ˆ ∗ − p) and consider a point p0 such that p0 ∈ M (p∗ , α) = V (p) = (p∗ − p)T B(p ˆ ∗ − p) < α} ⊂ U (p∗ , δ) for some α > 0. The function {p : pn = 1, (p∗ − p)T B(p ˆ is positive definite. Further it is continuous V (p) is non negative, because B

4. Stability of competitive equilibrium

34

and V (p) = 0 if and only if p = p∗ . Denote p(t, p0 ) solution to the equation p0 = f (p) with initial condition p(0) = p0 . Taking the first derivative with respect to time of the function V (p(t, p0 )) we get the following: ˆ (p(t, p0 )) < 0, V 0 (p(t, p0 )) = −2(p∗ − p(t, p0 ))T Bf if p(t, p0 ) ∈ M (p∗ , α) and p(t, p0 ) 6= p∗ . Then V (p(t, p0 )) is non increasing (it has non positive first derivative), the derivative is zero if and only it p(t, p0 ) is equilibrium and vV (p(t, p0 )) ≤ V (p0 ) in M (p∗ , α), so p∗ is asymptotically stable point of the system p0 = f−n (p). Overall we derived that if the system p0 = f−n (p) is linear approximation stable, then the generalised version of WARP holds, therefore it is a necessary condition for the system to be linear approximation stable. Moreover the generalised weak axiom of revealed preference is sufficient condition for local asymptotic stability of the equilibrium of the system p0 = f−n (p), as was shown in the past two propositions. We now get to the question, what economic and mathematical assumptions are needed for the matrix’s A eigenvalues to have negative real parts. One of the core assumptions will be the gross substitutability, i.e. the matrix A has all non diagonal elements strictly positive. The second condition assuring us that the matrix will be negative definite is that it has dominant negative diagonal. By having dominant negative diagonal, we mean that the terms on the diagonal are negative and satisfy the following assumption. There exists posin−1 P ci |aij | tive numbers c1 , ..., cn−1 such that for all j the inequality cj |ajj | > i=1,i6=j

holds. It can be shown, that it is equivalent to following: there exists positive n−1 P di |aji | holds for numbers d1 , ..., dn−1 such that the inequality dj |ajj | > i=1,i6=j

all j. This definition is more general than the standard one for the dominant diagonal matrix, in which ci = di = 1 for all i. Having such generalization means, broadly speaking, that if the diagonal term is really dominant in one row/column, it does not have to be dominant in other one. In the following proposition, we show the relation between these two conditions. Proposition 4.6. If the (WL) and (GS) holds, then A has a dominant negative diagonal with ci = di = p∗i , i = 1, ...n − 1. Proof. We take an arbitrary j ∈ {1, ..., n − 1}, differentiate the Walras’ law with respect to the j-th price and evaluate it at equilibrium. We get that

4. Stability of competitive equilibrium n P

i (p p∗i ∂f∂p j

i=1 n P

i=1,i6=j

p∗i

∗)

35

= 0. We can rewrite this equation subsequently: −p∗j

∂fj (p∗ ) ∂pj

>

n−1 P

p∗i

i=1,i6=j

∂fj (p∗ ) , ∂pj

because p∗n = 1 and

∂fn (p∗ ) ∂pj

∂fj (p∗ ) ∂pj

=

> 0 from gross

substitutability. All the terms in the last sum are positive and so is the term ∂fj (p∗ ) on the left hand side of the equation, so we can rewrite it as p∗j | ∂p | > j n−1 P ∗ ∂fj (p∗ ) pi | ∂pj | and we are done. i=1,i6=j

Now we use this theorem to derive some results about WARP and subsequently about local stability. Proposition 4.7. If the (WL) and (GS) holds, then A is quasi-negative definite (A + AT is negative definite) and WARP is satisfied locally, so there exists δ > 0 such that ∀p ∈ U (p∗ , δ), p 6= p∗ we have (p∗ − p)T f (p) = (p∗ )T f (p) > 0. Proof. By the previous theorem, we know that A has a dominant negative T diagonal with ci = di = p∗i for all i. Define B = (bij )n−1 i,j=1 = A + A . We n−1 n−1 P ∗ P ∗ pi |aji |. Adding up these pi |aij | and p∗j |ajj | > know that p∗j |ajj | > i=1,i6=j

i=1,i6=j

two expressions, we get that p∗j |bjj | >

n−1 P i=1,i6=j

p∗i (|aij | + |aji |) ≥

n−1 P

p∗i |bij |

i=1,i6=j

for all j. The matrix B is symmetric, which follows immediately from the definition and has dominant negative diagonal. Hence it is negative definite. In turn we have that A is quasi-negative definite. For the part about WARP, we again define g(p) = (p∗ −p)T f (p) and we take notice that g(p∗ ) = 0 and also ∇g(p∗ ) = 0. Similarly like in the proof of the Liapunov’s theorem for excess demand, we have that the Hessian matrix of g(p) evaluated at p∗ is equal to matrix −(A + AT ) = −B, which is positive definite, hence g attains a strict local minimum at p∗ , so there exists δ > 0 such that WARP holds for each p ∈ U (p∗ , δ), p 6= p∗ . We are now just a step away from putting the propositions from this sub chapter into context and deriving the main result about local stability. Theorem 4.1. (Local stability of the competitive market equilibrium) The equilibrium price p∗ = (p∗1 , ..., p∗n−1 , 1) of the system of the system p0 = f−n (p) is locally asymptotically stable if one of the following conditions is satisfied: i) the (WL) and the (GS), ii) (GWARP). Further the assumption of generalized WARP is necessary for the equilibrium to linear approximation stable.

4. Stability of competitive equilibrium

36

The results should give us a good insight what will be needed for global stability to hold.

4.2.3

Global stability of the general model

In this section, we will derive under what assumptions is the market equilibrium globally asymptotically stable. Let us recall that we are interested in global stability of the equilibrium p∗ ∈ Ω of the competitive market model p0i = fi (p) = Di (p) − Si (p), i = 1, ..., n − 1 and pn = 1, where the price adjustment process is given by the tatonnement process and we consider arbitrary initial price p0 . The first theorem on our way to prove global stability of the general model states, that the equilibrium is unique. This justifies that the concept of global stability is well defined, because if there are more equilibrium points, the market cannot be globally stable. Proposition 4.8. Under the (GS) and zero homogeneity in prices, the equilibrium price is unique. Proof. We first prove, that when we do not consider pn = 1, the equilibrium price is unique up to scalar multiples. Suppose that the converse is true, so there are at least 2 equilibrium prices p1∗ ∈ Rn and p2∗ ∈ Rn such that p1∗ 6= αp2∗ 1∗ p1∗ p1∗ for any α > 0. Define β = max{ p12∗ , ..., ppn2∗ } = pk2∗ . From the definition, we n 1 k 1∗ 2∗ 1∗ have that βp2∗ ≥ p for all i, βp = p for i = k and the inequality has to i i k k be strict for some i, say i = l. Without loss of generality, we can assume that l 6= n, otherwise we would relabel the prices. By homogeneity of the excess demand function, we obtain fk (βp2∗ ) = fk (p2∗ ) = 0. We expand the excess demand function into the Taylor’s polynomial of first order at the point p1∗ with Lagrange’s reminder (ˆ p lies on the line connecting p1∗ and βp2∗ ). From the assumption of gross substitutability we have fk (βp2∗ ) = fk (p1∗ ) +

n X ∂fk (ˆ p) j=1

∂pj

1∗ (βp2∗ j − pj ) ≥

∂fk (ˆ p) 1∗ (βp2∗ l − pl ) > 0, ∂pl

which is a contradiction, so the equilibrium is unique up to scalar multiples. The first inequality comes from the fact that for the k-th commodity, fk (ˆ p) 1∗ (βp2∗ k − pk ) = 0 and the rest is non negative from gross substitutability. ∂pk All together we can conclude, that the equilibrium price with pn = 1 is uniquely determined.

4. Stability of competitive equilibrium

37

The idea of the proof of global stability is to find a Lyapunov’s function, which should naturally arise from the economic assumptions and apply the Lyapunov’s theorem about stability. Define sets P (p) = {i ∈ {1, ..., n − 1} : fi (p) ≥ 0} ∪ {n} and N (p) = {i ∈ {1, ..., n − 1} : fi (p) < 0}. Next we P define V (p) = pi fi (p) and in the following few paragraphs we will show, i∈P (p)

that the function V (p) is truly Lyapunov’s function. If we rewrite Walras’ law P P as pi fi (p) + pi fi (p) = 0, then if the set N (p) is empty, V (p) = 0 i∈P (p)

i∈N (p)

and that happens only in the equilibrium. In case that N (p) is not empty, then V (p) > 0, so overall V (p) ≥ 0, ∀p and V (p) = 0 if and only if p is an equilibrium price. We would like to know, whether the function V (p(t, p0 )) is non increasing for every solution p(t, p0 ). Unfortunately we cannot directly differentiate V (p(t, p0 )), since the derivatives does not have to exists for some p for which fi (p(t, p0 )) = 0, for certain i. The following theorem tells us something about the first derivative of function V (p(t, p0 )) if it exists. Proposition 4.9. Let p(t, p0 ) be a solution to the system p0 = f (p) with initial condition p(0) = p0 and denote V (t, p0 ) = V (p(t, p0 )). Then V 0 (t, p0 ) ≤ 0, whenever the derivative exits and moreover if V (t, p0 ) > 0, then V 0 (t, p0 ) < 0. Proof. Write V 0 (t, p0 ) = (

X

pi (t, p0 )fi (p(t, p0 ))0 =

i∈P (p(t,p0 ))

X

=

[p0i (t, p0 )fi (p(t, p0 ))

+ pi (t, p0 )

j=1

i∈P (p(t,p0 ))

=

X

[−p0i (t, p0 )

i∈P (p(t,p0 ))

n−1 X ∂fi (p(t, p0 ))

∂pj

p0j (t, p0 )] =

n−1 X

n−1 X ∂fj (p(t, p0 )) ∂fi (p(t, p0 )) 0 pj (t, p0 ) +pi (t, p0 ) pj (t, p0 )], ∂pi ∂pj j=1 j=1

where the identity fi (p(t, p0 )) = −

n−1 X j=1

pj (t, p0 )

∂fj (p(t, p0 )) ∂pi

comes from the first derivative of the equation in Walras’ law with respect to pi . Next we split the sums over j into two separate sums - one over j ∈ P (p(t, p0 ))

4. Stability of competitive equilibrium

38

and the second over j ∈ / P (p(t, p0 )) and leave out cancelling terms. So X

V 0 (t, p0 ) = −

X

[−p0i (t)

i∈P (p(t,p0 )) j ∈P / (p(t,p0 ))

X

X

pi (t)

i∈P (p(t,p0 )) j ∈P / (p(t,p0 ))

∂fj (p(t, p0 )) pj (t, p0 )+ ∂pi

fi (p(t, p0 )) 0 pj (t)] ≤ 0, ∂pj

because the first term in the sums is positive, thus with minus sign negative and the second term is also negative. The above theorem tells us, that if the derivative exists, the function V (t, p0 ) is non increasing. We would like to show that the function is non increasing even in the points, where the derivatives does not exits. Note that in these points, there exists the right hand derivative and the left hand derivative, but they are not equal. The proof of this claim is somewhat technical. Proposition 4.10. For small h > 0, V (t + h, p0 ) ≤ V (t, p0 ). Proof. For simplicity, we write p(t) = p(t, p0 ) and V (t) = V (t, p0 ). We already solved this problem for the times t, where the function V (t, p0 ) has own derivative. Now define the sets Q(t) = {i : i 6= n, fi (p(t)) ≥ 0}, Q1 (t) = {i ∈ Q(t) : fi (p(t)) > 0} and finally Q2 (t) = Q(t)\Q1 (t). From the continuity of the excess demand function, we know that Q1 (t) ⊂ Q1 (t + h) for some small h > 0. The functions V (t) may lack derivatives, because there may be i ∈ Q1 (t + h) such that i ∈ / Q1 (t) ⇒ i ∈ Q2 (t). Suppose that the derivative does not exist at time tˆ. We can write V (tˆ + h) − V (tˆ) = fn (p(tˆ + h)) − fn (p(tˆ))+ +

X

pi (tˆ + h)fi (p(tˆ + h)) −

i∈Q(tˆ+h)

X

pi (tˆ)fi (p(tˆ)).

i∈Q(tˆ)

By defining Q3 (t) = {i : i ∈ Q2 (t) ∩ Q1 (t + h)} for small h > 0, we can further rewrite the right hand side of the equation in a following way: fn (p(ˆ(t) + h)) − fn (p(tˆ) +

X

[pi (tˆ + h)fi (p(tˆ + h)) − pi (tˆ)fi (p(tˆ))]+

i∈Q1 (tˆ+h)

+

X

[pi (tˆ + h)fi (p(tˆ + h)) − pi (tˆ)fi (p(tˆ)] −

i∈Q3 (tˆ)

X i∈Q2 (tˆ)\Q3 (tˆ)

pi (tˆ)fi (p(tˆ)).

4. Stability of competitive equilibrium

39

From the definition of Q2 , the last term equals zero. After simplifying the last term, we get that: X

V (tˆ + h) − V (tˆ) =

[pi (tˆ + h)fi (p(tˆ + h)) − pi (tˆ)fi (p(tˆ))]

i∈Q1 (tˆ)∪Q2 (tˆ)∪{n}

and after dividing the equation by h and making a limit passage for h → 0+ , it can now be seen that V+0 (tˆ)

=

X

[p0i (tˆ)fi (p(tˆ))

+ pi (tˆ)

n−1 X ∂fi (p(tˆ)) j=1

i∈Q1 (tˆ)∪Q2 (tˆ)∪{n}

∂pj

p0j (tˆ)].

As was shown in the previous theorem, the right hand side is negative, when the price is different from equilibrium and it is non positive in general. Hence V (t) is monotone and non increasing ∀t ∈ [0, ∞). Because the function V (p(t, p0 )) is monotone, non increasing and bounded from below, we know that there exists a limt→+∞ V (p(t, p0 )). Since prices and excess demand functions are continuous V (p) is also continuous. We derived that the function V (p(t, p0 )) satisfies all the properties of a Lyapunov function. One of the last theorems on the way to prove global stability of general model tells us, that the solution of the system p0 = f−n (p) is bounded. Proposition 4.11. The solution p(t, p0 ) to p0 = f (p) is bounded and further there exists δ1 , ..., δn > 0 such that pi (t, p0 ) ≥ δi > 0 for all i ∈ {1, ..., n}. Proof. Because pn (t, p0 ) = 1 for all t, so the conditions are trivially satisfied. For the rest suppose that the converse is true, so kp(t, p0 )k → +∞ as t → +∞. This implies that there exists i ∈ {1, ..., n − 1} and sequence {ts }∞ s=1 p(t,p0 ) such that pi (ts , p0 ) → +∞. Define g(t, p0 ) = pi (t,p0 ) , so gi (t, p0 ) = 1 for all t ∈ [0, +∞). Having pn (t) = 1, ∀t ∈ [0, +∞) implies gn (ts , p0 ) → 0. By n n P P assumptions (E3) and (E4), the sum fi (q(ts , p0 )) = fi (p(ts , p0 )) → +∞. i=1

i=1

Then V (t, p0 ) → +∞, but at the same time V (t, p0 ) ≤ V (0, p0 ) < +∞ and that is a contradiction. The second part is done in almost the same manner as the first part. Suppose it does not hold. Take arbitrary fixed i ∈ {1, ..., n − 1} and suppose that there exists a sequence {ts }∞ s=1 such that pi (ts , p0 ) → 0. Define q(t, p0 ) = pi (t, p0 )p(t, po ), so qn (ts , p0 ) → 0 as s → +∞. The rest is the same, so the prices are bounded away from zero.

4. Stability of competitive equilibrium

40

One can now see, that we are almost done, because the function V (p(t, p0 )) = V (t, p0 ) satisfies all the properties of the Lyapunov’s function, which was described at the end of the third chapter. Further we know that the solution is bounded by the last proposition, so the condition in Lyapunov’s theorem about stability is satisfied. The theorem implies, that the system is quasi globally stable, but since the equilibrium is isolated, it follows from the Proposition 3.3, that it is globally asymptotically stable. We now state the result about global stability of the system p0 = f−n (p) as a separate theorem. Theorem 4.2. (Global stability of market equilibrium) Under the assumptions of (GS), zero homogeneity, (BC) and (WL), the equilibrium point p∗ = (p∗1 , ..., p∗n−1 , 1) (which is unique) of the system p0 = f−n (p) is globally asymptotically stable. By this theorem, we can conclude our analysis of stability of competitive equilibrium. To sum it up, at first we explained why it is appropriate to analyse stability with one good as numeraire. Then we showed what the necessary conditions for local asymptotic stability are and derived the sufficient conditions. Finally we proved that the equilibrium is uniquely determined under the economic assumptions of gross substitutability and zero homogeneity and then we gave the proof of global asymptotic stability of the equilibrium. It turns out, that the weak axiom of revealed preference, Walras’ law, gross substitutability and zero homogeneity of excess demand function plays a major role in the analysis of stability.

Chapter 5 Conclusion In this bachelor thesis, we dealt with stability of the equilibrium of competitive economy under the price adjustment process of tatonnement. Our main question was whether and under which assumptions is the equilibrium locally and globally asymptotically stable. Since the topic has a deep mathematical background, we tried to explain all aspects of the analysis in detail and link it to the economic theory and intuition. After short introduction to the topic, we presented the general model and the pure exchange model in the second chapter and we went through the economic and mathematical background needed for a proper treatment of stability of the equilibrium in the third chapter. In the fourth chapter, we made a brief literature review and then moved on to the analysis of stability. The necessary assumptions for local stability were derived and further we proved under which assumptions the equilibrium is locally asymptotically stable. Finally, the sufficient conditions for global stability of the market are presented and stability of the general model of competitive economy is proved under certain microeconomic assumptions. The thesis can have a few possible extensions. We could try to take another approach to global stability, than the one we made in the thesis. Stability of non competitive markets can be studied or we can focus our attention on the non-tatonnement pure exchange models. We may conclude, that the analysis of stability of the equilibrium of competitive economy is nevertheless an interesting topic of microeconomic analysis.

Bibliography Arrow, K. & L. Hurwicz (1958): “On the Stability of the Competitive Equilibrium, I.” Econometrica 26: pp. 522–552. Hahn, F. (1958): “Gross Substitutes and the Dynamic Stability of General Equilibrium.” Econometrica 26: pp. 169–170. Hicks, J. (1946): Value and Capital. Oxford, United Kingdom: Claredon Press. Kenneth J. Arrow, H. D. Block, L. H. (1959): “On The Stability of the Competitive Equilibrium.” Econometrica 27: pp. 82–109. Mukherji, A. (2003): “Competitive equilibria: Convergence, cycles or chaos.” Monograph p. 277. Mukherji, A. (2008): “Stability of a competitive economy: A reconsideration.” International Journal of Economic Theory 4(2): pp. 317–336. Mukherji, A. (2010): “Stability of the Market Economy in the Presence of Diverse Economic Agents.” JICA-RI 8: pp. 1–41. Negishi, T. (1958): “A Note on the Stability of an Economy Where All Goods are Gross Substitutes.” Econometrica 26: pp. 445–447. Negishi, T. (1962): “The stability of a competitive economy: a survey.” Econometrica 30: pp. 635–669. Rudin, W. (1976): Principles of Mathematical Analysis. United States of America: McGraw-Hill, Inc. Samuelson, P. (1946): Foundations of Economic Analysis. Cambridge, United States of America: Harvard University Press.

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