## Statics and Dynamics in Economics

## Statics and Dynamics in Economics

# Statics and Dynamics in Economics

Since the end of the nineteenth century, economic analysis has been fairly rigidly compartmentalized into statics and dynamics. Various, sometimes conflicting definitions of these terms have appeared in the literature; some have based the distinction on the nature of the subject matter studied, while others have emphasized the difference in analytic approach. Utilizing the first of these viewpoints, we may distinguish between a *stationary* economic phenomenon—that is, one that does not change with the passage of time— and a *developing* or *changing* phenomenon. But no matter which type of phenomenon we study, we may focus upon it an analytic apparatus which we describe as *dynamic*—that is, one that takes explicit account of the role of the passage of time in the structure of the subject—or we can subject it to a static analysis, which deals with its mechanism at a given moment and abstracts from the effect of past events on the present and future. For example, an investigation of the determination of the level of employment at a given moment and its dependence on current consumption, investment, and governmental demand may be considered static in character, but a discussion of the same problem that considers how the relationship between today’s supply of capital equipment and its growth affects tomorrow’s investment demand and how this process can generate a time path of investment demand can be considered a dynamic analysis. (For further discussion of the definitions, see Samuelson 1947, pp. 315-317; Frisch 1935-1936.)

In earlier periods of economic analysis, the distinction between statics and dynamics was not very clear-cut. Characteristically, the object of the economist was to provide useful advice to the rulers of the body politic rather than systematic studies defensible under the canons of scientific analysis. Such a pragmatic concern justified the construction of a pastiche composed of both static and dynamic relationships that switched from one approach to the other whenever it became convenient. For example, monetary theorists, in their discussions of the effects of an increase of the supply of currency, turned rather early to an examination of the sequence of events whereby the initial stimulation of production and prosperity resulting from an injection of funds may gradually be dissipated by translation into a rise in price levels that is ultimately diffused throughout the economy. Cantillon and Hume were among those who described this process perspicaciously and in considerable detail.

The marriage of the two analytic approaches attained its apex in the work of the early classical economists, including Malthus and Ricardo, and culminated in the work of J. S. Mill. In these writings a purely static theory of wages, profits, and rent was wedded to a mechanism that accounted for the rates of growth of capital and population. In rough outline the model may be described as follows: In the short run wages are determined by the demand for labor, which increases with the supply of capital. But if the demand for labor is high, causing high wages, population growth is encouraged; and this gradually increases the supply of labor and reduces wages. Profits are thereby increased, stimulating investment both by enhancing its profitability and by augmenting the supply of funds that are normally used for this purpose. Thus, the levels of capital supply, population, and labor supply will grow with the passage of time, and in this process wages will frequently be kept up by high demand for labor. But, in the classical model, this process cannot continue indefinitely. The supply of natural resources, including land, being fixed, a growing population must result in an ever increasing ratio of labor supply to resources. Here enters the principle of diminishing returns, which states that if the supply of natural resources remains constant, increased labor supply will, at least after some point, yield diminishing marginal returns; that is, further increments in the labor supply will add less to the national product than did previous increments. Ultimately, then, the capital-population growth process will reduce per capita output and hence will decrease either wages or profits or both. Reduced profits will cause decreased investment, decreased demand for labor, and decreased wages. And when wages have been reduced to a minimal level representing either the amount necessary to provide subsistence or some habitual minimum living standard, population growth will cease. In sum, in this analysis the “law” of diminishing returns ultimately brings growth to a halt and then, finally, takes the economy into a stationary state where wages and profits are low and the general welfare is in a sorry condition.

From this model the classical economists derived a number of policy conclusions, among them their approbation of such measures as late marriage, which reduces the rate of growth of population, and their general denigration of activities designed to enforce increases in wage rates, which in their view could only temporarily ameliorate the lot of the laboring classes and would do so at a heavy cost—the increased imminence of the unhappy stationary state.

From our point of view, this construct is significant because it shows how an unsophisticated mélange of statics and dynamics can sometimes serve the needs of policy more effectively than a structure that is analytically pure. By bringing the static relationship between wages and the demand for labor together with the dynamics of population growth and marginal productivity, the classicists were able to derive conclusions which, right or wrong, were at least relevant for public decisions. (For further information on the classical and other earlier dynamic models, see Adelman 1961, chapters 3-5; Baumol [1951] 1959, chapters 2, 3.)

Toward the end of the nineteenth century, when economic analysis began more self-consciously to follow the precepts of scientific method, statics and dynamics were more rigidly compartmentalized. At first this meant in practice that systematic dynamic analysis was largely abandoned and was confined primarily to some casual observations about the time processes that underlay the well-developed static constructs. These observations ranged from the apologetic comments of John Bates Clark, who commended dynamic analysis to the care of the future economists, to Walras’ suggestive discussion of *tátonnements* and Edgeworth’s recontracting model. Both of the latter economists were fully cognizant of the difficulties besetting the equilibrium assumptions of static price theory, recognizing that in a world of imperfect knowledge initial pricing and supply bids were unlikely to approximate their equilibrium levels. They therefore described, in rather vague terms, intertemporal processes whereby the market, by trial and error, might gradually approach the price and output levels predicted by static theory.

Only in the last few decades has dynamic analysis developed into the independent and complex body of theoretical matter to be described presently. However, before that development occurred, a related analytic approach, *comparative statics*, provided a substantial contribution to economics.

**Comparative statics** . Comparative statics is a method of theoretical investigation of the consequences of a change in some datum in an economic model. For example, to determine the effects of a tax rate change on the level of output of a firm, a comparative statics study examines the two relevant equilibrium outputs—the equilibrium output under the old tax rate and the modified equilibrium output that would result from the imposition of the alternative tax arrangement. In effect, the comparative statics approach starts from an equilibrium position, then imposes an experimental variation on one or more of the parameters of the model and examines what has happened after the smoke has cleared away and the system has had time to settle down to a new equilibrium. But this method abstracts entirely from the time path of the variables of the model during the adjustment process. It considers a dynamic process (the time-consuming adjustment of the system to the change in parameters) but disregards everything but its end points—the initial equilibrium from which the process of change started out and the terminal equilibrium point at which the process comes to rest. This is why contemporary terminology distinguishes sharply between comparative statics analysis and dynamic analysis. However, the two are not unrelated. Implicit in every theorem of comparative statics is an adjustment process for which only dynamics can account. This, in part, constitutes what Samuelson (1947, chapter 9) has called the *correspondence principle.* He has also shown, by investigating the mathematics of the dynamic process, that if this process is unstable (if its time path does not converge toward an equilibrium value), the corresponding comparative statics analysis is likely to yield nonsense results. This principle will be explained and illustrated below, after some of the necessary tools of dynamic process analysis have been introduced.

It is probably no exaggeration to say that comparative statics has yielded some of the most fruitful portions of economic analysis. Both in terms of our insight into the workings of the economy and in its guidance for policy, comparative statics may well have contributed more than either pure statics or pure dynamics. Almost every branch of economic theory has made effective use of the approach. Two typical results of comparative statics are the macroeconomic conclusion, from the Keynesian model, that a fall in the supply of money will reduce the (equilibrium) level of employment, and the microeconomic result that a rise in price must decrease the (equilibrium) quantity of a commodity demanded by a utility-maximizing consumer if the item is not an inferior good (a commodity whose consumption declines when income rises).

*Methodology of comparative statics.* In recent decades comparative statics has developed an effective and fairly standardized mathematical methodology. (For descriptions of the method, see Samuelson 1947, chapter 9; Henderson & Quandt 1958, chapters 2, 3; Bushaw & Glower 1957.) The analysis involves the following steps:

(1) Formulate an expression describing the determination of the value of the variable that constitutes the objective of the person(s) directly involved and that therefore is maximized in equilibrium; for example, write out the functional relationship between the level of profit of a firm and the length of time for which an investment is retained.

(2) Write out the first-order and second-order maximum conditions of this expression, for example, the condition that the first derivative of the profit function with respect to the length of the investment period be zero and that the second derivative be negative.

(3) Vary a parameter in the problem (for example, allow for a change, *di*, in the rate of interest, *i*, that the firm confronts) and use the first-order maximum condition to determine what compensating change in the other variables (for example, in the period of investment, dt) is necessary to bring the situation back to equilibrium.

(4) Use the preceding step to determine the marginal effect of the change in the parameter on the other variables of the system, for example, to determine *dt/di.*

(5) Use the second-order condition and any other economic information to determine the sign of this marginal yield figure, for example, to determine whether a rise in the rate of interest will cause a profit-maximizing firm to lengthen its investment period, that is, to find whether *dt/di >* 0.

A relatively simple illustration should make these steps clearer. Suppose that the firm in question is growing timber and that the revenue it gets when it sells the product depends upon the length of time, *t*, a tree has been permitted to grow. Revenue, R(t)> will then be a function of time. But this amount will be available only *t* periods in the future, and its present value will be smaller by a discounting factor that represents the loss of earnings resulting from not having the money during the interim. Suppose an investment. Í, could grow continually at a constantly compounded rate of interest, *ρ* so that *dl/dt = ρI* (where we no longer use the symbol *i* to represent the interest rate, because *i* denoted interest compounded only at discrete points in time). Then we must have I(t) = I(0)e^{pt}. Therefore, the present value, I(0), of R(t) s I(t), obtainable *t* periods in the future, must be given by *I(0)e ^{pt} =* R(t), or

(1) *I(0)* = e^{-pt}R(t).

We have now completed step 1 of our comparative statics analysis; that is, we have found the function expressing the determination of the variable, I(0), whose value is to be maximized.

Step 2, giving first-order and second-order maximum conditions, yields

(2) *dI(0)/dt* = *−pe ^{-pt}R(t) + e^{-pt}dR(t)/dt = 0*

and

(3) *d ^{2}I(0)/dt^{2} < 0.*

In step 3 we allow for a change in the interest rate, *p*, and determine the compensating change in the investment period, *t*, necessary to maintain the maximum condition (2). To do this we calculate the total differential of (2), d[dI(0)/dt], and require that it be equal to zero, that is, that there be no change in the first derivative from its optimal (zero) value. We thus obtain

*d[dI(0)/dt]* = [d^{2}I(0)/di^{2}]dt + *[∂(dI(0)/dt)/∂ _{ρ}]dρ* = 0.

By direct differentiation of (2) with respect to *p*, we see that

*∂[dI(0)/dt]/∂ρ = -e ^{-pt}R(t) - ρ[dI(0)/dt] = -e^{-pt}R(t)*,

since, by (2), *dI(0)/dt* = 0. Thus, we have

(4) *d[dI(0)/dt]* = [d^{2}I(0)/dt^{2}]dt - *[e ^{-pt}R(t)]d_{p} =* 0.

Step 4 requires us to solve (4) for *dt/dp*, the marginal effect of the change in *p* on the equilibrium value of t. We obtain directly

Our final step consists in evaluation of the sign of the preceding expression. Now we know that *e ^{-pt}* is positive for any value of

*t.*Moreover, in any economic context we expect our tree to have a positive value; that is, R(t) > 0. Finally, by (3), the denominator of our fraction is negative. Hence, we conclude that

*dt/dp <*0; that is, a rise in the interest rate must reduce the optimal life of a tree grown for lumber. In our simple illustration the conclusion is not very startling, but when applied to more complex problems this same comparative statics procedure has provided very helpful and far less obvious results.

It should be noted that much of this analysis might be described as qualitative, in that only the sign of *dt/dp*, not its magnitude, is evaluated. But for many policy problems even this limited information can be very helpful. For example, evidence that a tax cut will stimulate employment can be useful even if the extent of the increase in employment is not clear. Thus, it is desirable to develop this sort of qualitative calculus, whereby from qualitative information (such as the signs of the second derivatives) one derives qualitative results. However, it is not always possible to do so. For example, we cannot determine the sign of the difference between two terms simply from the information that each term, by itself, is positive. Only when we know their relative magnitudes can we hope to determine the sign of the term that results when we subtract one from the other. Samuelson (1947) and Lancaster (1965) have explored the theory of qualitative comparative statics analysis and have characterized the sorts of results that can and cannot be derived with the aid of these procedures. Meanwhile, the applications of comparative statics analysis have continued to grow, with some of the most interesting contributions being those of Metzler (1945), Patinkin (1956), and Samuelson.

*Stock-flow analysis.* A number of writers, notably Bushaw and Glower (1957), have also produced comparative statics analyses of the important relationship between stocks and flows. They have pointed out that in a general equilibrium model the market for a typical commodity is characterized by *two* equilibrium conditions, one of them requiring that the excess demand for stocks be zero and the other specifying the same requisite for flows. Thus, even if the demand for stocks of some item exceeds the supply, flows may be in equilibrium (at a nonzero level) because any further speeding up of production of the good would raise its cost prohibitively. These observations and the more complex model to which they give rise offer illumination on such issues as the relationship between the liquidity preference (money stock equilibrium) and loanable funds (flow equilibrium) models of interest rate determination and Lerner’s (1944). distinction between the marginal efficiency of investment (flow) and the marginal efficiency of capital (stock). *[See* Stock-Flow Analysis.]

**Methods of dynamic analysis** . Dynamic analysis proper has developed a totally distinct methodology. By definition, these procedures must take explicit account of the passage of time and its economic influence.

*Period analysis.* The nonmathematical approach to dynamics took the form of period analysis, which was used frequently in business cycle theory, particularly in the work of Dennis Robertson (1926) and in the Swedish works of the 1930s, for example, Lindahl (1939). In period analysis, time is artificially partitioned into discrete intervals, sometimes referred to as “weeks” or “days” (which, of course, do not correspond to the calendar intervals bearing similar names). The developments in any particular period were taken to depend on the events that preceded it, and the consequences of these interconnections were then determined. The multiplier process, whereby an injection of funds into the expenditure stream can provide a convergent sequence of additions to national income, is an illustration of period analysis. Suppose 90 per cent of additions to income are spent during the period after they are received and that somehow an autonomous $1,000 increase in income occurs during the first “day” of our study. As a consequence of this increase in income, an additional $900 will be spent during the second day, and that amount will constitute added income for those who receive these payments. During the third day, these persons will, in turn, spend 1̅^{9}0̅ of this amount, that is, 1̅^{9}0̅ ($900) = $810, and so on, ad infmitum. (Some standard references on period analysis are Robertson 1926, appendix to chapter 5; Lindahl 1939, part 1. For a summary discussion, see also Baumol [1951] 1959, chapter 8.)

*Difference equations.* Period analysis has also adopted an extremely suitable mathematical technique—the difference equation. This is an equation in which the value of some variable during period *t* is a function of the values of this and perhaps other variables during earlier periods. For example, if I(t), P(t), and S(t) represent the values of inventory, production, and sales of some commodity during period t, then we have

*I(t+1) = I(t) + P(t) − S(t)*

This identity is a very simple first-order linear difference equation. It is said to be linear because none of the terms contain anything but the first power of a variable; that is, the terms do not contain [*I(t)*]^{2} or log P(*t*) or any other more complex expression. Moreover, the relationship in the preceding equation is said to be first order because it contains no time difference greater than one period; that is, only variables at periods t and t + 1 enter the equation. The following is an illustrative second-order linear difference equation in one variable, A(t), advertising expenditure at time t:

*A*(*t* + l) = 0.9*A*(*t*) + 0.1[*A(t)* - *A(t* - 1)].

This equation states that next period’s advertising expenditure, A(t + 1), is a weighted average of the current advertising outlay, A(t), and the current rate of growth of advertising outlay, A(t) - A(t - 1). The most remarkable feature of such a relationship is the long-term prediction of the future that it allows. Given a limited amount of information about advertising expenditure in the past, the equation can be used to make a series of predictions of advertising expenditures that *must* hold as long as the equation retains its validity, that is, as long as it is legitimate for *any* year, *t*. For example, suppose we take *t –* 1 = 1960, t = 1961, and *t* + 1 = 1962. If statistics indicate that *A(t -* 1) = A(1960) = 150 and *A(t)* = *A*(1961) = 200, our equation becomes

*A*(1962) = 0.9*A*(1961) + 0.1[*A*(1961) - *A*(1960)]

= 0.9(200) + 0.1(50) = 185.

Now, utilizing the information A(1962) = 185 and A( 1961) = 200, we can employ the same equation to make a prediction about advertising in 1963 (this time setting *t =* 1962):

*A*(1963) = 0.9*A*(1962) + 0.1[*A*(1962) - *A*(1961)]

= 0.9(185) + 0.1(-15) = 165.

In the same way our difference equation can be used in turn to make forecasts for 1964, 1965, etc., going on into the indefinite future. Of course, if the history of advertising follows no such readily predictable pattern, then the facts can only be represented adequately by a relationship more complicated than the second-order, single-variable difference equation given above.

Where, as in our first illustration, difference equations contain several distinct variables, such as *I(t)*, *S(t)*, and *P(t)*, it will take several simultaneous equations to constitute a determinate system capable of making predictions about the time path in question. Normally, we expect the system to contain as many equations as there are variables. Such a linear difference equation system, whether it contains one or several equations, can usually be solved. The solution of such a system consists of a set of relations expressing each variable as an explicit function of time, for example, *I(t)* = *f(t)*. These relations give us the value of each variable for any given calendar date, t; that is, the solution gives explicitly the time path of each variable.

The solution also enables us to characterize qualitatively the time path of each variable. In the linear case, these time paths are of four different types: stable, explosive, stable (or damped) oscillatory, and explosive oscillatory. The time path of a variable, A(t), is said to be *stable* (or *damped)* if, with the passage of time, the value of A(t) approaches some equilibrium value, *A _{c}*, as a limit, that is, if after some interval of time the value of A(t) becomes and remains as close to

*A*as we may pre-specify. The time path of a variable is called

_{c}*explosive*if the value of the variable either increases or decreases indefinitely and without limit, that is, if the value of the variable eventually exceeds any prespecified positive number or eventually falls below any prespecified negative number.

A time path is termed *oscillatory* if it is characterized by periodic increases and decreases in the value of the variable, that is, if the variable falls, then rises, then falls, etc., and does so at regular repetitive intervals of time. It is termed *nonoscilla-tory* or *monotonic* if the value of the variable continually rises, continually falls, or remains absolutely constant. Thus, for example, a time path is called *stable oscillatory* if the variable alternately takes values above and below its equilibrium level but its range of oscillation about that value grows constantly smaller.

To illustrate the concept of a solution, consider the very simple first-order linear difference equation *A _{t} =* CA

_{t-1}. This gives us successively

*A _{1} = CA*

_{0},

*A _{2} = CA*

_{1}=

*C*(

*CA*

_{0}) =

*C*

^{2}

*A*

_{0},

*A _{3} = CA*

_{2}= C(

*C*

^{2}

*A*

_{0}) = C

^{3}

*A*

_{0},

which yields the general solution

*A _{t}* =

*C*

^{t}A_{0}.

We see that if 0 < C < 1, this solution will be stable and nonoscillatory because each *A _{t}* will be successively smaller than the preceding value of A, and, indeed, A, will approach zero asymptotically. Similarly, if C > 1, the time path will be explosive— At will grow without limit.

Moreover, if C is negative, our time path will be oscillatory. If A_{0} is positive, then A_{1}, will be negative, A_{2} will be positive, etc. (It will be stable and oscillatory if 0 > C > -1 and explosive oscillatory if C<-1.)

Thus, we conclude that even with a given model different values of the parameters can lead to different types of time paths. A model may be stable for some more or less wide ranges of parameter values and unstable for others, and it is often possible to specify these ranges from the structure of the model, as has just been done in our very simple case. It should be noted that the solution is also influenced by the initial value of our variable, A_{0}, although in a *linear* difference equation model the long-run characteristics of the time path are unaffected by such initial conditions.

We can use our difference equation analysis to illustrate a subject that was mentioned earlier— Samuelson’s correspondence principle. Consider the following simple dynamic model describing a specific economic problem—the economic decline of a city. Let us suppose that the lower the per capita income in the city (and hence, the poorer its amenities), the more middle-class inhabitants will move to the suburbs in a given period, but the more rapid this exodus, the more quickly will per capita income decline. Thus, we have a cumulative process that can be described by the following equations:

*ΔY _{t}* =

*G*—

*kM*and

_{t}*M*=

_{t}*—wΔY*,

_{t-1}where ΔY_{t}, is the rate of growth of per capita income in the city in period *t, G is* the exogenous growth rate per period, reflecting perhaps a rising national output per capita, M_{t} is the number of middle-class migrants in period t, and *k* and w are positive constants. The first of our equations states that per capita income growth will be reduced below its exogenously determined rate by an amount proportional to migration from the city. The second (behavioral) equation alleges that migration is strictly proportional to the rate of decrease of per capita income in the previous period. Combining these equations, we obtain our basic difference equation,

ΔY_{t} = G + kwΔY_{t-1},

This equation can yield a stable time path only if few < 1, for otherwise a change in ΔY_{t-1} will lead to a still larger change in ΔY_{t}, and so on, ad in-finitum. This is analogous to the stability requirement C < 1 in our previous model.

Let us now find the equilibrium value of ΔY_{t}. This is a value, *E*, such that if ΔY_{t-1} = *E*, then we will also have (for the next period) ΔY_{t}, = E. Hence, substituting E for both ΔY_{t}, and ΔY_{t-1} in our difference equation, we have

E = G + kwE, or E = G/( 1 − kw).

**This shows that if the** . system is stable (few < 1), it will have reasonable comparative statics properties; a rise in G, the exogenous growth rate, will also increase *E*, the equilibrium growth rate of per capita income. But if the system is unstable, so that (1 – *kw*) is negative, it will no longer yield obviously sensible comparative statics results. A rise in the exogenous growth rate will then reduce the equilibrium rate of growth, and, indeed, only a negative exogenous growth rate can yield a positive equilibrium growth rate!

This, then, illustrates graphically the substance of the correspondence principle showing that an unstable system is likely to yield nonsensical comparative statics results. This is so because of the way in which the value of few enters both the dynamic and comparative statics solutions, a relationship that can be extended to much more complex systems.

In general, after some initial period of time, the path of a variable generated by a linear difference equation system will either be stable or explosive, and it will either be oscillatory or monotonic. Furthermore, it will retain its pattern throughout the remainder of its history, since more complex time paths, even time paths that switch from one of these characteristics to another, are ruled out by linearity. This restricts considerably the uses to which linear difference equation models can be put in economic analysis. For example, because they can generate oscillatory behavior, a number of linear business cycle models have been constructed. But, as Hicks (1950) and Goodwin (1951) have pointed out, these models suffer from a serious defect. For, aside from what may be considered the very exceptional case, on the “razor’s edge” borderline between stability and instability (the special case C = –1 in our model A_{t} = CA_{t-1}), the cycles generated by linear models must either gradually disappear (if their time path is stable) or the amplitudes of the cycles must increase without limit (the explosive case). Only by the use of nonlinear models can we generate cycles with any degree of constancy in their amplitude.

*Differential equations.* Another mathematical tool that has proved particularly useful in dynamic analysis is the differential equation. A simple example is the equation describing the behavior of a variable, *y*, whose rate of change, *dy/dt*, is proportionate to its own value, so that *dy/dt = ky.* Such a relationship would hold, for example, where interest on a bank deposit was compounded continuously, so that at every moment of time the principal would be increasing at a constant *k* per cent. We know that a function that satisfies this relationship is *y = ae ^{kt}*, where a is a constant. This last function is the solution to the preceding differential equation, and it obviously specifies the time path of the variable

*y.*In particular, we see that if k is positive,

*ae*must increase indefinitely as

^{kt}*t*becomes larger, so that the time path of

*y*will be explosive (without oscillation). On the other hand, if fe is negative, so that -k = c> 0, then

*ae*Hence,

^{kt}= ae-^{ct}= a/e^{ct}.*y - a/e*will get closer and closer to zero as t increases. Thus, if fe is negative, the time path of

^{ct}*y*will be stable (and nonoscillatory).

The preceding differential equation is said to be of first order because it contains only a first derivative of *y.* An example of a third-order linear differential equation with constant coefficients is

*d ^{3}y/dt^{3} = αd^{2}y/dt^{2} + βdy/dt + γy +* δ.

Such an equation will also yield a determinate time path for *y.* While the resulting time path may be more complex than those that can arise in the first-order case, it will still fall within one of the four types that characterize linear difference equations. And, once again, one may have recourse to multivariable differential equation systems composed of several simultaneous equations, or it may be appropriate to employ nonlinear differential equations that are more flexible in the time paths they generate. Mixed difference-differential equation models involving both derivatives and lagged variable values have also occurred in the literature. (For further materials on difference and differential equations and their applications in economic dynamics, see Samuelson 1947, chapters 10, 11, appendix B; Goldberg 1958; Allen 1956, chapters 1-4; Baumol [1951] 1959, chapters 9-16.)

**Stochastic dynamic models** . A number of dynamic (difference and/or differential equation) models in which stochastic (random) elements play an important role have been used in both theoretical and empirical applications. For example, Frisch (1933) proposed a business cycle model in which the dynamic equation generated a damped oscillatory time path, so that if left to itself, the model produced cycles that would tend to disappear. However, he superimposed on this structure a “shock variable”—a random element that caused unpredictable and sometimes sharp displacements in the time path. These shocks were meant to correspond to exogenous events, such as wars and crop failures, that can affect significantly the workings of the economy. Even with a linear model it was shown that such shocks can prevent the system from settling down to an equilibrium. Thus, the random shocks yield a time path characterized by fluctuations of varying initial amplitude and timing.

Dynamic stochastic models are found extensively in the econometrics literature dealing with specification and estimation of simultaneous equation systems. Here, following the pioneering work of Koopmans (1950) and his colleagues at the Cowles Commission, random shock variables in each of the relevant equations have provided the basis for probabilistic analysis of the statistical properties of the systems.

**Evaluation of dynamic models** . Unfortunately, the dynamic mathematical models that have so far been developed have suffered from two serious shortcomings. First, there has been little empirical evidence supporting the choice of the values of the coefficients or even the types of functional relationships utilized (linear or nonlinear, difference or differential, of first or of some higher order). Second, and perhaps even more serious, it is possible to show that the predicted behavior of the variables of such models is often highly sensitive to changes in functional relationships and coefficient values. For example, a small change in the value of a coefficient can lead to drastic qualitative changes in time paths; it can transmute stability into explosion, or monotonicity into oscillation. Since econometrics has up to this point provided so little empirical basis for the choice of dynamic models and since the implications of these models are so highly dependent on the assumed structure, we can as yet have little confidence in the applicability of the results of most dynamic analyses, either as descriptions of reality or as satisfactory bases for policy.

So far these models have mainly served negative purposes. They have functioned most effectively as counterexamples indicating cases where persuasive arguments were not in fact conclusive and where apparently attractive policy measures might fail to serve the purposes for which they were designed. For example, it has sometimes been alleged that profitable speculation must always mitigate fluctuations (they must always be stabilizing). The argument is that if a speculator is to make a profit, he must purchase when price is low and sell when price is high. Hence, his actions will raise prices in the one case and reduce them in the other, on both counts making for a more stable price level. A somewhat more explicit set of dynamic models was able to show that this argument was oversimplified and that when price trends as well as price levels were taken into account the profit-making speculator might easily aggravate any cyclical behavior that was present in the system.

Similarly, it can be shown that a policy of raising net government expenditure when national income is low and decreasing net government outlays when income is high may well worsen business cycles unless the changes in government expenditure are perfectly timed. Moreover, this defect may plague discretionary and other types of contra-cyclical policies as well as automatic contracyclical policy measures of the sort just described. Thus, even if dynamic models have been most effective in providing such negative conclusions, they clearly can be extremely important both for understanding the workings of the economy and for the formulation of economic policy. (For discussions of the effectiveness of contracyclical measures, see Friedman 1953; Phillips 1954; Baumol 1961.)

Despite the unsolved problems of dynamic analysis, there is a sense in which it may be considered indispensable for the progress of economics. In reality the economy is never fully in equilibrium. Therefore, the applicability of static and comparative statics analyses is necessarily somewhat limited. When things are in flux, only a dynamic model can account for everything that is going on. However, this argument should not be pushed too far, for in no science do theoretical constructs correspond perfectly to reality. One must always work with models that provide approximative illumination, and there can be little doubt that the economists’ static models have often successfully served this purpose.

**Recent developments—growth models** . In recent years the major focus of dynamic analysis has shifted from business cycles to longer-term economic growth. Recent writings in growth theory have ranged from illuminating observations firmly based on empirical evidence, an approach most effectively developed in the work of W. A. Lewis (1955) to the highly abstract growth model of von Neumann (1938). In between lies a spectrum of work running the full gamut of applicability and technical sophistication. Leibenstein (1954; 1957), Kaldor (1957; 1961), and Robinson (1956) have produced growth analyses whose mathematical complexity is not very great but which are extremely suggestive. For example, one of the Leibenstein models (1954, chapters 4-6) shows that an economy whose population grows at a fixed rate must somehow surpass some minimum increase in per capita income before it will “take off” and embark on a long-term course of growth. Any smaller increase in per capita income must, in his model, prove transitory, with the economy soon reapproaching a subsistence output level.

Kaldor’s model seeks to describe the interactions among the distribution of income, the development of technology, and economic growth. His argument, roughly speaking, is that the level of investment is largely determined by technical progress in an economy that tends in the long run to approximate full employment. But in such a system, the distribution of income must adapt itself to permit desired savings to equal investment demand. Thus, for example, if the latter exceeds the former, there will be excess demand, causing prices to rise and the share of profits in national income to increase. Since profit earners’ collective propensity to save exceeds that of wage earners, the resulting redistribution of income will increase the level of desired savings until it becomes equal to investment demand.

There has also been considerable interest in the model developed by Harrod (1948) and Domar (1957), in which the acceleration principle of investment demand and the multiplier theorem of saving analysis are employed to investigate the characteristics of a growth path that balances supply and demand. Savings, S (the supply of resources available for investment), is assumed to be proportional to the level of income, *Y*, so that S = *KY.* The demand for capital, C, is also assumed to be proportional to output (income), *Y*, so that the demand for net investment, I (which is the rate of growth of capital), becomes proportionate to the *rate of growth* of *Y*, that is, *I = dC/dt = adY/dt.* The two equations together then tell us that if demand for and supply of investment are to be equal, we must have *Ky = adY/dt.* The time path of income that satisfies this first-order differential equation is said to exhibit the “warranted rate of growth.” Several paradoxical conclusions follow. The equation shows, for example, that if there is overproduction (if more goods are saved or unconsumed than are wanted for investment), the way to remedy the situation is to increase production even faster (increase dY/dt)! For in this way the demand for new equipment can be stimulated sufficiently to take up the otherwise undemanded output. The time path that satisfies the Harrod-Domar requirements can readily be shown to involve a geometric rate of growth of income.

The highly mathematical growth models of recent years all have their origin in von Neumann’s original construct (1938). His magnificent piece of reasoning also foreshadows important portions of game theory and of the duality theory of linear programming. The von Neumann economy is a highly artificial construct in which the production function is characterized by constant returns to scale and perfectly fixed input-output coefficients. It is assumed that there are no scarce resources to limit the expansion of the economy, and the model is perfectly closed, with consumers appearing only as factors of production who use up outputs in the course of their own productive efforts. Finally, it is assumed that some positive amount of each and every input is employed in the production of each output. Using these highly restrictive assumptions, which have been considerably relaxed in later work by Kemeny, Morgenstern, and Thompson (1956), von Neumann was able to show, among other things, that there is a unique (maximal) growth rate for the entire economy and that this growth rate is the appropriate discounting rate of interest for the economy.

Dorfman, Samuelson, and Solow (1958) have developed what they call the turnpike theorem, which asserts that in the long run an optimal growth path for an economy must approximate the time path called for in the von Neumann model. Although short-run output maximization may be achieved through time paths that differ very significantly from the von Neumann path, the longer the period considered, the more closely will the optimal path approximate von Neumann’s. The authors suggest that their result is analogous to the routing of a trip, which, if it is very short, should often follow the side roads that lead most directly to one’s destination. But on a long journey, time will be saved by going out of one’s way to take the turnpike (the von Neumann route). Harold W. Kuhn has shown (in unpublished work) that the analogy is somewhat misleading because the optimal time paths only approach the “turnpike,” as it were, asymptotically and may in fact never coincide with the von Neumann path at any point in their history. Moreover, Kuhn has shown that the turnpike theorem does not hold for as general a set of circumstances as the authors originally believed. Nevertheless, the theorem remains an illuminating and substantial contribution to the literature of economic growth theory. Since the appearance of the Dorfman-Samuelson-Solow volume, the number of abstract growth models has grown enormously, broadening the analysis to include a wider range of technological circumstances and production relationships. *[See* Economic Growth, *article on* Mathematical Theory; *see also* Solow 1956; Phelps 1961; Uzawa 1964.]

William J. Baumol

*[Directly related are the entries* Business Cycles, *article On* Mathematical Models; Economic Equilibrium

## BIBLIOGRAPHY

Adelman, Irma (1961) 1964 *Theories of Economic Growth and Development.* Stanford Univ. Press.

Allen, R. G. D. (1956)1963 *Mathematical Economics.* 2d ed. New York: St. Martins; London: Macmlllan.

Baumol, William J. (1951) 1959 *Economic Dynamics: An Introduction.* 2d ed. New York: Macmillan.

Baumol, William J. 1961 Pitfalls in Contracyclical Policies: Some Tools and Results. *Review of Economics and Statistics* 43, Feb.:21-26.

Bushaw, D. W.; and Clower, R. W. 1957 *Introduction to Mathematical Economics.* Homewood, Ill.: Irwin.

Domar, Evsey 1957 Essays *in the Theory of Economic Growth.* Oxford Univ. Press.

Dorfman, Robert; Samuelson, Paul A.; and Solow, Robert M. 1958 *Linear Programming and Economic Analysis.* New York: McGraw-Hill.

Friedman, Milton (1953) 1959 *Essays in Positive Economics.* Univ. of Chicago Press. → See especially “The Effects of a Full-employment Policy on Economic Stability: A Formal Analysis,” pages 117-132.

Frisch, Ragnar (1933)1965 Propagation Problems and Impulse Problems in Dynamic Economics. Pages 155-185 in American Economic Association, *Readings in Business Cycles.* Homewood, Ill.: Irwin.

Frisch, Ragnar 1936 On the Notion of Equilibrium and Disequilibrium. *Review of Economic Studies* 3, Feb.: 100-105.

Goldberg, Samuel 1958 *Introduction to Difference Equations: With Illustrative Examples From Economics, Psychology, and Sociology.* New York: Wiley.

Goodwin, R. M. 1951 The Nonlinear Accelerator and the Persistence of Business Cycles. *Econometrica* 19: 1-17.

Harrod, R. F. (1948) 1960 *Towards a Dynamic Economics: Some Recent Developments of Economic Theory and Their Application to Policy.* London: Macmillan; New York: St. Martins.

Henderson, James M.; and Quandt, Richard E. 1958 *Microeconomic Theory: A Mathematical Approach.* New York: McGraw-Hill.

Hicks, J. R. 1950 *A Contribution to the Theory of the Trade Cycle.* Oxford: Clarendon.

Kaldor, Nicholas 1957 A Model of Economic Growth. *Economic Journal* 67:591-624.

Kaldor, Nicholas 1961 Capital Accumulation and Economic Growth. Pages 177-222 in International Economic Association, *The Theory of Capital: Proceedings.* New York: St. Martins.

Kemeny, J. G.; Morgenstern, Oskar; and Thompson, G. L. 1956 A Generalization of the von Neumann Model of an Expanding Economy. *Econometrica* 24: 115-135.

Koopmans, Tjalling C. (editor) 1950 Statistical *Inference in Dynamic Economic Models.* New York: Wiley.

Lancaster, K. J. 1965 The Theory of Qualitative Linear Systems. *Econometrica* 33:395-408.

Leibenstein, Harvey 1954 *A Theory of Economic-Demographic Development.* Princeton Univ. Press.

Leibenstein, Harvey (1957) 1963 Economic *Backwardness and Economic Growth: Studies in the Theory of Economic Development.* New York: Wiley.

Lerner, Abba P. 1944 *The Economics of Control: Principles of Welfare Economics.* New York: Macmillan. → See especially “Capital, Investment, and Interest,” pages 323-345.

Lewis, W. Arthur 1955 *The Theory of Economic Growth.* Homewood, Ill.: Irwin.

Lindahl, Erik 1939 *Studies in the Theory of Money and Capital.* London: Allen *&* Unwin.

Metzleh, Lloyd A. 1945 The Stability of Multiple Markets: The Hicks Conditions. *Econometrica* 13: 277-292.

Patinkin, Don (1956) 1965 *Money, Interest, and Prices: An Integration of Monetary and Value Theory.* 2d ed. New York: Harper.

Phelps, Edmund S. 1961 The Golden Rule of Accumulation: A Fable for Growthmen. *American Economic Review* 51:638-643.

Phillips, A. W. 1954 Stabilization Policy in a Closed Economy. *Economic Journal* 64:290-323.

Robertson, Dennis Holme (1926) 1949 *Banking Policy and the Price Level.* Rev. ed. New York: Kelley.

Robinson, Joan 1956 *The Accumulation of Capital.* Homewood, Ill.: Irwin; London: Macmillan.

Samuelson, Paul A. (1947)1958 *Foundations of Economic Analysis.* Harvard Economic Studies, Vol. 80. Cambridge, Mass.: Harvard Univ. Press. → A paperback edition was published in 1965 by Atheneum.

Solow, Robert M. 1956 A Contribution to the Theory of Economic Growth. *Quarterly Journal of Economics* 70:65-94.

Uzawa, Hirofumi 1964 Optimal Growth in a Two-sector Model of Capital Accumulation. *Review of Economic Studies* 31, no. 1:1-24.

Von Neumann, John (1938) 1945 A Model of General Economic Equilibrium. *Review of Economic Studies* 13, no. 1:1-9. → First published in German.