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Learning by doing and optimal factor demand
International Journal qfProduction Economics, 22 ( 1991 ) 67-79
67
Elsevier
Learning by doing and optimal factor demand* Kofi Kissi Dompere Department of Economics, ttoward University, Washington, DC 20059, USA
Kofi O. Nti Department of Management Science, The Mary Jean and I~¥ank P. Smeal College of Business Administration, The Pennsylvania State University, University Park, PA 16802, USA ( Received March 1. 1988: accepted in revised form March 19, 1991 )
Abstract In this paper we consider the problem of the firm's optimal plans for factor usage under conditions of profit maximization. Two factors of capital and labor are considered in a dynamic choice system where labor is endowed with a capacity to learn on the job but capital is not. By specifying the problem as an optimal control process, two shadow prices of capital and labor services are derived. We then advance a theory of investment demand and labor employment as a joint process. The effects of learning by doing and labor turnover rate on the paths of optimal investment and employment plans are investigated. Using the method of sensitivity analysis we also derive testable hypotheses about the effects of variations in the market wage rate and the interest rate on the optimal investment and employment plans of the firm.
1. Introduction
It is well known that learning by doing is common in social production. A study by Hirsch [ 1 ] reported that assembly workers discover many time-saving techniques over time. Management, too, may introduce improvements in manufacturing techniques and production controls to economize on materials. Learning by both labor and management reinforces each other and stimulates the growth and development of the learning process. Thus over time the real cost of doing certain repetitive tasks decreases. A basic problem in production economics is to conceptualize and integrate the phenomenon of learning by doing into the various theories of the production process. The main body of economic theory distinguishes between capital and labor largely on the grounds that capital is fixed in the short run but labor can be adjusted. But we think it is also necessary to distinguish between labor and capital with respect to capacity for learning in order to integrate learning by doing into the theory of production. Labor has a built-in potential and capacity of self-appraisal and improvement due to experience with an increasing frequency of a repetitive operation. Productive capital does not learn. In this paper we consider the effect of learning by doing on the firm's optimal plans for factor usage under conditions of profit maximization. Two factors of capital and labor are considered in a dynamic choice system where labor is endowed with a capacity to learn on the job but capital is not. We specify the problem as an optimal control process and derive shadow prices for capital and labor services. It turns out that learning by doing introduces an increasing disparity between the shadow wage rate for labor services and the market wage rate. The paper also develops a theory of investment demand and labor employment as a joint process. We use the method of dynamic sensitivity analysis to study the effects of variations in the market wage *The authors thank anonymous referees for their helpful comments and suggestions.
0925-5273/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
68
rate and the rate of interest on investment and employment plans of the firm. We show that an increase in the market wage rate affects investment positively when capital and labor are substitutes; however, investments are affected negatively when capital and labor are complements. Employment demand decreases with an increase in the market wage rate. The effect of an increase in interest rate on investment and employment is quite complex. When capital and labor are complements, we show that both investment and employment decline as the rate of interest increases. But if capital and labor are substitutes both employment and investment will increase with interest rate if the "wage effect" dominates the"investment price effect"; investment and employment will decrease if the investment price effect dominates the wage effect. The rest of the paper is organized as follows. Section 2 develops the mathematical structure of learning by doing used in this paper. In Section 3 we formulate the firm's decision problem as an optimal control process. Investment and employment demand functions are derived in Section 4. Section 5 contains the sensitivity analyses and a summary of the main propositions. This is followed by some concluding remarks.
2. On the mathematical structure of learning by doing It has been suggested in the economics literature that cumulative output or cumulative investment can serve as an index of the progress of learning by doing. (See, for example, Arrow [2], Fellner [3] and Sheshinski [4]). While these indices may be appropriate in some contexts, they do not capture certain aspects of learning by doing which we wish to focus on in this paper. First, we would like to relate the index of learning to the individual's experience on the job, especially the frequency of successful operation of a task. Hirsch [ 1 ] has stressed the importance of lot frequency and the frequency of successful operation of a task in individual learning. Second, we would like to relate our index of learning to the "Horndal effect" phenomenon described in Lundberg [5]. According to Lundberg, the Horndal Iron Works in Sweden achieved an average productivity growth of 2% per annum over a period of 15 years without any new investments. Cumulative investment may be an inadequate measure of learning in the presence of the "Horndal effect" phenomenon. Third, we note that cumulative output may increase for reasons other than learning by doing. For example, output increases may be due to the firm's decision to replicate its operations. Similarly, output per worker may increase because the firm has changed its production technique. For the above reasons, we will choose cumulative frequency of successful completions of a given repetitive task at a constant technique as our index of learning by doing. The structure of learning by doing presented here is based on the following assumptions: (a) The state of technique will remain constant throughout the analysis. (b) Labor is homogeneous for a given job specification. (c) Learning takes place through the successful completion of a given cycle of an operation and through the passage of time. (d) There is a probability distribution for the cumulative frequency of successes for the given job cycle. Let X(t) be a random variable representing the cumulative frequency of successes in a given repetitive operation at a constant technique over time. suppose X(t) assumes values in the set {x~ (t), x2 (t) ..... x~ (t) }, where xi (t) is the time path of the cumulative frequency of successes for employees of type i, 1 ~
69
ol(t) = ~ f A ( x , (t) ) ,=J
The function o~(t) may be assumed to be continuous with a finite number of discontinuities. The value of c~(t) at each time point may be viewed as a measure of the average state of efficiency gains within the firm's workforce due to the progress of learning by doing. The discontinuities in the function may be attributed to interruptions such as holidays, strikes, and other shut-downs. Note that if the operation stops for an extended period the worker may begin to forget some of what was learned; however, we do not consider this aspect of learning in this paper. It is well known that learning by doing cannot proceed forever. For example, Hirsch [ 1 ] observed that the rate of improvement usually declines as the frequency of successful operations increase. Eventually the progress of learning levels off. Therefore, we will assume that after some time improvements in the function o~(t) eventually tails off. We will also normalize so that oL(0) = 1. In view of the above discussion, it will be assumed that under continuous operation (no interruptions) the learning curve has the following properties: (a) o~(0) = 1, o~(t) > 1 for all t> 0; (b) &(t) >10 and lira & ( t ) = 0 ; t~oc,
(c) lim a ( t ) =c~, where ~ is a constant. I
~oo
A sketch of the learning function a (t) under continuous operation is illustrated in Fig. 1. In order to exploit the structure of learning by doing discussed above, we shall distinguish between "raw" and "efficient" factor services supplied by an input. At any moment in time the services provided by a factor may be measured either in raw units or in efficiency units. When a factor does not learn from experience the raw and efficient factor services will be the same at every time point. But when a factor learns from experience raw and efficient factor services will differ as operations proceed in time. Productive capital does not learn. The technical structure of a capital equipment fixes the flow of services that it can deliver per unit time. Hence the raw services supplied by a capital equipment is equal to the efficient capital services at every time point. As noted earlier labor is endowed with a capacity to learn. Consequently, at every time point raw and efficient labor services will differ due to the progress of learning by doing. Suppose the state of learning a(t)
Maximum learning distance
{ Initial level of all inexperienced workers.
Fig. 1. The learning function.
70
by doing within the firms workforce at time t is measured by the learning function o~(t). Then we can use the learning function a (t) to convert raw labor services into efficient labor services. Since a (t) >/1, efficient labor services is always greater than or equal to raw labor services. 3. The model Consider a firm which uses capital services K(t) and labor services L(t) to produce a single output flow Q(t) at time t. Throughout this paper (K(t) and L(t) are measured in efficiency units. Let I(t) be the investment goods acquired by the firm at time t and N ( t ) be the firm's employment of labor at time t. Let the price system be described by {p (t), w (t), q (t), r (t) }, where p (t) is the time path of price per unit output, w(t) is the time path of wage rate for labor, q(t) is the time path of price of capital goods, and r(t) is the time path of interest rate. All time dependent variables are assumed to be differentiable functions of time. We assume that output flow Q (t) is determined by the production function
Q(t) =F(K(t),L(t) )
( 1)
where F is twice differentiable and satisfies the capital and labor indispensability condition F(K,O) = F ( 0 , L ) =0. The state of the system at any time point is specified by K(t) and L (t). The evolution of the system is governed by two laws of motion, one for the flow of capital services and the other for the flow of labor services. The equation of motion for the flow of capital services is assumed to be K(t) = I ( t ) -c~K(t)
(2)
where ~, satisfying 0 ~<~< 1, is the constant rate of depreciation. The flow of labor services is assumed to satisfy the equation of motion
L=o~( t)N( t) -fl( t)L( t)
(3)
where a (t) is the learning function associated with learning by doing and fl (t), satisfying 0 ~/0. Since o~(0) = 1, the initial employment will supply L ( 0 ) = N o of efficient labor services. We can then relate the time path of efficient labor services L (t), given the initial employment No, to new employment N(t) by means of the equation
L(t) = o~(t) [No +N(t) ]
(4)
where N ( 0 ) = 0 . Substitution of (4) into (3) yields L = ce(t) [ l -fl(t)]N(t) -a(t)fl(t)No
(5)
Thus the equation of motion for efficient labor services depends on the time paths of cumulative learning c~(t), the employment of raw labor N(t), the rate of labor turnover fl(t) and the initial employment
No.
71 At any m o m e n t in time the performance of the enterprise is measured by the profit index R(t) =p(t)Q(t) - w(t)N(t) -q(t)I(t)
(6)
Given the price system and the state of technology, the problem of the firm is to choose the time paths of employment N(t) and investment I(t) to maximize the discounted present value of the enterprise. In the absence of internal adjustment costs (Gould [6] or Treadway [7 ] ) the problem may be set up mathematically as max i e - ' R ( t ) d t
l(t),N(t)
0
Subject to Q = F ( K,L ) k=I-aK
L = o~(t) [1 -fl(t) ] N ( t ) -o~(t)fl(t)No
(7)
d~-a(t)>~O N(t)>~O I(t)>~O R(t) =p(t)Q(t) - w(t)N(t) -q(t)I(t) L(O) ---No,
K(O) =Ko
The problem described by ( 7 ) is an optimal control process with constraints on the control variables. It is straightforward to verify that the constraint qualification is satisfied. The Hamiltonian for the problem is H(K,L,N,I,2~ ,22) = e-r'R (t) +21 ( I - S K ) +22 {oe(t) [ 1 - f l ( t ) ]N(t) - ~ ( t ) f l ( t ) N o }
(8)
And the Lagrangian is G(H, yl ,Y2,Y3) = H + y l [c~-- ol(t) ] + y 2 N + y3I
(9)
Applying the Pontryagin maximum principle and a theorem of Hestenes [ 8 ], the necessary conditions for an optimum are: OG/ON= -we-r'+22o~( 1 - f l ) +~'2 =0
(10a)
OG/OI= - q e - r t + 2 1 +73 = 0
(10b)
k=I-6K
(10c)
L = o~( 1 - f i ) N - o~flNo
(lOd)
-- ~ 1 =
OG/OK=pFKe-
~ t _ (~,1
(10e)
-J,2 = OG/OL=pFL e -~t
(lOf)
y, >/0,
y,[~-a(t)]=O
(10g)
72 >/0,
y2N=0
(lOh)
72 Y3>/O,
(lOi)
~'31= 0
We shall now solve for the values of the variables in the system of equations (10). We will study interior solutions where I ( t ) > 0, N ( t ) > 0 and o~( t ) < d. These conditions imply that the Lagrangian multipliers Yl = 72 = 73 = 0. Note that these conditions are important economically. Efficient capital and labor are indispensable in production yet there is instantaneous capital depreciation and labor turnover. It is, therefore, reasonable to investigate a situation where the enterprise acquires new investments and labor for at least replacement purposes. Due to the turnover of some experienced workers, it is unlikely that maximum learning on the average will be attained within the enterprise. Hence the condition o~(t) < ~ is appropriate. From equation (lOa) we have 72=e-"'w/c~( 1-fl) and, therefore,
J.2=e-"[--rwa(1--fl)+o~(1--fl)~i'--{(~(1--fl)--o@}w]/o~ 2 ( 1 - - f l ) 2
(11)
Eliminating ,~2 from ( I Of) and ( 11 ) we have
PFL =w[r+&/o~--fl/(1 --fl) -- vi,/w]/c~ ( 1 --fl)
(12)
or (13)
Fir = W ( t ) / p ( t )
where
W(t) =wtr+&/o~--fl/ ( 1 - - f l ) - w / w l / ~ ( l--fl)
(14)
Combining equation (10b) and (10e) it may easily be established that
PF~,.=q( d+ r)-it
(15)
or
F~, =C(t)/p(t)
(16)
where
C( t )=q( ~+ r)-[t
(17)
The complete model of optimal factor usage consists of (a) the production function
Q=F(K,L ) (b) two dynamic marginal productivity conditions
F~=W/p
and
F~:=C/p
and (c) four side conditions consisting of two equations of motion involving the time paths of investment and employment given by
I(t) = k + O K and
U(t)= [ L + a ( t ) fl(t) No]/a(t) [ 1 - f l ( t ) ] and two imputed cost functions for the flow of unit services of capital and labor given by
C(t)=q(r+6)-O and
W(t) =w[r+dUa-fl/ (1 -fl)-;v/wl/o~(1 -fl)
73 All of the above relationships are dynamic conditions which must hold at every point of time. Together, they combine to determine the output level, the levels of efficient labor and capital inputs, the levels of investment and employment and the shadow prices for efficient capital and labor services.
4. The investment and employment demand functions From the theory of optimal factor requirement we may derive differentiable factor demand functions for capital and labor services and a differentiable output supply function. These functions which will depend on the time paths of output price and the imputed factor costs may we written as
L=L( W,C,p)
(18)
K=K( W,C,p)
(19)
Q=Q(w,C,p)
(20)
Starting with the demand function (19) for efficient capital services and under the hypothesis that the firm follows an optimal path for the usage of capital services, the demand functions for investment goods can be obtained as a function of the time paths of prices through the method of comparative dynamics (see, for example, Dompere [9] and Jorgenson [ 10] ). We may obtain the demand function for investment goods by first differentiating the demand function for efficient capital services with respect to time to obtain
K= (OK/OW) (OW/Ot) + (OK/OC) (OC/Ot) + (OK/Op) (Op/Ot)
(21)
Equation (21 ) requires that we differentiate the imputed costs of capital and labor services with respect to time. Let/1 = & / a , a = 1 - fl and 0 =fl/a. Then it is straightforward to show that
~V= ( 0 W/OI ) = [ W~bt( O~0-1~/W -- &~r -]- OLO'~,g/~ff) + I~ -- O£0-W]/O~ 20"2
( 22 )
where ~u= r + / t - ~. Similarly,
C= (OC/Ot) = q ( 6 + r ) + q ? - #
(23)
Substituting (22) and (23) into (21) we obtain
K= (OK/OW) [w~( o~afi~/w-&6+ a0"(u/~,) "Jt-I~--OlO'l~]/a20 + (OK/OC) [O(6+r) + q k - / ] ] + (OK/Op)[~
"2
(24)
It follows from the side condition for the equation of motion of capital that
I=K+6K
(25)
By substituting k from eqn. (24) into (25) we obtain the demand function for investment goods as
I( t) = (OK/OW) [w~,{c~0-(&/w+(u/~u) -&6} + & - o~o~]/o:2a 2 + (OK/OC) [ 0 ( 6 + r ) + q f - q ] + (OK/Op) [~+OK(W,C,p)
(26)
The investment demand function may be written in implicit form as
I(t) = I ( W,C,p, (V,C,[9)
(27)
where
W= W(w,r;o~,fl)
and
C=C(q,r;g)
The investment demand function defined by equation (26) depends on (a) the price system {p(t),
w(t), q(t), r(t)} and its rate of change {b(t), &(t), 0(t), k(t)}, (b) the imputed cost of capital services
74 C( t ), (c) the imputed cost of labor services W( t ), ( d ) the progress of learning by doing a ( t ), and ( e ) the labor turnover rate fl(t). We now derive the employment demand function. Again, we will assume that the firm follows an optimal path for factor usage. The method of derivation is similar to the one used to obtain the investment demand function. Here, we combine the optimal path for the use of efficient labor services with the side conditions for the equation of motion for employment and the imputed cost of labor services. Differentiating the labor demand function ( 18 ) with respect to time, we obtain
L= (OL/OW) (OW/Ot) + (OL/OC) (OC/Ot) + (OL/Op) (Op/Ot) Substituting for (0 W/Ot) and (OC/Ot) eqns. (22) and (23 ), respectively, we obtain
L = (oL/o w ) [w~,{ a~( w/w+ ~,/~,) - ,~} + ~ - a o ~ ] / a 2~2 + (OL/OC) [ q ( r + 6 ) + q i ' - q ] + (OL/Op)p
(28)
The side condition for the equation of motion is given by
N= [ L + a ( t ) t ( t ) N o ] / a ( t ) [ 1 - i ( t ) ]
(29)
Substituting L from (28) into (29), we obtain the employment demand function
N( t ) = ( OL/OW) [ W~l{aa( l~/W'~-~l/~ff--C)Lb}q- ~l;--aO~ ]/a3a 3 + (OL/OC) [~( r + 6) + q k - / ] ] / o u r + (OL/Op)p/aa+ flNo/tr
(30)
From (4), No= ( L / a ) - N . Therefore, we can eliminate No from (30) and rewrite the employment demand function, after some rearrangement and simplification, as
N( t) = (OL/OW) [ w~{atr( fv/w+ ~//g/)--~}"~ I~--OLOW]/OL30"2(O"~-t) + (OL/OC) [ q ( r + 6 ) + q ~ - # ] / a ( a + f l ) + OL/Op)b/a(a+fl)
(31)
+ilL(W,C,p)/a(a+fl) The employment demand function may also be written in implicit form as
N(t) = N ( W,C,p, I;V,C,[7) where
W=W(w,r;a,fl)
and
C=C(q,r;6)
The employment demand function defined by ( 31 ) also depends on (a) the price system {p (t), w (t),
q(t), r(t)} and its rate of change {/)(t), if(t), q(t), k(t)}, (b) the imputed cost of capital service C(t), (c) the imputed cost of labor services W(t), (d) the progress of learning by doing a(t), and (e) the labor turnover rate fl(t). 5. Sensitivity functions and sensitivity analysis The investment and employment demand functions can be used to derive some testable hypotheses. Our objective here is to investigate how changes in certain time dependent parameters of the model affect the optimal paths of investment and employment. We will conduct the sensitivity analysis under the following assumptions: Parallel variations - Time dependent variables are subjected to parallel shifts without altering the slope at any time point;
75
Variationally static expectations - A variation in one component of the price system does not induce changes in other components of the price systems; and Stabilizing conditions - Learning by doing and the labor turnover rate have stabilized at o~(t) = & and fl(t) =p, respectively, so that & and fl are practically zero as the number of successful operations and time get very large. For further details on the methodology of dynamic sensitivity analysis, including a discussion of the concepts of parallel variations and variationally static expectations, see Oniki [ 11 ] and Tomovic and Vukobratovic [ 12 ]. A short explanation of the stabilizing conditions may be useful. Note that the stabilizing conditions apply after the firm has been operating for an arbitrary large time. In this case, ~ (t) will be arbitrarily close to c~. Also note that the stabilizing value of the labor turnover rate p must satisfy 0~
(32)
and
K=Bo + B 1 W+B2C+B3p
(33)
In the analysis given below, we consider variations with respect to the market wage rate and the interest rate. Although it is possible to consider variations with respect to the stabilizing values of the learning by doing and labor turnover parameters, we do not pursue that analysis in this paper.
5.1 Variation o f the market wage rate The effect of variations in the time path of the market wage rate on investment and employment may be obtained by differentiating the demand functions. Let us write the investment demand function (26) as
I( t ) = ( OK/OW)RI/ a2a: + ( OK/OC)R2 + ( OK/Op )b+ ~K( W,C,p ) = Bi R l / a2cr2 + Bz R 2 + B 3 / ) + S K where
B~ = (OK/OW),
B2 + (OK/OC),
83 = (OK/Op)
R, = w~u[ ~cr( Cv/w+ ~,/~u) - &a] + ~ - o~o-~
and
Rz =O( 8+ r) + qi'-~] With variationally static expectations, it is straightforward to show that the effect of a parallel shift in market wage rate on investments is given by
76 (OI/Ow) =B, ( OR,/ Ow) /t~202 + B2( OR2/ Ow) +B3(@lOw) d-(~B, (OW/Ow) ---B, [atr~/(0#/aw) + a a ~ - &b~+ (0#/aw) - atr(0f~/O#) (aCv/Ow) ]/a 202 +6B, [ r - (0~/aw) ] lacr-JrB3(ap/0w) [ (ap/Op) -t-t~] = BI (ola~t- & t ~ t ) / a 2a2 + ~BI r/atr where we have used (a~/Ow) = (O~law) = (O~lOw)
=0,
(oR2 aw) =0
and
(aW/Ow) = [ r - (agv/Ow) ]/aa=r/cetr When we impose the stabilizing conditions a (t) = t2, ft.(t) =p, & =1~= 0 on learning by doing and labor turnover rates and use ~u= r + & / a - / ~ / ( 1 - fl) = r and ~u= k, we get
(allOw) =rB1 ( i / r + ~ ) / a ( 1- p ) =rBl Erla(1 - p ) where
Er = (k/r+6) If interest rates are steady or rising then i>/0 and Er> 0. If interest rates are declining but IUrl < 6, then Er> O. This requires that the rate of change of interest rate be less than the rate of depreciation. Suppose either of the above conditions hold so that Er> 0. Then the sign of (OI/Ow) is determined by the sign of BI. However, the sign of Bl depends on whether capital and labor are substitutes. If capital and labor are substitutes then (OK/Ow) > 0. But (OK/Ow) = (OK/OW) (OW/Ow) =B~ r/aa. Therefore, B~ = (OK/OW) > 0 if capital and labor are substitutes. Similarly, B1 < 0 if capital and labor are complements. Hence (OI/Ow) is positive or negative according as capital and labor are substitutes or complements. We now consider the effect of market wage rate on employment. The time path of optimal employment by the firm is given by
N ( t ) =At R~/ [o~3a2( a + fl) ] + A2Rzl a( a+ fl) "bA3fil ~( c~d-fl) + fl(Ao +A1 W+AzC+A3p)/o~(a+ fl)
(34)
The sensitivity function under conditions of variationally static expectations may be computed as
( ON/Ow) =A, (aR,/Ow)la3az(a+fl) + flA, (OW/Ow)/c~(a+fl) Substituting for the partial derivatives (ORt/Ow) and (aW/aw) in (34) and using the stabilizing conditions it can be established that
(ON/Ow) =rAt ( i/r+6) /~2( 1- p ) = rA, Er/~2( 1- p ) where
Er= (Ur+~) We shall assume a normal demand curve for factor services. Then (OL/O W) =A~< 0. Again, if interest rate are steady or rising then i>/0 and Er> O. On the other hand, if interest rates are declining then E~> 0 if ]i/r] < 5. This requires that the rate of change of interest rate be less than the rate of depreciation. If either of the above conditions holds then (ON/Ow) < O.
77
5.2 Variation of interest rate The effect of a variation of interest rate on investments is given by
(OI/Or) =B~ ( OR,/Or)/a2a2+B2(OR2/Or) +8[B~ (OW/Or) +B2(OC/Or) ] Substituting for the partial derivatives and using the stabilizing conditions we obtain
(OI/Or) = wB~ (~v/w+8)/6~(1 - p ) + q B 2 ( q / q + 8 )
(35)
It is interesting to note from (35) that the effects of variations in the time path of the rate of interest on the time path of investment consists of a wage effect wB~ (~v/w+8/6~( 1 - p ) and an investment-price effect qB2 ( (i/ q + 8). We will assume, as before, that Ew = ( ~v/ w + 8) and Eq = ( (i/ q + 8) are both positive. Clearly these conditions will hold if wages and investment price are both steady or increasing (~i,>~0 and ~ >t 0). But if wages are decreasing then E,, > 0 if I ~i,/w I < 8 that is, if the net effect of wage deflation and depreciation is positive. Similarly, if investment price is decreasing, then the assumption Eq> 0 requires that I(i/qf < 8; the net effect of investment price deflation and depreciation is positive. Under these assumptions the sign of (OI/Or) is determined by the sign of B~ and B2. However, the signs of Bt and B2 depend on whether capital and labor are substitutes or complements. If capital and labor are complements then (OK/Or) 0 and 8 2 < 0. In this case (OI/Or) > 0 if the wage effect dominates the investment-price effect; and (OI/Or) < 0 if the investmentprice effect dominates the wage effect. Similarly, the effect of a variation o.f interest rate on employment can be written as
(ON~Or) =31 w( ~v/wq- 8) / 6~2 ( 1 --p ) +A2 q( ~t/ q + 8) / a If we assume, as in the previous analysis, that ((t/q+ 8) and (;v/w+ 8) are both positive, then the sign of (ON~Or) also depends on whether capital and labor are complements or substitutes. If capital and labor are complements then A: < 0; however, A2 > 0 if capital and labor are substitutes. But (0L/0w) < 0 if demand for labor services is normal. And (0L/0w) = (OL/O W) (OW/lOw) = A ~r/aa. Therefore, A~ < 0. Hence, (ON~Or) < 0 if capital and labor are complements. But if capital and labor are substitutes then we can deduce, as before, that (ON~Or) > 0 if wage effect dominates the investment-price effect, and (ON~Or) < 0 if the investment-price effect dominates the wage effect.
5.3 Summary of main propositions In this subsection, we summarize and interpret the main propositions concerning the effects of parameter variations on the time paths of optimal investment and employment plans. Several of the resuits depend on whether capital and labor services are considered as substitutes or complements. It is, therefore, useful to clarify this issue. Let us recall that in a two factor model, capital and labor are substitutes if the technology has the variable proportions property. In this case, the two factors are limitative; that is, an increase in the relative usage of a factor is both necessary and sufficient for output to increase. And the inputs may be smoothly substitutable in the production of the output. But if the production technology is of the fixed proportions type then the inputs are limitational; that is, an increase in the usage of an input is necessary but not sufficient for the output to increase. In this case, the two inputs are complements in production (see Georgescu-Roegen [ 13,14 ], and Ferguson [ 15 ] for details). In view of the above discussion, it is necessary to consider situations where capital and labor services are either substitutes or complements. We, therefore, distinguish between these two cases in the statement of our main propositions. Also, we note that the conditions Er> 0, Ew> 0 and Eq> 0 used in the propositions were defined and interpreted in Sections 5.1 and 5.2.
78
Proposition I. If Er> 0 and capital and labor services are complements (substitutes) in production then the time path of the volume of optimal investment plans is inversely (positively) related to the time path of the market wage rate. Proposition 2. I f E r > 0 then the time path of the firm's optimal employment plans is inversely related to the time path of the market wage rate. Note that in Propositions 1 and 2 the variations in the time path of the market wage rate on the optimal paths of investment and employment also affect the imputed (shadow) wage rate.
Proposition 3. IfEw> O, Eq> O, and capital and labor services are complements, then the time path of the firm's investment and employment plans are inversely related to the time path of the rate of interest. Proposition 4. IfEw> 0, Eq> O, and capital and labor services are substitutes, then the time paths of volumes of optimal investment and employment plans will be inversely (positively) related to the time path of the rate of interest according to whether the investment price effect dominates (is dominated by) the wage effect. The intuition behind Proposition 4 is related to the part played by the relative magnitude of capital and labor in production. If the labor component is small relative to the capital component, then a rise (fall) in the interest rate will inversely (positively) affect investment decision because it increases (reduces) the burden of the interest cost on sunk capital. Labor services will, therefore, be substituted for capital services and thus employment will increase. 6. Conclusion
The model developed in this paper has been used to analyze the role of learning by doing on the firm's optimal demand for investment and employment. A divergence between the market wage rate and the shadow wage rate (the Horndal effect) was established. The effects of variations of market wage rate and interest rate on investment and employment demand were investigated. Several results were dependent on whether capital and labor services are substitutes or complements. We must point out that the theory developed here is incomplete and has certain shortcomings. Technological progress in the investment goods sector as well as internal adjustment costs were ignored. The sensitivity analyses were restricted to variationally static expectations and linear approximations for the investment and employment demand functions. These simplifications facilitated the derivations and the interpretations of the results. We plan to relax some of the assumptions used in this paper and explore further generalizations of the model in future research. References 1 Hirsch, W.Z., 1952. Manufacturing progress functions. Review of Economics and Statistics, 34: 143-155. 2 Arrow, K.J., 1962. The economic implications of learning by doing. Review of Economic Studies, 29:155-172. 3 Fellner, W., 1969. Specific interpretations of learning by doing. Journal of Economic Theory, 1:119-140. 4 Sheshinski, E., 1967. Tests of the learning by doing' hypothesis. Review of Economics and Statistics, 49: 568-578. 5 Lundberg, E., 1961. Produktivitet och R~intabilitet. P.A. Norstedt and S6ner, Stockholm. 6 Gould, J.P., 1968. Adjustment costs in the theory of investment of the firm. Review of Economic Studies, 35: 47-55. 7 Treadway, A.B., 1970. Adjustment cost and variable inputs in the theory of competitive firm. Journal of Economic Theory, 2: 329-347. 8 Hestenes, M.R., 1965. On variational theory and optimal control theory. Siam Journal of Control, 3: 23-48. 9 Dompere, K.K., 1980. Sensitivity, optimality, and stability conditions in a pure theory of investment behavior. Ph.D. Dissertation, Temple University, Philadelphia.
79 10 Jorgenson, D.W., 1967. The theory of investment behavior. In: R. Ferber (ed.), Determinants of Investment Behavior. National Bureau of Economic Research, pp. 129-155. 11 Oniki, H., 1973. Comparative dynamics (sensitivity analysis) in optimal control theory. Journal of Economic Theory, 6: 265-283. 12 Tomovic, R. and Vukobratovie, M., 1972. General Sensitivity Theory. American Elsevier, New York. 13 Georgescu-Roegen, N., 1935. Fixed coefficients of production and marginal productivity theory. Review of Economic Studies, 3: 40-49. 14 Georgescu-Roegen, N., 1955. Limitationality, limitativeness and economic equilibrium. In: H.A. Antosiewiez (ed.), Proceedings of the 2nd Symposium in Linear Programming, Vol. 1. National Bureau of Standards, Washington, DC, pp. 295330. 15 Ferguson, C.E., 1971. The Neoclassical Theory of Production and Distribution. Cambridge Univ. Press, Cambridge.
67
Elsevier
Learning by doing and optimal factor demand* Kofi Kissi Dompere Department of Economics, ttoward University, Washington, DC 20059, USA
Kofi O. Nti Department of Management Science, The Mary Jean and I~¥ank P. Smeal College of Business Administration, The Pennsylvania State University, University Park, PA 16802, USA ( Received March 1. 1988: accepted in revised form March 19, 1991 )
Abstract In this paper we consider the problem of the firm's optimal plans for factor usage under conditions of profit maximization. Two factors of capital and labor are considered in a dynamic choice system where labor is endowed with a capacity to learn on the job but capital is not. By specifying the problem as an optimal control process, two shadow prices of capital and labor services are derived. We then advance a theory of investment demand and labor employment as a joint process. The effects of learning by doing and labor turnover rate on the paths of optimal investment and employment plans are investigated. Using the method of sensitivity analysis we also derive testable hypotheses about the effects of variations in the market wage rate and the interest rate on the optimal investment and employment plans of the firm.
1. Introduction
It is well known that learning by doing is common in social production. A study by Hirsch [ 1 ] reported that assembly workers discover many time-saving techniques over time. Management, too, may introduce improvements in manufacturing techniques and production controls to economize on materials. Learning by both labor and management reinforces each other and stimulates the growth and development of the learning process. Thus over time the real cost of doing certain repetitive tasks decreases. A basic problem in production economics is to conceptualize and integrate the phenomenon of learning by doing into the various theories of the production process. The main body of economic theory distinguishes between capital and labor largely on the grounds that capital is fixed in the short run but labor can be adjusted. But we think it is also necessary to distinguish between labor and capital with respect to capacity for learning in order to integrate learning by doing into the theory of production. Labor has a built-in potential and capacity of self-appraisal and improvement due to experience with an increasing frequency of a repetitive operation. Productive capital does not learn. In this paper we consider the effect of learning by doing on the firm's optimal plans for factor usage under conditions of profit maximization. Two factors of capital and labor are considered in a dynamic choice system where labor is endowed with a capacity to learn on the job but capital is not. We specify the problem as an optimal control process and derive shadow prices for capital and labor services. It turns out that learning by doing introduces an increasing disparity between the shadow wage rate for labor services and the market wage rate. The paper also develops a theory of investment demand and labor employment as a joint process. We use the method of dynamic sensitivity analysis to study the effects of variations in the market wage *The authors thank anonymous referees for their helpful comments and suggestions.
0925-5273/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
68
rate and the rate of interest on investment and employment plans of the firm. We show that an increase in the market wage rate affects investment positively when capital and labor are substitutes; however, investments are affected negatively when capital and labor are complements. Employment demand decreases with an increase in the market wage rate. The effect of an increase in interest rate on investment and employment is quite complex. When capital and labor are complements, we show that both investment and employment decline as the rate of interest increases. But if capital and labor are substitutes both employment and investment will increase with interest rate if the "wage effect" dominates the"investment price effect"; investment and employment will decrease if the investment price effect dominates the wage effect. The rest of the paper is organized as follows. Section 2 develops the mathematical structure of learning by doing used in this paper. In Section 3 we formulate the firm's decision problem as an optimal control process. Investment and employment demand functions are derived in Section 4. Section 5 contains the sensitivity analyses and a summary of the main propositions. This is followed by some concluding remarks.
2. On the mathematical structure of learning by doing It has been suggested in the economics literature that cumulative output or cumulative investment can serve as an index of the progress of learning by doing. (See, for example, Arrow [2], Fellner [3] and Sheshinski [4]). While these indices may be appropriate in some contexts, they do not capture certain aspects of learning by doing which we wish to focus on in this paper. First, we would like to relate the index of learning to the individual's experience on the job, especially the frequency of successful operation of a task. Hirsch [ 1 ] has stressed the importance of lot frequency and the frequency of successful operation of a task in individual learning. Second, we would like to relate our index of learning to the "Horndal effect" phenomenon described in Lundberg [5]. According to Lundberg, the Horndal Iron Works in Sweden achieved an average productivity growth of 2% per annum over a period of 15 years without any new investments. Cumulative investment may be an inadequate measure of learning in the presence of the "Horndal effect" phenomenon. Third, we note that cumulative output may increase for reasons other than learning by doing. For example, output increases may be due to the firm's decision to replicate its operations. Similarly, output per worker may increase because the firm has changed its production technique. For the above reasons, we will choose cumulative frequency of successful completions of a given repetitive task at a constant technique as our index of learning by doing. The structure of learning by doing presented here is based on the following assumptions: (a) The state of technique will remain constant throughout the analysis. (b) Labor is homogeneous for a given job specification. (c) Learning takes place through the successful completion of a given cycle of an operation and through the passage of time. (d) There is a probability distribution for the cumulative frequency of successes for the given job cycle. Let X(t) be a random variable representing the cumulative frequency of successes in a given repetitive operation at a constant technique over time. suppose X(t) assumes values in the set {x~ (t), x2 (t) ..... x~ (t) }, where xi (t) is the time path of the cumulative frequency of successes for employees of type i, 1 ~
69
ol(t) = ~ f A ( x , (t) ) ,=J
The function o~(t) may be assumed to be continuous with a finite number of discontinuities. The value of c~(t) at each time point may be viewed as a measure of the average state of efficiency gains within the firm's workforce due to the progress of learning by doing. The discontinuities in the function may be attributed to interruptions such as holidays, strikes, and other shut-downs. Note that if the operation stops for an extended period the worker may begin to forget some of what was learned; however, we do not consider this aspect of learning in this paper. It is well known that learning by doing cannot proceed forever. For example, Hirsch [ 1 ] observed that the rate of improvement usually declines as the frequency of successful operations increase. Eventually the progress of learning levels off. Therefore, we will assume that after some time improvements in the function o~(t) eventually tails off. We will also normalize so that oL(0) = 1. In view of the above discussion, it will be assumed that under continuous operation (no interruptions) the learning curve has the following properties: (a) o~(0) = 1, o~(t) > 1 for all t> 0; (b) &(t) >10 and lira & ( t ) = 0 ; t~oc,
(c) lim a ( t ) =c~, where ~ is a constant. I
~oo
A sketch of the learning function a (t) under continuous operation is illustrated in Fig. 1. In order to exploit the structure of learning by doing discussed above, we shall distinguish between "raw" and "efficient" factor services supplied by an input. At any moment in time the services provided by a factor may be measured either in raw units or in efficiency units. When a factor does not learn from experience the raw and efficient factor services will be the same at every time point. But when a factor learns from experience raw and efficient factor services will differ as operations proceed in time. Productive capital does not learn. The technical structure of a capital equipment fixes the flow of services that it can deliver per unit time. Hence the raw services supplied by a capital equipment is equal to the efficient capital services at every time point. As noted earlier labor is endowed with a capacity to learn. Consequently, at every time point raw and efficient labor services will differ due to the progress of learning by doing. Suppose the state of learning a(t)
Maximum learning distance
{ Initial level of all inexperienced workers.
Fig. 1. The learning function.
70
by doing within the firms workforce at time t is measured by the learning function o~(t). Then we can use the learning function a (t) to convert raw labor services into efficient labor services. Since a (t) >/1, efficient labor services is always greater than or equal to raw labor services. 3. The model Consider a firm which uses capital services K(t) and labor services L(t) to produce a single output flow Q(t) at time t. Throughout this paper (K(t) and L(t) are measured in efficiency units. Let I(t) be the investment goods acquired by the firm at time t and N ( t ) be the firm's employment of labor at time t. Let the price system be described by {p (t), w (t), q (t), r (t) }, where p (t) is the time path of price per unit output, w(t) is the time path of wage rate for labor, q(t) is the time path of price of capital goods, and r(t) is the time path of interest rate. All time dependent variables are assumed to be differentiable functions of time. We assume that output flow Q (t) is determined by the production function
Q(t) =F(K(t),L(t) )
( 1)
where F is twice differentiable and satisfies the capital and labor indispensability condition F(K,O) = F ( 0 , L ) =0. The state of the system at any time point is specified by K(t) and L (t). The evolution of the system is governed by two laws of motion, one for the flow of capital services and the other for the flow of labor services. The equation of motion for the flow of capital services is assumed to be K(t) = I ( t ) -c~K(t)
(2)
where ~, satisfying 0 ~<~< 1, is the constant rate of depreciation. The flow of labor services is assumed to satisfy the equation of motion
L=o~( t)N( t) -fl( t)L( t)
(3)
where a (t) is the learning function associated with learning by doing and fl (t), satisfying 0 ~
L(t) = o~(t) [No +N(t) ]
(4)
where N ( 0 ) = 0 . Substitution of (4) into (3) yields L = ce(t) [ l -fl(t)]N(t) -a(t)fl(t)No
(5)
Thus the equation of motion for efficient labor services depends on the time paths of cumulative learning c~(t), the employment of raw labor N(t), the rate of labor turnover fl(t) and the initial employment
No.
71 At any m o m e n t in time the performance of the enterprise is measured by the profit index R(t) =p(t)Q(t) - w(t)N(t) -q(t)I(t)
(6)
Given the price system and the state of technology, the problem of the firm is to choose the time paths of employment N(t) and investment I(t) to maximize the discounted present value of the enterprise. In the absence of internal adjustment costs (Gould [6] or Treadway [7 ] ) the problem may be set up mathematically as max i e - ' R ( t ) d t
l(t),N(t)
0
Subject to Q = F ( K,L ) k=I-aK
L = o~(t) [1 -fl(t) ] N ( t ) -o~(t)fl(t)No
(7)
d~-a(t)>~O N(t)>~O I(t)>~O R(t) =p(t)Q(t) - w(t)N(t) -q(t)I(t) L(O) ---No,
K(O) =Ko
The problem described by ( 7 ) is an optimal control process with constraints on the control variables. It is straightforward to verify that the constraint qualification is satisfied. The Hamiltonian for the problem is H(K,L,N,I,2~ ,22) = e-r'R (t) +21 ( I - S K ) +22 {oe(t) [ 1 - f l ( t ) ]N(t) - ~ ( t ) f l ( t ) N o }
(8)
And the Lagrangian is G(H, yl ,Y2,Y3) = H + y l [c~-- ol(t) ] + y 2 N + y3I
(9)
Applying the Pontryagin maximum principle and a theorem of Hestenes [ 8 ], the necessary conditions for an optimum are: OG/ON= -we-r'+22o~( 1 - f l ) +~'2 =0
(10a)
OG/OI= - q e - r t + 2 1 +73 = 0
(10b)
k=I-6K
(10c)
L = o~( 1 - f i ) N - o~flNo
(lOd)
-- ~ 1 =
OG/OK=pFKe-
~ t _ (~,1
(10e)
-J,2 = OG/OL=pFL e -~t
(lOf)
y, >/0,
y,[~-a(t)]=O
(10g)
72 >/0,
y2N=0
(lOh)
72 Y3>/O,
(lOi)
~'31= 0
We shall now solve for the values of the variables in the system of equations (10). We will study interior solutions where I ( t ) > 0, N ( t ) > 0 and o~( t ) < d. These conditions imply that the Lagrangian multipliers Yl = 72 = 73 = 0. Note that these conditions are important economically. Efficient capital and labor are indispensable in production yet there is instantaneous capital depreciation and labor turnover. It is, therefore, reasonable to investigate a situation where the enterprise acquires new investments and labor for at least replacement purposes. Due to the turnover of some experienced workers, it is unlikely that maximum learning on the average will be attained within the enterprise. Hence the condition o~(t) < ~ is appropriate. From equation (lOa) we have 72=e-"'w/c~( 1-fl) and, therefore,
J.2=e-"[--rwa(1--fl)+o~(1--fl)~i'--{(~(1--fl)--o@}w]/o~ 2 ( 1 - - f l ) 2
(11)
Eliminating ,~2 from ( I Of) and ( 11 ) we have
PFL =w[r+&/o~--fl/(1 --fl) -- vi,/w]/c~ ( 1 --fl)
(12)
or (13)
Fir = W ( t ) / p ( t )
where
W(t) =wtr+&/o~--fl/ ( 1 - - f l ) - w / w l / ~ ( l--fl)
(14)
Combining equation (10b) and (10e) it may easily be established that
PF~,.=q( d+ r)-it
(15)
or
F~, =C(t)/p(t)
(16)
where
C( t )=q( ~+ r)-[t
(17)
The complete model of optimal factor usage consists of (a) the production function
Q=F(K,L ) (b) two dynamic marginal productivity conditions
F~=W/p
and
F~:=C/p
and (c) four side conditions consisting of two equations of motion involving the time paths of investment and employment given by
I(t) = k + O K and
U(t)= [ L + a ( t ) fl(t) No]/a(t) [ 1 - f l ( t ) ] and two imputed cost functions for the flow of unit services of capital and labor given by
C(t)=q(r+6)-O and
W(t) =w[r+dUa-fl/ (1 -fl)-;v/wl/o~(1 -fl)
73 All of the above relationships are dynamic conditions which must hold at every point of time. Together, they combine to determine the output level, the levels of efficient labor and capital inputs, the levels of investment and employment and the shadow prices for efficient capital and labor services.
4. The investment and employment demand functions From the theory of optimal factor requirement we may derive differentiable factor demand functions for capital and labor services and a differentiable output supply function. These functions which will depend on the time paths of output price and the imputed factor costs may we written as
L=L( W,C,p)
(18)
K=K( W,C,p)
(19)
Q=Q(w,C,p)
(20)
Starting with the demand function (19) for efficient capital services and under the hypothesis that the firm follows an optimal path for the usage of capital services, the demand functions for investment goods can be obtained as a function of the time paths of prices through the method of comparative dynamics (see, for example, Dompere [9] and Jorgenson [ 10] ). We may obtain the demand function for investment goods by first differentiating the demand function for efficient capital services with respect to time to obtain
K= (OK/OW) (OW/Ot) + (OK/OC) (OC/Ot) + (OK/Op) (Op/Ot)
(21)
Equation (21 ) requires that we differentiate the imputed costs of capital and labor services with respect to time. Let/1 = & / a , a = 1 - fl and 0 =fl/a. Then it is straightforward to show that
~V= ( 0 W/OI ) = [ W~bt( O~0-1~/W -- &~r -]- OLO'~,g/~ff) + I~ -- O£0-W]/O~ 20"2
( 22 )
where ~u= r + / t - ~. Similarly,
C= (OC/Ot) = q ( 6 + r ) + q ? - #
(23)
Substituting (22) and (23) into (21) we obtain
K= (OK/OW) [w~( o~afi~/w-&6+ a0"(u/~,) "Jt-I~--OlO'l~]/a20 + (OK/OC) [O(6+r) + q k - / ] ] + (OK/Op)[~
"2
(24)
It follows from the side condition for the equation of motion of capital that
I=K+6K
(25)
By substituting k from eqn. (24) into (25) we obtain the demand function for investment goods as
I( t) = (OK/OW) [w~,{c~0-(&/w+(u/~u) -&6} + & - o~o~]/o:2a 2 + (OK/OC) [ 0 ( 6 + r ) + q f - q ] + (OK/Op) [~+OK(W,C,p)
(26)
The investment demand function may be written in implicit form as
I(t) = I ( W,C,p, (V,C,[9)
(27)
where
W= W(w,r;o~,fl)
and
C=C(q,r;g)
The investment demand function defined by equation (26) depends on (a) the price system {p(t),
w(t), q(t), r(t)} and its rate of change {b(t), &(t), 0(t), k(t)}, (b) the imputed cost of capital services
74 C( t ), (c) the imputed cost of labor services W( t ), ( d ) the progress of learning by doing a ( t ), and ( e ) the labor turnover rate fl(t). We now derive the employment demand function. Again, we will assume that the firm follows an optimal path for factor usage. The method of derivation is similar to the one used to obtain the investment demand function. Here, we combine the optimal path for the use of efficient labor services with the side conditions for the equation of motion for employment and the imputed cost of labor services. Differentiating the labor demand function ( 18 ) with respect to time, we obtain
L= (OL/OW) (OW/Ot) + (OL/OC) (OC/Ot) + (OL/Op) (Op/Ot) Substituting for (0 W/Ot) and (OC/Ot) eqns. (22) and (23 ), respectively, we obtain
L = (oL/o w ) [w~,{ a~( w/w+ ~,/~,) - ,~} + ~ - a o ~ ] / a 2~2 + (OL/OC) [ q ( r + 6 ) + q i ' - q ] + (OL/Op)p
(28)
The side condition for the equation of motion is given by
N= [ L + a ( t ) t ( t ) N o ] / a ( t ) [ 1 - i ( t ) ]
(29)
Substituting L from (28) into (29), we obtain the employment demand function
N( t ) = ( OL/OW) [ W~l{aa( l~/W'~-~l/~ff--C)Lb}q- ~l;--aO~ ]/a3a 3 + (OL/OC) [~( r + 6) + q k - / ] ] / o u r + (OL/Op)p/aa+ flNo/tr
(30)
From (4), No= ( L / a ) - N . Therefore, we can eliminate No from (30) and rewrite the employment demand function, after some rearrangement and simplification, as
N( t) = (OL/OW) [ w~{atr( fv/w+ ~//g/)--~}"~ I~--OLOW]/OL30"2(O"~-t) + (OL/OC) [ q ( r + 6 ) + q ~ - # ] / a ( a + f l ) + OL/Op)b/a(a+fl)
(31)
+ilL(W,C,p)/a(a+fl) The employment demand function may also be written in implicit form as
N(t) = N ( W,C,p, I;V,C,[7) where
W=W(w,r;a,fl)
and
C=C(q,r;6)
The employment demand function defined by ( 31 ) also depends on (a) the price system {p (t), w (t),
q(t), r(t)} and its rate of change {/)(t), if(t), q(t), k(t)}, (b) the imputed cost of capital service C(t), (c) the imputed cost of labor services W(t), (d) the progress of learning by doing a(t), and (e) the labor turnover rate fl(t). 5. Sensitivity functions and sensitivity analysis The investment and employment demand functions can be used to derive some testable hypotheses. Our objective here is to investigate how changes in certain time dependent parameters of the model affect the optimal paths of investment and employment. We will conduct the sensitivity analysis under the following assumptions: Parallel variations - Time dependent variables are subjected to parallel shifts without altering the slope at any time point;
75
Variationally static expectations - A variation in one component of the price system does not induce changes in other components of the price systems; and Stabilizing conditions - Learning by doing and the labor turnover rate have stabilized at o~(t) = & and fl(t) =p, respectively, so that & and fl are practically zero as the number of successful operations and time get very large. For further details on the methodology of dynamic sensitivity analysis, including a discussion of the concepts of parallel variations and variationally static expectations, see Oniki [ 11 ] and Tomovic and Vukobratovic [ 12 ]. A short explanation of the stabilizing conditions may be useful. Note that the stabilizing conditions apply after the firm has been operating for an arbitrary large time. In this case, ~ (t) will be arbitrarily close to c~. Also note that the stabilizing value of the labor turnover rate p must satisfy 0~
(32)
and
K=Bo + B 1 W+B2C+B3p
(33)
In the analysis given below, we consider variations with respect to the market wage rate and the interest rate. Although it is possible to consider variations with respect to the stabilizing values of the learning by doing and labor turnover parameters, we do not pursue that analysis in this paper.
5.1 Variation o f the market wage rate The effect of variations in the time path of the market wage rate on investment and employment may be obtained by differentiating the demand functions. Let us write the investment demand function (26) as
I( t ) = ( OK/OW)RI/ a2a: + ( OK/OC)R2 + ( OK/Op )b+ ~K( W,C,p ) = Bi R l / a2cr2 + Bz R 2 + B 3 / ) + S K where
B~ = (OK/OW),
B2 + (OK/OC),
83 = (OK/Op)
R, = w~u[ ~cr( Cv/w+ ~,/~u) - &a] + ~ - o~o-~
and
Rz =O( 8+ r) + qi'-~] With variationally static expectations, it is straightforward to show that the effect of a parallel shift in market wage rate on investments is given by
76 (OI/Ow) =B, ( OR,/ Ow) /t~202 + B2( OR2/ Ow) +B3(@lOw) d-(~B, (OW/Ow) ---B, [atr~/(0#/aw) + a a ~ - &b~+ (0#/aw) - atr(0f~/O#) (aCv/Ow) ]/a 202 +6B, [ r - (0~/aw) ] lacr-JrB3(ap/0w) [ (ap/Op) -t-t~] = BI (ola~t- & t ~ t ) / a 2a2 + ~BI r/atr where we have used (a~/Ow) = (O~law) = (O~lOw)
=0,
(oR2 aw) =0
and
(aW/Ow) = [ r - (agv/Ow) ]/aa=r/cetr When we impose the stabilizing conditions a (t) = t2, ft.(t) =p, & =1~= 0 on learning by doing and labor turnover rates and use ~u= r + & / a - / ~ / ( 1 - fl) = r and ~u= k, we get
(allOw) =rB1 ( i / r + ~ ) / a ( 1- p ) =rBl Erla(1 - p ) where
Er = (k/r+6) If interest rates are steady or rising then i>/0 and Er> 0. If interest rates are declining but IUrl < 6, then Er> O. This requires that the rate of change of interest rate be less than the rate of depreciation. Suppose either of the above conditions hold so that Er> 0. Then the sign of (OI/Ow) is determined by the sign of BI. However, the sign of Bl depends on whether capital and labor are substitutes. If capital and labor are substitutes then (OK/Ow) > 0. But (OK/Ow) = (OK/OW) (OW/Ow) =B~ r/aa. Therefore, B~ = (OK/OW) > 0 if capital and labor are substitutes. Similarly, B1 < 0 if capital and labor are complements. Hence (OI/Ow) is positive or negative according as capital and labor are substitutes or complements. We now consider the effect of market wage rate on employment. The time path of optimal employment by the firm is given by
N ( t ) =At R~/ [o~3a2( a + fl) ] + A2Rzl a( a+ fl) "bA3fil ~( c~d-fl) + fl(Ao +A1 W+AzC+A3p)/o~(a+ fl)
(34)
The sensitivity function under conditions of variationally static expectations may be computed as
( ON/Ow) =A, (aR,/Ow)la3az(a+fl) + flA, (OW/Ow)/c~(a+fl) Substituting for the partial derivatives (ORt/Ow) and (aW/aw) in (34) and using the stabilizing conditions it can be established that
(ON/Ow) =rAt ( i/r+6) /~2( 1- p ) = rA, Er/~2( 1- p ) where
Er= (Ur+~) We shall assume a normal demand curve for factor services. Then (OL/O W) =A~< 0. Again, if interest rate are steady or rising then i>/0 and Er> O. On the other hand, if interest rates are declining then E~> 0 if ]i/r] < 5. This requires that the rate of change of interest rate be less than the rate of depreciation. If either of the above conditions holds then (ON/Ow) < O.
77
5.2 Variation of interest rate The effect of a variation of interest rate on investments is given by
(OI/Or) =B~ ( OR,/Or)/a2a2+B2(OR2/Or) +8[B~ (OW/Or) +B2(OC/Or) ] Substituting for the partial derivatives and using the stabilizing conditions we obtain
(OI/Or) = wB~ (~v/w+8)/6~(1 - p ) + q B 2 ( q / q + 8 )
(35)
It is interesting to note from (35) that the effects of variations in the time path of the rate of interest on the time path of investment consists of a wage effect wB~ (~v/w+8/6~( 1 - p ) and an investment-price effect qB2 ( (i/ q + 8). We will assume, as before, that Ew = ( ~v/ w + 8) and Eq = ( (i/ q + 8) are both positive. Clearly these conditions will hold if wages and investment price are both steady or increasing (~i,>~0 and ~ >t 0). But if wages are decreasing then E,, > 0 if I ~i,/w I < 8 that is, if the net effect of wage deflation and depreciation is positive. Similarly, if investment price is decreasing, then the assumption Eq> 0 requires that I(i/qf < 8; the net effect of investment price deflation and depreciation is positive. Under these assumptions the sign of (OI/Or) is determined by the sign of B~ and B2. However, the signs of Bt and B2 depend on whether capital and labor are substitutes or complements. If capital and labor are complements then (OK/Or)
(ON~Or) =31 w( ~v/wq- 8) / 6~2 ( 1 --p ) +A2 q( ~t/ q + 8) / a If we assume, as in the previous analysis, that ((t/q+ 8) and (;v/w+ 8) are both positive, then the sign of (ON~Or) also depends on whether capital and labor are complements or substitutes. If capital and labor are complements then A: < 0; however, A2 > 0 if capital and labor are substitutes. But (0L/0w) < 0 if demand for labor services is normal. And (0L/0w) = (OL/O W) (OW/lOw) = A ~r/aa. Therefore, A~ < 0. Hence, (ON~Or) < 0 if capital and labor are complements. But if capital and labor are substitutes then we can deduce, as before, that (ON~Or) > 0 if wage effect dominates the investment-price effect, and (ON~Or) < 0 if the investment-price effect dominates the wage effect.
5.3 Summary of main propositions In this subsection, we summarize and interpret the main propositions concerning the effects of parameter variations on the time paths of optimal investment and employment plans. Several of the resuits depend on whether capital and labor services are considered as substitutes or complements. It is, therefore, useful to clarify this issue. Let us recall that in a two factor model, capital and labor are substitutes if the technology has the variable proportions property. In this case, the two factors are limitative; that is, an increase in the relative usage of a factor is both necessary and sufficient for output to increase. And the inputs may be smoothly substitutable in the production of the output. But if the production technology is of the fixed proportions type then the inputs are limitational; that is, an increase in the usage of an input is necessary but not sufficient for the output to increase. In this case, the two inputs are complements in production (see Georgescu-Roegen [ 13,14 ], and Ferguson [ 15 ] for details). In view of the above discussion, it is necessary to consider situations where capital and labor services are either substitutes or complements. We, therefore, distinguish between these two cases in the statement of our main propositions. Also, we note that the conditions Er> 0, Ew> 0 and Eq> 0 used in the propositions were defined and interpreted in Sections 5.1 and 5.2.
78
Proposition I. If Er> 0 and capital and labor services are complements (substitutes) in production then the time path of the volume of optimal investment plans is inversely (positively) related to the time path of the market wage rate. Proposition 2. I f E r > 0 then the time path of the firm's optimal employment plans is inversely related to the time path of the market wage rate. Note that in Propositions 1 and 2 the variations in the time path of the market wage rate on the optimal paths of investment and employment also affect the imputed (shadow) wage rate.
Proposition 3. IfEw> O, Eq> O, and capital and labor services are complements, then the time path of the firm's investment and employment plans are inversely related to the time path of the rate of interest. Proposition 4. IfEw> 0, Eq> O, and capital and labor services are substitutes, then the time paths of volumes of optimal investment and employment plans will be inversely (positively) related to the time path of the rate of interest according to whether the investment price effect dominates (is dominated by) the wage effect. The intuition behind Proposition 4 is related to the part played by the relative magnitude of capital and labor in production. If the labor component is small relative to the capital component, then a rise (fall) in the interest rate will inversely (positively) affect investment decision because it increases (reduces) the burden of the interest cost on sunk capital. Labor services will, therefore, be substituted for capital services and thus employment will increase. 6. Conclusion
The model developed in this paper has been used to analyze the role of learning by doing on the firm's optimal demand for investment and employment. A divergence between the market wage rate and the shadow wage rate (the Horndal effect) was established. The effects of variations of market wage rate and interest rate on investment and employment demand were investigated. Several results were dependent on whether capital and labor services are substitutes or complements. We must point out that the theory developed here is incomplete and has certain shortcomings. Technological progress in the investment goods sector as well as internal adjustment costs were ignored. The sensitivity analyses were restricted to variationally static expectations and linear approximations for the investment and employment demand functions. These simplifications facilitated the derivations and the interpretations of the results. We plan to relax some of the assumptions used in this paper and explore further generalizations of the model in future research. References 1 Hirsch, W.Z., 1952. Manufacturing progress functions. Review of Economics and Statistics, 34: 143-155. 2 Arrow, K.J., 1962. The economic implications of learning by doing. Review of Economic Studies, 29:155-172. 3 Fellner, W., 1969. Specific interpretations of learning by doing. Journal of Economic Theory, 1:119-140. 4 Sheshinski, E., 1967. Tests of the learning by doing' hypothesis. Review of Economics and Statistics, 49: 568-578. 5 Lundberg, E., 1961. Produktivitet och R~intabilitet. P.A. Norstedt and S6ner, Stockholm. 6 Gould, J.P., 1968. Adjustment costs in the theory of investment of the firm. Review of Economic Studies, 35: 47-55. 7 Treadway, A.B., 1970. Adjustment cost and variable inputs in the theory of competitive firm. Journal of Economic Theory, 2: 329-347. 8 Hestenes, M.R., 1965. On variational theory and optimal control theory. Siam Journal of Control, 3: 23-48. 9 Dompere, K.K., 1980. Sensitivity, optimality, and stability conditions in a pure theory of investment behavior. Ph.D. Dissertation, Temple University, Philadelphia.
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