## Comparative Statics

## Comparative Statics

# Comparative Statics

*Comparative statics* is a methodological concept of economic theory and is related to *economic models*. An economic model consists of a set of relations among economic variables. These relations may be definitional, behavioral, or technological in nature. For instance, in the model of a competitive market there are three relations. The first two are behavioral relations describing how the buyers and sellers behave in the market. The behavior of the buyers is summarized in a demand relation in which the quantity of a good demanded in the market depends on its price and the total income of the buyers. The supply relation states that the quantity of the good supplied in the market depends on its price and the number of sellers supplying the good. The third relation of the model defines equilibrium as a situation in which the quantity demanded is equal to the quantity supplied. The market model involving these three relations is designed to explain the values of three *endogenous* variables: quantity demanded, quantity supplied, and the price of the good. Under certain conditions the model can be solved to get the values of endogenous variables in a static equilibrium. But there are two *exogenous* variables in the model—the total income of the buyers and the number of sellers—and the model is not designed to explain the values of these variables. A static equilibrium may be disturbed when there is a change in number of sellers, for instance. An increase in the number of sellers may occur if new firms enter the market where the existing firms are earning economic profits. In this case, a new static equilibrium will emerge. The methodology of comparative statics allows the theorist to compare these two static equilibria.

Comparative statics was introduced by the British economist Alfred Marshall (1842–1924) in his book *Principles of Economics* (1890), in which he discussed the effect of entry or exit of firms on market equilibrium. Later, Paul Samuelson developed the concept of comparative statics systematically in his book *Foundations of Economic Analysis* (1947), along with the related concept of comparative dynamics. Comparative static methodology is used in all branches of economic theory because it enables the theorist to derive a set of hypotheses that can be empirically tested. Theoretical econometrics and

applied econometrics owe their existence largely to the development of comparative static methodology, and the contribution of these two allied fields of economics to economic policy analysis is tremendous.

To illustrate comparative statics in a market model let us state the demand and supply relations as:

*Q*=_{D}*f*(*P, Y*)*Q*=_{S}*g*(*P, N*)*Q*=_{D}*Q*_{S}

The endogenous variables in this three-equation market model are *Q _{D}* (quantity demanded),

*Q*(quantity supplied) and

_{S}*P*(price). The exogenous variables are

*Y*(income) and

*N*(number of sellers).

The figure above shows an inverse relationship between price and quantity demanded for a given level of income, *Y* _{0}. The relationship between price and quantity supplied is a positive one, and the diagram shows four supply curves for the number of sellers ranging between *N* _{1} and *N* _{4}. The price quantity combination (*P*,Q** ) is one static equilibrium when the number of suppliers is *N* _{1}. Another static equilibrium is represented by the price quantity combination (*P′, Q′* ) when the number of suppliers increases to *N* _{4}.

It is possible to derive a testable hypothesis by using comparative static methodology, provided that certain conditions are satisfied. First, the two static equilibria that we are comparing must exist and be stable. If the demand and supply curves do not intersect at all or if they do not intersect in the positive orthant, market equilibrium does not exist. Second, we must have an established theory connecting the number of firms that is exogenous to the model with the quantity supplied that is endogenous. Because the theory of supply predicts a positive relationship between the quantity supplied and the number of firms, the testable hypothesis that emerges is: as the number of firms increases in the industry, price falls and more quantity of the good is bought and sold in the market.

Comparative static methodology is also used in optimization models, where the conditions under which it works are somewhat different. In economic theory, consumers maximize utility subject to their budget constraint, or the producers maximize profits subject to technological constraints. These models also have a set of endogenous and exogenous variables. If we take the case of a competitive firm, the quantity of output produced or the quantities of inputs used in production are endogenous variables, but the prices of goods produced or the prices of inputs are exogenous variables. In the consumer model, the quantities of goods purchased are endogenous variables, but prices paid for these goods as well as amount of money spent are exogenous variables. A typical testable hypothesis is the law of demand that says that if the price of a good decreases, the consumer tends to purchase more quantities of that good. This hypothesis can be analytically derived by the use of comparative static methodology, provided that a *maximum* of the consumer’s utility exists subject to the budget constraint. Similarly, for deriving any hypothesis related to the behavior of producers it is necessary that the profit function have a maximum. Thus, the comparative static methodology fails in the competitive model if there are increasing returns to scale, because profit function does not have a maximum. Under increasing returns to scale, the firm can always reduce unit cost by expanding the scale of output that can be sold at a fixed price.

Many economic variables such as quantity demanded or supplied are nonobservable. Econometricians can estimate both demand and supply functions only under certain conditions. In the figure above, demand is stable, but supply curve shifts as the number of firms changes. In this case, the demand curve can be estimated from data on price and quantity, but the supply curve cannot. Empirical estimation is crucial for predicting the effects of a tax or subsidy on welfare.

**SEE ALSO** *Comparative Dynamics*

## BIBLIOGRAPHY

Kogiku, Kiichiro C. 1971. *Microeconomic Models*. New York: Harper.

Kuska, Edward A. 1973. *Maxima, Minima, and Comparative Statics*. London: Weidenfeld and Nicolson.

Marshall, Alfred. [1890] 1997. *Principles of Economics*. Amherst, NY: Prometheus.

Samuelson, Paul A. 1947. *Foundations of Economic Analysis*. Cambridge, MA: Harvard University Press.

*Monica Das*

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**Comparative Statics**