# Mathematics in the Social Sciences

# Mathematics in the Social Sciences

Methodologically, scientists think that the social sciences differ from the physical sciences in degree but not in kind. The commonality among the sciences is best expressed by Paul A. Samuelson: “All sciences have the common task of describing and summarizing empirical reality.… There are no separate methodological problems that face the social scientist different in kind from those that face any other scientist” (Samuelson 1966, Vol. 3, p. 1756). Another common feature among the sciences is that they use mathematics as a language.

Joseph Schumpeter (1954, p. 955) asserted that in studying quantitative relationships, the knowledge of a handful of simple concepts in calculus—variables, functions, limits, continuity, derivatives, differentials, maxima, minima, system of equations, determinateness, and stability—change one’s attitude to the problem. The applied interest of social scientists is piqued by advances in the frontiers of mathematics and includes, besides calculus, linear programming; information, game, and network theories; and Markov processes, which have industrial, political, social, economic, and military applications (Tucker 1963).

Some have identified stages in the development of mathematics in economics. According to John Hicks, “All through the first period of mathematical economics—the age of Marshall, of Pareto, of Wicksell, even that of Pigou and Keynes—the economist’s main mathematical tool was the differential calculus.… Most economics problems were problems of maxima and minima” (Hicks 1983, pp. 247–248). As in the physical sciences, the approaches to mathematical modeling in the social sciences follow broad methodologies. One simple approach is reductionism, where the laws of one discipline are transplanted to that of another. The physicist Stephen Hawking abides by that rule (Hawking [1997] 2000, p. 169), and so does Samuelson (1966, Vol. 3, p. 1755).

In the social sciences, an abstraction from reality is made in the form of a model that can be expressed in mathematical form. Social scientists tend to divide into two camps in their fundamental belief in the use of mathematical models: Some build models that follow the principle that nature does not make any leaps, and others believe in a dialectical or chaotic process. In either case, a good grasp of the history, statistics, and theory required to function in the social sciences in turn depends on a good understanding of mathematics (Schumpeter 1954, p. 14). In editing mathematical models of the social sciences, James R. Newman (1956) selected classical contributions from the fields of psychology, economics, and sociology. A later book by Kenneth Arrow, Samuel Karlin, and Patrick Suppes (1960) tackles topics in economics, management science, and psychology.

## MODELING IN PSYCHOLOGY

An early form of an inner observation model in psychology is Fechner’s Law, after Gustav Theodor Fechner (1801–1887). This law is traced to the work of Ernst Heinrich Weber (1795-1878), who had conducted several experiments giving subjects weights to hold and asking them to report when they sensed an increase in weight (sensation, *S* ). Because sensations cannot be measured, Fechner proposed to observe noticeable differences in sensation. (The expressions that follow use the symbols of Edwin Boring [1956, pp. 1159–1160]). One can measure the stimulus, *R*, as so many pounds of weight, and then record the constant point, *c*, where the stimulus has created a noticeable difference as *dR/R* = *c*. This relationship takes the form *dS* = *c(dR/R)*. The implication here is that all changes in stimulus are homogeneous and can be aggregated. One can, therefore, integrate the equation to measure sensation. The integration constant can be approximated by choosing a threshold value of the stimulus, *r*, such that when *R* = *r*, then *S* = 0. This will yield the expression *S* = *c* log* _{e} (R/r)*. By moving to a different base of logarithms and by measuring stimuli in units

*r*, Fechner’s Law is expressed in a more compact form as

*S*=

*k*log

*R*. The equation states that as stimulus multiplies in strength, such as in tripling (3 × 1), the corresponding perception will increase additively, say twice stronger (1 + 1).

By the use of combinatorial mathematics, Kenneth Arrow developed a possibility theorem to show that consistent voting behavior is impossible, even in a democracy. He was preceded by a long line of political theorists who were trying to aggregate individual preferences for a group or society. Attempts by Jean-Charles de Borda (1733-1799) and Marie Jean Antoine Nicolas Caritat (1743–1794) established that the law of transitive reasoning does not hold up when voting preferences are aggregated. A set of voters may choose the candidate A over B, and B over C, but not A over C. Arrow postulated a set of consistent axioms that cannot all be met in the aggregation of social preferences. Besides the transitivity axiom, he required that there be no dictator, that when two choices are ranked their outcome should not depend on the third choice, and finally, that if everyone prefers one outcome over another, the preferred outcome should dominate.

To illustrate Arrow’s model, consider two persons, X and Y, who are voting on three candidates, A, B, and C. We can create thirty-six profiles of their ranking, because two persons can rank three outcomes in 6 × 6 = 36 ways. Of these thirty-six profiles, X and Y can make only six unanimous rankings, where their first, second, and third rankings can be {A,B,C}, {A,C,B}, {B,A,C}, {B,C,A}, {C,A,B}, and {C,B,A}. In many instances, however, X and Y will not be able to rank the candidates the same. Although they both may rank A first, X might rank B second, and Y may rank B third. The aggregation of these preferences will not, therefore, be unanimous, and this may require a benevolent dictator to pick the final candidate.

## MODELING IN SOCIOLOGY

As an example of mathematical models from sociology, we take one developed by Arnold Faden (1977, p. 266) as in Alfred Lotka’s predator-prey model, which explains how crime spreads. Loosely speaking, the density of crime at a location has a three-population functional relationship *f [v(s), c(s), p(s)],* where *v, c, p, s* represent victims, criminals, policemen, and location, respectively. A particular explicit relation of this model can take the form of *f (v, c, p) = vce ^{–p}.* One implication of the model is that a crime is possible when a potential victim meets a criminal, assuming no policing,

*p =*0.

From the game perspective, several implications can be modeled. Faden examined cases where victims and criminals follow their individual rationality, where they collude in the absence of police, and where the police and criminals collude, given the distribution of victims. Such modeling represents only the beginning of a complex social problem. The assumption about the size and distribution of the three populations, the functions that relate them, and their motivations can vary. For instance, if we focus on the struggle for existence between two populations, say victims and criminals, the increment of the former will add to the possibility of the latter, yielding a simple system of differential equations such as (*dv/dt* ) 1/*v* = *f (c)* and (*dc/dt* )1/*c* = *f(v)*. As Samuelson pointed out (1972, p. 474), solutions for these types of equations can lead to perpetual movement in a circle around an equilibrium value. However, the potential for such modeling is on the increase. In the global economy, a country enforcing international trade law can represent the police variable. Consumers, countries, or agents may be the victims, and criminals may be engaged in international piracy of intellectual property rights.

## MODELING IN ECONOMICS

The classical economists were concerned with how the economy is interconnected. Prices of commodities depend on cost of production (the price of inputs), and output is what is demanded at existing prices. Antoine Augustin Cournot ([1838] 1956, p. 1215) was the first to use calculus in determining both prices and quantities in his model. He formulated quantity demanded as a function of price *D = F(p),* and foreshadowed the development of the Marshallian elasticity concept by advancing that Taking the differential of demand to yield *F(p)* + *pF* ′(*p* ) = 0, Cournot proceeded to analyze maximum problems.

Cournot has bequeathed a behavioral assumption that makes possible the solution of rivalry problems in modern economics. Each firm assumes that the other firms’ output is fixed. The expected output of the other firms now enters into the one firm’s own demand curve. If costs are given, the calculus method can be used to find optimal price and output. Variations in demand and cost curves are prominent exemplars in intermediate microeconomics and managerial textbooks.

The founders of the marginal revolution in economics, Léon Walras, Stanley Jevons, and Karl Menger, were influenced by Cournot. They extended Cournot’s model into the theory of the firm, consumer behavior, and general equilibrium analysis. Walras (1954, pp. 103, 110–111) indicated equilibrium where supply and demand curves cross. The markets are related because a change in price in one market affects changes in other markets. A general equilibrium is sought for all markets through the process of *tatonnement* —raising the price in cases of excess demand and lowering it in cases of excess supply until equilibrium is attained.

Paul Samuelson (1966, Vol. 1, p. 544) explained how stability in a Walrasian system can be achieved using differential equations. Solving the equation *dp/dt* = *H(q _{D} – q_{s})* the term on the left is the rate of change of prices,

*dp*, with respect to changes of time,

*dt*. Given that

*H*is a proportional constant,

*q*is quantity,

*S*is supply, and

*D*is demand, one can find the time path. Stability occurs when, as time goes to infinity, the solution of the differential equation breaks down, which in economic terms means that the supply curve must cut the demand curve from below. Solving for quantity in terms of price, one can also find the Marshallian stability conditions under which quantity adjusts to clear the market. Malthus’s demographic model used differential equations to show stability. If a population,

*x*, grows at an exponential rate,

*c*, over time,

*t*, then

*dx/dt*=

*cx*. If food supply,

*y*, grows at a constant rate,

*k*, over time,

*t*, then

*dy/dt*=

*k*. Thus, the ratio of

*y*/

*x*= (

*y*+

_{0}*kt*)/(

*x*) will break down as

_{0}e^{ct}*t*approaches infinity, implying misery (Hirsch 1984, p. 14).

Mathematical modeling in economics climaxed with the Arrow-Debreu theorem (1954). In this system, a fixed-point mapping method was used instead of counting equations and unknowns in the Walrasian system. The idea of a fixed point is that, in the process of transforming a rubber sheet, say by stretching it, one point will not move. It will be mapped onto itself. William Baumol (1965, p. 494) explained this process for the case of the demand and supply curves. Starting with demand and supply curves on two separate graphs, let *Q _{s}* be the quantity supplied, and

*Q*the quantity demanded. Then, tracing up from the two-quantity axes to the demand and the supply curves, one finds two prices, say a supply price,

_{d}*P*and a demand price,

_{s},*P*. If we can find a function, a map of

_{d}*f(P*) =

_{s}*P*or

_{d}*f(Q*=

_{s})*Q*, then we have a fixed point mapping for the equilibrium problem.

_{d}The application of the fixed point theorem to the Arrow-Debreu general equilibrium required that convex assumptions be applied on consumers’ and producers’ sets, as well as special ideas of continuity, and the idea of separating planes in Euclidean space. For instance, a commodity has spatiotemporal and physically differentiated characteristics, which can be represented by a point in space. If there are *L* amounts of commodities, one can imbed them in the Euclidian space, *R ^{L}.* As Debreu put it, “By focusing attention on changes of dates one obtains … a theory of savings, investment, capital, and interest. Similarly by focusing attention on changes of locations one obtains … a theory of location, transportation, international trade and exchange” (Debreu 1959, p. 32). In his

*Theory of Value*(1959) Debreu relied on a lemma of Hukukane Nikaido (1956) to prove equilibrium. The lemma assumes a referee that sets the price,

*P.*Consumers will maximize their utility,

*U*subject to

_{i}(x),*PX = PA,*where

*A*is endowment and goods

*X*is in space

*E =*(

*X*ǀ0 ≤

*X*≤

*C), C*being an arbitrary bundle such that

*C> A.*All acceptable bundles with respect to

*P*are labeled

*φ*, the i

_{i}(P)^{th}individual demand function. Its sum is just

*φ(P).*If total demand,

*X,*does not match total available bundles,

*A,*the referee must make an adjustment. The difference is

*X– A.*Its value is

*P(X– A).*The referee’s objective is to pay a person a value

*PX*that is greater than the endowment value

*PA.*In other words, choose a price,

*Q*that will maximize the price-manipulating function

*θX =*(

*P\P*(

*X– A)*) = max(

*Q*(

*X*–

*A)*) for all

*Q*in

*S*, where

^{k}*X*is total demand lying in

**. We now have the following demand function and a price-adjustment function:**

*Γ**S ^{k}* ∍

*P*→ φ(

*p*) ⊂

*Γ*(Demand function)

*Γ* ⋐ *X* → *θ(X* ) ⊂ *S ^{k}* (Price-manipulating function).

We want to choose (*X, P* ) in this demand and price adjustment space, *Γ* ×*S ^{k}*, such that

*φ(p)*

**×**

*θ(X*) is contained in

*Γ × S*. This is possible because the mapping is upper semicontinuous. Therefore, the equilibrium price exists.

^{k}Upper semicontinuity is easy to show for *θ* (X). Given *P _{n}* →

*P*in S

^{k},

*X*→

_{n}*X*in Γ and

*P*∈

_{n}*θ(X*then

_{n}),*P∈θ(X)*. The proof is that, for any

*Q*price-constellation,

*P*

_{n}(X_{n}– A)

*≥**Q(X*and, for the whole sequence, as

_{n}– A),*n*→ =∞, we get

*P(X– A)*

*>**Q(X– A).*Therefore, the existence of

*P∈ θ(X)*is demonstrated.

A model developed by Karl Marx has been used as a material balance approach to modeling the economy. Total value is the sum of *c* = constant capital (depreciation), *v =* variable capital (wage bill), and *s =* surplus, and the cost price of total capital to produce the commodity is *c + v.* Assuming turnover rates of 1 for both *c* and *v,* let the total invested capital in one department be 100 units of capital distributed as 80*c* + 20*v,* and 70*c* + 30*v* in the other. The average rate of profits is *s/(c+ v),* and the average rate of surplus value is *s* /*v*. For the first department, 80*c* + 20*v*, *s/v* = 20, or 100 percent of variable capital, and *s/(c + v)* is 20 percent. For the second department, 70 *c* + 30 *v*, the rate of profit becomes 30 percent.

Because of competition, the sum of the profits, 20 + 30 = 50, will have to be distributed evenly across the two departments discussed above, in this case 25 each. Adding this average profit of 25 to the cost price of total capital (*c* + *v* ) yields prices of $125 for each department. Adding *s* to the cost price of total capital yields values of 120 for the first department, and 130 for the second department. The price-value differences are $5 and $–5, respectively, whereas the total values and total prices are the same, each equal to 250. This difference between price and value has been called the “transformation problem” in Marxian economics. Tremendous efforts have been made to solve this problem, and to extend this model to show growth. All in all, the mathematization of the social sciences deepens their ability to probe the human condition.

**SEE ALSO** *Arrow Possibility Theorem; Arrow-Debreu Model; Economics; Economics, Marxian; Economics, Neoclassical; General Equilibrium; Hicks, John R.; Marginalism; Marxism; Models and Modeling; Psychology; Samuelson, Paul A.; Schumpeter, Joseph Alois; Social Science; Statistics in the Social Sciences; Walras, Léon*

## BIBLIOGRAPHY

Arrow, Kenneth J. 1963. *Social Choice and Individual Value*. New York: John Wiley and Sons.

Arrow, Kenneth J., and Gerard Debreu. 1954. Existence of an Equilibrium for a Competitive Economy. *Econometrica* 22: 265–290.

Arrow, Kenneth J., Samuel Karlin, and Patrick Suppes, eds. 1960. *Mathematical Methods in the Social Sciences*. Stanford, CA: Stanford University Press.

Baumol, William J. 1965. *Economic Theory and Operations Analysis*. 2nd ed. Englewood Cliffs, NJ: Prentice Hall.

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Marshall, Alfred. [1890] 1920. *Principles of Economics*. 8th edition. London: Macmillan.

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Nikaido, Hukukane. 1956. On the Classical Multilateral Exchange Problem. *Metroeconomica* 8: 135–145.

Samuelson, Paul A. 1966. *The Collected Scientific Papers of Paul A. Samuelson*. 3 vols., eds. Joseph E. Stiglitz and Robert C. Merton. Cambridge, MA: MIT Press.

Schumpeter, Joseph A. 1954. *History of Economic Analysis*, ed. Elizabeth Boody. New York: Oxford University Press.

Tucker, A. W. 1963. New Directions in Applied Mathematics. In *New Directions in Mathematics*, eds. John G. Kemeny, Robin Robinson, and Robert W. Ritchie. Englewood Cliffs, NJ: Prentice Hall.

Walras, Léon. [1874] 1954. *Elements of Pure Economics, or the Theory of Social Wealth*. Trans. William Jaffe. Homewood, IL: Richard D. Irwin.

*Lall Ramrattan*

*Michael Szenberg*

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