Role of Mathematics in Economic Theory and Modeling

Rakesh Sharma

Assistant Professor of Economics, Govt. College, Sangrah, District Sirmour, H.P.– 173 023.

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The introduction of Mathematical methods into Economics has been a long process passed through various stages. Initially, it began with a total rejection, followed by a partial acceptance. Pieces of mathematical reasoning applied to economic problems have been found as far back in history as in Aristotle’s work. In 18^th and 19^th Century, Bernoulli, Gauss, Laplace and Poisson developed mathematical models to discuss economic problems. However, Economics became more mathematical as a discipline throughout the first quarter of the 20th century, but introduction of new and generalized techniques in the period around the Second World War, broaden the use of mathematical formulations that led to the major development of mathematization of the Economics. Now, it is being widely accepted as a tool to be very useful in improved Economic Theory and Modeling for policy formulation and providing corrective measures for deficiencies or shortcomings of existing policies with better sense and sound judgment. Having a fair idea of economic problems with expert knowledge supplemented by appropriate mathematical techniques do provide a better formal intuitive insight into the problem and can promote one’s understanding in a systematic and consistent form. Empirical support and mathematical logic makes the theory more easily understandable so that some otherwise confused and ambiguous observations can be visualized within a proper perspective. Economic reasoning based on mathematics has been a fundamental factor in the development of economic theory as a science. Some challenges of mathematics and certain prerequisites and concepts may push the readers away from modern Economics, but knowledge of some important tools of mathematics provides a convenient alternative between intensive and extensive points of view and may end up providing a grasp of the both.

Schumpeter says that ‘even the restatement in algebraic form of some results of non-mathematical reasoning does not constitute mathematical economics, a distinctive element enters only when the reasoning itself that produces the result is explicitly mathematical’.

Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects.

Economic Modeling:

Modeling suggests what will happen if certain actions are taken. Simulation of real world situations is possible with economic analysis and modeling and would not be possible without mathematics. An Economic Model is a simplified representation of a real world economic process and can be put in the form of a set of equations, simplified mathematical relationships asserted to clarify assumptions and implications, which together explain the system which is being examined in a given study. In addition, there can be identities, technical and behavioural equation expressing hypothesis about behaviour and equilibrium conditions implying balancing of forces. The intention behind the creation of a Model is to evolve a tool of deductive reasoning relevant to economic policy and also to pave the way for more realistic and better approximation. If handed with reasonable care, an Economic Model can be approximated to reality sufficiently closed. There is no question here of attaining perfectly realistic model because that would be an almost impossible task in Economics; the more relevant issue is their workability, predictive accuracy and usefulness in the specification of empirical relationships. The ultimate success of a Model rests on its pragmatic utility and relevance. The ability of elegant model formulation with sensible use of mathematics is an asset in a Social Science. Economic models have practical and computational roles as well as theoretical and educational roles while allowing one to go intensively into the depth of an economic problem. In fact, the search for a good model is never complete in empirical work.

A Mathematical Model of the economy is a formal description of certain relationships with the purpose to analyze their logical implications. Some of those are derived from empirical observations; others deduced from theoretical assumptions, the behaviour of a ‘rational’ economic agent whose decisions depend on past events and their expectations about the future. Assuming that no mathematical mistake is made, the relevance and importance of the conclusion of the analysis depend upon the validity of the premises of the Model and on the ability to find out all their consequences. It does not matter how sophisticated mathematical methods are being employed in the analysis, the value of its final results heavily depend on the basic hypothesis of the Model. Thus Mathematical Models are recognized in providing a rational approach to solving many of the problems in decision making, allocation, forecasting, help to understand cause-and-effect relationships in the economy.

Important contribution of Mathematics techniques brought into Economics comes from the Economists; Leon Walras, Slutsky, W. Leontief, Cobb-Douglas, Cobweb, Von Newmann, Samuelson and Kaldor, in providing solution to many economic problems.

Mathematics in Theory of Consumer Behaviour:

An important usage of Mathematics in Economics is the one related to the interpretation of the decisions made by consumers, explaining this way the consumer’s behavior, based on practical applications.

Let U represents the Total Utility derived from consumption of goods X₁, X₂, X₃,……X_n goods then ∂U/∂X₁ depicts the rate of change in Total Utility U with respect to change in their amount of consumption of X₁ when the amount of consumption of other goods remains unchanged.

For a hypothetical Utility Function of two variables U = f (x, y), we have to find how much of x and y should the consumer buy with the given purchasing power to maximize his Utility. With the given purchasing power, if the consumer buys more of x, he will have to buy less of y and vice versa and, therefore, the amount of x and y is not independent of each other. Thus the problem of Utility Optimization should also take in to consideration the purchasing power of the consumer i.e. Budget Constraint. Most of the economic problems concerning Maxima and Minima are of this nature. There is always a constraint on the variables and as such the variables x and y are not independent. Therefore, Utility Optimization is attained by using Lagrange Multiplier and Bordered Hessian Determinant.

To maximize the Utility Function u (x₁, x₂) subject to the budget constraint y = p₁x₁ + p₂x₂

z = u (x₁, x₂) – λ (p₁x₁ + p₂x₂– y)

∂Z/∂X₁ = u₁ - λ p₁= 0

∂Z/∂X₂ = u₂ - λ p₂= 0 Now it follows that λ = = i. e. Marginal Utilities should be proportional to prices for consumer equilibrium and hence for maximum utility.

With the help of Partial Derivates, Slutsky divided Price Effect into Income effect and Substitution effect of various goods. Generalized form of Slutsky Equation is:-

∂q_i/∂p_i= (∂q_i / ∂p_i) _{U - constant} - q_i(∂q_i/∂y)_{p - constant}

Commodity q_i is defined as Normal good if ∂q_i/∂p_i< 0, Giffen good if ∂q_i/∂p_i> 0 and Inferior good if ∂q_i/∂y < 0

Given the Demand Function: P = a – bX, where P is Price and X is quantity demanded , then Consumer’s Surplus is calculated as

Mathematics in Optimization Theory:

In mathematics optimization refers to the selection of a best element from some set of available alternatives. In the simplest case, an optimization problem involves maximizing or minimizing a real function by selecting input values of the function and computing the corresponding values of the function. The solution process includes satisfying general necessary and sufficient conditions for optimality.

Economics is closely enough linked to optimization by agents in an economy that an influential definition relatedly describes economics qua science as the "study of human behavior as a relationship between ends and scarce means" with alternative uses.

In any widely and most accepted Economic system i.e. Capitalist System, whole economic chain; that consumers maximize their utility, subject to their budget constraints and that firms maximize their profits, subject to their production functions, input costs and market demand, is influenced by Mathematics.

Profit (π) = Total Revenue (TR) – Total Cost (TC)

In order to Maximize Profits, First order condition is d π/dQ = 0 and Second Order Condition is d²π /dQ² < 0

However, in case of Profit Maximization or Cost Minimization subject to Budget Constraint, we make use of Lagrange Multiplier and Bordered Hessian Determinant.

Mathematics in Production Function:

Production itself can be described as a conversion of some variables (Inputs) in others (outputs).

C.W. Cobb and P.H. Douglas proposed a general production Function as a law of production for manufacturing Industries: Q = A K^α L^β

Here Q represents total output, K is Capital input and L is Labour input. A is efficiency parameter and α, β are output elasticity’s. The main feature of this function is that it is homogeneous of degree (α + β). In special case of α + β =1, the function turns into linearly homogeneous function although the function itself is not linear, depicting constant returns to scale.

There is another C.E.S. Production Function which is linearly homogeneous, though possesses the constant elasticity of Substitution but may take a value other than unity.

Q = A[ α K^-β+ (1- α) L^-β] ^-1/β

As thisfunction is linearly homogeneous, it depicts constant returns to scale and qualifies for the application of Adding-up Theorem/ Euler’s Theorem, describing that under the condition of constant returns to scale, the total product will get exhausted by the shares distributed among all the input factors if each factor is paid according to its marginal product, thereby providing a solution to the distribution aspect of National Product in the economy. That is, where Total Product is maximum, Marginal Product is zero, while the Marginal Product of other factor of production is equal to average product.

Mathematics in Capital Formation:

Capital Formation is the rate of net investment flow (t), in other words it is the process of adding to a given stock of capital, may thus be expressed as dk / dt where k is a function of time t.

Therefore, dk/dt = (t) or k(t) = dt =

Income from Investment Y = ^t/ (1+r)^t

Present Value (a^t) of any Investment depends upon the rate of interest (r) and its frequency of being added up in n years.

Present value of continuous equal Income stream for n years: Po = ^{– r t}

Mathematics in Market Equilibrium, fluctuations and Growth Path:

Differential Equations have a wide application in Economics by providing solution to Demand and Supply functions resulting in Equilibrium Price. Further, Differential Equations provide Model of Dynamic Multiplier when an autonomous/Induced Investment has a multiplier effect on the level of Income, output and employment in the economy. It further provides solution to Growth Path of Herrod-Domar and Solow Model (describes that capital grows at a rate of g = s / v where s is the propensity to save and v is capital –output ratio).

First Order Difference Equations provide solution to Dynamics of the Equilibrium i.e. how fluctuations converge or diverge to Equilibrium and paves the way for understanding a Business Cycle over a period of time. Herrod and Domar propounded their Multiplier-Accelerator Model describing that for full employment equilibrium, n = s / v , rate of growth of Labour force termed as natural rate of growth should be equal to the relative value of s and v.

By using Second Order Difference Equations, Paul Samuelson developed his Multiplier-Accelerator Interaction Model describing that different combinations of s (propensity to save) and β (accelerator) will generate different time-paths of national Income leading to convergent/explosive non-cyclic/explosive cyclic /damped or regular time path.

The Cobweb Model assumes that today’s demand for any commodity is a function of the present price while today’s supply depends upon yesterday’s decision about output. To put this in mathematical terms:

Demand Function: X _Dt = a + α P_t

Supply Function: X _St = b + β P_{t - 1}

Ultimate solution describes that growth path over time must always be oscillatory in terms of explosive oscillations, damped oscillations or regular oscillations.

Mathematics in Input-Output Analysis:

With the use of the Technology Coefficient Matrix, the Russian born Economist Wassily W. Leontief developed Input-Output Analysis, a solution to the problem that what level of output of each producing sector in an economy can bring about equilibrium for its product in the economy as a whole. Since inputs of one industry are the outputs of another industry and vice versa, ultimately their mutual relationship must lead to equilibrium between supply and demand in the economy consisting of n industries, so as to avoid any bottlenecks anywhere in the economy. The main crux of Input-Output Analysis is that, given certain technological coefficients and final demand, each endogenous sector would find its output uniquely determined as a linear combination of multi-sector demand.

Mathematics in Linear Programming:

Linear programming was developed to aid the allocation of resources in firms and in industries during the 1930s in Russia and during the 1940s in the United States. During the Berlin airlift (1948), linear programming was used to plan the shipment of supplies to prevent Berlin from starving after the Soviet blockade. Linear programming helps to find out maximum or minimum values in problems confronting the decision-making authorities subject to certain constraints when side conditions are inequalities and not equations.

The problems in Linear Programming deal with those Optimization problems which possess:

a) The nature of Optimizing process is either of maximizing or of minimizing but never any mixture of two

b) Constraints are in the form of inequalities involving side-conditions of 1^st degree

c) A single decision maker is involved

In Simplex method, the original problem is referred as a Primal Problem. If the Primal Problem requires maximization, the Dual problem is one of minimization. However, value of Dual Objective function will always be greater than or equal to the value of the Primal Objective function called as Kuhn Tucker Theorem. To find out optimal solution of the objective function, we move from one basic feasible solution to another basic feasible solution until the Optimal Solution is achieved. Extensions to Non-linear Optimization with inequality constraints were developed by Albert W. Tucker and Harold Kuhn in 1951.

Mathematics in Game Theory:

Game theory propounded by John Von Newmann introduces the possibility of more than one decision –maker, wherein optimum value of the Objective function for any one decision maker depends not only on his choice but also on the choice of the others. The decision makers are called players and the objective function is called Payoff Function. These players confront one another in pursuit of certain conflicting objectives. Being in conflict, all players cannot realize their objectives. Some players may be risk averters while the others may be risk lovers. Value of the game is just the outcome when both/all the players follow their best strategy. Thus in Game Theory, optimization is that of finding either the maximum among the set of minima (maximin) or the minimum among the set of maxima (minimax), thereby reaching to a Saddle Point Solution through pure strategy, mixed strategy or dominated strategy.

Mathematical Plan Models:

In Russia, the mathematician Leonid Kantorovich developed Economic Models in partially ordered vector spaces. In India, First Five Year Plan was based on Herrod-Domar Growth Model in the form of differential equation easily solved to give the time profile of capital stock and output. Second Five Year Plan was based on Two and Four Sector Model prepared by a Russian Economist P.C. Mahalanobis which marked a shift in the favour of Capital goods industries. Third Five Year Plan was based on Two-Sector consistency Model casted in principle, not in terms of difference equations but in the form of a set of algebraic equations in which initial and final values were related. Fourth Five Year Plan Model was based on Vector and Matrix Analysis propounded by Maane and Rudra. Fifth Five Year Plan was based on open ended static Leontief inter-industry macroeconomic growth Model. The Model structure of sixth Five Year Plan was an extension of the Fifth Plan Model, comprising a core Model with several Sub-Models, covering a period of 15 years from 1980-81 to 1994-95. However, with the introduced of LPG policies in India from 1991 onwards, the role of Mathematical Models has reduced and Planning Commission has been named as ‘Neeti Aayog’.

It may be concluded that in order to explain the rules of economic theory, mathematical methods based on economic thinking have been widely used in providing solution to a number of economic problems as it is reflected from the work of many economists, thereby this way it proves, in practice, that mathematics is not a substitute but a complement for the economic sciences, offering new sources for theories, postulates, axioms, theorizations, and explanations. Any new theorization, new concept and axiomatiszation in Economics, passed through the filter of Mathematics will prove a high level of credibility and its impact in the Society. Economics and Mathematics obtaining this way, has propelled both Mathematics and Economics amongst the noble sciences.

REFERENCES:

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Received on 24.08.2016 Accepted on 09.09.2016

Int. J. Tech. 2016; 6(2): 170-174.

DOI: 10.5958/2231-3915.2016.00026.2