Further Implications of Learning by Doing

The Review of Economic Studies, Ltd.

Further Implications of Learning by DoingAuthor(s): David LevhariSource: The Review of Economic Studies, Vol. 33, No. 1 (Jan., 1966), pp. 31-38Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2296638 .

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Further Implications of Learning

by Doing1 The following discussion tries to explore further and extend Arrow's 2 analysis of

learning by doing; in particular, it tries to clarify the nature of the divergence between social and private returns and to determine explicitly the " golden rule " path of maximum maintainable consumption.

In Arrow's model productivity is related to cumulative gross investment. Technical progress is fully embodied in machines. The production function discussed is that of fixed coefficients.

We use Arrow's notation: G(t)-cumulative gross investment or the serial number of machines.

a-output capacity of machines. bG-n-labor requirement per unit of time for operating a machine with serial number

G and 0 < n ? 1. L-total employment. x-total gross output.

G'-the serial number of the oldest machines used at a given time. Total output is

(1) x = a(G - G'). We find G' by equating the labor requirement for the operation of machines of serial numbers [G', G] with labor supply:

(2) L = b jGndG= 1 (Gl-n G"-n) n I 1

b logvG n 1.

For G' we easily obtain from (2):

(3) G' =(G1l _ ) ,wherec= 1 n#1

Ge-Llb n = 1. Substituting in (1) we get

L 1n

(4) XX= aG[I - - n #A1

-aG(l - e-Llb) n 1.

The oldest machines of serial number G' must earn zero quasi-rents so that a - wbG'-n = 0 if the wage w is measured in units of x.

We find then n

a a L 1)n 1 (5) W i ln-- n#=41.

b b \ I I am indebted to R. M. Solow, F. M. Fisher, and P. A. Samuelson for many helpful comments, and

to Professor Arrow for reading this manuscript. I am responsible, of course, for any errors remaining. 2 K. J. Arrow, " The Economic Implication of Learning by Doing, " Review of Economic Studies,

XXIX (1962), 155-73. 31



32 REVIEW OF ECONOMIC STUDIES

Social and Private Returns

If society saves an extra unit of output at time v then in all future time society can

enjoy an extra a- units of output: aG n

ax _ ~ L\ 1~n1 (6) x = a[1 \Gl-n -_) nGj

aG [i (G )]

On the other hand a private investor who saves a unit of output at v when the serial number of capital is G(v) will get a stream of rentals of r(v, t) = a - w(t)bG-n(v) or, using (5),

( [ (~~~~~~~~G(v)) The marginal social product of capital is thus r(t, t)-the private rental of a new

machine. It is obvious that uniformly with t

(8) a[1-(G((t)) >a 2 [-G(()) J

with equality only at t = v. G'(t) is increasing and G'(t) G(v) at some t > v, and the capital produced at v is scrapped and its private rental from then on is zero.

It is easy to find out what happens in the exponential world described by Arrow. Here labor is growing at a rate a and accumulated gross investment and output are both

aG' growing at a rate 1 _ is constant in this case, and if T* is the length of life of the

capital, then

-n(t- T*) _( '

G' Goe 1-n

(9) G a

t G

Goe -~nT*

The marginal social product is then a(l - e 1-n )-a constant independent of time. The stream of quasi-rent is

(10) r(v, t) = a - - e I-G ]

At t = v it is the same as the marginal social product a(l-e e-n and it then declines till at t = v + T* it is zero.

Similar results are easily derived for n = 1:

ax = a(l-eL/b) a(

Denote by pS(t) the social rate of return, and by pp(t) the private rate of interest. Assuming perfect foresight and competitive market we find the profile of pp(t) by the

condition that the present value of the stream of rentals is 1. Similarly p,(t) is such that the present value of the stream of additional social product is 1. Remembering the identity

u

00 -ff(x)dx 00

f(u)e t du = 1 (if f(x)dx = oo) we find that at each t the instantaneous social rate t .t

1 For this case, R. Solow reached the same result by a different approach (unpublished lectures, 1962).



FURTHER IMPLICATIONS OF LEARNING BY DOING 33

of return is p,(t) = a[l - (G ))]. The only case of constant social rate of return is

GI G' of constant -, which occurs in the exponential case. As - increases ps(t) decreases;

this is clear intuitively since we transfer labor from " not very old " capital to new capital. If T(v) is the economic life of machines produced at v, and assuming perfect foresight, pp(v) should satisfy the functional equation

v+ T(v)

af&Ij(u)du [ G

- (t) a e-vP)d [1-G( ) dt=1

and G'[v + m(v)] = G(v). In the exponential case where the life of capital is a constant T* the equation takes

the form

-fp (u)du iv+T*

a (tT*) v P

Ia[1 - e1n ]e dt =1. v

Or by using = t - v

-fp (u)du T - C--n(t -T*) V

a[l - e (1-n ]e dr= 1. 5 Trying pp-a constant-we see that, as Arrow shows, this equation possesses one and only one constant solution since the function on the left is a monotonic decreasing function of pp with a range (0; co). It is possible, moreover, to prove that this is the only solution for this functional equation.' In this exponential case when both ps, pp are constants it is easily seen from the dominance of marginal social product over private rentals that Ps > pp.

Share of Capital and Labor

The pseudo-production function x = aG[l (1I- is of increasing

returns to scale. Multiplying G by X and L by xi-n, output is multiplied by X. It is clear that both capital and labor cannot get their marginal social products. Finding

n ax I a (lL\=\a In =G,

(aL I-nc ( cG1n b

we see that labor gets its marginal product. Since labor gets its marginal social product, capital, as we already know, gets less

than its marginal product. If capital did receive its marginal social product, the share of capital would be

n

ax LF1/ DG [I - (Gl-n --G-n I

(12) X G

-fp (u)du

'We can rule out all other solutions by using the non-negativity of e . This is proved for an analogous case in a forthcoming paper by Solow and others " A Model of Fixed Capital without Sub- stitution. "




-I (n Remembering that 0 < n < 1 and G <1, 0< G < 1.

G G 1-G

The implied labor share is

(13) 1- ~3G - \ G, (G /

G But labor really gets

(14) wL _ c (G1-n - Gl-n)G'n 1 (G) -(G x b G-G' ]-n GI

I G

The labor share is inflated by 1 , where 1 > 1, compared with its share if capital

received its marginal product. Labor offers itself inelastically and its income is rent; the only allocation problem in this model is allocation over time. If, under certain assumptions on saving behavior there is misallocation if capital does not get its marginal product government may interfere in the competitive process. Government can impose a tax on wages at a rate n and then transfer the proceeds to all persons who have ever invested regardless of whether the capital they invested is still being used. The subsidy will increase through the life of a machine and eventually, when the machine is scrapped, will reach the level of the marginal social product of capital. Even though the machine has been scrapped its social product remains the same since it has raised the serial numbers of subsequent machines. Thus in the exponential case where the marginal social product of capital is

a(1 - e-gnT*) (where g = 1 - ) and rental of capital is r(v, t) = a[1 - eng(t-v-T*)] each

unit of capital gets a subsidy of a(l - e-gnT*) - a[ -eng('-V-T*)] ae-gnT*[eng(' _ 11 for v ? t ?<E - v + T* and a(l - e-gnT*) for t v + T*.

subsidy a(t] -e- g,T*)

/ae-gtT (e.? 1 ) I

0 t=-T* t

Alternatively, the following tax-subsidy system would accomplish the same result: tax all profits and a proportion n of wages; then give back a subsidy to all investors so that each receives payment according to his share of total accumulated gross investment.




Our shares calculation shows that in this way capital would get its marginal social product. To find the welfare effect of our tax-subsidy, we must assume some welfare function

over time and some functional relationship between savings and the rate of return. We

shall follow Arrow and assume that society wants to maximize U e-5tC(t)dt where

C(t) is the consumption society enjoys at t. Individuals have a rate of time preference of r; the supply of capital is infinitely elastic at a private return of P. Society will take all investment at rates above P, none at rates below P. So the private return must be P in a case where some, but not all, income is saved. With the subsidy there is no divergence

a9x between the private and the social return and p, = ps = G = 3.

We shall later show that x= implies that we are on the optimal path calculated

by Arrow and that the capital-output ratios in the competitive case with the tax-subsidy system and those on the optimal path calculated by Arrow are the same.

Stability of Exponential Growth

Even though Arrow pays close attention to exponential growth he does not discuss the question of relative stability. Assume a constant saving rate s and labor growing at a constant rate a. The system is described by

(15) G = sx(t) = saG[ (1 - cG'-fn)

Trying the exponential solution G(t) = Goel-n we get

a sa Lo 1-n1 1 -n sa[1 - cG -n,IJ

Solving for Go we obtain

(16) Go(s)= ( 1-n

1~ ~ ~ ~~~~~~~- [ ( (I-~n) as)

if s > 1 -nl)

'It is easiest to see this restriction on the savings rate by using Arrow's (41) T* =-

= G(t) ~ ~~~~~s s1 log (I--) where p = On the exponential path ta = - = and T* a~~t x(t) 9 ~~~~ a/I -n a/i-n

log I n( as and the condition for T* > 0 is s > n)a If s < na e savings a

rate is too small to maintain steady growth at a rate 1- while length of life is constant.

Go(s) is a monotonic increasing function of s. To show that the solution of (15) tends a

asymptotically (in the relative sense) to this exponential solution define y = log G - ~ t.




With Y = - 1a, (15) is then transformed to

(17) j=sa[1 -(1 Le-(l-n)Y)] 1 a

Call the function on the right-hand side g(y). It is easy to see that g(y) is a monotonic

decreasing function of y, with lim g(y) and g log :o = as-1

> 0. So there is a unique y* such that g(y*) = 0. By the monotonicity of g(y) we see that j'> 0 if y < y* and y < 0 if y > y* and hence y -+ y*. Solving g(y*) = O we obtain

Lo

(18) y 1 log_c_l

= log Go(s). -n * 1 1gn (1

a~~~~~~~~~)a

Thus, G(t) - Goe1-n in the relative sense.

Optimal Saving in Arrow's Model

Here we interpret Arrow's optimal growth path and show that the tax-subsidy we have devised guarantees that we are-if competitive conditions prevail-on this path. The

welfare function is U= {e-PtC(t)dt. Note that Arrow does not take into account that

the initial serial number G(O) is given. Thus, in Arrow's (49), U = U1 - lim e-tG(t) +

G(O) but then in the discussion on optimal behavior he derives optimal capital expansion a

of the form G(t) - Ge1-n where G is the constant maximizing the expression

G[a-r3--a(\i- - n ]

For t = 0 we see that G(O) = G and unless this is satisfied we have discontinuity at 0.

Since the integrand is of the type e-tC(t), we should invest everything so long as ax or G~~~~~~~~~~~~~~~~a

aG (the own rate of interest) is above r; then when the marginal product of capital is below

ax 3 we should consume everything. Eventually we reach a path where -= 3. It is easy

ax to see that = 3 implies the path calculated by Arrow:

- X=a

I (G

= P

n

and ( I) = 1 - = y; but on the exponential path (Gs) = (I -_o 1n (which




Arrow denotes by v) and thus v y which is the solution derived by Arrow.' So the a9x a9x

optimal path is of the following type: if G> E invest everything until the path G = f

is reached; if ax < 3 consume everything until the same path is reached; then invest

exactly the amount required to remain on ax= . To show this pattern of the optimal path

we can apply Bellman's " Principle of Optimality ".2 Assume that we have a feasible path

of consumption C(t) and at a certain neighbourhood of to, a < 3 and C(t) > 0. Now

if we save an extra AC in the period (to -2 t0 + we lose "utility " of (approxi-

mately) e-4t. AC. A. In the next A-period we increase consumption uniformly so that at A

to + 3 2 we return to C(t). We gain by consuming what we saved before and by the extra

production with the additional capital AC. A. The gain amounts to (approximately)

e-p(to+A) LAC. A + ax(AC. A)AJ. The net gain is AC. A. e-Pto e-Po" + ax . A-1) =

AC. Ae-t a

- )A > 0. Similarly, we can show that if ax < P we have to consume

everything; otherwise we can gain by a small perturbation.3 Eventually we reach the path

ax= , and we cannot gain by perturbation of this path. The tax-subsidy that we devised

ax guarantees that ps = pp, hence when the rate of interest is P, Ps =G = 3.

aG If society's rate of time preference P is not greater than we can still find among

all exponential patterns one which dominates the others. We call this the " golden rule a

path ". On the exponential path production is growing like xoel-n and consumption like

(1-s)xoe'- . We want to maximize the level (1- s)x. On this path x = )

and xo(I - s) a Gn(s) (1 - s), or, using (16) andxo1 s ~ -n s

1 s 1 Max

a(1-n)- -

'An alternative presentation that Lx= implies Arrow's optimal path, is to use(G)=(I )

[Arrow's (40)], where , ; from this we obtain - I )

Using the notation of Arrow's (57) and (60), [y = 1- and W = - - ] we see that y = W. aat

Using y = (1-a-g) [Arrow's (59)], this implies that the capital-output ratio G/x on the path calculated

by Arrow is the same as on the path ps =

2 R. Bellman, Dynamic Programming, Princeton University Press, 1957.

a We can also apply the Pontriyagin Maximum Principle to prove all this.




Looking at the logarithm of this function, the maximum is attained at the same s and we then want

1s 11r__ Max log s 5- 1 log I[ - _ sa I ]

~~5 s 1-nI a(I -n)

Let(1 k and define u= 1-- so that k-- 1-k-u. The range of the a(l-n) s s

savings rate is [k, 1], hence that of u is [o, 1 k]. u is a monotonic increasing function

of s and we can maximize log (I-k-u)- 1n log (1- ul-n) with respect to u.

The necessary condition for this is

1 U-__ _

1 - k -u + 1 - u1-n or

u-n-i -k and u=(1 k)n.

It is easily seen that the second derivative is negative and we have a true maximum. Using the definition of k and u, we get

a

(19) S* k a(l-n)

1 -(Il k)n ln

We see that the optimal savings rate is independent of Lo and b which give only the scale of the economy and depend only on k. On the path with the savings rate s*, the rate of growth and the social rate of return are the same. To see this, note that on an exponential

path s = 4 1-n [Arrow-'1-s (39) and G) W= I - [Arrow's (40)]. The

social rate of return is then ax=a1 - (i- n!~) {I (- I a n) aG aal sa(l n)

a(l - u) - ak = 1 - The share of capital, if capital receives its marginal product,

_x

-G. G i x = 1n =s* with this saving rate. So on the golden rule path the share of

profit is equal to the rate of saving.

It is clear that Arrow's optimal path aa > a

is below the golden rule path

with - = -n Because of its time preference society is not ready to do the extra saving

to move to the higher path. If society is above the path a- = l it prefers the short burst

ax~ a

of consumption and the return to the path LG 3.

Hebrew University, DAVID LEVHARI. Jerusalem.



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