# Selected Research Results

Research at the Laboratory is organized in three major directions:

1. Development of economical and mathematical tools, including optimization theory, the theory of general economic equilibrium and game theory;

2. Development of the general theory of institutional change in a catching-up economy;

3. Investigation of the evolution of institutions and evaluation of reforms in different sectors of the Russian economy.

Additionally, several staff members conduct research in pure mathematics. Some results directly related to economic problems are described below.

## 1. Development of economical and mathematical tools

Since 1970s, older generation of laboratory researchers has been working on development and refinement of economical and mathematical tools, including optimization theory, the theory of general economic equilibrium and game theory.

Staff members
developed an original theory of resource redistribution (Polterovich, 1970, in
Russian); a new axiomatic basis of competitive equilibrium as a principle of efficient
distribution of resources was given (Polterovich, 1973, in Russian; Sotskov ^{1},
1987, in Russian); a number of general theorems on the existence of equilibrium
and core were proved (Danilov, Sotskov,1990, in Russian); a synthesis of classical
economic equilibrium and optimal growth theories was suggested (Polterovich, 1986);
new conditions for stability of a competitive market were obtained (Polterovich,
Mityushin, 1978, in Russian; Polterovich, Spivak ^{2},
1983). The notion of generalized economic equilibrium was suggested, a theorem
on its existence was proved (Danilov, Sotskov, 1990). Conditions for the existence
and stability of equilibrium in nonlinear models of individuals' interaction in
social groups were found (Efimov (with co-author), 1998, in Russian). Significant
contributions to the theory of equilibrium under non-equilibrium prices were made
(Braverman, M. Levin, 1981, in Russian; Polterovich, 1980, 1990, in Russian; Polterovich,
1993). On the basis of this theory, a model of a mixed economy was developed.
The model explained facts of differentiation in sales tax rates and in sectoral
efficiency rates and provided a rationale for the "Chinese model" of
gradual transition to market (Polterovich, 1990, in Russian).

A theory of discrete convexity, which allows analyzing equilibria on markets of intellectual products and on markets of indivisible goods, was developed. A general discrete fixed point theorem, which is based on the notion of "proximity" that substitutes topology in a discrete case, was proved. For models with indivisible goods, it was shown that if utility functions are submodular or induce demand, which has a special property of polymatroidity, then the equilibrium exists. Polymatroidity is closely related to gross substitutability of demand, and submodularity of utility functions results in complementarity of goods. These facts allowed to derive known theorems and to prove a number of new statements on the existence and uniqueness of equilibrium prices with a single method (Danilov, Koshevoy, Sotskov, 1993, in Russian; Danilov, Koshevoy, Murota, 2001; Danilov, Lang, 2001, in Russian; Danilov, Koshevoy, 2004; Koshevoy, Talman, 2006; Danilov, Koshevoy, 2007, in Russian).

Duality theory for the Monge-Kantorovich problem în mass transfer was developed (V. Levin, 1975, 1985, 1997, in Russian); its applications to demand and utility theories, economical dynamics, etc., were considered. In particular, a generalization of the Debreu theorem on the existence of a continuous utility function for a wide class of preference relations without traditional assumption on their completeness was proved. A general theorem on universal continuous utility was also derived (V. Levin, 1983, in Russian; V. Levin, 1990). It was shown that rationalizability of demand by concave utility functions is equivalent to non-emptiness of a set of constraints of a dual Monge-Kantorovich problem with a special cost function and with a fixed difference of marginal measures. The specified criterion of concave rationalizability generalizes the Afriat-Varian nonparametric method. A general form of concave utility functions, which rationalize a given demand function, was described. Conditions for rationalizing demand functions by strictly concave and positively homogeneous functions were obtained (V. Levin, 2005). On other applications, see V. Levin, 2001, in Russian; V. Levin, 2009.

A condition on the blocking power of coalitions (almost-additivity), which guarantees non-emptiness of the core under any profile of agents' preferences and results in a hierarchical organization of blocking coalitions, was suggested (Danilov, Sotskov, 1988, in Russian; Danilov, Sotskov, 2002). General conditions for implementing social choice mechanisms by Nash equilibria were obtained (Danilov, 1991). A game-theoretical model of forming opinions of individuals in a reference group was developed (Efimov et al., 2000, in Russian).

A new approach for describing evolution of industries
was suggested. It was based on considering technological shifts as results of
interaction of processes of creation and adoption of technologies. The base model
represents a difference-differential analogue of Burgers' equation. Its solution
can consist of one or several nonlinear waves, moving at different velocities,
which reflects disintegration of the industry on subindustries with different
efficiencies. Description of a stationary system of waves was obtained; stability
of the system was proved. Conditions for the uniqueness of a wave were derived,
and wave's properties were determined (Polterovich, Henkin, 1988, in Russian;
Henkin, Polterovich 1991, 1999; Henkin, 2007).

Multidimensional generalization
Lorenz curve, which is widely used for measuring inequality, was suggested and
studied. Developed theory allows constructing inequality indices, which take into
account different characteristics of individuals in the studied population (Koshevoy,
2005).

In a "principal - agent" model with a set of agents, contract schemes stable relative to coordinated actions of agents were analyzed. Conditions for existence of strong equilibrium in supermodular games were obtained as corollaries; a characterization of rational strategies of fighting corruption and tax enforcement was given (Savvateev, 2003, in Russian).

Several game-theoretical models of forming a system of club goods (jurisdiction formation) were studied. For the case of one-dimensional differentiation, full description of Nash stable structures and partial description of coalitionally stable structures were obtained (Savvateev et al., 2008).

## 2. Development of the general theory of institutional change in a catching-up economy

The central task in this research direction is the development of conceptual and mathematical tools for studying the dynamics of economic institutions and policies and for elaborating the principles of their reforming. A great deal of results on this topic is presented in a monograph by Polterovich (2007, in Russian). Within the framework of the developed approach, reform is considered not to be a single action or even a package of laws, but a realization of an institutional trajectory, a sequence of intermediate institutions, which leads to the formation of more efficient system of interaction between economic agents. A classification of institutional dysfunctions was suggested, and a theory of institutional traps, which explains the formation of stable equilibrium, but inefficient, norms of behavior, and considers mechanisms of transition to efficient norms, was elaborated (Polterovich, 1999, in Russian; Polterovich, 2008). Another important element of the developed approach is the theory of institutional transplantation, which studies principles and technologies of transferring institutions from one institutional environment into another (Polterovich, 2001, 2007, in Russian). Its main hypotheses found support in empirical study of 125 episodes of transplantation of mortgage institutions around the world during 230 years (Starkov, 2004, in Russian; Polterovich, Starkov, 2007, in Russian).

A concept of relative stages of development was elaborated. Corresponding theory attempts to answer the question on how instruments and methods of stimulating economic growth should change as the quality of institutions and country's welfare approach the level of advanced economies. Evolution of economic policies used by "economic miracle" and western countries during the period of their catching up development was described (Polterovich et al., 2006à, b, in Russian).

A study of peculiarities of development of resource-based economies was conducted. A concept of "conditional curse", according to which an increase in the volume of concentrated resources can have either positive or negative effect on economic growth depending on the quality of institutions and economic policy, was grounded. Mechanisms generating inefficiency in the use of resources were considered (Polterovich et al., 2007, in Russian). Hypothesis that in resource-abundant economies "resource curse" can arise because of underdevelopment of human capital was supported empirically (Volchkova, 2007).

## 3. Investigation of the evolution of institutions and evaluation of reforms in different sectors of the Russian economy

A strategy of forming a mortgage market in Russia was suggested on the basis of transplantation theory (Polterovich, Starkov, 2007, in Russian).

An empirical study (Ì. Levin, Satarov, 2000) evaluated the scale of corruption in Russia and outlined measures required for fighting corruption. Special attention was paid to the problem of corruption on the market of educational services (M. Levin et al., 2006, in Russian).

Computable general equilibrium and simulation models were developed to analyze tax reform, monetization of social benefits and housing and communal services reform (Volchkova et al., 2004, 2006, in Russian). The level of technological complexity of the Russian export was estimated, possible ways of export diversification were explored (Volchkova et al., 2008, in Russian).

Retraining programs for the unemployed were evaluated, recommendations for their improvement were made. Relationship between the quality of education and income level on the Russian labor market was analyzed (I.A. Denisova et al., 2002, 2006, in Russian).

Modern industrial policy trends in new market economies were investigated. An analysis of development of special economic zones was given (Andryushkevich, I.M. Denisova, 2004, in Russian; Andryushkevich, 2008, in Russian). Dynamics of PC components market was studied (Varshavsky, 2004, in Russian).

^{1} Aleksander Sostkov
was a Laboratory researcher since its establishment. He died tragically in a car
accident in 2004.

^{2 }Vladimir
Spivak worked at the Laboratory since its establishment until 2007.