Poznan University of Economics
Faculty of Informatics and Electronic Economy
Department of Mathematical Economics
PHD Dissertation
ABSTRACT
Equilibrium and stability in the competitive economy
with stocks
Monika Naskręcka
Advisor:
dr hab. Piotr Maćkowiak
Poznań 2015
2
General equilibrium theory for many years remained constantly in a focus of attention for economists
working on the mathematical theory of economy. L. Walras is recognized as precursor of this theory.
In his book, „Éléments d’économie politique pure” from 1874, he developed foundations of the production and exchange theory in the competitive economy and proposed mathematical model of such an
economy. He also defined the concept of equilibrium and raised the question whether the equilibrium
state in such an economy exists. He was looking for an answer by constructing simple mathematical
exchange model with two producers and three consumers. Walras called the process of price adaptation to their equilibrium level as tâtonnement process. He considered also the problem of stability of
equilibrium prices and he formulated the paradigm stating that the value of total demand is always
equal to the total value of the supply (known today as Walras’ law).
Meanwhile, many elements of the general equilibrium theory were developed independently by
many prominent economists – F. Edgeworth, S. Jevons, V. Pareto or A. Marshall.1 The interest in
that subject increased significantly in the twentieth century. In 1941, P. Samuleson criticized in his
article2 most of the existing papers on dynamic processes and stability of equilibrium price vector and
he presented mathematical models of price dynamics in which he used differential equations. Yet, there
still remained the crucial question - if there exists the equilibrium price vector in exchange economy
model. In 1954 there were published two papers at the same time - the one of K. Arrow and G. Debreu3
and the second written by L. McKenzi4, in which the authors presented (using mathematical models)
proofs of existence of equilibrium price vector.
Further research on general equilibrium theory focused primarily on the stability of equilibrium. The
first successful attempts were undertaken by K. Arrow and L. Hurwicz. In their papers5 they proposed
proofs of stability of the price trajectory in few exchange economy models. However, H. Scarf in
his article6 pointed out that those proofs contained some errors. As a response, Arrow and Hurwicz
published in 1962 an article7 in which they indicated some supplementary conditions under which their
proofs were correct. In their papers they were using Lyapunov stability theory.
1
F. Y. Edgeworth. Mathematical Psychics. C. Kegan Paul and Co., 1988.
A. Marshall. Principles of Economics. Macmillan and Co., London, 1920.
2
3
P. A. Samuelson. The stability of equilibrium: Comperative statics and dynamics. Econometrica, (9):97–120, 1941.
K.J. Arrow and G. Debreu. Existence of an equilibrium for a competitive economy. Econometrica, 22(3):265–290,
1954.
4
L. W. McKenzie. On equilibrium in Graham’s model of world trade and other competitive systems. Econometrica,
22(2):147–161, 1954.
5
K.J. Arrow and L. Hurwicz. On the stability of the competitive equilibrium I. Econometrica, 26:522–552, 1958.
K.J. Arrow, H.D. Block, L. Hurwicz. On the stability of the competitive equilibrium II. Econometrica, 27:82–109, 1959.
6
H. Scarf. Some examples of global instability of the competitive equilibrium. International Economic Review,
1(3):157–172, 1960.
7
K.J. Arrow and L. Hurwicz. Competitive stability under weak gross substitutability: Nonlinear price adjustment
and adaptive expectations. International Economic Review, 3(2):233–255, 1962
3
Assumptions introduced to the model by K. Arrow and L. Hurwicz in their last paper were, however,
extensively criticized by many economists. They were regarded to be not entirely correct, since the
exchange was allowed in them only at equilibrium prices and they did not take into consideration the
adjustment process of the prices. That gave rise to the new class of general equilbirium models - nontâtonnement processes, which described the exchange between consumers and producers in real time,
in particular also when prices were distant from equilibrium prices or they approached their equilibrium
level. First papers that pursued this topic were written by T. Negishi8 from 1961 and 1962, F. Hahn
and T. Negishi9 from 1962 and H. Uzawa10 from 1962. Authors considered simple exchange model, in
which consumers are the only participants and they want to exchange goods. Transactions occurred
at any time, regardless of whether there were equilibrium prices or not. These economists pointed out
that there are no unstable points in their models (which occurred in previous works). They proposed
simple exchange models, in which they described the dynamics of prices and the quantity of goods
owned by consumers. What is more, they conducted a proof of stability of the equilibrium price vectors
using Lyapunov method.
In the subsequent years the general equilibrium theory developed, however, at a slower pace. Several
economists, such as H. Scarf, F. Ackerman, A. Kirman, regarded that general equilibrium theory
featured the exchange process in a way far from reality, that assumptions used in it were too strong
and that without them there was no proof of the stability of equilibrium price vector. At the turn
of the twentieth and the twenty-first century the problem of general equilibrium theory was resumed
once more. Economists saw a great potential in non-tâtonnement models. F. Fisher in his papers11
tried to construct the exchange model with both consumers and producers. He assumed, though, that
producers start the production only when the economy is in equilibrium. Only then consumers get
goods that they bought in the past. However, Fisher has changed commonly accepted definition of the
equilibrium and did not provide any proof of existence or stability of the equilibrium price vector.
8
T. Negishi. On the formation of prices. International Economic Review, 2(1):122– 126, 1961.
T. Negishi. The stability of a competitive economy. A survey article. Econometrica, 30(4):1635–669, 1962.
9
F. Hahn and T. Negishi. A theorem on non-tâtonnement stability. Econometrica, 30(3):463–469, 1980.
10
H. Uzawa. On the stability of Edgeworth’s barter process. International Economic Review, 3(2):218–232, 1962.
F. M. Fisher. On price adjustment without an auctioneer. The Review of Economic Studies, 39(1):1–15, 1972.
11
F. M. Fisher. The Hahn process with firms but no production. Econometrica, 42(3):471–486, 1974.
F. M. Fisher. A non-tâtonnement model with production and consumption. Econometrica, 44(5):907–938, 1976.
4
In the twenty-first century many papers were published with various non-tâtonnement models, e.g.
G. Giraud12, D. Katzner13, M. Kitti
14
, Y. Balasko15 or E. Panek16. However, none of those papers
contained a complete description of exchange between consumers and producers in the economy. They
were usually focused on the simple exchange between consumers and did not present any proofs of
existence or stability of the equilibrium price vector.
In this Ph. D. thesis we present a new approach to the concept of a competitive economy with
non-tâtonnement models. We present an exchange model in which producers can accumulate stocks if
they produce more goods than consumers want to buy. Those stocks can be then used in the future,
when demand will exceed their current production.
The main aim of this thesis is to introduce new concept of competitive economy model using nontâtonnement approach. The following subsidiary aims are essential to the realisation of the main
goal:
(1) Construction of models of competitive economies with stocks – we want to present mathematical models that would describe the dynamics of prices in the competitive economy. In those
models, when the supply exceeds demand, producers can accumulate stocks (which are not
included in classical models). We present models with stocks, which are described also with
the system of differential equations.
(2) Equilibrium and stability study – the proposed models were examined from the point of view
of the existence of price equilibrium and conditions for stability of competitive economy with
stocks. Our goal is to adequately describe processes in competitive economy in a way that
would be closer to reality and based on dynamic systems theory. It is also meant to show that
such an extension, proposed on the basis of new generation of models, does not deprive the
economy of its two basic attributes: equilibrium and stability.
(3) Simulations for the propose competitive economy model – on the basis of the proposed model
without costs of storage, simulations were performed in order to define a trajectory for prices of
goods, demand, supply/productions and stocks. The goal behind this part of the project was
to create a visualization of the trajectory for the above mentioned variables and to compare
consumers’ expectations with reality.
12
G. Giraud. From non-tâtonnement to monetary dynamics within general equilibrium theory: the limit-price ex-
change process. CNRS, 2009.
13
D. W. Katzner. The current non-status of general equilibrium theory. Review of Economic Design, (14):203–219,
2010.
14
M. Kitti. Convergence of iterative tâtonnement without price normalization. Journal of Economic Dynamics and
Control, 34:1077–1091, 2010.
15
Y. Balasko. Out of equilibrium price dynamics. Economic Theory, 33(3):413–435, 2007.
16
E. Panek. System Walrasa i zapasy. Przeglad statystyczny, (58):195–204, 2011.
5
We present two models, in which we describe the behaviour of consumers and producers in the
competitive economy. In the first of these models producers do not bear the costs of accumulating
stocks. Consumers’ incomes come from their share in companies’ profits and from labour that they
offer to producers. In the second model producers can accumulate stocks, yet they have to pay for
it. Introduction of those costs results in the change of producers’ behaviour. They are more inclined
to sell stocks in the first place and only then to produce goods. It affects the form of the current
supply function. The dynamic model, nevertheless, remains not changed. For both models we present
proofs of existence of demand and current supply function, their continuity and some of their additional
properties. We prove that the equilibrium price vector exists, we indicate the assumptions under which
the prices are uniquely determined (with accuracy to the vector structure) and their properties, such
as Walras’ Law or theorems of welfare economics.
Described models differ significantly from other models presented so far in the literature. We
describe in both of them prices and stocks dynamics. The price dynamics is very similar to the
one introduced in classical models. The stocks dynamics is describes with the system of differential
equations with discontinuous right–hand side:
ṗi (t) = σFi (p(t), φ(t)) ,
φ̇i (t) = Hi (p(t), φ(t)),
where i = 1, 2, . . . , n + 1 and
• t - a moment in time (time is continuous), t ∈ R+ ;
• σ > 0 - positive coefficient of price reaction to disequilibrium;
• p(t) = (p1 (t), . . . , pn (t), pn+1 (t)) - price vector in economy at time t;
• φ(t) = (φ1 (t), φ2 (t), . . . , φn (t), 0) - aggregated stock of the ith good at time t;
• Fi (p(t), φ(t)) = fi (p(t), φ(t)) − gi (p(t)) − φi (t) - excess demand function of the ith good in the
economy at time t, i = 1, 2, . . . , n, n + 1;
• f (p(t), φ(t)) = (f1 (p(t), φ(t)) , . . . , fn (p(t), φ(t)) , fn+1 (p(t), φ(t))) - aggregated demand for
ith good in the economy at time t;
• g(p(t)) = (g1 (p(t)), . . . , gn (p(t)), gn+1 (p(t))) - aggregated production of the ith good in the
economy at time t;
6
and
Hi (p(t), φ(t)) =








 −Fi (p(t), φ(t)) − (1 + β)φi (t),
when φi (t) ∈ (0, Ai ) or
(φi (t) = 0 i Fi (p(t), φ(t)) + φi (t) ≤ 0)
or (φi (t) = Ai








and Fi (p(t), φ(t)) ≥ −(1 + β)φi (t))
0
in other cases,
i = 1, 2, . . . , n, β ∈ (0, 1) is the stock depreciation rate i Hn+1 (p(t), φ(t)) = 0.
In t = 0 goods’ prices and stocks quantities are given
pi (0) = p0i > 0, i = 1, 2, . . . , n + 1,
φi (0) = φ0i ∈ [0, Ai ], i = 1, 2, . . . , n,
φn+1 (0) = 0,
where Ai > 0 is the maximum allowable level of i good’s in economy.
The level of stocks is affected by the difference between the supply of goods (reduced by stocks
depreciation) and the demand. Proposed models are, to our knowledge, the first in the literature
that describe the stocks dynamics with a specific function with a clear economic interpretation. We
prove the several lemmas and theorems that establish necessary and sufficient conditions for the global
asymptotic stability of a competitive economy. We illustrate both models with simple examples of
the economy with two producers and three consumers and we present the results of simulations of the
model without costs of storage.