learning by doing and optimum savings

Learning by Doing and Optimum SavingsAuthor(s): J. BlackSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 2, No. 4(Nov., 1969), pp. 604-612Published by: Wiley on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/133848 .

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Reallocation of the economy's resources towards X and away from Y, main-

taining the capital-labour ratios required by the minimum wage law, releases more labour from the Y industry than can be absorbed in the X industry, given the relative capital-intensity of the X industry, and necessarily creates unem-

ployment (or, where there is assumed to be a subsistence sector, forces part of the labour force to retreat into that sector). The resulting equilibrium of the economy is depicted by the point Q (which may involve more or less production of X than point P, since the relative price of X as the capital-intensive good must have fallen by comparison with Y, the labour-intensive good). O0,0/, the shift of origin for the production of X, represents the amount of

unemployment created by the minimum wage law.

LEARNING BY DOING AND OPTIMUM SAVINGS*

J. BLACK University of Exeter

I / A simplified presentation of learning by doing

This section presents the completely aggregated single sector model of learning by doing originated by Arrow' and developed by Levhari2'3, but treats it as a purely neoclassical growth model. No essential use is made of the marginal productivity or any other theory of income distribution. No consideration is given to the rate of discount required to make investors expect to break even. Except when the economy is on a path of steady growth consideration of the rate of discount raises great economic and mathematical difficulties, which are avoided by the approach used here. Steady growth paths are examined, but interest centres on the path which will be followed by an economy starting from some arbitrarily stipulated combination of labour force and stock of machines.

As in Arrow, G is the "serial number" of a machine and y(G) is the output produced by "the Gth machine." G is, however, treated as a continuous variable, and equation 1 gives the total output per unit of new machines when a cumulative total of G machines have been built. This output is assumed to be constant.

(1) y(G) = a.

*The author is indebted for comments and suggestions to C. J. Bliss and F. H. Hahn. Any remaining errors are his own.

1K. J. Arrow, "The Economic Implications of Learning by Doing," Review of Economic Studies, 29 (1962). 2D. Levhari, "Further Implications of Learning by Doing," ibid., 33 (1966). 3D. Levhari, "Extensions of Arrow's 'Learning by Doing'," ibid.

Canadian Journal of Economics/Revue canadienne d'Economique, II, no. 4 November/novembre 1969. Printed in Canada/Imprime au Canada.

Reallocation of the economy's resources towards X and away from Y, main-

taining the capital-labour ratios required by the minimum wage law, releases more labour from the Y industry than can be absorbed in the X industry, given the relative capital-intensity of the X industry, and necessarily creates unem-

ployment (or, where there is assumed to be a subsistence sector, forces part of the labour force to retreat into that sector). The resulting equilibrium of the economy is depicted by the point Q (which may involve more or less production of X than point P, since the relative price of X as the capital-intensive good must have fallen by comparison with Y, the labour-intensive good). O0,0/, the shift of origin for the production of X, represents the amount of

unemployment created by the minimum wage law.

LEARNING BY DOING AND OPTIMUM SAVINGS*

J. BLACK University of Exeter

I / A simplified presentation of learning by doing

This section presents the completely aggregated single sector model of learning by doing originated by Arrow' and developed by Levhari2'3, but treats it as a purely neoclassical growth model. No essential use is made of the marginal productivity or any other theory of income distribution. No consideration is given to the rate of discount required to make investors expect to break even. Except when the economy is on a path of steady growth consideration of the rate of discount raises great economic and mathematical difficulties, which are avoided by the approach used here. Steady growth paths are examined, but interest centres on the path which will be followed by an economy starting from some arbitrarily stipulated combination of labour force and stock of machines.

As in Arrow, G is the "serial number" of a machine and y(G) is the output produced by "the Gth machine." G is, however, treated as a continuous variable, and equation 1 gives the total output per unit of new machines when a cumulative total of G machines have been built. This output is assumed to be constant.

(1) y(G) = a.

*The author is indebted for comments and suggestions to C. J. Bliss and F. H. Hahn. Any remaining errors are his own.

1K. J. Arrow, "The Economic Implications of Learning by Doing," Review of Economic Studies, 29 (1962). 2D. Levhari, "Further Implications of Learning by Doing," ibid., 33 (1966). 3D. Levhari, "Extensions of Arrow's 'Learning by Doing'," ibid.


604 604 J. BLACK J. BLACK



Notes

The total output which could be produced using all the machines built up to G if only there were sufficient labour available to man them is r(G).

(2) r(G) = aG

It is assumed that once machines are built there is no variation in the amount of labour required to man them. X(G) is the labour required to man the Gth machine, i.e. the labour required per unit of new machines.

(3) X(G) = bG-n

The total labour required to man all the machines built up to G is A(G); integrating equation 3, and letting c = b/(l - n),

(4) A(G) = cG1-n

It is thus assumed that the capital-output ratio on new machines is constant, and the labour required to man the Gth machine is a decreasing function of G. Technical progess is thus purely embodied, purely labour-saving, and is a function only of cumulative gross investment.

It will be assumed that 0 < n < 1, since this is the only plausible case (the rate of growth of cumulative gross investment in modern economies is surely greater than the rate of increase in the productivity of labour manning new equipment). To make the assumption once and for all greatly simplifies the presentation of the model.

As in Levhari, use is made of a gross savings ratio,

s = G/x,

where x denotes output. It is assumed that each unit of G is produced at a constant opportunity cost of one unit of x not available for consumption. s is used in preference to Arrow's ,u = G/x. When comparing steady growth paths, s and ,z are uniquely related, but we wish to examine the behaviour of the economy when s is held constant, or varied in accordance with optimality criteria, while the economy is not on a steady growth path. No attempt is made to consider net savings ratios, since this involves the concepts of net income and net investment, which both raise grave conceptual and mathematical difficulties except when the economy is in a state of steady growth. Depreciation is neglected, as is the possibility that capital goods may have a technically determined finite life. The neglect of depreciation has the great mathematical advantage that it allows the convergence of the economy to a steady growth path when any constant savings ratio is maintained to be proved very easily. This is in marked contrast to the difficulties encountered in estab- lishing the convergence of the analogous no-substitution vintage model where technical progress is assumed to be constant over time (see Solow et al.4). The introduction of time-based depreciation would raise all the same difficulties.

As in Arrow, it is assumed that the labour supply L grows at a constant rate o. It is assumed that the economy starts at t = 0 with L(O) and G(0) set arbitrarily. Wherever this raises no ambiguity time is shown implicitly rather than explicitly. 4R. M. Solow, J. Tobin, C. C. von Weizsacker, and M. Yaari, "Neoclassical Growth with Fixed Factor Proportions," ibid.

605



We first consider how the economy will grow if the gross savings ratio is held constant. If initially L > cG1-n there must be structural unemployment; all machines can be manned and

(5) x = aG.

x and G thus grow at the same rate sa, and Gl-n grows at the rate sa(l - n). If sa(l - n) <5 a, i.e. if s ? a/[a(l - n)], the supply of labour grows at least as fast as the demand, and unemployment persists for ever. While labour remains redundant there is no gain in productivity per unit of gross capital from the purely labour-saving technical progress, and the production per head of those actually employed grows at the rate san.

The ratio o/[a(l - n)] occurs so often that it is convenient to follow Levhari and give it the symbol k.

(6) k = o-/[a(l - n)].

It should be noted that k incorporates the effects in the model of all the param- eters of the system which are not policy variables.

If s > k the demand for labour grows faster than the supply, so that full employment will at some time become possible. In a neoclassical model we assume that when it is possible it is in fact attained.

If initially L < cG1- there is full employment. Not all machines can be manned, and efficient allocation of labour requires that the newest machines are manned. Where G' is the serial number of the oldest machine in use,

(7) x = a(G -G'),

(8) L = A(G) - A(G'),

c(G')1-n = c G1 - L, and

(9) G' = [G1- (L/c)](l-) = G[1 - (L/(cG1-'))1/(l-)], and

(10) x = a G[1 - (1 - [L/(c G1-n])1/1-n)].

It is convenient to abbreviate the expressions in equations 9 and 10; let

(11) q = L/[c G-"]

and

(12) z = (1 - L/[cG1-1])1/(l-) = (1 - q)1(1-n

Note that q is the ratio of the supply to the potential demand for labour, and that z = G'/G, the ratio of the serial number of the oldest machine in use to that of the newest. Also,

x = aG(1 - z).

We can examine the effect of maintaining a constant gross savings ratio s under full employment;

G = sx, so

(14) O/G = sa(1 - z).

606 J. BLACK



Since L grows at the rate o,

(15) /q= a - (1 - n)(/G) = o- (1 - n)sa(l - z) = a(1 - n)[k - s(1 - z)].

If s < k, let k - s = a > 0. (q/g) a(l - n) so q reaches 1 in finite time, and even then the supply of labour grows faster than the demand and so full employment cannot continue. If s > k, however, full employment will persist even if initially the demand for labour is rising more slowly than the supply.

Z 2 0 as q S 0, and q I 0 as s(1 - z) I k.

As z falls, equal rates of growth of the supply and demand for labour are approached as s(l - z) -- k, i.e. before z = 0. If s = k but we start from full employment, it can be shown that q -* 1 and z -> 0, but that q = 1, z = 0 is not reached in finite time. This is proved as follows: when s = k, (q/q) = a(l - n)kz = az > 0. q thus increases in finite time to above any finite limit short of 1. If we assume the contrary, i.e. that 1 - q _ 6 > 0 at all t, then z _ e > 0 at all t, and (g/q) oce > 0, so that in finite time q exceeds any given limit, contrary to hypothesis. q never reaches 1, and z zero, in finite time, however, since

(16) z = [1/(1 - n)](1 - q)n/(l-n)(_-) = -Zn(4)/(l - n).

When s = k, by equation 15, q = a(l - n)kzq = azq, so

(z/z) = -a-zn(1 - z-n)/(1 - n) = -_ (z - z)/(1 - n).

(e - Z) is finite at all t, so x > 0 at all t.

Thus whatever the initial state of the economy, full employment is eventually reached and maintained if s > k; unemployment is eventually reached and maintained if s < k; and if s = k, the economy remains in its initial state whatever this was. The relation between s and k required for the maintenance of full employment is analogous to that found in the no-substitution vintage model with technical progress dependent only on time (Solow et al.6) ; a/(1 - n) is the "natural growth rate" of the learning by doing model.

The stability of the steady growth path with growth rate </(l -- n) and levels of x and G dependent on the constant gross savings ratio s is easily established.

z I 0 as s(l - z) - k, i.e. as z 1 - (k/s);

thus z -> 1 - (k/s). We check that as z -> 1 - (k/s),

(G/G) = sa(l - z) --> sa(k/s) = a/(1 - n).

Thus x grows at the same rate as G when z is constant, and per capita output (x/L) grows at the rate an/(l - n).

The feature of the learning by doing model which makes this simple convergence proof possible, in place of the elaborate proof necessary in the model of Solow et al.,6 is that in the learning by doing case there is only one possible structure which can be taken by any initial capital stock G(O). Ibid. 6Ibid.

Notes 607



It is of interest to examine what happens in the learning by doing model if

wages are equal to the marginal product of labour, and capitalist incomes to the rest of total output, and there are constant but not equal savings ratios out of

wages and profits, with s, < s,. It can be shown that the steady growth path with growth rate ol/(l - n) is a fortiori stable in this case, provided that the income distribution on this path yields an average s such that s > k. This occurs because the form of the production function makes the share of wages in total output, (wL/x), a decreasing function of g, and thus an increasing function of z. Thus with s, < s,, a rise in z lowers the average savings ratio and z is a fortiori stable. That d(wL/x)/dz > 0 is proved as follows, x = aG(1 - z) and thus

(17) w= (Ox/L) = -aG(dz/dq)(dq/dL). z = (1 - q)l/(1-f) thus (dz/dq) = -[(1 - q)'(-n)]/(l - n) = -z/(l -n).

(a/aL) = (q/L), so

(18) w = aGznq/[(l - n)L], and

(19) (wL/x) = zeq/[(l - n)(1 - z)] = z(1 - z-n)/[(l - n)(1 - z)] = (- z)/[(1 - n)(1-z)].

To evaluate d(wL/x)/dz, set v = (1 - z); (zn-z)/( l-) = [(1-V)n- (I -v)]/v

= [v -1 + 1 - nv - [n(l - n)/21]v2 - ...]/v =1- n - n(- n)/2 - [n( 1-n) (2- n)/3!]2 - .

Since all powers of v appear with negative co-efficients, evidently d[zn -)/ (1 - z) ]/dv < O and so d(wL/x)/dz > O and d(wL/x)/dq < 0.

II / Comparison of steady growth paths

When the economy is growing under full employment at the steady rate

o/(l - n), the relation between s and Arrow's ,i = (G/x) is given by

(20) s = ((/x) = cr(G/x)/(l - n) = c-r/(l - n).

Thus s and A are proportional. Preparatory to a consideration of optimal growth paths from an arbitrary

initial point, we shall consider the various principles of optimality which can be applied to steady growth paths. In doing this it will be useful to consider r, the marginal product of capital. This marginal product is the output per unit of new investment, a, less, under full employment, the output forfeited on the oldest machine in use by taking the labour manning it to operate the new machine. If labour is paid its marginal product, the quasi-rent on the oldest machine in use, G', must be zero, so

(21) a - wb(G')- = 0;

thus the marginal product of capital is

(22) r = a - wbG8- = a - a(G')G- = a(l - ).

If labour is not paid its marginal product, the same result still holds when w is

regarded as the shadow price of labour required for efficient resource allocation.

608 J. BLACK



Notes

GOLDEN RULE OPTIMALITY

On a steady growth path, consumption, C, is given by

(23) C = x - G = aG(1 - z) - [aG/(l - n)] = aG[l - z - k].

By equation 12, G is proportional to (1 - zl-n)-l/(l-n), and C is maximized when log C is maximized, i.e. when z is chosen so as to maximize

log(l - z - k) - {[log(l - zl-")]/(l - n)}, i.e. when

[-1/(1 - z - k)] + [-1/(1 - n)][-(1 - n)z-/((l - z)-)] = 0,

i.e. when

1 - i-n = (L - z - k)z-n, so that

(24) 1 - zn = k.

This result can be reached by an alternative argument; for consumption to be maximized, the marginal product of capital when first built, i.e. the output on new machines less the opportunity cost of using labour to man new machines, or r, must equal the rate of growth. Thus

r = a(1 - Z) = o-/(1 - n), so

1 - n = k.

ARROW OPTIMALITY

Arrow applies a discount rate 1 to consumption, i.e. seeks to maximize the present value of consumption at all t, discounted at this rate. To achieve this the marginal product of capital must equal the discount rate A. Thus

(25) 3 = r =a(l - Z), and

(26) 1 - z= (/la).

RAMSEY OPTIMALITY

Let there be an iso-elastic utility function, with the marginal utility of per capita consumption, y, given by

(27) u = y-.

On any steady growth path, C grows at the same rate as G or x, and y grows at the same rate as (x/L), i.e.

(y/y) = (x/x) - o = [o-/(1 - n) - a = c-n/(l - n) = akn.

Assuming no pure time-discounting, Ramsey optimality of a steady growth path requires

(28) r = a(l - n) = aakn, so

(29) 1 - z= akn.

The conditions under which the optimality conditions on steady growth paths coincide are thus as follows. The Golden Rule path is Arrow-optimal if (if and only if) 3 = ak. The Arrow path is Ramsey-optimal if (f/a) = akn. The Golden Rule path is Ramsey-optimal if an = 1. If a > (1/n), there will be a Ramsey-optimal steady growth path with G, x, and C all below those on

609



the Golden Rule path. On this Ramsey-optimal path, the extra savings which would be needed to catch up with the Golden Rule path will never appear worth while. Even if the economy started with sufficient capital to make steady growth along the Golden Rule path possible without any initial extra savings being needed to attain it, it is shown in the next section that a Ramsey-optimal path from this initial point will involve convergence onto the Ramsey-optimal steady growth path from above.

If a < (1/n), however, the Ramsey optimality conditions break down; mechanical application of formulae 24 and 29 gives

1 - () = an, and 1- (ZG)n = k; if an < 1, 1 - (R)n < 1 - (zG)n, so ZG < ZR and SR > SG. The resulting steady growth path however gives CR < CG at all t, so that the policy resulting from mechanical application of the Ramsey Rule cannot be optimal. No Ramsey- optimal growth path exists in this case, since once any positive consumption stream is embarked upon, Ramsey optimality conditions upon its time-shape require it to grow faster than is possible on any steady growth path.

III / Ramsey-optimal time paths from arbitrary initial L(O), G(O)

In this section we assume that a > (1/n), so that a Ramsey-optimal steady growth path exists, and show that the Ramsey-optimal path from any initial position converges onto this steady growth path. For Ramsey-optimality,

r = a(1 - ^) = a(y/y);

(y/y) = (C/C) - ;

and C = x(l - s) so

(C/C = (x) - [s/(I - s)]. x = aG(l - z) so

(x/x) = (G/G) - [/(1 - z)]; and

(G/G) = sa(l - z).

Thus for Ramsey-optimality,

(30) (a/a)(1 - z") = (y/y) = sa(l - z) - [/(1 - s)] a - [/( -z)],

and

(31) [s/(1 - s)] = sa(l - z) - (a/a)(l - z") - a - [/(l - z)].

Given z and s at any time, z can be found as follows:

by equation 16, Z = -_zn()/(l - n),

and by equation 15, q = [ - (1 - n)sa(l - z)]q. Also,

q = 1 - zl-, thus

= -Z"[o - (1 - n)sa(l - z)](1 - z-")/(1 - n), so

(32) z = a(zn - )[s(l - z) - k], and

/(1 - z) = a[(z - z)/(l - z)][s(l - z) - k], and

s/(l - s) = sa(l - z) - (a/a)(l - z") - a - a[(zn - z)/(l - z)]

X [s(l - z) - k], so

610 J. BLACK



(33) s/(1 - s) = a{(l - z-)[s - (1/a)] + k[(z - z)/(l - z)l - k(l - n)}.

Equation 32 follows from the causal mechanics of the model; equation 33 from these plus the application of the Ramsey-optimality condition. We are now in a position to draw a phase diagram for z and s when they obey equations 32 and 33. In the figures, E represents the Ramsey-optimal steady growth path, where

* = (1 - akn)l'I and s* = k/(1 - s*).

The locus z = 0 passes through E, with a positive gradient, since on this locus s = k/(l - z) and so (ds/dz), measured along the locus, is k/[(l - z)2] > 0. With given z,

(az/8s) = a(z4 - z)(1 - z) > 0;

thus above the locus z = 0, z rises, and below it z falls. By equation 33, the locus s = 0 is given by

(34) s - (1/a) = k[(l - z)(l - n) - (Zn - Z)]/[(1 - Z)(l - z)]I.

The LHS of equation 34 evidently increases with s. Behaviour of the RHS as z increases depends on the numerical values of n and z. If the RHS of equation 34 increases as z rises, then the locus s = 0 slopes upwards, and vice versa. s = 0 passes through E, and can be shown to lie above z = 0 to the left of E and below z = 0 to the right of E; the slope of s = 0 at E is less than that of z = 0. This is proved as follows:

by equation 33, with given z, d[s/(1 - s)]/ds = a(l - zn) > 0;

thus below s = 0, s falls, and above s = 0, s rises. To find d[s/(1 - s)]/dz along the locus z = 0, consider the RHS of equation 31; as s(1 - z) = k and the last term = 0,

d[s/(l - s)]/dz = (a/a)nzn-, > 0.

Figures 1 and 2 show the phase diagram of s, and z, with s = 0 downward and upward sloping respectively. With either phase diagram, E is a saddle point. If z(0) > z*, there will be one choice of s(O) which gives z -> z* and s -> s*. Any higher s(O) would give a path of s and z which diverged upwards, leading to pointless continued accumulation bringing the economy above the Golden Rule output path, but with too little consumption. Any lower s(O) would give a path of s and z which diverged downward, involving commence- ment of a time-path of consumption too high to be permanently sustained, so that the capital stock would ultimately cease to grow. The initial s(O) must be below that required to maintain the initial level of z; whether s(O) is above or below s* depends on the value of n. Similarly, if z(0) < z*, there will be only one choice of s(O) which will lead to a time path converging on s* and s*; this s(O) will exceed that required to maintain the initial value of z. The chosen s(O) must always lie between s* and that required for s = 0.

The results obtained here are very similar to those obtained by Sheshinski7

7E. Sheshinski, "Optimal Accumulation with Learning by Doing," in K. Shell, ed., Essays on the Theory of Optimal Economic Growth (Cambridge, Mass., 1967).

Notes 611



ALTON D. LAW ALTON D. LAW

FIGURE 1 FIGURE 1

S S I=o FIGURE 2

I=o FIGURE 2

for a learning by doing model with disembodied technical progress. The present article applies Ramsey-optimality criteria to the vintage learning by doing model originated by Arrow. It shows also that these results can be reached with less sophisticated mathematical methods than those employed by Sheshinski.

INTERNATIONAL COMMODITY AGREEMENTS TO PROMOTE AID

AND EFFICIENCY: THE CASE OF COFFEE:' A COMMENT

ALTON D. LAW Western Maryland College

In the May 1968 issue of this JOURNAL Irving B. Kravis produced a provocative article on commodity agreements of the International Coffee Agreement type.' He gave some history and analysis of that agreement and put forward a sug-

*The author wishes to express appreciation to James H. Street and Tony Killick for comments and encouragement. lIrving B. Kravis, "International Commodity Agreements to Promote Aid and Efficiency: The Case of Coffee," this JOURNAL, I, no. 2 (May 1968), 304.


for a learning by doing model with disembodied technical progress. The present article applies Ramsey-optimality criteria to the vintage learning by doing model originated by Arrow. It shows also that these results can be reached with less sophisticated mathematical methods than those employed by Sheshinski.

INTERNATIONAL COMMODITY AGREEMENTS TO PROMOTE AID

AND EFFICIENCY: THE CASE OF COFFEE:' A COMMENT

ALTON D. LAW Western Maryland College

In the May 1968 issue of this JOURNAL Irving B. Kravis produced a provocative article on commodity agreements of the International Coffee Agreement type.' He gave some history and analysis of that agreement and put forward a sug-

*The author wishes to express appreciation to James H. Street and Tony Killick for comments and encouragement. lIrving B. Kravis, "International Commodity Agreements to Promote Aid and Efficiency: The Case of Coffee," this JOURNAL, I, no. 2 (May 1968), 304.


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learning by doing and optimum savings

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