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					Balanced Growth and Stability in the Johansen Vintage Model

         E. Sheshinski

         The Review of Economic Studies, Vol. 34, No. 2. (Apr., 1967), pp. 239-248.

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Balanced Growth and Stability in the 

     Johansen Vintage Model1 

                                     1. INTRODUCTION
     The purpose of the present note is to analyze the structure of the model of economic
growth introduced by Johansen in a well-known article [2], and later discussed by Solow
[7], [8], Phelps [6], Kurz [5], Inada [I], Kemp and Thanh [3], and Kemp, Sheshinski
and Thanh [4]. These studies have concentrated on (1) balanced growth and comparative
dynamics when the ex-ante production function is Cobb-Douglas and technical progress
is embodied in capital; (2) balanced growth and stability when technical progress is
excluded altogether; (3) simulations which were intended to reveal the short-run char-
acteristics of the model.
     Our primary interest here is to prove that if technical progress is embodied in capital
equipment and neutral in Harrod's sense, then any path of growth equilibrium asymp-
totically approaches an equilibrium balanced growth path in which the economic lifetime
of capital equipment is fixed and its vintage distribution remains stationary.
     The model-economy is visualized as composed of homogeneous labourers and
capital equipment of various vintages which are engaged in producing a homogeneous
output. A constant portion of output is instantaneously consumed while the rest is
invested in capital equipment of the current vintage. Capital equipment is assumed to
be completely non-malleable. Thus, while there is scope for substitution between labour
and capital along the production function before the capital equipment is constructed
(ex-ante), no substitution is possible thereafter (ex-post). In other words, capital equipment
embodies the technology of its date of construction in the sense of embodied technical
progress and in the sense that the number of labourers working with the equipment (if
at all) is fixed by design. Investors choose the optimal labour intensity by maximizing
expected future discounted quasi-rents. Under zero foresight, the optimal intensity equates
the marginal product of labour to the current wage rate.' The wage rate is determined
by the supply and demand for labour. With fixed coefficients ex-post, there is a maximal
employment-capacity of a given capital stock. While this is usually sufficient to accom-
modate the available labourers, unemployment due to insujficient capacity cannot be ruled
out entirely.

                                        2. THE MODEL
     Let L(t) be the number of labourers available at time t, and K(u, t) be the quantity of
capital equipment of vintage v existing at time t. It is assumed that the rate of growth
of labour is constant, say n; i.e.



     1 I am indebted to F. M. Fisher, M. C . Kemp, R. M. Solow and M. E. Yaari for helpful comments.
Errors, of course, are mine.
     2 The conceptually awkward assumption of zero foresight is adopted for mathematical convenience.
The treatment of dynamic expectations out of balanced growth is quite complicated and outside the scope
of this paper.
240                        REVIEW OF ECONOMIC STUDIES
    A depreciation formula p(t-v) connects K(v, t ) with the amount K(v) of capital
equipment constructed at time v


where p(t -0) is assumed to satisfy the conditions
                                p(0) = 1 , p(t)   =    0 for t 2 z,
where z is a positive number or z =      + co ;
                           p(t) > 0 , pl(t) 5 0 , for all 0 < t < z.
     It is assumed that part of technical progress is embodied in capital equipment while
the rest of progress is disembodied, i.e. it affects the efficiency of all the vintages equally.
All technical progress is postulated to be Harrod-neutral and exponentially increasing.
    The output Q(v, t ) produced at time t by capital equipment of vintage u is determined
by K(v, t ) and the number of labourers allocated L(u, t )
                           Q(v, t ) = F[K(V, t), e a u + B t ~ ( v , v 5 t.
                                                               t)]                     ...(3)
eav+gt
       is the efficiency of labour at time t associated with capital of vintage v. ct>O and
p 2 0 stand for the rates of increase of embodied and disembodied technical progress,
respectively. The production function F is assumed to exhibit constant returns to scale
and diminishing marginal rates of substitution; hence we can write

where



and F[1, A(v, t)] is replaced by the shorthand notation g(A(v, t)),
                            g(A) is twice continuously differentiable,                        ...(6)
                         g(A)>O,g'(A)>O,g"(A)<O for all A>O,                                  ...(7)
                         g(0) = 0 , g(co) = co.                                               ...(8)
     It is assumed that the number of labourers allocated to capital equipment is fixed
as long as the equipment is utilized. This assumption can be interpreted in two ways;
one can postulate that the number of labourers in terms of the efficiency of labour at
time t associated with the capital equipment of vintage v, A, is fixed; i.e.
                                                  --
                            A(u, t ) = A(v, V ) A(v) for all v< t.                ...(9)
The alternative interpretation is that the number of labourers in terms of the efficiency of
labour at time v associated with the capital equipment of vintage v is fixed; i.e.
                               I(v, t ) = A(~)ep(~-") all u< t.
                                                        for                         ...(lo)
    In line with Johansen's original formulation [2], we shall follow assumption (9), but
conclusions derived from (10) will not be much different.2
     Investors choose the optimal labour intensity on the currently constructed equipment
A(t) by maximizing discounted expected quasi-rents. With stationary expectations, the
optimal intensity is determined so as to equate the marginal product of labour to the
prevailing wage rate W ( t ) ; namely
                                             aeo7
                                       W ( t )= - -
                                                  -
                                                      t)
                                                aL(t, t)'

     1 The stationary depreciation pattern p(t-v) implies that improvements in capital equipment are
not reflected in increases in durability.
     2 With no disembodied technical progress (j? 0) the two cases are, of course, identical.
                                                 =
         BALANCED GROWTH IN THE JOHANSEN VINTAGE MODEL                                                     241



where w(t) is the wage rate measured in terms of the efficiency of labour:


    At each instance of time t 2 0 we define a set V(t, w) of past vintages for which,
given the wage rate w(t), the capital equipment so dated can be operated without loss.
The quasi-rent of any vintage u at t is g(l(v))-l(v)e-a't-"w(t). Hence


                                      {
                         V(t, w ) = t-z      s u<t      1   gg;) e
                                                            -a(u-t)     - W(t) 2 0
                                                                                -    I.
In particular, if A(v), v< t, is continuous then V(t, w) consists of one or more time-intervals,
for each of which the condition in (12) is satisfied.'
     Total demand for labour is the sum of demand for the vintages utilized


                                             S V(t,W)
                                                        L(u, t)dv.

Demand is a decreasing function of w ranging from zero to the maximum possible employ-
ment when all the existing capital equipment is utilized; that is, when


    Until the final section we shall assume that labour is a non-redundant factor. In other
words, we assume that for all t 2 0



    The economy is assumed to be in equilibrium at any moment of time. In particular,
the wage rate is set so as to equate supply and demand for labour



     We assume that savings are a constant fractions, O<s 5 1, of total output. The amount
of new capital equipment produced at time t is then given by



    Let k(t) be the ratio of vintage t capital equipment to the labour force at time t,
measured in terms of efficiency units



     In view of (I), (2), (4), (5), (9) and (16), equations (14) and (15) can be conveniently
written in terms of intensive variables



        1 Notice that there is no unique relation between h(t) and the maximal h that is included in the set
V ( t , w). If there were no embodied technical progress, however, then the maximal A, denoted A,,&),
would be given by w(t) = gt(h(t)) =                and the set Vwould consist of all those A(v) that satisfy the
                                         h,,,(t)
condition h(v) 2 h,,,(t).      The presence of embodied technical progress requires one to take account, in
determining V, not only of the labour intensity but also of the date of construction of each equipment.
242                       REVIEW O F ECONOMIC STUDIES
and
                      k(t) = s   j
                                 V ( t ,W )
                                              g(*(v))e-(a+P+n)(t-.'          p(t - v)k(v)du.   ...(18)
      Given a capital stock profile (k(u), /Z(v)),u<t, equilibrium condition (17) uniquely
determines the wage rate w(t) and, by ( l l ) , the intensity /Z(t). k ( t ) is then uniquely deter-
mined by (18).
      A triple (k(t), A(t), w(t)) of non-negative real-valued functions which are defined for
all t 2 0, and which satisfy equations ( 1 I ) , (17), and (18) everywhere, will be referred to
as an equilibrium solution of the system.


                      3. THE BALANCED GROWTH SOLUTION
    It is natural to look first for constant solution functions to (17)and (18), i.e. equilibrium
solutions (k(t), A(t), w(t)) for which k(t), A(t) and w(t) are all constant functions. Any
such solution is called a balanced growth solution.
     Let A and m be a pair of positive numbers which satisfy the relation

                                 1=S g ( ~ )
                                  

                                                   1
                                                   :       e-(a+8+n)u
                                                                        P(u)~u.

    In view of (7) and (8), this relation gives m as a decreasing function of A, ranging
from infinity to zero and defined for all A >Ami,, where Amin is implicitly defined by



      Consider now the expression

                                                gl(A) = g(A> e-am
                                                        ---              ,
                                                              A
which gives m as an increasing or decreasing function of A depending upon whether the
elasticity of substitution is less than or greater than one, respectively. Suppose that the
elasticity of substitution does not exceed one. m is then a non-decreasing function of A. In
this case, there always exist unique positive numbers A and m* which satisfy
                                                         *

                                 1 = sg(A*)           J
                                                      :      e - ( a + ~ + n ) o ~(v)dv7       .. .(19)
and



      Corresponding to A and m* define k* and w* by
                        *




and
                                                    11'"   = g'(A*).                            ..   ,


      The triple (k(t),A(t), w(t)) = (k*, A*, I V * ) is seen to be an equilibrium solution of the
 system. Along this path the economic lifetime of capital equipment is the constant m*
 and the equilibrium set V* comprises the newest equipment available:
                               V* = ( t -m* 5 v< t ) for all t,
         BALANCED GROWTH IN THE JOHANSEN VINTAGE MODEL                                                           243
The strict concavity (7) of the production function guarantees that this solution is unique.
Thus, we shall refer to (k*, A*, w*) as the balanced growth so1ution.l
     In order to ensure the existence of a balanced growth solution we shall assume here-
after that the elasticity of substitution of the production function g, denoted by a, does
not exceed unity; namely
                                         O<o 5 1.                                     ...(23)

                                   IV. STABILITY ANALYSIS
     Starting at time t = 0, the economy begins to evolve according to equilibrium condi-
tions (17)-(18). At that instance of time we visualize it as endowed with an arbitrary
capital profile, (k(u), A(v)), -z I_ v<O, that reflects the amount and the labour intensity
of the inherited capital equipment. These are given data that determine the future course
of the economy. We shall prove that for a wide class of initial conditions, any equilibrium
solution converges asymptotically to the balanced growth solution.
     I t is assumed that the initial capital profile is given by two arbitrary, positive and
uniformly bounded functions k(u) and A(v), defined for all - z 5 v < 0:
                                    _ko   5 k(u) 5 Eo, 2, 5 I(v) 5 Xo                                        . ..(24)
where ko, E,, do and 2, are positive real numbers, _kO I_ k* 5 Eo and 4, 5 A* 5 2,.
It is also assumed that k(v) and A(u) are integrable on [-z, 01.
      It can be observed that, under (24), k(t) and A(t) are positive and continuous for all
t2o . ~
      We shall first put upper and lower bounds on the range of all possible equilibrium
solutions, and then we shall show how this range can be gradually decreased as time
passes.
      Let y = y(A, k) be a function defined by



where A and m are positive numbers obeying the restriction



                                                      8 .
                                                        Y   ay
for any given positive number k (Fig. 1). Clearly, - <O and - >O.                                One can readily
                                                      an    ak
show that for any k, there exists a unique I! such that
                                Y = ~ ( 2 k) = gl(A).
                                          ,                                                                  ...(27)
     Let E be a number, E 2 Eo, and let be the solution to (27) when k                             =   E; namely



    1 If the lifetime of capital equipment z is finite this imposes the restriction that m* I Whenever
                                                                                             z.
the solution nz* exceeds z, the no-profit condition (20) has to be dropped (since there is no zero-profit


                                                                    J:
extensive margin), in which case A is given by 1 = sg(A*)
                                  *                                      e-'"+u+")vp(v)dv,and k* is defined by
                                                                1
                                           k*   =
                                                    A*
                                                     J
                                                         rZ
                                                          0
                                                                     .
                                                              e-(o+n)v~(v)dv
                                                                      .,
We shall assume throughout that z is not binding.
   2    = - s'(A)[g(A) -As'(A) I
                 Xg(A)sm(A) '
   3 'The analysis in this section is based to a large extent on that in [9].
   4 See 191.
244                       REVIEW O F ECONOMIC STUDIES
where _m is given by
                                  I = dE   j
                                           :     e-(pn)"p(u)dv.                        ...(29)
     E is taken to be sufficiently large so that        ,
                                                     5 4. Similarly, let X be the solution of
(27) when k = _kSk,; i.e.




                                                 Figure 1


where E is given by
                                 1 = 2k
                                        J0
                                           rrn
                                            -

                                            e-(bin)"p(u)du.                        .. .(31)
      Again, _k can be chosen so that 2 2 2,. Since 2 S A* 5 2 we have, by (23), that
                                       m
                                       - 5 m* E. 

      Lemma. _k   5 k(t) 5 E and 2 5 A(t) 5 2 for all t 2     0.

        BALANCED GROWTH IN THE JOHANSEN VINTAGE MODEL                                        245
    Proof. Take any to 2 0, and suppose that _k 5 k(t) 5 E and t?, 5 A(t) 5           X   for all
t <to. We shall prove that in this case _k < k(to) < E and < A(to)<2.
      Consider the following maximization problem:

                         man J($, 6) = s
                         &. a
                                            js                           ~)~
                                                g ( 6 ( ~ ) ) e - ( " ~ "P(V)+(V)~V

over all the integrable functions $ and 6 defined on (0, z] such that _k 5 $(v) 4 E and
d 5 6(v) 5 X and,
                                       P



                                       Js
where S is any regular subset of (0, z].
    It can be shown that the maximizing functions ** and 6* are: $*(v) = E and
6*(v) = 2,O < v 5 z. Correspondingly, S = S* = {O<v 5 _m), where _m is given by (29).
     By (19) and the fact that < A * and _m 5 m*, the following inequality holds for J
at $* and 6*


      This proves that k(to) 5 J(+*, 6") < E and m(to) 2 _m. Using a similar argument, one
can also prove that k(to) > _k and m(to) 5 m. From the restrictions on k(t) and A(t) for
all t <to, (ll), (28) and (30), it also follows that



which implies that < A ( t o ) <A. Since to is arbitrary and (k(t), A(t)) are continuous, this
completes the proof.
    Since k(t) and A(t) are positive and uniformly bounded for all t, so is the set V(t, w).
The longest possible lifetime of capital equipment f i is given by

                                 1 = A&     j
                                            :   e-(a'n)up(v)dv.

     For all t 2 0, we have that V(t, w)e[t-fi, t). In the course of the following proof
we shall look at successive intervals of the form [t,-,, t,), n = 1, 2, with to = 0 and
t, = t,- ,+2fi.
     Stability Theorem. Let (k(t), &t), ~ ( t ) ) ,0 5 t 5 a, an equilibrium solution o
                                                             be                        f
(17)-(18). Then lim k(t) = k*, lim A(t) = A and lim w(t) = w*.
                                           *
                  t'rn           t+m                    t-r m

      Proof.   ( ) sup k(t) 2 k* and lim inf A(t) 2
                Iim
                 l                                          A*. Suppose one can find numbers
                   t                        t
a,-, and b,-, such that k(t) 5 a,-,         and A(t) 2 b,-, for all t 2 t,-,, where a,-, and
    ,
b,- satisfy the conditions:


and


                         ,
    Take any t 2 t,- + fi. We have already proved in the Lemma that the maximum
value of k(t), denoted by it,, is given by
246                     REVIEW O F ECONOMIC STUDIES
    Clearly a,-, >k* and b n - , <A* or otherwise there is nothing to prove. Consequently
Cn - 5 m*. It follows that ri,<a,_,.
    Consider now any t 2 t,. Given that k(v) 5 ci, and A(v) 2 bn-, for t-riz 5 u < t,
the minimal value of A(t), denoted b, (see Fig. 2), is given by




                                                                          -           J'(/.. a,)




where



                       ,                    ,
    Now, since 4 <a, - we have that 2, > c,- and b, >b,-   ,.   Finally, let a,, d, 5 a, <a,-,
and c,, en-, <en 5 1, be the numbers defined by
                    ,
               BALANCED GROWTH IN THE JOHANSEN VINTAGE MODEL                                                              247
    We have that k(t) 5 a, and i ( t ) 2 b, for all t 2 t,. {a,) is a decreasing sequence
while {b,) and (c,) are increasing sequences. From (34)-(37) one finds that lim a, = k*,
                                                                                                                 n-rm
lim b,        =   A and lim c,
                   *                      =   m*. For initial values a,, b, and c0 we take, by the Lemma,
n
='    03                    n-+m
k , 2- and _m respectively. This proves part ( I ) .
       (11) lim inf k ( t ) 2 k* and lim sup A(t) S *.                            The proof of this part is completely
                  t-03                             t+ m
symmetrical to the previous one and we shall not give it here.
   Combining parts (I) and (11), we have that lim k(t) = k* and lim i,(t) = A*. From
                                                                                  t+m               t'cu
( 1 1 ) it then follows that lim w(t) = w*. This completes the proof.
                                          t+ m

                                          6. UNEMPLOYMENT O F LABOUR
    Up to this point we have assumed that labour is fully employed at each instant of time.
With fixed coefficients ex-post, however, the economy may lack, temporarily or per-
manently, the capacity to employ all the labourers available.
    Suppose there is a subsistence wage rate w, such that w(t) 2 w for all t 2 0. Corres-
ponding to w, the highest value that A(t) can take, 2, is given by
                                                9'(2),            w    =
 since by ( 1 l ) , ~ ( t2 w implies that A(t) 5 2. Clearly, if
                          )                                                                   w >lo*,      balanced growth is 

 impossible. Furthermore, let 2 and $ be defined by 




            We shall prove that if               w>$      then lim k(t) = 0. This implies that in this case per-
                                                                            t+m
 manent unemployment of labour is inevitable.'                                    From (18) we have that

 k(t)e(crfB + n ) f = S
                             S  V(t,W )
                                                          - u)k(v)dv
                                          g(/2.(v))e("+B+n)"p(t

                                                                           e(afB+n)"k(u)dvk , +sg(X)
                                                                                        =                   j
                                                                                                            l        k(v)dv,
                                                                                                               e(a+B+n)"
     where k , is given by
                                                                      PO
                                                  k o = sg(2) J             o(at"n)vk(u)dv.
                                                                       -m
     Hence,
                                                        ~~(X)k(t)e(~+~+~)~
                                                             S:
                                                 k , + sg(2) e(a+8f
                                                                 ")'k(v)dv
                                                                           5 ss(4.

             Integrating both sides over [O, t ] , we have

                            [     +
                         log k o sg(i)         :
                                               j        k(v)dv] -log k , 5
                                                  efa+B+n)"                              d
                                                                                         j   sg(2)du = sg(Z)r,



     since sg(2)- ( x       +p+ n) <0, we have that lim k(t) = 0.
                                                      m           t+


           Massachusetts Institute of Technology                                            E. SHESHINSKI.
            1 If _w 1 it is possible that temporary unemployment will exist for one or more periods. Eventually,
                     w,
     however, full employment will be restored after each unemployment period. A completely general analysis
     of the Johansen model should, of course, try to tie together all these possible regions, i.e. periods of
     unemployment followed by periods of full employment and uice uersa.
248                    REVIEW O F ECONOMIC STUDIES

REFERENCES
[I] 	 Inada, K. " Economic Growth and Factor Substitution      ", International Economic
      Review, 5 (1964), 318-327.
[2] 	 Johansen, L. " Substitution Versus Fixed Production Coefficients in the Theory of
      Economic Growth ", Econometrica, 29 (1950), 157-176.
[3] 	 Kemp, M. C . and Thanh, P. C.   "   On a Class of Growth Models ", Econornetrica,
      34 (1966), 257-282.
[4] 	 Kemp, M. C., Sheshinski, E. and Pham Chi Thanh. " Economic Growth and
      Factor Substitution: A Comment ", International Economic Review (forthcoming).
[5] 	 Kurz, M. " Substitution Versus Fixed Production Coefficients: A Comment           ",
      Econornetrica, 31 (1963), 209-217.
[6] 	 Phelps, E. S. " Substitution, Fixed Proportions, Growth and Distribution   ", Inter-
      national Economic Review, 4 (1963), 265-288.
[7] 	 Solow, R. M. " Substitution and Fixed Proportions in the Theory of Capital ",
              f
      Review o Economic Studies, 29 (1962), 207-218.
[8] 	 Solow, R. M. " Heterogeneous Capital and Smooth Production Functions: An
      Experimental Study ", Econornetrica, 31 (1963), 523-545.
[9] 	 Solow, R. M., Tobin, J., von-Weizsacker, C. C., and Yaari, M. E. " Neoclassical
      Growth with Fixed Factor Proportions ", Review o Economic Studies, 33 (1966),
                                                        f
      79-116.
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You have printed the following article:
        Balanced Growth and Stability in the Johansen Vintage Model
        E. Sheshinski
        The Review of Economic Studies, Vol. 34, No. 2. (Apr., 1967), pp. 239-248.
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[Footnotes]

    3
     Neoclassical Growth with Fixed Factor Proportions
    R. M. Solow; J. Tobin; C. C. von Weizsäcker; M. Yaari
    The Review of Economic Studies, Vol. 33, No. 2. (Apr., 1966), pp. 79-115.
    Stable URL:
    http://links.jstor.org/sici?sici=0034-6527%28196604%2933%3A2%3C79%3ANGWFFP%3E2.0.CO%3B2-E

    4
     Neoclassical Growth with Fixed Factor Proportions
    R. M. Solow; J. Tobin; C. C. von Weizsäcker; M. Yaari
    The Review of Economic Studies, Vol. 33, No. 2. (Apr., 1966), pp. 79-115.
    Stable URL:
    http://links.jstor.org/sici?sici=0034-6527%28196604%2933%3A2%3C79%3ANGWFFP%3E2.0.CO%3B2-E


References

    1
     Economic Growth and Factor Substitution
    Ken-Ichi Inada
    International Economic Review, Vol. 5, No. 3. (Sep., 1964), pp. 318-327.
    Stable URL:
    http://links.jstor.org/sici?sici=0020-6598%28196409%295%3A3%3C318%3AEGAFS%3E2.0.CO%3B2-F




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    2
     A Note on "Aggregation in Leontief Matrices and the Labour Theory of Value"
    Leif Johansen
    Econometrica, Vol. 29, No. 2. (Apr., 1961), pp. 221-222.
    Stable URL:
    http://links.jstor.org/sici?sici=0012-9682%28196104%2929%3A2%3C221%3AANO%22IL%3E2.0.CO%3B2-3

    3
     On a Class of Growth Models
    Murray C. Kemp; Ph#m Chí Th#nh
    Econometrica, Vol. 34, No. 2. (Apr., 1966), pp. 257-282.
    Stable URL:
    http://links.jstor.org/sici?sici=0012-9682%28196604%2934%3A2%3C257%3AOACOGM%3E2.0.CO%3B2-A

    5
     Substitution versus Fixed Production Coefficients: A Comment
    Mordecai Kurz
    Econometrica, Vol. 31, No. 1/2. (Jan. - Apr., 1963), pp. 209-217.
    Stable URL:
    http://links.jstor.org/sici?sici=0012-9682%28196301%2F04%2931%3A1%2F2%3C209%3ASVFPCA%3E2.0.CO%3B2-%23

    6
     Substitution, Fixed Proportions, Growth and Distribution
    Edmund S. Phelps
    International Economic Review, Vol. 4, No. 3. (Sep., 1963), pp. 265-288.
    Stable URL:
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    7
     Substitution and Fixed Proportions in the Theory of Capital
    Robert M. Solow
    The Review of Economic Studies, Vol. 29, No. 3. (Jun., 1962), pp. 207-218.
    Stable URL:
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    8
     Heterogeneous Capital and Smooth Production Functions: An Experimental Study
    Robert M. Solow
    Econometrica, Vol. 31, No. 4. (Oct., 1963), pp. 623-645.
    Stable URL:
    http://links.jstor.org/sici?sici=0012-9682%28196310%2931%3A4%3C623%3AHCASPF%3E2.0.CO%3B2-6




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    9
     Neoclassical Growth with Fixed Factor Proportions
    R. M. Solow; J. Tobin; C. C. von Weizsäcker; M. Yaari
    The Review of Economic Studies, Vol. 33, No. 2. (Apr., 1966), pp. 79-115.
    Stable URL:
    http://links.jstor.org/sici?sici=0034-6527%28196604%2933%3A2%3C79%3ANGWFFP%3E2.0.CO%3B2-E




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