Learning, experience, and firm size

J ECO BUSN 1989; 41:283-296

283

Learning, Experience, and Firm Size Spiro E. Stefanou

The process of producers' learning by experience is characterized as a parameteradaptive process, rather than an ad hoc specification, that includes the accumulated volume of output in the firm's cost function. Deviations from the neoclassical optimization conditions can be decomposed into learning and production variation components. In a given time period, the production level may be greater or less than the level associated with price equals marginal cost, depending on the learning value and the direction of the marginal risk of production. The uncertainty-based measure of elasticity of scale is presented and indicates that the traditional view that the presence of uncertainty can discourage the managerial decision to expand is restrictive.

I. Introduction Uncertainty in production decision making has been addressed at length in the literature, most frequently focusing on price or output uncertainty in static settings (e.g., Sandmo 1971, Batra and Ullah 1974, Just and Pope 1978, Pope and Kramer 1979) or the role of price uncertainty on investment behavior (e.g., Hartman 1972, Pindyck 1982, Abel 1983). Little formal analysis has been undertaken to assess the impact of uncertainty in the economic or physical environment on the size of firms. When firms are observed to deviate from expected profit-maximizing behavior, the standard conclusion is that this deviation is attributable to the manager's response to risk. In turn, this suggests that the firm is maximizing the expected utility of profit. In a static setting, output-price uncertainty for the expected-utility maximizing firm may encourage a firm to produce less than it would in the absence of uncertainty; conversely, price stabilization may encourage an increase in output and firm size (Sandmo 1971). On the other hand, price uncertainty may encourage the expected-utility-maximizing firm to expand (Chambers 1983). These conclusions imply that the price and other kinds of uncertainty that the expected-utilitymaximizing manager may face can lead to ambiguous conclusions regarding the impact of uncertainty on firm size. An alternative perspective concentrates on how uncertainty influences firm size decisions. With this focus on the manager's information-processing ability, the deviation between observed and expected profit-maximizing behavior can be viewed as a response to uncertainty in learning and in production. This allows the

This study is Journal Series Article No. 7762 of the Pennsylvania Agricultural Experiment Station. Address reprint requests to Spiro E. Stefanou, Associate Professor of Agricultural Economics, Weaver Building, Pennsylvania State University, University Park, PA 16802. Journal of Economics and Business © 1989 Temple University

0148-6195/89/$03.50

284

S . E . Stefanou

development of an uncertainty-based measure of the elasticity of scale, related to the static, deterministic measure of elasticity of scale. Uncertainty concerning the technical relationships of production can take the form of the uncertain evolution of the state of technical knowledge or the uncertain technical input substitution and input-output response relationships of a particular production technology. The former type of uncertainty refers to a shift to a new technology; the latter concerns the specifics of a particular technology that are taken as an exogeneous restriction (Mirman 1973, Balcer and Lippman 1984, Mamer and McCardle 1987, Stefanou 1987). In addition to technical change, some of the increase in productivity and the changes in the size distribution of firms can be attributed to learning a particular technology. This article focuses on the role of information after a technology has been adopted. Such information can mitigate the effects of technological uncertainty. One source of information available to all managers is their production experience. Viewing experience as information, a model is developed that views learning as information and production uncertainty as part of the decision-making process. The manager must learn a number of things in order to thrive over the long run: market behavior and demand for the product(s) and services the operation produces, the market for inputs, and the production process. Traditionally, studies of the economies of firm size have focused on the production process and the technical relationship between the firm's output and its average cost or its efficiency. These studies necessarily assume that the manager has perfect information concerning the production, purchasing, and marketing domains. Specifically, it is assumed that the manager possesses and effectively employs accurate information that is germane to the firm's choice of technology, input mix, output levels, buying and selling strategies, and the overall profitability of the firm. In reality, however, the level of production attained is strongly influenced by the amount of information available to the manager. Not all resources are homogeneous and not all managers develop the same information set. While a certain proportion of the information set is theoretically available to all managers (e.g., via trade publications), individual managers have unique ways of processing information and unique problems in adapting the information to their resource situation. Hence, the stock of entrepreneurial ability, the range of resources available to the firm, and the firm's objective function influence how effectively the collected information is employed in production decision making. In what follows, managers' learning by experience is characterized as a parameteradaptive process over time using a system of stochastic differential equations. The optimal input demands and output supply characteristics of expected cost minimization and profit maximization over time are developed and contrasted with the results of the traditional static theory of the firm. In order to assess firm size relationships the elasticity of scale is developed in the presence of stochastic learning by experience and production. Finally, the managerial implications of stochastic learning by experience are discussed and some concluding comments are offered.

II. The Learning Process Some of the basic objectives of production analysis concern the degree of input substitution and the output response to input applications for a given technology. Identification of the substitution and output response relationships involves learning how to manipulate the production technology over time in response to stochastic events. The manager can

Experience and Firm Size

285

accumulate information about the input substitution and output response relationships from a number of sources, such as his own experience, free services (e.g., in agriculture, the extension service and free services available from salesmen), independent consultants, and conversations with other managers. The model developed below focuses on the manager's learning more about the production process as more production experience is acquired. The information the manager obtains is typically imperfect and must be screened and adapted to the operation's specific production characteristics (e.g., labor supply, managerial skill). In general, the learning process involves three mechanisms: an input mechanism for acquiring information, a filtering mechanism for interpreting and adapting the information acquired, and a decision, or executive, mechanism for making choices. The traditional approach to determining the impact and value of information relies on Bayesian updating and requires (a) that the problem be highly structured and (b) that the manager's choice subsequent to receiving information be predictable, given the structure of the problem (Raiffa 1968). A large number of studies in cognitive psychology suggest that decision makers typically do not process (or filter) and act on information in a manner consistent with the Bayesian decision analytic framework. Decision makers often employ a number of biases and heuristics in evaluating information on decision making (for a review see Sage 1981). The learning-by-experience literature typically assumes that a learning (or progress) function is a well-defined, deterministic feature of managerial behavior (Arrow 1962, Spence 1981, Brueckner and Raymon 1983, Devinney 1987). Such studies include the ad hoc specification of the accumulated volume of output in the firm's cost function. The production-experience phenomenon implies the existence of a spillover or by-product effect from the firm's production of goods and services--namely, the production of information (Rosen 1972). Hence, production is both an intermediate and a final output.

A Model o f Production Experience Processing The consideration of stochastic learning by experience, however, has drawn scant attention in the literature. Grossman et al. (1977) considered learning within a dynamic, Bayesian expectations revision framework; however, their analysis concerns a constraining decision set. Cross (1973, 1983) has focused on how psychological theories of learning can supplement conventional economic theory. However, the lack of generality of the models developed from the learning theory does not allow broad consideration of issues concerning the firm's determination of its optimal size. In a particular time period s, the firm attempts to minimize its variable costs of production subject to the expected production target. The expected production target, y(s), is a function of m inputs selected at time s, x(s) = [Xl(S) . . . . . Xm(S)], and a vector of parameters, A(s) = [A 1(s) . . . . . A n (s)] that characterize the production technology. This production technology is denoted F(x(s), A(s)). The analysis that follows can be extended to the multiple-output case, but it does not significantly add to the discussion. As time goes on the cumulative volume of actual (or realized) production as of time t can be expressed as

tp(

V(t) =

f0

s) ds,

(1)

286

S . E . Stefanou

where j~(s) is the actual rate of production in time s. The change in the volume of output is assumed to evolve stochastically according to the Ito process:

dV(t) = y(t)dt + a[x(t), A(t), V(t)] dW(t),

(2)

where W(t) is a Weiner process characterized by E { d W } -- 0 and E{(dW) 2} = dt. l The rate of expected change in the cumulative volume of output in time t is the expected current production target, y(t). The variation in the rate of production depends on the input decisions, the current estimates of the production technology, and the volume of actual output to date (the production history). The variation in production is influenced by the input levels. Some inputs may be variation-increasing and some inputs may be variation-decreasing (Just and Pope 1978). A useful production function should possess enough flexibility so that the effects of inputs on the deterministic component of production is different than on the stochastic component. The specification introduced in equation (2) allows for inputs to exhibit positive and negative marginal risk effects, where risk is defined as the variance in the rate of the accumulated volume of production, var(dV) = a( )dr. Such a model has been econometrically estimated by Just and Pope (1979), Griffiths and Anderson (1982), and Buccola and McCarl (1986). Over time, the firm learns more about its production technology; that is, the firm learns more about the parameters A. Thus, the state of knowledge regarding the parameter estimates evolves in a stochastic manner over time. This process can be expressed in continuous time form as a set of Ito equations:

dAj(t) = gj[x(t), A(t), V(t)] dt + ~'flx(t), A(t), V(t)] dBj(t)

(3)

for j = 1, 2 . . . . . n, where (Bt(t) . . . . . Bn(t)) is a vector Of independent Weiner processes, E{dBj(t)} = 0 and E{(dBi(t))(dBj(t))} = pijdt with pjj = 1, that are correlated with W(t), E{[dW(t)][dBj(t)]} ~ ojdt, for j = 1. . . . . n. Equations (2) and (3) constitute a stochastic system. 2 The specification of the Ito diffusion processes implies that the state variables possess the Markov property. In the case of equation (3), it is implied that the changes in the perceived parameter values are subject to an error that is serially uncorrelated. The history of the past changes in the vector of parameters is embodied in the current value of A. The rate of expected change in the parameter estimates (or the filtering process), gj(x, A, V), depends on the input decisions, the current estimates of the production technology, and the production history. The filter gj( ) is a rule for updating the coefficient estimates of the production function. A candidate specification is the filter that is linear in A known as the Kalman filter (Schuss 1980, pp, 260-261). The parameters of the functions gj( ) and ~'j( ) are specific to the commodity produced and the individual manager. While no a priori assignment of sign can be made concerning the partial derivatives (Ogj/Oxi) and (Ogj/OAi), one can reasonably propose that as the firm accumulates more and more production experience the rate of expected absolute value of the change in the parameter estimates decreases; ~For an introduction to stochastic calculus and stochastic differential equations see Schuss (1980) or Malliaris (1982, Chapter 2). 2 Kozin and Promdromou (1971) and Ludwig (1975) discuss the stochastic analog to system stability for stochastic differential equations systems.

Experience and Firm Size

287

that is, IOgj I/0 V < 0 for j = 1 . . . . . n. The reduction in uncertainty regarding changes in parameter estimates is assumed to be precipitated by increased production experience; that is, (O~j/OV) < 0 for j = 1. . . . . n. However, as the firm gains more and more production experience the parameter estimates change less and less. Specifically, it is assumed that lira g j (x, A, V) = 0 (4.1) V----~_v

lim ~'j(x, A, V) -- 0

(4.2)

lim tr(x, A, II) = ~r(x)

(4.3)

V---*v V--~v

for j ---- 1 . . . . . n. Equations (4.1) and (4.2) suggest that as production experience approaches a sizable level _v, the expected rate of change and the variation in the change in the production parameters is negligible. Equation (4.3) suggests that as production experience approaches _v the variation in the production level is influenced only by the input application levels. Modeling the information role of the production experience in this way can be viewed as consistent with the behavioral approach (Cross 1983). The existence of a feedback mechanism that directs the manager on to the behavioral paths through the use of historical production information is assumed in both the deterministic, g j ( ) , and random drift, ~'y(), components.

III. Learning, Uncertainty, and the Elasticity of Scale Production Decision Making

at the Margin

Consider the firm with a given technology seeking to optimally produce over time. The appendix presents the details of expected intertemporal cost minimization and profit maximization under stochastic learning by experience. The intertemporal linkage of one period's input level decision to another period's is attributed to the feature that as the manager accumulates production experience, the parameters of the production technology are revised and, thus, alter the variation in the rate of production. A more complete description of producer behavior and the cost function involves the minimization of the discounted stream of costs. Such a characterization is offered by Alchian (1959) and elaborated by De Alessi (1967). The value function, J ( a , v, y(t)), is the expected discounted stream of cost over the learning horizon. This formulation of the stochastic production experience model results in a cost function that includes a production experience variable with two components: (a) the actual production history to date, characterized by the accumulated volume of production, and (b) the planned production path. In the absence of production uncertainty and learning by experience, the static theory first-order condition emerges; namely, input price equals the Lagrangian (marginal cost) times the marginal physical product. The deviation from this optimization condition can be attributed to two effects. 3 The first effect involves the role of learning by experience. 3 The role of learning by experienceis represented by the derivativesof gj(). The direct impact of input use on production uncertainty is represented by the derivativesof tr(). Since ~'j( ) is independent of the input application levels, the correlations pij (i, j = 1.... , n) do not influence the optimizedinput applicationlevels. Further complicatingthe identification of the signs and relative magnitudes of the terms that deviate from the static deterministic condition is the indeterminate nature of the signs of the higher-order derivatives of J( ) with respect to v and the aj's.

288

S . E . Stefanou

The second effect involves the role of production uncertainty and has two components; (a) the impact of input use on production uncertainty and (b) the interaction of the change in the input use and the variation of parameter learning. In order to evaluate the effect of learning on the choice of production targets, the firm seeks to choose production targets over time to maximize the expected discounted stream of net revenues. The first-order condition deviates from the deterministic profit maximization result by the wedge n

(5)

W(t) = r - l J o + Go + ~_~(Oxk/Oy(t))Qt, k where n

Qk = Z ( O g j ( J

)/OX~c)Gaj 4-(Oa( )/Ox~)a( )Guy n

n

J

J

(6)

No a priori assignment of direction can be made for W(t) for a particular time period. In the absence of production uncertainty (i.e., pj = O, a( ) = 0 ~ Jo = Go = 0), the wedge simplifies to

W'(t) = Z ( O x k / O y ( t ) ) k

(Ogj( )/Oxk)Gaj

•

(7)

The two-period, deterministic case of learning by experience implies that production is not less than the nonlearning case (Rosen 1972). The intuition behind this result is obvious; namely, the additional cost of producing beyond the output level associated with price equal to marginal cost equals the learning value of the added production. While Ox~/Oy(t) > 0 can be safely assumed (i.e., all factors of production are normal), the direction of the bracketed term in equation (7) cannot be unambiguously assigned. For W'(t) < 0, producing less than the nonlearning case in period t implies that the future value of experimenting with producing less offsets the foregone instantaneous profit that can be realized by producing up to the output level consistent with price equals marginal cost. The observation of production at less than the profit-maximizing level at a particular time may not necessarily justify the assumption that the firm is maximizing the expected utility of profit. Hence, the negative deviation from expected profit-maximizing behavior may not be a response to risk, as Sandmo (1971) suggests, but the risk-neutral firm's response to technological uncertainty.

Elasticity o f Scale Observed increases in the size of firms (where size remains an unresolved issue) is considered evidence that firms are exploiting economies of scale. In fact, a number of factors unrelated to the technological and physical forces traditionally maintained as

Experience and Firm Size

289

reasons for a firm's expansion may be in force. The impact of uncertainty and learning can influence the measurement of firm size, as it is characterized by elasticity of scale. Treatment of the role of uncertainty in elasticity of scale is nearly absent from the literature. An exception is the contribution of Chambers (1983), who develops the scale elasticity for the expected-utility-maximizing firm facing uncertainty of output price. Devinney (1987) concluded that the scale effect is indeterminate in the presence of learning by experience. This result is based on using the shape of the average cost curve to identify the regions of increasing, constant, and decreasing returns to scale. While the shape of the average cost curve can identify these regions for the traditional static cost minimization problem (Hanoch 1975), one must derive the elasticity of scale when a nontraditional cost minimization problem is posed. The elasticity of scale is defined (Ferguson 1975, pp. 81-83) as E(x, t) = 0 In F(Izx, A = a)/0 In tzlu_, m

= F-' Z(OF/Oxk)xk.

(8)

k

Using the first-order condition from intertemporal costminimization, the marginal physical product of input i can be rewritten as n

OF/Oxk = X-'

)/OXk)Jai

wk + ~(Ogj( J

+ (&()/Ox~)~( )J~ + (&()/Oxk)~__,fj( )pj&., J

•

(9)

Substituting equation (9) into equation (8), the elasticity of scale can be expressed as

m[ E(x, t) = (~)-'~ k

.

WkXk +

y~(Ogj()/OXk)XkJ~, j'

+ Xk(&r( )/OXk)o( )J~ +xk(Oo( ) / O x k ) ~ f j ( )PjJ~aj • J

(10)

Thus, E(x, t) involves both deterministic and uncertainty components. The deterministic component, ~ n WkXk/XF, differs from the static, deterministic measure by the interpretation of X found in the appendix. The uncertainty component involves the variation in the learning by experience and production-accumulation processes. Even though the variation in parameter adjustments via learning, ~'j(), is independent of the choice of optimal input levels, parameter adjustments still influences E(x, t). The elasticity of scale is time-specific, because it depends on the starting values of the stochastic processes involved. If variation in production is independent of the production history, then v plays no role in the input application decision process; that is, Jo = Joo = J~aj = 0 for all

290

S.E. Stefanou j = 1. . . . . n. In this case the Lagrangian is the marginal cost of production; that is, the deterministic component of E(x, t) is equivalent to the static, deterministic model of production and n

(Oa( )/Oxk)E~j( )PjJ~aj = O. J Allowing the stock of production experience to increase toward some very large level _v implies

imE(x, t): -

z

WkXk+

k

(Ogj( )/Oxk)x Jo j

m

+ o( )yvv~l~(o~( )/Oxk)xk

(11)

k

Thus, the elasticity of scale fluctuates over time in response to management's learning the production technology, represented by the change in the deterministic component of the parameter filter arising from changes in input levels, and to the variation in the production process. In the absence of production uncertainty (i.e., q( ) -- 0) the limiting case in equation (11) still deviates from the static, deterministic elasticity of scale (the ratio of average cost to marginal cost) by the influence of changes in the input levels on the filtering process, (XF)- 1 ~ ~ (Ogj ( )/cgxk )x kJa r • The input application decision process is still dynamic, because input decisions in one period are linked to future periods by the influence of the decisions on parameter learning. When Ogj( )/OXk is a function of (v, a) alone, the limiting case of q( ) = 0 coincides with the static, deterministic elasticity of scale, which implies that learning is no longer taking place. The derivatives of gj( ) with respect to a given input cannot be signed a priori. Much can be gained by perturbing the production process in order learn more about the technology. For example, in agriculture no (or a very high) application of pesticide can provide significant information on the output and substitution responses of the pesticide input. Thus, Ogj( )/Oxi may be very large in either direction. In the absence of learning, this limiting case still deviates from the deterministic elasticity of scale by the influence of changes in the input levels on the variation in the rate of production, _a( )Jov(Oq( )/OXk)Xk. The input application decision process is no longer dynamic. Assuming that gj(), a ( ) , and ~'j(), j -- 1 . . . . . n are concave in v, J( ) is also concave in v (Brock and Magill 1979). The direction of the deviation from the deterministic static measure of the elasticity of scale depends on the weighted marginal risk of production, ~-~ (0 q ( )/OXk)X~. When positive (negative) marginal risk dominates, ~--~n(0q( )/OXk)Xk > 0 (< 0), the constant-returns-to-scale output level is greater (less) than the deterministic case.

IV. Implications and Conclusions The process of producers' learning by experience developed here significantly departs from past work by characterizing learning as a parameter-adaptive process, rather than an ad hoc specification, that includes the accumulated volume of output in the firm's cost

Experience and Firm Size

291

or profit function. The cost function is now a stock concept and includes the volume of production to date and the stream of planned production as arguments. The discrepancy between the neoclassical theory first-order conditions and those associated with stochastic learning by experience can be specifically decomposed into learning and variation components. Observed deviations from the neoclassical optimization conditions, typically ascribed to risk response on the part of an expected-utility-maximizing producer, are shown to be a response to technological and production uncertainty on the part of a risk-neutral producer. The phenomenon of stochastic learning by experience suggests some thorny measurement issues. The first concerns the oscillating patterns of input use and output supply over time. Although the practicing econometrician may attribute these patterns to heteroskedastic or autocorrelated error processes, or both, the oscillations may be a rational systematic response by a firm that is relying on its production experience to learn more about its production process. A second issue is that the empirical measurement of the impact of technical change on changes in productivity and the size distribution of firms tends to lump technical change and learning together in the same measure. While both technical change and learning involve a shift in the production function, it is important to distinguish between the returns to research and development (the technical change) and the returns to more precise knowledge of the boundaries of a given technology. The stochastic learning by experience model developed here suggests that the firm need not attempt to quickly converge to an optimal size. There is value in probing the flexibility of the production process. Of course, the gains of probing (measured as cost savings into the future) must be balanced against the immediate cost of probing. As the firm maintains the same technology over time, more and more is learned of the production process. Furthermore, as the volume of production approaches ~, the elasticity of scale measure presented here indicates that learning and production uncertainty effects result in deviations from the deterministic measure of scale elasticity. The direction of the deviation is ambiguous at any given time. The learning effect depends on how input changes influence the parameter filtering mechanism. The production uncertainty effect depends on whether positive or negative marginal risk with respect to the applied inputs dominates. The traditional view that the presence of uncertainty can discourage the managerial decision to exploit the firm's economies of scale is shown to be restrictive for the risk-neutral firm that is minimizing expected cost, or maximizing expected profit.

Appendix: Expected Intertemporai Optimization Cost Minimization Consider the firm with a given technology employing the input vector x(t) at prices w(t) = [wl(t) . . . . . win(t)] that seeks to minimize the expected discounted stream of cost over the learning time horizon (t, o0). The form statement of this optimization problem is

J(a'v'Y(t))=r~xnEt{f ~e-rsw'dS }

(A.1)

subject to

d V(t) = y(t) dt + ~[x(t),

Aft), V(t)] dW(t),

V(t) = v

292

S . E . Stefanou

dAj(t) = gj[x(t), A(t), V(t)] dt + ~'j[x(t), A(t), V(t)] dBj(t), y(r) = F(x(r), A(r))

Aj(t)=aj

forj=l

..... n

for all t < r < c~,

where a = (al . . . . . an), y(t) is the set of production targets over the planning horizon starting at time t, and r is the constant rate of discount. Assuming that J( ) is twice differentiable and letting subscripts of J( ) indicate partial differentiation, the dynamic programming equation (DPE) is (Malliaris 1982, Mangel 1985)

[ rJ(a, v, y(t)) = min

X(t), X(t)

Iw(t)'x(t) + y(t)Jo

L

n

+ Z g j ( x ( t ) , a, v)Jaj + (1/2)a2(x(t), a, v)J~u J n

+ a(x(t), a, v)E~j(x(t), a, v)pjJoaj J

+ X(t)(y(t) - F ( x ( t ) , a))] ,

(A.2)

where k(t) >_ 0 is the Lagrangian multiplier for the technical constraint at time t. Assuming an interior solution, the first-order maximization conditions are

w k - X(t)(OF/Oxk)+

(Ogj( )/Oxk)Jaj

][

+ (Oo( )/Oxk)a( )Joo

+ ( O o ( ) / O x k ) Z t j ( )PjJoaj + o( )ZO~j( )/OxkpjJvaj J J

=-0

(A.3)

for k = 1. . . . . m, and

y(t) = F(x(t), a).

(A.4)

The partial derivatives Jo and Jaj are interpreted as shadow values. The term Jo measures the change in the value function, given a change in the firm's initializing level of production experience; that is, the marginal value of experience or, alternatively, the value of information because information is the change in experience, Because more information is better, Jo should not increase costs and is expected to be nonpositive. When o( ) is independent of V, Jo is zero, because volume of production to date does not contribute to the decision concerning the current and future rates of production. The term Jar measures the change in the value function, given a change in the starting

Experience and Firm Size

293

estimate of the jth production parameter. Even though these production estimates may incorporate past information, no a priori direction can be assigned to this derivative. The solution to this problem yields the optimal input levels at time t, x* = x*(a, v, y(t)) and the Lagrangian ~* = ),*(a, v, y(t)). Inserting the optimal values x*( ) and ),*( ) in the DPE yields t/

rJ(a, v, y(t)) = w'x* +y(t)J~ + Z g j ( x * , a, V)Jaj J n

+ (1/2)aZ(x *, a, v)Joo + a(x*, a, v ) ~ ' i ( x ( t ) , a, v)ojJu,j )

(A.5)

+ h*(y(t) - F(x*, a))].

The expression r J( ) can be viewed as an imputed cost function at time t because the shadow values serve as the valuation of the stock variables. An alternative characterization of equation (A.5) can be developed with the Ito calculus. By Taylor-expanding J(a, v, y(t)), dividing through by dr, and taking the expectation starting at time t as dt ~ O, an expression for the expected change in the value function can be expressed as

n

(1/dt)Et {d J } = y(t)Jo + Z g j ( x * , a, v)Jaj J n

+ (1/2)aZ(x *, a, v)Jou + a(x*, a, v ) Z ~ j ( x ( t ) , a, v ) p j J v a j . J The DPE can now be more compactly written in terms of the expected change in the value function as

r J( ) = w'x* + (1/dt)Et{dJ}.

(A.6)

Thus, equation (A.6) implies that the opportunity cost of the stock of production experience, r J, is equal to the instantaneous cost, wtx *, plus the instantaneous change in the value of production experience, (1/dt)Et {dJ }. The Lagrangian, X*/r is the marginal cost of production in the static, deterministic setting. Viewing the value function J( ) as the expected long-run cost function, equation (A.5) can be differentiated with respect to y(t) and (using the envelope theorem) it yields the expected long-run marginal cost

Jy(t) :- (Jr + X*)/r.

(A.7)

The marginal value of production experience, Ju, is expected to be nonpositive, because more information is better (or, at least, not worse). The derivative Ju > 0 is admissable as long as rJy(t ) > Jo, implying that the increment in production experience is misinforming. This is a possibility due to the stochastic production specification; that is, the realization of a random effect may suggest a misinforming output response to a

294

S . E . Stefanou

particular level of input. Since X* _> 0, given a nonnegative marginal cost of production in time t, Jy(t) >- O, and Jo <_ O, the presence of learning by experience serves to shift the marginal cost curve to the right. The Lagrangian is interpreted as the present opportunity value of the marginal cost of production less the present value of the marginal cost of production experience. Profit Maximization In order to evaluate the effect of learning on the choice of production targets, let the firm seek to choose production targets over time to maximize the expected discounted stream of net revenues. The expected profit maximization problem can be stated in terms of the solution to the expected intertemporal cost minimization problem; that is, G(a, v) = maxEt

e-rS[py(s) - J(A, V, y(t))] ds

y(t)

(A.8)

subject to d r ( t ) = y(t) d t + a[x(t), A(t), V(t)] d W ( t ) ,

V(t) = v

(A.9)

d A j ( t ) = gj[x(t), A(t), V(t)] dt + ~'j[A(t), V(t)] d B j ( t ) ,

A j ( t ) = aj

for j = 1. . . . . n,

(A.10)

recognizing that x(t) = x*(a, v, y(t)) in equations (A.9) and (A.10). Assuming that G(a, o) is twice differentiable and letting subscripts of G( ) indicate partial differentiation, the DPE is

r L

rG(a, v) = max IPy(t) - J(a, v, y(t)) + y(t)Go y(t)

n

+ ~--~gj(x*(a, o, y(t)), a, o)Gaj + (1/2)a2(x*(a, v, y(t)), a, o)Goo J n

+ O'(X* ( a , o, y(t)), a, v ) ~ ' j ( x * ( a , o, y(t)), a, v)pjGoaj J

•

(A.11)

Assuming an interior solution, the first-order maximization condition is n

P - J.y(t)( ) + Go + Z ( O x ~ / O y ( t ) ) Q k = O, J

(A.12)

Experience and Firm Size

295

where n

Qk = Z ( O g j ( J

)/Ox~)Gai + (Oo( )/Ox~)o( )Go~ n

tt

+ (oo()lOx;~l~_,rj( )ojOvo, + (o()~_,(orj( )IOxDojGvoj. J

(A.13)

J

Since Jy(t)( ) = r -1 (J~ +h*), where r - I h* is the discounted marginal cost, the presence of production uncertainty and the impact of learning result in a deviation from the static, deterministic optimization condition of (p - r - I X * ) .

References Abel, A. B. 1983. Optimal investment under uncertainty. American Economic Review 73:228-233. Alchian, A. A. 1959. Costs and outputs. In The Allocation of Economic Resources: Essays in Honor of Bernard F. Halley (M. Abramovitz et al., eds.). Palo Alto: Stanford University Press. Arrow, K. J. 1962. The economic implications of learning by doing. Review of Economic Studies 29:155-173. Balcer, Y., and Lippman, S. A. 1984. Technological expectations and adoption of improved technology. Journal of Economic Theory 34:292-318. Baron, D. P. 1970. Price uncertainty, utility and industry equilibrium under pure competition. International Economic Review 11:463-480. Batra, R. N., and Ullah, A. 1974. Competitive firm and the theory of input demand under uncertainty. Journal of Political Economy 82:537-548. Brock, W. A., and Magill, J. P. G. 1979. Dynamics under uncertainty. Econometrica 47:843-868. Brueckner, J. K., and Raymon, N. 1983. Optimal production with learning by doing. Journal of Economic Dynamics and Control 6:127-135. Buccola, S. T., and McCarl, B. 1986. Small-sample evaluation of mean-variance production function estimators. American Journal of Agricultural Economics 68:732-738. Chambers, R. G. 1983. Scale and productivity measurement under risk. American Economic Review 73:802-805. Cross, J. G. 1973. A stochastic learning model of economic behavior. Quarterly Journal of Economics 87:239-266. Cross, J. G. 1983. A Theory of Adaptive Economic Behavior. Cambridge: Cambridge University Press. De Alessi, L. 1967. The short run revisited. American Economic Review 57:450-461. Devinney, T. M. 1987. Entry and learning. Management Science 33: 706-724. Ferguson, C. E. 1975. The Neoclassical Theory of Production and Distribution. Cambridge: Cambridge University Press. Griffiths, W. E., and Anderson, J. R. 1982. Using time-series and cross-section data to estimate a production function with positive and negative marginal risks. Journal of the American Statistical Association 77:529-536. Grossman, S. J., Kihlstrom, R. E. and Mirman, L. J. 1977. A Bayesian approach to the production of information and learning by doing. Review of Economic Studies 44:553-547. Hanoch, G. 1975. The elasticity of scale and the shape of average cost. American Economic Review 65:492-497. Hartman, R. 1972. The effects of price and cost uncertainty on investment. Journal of Economic Theory 5:258-266. Kozin, F., Promdromou, S. 1971. Necessary and sufficient conditions for almost sure sample stability of linear Ito equations. S l A M Journal of Applied Mathemativs 21:413-424.

296

S . E . Stefanou

Ludwig, D. 1975. Persistence of dynamical systems under random perturbations. SlAM Review 17:605-640. Just, R. E., and Pope R. D., 1978. Stochastic specification of functions and economic implications. Journal of Econometrics 7:67-86. Just, R. E., and Pope, R. D. 1979. Production function estimate and related risk considerations, American Journal for Agricultural Economics 61:276-284. Malliaris, A. G. 1982. Stochastic Methods in Economics and Finance. New York: Elsevier. Mamer, J. W., and McCardle, K. F. 1987. Uncertainty, competition and the adoption of new technology. Management Science 33:161-177. Mangel, M. 1985. Decision and Control in Uncertain Resource Systems. Orlando, Fla.: Academic. Mirman, L. J. 1973. The steady-state behavior of a class of one-sector growth models with uncertain technology. Journal of Economic Theory 6:219-242. Pindyck, R. S. 1982. Adjustment costs, uncertainty and the behavior of the firm. American Economic Review 72:415-427. Pope, R. D., and Kramer, R. A. 1979. Production under uncertainty and factor demands for the competitive firm. Southern Economic Journal 46:489-501. Raiffa, H. 1968. Decision Analysis. Reading, Mass.: Addison-Wesley. Rosen, S. 1972. Learning by experience as joint production. Quarterly Journal of Economics 86:366-382. Sage, A. P. 1981. Behavioral and organizational considerations in the design of information systems and processes for planning and decision support. IEEE Transactions on Systems, Man, and Cybernetics SMC-11:640-678. Sandmo, A. 1971. On the theory of the competitive firm. American Economic Review 61:65-73. Schuss, Z. 1980. Theory and Applications of Stochastic Differential Equations. New York: Wiley. Spence, A. M. 1981. The learning curve and competition. Bell Journal of Economics 12:49-70. Stefanou, S. E. 1987. Technical change, uncertainty and investment. American Journal of Agricultural Economics 69:158-165.