technological condition for balanced growth: a criticism and restatement

JOURNAL OF ECONOMIC THEORY 9, 171-184 (1974)

Technological Condition for Balanced ~r~wt

A Criticism and Restatement*

ALBERTO C~r~osr

Department of Economics, Umkersity of Piss, Piss, Italy

AN0

STANISLAW GOMULKA

London School of Economics, London, Englaalzd

Received July 1, 1973

According to prevailing opinion, only the neutral form of technological progress in the Harrod sense is consistent with balanced growth in a one-sector constant returns-to-scale economy. Though various definitions of balanced growth are in use in the literature, the above highly restrictive technological condition is believed to hold for all of them. The paper demonstrates that this belief is not correct. The condition is shown to be false if the definition of balanced growth (i) does not require the constancy of the marginal product of capital (or the interest rate), and (ii) permits the time semiinfinite or indeed any finite balanced growth path. More specifically, under (i) and (ii) there exists a balanced growth path consistent with a significantly wide class of technological changes of the capital-using (labour-saving) form in the Harrod sense. Alterna- tively, this condition is correct if either (i) the interest rate is required to be constant or (ii) growth is balanced if it is such for all time-mat is, for both past and future. The condition is also correct if the socioeconomic institutions are such that the constancy of the savings ratio implies the constancy of the capital share.

1. INTRODUCTION

According to prevailing opinion, in a one-sector constant returns-to- scale economy the only form of disembodied technological progress consistent with an equilibrium-balanced growth path, on which both

* The authors would like to thank Professors R. 6. D. Allen, C. Bhss, W. Fuknda, F. Hahn, K. Laski, M. Morishima, S. O&a, R. Solow, and especially T. Fujimoto for conversations, cements, and encouragement. Their suggestions have helped us a great deal to prepare the f&l version.

171 Copyrighr 0 1974 by Academic Press, Inc. AU rights of reproduction in any form reserved.

172 CWILOSI AND GOMULKA

capital and output are growing at a common constant rate and labor at the same or another constant rate, is a Harrod-neutral progress. This proposition was first formulated by Swan [7] in 1960. Since then it has been quoted by many other authors, often with independent proofs. By now it has become a part of conventional wisdom in growth theory.

The primary purpose of this paper is to show that the proposition above is not correct. Since different definitions of balanced growth are used in the literature, the framework of analysis offered below will be general, so that one can test quickly whether, under a given definition, the standard condition is in fact correct. In particular it will be shown that if the standard condition is to remain valid, the definition quoted above must be modified to require also the constancy of the interest rater or, alter- natively, to describe as a balanced growth path a situation that must exist forever and must have existed forever. The paper includes also a brief comment on the literature on the subject.

As a by-product of the analysis, two general results concerning one- sector production functions are given. They state (i) that any constant returns-to-scale well-behaved production function can be rewritten in a “quasi-labor-augmenting” form and (ii) that the classification of technical change by Harrod is equivalent to that by Kalecki.

2. MODEL

In a self-explanatory notation, the economy in question is described by the following three equations:

WI = 10(K(~), w; t>, (1)

L(f) = nL(t), (2)

K(t) = sY(t), (3)

where JZ > 0 and 0 < s < 1, and where F(e) exhibits constant returns to scale. F(a) is also concave in capital K (expressed in terms of Meccano sets) and employment L (in terms of man-years), and satisfies the Inada conditions. That is to say, it is an ordinary neoclassical, well-behaved production function. The savings rate s is in this model treated as a “control” param-

1 This additional requirement is included in the definition given by Burmeister and Dobell [2, pp. 77-781, but their proof makes no use of it. For a comment on the proof see Section 9, part (iii). Section 9, part (iv) constains also a critical comment on the proof offered by Swan.

CONDITION FOR BALANCED GROWTH 173

eter, independent of the distribution of income between wage earners and profit earners, as this is a characteristic feature of the Swan-Solow type of analysis.2

3. BASIC DEFINITIONS

DEFINITION 1. The economy is said to enjoy balanced (or steady-state) growth if K and Y are both growing at a common, constant rate and L at the same or another constant rate.

This definition, which is equivalent to that given by Allen [19 p, 1741 and by Hahn and Matthews [5, p. 8291, implies that the saving-investment ratio s and the capital/output ratio v must both be constant over time. definition does not specify the time interval. This should be interpreted to mean that growth is balanced as lon g as the specified conditions are satisfied. However, it will be shown later that the length of the time interval is not always a matter of indifference for the technological condition The following two specifications appear to be particularly important; growth is balanced if it is such:

(la) for all t E (-00, +co), or (1 b) for all 2 E (0, + co).

Accordingly we have, apart from Definition 1, two other definitions balanced growth, referred to as Definitions la and lb. If not specifi otherwise, we use the notion of balanced growth (or steady state) in the sense of Definition lb.

DEFINITION 2. Technological change is neutral in the I-Iarrod sense if a constant ZI = K/Y implies the income elasticity with respect t,o capital

?T = vF~ to be constant over time. It is capital using (saving) if a constant u implies 7r to be increasing (decreasing) over time.

* The institutional arrangements which would justify such treatment are not obvior?s. In a one-sector perfectly competitive economy with no public sector the income elasticity with respect to capital rr, defined as VF K, where 21 = K/Y, equals the capital share in national income. If we accept the Kalecki-Robinson assumption that in such an economy the wage-earners consume what they earn and profit-earners save all they get, then s must be equal to V, Of course, s = n is an extreme case. On the other hand, the postulate of s being completely independent of 71 is clearly another extreme, since it requires the propensities to save of both capitalists and workers to ‘be the same. However, this postulate might not be very remote from reality in a planned economy with a centrally controlled savings ratio. It is also possible to conceive a “centrally regulated” capitalist economy with a government policy of taxation geared to keeping the savings-investment ratio constant.

174 CHILOSI AND GOMULKA

DEFINITION 3. Technological change at time t is globally neutral or globally capital using (saving) if it is, respectively, neutral or capital using (saving) for all v > 0.

4. FIRST NECESSARY CONDITION

Equation (1) implies that

g=ri~+(l--)n+h=~~~(~-g)+n+~, (4)

where g = p/Y and h - (l/Y)(aY/at). Along a balanced growth path, z/K = g. Hence

x g=n+l-n =n+a.

Because h and n are, in general, functions of v and t, LX in (5) may also depend on z, and t. However, balanced growth requires g to be constant over time. Hence, in view of y1 being a fixed number,

for every t (6)

is a necessary condition for the path associated with a given v = v* to be a balanced growth path. This is obviously a constraint on the nature of technological progress. Yet, the dependence of 01 and g on capital/output ratio, as being compatible with the definition of balanced growth, still remains a possibility.

5. TVPE OF TECHNOLOGICAL PROGRESS AND 01 =~(v,t)

To find what type of progress is consistent with the steady-state configuration, we need to know first what form of production function is generated by a particular 01 = CX(V, t). Here two general results will be helpful (proof in Appendix).

LEMMA 1. Let both F(K, L; 0) and 01 = ol(v, t) be given, The production function Y = F(K, L; t) thus generated is implicitly defined by the following functional form:

Y = G(K, Les~ar(u,7)dT), where v = K/Y.

GONDITION FOR BALANCED GROWTH 175

According to this lemma, any form of disembodied technological progress in a one-sector constant return-to-scale economy is of “quasi-labor- augmenting” type, with the rate of augmentation depending generahy upon capital/output ratio and time.

LEMMA 2. Technological progress at time t and for a given u is ~e~tr~~ in the Harrod sense if and only if 01, = &(v, t>/av vanishes. It is capstan using (saving) if and only if 01~ is positive (tiegative). That is to say,

sign t ant t> ) = sign ( aaFi t’ j.

A classification of technological progress in which the different forms of technological change are distinguished by the sign of &(v, t)/av is due to Kalecki [6]. According to this classification, technological progress is capital using (saving) if a higher rate of growth of labor productivity a is associated with a higher (lower) level of the capital/output ratio v; it is neutral if LY is independent of v. This classification is believed to be particularly convenient in the framework of a centrally planned economy; the growth-oriented central planners may indeed be expected to adopt a more capital intensive technique if the growth rate of labor productivity can be increased by switching to a higher capital/output ratio. y Lemma 2 the Kalecki classification appears to be equivalent to tha arrod.3

6. CONCAVITY OF F AND THE FORM OF 01 = ol(v,t): Two NECESSARY CONDITIONS

According to (6), only those v’s at which SE/at vanishes for all t > 0 will be compatible with balanced growth. Suppose the rate a* = ol(v*) is constant over time, but CX,(V*) may or may not vanish. Hf it does not, then by Lemma 2 the technological progress is nonneutral in the Nevertheless, the growth rate of income and capital, g” = E + cz(v*), is constant, and this rate calls for the savings-investment ratio s* = v* D g*$ also a constant.

This is, however, not the end of the story. For the required savings ratio s* might be too great, even greater than 1, and CL,(V*) # 0 may sooner or later violate the concavity of the production function.

3 This result was first published by these authors in a Polish journal, Ekonomista (January 1969). [4] is an English version of that article. Notice that in a special case of (globally) Harrod-neutral technological change cr, = 0 for all v (Lemma 2), and the rate of labor augmentation depends then solely on time t (Lemma l), which is a well- known result due to Uzawa [S].


The implications of ol,(v*) + 0 for concavity of F can easily be traced out. According to (A.6) and by (A.9) (see Appendix),

q(v”, t) = v* (pJ- + ta,cu*,) = 1 “‘$f Q 3

where t < 0 if it refers to past and t 3 0 otherwise. From (A.10) it is clear that q must be positive for any finite t. Therefore, when either past or future (but not both) is of our concern, the permissible form of ~(0, t) must satisfy ta, > 0 for z, = a* (and 01~ = 0 when both). Accordingly, we have

20 t E (0, m> o!,(v*, t) = 0 for tE(-co, co) (7)

GO tE(-co,O)

as our second necessary condition for 01 = ol(v, t). Interpretation of this condition in terms of the form of technological progress, in view of Lemma 2, is self-evident.4

Condition (7) guarantees only that both FK and FL are positive as the economy moves along v = v*. But concavity of F requires also FKK and FLL to be nonpositive.

Since ‘IT = vFK = q/(1 + q), FK = v/(1 + v), where 7 = q/v, and therefore sign FKK = sign(q, - 73. Thus, in addition to (7), we also require rlV < r2. Because 7 = Q + j’” ,, 01~ do, where Q, = yuo/yo, the latter in- equality reads qou + $01,~ do < (q. + ji 01, d~)~. Along a balanced growth path, 01 is time invariant (condition (6)). Hence it is required that

G2t2 + @70% - %,) t + To2 - qbv > 0 (8)

for v = v* and for all t from a specified range. This is our third necessary condition to be satisfied by 01 = OI(V, t). From (8) it is clear that here again the time specification is important. We may note that since vo2 - yoV > 0 (as at t = 0, FKK is assumed negative), (8) is certainly satisfied if c+o% - 01~~) t > 0; that is, when

30 t E (0, a> 2rjoa, - a,, = 0 for tE(--CO, a>, (9)

GO tE(-q0)

v = v*. One may check that FLL < 0 also if (8) is satisfied.

4 If the definition of balanced growth is modified to require gr = go , instead of gr = gK , then ol(v, t) must satisfy fol, Q 0, instead of tol, > 0. As a result we have a condition which is dual to (7). In particular it permits technical progress of capital- saving form for t E (0, co). We owe this observation to Professor M. Morishima.


EXAMPLE 1. Suppose we are concerned with t E (0, co). Take a. = a0 + E 0 v, where both c+, and E are positive constants. Sin.ce 01, = E and 01,~~ = 0, both (7) and (9) are satisfied (we note that Q > 0); the production function generated by this particular OL is concave as required. Indeed, under this time specification any concave CY = E(G) generates a concave production function.

7. FOURTH NECESSARY CONDITION

Let us now turn to the question of having sufficient savings to support balanced growth of capital. Here the Harrod-Domar consistency condition has to be satisfied. According to this condition, a capital/output ratio z; could be maintained constant only if

s = ?I(?2 + a(u, t)). (101

We may note that (10) permits situations when technological change is of nonneutral form. For example, let 01 depend on u alone and be swh that c+, > 0 (the technological change of capital-using form). Since y1 T a(v) is positive for all z)‘s, the right-hand side of (10) is a monotonically increasing function of V, from zero for u = 0 up to infinity for 2, = Go, so that to any given n > 0 and s E (0, 11 we can find zi satisfying (10) permanently.

However simple this seems to be, the above example clearly contradicts the following very firm statement by Solow [7, pp. 34-381: ‘“The Harrod-

omar condition is particular about the sorts of production conditions it will permit... . The particular form that technological progress must take is called labor-augmenting [that is Harrod-neutral]... ~ If this is not the case, there is no steady-state configuration. The Harrod-Domar consistency condition cannot be permanently satisfied.” This statement is of general type in the sence that it refers to all possible forms of technological progress. Unfortunately, its author offers a proof only for the case of factors’ augmenting technological progress. We accept the proof, but the generalization of the result is clearly not always correct.

8. NECESSARY AND SUFFICIENT CONDITION: A SUMMARY

We thus have four conditions for cy. = 01(0, t). Two of them, (6) and (IO), are to meet requirements of the definition of balanced growth, and a further two, (7) and (8), are to preserve concavity of the pr~d~~ti~~ function. These are all the conditions that OL = a(~, t) must satisfy. So together they constitute four parts of the necessary and sufhcient condition


for the technological change to be compatible with both balanced growth and the neoclassical assumptions concerning marginal products.

This is perhaps the right moment to notice that, if we include the constancy of the marginal product of capital (or interest rate) to the definition of balanced growth, then from (A.9) and (A.10) it follows that 01 must be independent of ~1. This is just the same as under the time specification: t E (- co, co). So that under those two more restrictive definitions of balanced growth the standard condition holds true.

EXAMPLE 2. Let t E (0, co) as in the original proof by Swan. Let 01 = CX,, $- EV, where E > 0. It is clear that this particular form of (y. is consistent with a balanced growth path (in the sense of Definition 1 b). And since technological progress is of the capital-using form, the example runs counter to the standard technological condition. To complete the discus- sion of the this counterexample, we shall also show asymptotic (global) stability of the equilibrium capital/output ratio in the case when o( = CX(V) and 01, > 0 (see Appendix).

9. A COMMENT ON THE LITERATURE ON THE SUBJECT

It might be useful to contrast these results with those of some other writers.

(i) In their survey article Hahn and Matthews wrote: “If technical progress is nonneutral in the Harrod sense it is impossible for the economy to stay on a path of steady growth in which Y and K are both growing at the same, constant, rate. If K and Y were growing at the same rate the distribution of income would then be altering. The rate of growth of output per head is equal to the exogenous time term plus capital’s share multiplied by the rate of growth of capital per head. If the exogenous time term is constant [our underlining], and capital’s share is changing, then evidently the overall rate of increase of output and capital cannot both remain constant, and steady growth is not maintainable” [5, p. 8291.

If the exogenous time term is constant, then indeed technological progress of nonneutral form in the Harrod sense is incompatible with steady growth. Notice that by definition of 01 (see Eq. (5)) this time term, denoted in this paper by A, is equal to (1 - Z-) 01. Since 01 must be constant along a path of balanced growth, the assumption of constant A is therefore equivalent to assuming v to be a constant-that is, assuming the Harrod- neutral form of technological progress along this path. However, there are situations in which at a constant capital/output ratio a change in the growth rate of output and capital, brought about by changes in the factor


shares, is exactly matched by the opposite change brought about variations in the exogeneous time term.

(ii) In a paper by Bliss, we have a theorem according to which ‘“Balanced growth is technically possible, given sufficient labour supply, if and only if technical progress is Harrod-neutral at a constant positive exponential rate” [2, Theorem 21.

We question only the necessity part of the proof of this theorem.” Let Y = G(K, L, t). Without loss of generality write the production function in the form

Y = G(K, ewtL, b). 00

Let o( in (11) be equal to Y/Y - i/L = g - n. Since by assumption the economy is enjoying balanced growth, the growth rates YI, g, and (therefore) a: are all constant over time. Equation (11) implies that

Y . -= m;+(l -%-)(Pz+~)+~~. Y

Because K/K = Y/Y = iz + 01, the partial derivative aG/;jat in (12) must equal zero. Therefore the assumption of balanced growth implies that

Up to this point the proof is correct, but from (13) Bliss concludes that technological progress must be Harrod-neutral. We may now see that this conclusion is not generally true. Notice that D: is de&& here as a difference between n and g. The growth rate g must be constant over time, but may in general vary with the capital/output ratio. So may a. Therefore, by Lemma 2 the conclusion by Bliss is correct only in such situations of balanced growth when g is independent of the capital/output ratio. In particular, for t > 0 the conclusion is false when 01 is positively related to v.

(iii) Burmeister and Dobell offer a slightly diRerent proof [3, p. 7X]. They observe that constancy of the capital/output ratio implies (in our notation)

where f = F(K/L, 1, t) = f(k, t). Indeed, the left-hand side is the i&s residual, denoted in this paper by A. Since h/(1 - 7r) = M, Eq. ( read as a = g - n, which is Eq. (5). The authors refer then to the diamond

5 The original proof is for a putty-clay model, but for our criticism this is not relevant.


result, which states that, if a: is a function of time alone, then technological progress is Harrod-neutral (a part of our Lemma 2). Thus again g and 01 are implicitly assumed to be independent of the capital/output ratio.

The ultimate conclusion of Burmeister and Dobell is nevertheless correct because their definition of balanced growth requires rr to be constant over time. This requirement implies 01, = 0. This implication, however, is missing in their proof.

(iv) Also, Swan in his original proof [7, Appendix] assumes that the common (constant) rates of growth of capital and income are the same for all (constant) capital/output ratios. He shows correctly that a general class of production functions capable of permitting balanced growth from some initial state (K, , L,) at t = 0 has to be of the following form (see his Eq. (A 10); the following notation is ours)

Y = F(K, Loe(Bn+a)t), (15)

where ,6 is a scale parameter. The balanced growth rate of income (and capital) g is equal to @ + 0~. The term 01 (and p for that matter) is constant over time, but in general may vary with ZJ~ = KO/ Y,, = K,/ Y, .

Swan treats g as a given number, independent of z+, . By our Lemma 2 Swan in this manner eliminates any technological change of the nonneutral form in the Harrod sense, including those changes which could otherwise be consistent with the balanced growth path in his sense.

10. Two CLOSING DIGRESSIONS

(i) It is true that in a realistic model of the capitalist economy it would probably be absurd to assume a complete independence between savings and the capital share. A more realistic assumption about the form of the savings function, e.g., assumption of the Kaldor-Pasinetti type, is likely to imply that along a balanced growth path rr must also be a constant. This would have been an additional requirement which could be met only if the technological progress along this path is of the very special neutral form in the Harrod-Kale&i sense. The “technological” condition may thus depend heavily on the institutional setup, on the form of the link (if any) between the savings ratio and the capital share.

(ii) In the situations when the savings ratio could be kept constant independently of z-, a balanced growth path is allowed by a wide class of technological change of capital-using form, but the capital share tends then to unity as time goes to infinity. Burmeister and Dobell would probably object to calling such a growth path a balanced one

CONDITION FOR BALANCED GROWTH asn

[2, pp~ ‘73-78-J. This is, of course, purely a matter of definition, a matter of the bounds imposed on changes in r that one is prepared to accept. A “practical” argument for excluding any change in rr over time along a balanced growth path is seeking support in the so-callled “stylized facts.” This cannot be definitive since any ‘“stylized fact” (saying, for instance, that rr is constant over time) is actually far from being a fact. Pt is possible, for instance, that rr, apart from fluctuating, is slowly increasing with time, by say 5 % over 100 years. Solow’s belief that the technological change we face includes, apart from a labor-augmenting component, a small component of capital-augmenting form [2, p. ix], which can also be called a “stylized fact,” does not contradict that posibility.

APPENDIX

Proof of Lemma 1. The assumption of constant returns to scale implies that Y = LF((K/Y) * (Y/L), 1; t). Define Y/L = y. Tlpus

y = F(yv, 1; t). (A-l>

With our assumptions about F(s), the right-hand side of this equation is

for a given positive o and at any t a concave and increasing function of y. Moreover, lim,,, (W/Q) = $-co and F(0, 1; t) = 0. ence it follows that for every given pair (v, t), where u > 0, and for positive y’s the straight line y = y (the left-hand side of (A.l)) has one and only one point in common with F(yv, 1; t). And since also at 1) = 0 Eq. (A.1) is satisfied for all t by y = 0 only, a pair (v, t) determines uniquely y for all u > 0 and any t. Let

y = h(v, t)~ (A.8

ecause g - y1 = j/y and g/K - g = ti/a, Eq. (4) may be rewritten in the following form?

But from Eq. (A.2) it follows that

(A.3)

(A.4)

6 We can get this form also by implicit differentiation of (A~lf.


The fact that (A.3) and (A.4) must hold for any z1 = u(t) and for all t implies that

1 ah --zT$ Y at

v ah 5r ---.-=-= yau I-~ 43 say.

When the production function (1) is known, then h(v, t), ol(v, t), and ~(v, t) are known as well, and (A.5) and (A.6) are then simply identities. However, when (1) is not given, and when h(v, 0) and OI(D, t) are given instead, then (A.5) should be regarded as a partial differential equation defining h(v, t), and thereby also the production function (l), in terms of ol(v, t) and h(v, 0). The solution to it is as follows:

y = h(v, 0) exp (I (Y(v, T) &) = h(v, 0) 3 A(v, t) = h(v, t), (A-7)

where h(v, 0) is of course defined by F(K, L; 0). Since z, and t are in (A.7) independent variables, the equation holds true

for any admissible values of D and t. In particular it holds for u given in a parametric form as a function of time, which could be of any kind, provided it is nonnegative and differentiable. In this particular case

y(t) = hW, 0) 440, Q 64.8)

Because y = Y/L and v = K/Y, by (A.8) we have that, in general, Y = h(K/Y, 0) AL. The latter equation can always be solved with respect to Y, giving income as a function of capital K and “augmented” labor, AL. Hence it follows that Y = G(K, AL). This concludes the proof.’

Proof of Lemma 2. From (A.7)

where y” stands for h(v, 0) and yuo is its derivative. Hence

=$ = v (+ + JotvdT) E q(v, t).

’ Since v = k/f (k, t), where k = K/L and f (k, t) is the per capita production function, we have that k = k(v, t). Also, for t = 0, Iz(v, 0) = f(k(v, 0), 0). But for all t, v = k/y. Hence k(v, O)/h(v, 0) = k(v, t)/k(v, t) = k(v, t)/A . k(v, 0). This implies k(v, 0) = k(v, t)/A = K/AL. Therefore Y = k(v, 0)AL = f (K/AL, 0)AL = G(K, AL). We wish to thank W. F&da for noting that our previous proof of this section was wrong, and T. Fujimoto for suggesting the above alternative proof.

CONDITION FOR BALANCED GROWTH %83

Equation (AX) links q and rr:

From (A.9) the partial derivative with respect to time qt = ~a, , and from (A.lO) we have that

?I. = (1 - Try qt = (1 - Tr)” tq; . at

The latter expression gives the lemma, since from (All) it follows that sign (a7hjat) = sign (&/a~).

Asymptotic Stability.

Since s = K/Y and because v = K/Y, we have that, in general, s = g . v + 8. But by (A.4) g = qd/v + a(v) + n. Hence

s = (1 + q) d f (a(v) + n) v. (A.2)

THEOREM. If 01 is an increasing finctioiz of v alone and n f a(ti) is positiue for every v, then v(t) satisfying (A.12) subject to the initial condition v(O) = vg > 0 converges, at a given constant savings ratio s E (0, 11 and at another constant growth rate of employment n, to uniquely determined v’* = v*(s), irrespective of the value of v, .

Proo$ From (A.12)

d = s - (a(v) + 11)u 1 + dv, t> 9

(A.13)

where by (A.9) q(v, t) = ~(y,~/y~ + tol,(v)). Under our assumptions the solution to s = (o(v) + 8) v, say v*, is a single-valued function of s. Also, q(v, t) > 0 for all v > 0 and for every t > 0. Thus Eq. (A.13) implies that

sign 6(t) = sign(v* - v(t))” (A. 14)

Let v0 < v* (the opposite case is exactly symmetrical). It follows from (A.44) that v(t) is with time diminishing its distance to Use We shall show that v(t) -+ v*-that is to say, for any given E > 0 there exists time T, such that for every t > T, the distance / a* - u(t)\ < E. Suppose the contrary. Then v(t) -+ v** < v*. By virtue of (A.14)

vg < v(t) < v** for all t > 0. (A.lYJ


But from (A.13), d = a/(b + et) = H(u(t), t), where a, b, and c are self- defined functions of o alone, all of them positive for z, satisfying (A.15). Hence

v(t) = ZIO + jt H@(T), T) dT. (A.16) 0

But His bounded from below:

where both 6 = min, (a/b) and y = max, (c/b) are finite and positive. This implies that

do = o. + $In(l + rt). (A.17)

Thus there exists a number T such that for every t > T the right-hand side of (A.17) will be greater than ZJ**, in contradiction to (A.15).

REFERENCES

1. R. G. D. ALLEN, “Macro-Economic Theory,” Macmillan, London, 1967. 2. C. J. BLISS, On putty-clay, Rev. Econ. Stud. (April 1968). 3. E. BURMEISTER AND A. R. DOBELL, “Mathematical Theories of Economic Growth,”

Macmillan/Collier Macmillan, London, 1970. 4. A. CHILOSI AND S. GOMLJLKA, Technical progress and long-run growth, Riv. Polit.

Econ. (1970), Selected Papers of 1969. 5. F. H. HAHN AND R. C. 0. MATTHEWS, The theory of economic growth: A survey,

Econ. J. 73 (1964). 6. M. KALECKI, “Introduction to the Theory of Growth in a Socialist Economy,”

Warsaw, Poland, 1963. The English translation is published by Blackwell, London, 1969. Also available as an essay in “Selected Essays on the Economic Growth of the Socialist and the Mixed Economy,” Cambridge Univ. Press, Cambridge, England, 1972.

7. T. W. SWAN, Golden ages and production functions, in “Growth Economics” (A. K. Sen, Ed.), Penguin, Baltimore, MD, 1970.

8. R. SOLOW, Growth theory: An exposition, Oxford, England, 1970. 9. H. UZAWA, Neutral inventions and the stability of growth equilibrium, Rev. Econ.

Stud. 28 (1961).

technological condition for balanced growth: a criticism and restatement

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