Mathematical economics

The mathematical discipline whose subject concerns models of economic objects and processes, and methods for investigating them. However, the concepts, results and methods of mathematical economics are conveniently and commonly expounded in close connection with their economic derivations, interpretations and practical applications. Of particular significance is the connection with the science and practice of economics.

Mathematical economics, as a part of mathematics, began only in the 1920's. Earlier there was only sporadic research which cannot be strictly attributed to mathematics.

Peculiarities of economic-mathematical modelling.[edit]

A peculiarity of economic modelling is the exceptional variety and diversity of the objects being modelled. In economics there are elements of controllability and spontaneity, rigid determinacy and essential ambiguity and freedom of choice, processes of technical nature, and social processes where human behaviour comes to the forefront. Different levels of economics (for example, shop economics and households) require essentially different descriptions. All this leads to a great diversity, in the models, of the mathematical apparatus. A delicate question is how to express the type of socio-economic systems which are modelled, taking account of the social structure. It often happens that an abstract mathematical model of some economic process or object can be successfully applied to both capitalist and socialist economies. This is accommodated in the method of utilization and interpretation of the results of analysis.

Production. Efficient production.[edit]

Economics deals with wealth, or products, which are understood in an extremely broad sense in mathematical economics. For this one applies the general terminology of ingredients (goods or commodities). Ingredients are services, natural resources, the unfavourable influence on man of environmental factors, characteristics of the comfort of a present security system, etc. It is usually assumed that the number of ingredients is finite and the space of products is ${\mathbf{R}}^l$, Euclidean space, where $l$ is the number of ingredients. A point $z$ from ${\mathbf{R}}^l$, under appropriate conditions, can be considered as a "production" method; positive components indicate the volume of output of the corresponding ingredients, and negative components the inputs. The word "production" is put between quotes because it is to be understood in a very broad sense. The set of available (given, existing) production possibilities is $Z\subset {\mathbf{R}}^l$. A method of production $\bar{z}\in Z$ is efficient if there is no $z\in Z$ such that $z\ge \bar{z}$, with strict inequality in at least one component. The problem of discovering efficient methods is one of the most important in economics. Usually it is assumed, and in many cases this agrees well with reality, that $Z$ is a compact convex set. By expanding the space of products the problem of the analysis of the efficient methods here may be reduced to the case where $Z$ is a closed convex cone.

A typical problem is the fundamental problem of production planning. Given a set of production methods $Z\subset {\mathbf{R}}^l$ and a vector of requirements and resource limitations $b\in {\mathbf{R}}^{l-}1$, it is required to find a method $\bar{z}=(b,\bar{\mu })\in Z$ such that $\bar{\mu }\ge \mu$ for all $(b,\mu )\in Z$. If $Z$ is a closed convex cone, then this is the general problem of convex programming. If $Z$ is given by a finite number of generators (the so-called basic methods), then this is the general problem of linear programming. A solution $\bar{z}$ lies on the boundary of $Z$. Let $\pi$ be the coefficients for the supporting hyperplane for $Z$ at the point $\bar{z}$, that is, $\pi z\le 0$ for all $z\in Z$ and $\pi \bar{z}=0$. The fundamental theorem of convex programming gives conditions under which ${\pi }_l>0$. For example, a sufficient condition is: There is a vector $(b,\mu )\in \mathrm{int}Z$( the so-called Slater condition). The coefficients of $\pi$, which characterize the efficient method $\bar{z}$, have an important economic meaning. They can be interpreted as prices commensurate with the efficiency of the inputs and outputs of the different ingredients. A method is efficient if and only if the cost of the outputs is equal to the cost of the inputs. The given theory of efficient production methods and their characterization using $\pi$ has exerted a revolutionary influence on the theory and practical planning of socialist economics. It has underpinned objective quantitative methods for the determination of prices and the social evaluation of resources, giving the possibility of choosing more efficient economic solutions under the conditions of a socialist economy. The theory generalizes naturally to an infinite number of ingredients. Then the space of ingredients is a suitably chosen function space.

Efficient growth.[edit]

Ingredients relating to different times or time intervals can be formally regarded as distinct. Therefore the description of production in dynamic form, in principle, is contained in the above scheme, which consists of objects $\{X,Z,b\}$, where $X$ is the space of ingredients, $Z$ is the space of production capacities and $b$ is the specification of requirements and restrictions on the economy. However, the study of the truly dynamical aspects of production requires a more special form of the description of the production capacities.

The production capacities of a sufficiently general model of economic dynamics are given via a point-to-set mapping (many-valued function) $a:{\mathbf{R}}_{+}^l\to 2^{{\mathbf{R}}_{+}^l}$. Here ${\mathbf{R}}_{+}^l$ is the (phase) space of the economy, $x\in {\mathbf{R}}_{+}^l$ is interpreted as the state of the economy at some time and $x_k$ is the available quantity of the product $k$ at that time. The set $a(x)$ consists of all states of the economy into which it may pass in unit time from $x$. One calls

$$Z=\{(x,y)\in {\mathbf{R}}_{+}^{2l}:y\in a(x)\}$$

the graph of the mapping $a$. A point $(x,y)$ is an admissible production process.

Different versions of the specification of possible trajectories of development of the economy have been considered. In particular, consumption by the population is allowed for either in the mapping $a$ itself, or explicitly. For example, in the second case an admissible trajectory is a sequence $(X,C)=(x(t),c(t+1){)}_{t=0}^{\mathrm{\infty }}$ such that $x(t+1)+c(t+1)\in a(x(t))$, $c(t)\ge 0$ for all $t$. Different concepts of efficiency of trajectories have been studied. A trajectory $(\bar{X},\bar{C})$ is efficient relative to consumption if there does not exist an admissible trajectory $(X,C)$ beginning from the same initial state for which $C\ge \bar{C}$. A trajectory $(\bar{X},\bar{C})$ is intrinsically efficient if there is no other admissible trajectory $(X,C)$, beginning from the same initial state, a time $t_0$ and a number $\lambda >1$, such that

$$\lambda \bar{x}(t_0)=x(t_0).$$

Optimality of a trajectory is usually defined depending on a utility function $u:{\mathbf{R}}_{+}^l\to {\mathbf{R}}_{+}^l$ and a coefficient for discounting utility in time $\mu \ge 1$( see below, and also Utility theory, for something on utility functions). A trajectory $(\bar{X},\bar{C})$ is called $(u,\mu )$- optimal if

$$\underset{\overline{t\to \mathrm{\infty }}}{lim}(\sum _{\tau =0}^{t}u(\bar{c}(\tau )){\mu }^{-\tau }-\sum _{\tau =0}^{t}u(c(\tau )){\mu }^{-\tau })\ge 0$$

for any admissible trajectory $(X,C)$ starting from the same initial state. There is a number of quite general existence theorems for the corresponding trajectories.

Trajectories which are efficient in different senses are characterized by a sequence of prices in exactly the same way as an efficient method $\bar{z}$ is characterized by the prices (the coefficients of the supporting hyperplane) $\pi$. That is, if for an efficient method the cost of input is equal to the cost of output at optimal prices, then on an efficient trajectory the cost of a state is constant and maximal, and on all other admissible trajectories it cannot increase.

All of these definitions are easily generalized to the case when the production mapping $a$, the function $u$ and the coefficient $\mu$ depend on time. Time itself may be continuous or, more generally, the parameter $t$ may run through a set of quite arbitrary form.

From the economic point of view the interest is in trajectories which attain the maximum possible rate of economic growth and which can be sustained for an arbitrarily long time. It turns out that for $a$ and $u$ which do not vary in time such trajectories are stationary, that is, have the form

$$x(t)=x(0){\alpha }^t,c(t)=c(0){\alpha }^t,$$

where $\alpha$ is the rate of growth (the expansion) of the economy. Stationary efficient, in some sense, and also stationary optimal, trajectories are called turnpike trajectories.

Under very broad assumptions, a theorem on turnpike trajectories asserts that every efficient trajectory, independent of the initial state, as time goes on approximates a turnpike trajectory. There is a large number of different theorems on turnpikes, which differ in their definitions of efficiency and optimality, the means of measuring the distance from a turnpike, the type of convergence, and, finally, on whether finite or infinite time intervals are involved.

The model of economic dynamics, in which production capacities are given by a polyhedral convex cone, is called the von Neumann model. A particular case of the von Neumann model is the closed Leont'ev model, or (in other terminology) the closed dynamical interdepartmental balance model (the term "closed" is used here as a characteristic property of economics without non-reproducible products), which is given in terms of matrices $\mathrm{\Phi }$, $A$ and $B$ of order $l\times l$ with non-negative entries. A process $(x,y)\in Z$ if and only if vectors $v,w\in {\mathbf{R}}_{+}^l$ can be found such that

$$x\le v\mathrm{\Phi },v\ge vA+wB,y\le v\mathrm{\Phi }+w.$$

The model of interdepartmental balance is more widespread because of the convenience in obtaining the initial information for its construction.

Models of economic dynamics are also discussed in continuous time. In fact, the first models to be studied were precisely models with continuous time. In particular, several works were devoted to the simplest one-product models, given by an equation

$$\dot{x}=f(x)-c,$$

where $x$ is the volume of stock per unit of labour resource, $c$ is the requirement per head of population and $f$ is the production function (increasing and concave). Non-negative functions $(x(t),c(t){)}_{t=0}^{\mathrm{\infty }}$ satisfying this equation characterize an admissible trajectory. For a given utility function $u$ and discount coefficient $\mu$ an optimal trajectory can be determined. Optimal trajectories (and only they) satisfy an analogue of the Euler equation:

$$u(x)\dot{x}=u(\bar{c})-u(c),$$

where $\bar{c}$ is the largest number satisfying the condition $f(x)-c=x$.

The Leont'ev model was also initially formulated in continuous time as a system of differential equations

$$X=AX+B\dot{X}+C,$$

where $X$ is the stream of products, $A$ and $B$ are the matrices of current and capital expenses, respectively, and $C$ is the stream of finite requirements.

Efficient and optimal trajectories in models with continuous time are studied with the help of the methods of variational calculus, optimal control and mathematical programming in infinite-dimensional spaces. Models whose admissible trajectories are given by differential inclusions of the form $\dot{x}\in a(x)$, where $a$ is the production mapping, are also discussed.

Rational behaviour of consumers.[edit]

The tastes and goals of consumers, which determine their rational behaviour, are given in the form of some system of preferences in the space of products. Namely, for each consumer $i$ there is defined a point-to-set mapping $P_i:Z\to 2^X$. Here $Z$ is some subset of situations in which the consumer may find himself by the process of selection and $X$ is the set of vectors accessible to the consumer, $X\subset {\mathbf{R}}^l$. In particular, $Z$ may contain a subspace of ${\mathbf{R}}^l$. The set $P_i(z\mid x)$ consists of all vectors $\tilde{x}\in X$ which are (strictly) preferred to the vector $x$ in the situation $z$. For example, the mapping $P_i$ may be given by means of a utility function $u$, where $u(x)$ shows the utility to the consumer of the set of products $x$. Then

$$Z={\mathbf{R}}_{+}^l,P_i(z)=\{\tilde{x}\in {\mathbf{R}}_{+}^l:u(\tilde{x})>u(x)\}.$$

In the description of the situation $z$ the prices $\pi$ of all products and the monetary income $d$ of the consumer enter. Then $B_i(z)=\{x\in X:x\pi \le d\}$ is the collection of sets from which the consumer may choose in the situation $z$. This set is called the budget set. Rationality of consumer behaviour is to choose a set $x$ from $B_i(z)$ for which $P_i(z\mid x)\cap B_i(z)=\mathrm{\varnothing }$. Let $D_i(z)$ be the collection of sets of products chosen by consumer $i$ in situation $z$; $D_i$ is called the demand mapping (or demand function, if $D_i(z)$ consists of one point). There are many investigations devoted to clarifying the properties of the mappings $P_i$, $B_i$ and $D_i$. In particular, the case when $P_i$ is a function has been studied at length. Conditions under which the mappings $B_i$ and $D_i$ are continuous have been determined. Of special interest is the study of the properties of the demand function $D_i$. The fact is that sometimes it is more convenient to regard as primary only the demand functions $D_i$, and not the preferences $P_i$, since it is easy to construct them from the available information on consumer behaviour. For example, in an economy (a trade statistic) it is possible to observe quantities that approximate the partial derivatives

$$\frac{\partial D_{ik}(x)}{\partial {\pi }_{\rho }},\frac{\partial D_{ik}(x)}{\partial d},$$

where ${\pi }_{\rho }$ is the price of the product $\rho$ and $d$ is the income.

Bordering on the theory of rational behaviour of consumers is the theory of group choice (social choice), concerning, as a rule, discrete variants. It is usually assumed that there is a finite number of participants in a group and a finite number of, for example, alternatives. The problem lies in the choice of a group solution of the selection of one variant, given the preference between the alternatives of each participant. Group choice provides various voting schemes; here axiomatic and game-theoretic approaches are also used.

Agreement of interests.[edit]

The holders of interests are the individual parties of economic systems, and also society as a whole. As such parties one puts forward consumers (groups of consumers): enterprises, ministries, territorial organizations of administration, planning and financial organizations, etc. One distinguishes two mutually intertwined approaches to the problem of agreement of interests: the analytic, or constructive, and the synthetic, or descriptive. According to the first approach, initially there is a global criterion of optimality (a formalization of the interests of society at large). The problem is to derive the local (personal) criteria from the general one, taking account of personal interests. In the second approach there are just the personal interests, and the problem is to unify them into a single consistent system, the functioning of which leads to results which are satisfactory from the point of view of society as a whole.

Directly related to the first approach are the decomposition methods of mathematical programming. For example, in an economy let there be $m$ producers, and let each producer $j$ be given a set of production capacities $Y_j$, where $Y_j\subset {\mathbf{R}}^l$ is a compact convex set. Let there be given an objective function $V$ for the society at large, where $V:{\mathbf{R}}_{+}^l\to {\mathbf{R}}_{+}^l$ is a concave function. The economy must be organized in such a way that the following problem of convex programming results: Find $\bar{y}$ from the conditions $y\ge 0$, $y\in \sum _j{Y}_j$, $V(y)\to max$. From theorems on the characteristic behaviour of efficient production methods one may conclude that there are prices $p\in {\mathbf{R}}_{+}^l$ $(p\ne 0)$ such that

$$\bar{y}{}^{(}j)p=\underset{y^{(}j)\in Y_j}{max}{y}^{(}j)p\text{ for all }j,\bar{y}=\sum _j\bar{y}{}^{(}j).$$

The quantity $y^{(}j)p$ is interpreted as the profit of the $j$- th producer with prices $p$. Hence it follows that the criterion of maximizing the profits for each producer does not conflict with the common goal if the operating prices are defined in a corresponding form. The second approach has been strongly developed within the area of models of economic equilibrium.

Economic equilibrium.[edit]

Assume that the economy consists of individual parties having personal interests: producers listed by indices $j=1\dots m$, and consumers listed by indices $i=1\dots n$. Producer $j$ is described by a set of production capacities $Y_j\subset {\mathbf{R}}^l$ and a mapping $F_j:Z\to 2^{Y_j}$, giving his or her system of preferences. Here $Z$ is the set of possible states of the economy, made concrete below. Consumer $i$ is described by a collection of possible sets of products available for consumption, $X_i\subset {\mathbf{R}}^l$, an initial supply of products $w^{(}i)\in {\mathbf{R}}_{+}^l$, preferences $P_i:Z\to 2^{X_i}$ and, finally, functions ${\alpha }_i:Z\to \mathbf{R}$ of income distribution, where ${\alpha }_i(z)$ shows the amount of money available to consumer $i$ in the state $z$. The set of possible prices in the economy is $Q$. The set of possible states is $Z=\prod _i{X}_i\times \prod _j{X}_j\times Q$. The budget mapping $B_i$ is defined here as:

$$B_i(z)=\{\tilde{x}{}^{(}i)\in X_i:\tilde{x}{}^{(}i)p\le {\alpha }_i(z)+w^{(}i)p\}.$$

An equilibrium state of the described economy is a $\bar{z}\in Z$ satisfying the conditions

$$\sum _i\bar{x}{}^{(}i)=\sum _j\bar{y}{}^{(}j)+\sum _i{w}^{(}i),$$

$$\bar{x}{}^{(}i)\in B_i(\bar{z}),B_i(\bar{z})\cap P_i(\bar{z})=\mathrm{\varnothing },Y_j\cap F_j(\bar{z})=\mathrm{\varnothing }.$$

In essence, an equilibrium state of the economy is defined as a solution of a non-cooperative game with several players, in the sense of von Neumann–Nash, with the additional condition that there is a balance with respect to all products.

The existence of an equilibrium state has been proved under very general conditions on the initial economy. It is necessary to impose much stricter conditions to ensure that the equilibrium state be optimal, that is, is a solution to some global optimization problem with an objective function depending on the interests of the consumers. For example, let $P_i$ be given by a concave continuous function $u_i:{\mathbf{R}}^l\to {\mathbf{R}}_{+}$, and let $F_j$ be given by a function $p{y}^{(}j)$; let

$${\alpha }_i(z)=\sum _i{\theta }_{ij}{y}^{(}j)p,{\theta }_{ij}\ge 0,\sum _i{\theta }_{ij}=1,$$

$$Q=\{p\in {\mathbf{R}}_{+}^l:\sum _{k=1}^l{p}_k=1\},$$

where $Y_j$, $X_i$ are convex compact sets, $0\in Y_j$, $w^{(}i)\in \mathrm{int}{X}_i$. Any subset $S=\{i_1\dots i_r\}$ of indices of consumers forms a subeconomy of the initial economy, in which to each consumer $i_s$ from $S$ corresponds one (and only one) producer, the set of production capacities of which is

$${\hat{Y}}_{i_s}=\sum _{j=1}^m{\theta }_{i_s,j}{Y}_j.$$

The function of income distribution here has the form

$${\alpha }_{i_s}(z)=y^{(i_s)}p.$$

A state $z\in Z$ is called balanced if

$$\sum _i{x}^{(}i)\le \sum _i{y}^{(}j)+\sum _i{w}^{(}i).$$

One says that a balanced state $z$ of the initial economy blocks a coalition of consumers (cf. also Coalition) $S$ if in the subeconomy determined by the coalition $S$ there is a balanced state $\bar{z}{}^{(}s)$ such that $u_{i_s}(\bar{x}{}^{(i_s)})\ge u_{i_s}(x^{(i_s)})$, for $s=1\dots r$, and for at least one index the inequality is strict. The core of the economy is the set of all balanced states that do not block any coalition of consumers. For economies with these properties there is the following theorem: Every equilibrium state belongs to the core. The converse is false; however, there is a number of sufficient conditions under which the set of equilibrium states and the core are close to each other or coincide completely. In particular, if the number of consumers tends to infinity and the influence of each consumer on the states of the economy becomes ever smaller, then the set of equilibrium states tends to the core. The coincidence of the core and the set of equilibrium states holds in economies with an infinite (continuum) number of consumers (Aumann's theorem).

Let the economy be a market model (that is, there are no producers), the set of participants (consumers) of which is the closed unit interval $[0,1]$, denoted in the sequel by $T$. A state of the economy is $z=(x,p)$, where $p\in \{p\in {\mathbf{R}}_{+}^l:\sum _k{p}_k=1\}$ and $x$ is a function from $T$ into ${\mathbf{R}}_{+}^l$ each component of which is Lebesgue integrable over the interval $T$. The initial distribution of products among the participants is given by a function $w$, ${\int }_{T}w>0$, so that a balanced state $z$ is such that $\int x=\int w$. A coalition of participants is a Lebesgue-measurable subset of $T$. If the subset has measure zero, then the corresponding coalition is called null. The core is the set of all balanced states which do not block any non-null coalition. A state $\bar{z}=(\bar{x},\bar{p})$ is an equilibrium if for almost-all participants $t$,

$$u_t(\bar{x}(t))=max{u}_t(x(t)),$$

$$x(t)\in \{x:xp\le pw(t)\}.$$

Aumann's theorem asserts that in this economy the core and the set of equilibrium states coincide.

The question of the structure of the set of equilibrium states is particularly interesting when the set is finite or consists of one point. Here one has the theorem of Debreu. Let the set of market models be $W=\{(w^{(}i),D_i{)}_{i=1}^n\}$, where $w^{(}i)\in {\mathbf{R}}_{+}^l$ is the initial supply of products for the participants $i$ and $w=(w^{(}1)\dots w^{(}n))$ is a parameter defining a concrete model from the set $W$, $e\in {\mathbf{R}}_{+}^{n\cdot l}$. The mapping $D_i:Q\times M\to {\mathbf{R}}_{+}^l$ represents the demand function for the participant $i$. The functions $D_1\dots D_n$ are given (are fixed) for the whole set of economies $W$. Let $W_0$, $W_0\subset W$, be the collection of economies for which the set of equilibrium states is infinite. Debreu's theorem asserts that if the functions $D_1\dots D_n$ are continuously differentiable and if there are no points of saturation for at least one of the participants, then the closure of the set $W_0$ has (Lebesgue) measure zero in the space $W$.

On numerical methods.[edit]

Mathematical economics has a close connection with computational mathematics. Linear programming and linear economic models have exerted a great influence on the computational methods of linear algebra. Essentially because of linear programming, inequalities in computational mathematics have become as much used as equations.

The calculation of economic equilibria is a difficult problem, having many aspects. For example, much work has been devoted to conditions for convergence to an equilibrium for systems of differential equations

$$\begin{array}{}\text{(*)}& \dot{p}=F(p),\end{array}$$

where $p$ is the price vector and $F$ is the excess demand function, that is, the difference between the demand and the supply functions. Equilibrium costs $\bar{p}$ provide, by definition, equality of demand and supply: $F(\bar{p})=0$. The surplus demand function $F$ is given either directly, or by more primary concepts of the corresponding model of equilibrium. S. Smale [8] has studied a significantly more general dynamical system than (*), applied to a market model; alongside a variation in time of the prices $p$, a variation of the states $x$ is also considered; here an admissible trajectory $(p(t),x(t){)}_{t=0}^{\mathrm{\infty }}$ satisfies some differential inclusion of the form $\dot{p}\in K(p)$, $\dot{x}\in C(p)$, where $K(p)$ and $C(p)$ are the sets of possible directions of variation of $p$ and $x$, determined by the market model.

The economic equilibrium, the solution of a game, the solution of an extremal problem, all may be defined as a fixed point of an appropriate point-to-set mapping. Within the limits of research in mathematical economics, numerical methods for the computation of fixed points of various classes of mappings have been developed. The best known is Scarf's method, [6], which is a combination of the ideas of the Sperner lemma and the simplex method of solution of linear programming problems.

Related questions.[edit]

Mathematical economics is closely connected with many mathematical disciplines. Sometimes it is difficult to determine the boundary between mathematical economics and mathematical statistics or convex analysis, functional analysis, topology, etc. One only has to mention, for example, the development of the theory of positive matrices, positive linear (and homogeneous) operators and the spectral properties of superlinear point-to-set mappings, under the influence of the requirements of mathematical economics.

References[edit]

Comments[edit]

A classic on mathematical economics is [a8], and a useful general book from the optimization point of view is [a9]. Selected seminal papers on mathematical economics can be found in [a10]–[a12]. For more on dynamical systems as applied to economics, including (optimal) control and the calculus of variations, see also [a13]–[a15]. The books [a5]–[a7], [a16]–[a20] deal with prices, utility functions and general equilibrium theory. In its more advanced versions [a20] the latter makes sophisticated use of measure theory.

Besides the fields already mentioned, many other branches of mathematics can be fruitfully applied in economics, including bifurcation theory, Hamiltonian dynamical systems and the theory of Lie groups [a25].

The mathematical analysis of social choice and voting systems turned up a surprise in that a very reasonable sounding set of axioms for social choice (transitivity, independence of irrelevant alternatives, unanimity, no dictator) turns out to be contradictory. Such results (and there are many of them) are called Arrow impossibility theorems [a21]–[a24]. They have much to do with the Kondortsev paradox, which under simple majority voting gives a circular order in the case of three alternatives and three voters whose respective orderings of the three alternatives are $x>y>z$, $y>z>x$, $z>x>y$.

Scarf's method for the numerical calculation of Brouwer fixed points via Sperner's lemma developed into the homotopy methods for solving equations. In crude terms this amounts to deforming a given problem until a trivial problem is found, and then deforming back together with the solution, [a26]. This idea further developed into continuation methods for solving systems of equations (cf. Continuation method (to a parametrized family); Continuation method (to a parametrized family, for non-linear operators)). Many previously known solving methods turned out to be special cases of these ideas. A selection of references is [a26]–[a29].

The relation between decomposition methods in mathematical programming and centrally-guided economic systems is extensively discussed in [a30].

References[edit]