Karl Shell

English

Research Papers in Economics

American Economic Association
Toward A Theory of Inventive Activity and Capital Accumulation
Author(s): Karl Shell
Source: The American Economic Review, Vol. 56, No. 1/2 (Mar. 1, 1966), pp. 62-68
Published by: American Economic Association
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http://www.jstor.orgTOWARD A THEORY  OF INVENTIVE  ACTIVITY 
AND  CAPITAL ACCUMULATION* 
By  KARL  SHELL 
Massachusetts Institute of Technology 
I.  Introduction 
In at least two recent models of economic growth, the rate of techni- 
cal change depends upon other economic variables. The first, a model 
introduced by Kaldor [3] [4] [5], assumes a positive relationship (the 
technical progress function) between relative changes in productivity 
per worker  and relative changes in gross investment. The technical pro- 
gress function is an eclectic amalgam summarizing  basic technical and 
institutional forces in  a  free enterprise economy.  Kaldor takes  the 
Schumpeterian view that  the creation of new ideas largely occurs at 
an autonomous rate, but that the implementation of these new tech- 
niques by  entrepreneurs can be  explained by  economic phenomena. 
Obviously, if the implementation of a new technique requires  new capi- 
tal equipment as opposed to mere organizational  change, increased  pro- 
ductivity can only be transmitted through new (gross) investment. In 
addition, Kaldor argues that for a capitalist economy the higher the 
relative rate of gross investment the higher the degree of "technical 
dynamism." Technical dynamism is a mass measure of entrepreneurial 
psychology including the readiness  to adopt new methods of production. 
In  the second model with endogenous technical change, Arrow [1] 
concentrates upon the relation between learning and experience.  Eco- 
nomic learning results in higher productivity, while cumulative gross 
investment is the measure  of economic experience.  Therefore,  in refining 
the technical progress  function, Arrow  explicitly postulates that produc- 
tivity per worker  is determined by accumulated gross investment. The 
production  of new technical knowledge (invention) and the transmission 
and application of  that  knowledge (innovation)  are treated as  by- 
products in the production and adoption of new capital goods. 
While it is doubtlessly true that technical change is related to gross 
investment both as a by-product of capital goods production and as a 
vehicle for embodying new techniques in new capital equipment, it is 
also true that the rate of production of technical knowledge can be in- 
creased by  increasing the  allocation of economic resources explicitly 
devoted to inventive activity. 
*  Research for this  paper was  supported in part  by  National  Science Foundation  Grant 
GS-95, "Economic Growth of the United States." 
62 CAPITAL  THEORY  63 
At  least  two  peculiar properties of  technical  knowledge require 
special study.  First, technical knowledge can be used by  many eco- 
nomic units without altering its  character. Thus, for the economy in 
which technical knowledge is a commodity, the basic premises of the 
classical welfare economics are violated,  and  the  optimality  of  the 
competitive mechanism  is not assured.  Typically, technical knowledge  is 
very durable and the cost of transmission is small in comparison  to the 
cost of production. Second, at least on the microeconomic level,  the 
inventive process is characterized by extreme riskiness.' 
II.  The Model 
I have argued  that increases  in technical knowledge  are fundamentally 
related to the amount of resources explicitly devoted to inventive ac- 
tivity.  In order to study the respective roles of invention and invest- 
ment in economic growth, I assume that current aggregate output Y(t) 
is determined by the relation 
(1)  Y(t)  =  F[A(t),  K(t),  L(t)], 
where A (t) and K(t) denote the current  levels of the stocks of technical 
knowledge and physical capital, respectively; L(t) denotes the current 
size of the labor force inelastically offered for employment.2 
The growth in the stock of technical knowledge satisfies the differen- 
tial equation 
(2)  A(t)  =  a(t)Y(t)  -  pA) 
where 0 <?a(t)  < 1 is the fraction of output currently devoted to inven- 
tion and 0<cr<1  is the fraction of inventions that  are "successful." 
For the case where p is positive, equation (2) should be understood as a 
long-run approximation  to processes  not explicitly treated in the model. 
For example, decay in technical knowledge is observed because of the 
imperfect transmission of technical information from one generation of 
the labor force to the next. 
If capital is subject to evaporative decay at the given technical rate 
,u>O, then 
(3)  K s) =  s() (1-  a(t)) Y(t) -K(t)) 
where 0 < s(t) (I -a(t))  < 1 is the fraction of output currently devoted to 
investment. 
1 Cf., e.g., [8], especially Arrow's contribution on pp. 609-25. 
2  I treat the one-sector model for ease of exposition. This implies that the production possi- 
bility  frontier is a hyperplane in the consumption-investment-invention  space. If  the model 
is disaggregated to two or three sectors, then the frontier can possess greater curvature. Also 
notice that increases in efficiency are shared by all vintages of capital and labor. 64  AMERICAN ECONOMIC ASSOCIATION 
III.  A Stylized Economy 
In the  United  States,  intervention  in behalf of inventive  activity  has 
taken  two  basic  forms:  (1)  the  establishment  of a legal  device,  the  pa- 
tent,  designed  to bestow  property  rights on certain of the outputs  of the 
inventive  process;  (2)  direct  nonmarket  support  of  research  and  de- 
velopment.  The  universities  and  the  Department  of  Agriculture,  for 
example,  have  contributed  to  our  economy  in  the  second  role.  Re- 
cently  the  Department  of  Commerce  has  initiated  industrial  research 
programs  modeled  after  the  programs  of the  agricultural  research  sta- 
tions,  while the Department  of Defense  favors the device  of contracting 
research tasks  to private  enterprises  on a cost-plus-fixed-fee  basis. 
Consider  a  model  economy  in  which  production  is  undertaken  by 
many  individual  firms.  The  level  of  technical  knowledge  enters  each 
firm's production  function  as a pure public  good  of production.  Hence, 
the  competitive  price  of  invention  is zero  suggesting  the  desirability 
of social intervention  in the market process.3 Assume  that  the only form 
of intervention  is the  imposition  of a tax  upon  output  at  a given  con- 
stant  rate 0 <a  < 1, the proceeds of which are used to support invention. 
The private  sector saves  (and invests  in capital  accumulation)  the  con- 
stant  fraction  0 < s < 1 of disposable  income. 
I assume  that  the production  function  given  in  (1)  exhibits  constant 
returns  in capital  and labor  and  consequently  increasing  returns  in all 
three factors.  Then  if the special assumption  is made that  there are con- 
stant  returns  to  (Hicks-neutral)  technical  knowledge,4  then  (1)  can be 
rewritten  as 
(4)  y =  Af(k), 
where  lower-case  letters  denote  quantities  per  worker  and  f(k)  is 
shorthand  for  F(k,  1). f(.)  is  twice-continuously  differentiable  with 
f(k) >O, f'(k)  > O,ff"(k)  <O for 0 <k <  oo;  f(O)=O,  f(oo)  =  oo, f'(O) =  oo, 
f'(oo)  =0.  For simplicity,  I assume  that  there is no change  in the labor 
force,  thus  setting  L = 1 and  writing 
(5)  A = ao-y-pA, 
(6)  k =  s(l-a)y-  ik. 
From (4) and (5), A  =  0 if and only if 
8 In practical  applications,  the distinction  between  private  and public  goods  is fuzzy. What 
is considered  to be a public  good under  one set of legal and social  arrangements  may be con- 
sidered  to be a private  good  under  a different  set of arrangements.  In choosing  among  differing 
arrangements,  society  should  include  in its calculation  the buying  and selling  costs that they 
imply. 
4Qualitative  long-run  behavior  does  not depend  upon  this special  assumption.  It does  allow 
a simple  aggregation  congenial  to the competitive  hypothesis:  Y= 2j Yj  =AZj F(Kj, Lj) where, 
for example,  Kj is the quantity  of capital  employed  by thejth firm. CAPITAL  THEORY  65 
p 
(7)  f (k) =  - - 
ao- 
Call  the  unique  solution  to  (7)  k?. But  from  (4)  and  (6),  k0  if  and 
only  if 
,uk 
(8)  A  -, 
s(1  -  a)f(k) 
which yields 
(dA\  82(k)  -  kf'(k)]  >  O 
k  dk/k=O  s(l  -  a)[f(k)]2 
The  laws  of motion  for the  stylized  economy  are shown  in the phase 
diagram  below.  To  verify  that  (A0,  k?), the  unique  solution  to  (7)  and 
A 
A=O 
k? 
(8),  is  a  saddlepoint,  solve  the  characteristic  equation  for  the  linear 
Taylor  approximation  to  (5)  and  (6)  about  (A0,  k?). The  characteristic 
roots are real and opposite  in sign,  guaranteeing  that  (A0,  k?) is a local 
saddlepoint.  Twice-continuous  differentiability  of f(.)  then  guarantees 
that  (AO,  k?) is a global  saddlepoint.  Notice  that  for initial  endowments 
sufficiently  large  (small),  the  stylized  economy  explodes  (decays).5 
5 In the multisector model, it is possible that there are many equilibrium points. Some would 
be saddlepoints; the  others would be stable  or surrounded by  limit  cycles.  See  [10]. Global 
stability  is an interesting property for a descriptive growth model but should not be thought 
to be essential. In fact, Maruyama [7] argues that social systems are basically morphogenetic 
rather than morphostatic. 66  AMERICAN  ECONOMIC  ASSOCIATION 
IV.  The Controlled Economy 
In  what  follows,  it  will  be  convenient  to  assume  that  equation  (1) 
can be rewritten  as 
(9)  y  =  g(A,  k), 
where g(.)  is an increasing  strictly  concave  function  of A  and  k. This 
assumption  is made in order to avoid  complications  by insuring that  the 
usual  necessary  conditions  for optimality  are also sufficient  conditions. 
Suppose  that  the economic  planning  board desires to maximize  the sum 
of future  discounted  utility  (of  per  capita  consumption) 
rw 
(10)  f  U[c(t)]e-tdt. 
subject  to given  initial  endowments  A (0) = Ao, k(O)  =  ko, and technology 
given  by  (5),  (6),  (9).  U [c(t)]  is  the  utility  of  consumption  at  time  t, 
with  U'[c]>0,  U"[c]<0  for 0<c<oo,  U'[O]=co,  U'[o]o  0.  8>0  is 
the  (constant)  pure rate  of social  discount.  For simplicity,  set L(t)  1. 
Because  the  marginal  utility  of  consumption  is infinitely  large when 
consumption  is zero,  consumption  must  be everywhere  positive  on  the 
optimal  program.  Further  assume  that  the  initial  endowments  (Ao, ko) 
are sufficiently  small that  the optimal  program will never be specialized 
to consumption.  A feasible  consumption  program  {  c(t) :0 < t < oo } satis- 
fying  (5),  (6),  and  (9)  is optimal6 if, and only  if, there  exist  continuous 
functions  q(t) and v(t) such that 
(11)  q4  (  +,u)q  -  max  (q, vo)gk(A,  k), 
(12)  v =  (8 +  p)v -  max  (q, vo)gA(A,  k), 
(13)  lim q(t)e8t  =  lim v(t)e7t  =  0, 
t  -+ co 
where 0 <?a(t) < 1 and 0?  s(t) < 1 are chosen at each instant  to maximize 
(14)  U[(1  -  s)(1  -  a)g(A,  k)] +  {qs(l  -  a)  +  vaujg(A,  k). 
If utility  is the numeraire,  then  q and v are the  social  demand  prices 
for  investment  and  invention,  respectively.  Conditions  (11)  and  (12) 
imply  that  the  planning  board  has  perfect  foresight  with  respect  to 
marginal  products.  Transversality  condition  (13)  requires  the  present 
value  of  a unit  of future  investment  or invention  to  become  small  as 
the future date becomes  distant.  Expression  (14) is simply  the certainty 
I Cf. [91,  especially  theorem  7, p. 69, and pp. 188-91,  298-300. CAPITAL  THEORY  67 
equivalent of the imputed value of gross national product. Maximization 
of (14) implies that if U'=q>vvo then a= 0, and if U'=vor>q  then s=0. 
Further if the certainty equivalent net  marginal products are equal, 
oqgA-P=gk-  A,  then q=vo-. 
Notice that (11) and (12) have a special interpretation in a decen- 
tralizedeconomyinwhichfactorsarerewarded  by their marginal  products. 
From (11), for example, the change in the demand price of investment 
should be such as to  compensate the representative rentier for "ab- 
stinence" and depreciation  loss net of rewards  from the employment of 
his  capital. Of course, for a  simple decentralization of  the  economy 
treated above, condition (11)  will not  necessarily hold. If  output  is 
taxed to support invention, then private factors will not be paid their 
full marginal  products. At any rate, if an optimal program  exists in the 
fully controlled  economy, the stocks of technical knowledge  and physical 
capital will approach  limiting values (A*,  k*) such that ogA(A*,  k*)-p 
=  gk(A*, kV)-t-8.  The  limiting  value  of  consumption  c* is  given  by 
c*=g(A*,  k*)-tik*-pA*/f.. 
It may b&  that the planning board treats the process of private sav- 
ing and investment in physical capital as institutionally given leaving 
the  choice of  {  a(t) :0 <t<  oo  } as  the  remaining policy  instrument. 
Assume, for example, that  private  capital accumulation follows (6) 
with savings O<s<l  a given fixed fraction of disposable income. The 
planning board desires to maximize (10) subject to given initial endow- 
ments and subject to technological  and behavioral  relations (5), (6), (9). 
It is implicit in this formulation that the free play of the private capital 
market does not  necessarily yield  a  socially preferred result. There 
could be several reasons for this divergence, including the existence of 
conventional externalities and certain intrinsic impediments to borrow- 
ing and lending in a risky world.7 
In  the partially controlled economy, maximization of  (14)  implies 
that (1 -s)  U'+qs > v,  or with equality if a > 0. The optimal consump- 
tion program {  c(t): O  < t < oo } in  the  partially controlled economy is 
such that the stocks of technical knowledge and physical capital ap- 
proach  limiting  values  (A**,  k**)  where  TgA(A **  k**)  -  p  =  .  Notice 
that asymptotically the marginal products of technical knowledge are 
identical for the  partially and fully  controlled economies. This  is  a 
(long-run) dynamic generalization of the Rule of the Second Best. If, 
for example, long-run private savings are too low, then long-run inven- 
tive activity will be greater in the partially controlled economy than it 
would be in the fully controlled economy. 
7 Arrow in [81  refers to such impediments as "moral hazards." I have found  an exposition 
very similar to what follows in an unpublished paper by Arrow [2]. 68  AMERICAN  ECONOMIC  ASSOCIATION 
V.  Concluding  Commeents 
I have  argued  that  in the  study  of aggregative  models  it is useful  to 
think  of technical  knowledge  as a public  good  of production,  while  the 
level  of inventive  activity  (the  process  of production  of knowledge)  is 
dependent  upon  the  amount  of  economic  resources  devoted  to  that 
activity.  Of  course,  invention  is  a  particularly  risky  form  of  social 
investment.  In my mode], a given  fraction  of inventions  is "successful," 
thus  removing  the  difficult  decision  problems  associated  with  uncer- 
tainty.  Perhaps  this  is a legitimate  approximation  in a macroeconomic 
model.  On the other hand,  differential  equation  (2) represents  the most 
unsatisfactory  simplification  in  the  model.  A  complete  theory  should 
explicitly  treat  the  problems  of transmission  of knowledge,  e.g.,  educa- 
tion,  book publishing,  etc.,  and its effect upon the efficiency of different 
generations  of the labor force and different  vintages  of capital.  Finally, 
the  recent  contribution  of  Kennedy  [6]  warns  us  that  additions  to 
technical  knowledge  should  not  be  thought  of  as  increasing  efficiency 
in any  specified  wvay.  That  is, the  "bias of technical  progress,"  whether 
in a stylized  economy  or in a planned  economy,  should  be a subject  for 
economic  decision. 
REFERENCES 
1.  K. J. Arrow, "The  Economic Implications  of Learning by  Doing," Rev. of Econ. Studies, 
June, 1962, pp. 155-73. 
2.  , "Discounting  and Public Investment  Criteria," 1965 Western Resources Confer- 
ence, Seminar in Water Resources Research, Juily 6, 1965. 
3.  N. Kaldor, "A Model of Economic Growth," Econ. J.,  Dec.,  1957, pp. 591-624. 
4.  ,  "Capital Accumulation and Economic Growth," in F. A. Lutz and D.  C. Hague 
(eds.),  The Theory of Capital (St Martin's Press, 1961), pp.  177-222. 
5.  and J.  A.  Mirrlees, "A New  Model  of  Economic  Growth," Rev. of Econ. Studies, 
June,  1962, pp.  174-92. 
6.  C. Kennedy,  "Induced  Bias  in  Innovation  and the  Theory  of  Distribution,"  Econ. J., 
Sept.,  1964, pp. 541-47. 
7.  M.  Maruyama,  "Morphogenesis and Morplhostasis," Methodos, 12, 48  (1960). 
8.  The Rate and Direction of Iiventive Activity (N.B.E.R.,  Princeton Univ. Press, 1962). 
9.  L. S. Pontryagin, V. G. Boltyanskii,  R. V. Gamkriledze, and E. F. Mischenko, The Mat  he- 
miatical Thcory of Optimal Pr ocesses (Interscience Publishers, 1962). 
10.  K. Shell, "A Model of Inventive  Activity  and Capital Accumulation,"  Technical  Report 
No. 2 (NSF: GS-420) (Univ. of Chicago, Aug. 10, 1964). 

Toward A Theory of Inventive Activity and Capital Accumulation

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