Let us assume that there are "n" parameters describing the economy under consideration (n is also assumed to be constant). The values of these parameters will be labeled x1,(,xn. We assume that parameters have numerical values. The set (x1,(,xn) is said to be a state of economy or shortly a state. The space Rn is said to be generalized space of states. Let us denote by S( Rn the subset corresponding to feasible states. The set S is said to be a set of states of economy. Let us denote by T0 the initial time moment and assume that the economy is investigated during period from initial moment to the moment T. Let us consider the interval [T0, T] on the real axis R. Any mapping x(t) is said to be generalized trajectory of economy.
It is assumed that the current value of parameters can be explained using knowledge on the past (i.e. past values of parameters). More precisely one assumes that the time rate of change of a parameter depends on values of parameters in precedent time moment. Under technical assumptions on differential regularity of generalized trajectories, one can formally write the last condition in form of the ordinary differential equation. This equation is said to be the equation of state and solution of the equation of state is said to be trajectory of economy or its evolution. The set E of solutions of the equation of state is said to be the set of evolutions.
A mapping R:S(E is called the resolvent operator if J(s)(T0)=s, i.e. if it assigns to initial state the evolution starting in this state.
Example 1. Let us consider the world (and economy) in which only one limited asset is consumed according to the following rule: creatures which live in this world consume each time one tenth of what is left. The world is likely to disappear and the state of economy in time moment t is described by the number x(t). The state equation is therefore x'(t) =-0.1x(t). The resolvent operator R for this economy is to be obtained from the formula for the resolvent of the ordinary differential equation: R(x0)=x0exp[-0.1(t-T0)].
Example 2. Let us consider a mapping J:E-->R. Values of J will be called fitness of evolution. The resolvent operator R for this economy is to be obtained from the formula R (x0) = Arg max {J(e)}. This example is an allusion to modeling economic dynamics using optimal control theory. Approaches using generalization of the calculus of variation provide a variety of examples of evolution fitness. Resolvent operators are generally not given in a formula but as algorithm for resolving Euler-Lagrange type of equations.
This terminology allows to reformulate general problem of desire of description of parameters' change in time to the problem of definition of the space of evolutions of economy and finding resolvent operators. For the sake of simplicity a pair (E,R) is said to be the theory (of the economy).
Three examples of theories of economies in transition are presented in the paper. The first deals with normal state at the labor market. In 1945-1989 unemployment in Poland doesn't exist in official statistics. Temporary perturbations on labor market were considered to be momentary inaccuracies in management. In fact, a centrally planned economy has to cope with hidden unemployment. In the late 80's hidden unemployment reached approximately 25% of the total employment. There exists also reverse phenomenon (hidden employment). Employees of state-owned enterprises took up some extra work for private clients, often within working hours and with the use of materials and tools belonging to the enterprise. This market had its own its actors and prices and is referred to in the sequel as the gray zone. Since unemployment is related with waste of human capital considered to be the main factor of economic growth, and gray zone is related with loss of efficiency and ambiguity of relations underlying management, therefore the problem of description, understanding and forecasting the labor market dynamics plays fundamental role in creation successful policy of transformation. In particular identification of sources of unemployment and gray zone and investigation of possibilities to eliminate these phenomena seem of extreme importance. To this end the model of labor market is considered. This model fits the scheme presented above. The solution of the system of differential equations describes the changes in numbers of members within the group unemployed, legally employed and gray zone employees. It is proved that market trajectories asymptotically tend to point of rest called normal state. The model may serve as the tool for descriptive analysis aimed at studying microeconomic basis of the processes in the macro dimension. The model based upon the theory of shortage (in this particular case the shortage considers the demand for labor force) can be adopted to the research of problems of the market economy imbalance. On the other hand, the model can be used in predicting future tendencies on the labor market in terms of the volume of unemployment and black labor market.
The second example deals with the model of reaching equilibrium at the product and money market. Aggregated economy is considered and differential equations are employed. Demand equation constrains money, production, rate of interest, money supply and inflation. Model serves to explain reasons of existence of unequlibrium in 1989/1990. Demand for products was greater then supply and money supply was big and increasing. Simultaneously inflation and discount rates were growing. In January 1990 using restrictive monetary and fiscal policies the inflation was decreased and simultaneously market price mechanisms were introduced. Real money supply, discount rate and inflation decreased but growth was decreasing too. Observed later dynamics can be compared with forecast resulted from the model analysis. The interesting result is estimation of a stable equilibrium. Analysis of variables allows to understand their mutual interrelation. The explanation obtained is however not 100% precise since unemployment description is omitted. Even partial knowledge of solution allows to perform sensitivity analysis in terms of parameters that have operational interpretation. Another conclusion can be obtained from investigation of existence of (in)stable rest points interpreted as (un)equilibria. Since parameters are hardly available (short time series) one can try to fit data using computer simulation controlled by eigenvalues of characteristic equation.
The third example uses logistic curve to estimate dynamics of stock market in Poland. Instead of time series describing data from sessions, the change of price of NAV per unit in time for mutual fund Pioneer is investigated. Firstly, economic representation of the situation in terms of IS-LM model is performed. Two phases are identified. In the initial phase increase of money is small. The regular phase starts with increase in money supply which results in increasing of income, resigning from low-risk-small-return bank savings and increase of prices on stock market. Mathematical description uses variables addressing individuals and (logistic) differential equation model of time dependence of these variables is investigated. Two cases which include different perception of inflation are considered, and standard regression analysis is performed. There exists a surprisingly high compatibility of observed and forecasted data.
These three examples use in different ways the same methodology based on ordinary differential equations instead of econometric modeling. Thus, short data series do not disturb modeling. Partial descriptions of economy lead to acceptable compatibility with data. Also interesting research questions arise. Firstly, relatively simple models are investigated - still unresolved remains the problem of the creation of a unified model which will encompass money, product and labor markets (including sectors) and their interdependence. Secondly, models use nonhomogeneous concept of states. This, again, calls for unification. Thirdly, main results seem to be useful for forecasting equlibria. However, managerial meaning of equilibrium is not self-evident and requires addressing.
T.Szapiro; Division of Decision Analysis and Support,
Institute of Econometrics, Warsaw School of Economics (SGH)
Al. Niepodleglosci 162, 02-554 Warsaw, POLAND;
tel. +48-22-495094; fax +48-22 955312; e-mail: tszapiro@sgh.waw.pl