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Faculty Working Papers 
 
 THE IMPORTANT ECONOMIC IMPLia\TION'S OF LEAR.NING- 
 BY-DOING FOR POPULATION SIZE AND GROWTH 
 
 Julian L. Simon, Professor, Departments of Economics 
 and Business Administration 
 
 College of Commerce and Business Administration 
 
 Unlvtrsity of Illinois at Urbana-Champaign 
 
FACULTY WORKING PAPERS 
 College of Commerce and Business Administration 
 University of Illinois at Urbana-Champaign 
 August 15, 1980 
 
 THE IMPORTANT ECONOMIC IMPLICATIONS OF LEARNING- 
 BY-DOING FOR POPULATION SIZE AND GROWTH 
 
 Julian L. Simon, Professor, Departments of Economics 
 and Business Administration 
 
 //700 
 
 Summary 
 
 Arrow's (and Kaldor's) representation of learning-by-doing and the current 
 state of technique as a function of capital rather than of cumulated output — 
 the latter being the empirical basis for the learning-by-doing concept — leads to 
 major confusion. An obvious though not-very-important example is that a constant 
 production level implies no progress in such a formulation, though there is solid 
 evidence that learning-by-doing continues as cumulated output increases. 
 
 The most important implication of learning-by-doing for population policy — 
 with respect to immigration as well as natural increase — is that a population of 
 larger size implies faster growth of consumption per head than does a smaller 
 population size. Arrow's formulation does not reveal this implication, and 
 empirical work built on Arrow's model does not reach this conclusion. The paper 
 unravels an apparent paradox and shows that a corrected formulation of the learning- 
 by-doing model does indeed have this important implication, which is consistent 
 with relevant empirical data of several kinds. 
 
 Acknowledgment 
 
 I appreciate helpful readings of this paper by Mark Browning and Jonathan 
 Cave. 
 
THE IMPORTANT ECONOMIC IMPLICATIONS OF LEARNING-BY-DOING 
 FOR POPULATION SIZE AND GROWTH* 
 
 Julian L. Simon 
 
 I. INTRODUCTION 
 
 Following on Che empirical studies of learning-by-doing by Wright 
 (1936), Asher (1956) and others, and the summary and interpretation by 
 Hirsch (1956) and Alchian (1963), Arrow combined the effect of experi- 
 ence with a macromodel to study the economic implications. This taodel 
 followed Kaldor in making technical progress endogenous, though Kaldor's 
 point of departure was Verdoorn's macro and industry data (19A9) , rather 
 than the firm-level data on which Arrow built and which lend themselves 
 more obviously to a learning interpretation than do Verdoorn's data. 
 
 This note contends that Arrow did not derive the most important 
 implications of learning-by-doing, and that he also reached some wrong 
 conclusions, because he modeled learning-by-doing poorly. These 
 conclusions — that population growth, and even more importantly, popu- 
 lation size, have positive effects upon the rate of economic growth 
 through their positive effects on the rate of technical progress — have 
 major implications for the understanding of, and social policy with 
 respect to, population size and growth. 
 
 Section II criticizes the concept of experience Arrow used, to 
 make clear the necessity for a fresh start. Section III shows the 
 implications for evaluation of different sizes of population, which 
 requires unraveling an apparent paradox in Arrow's (and Kaldor's) 
 
 *I appreciate helpful readings of this paper by Mark Browning and 
 Jonathan Cave. 
 
-2- 
 
 analyses which leads to incorrect conclusions about this matter. Section 
 IV shows the implications of learning-by-doing for evaluation of the 
 effects on consumption of different rates of population growth on size. 
 Section V summarizes the work and the conclusions. 
 
 II. CRITICISM OF ARROW'S MEASURE OF EXPERIENCE 
 This section is in the nature of clearing of debris. 
 All the empirical studies on which Arrow relied, and also most of 
 those that have come afterwards (with exceptions to be noted below) re- 
 lated something like "the amount of direct labor required to produce an 
 airframe and the [cumulative] number of airframes produced" (Alchian, 
 1963, p. 680) . 
 
 (1) H^ = H^n"'^ = H^ - Hj_(1 - n"^) 
 
 where N = serial number of a particular unit of output 
 
 H = hours of work per unit of output for serial number N 
 b = constant. 
 Arrow is very clear on this: "The economic examples given above suggest 
 the possibility of using cumulative output (the total of output from the 
 beginning of time) as an index of experience" (p. 157) . 
 
 But then, because Arrow doubted the psychological generalization 
 implicit in that measurement (which also avoids the necessity of coming 
 up with a number for cumulated output since the beginning of time) , he 
 shifted to a very different concept, with entirely different implications. 
 "I therefore take instead cimiulative gross investment [the capital stock] 
 (cumulative production of capital goods) as an index of experience" 
 (p. 157) . That is, 
 
-3- 
 
 (2) ^t = H~= ^^V = '^t ' 
 
 where A = level of technique 
 
 K = capital stock 
 
 d = constant. 
 He also shifted from a serial-number concept to a time-period concept, 
 a difficult matter which will be taken up later. 
 
 Let us be clear on the possible confusions caused by taking the cap- 
 ital stock rather than cumulative gross output as the index of experience. 
 One such confusion is that, even using Arrow's own assumptions, cumulative 
 capital and cumulated output yield different conclusions, as may be seen 
 in the following: In monetary terms, the capital/output ratio may be 
 taken (and is so assumed by Arrow) to be roughly constant in this context. 
 Hence the capital stock and one year's production are roughly proportional. 
 So the capital stock can be considered as a proxy for incremental output 
 rather than cumulated output. And incremental output is generally not 
 proportional to cumulated output; if production in t and t+1 are equal, 
 cumulative output is higher in t+1 than in t even though capital is constant, 
 Formally, 
 
 t 
 
 (3) A = a Z Y 
 
 where Y = output in period t 
 
 which can be rewritten 
 
where the right-hand side is always positive. 
 
 Now consider Arrow's function (2) from which can be written 
 A - A , K^ - k\ 
 
 If the capital/output ratio is a constant c, as Arrow assunes it is, 
 then (2a) can be rewritten 
 
 This is unequal to (4), as can be seen from the fact that if Y _ = Y , 
 (4) will be positive but (5) will be zero. And (4) corresponds to the 
 facts that Arrow cites, including the Horndal effect, whereas (5) does 
 not. 
 
 Even worse are all the difficulties that flow from that most trouble- 
 some of all economic concepts to conceptualize and measure satisfactorily, 
 capital. Most fundamental here is that the capital concept used in the 
 model is physical, whereas the observed capital -output ratio that Arrow 
 builds on is a monetary notion, and there is no necessary or likely cor- 
 respondence between the two; more about that later. 
 
 Arrow's definition of the capital stock also is not analogous to 
 cumulated output in that it excludes old capital that has already 
 obsolesced. This is important because it means that Arrow's model does 
 not sum up experience since the very beginning of the learning process, 
 which causes a paradox in interpretation of rates of productivity 
 change that must be unraveled below. 
 
-5- 
 
 (Even cumulative output would have to be defined carefully because 
 of the difficulty of estimating it for an economy or society as a whole. 
 The entire transposition from, on the one hand, the firm or industry 
 level with technical measurements to, on the other hand, the national 
 economy with value measurements can — and in this case does — cause the 
 argument to lose its way, and later to be misinterpreted in empirical 
 work by others.) 
 
 In brief, the use of capital rather than cumulated output is one 
 source of erroneous interpretation of the implications of learning-by- 
 doing, though not the central issue here. 
 
 III. LEARNING-BY-DOING AND LABOR FORCE SIZE 
 This section is the piece de resistance of the paper, both because 
 of the necessary subtlety of the argument and because of the greater 
 relevance for policy of the conclusion about population size than of 
 conclusions about golden age growth patterns. It boils down to a dis- 
 tinction between equilibrium and nonequilibrlum analyses, with the latter 
 being appropriate for the question under discussion here. 
 
 Let us now return to the firm and industry level at which the 
 learning-by-doing phenomenon is actually observed, to investigate the 
 phenomenon's implications for various sizes of the labor force and 
 population. 
 
 Here we must be crystal-clear about the nature of the empirical 
 process on which the function is based, because a subtle shift in inter- 
 pretation has caused a fundamental misinterpretation. For the first unit 
 made, the measured number of hours spent is equal to H (the constant 
 a in Arrow's notation). The process is assumed to start at this point, 
 with K^ — one of the two key values in the system — given exogenously 
 
as a result of a variety of unnamed previous experiences. Successive 
 units require the amount of labor in the initial unit less a proportion 
 of that initial amount, the proportion depending on the serial number 
 of the successive unit. This makes clear that to portray a learning- 
 by-doing process one must either know, or be able to estimate from later 
 data, the original level of skill at which this particular process begins, 
 
 Assume that the good 1, whose cumulative quantity produced up to 
 period t is indexed by serial number N , , is a normal good. It is 
 reasonable to assume that the output of i in any period t is a propor- 
 tional function of total income 
 
 (6) N^ - Nj._^ = gY 
 
 and that total income is a proportional function of labor force size 
 
 (7) Y = hL. 
 
 where g and h are constants. The volume N - N = ghL produced in t 
 therefore is an increasing function of the population size. (It is 
 all-important here to distinguish between, on the one hand, the volume 
 produced in a given period t, and, on the other hand, the serial number 
 N, which might be produced in any period t.) 
 
 Consider the moment when a given production process N begins with 
 serial number N = 1. For convenience, this takes place on January 1 of 
 year t = 1 in country Alpha. Assume population size is P _ , labor 
 force size is !■_-,, ?-'nd total national output for that year is Y _ . For 
 those conditions, there will be X units of good i produced, so at the 
 end of the first year the serial number is N _ = X. 
 
-7- 
 
 Now assume instead that we are in countr>' Beta and population size 
 instead is 2P , labor force size is 2L ^^ , and consequently total 
 national output is 2Y . (Furthermore, the population and output have 
 
 always been twice as large in Beta as in Alpb^) . From (6) and (7) we can 
 
 g 
 expect that the end-of-year serial number will be N _^ = 2X rather than 
 
 n"^ = X. Hence n"^^ - N^^ = X for L and 2X for 2L . 
 
 Now consider the rate of change of productivity A over the first 
 
 \=2 " ^t=l 
 period. Whatever the percentage change , the rate will be 
 
 '^t=l 
 
 roughly twice as large for 2L , as for 1 -, (assuming that the change 
 
 is small relative to the initial value, which one can ensure by making 
 the period short, say a day rather than a year) . 
 
 To elucidate this point, let us write out the learn±ng-by-doing con- 
 cept in the greatest possible detail. Define the level of productivity 
 A as the inverse of work time per unit, and using (1) 
 
 N 
 
 = ^,-1 + ^^=1^^ - 1^ • 
 
 From (la) it is quite clear that for two firms or economies a and 2 that 
 produce, up to the same date, N and N > N units, the rate of 
 change of productivity from the beginning to that date is greater for 6 
 than for a, 
 
 V>N"-^?=il / v-\-=i ] 
 
 4 
 "i;=l 
 
 
 In Arrow's model we do not find that the rate of change of produc- 
 tivity is a function of output or labor force in anywhere near the 
 same way, because it is in the nature of any capital stock that it does 
 
-8- 
 
 not correspond in any simple additive way to all the capital produced 
 since the beginning of time. And this is even more true in Arrow's model 
 which includes obsolescence, and an arbitrary- obsolescence rule at that. 
 
 Even more important, Arrow's model invites one to write an experience 
 function such as 
 
 A , , - A aK^ , , - aK*' 
 
 , . t+1 t t+1 t 
 
 '»> —7; ^7— 
 
 where the dates include any given period. Though Arrow does not write 
 this function explicitly, he likens his model to Kaldor's, who writes 
 the similar function 
 
 ^t \ \-l j 
 
 In Arrow's words, "The production assumptions of this section are designed 
 to play the role assigned by Kaldor to his 'technical progress function,' 
 which relates the rate of growth of output per worker to the rate of 
 growth of capital per worker. I prefer to think of relations between 
 rates of growth as themselves derived from more fundamental relations 
 between the magnitudes involved" (p. 160), He also says that "Verdoorn 
 had also developed a similar simple model" (p. 160); Verdoorn differs 
 in using output per year, but his independent variable is also a rate 
 of change of the current magnitude. So there is little doubt that 
 function (8) is a fair representation of Arrow's model. And in his 
 empirical "Tests of the 'Learning By Doing' Hypothesis," Sheshinski (1967) 
 estimated exactly this model where "the increase in experience ... is 
 attributed to current gross investment" (that is, the rate of change of 
 
-9- 
 
 productivity = investment v capital stock),* or in an alternative 
 model, "the change in experience is attributed to the rate of output" 
 (that is, the rate of change of productivity = output -r "cumulated 
 gross output," where Sheshinki's empirical referent of the latter con- 
 cept is unclear but is certainly not cumulated gross output since the 
 beginning of the process, as required for a reasonable portrayal of the 
 leaming-by-doing process) . 
 
 Given that a country's situation at any given date is compared 
 to its own situation at some slightly earlier date, the rate of change 
 of the capital stock may be expected to be the same for larger country 
 Beta as for smaller country Alpha, ceteris paribus. Hence in Arrow's 
 model the rate of increase of productivity is essentially independent of 
 the scale of the economy and of the population, though it will reflect 
 short-run changes in output (through differences in investment and the 
 capital stock). That is, though this model may lend itself to estimating 
 the rate for a given process, it does not enable one to compare the re- 
 sults for two entities of different sizes. 
 
 To make the point as graphic as possible: If fathers cut their 
 male offspring's hair monthly, a father with twin sons will get twice 
 as much practice in any period, and his skill at any date will be higher, 
 than that of a father with only one son (assuming equal beginning skill) . 
 This does not imply that in each period the twins' father's rate of 
 increase of skill will be higher than that of the father of only one son; 
 
 "Actually, Sheshinski estimated Kaldor's model in which the r.h.s. is 
 
 ,'K - K '\ 
 ' t t-1^. 
 
 cl- 
 
 K^ , ' rather than Arrow's model in which the right hand 
 
 ' Ft - ^t-i 
 
 would be ; 
 
 learning-by-doing mod'el . 
 
 '\ , /K - K _ , 
 side would be ; 1, though he said that he was testing the 
 
-10- 
 
 if the learning curve has the same exponent at all points, the rates 
 will be the same in each period after the first. Nor need the absolute 
 increase in skill be greater in any period for the twins' father than 
 for the single-son's father. But the rate of increase of productivity 
 from the beginning to any given date will be greater for the father with 
 twins than for the father with one son.* 
 
 There is an apparent paradox here: If two countries of different 
 sizes of population and output are compared for a period starting after 
 a process has begun, at the same date in both, the rates of change of 
 productivity over the period can be the same, even though if the period 
 begins at the beginning of the process, the rate of change of productivity 
 of the larger country will be greater. For example, assume that Alpha pro- 
 duces one unit per period, and Beta produces two units per period. Be- 
 tween periods four and five. Alpha goes from serial 4 to serial 5, 
 whereas Beta goes from 8 to 10. If A_. — the level of technology' of the 
 first unit, the inverse of the numbers of hours required to make the 
 first unit — is equal to 10, if b = .5, and hence if A^ = ION* , then for 
 Alpha 
 
 ^t=5 ~ '^t=4 ^=5 " ■S}=4 lO/T - lO/T 
 
 ^t=4 'Sl=4 10V4' 
 
 and also for Beta 
 
 '^t=5 " '^t=4 •St=10 " "^=8 10/TO - lOvS" 
 
 .414 
 
 A 
 
 t=4 ^=8 lO.'B 
 
 .414, 
 
 *If one worries about the inequality in total employment of the two 
 fathers, the example could be changed to a society with one father- 
 barber versus a society with two father-barbers, all with one child 
 each. In the economy with two fathers, they exchange information on 
 what they learn, and hence increase the rate of their learning; this 
 is quite realistic. 
 
-11- 
 
 But from the beginning of the process to the end of period 5, 
 the calculations are for Alpha 
 
 "^t^S ~ '"^t^l ^ IQr^ - lO/T 
 '^t=l lOvT 
 
 and for Beta 
 
 '^t=5 " ^=1 ^ lO/lQ - lO/T . 
 
 \]=i 10 vT 
 
 So Beta's rate of increase of productivity is faster than Alpha's 
 when calculated from the beginning of the process. 
 
 We can see the economic importance of this population effect with 
 a simple simulation model, comparing the results for more-developed 
 technology-producing worlds of two sizes, one twice as large as the 
 other. Such a simple model has a Cobb-Douglas production function, a 
 savings function proportional to output (set at the much-lower-than- 
 steady-state value to avoid difficult arguments), and a one percent 
 labor-force growth rate per year. 
 
 The model is as follows: 
 
 Y = A L^K^ 
 t t t t 
 
 ^ = ^t-1 ^ ' \-V ' = -"^ 
 
 Initial values A = 1.0, L = 1000, K = 1000, Y = 500, a = .67, g = .33. 
 t t t t ' ' 
 
 An iterative program is used to make investment approximately a function 
 of current-period income rather than prior-period income, so that the 
 computer model would approximate the steady-state analytic model; the 
 results are nnjch the same with and without this refinement, however. 
 
-12- 
 
 The technological-change function, for both populations, is 
 
 t-1 , 
 
 M = m( Z Y ) b = .2 or .3 
 
 C Q t-1 
 
 The learning parameter in the model is initially set to produce 1% 
 productivity change per year in the smaller population. And the model 
 for both populations starts with the same cumulated amount of output 
 obtained by jobbing back from the initial year's value, which is unre- 
 alistically advantageous to the smaller population; the result is not 
 sensitive to differences in this initial condition, however. 
 
 The relevant comparison in a long-run context would seem to be one 
 in which capital per head has achieved the same level in both the worlds. 
 From this assumption it is obvious that the bigger world must have 
 faster growth, because there is no offset to the productivity effect 
 through capital dilution in this set-up (though later we shall see, 
 in the context of different rates of growth , how the tradeoff between 
 capital dilution and productivity effects plays itself out). 
 
 The relevant measure of performance in a policy context is a com- 
 parison of present values of future income streams under the different 
 population assumptions. Table 1 shows the present values for the two 
 initial populations at various rates of discount. Clearly the outlook 
 for the future standard of living is better with the larger initial 
 population. Of course the differences would be less with a lower 
 learning parameter, and greater with a higher learning parameter. 
 
 Table 1 
 
-13- 
 
 It is clear, then, that a larger population, market, and total out- 
 put imply faster economic growth due to learning-by-doing. But there 
 is no chance that this phenomenon will be observed if the Kaldor-Arrow 
 model is used, either theoretically as in Arrow's and Kaldor's work, or 
 empirically, as in Sheshinski's work. 
 
 Furthermore, this effect implies that the rate of growth of produc- 
 tivity will increase with time, rather than being steady-state growth 
 as Arrow concluded that it is. To show that this is so, we must return 
 our thinking to the company and industry level at which the learning- 
 by-doing phenomenon actually observed, rather than moving directly to 
 the macro level by simple analogy to the firm. 
 
 Once a learning-by-doing process begins, the rate of learning may 
 be constant with increasing production volume.* But another process that 
 is identical in all economic details with the one previously under dis- 
 cussion, except that it comes along later, will operate in an environment 
 of a bigger total market and total output (due in part to the earlier 
 innovation) even if population size remains constant. Therefore sales 
 will be greater if the good is a normal one, and the rate of learning 
 per unit of time will be faster for the later innovation than for the 
 earlier one. To the extent that these innovations are representative 
 of the economy as a whole, then, the rate of increase of productivity 
 for the economy as a whole will increase over time. 
 
 *0r it may decline with output (e.g., Barkai and Levhari, 1973; Levhari 
 and Sheshinski, 1973; Baloff, 1966, and references cited therein). 
 Though the rate of learning in a given product situation may decrease, 
 there may still be changes in the processes which restart a high-rate 
 learning process, and increase the overall rate of learning. 
 
-14- 
 
 An increasing rate of growth of productivity implies that there is 
 not a constant capital-output ratio, as may be deduced immediately from 
 the well-known 
 
 (10) s = -(n + A) . 
 
 where s = savings ratio 
 
 V = output-capital ratio 
 
 n = population growth rate 
 
 A = rate of change of A. 
 But this is in no way inconsistent with observed reality nor is it a 
 ground for concern, despite the supposed "stylized fact" of an observed 
 constant K/Y ratio, because the monetary estimates of the capital-output 
 ratio tell nothing about the physical capital-output ratio, as I show 
 elsewhere (Simon, 1980). The observed constant K/Y ratio in value terms 
 is simply an inevitable price response of competitive markets, and has 
 no significance for long-run production relationships.* The relevant K/Y 
 ratio — the physical ratio — probably is not constant in MDC's but rather 
 secularly falling, though this proposition cannot ever be examined rigor- 
 ously in its aggregate form. 
 
 One might wonder why larger economies do not grow faster than smaller 
 economies, in light of the above analysis. In fact, Chenery and associates 
 
 *It is still another problem with Arrow's model that despite its real- 
 resource nature, he bases it upon the assumption of a constant K/Y ratio, 
 despite the meaninglessness of the monetary K/Y ratio to describe the 
 physical K/Y ratio in the long-run and the fact that the physical K/Y 
 ratio is falling in the long run. 
 
-15- 
 
 (e.g., Chenery and Syrquin, 1975) do_ observe econoniies of size in cross- 
 national comparisons of growth rates in IDC's. And the Rostas-Clark data 
 show a strong effect in individual industries (Clark, 1967). But, addi- 
 tionally, to the extent that economies are open, there are forces that 
 reduce the observed differences in growth among countries. Specializa- 
 tion (e.g., Sweden in autos, Holland in communications) can lead to 
 larger markets than the national economy provides. And a considerable 
 proportion of technical progress is not locally tied and moved across 
 national boundaries. Hence there is no inconsistency between the ob- 
 served data and the analysis given here. The relevant conceptual unit 
 for analysis of the sort given above is the Western industrialized world 
 as a whole, with the relevant comparison being between imagined larger 
 and smaller sizes of it. 
 
 IV. LEARNING- BY-DOING AND LABOR FORCE GROWTH 
 It is reasonably easy to show that in a comparison of two populations 
 with different rates of labor-force growth (with optimum savings ratios) 
 that are already on a steady growth path, faster labor-force growth implies 
 higher consumption — as long as technical progress is an increasing func- 
 tion of the size of the labor of on total output, even to the slightest 
 degree . Consider a situation in which technical progress does not de- 
 pend upon either labor force or total output; if so, technical progress 
 will be the same, and the rate of growth of per-capita output will also 
 be the same , for every rate of population (labor force) growth, though 
 consumption will be lower with higher population growth due to the higher 
 warranted savings rate. But if technical progress _is faster with a higher 
 
-16- 
 
 population growth rate — as it is with function C6a) because faster popu- 
 lation growth implies faster increase in aggregate income and hence in 
 capital — then the rate of growth of per capita output must be faster 
 with higher population growth. And this must therefore eventually (no 
 matter how slight the dependence of technical progress on population 
 or output) lead to higher levels of per capita output and consumption. 
 This is the argument in a nutshell.* 
 
 Arrow's model does show increasing returns to scale. But 
 Arrow does not describe the effect of a change in only L or L. Further- 
 more, his returns to scale derive from an unusual sort of production 
 function — one with fixed coefficients rather than a neo-classical form — 
 which would not be appropriate for analysis of the effect of a change in 
 L or L. And Kaldor, using the same sort of technical progress function 
 but a different sort of growth model, concludes that an increase in L 
 depresses the rate of growth of output, or at best leaves it unchanged 
 (pp. 615-616). >iore generally, as Eltis puts it, "Both Kaldor and Arrow 
 wrote long and forceful passages to argue that there is likely to be a 
 strong endogenous relationship between investment and technical progress 
 in the real world. Unfortunately, their subsequent mathematical speci- 
 fication of the relationship precluded any endogenous connection in 
 equilibrium — in their growth models" (1973, p. 154). And this is even 
 
 *The comparison of steady growth paths is not the relevant comparison 
 for a given society's policy choice at a given moment, however. Rather, 
 the society wishes to evaluate its future streams of costs and benefits 
 with different rates of population growth, given its present endowment 
 of capital and level of income. That comparison is the subject of 
 ariDther paper. 
 
-17- 
 
 more true of Che connection between the rate of growth of the labor 
 force and the rate of growth of consumption. 
 
 In short, Learning-by-doing implies that the steady-state consump- 
 tion path inevitably comes to be higher with faster population growth 
 than with slower population growth, in contrast to standard growth theory's 
 conclusions. 
 
 Phelps' model (1966), too, implies that faster labor-force growth pro- 
 duces a higher golden-age consumption path. And Eltis (1973) arrived at 
 this conclusion explicitly. My C/m model (1977, Chapter 6), which has a 
 technical progress function closer to a learning-by-doing formulation 
 than that of Phelps or Eltis, also produced a conclusion that higher pop- 
 ulation growth implies higher consumption after an initial period during 
 which capital-dilution produces lower consumption. And a straightforward 
 neo-classical model with Arrow's own formulation of learning-by-doing also 
 produces this result under most assumptions. 
 
 Though it is outside the main purpose of this paper, the simulation 
 described earlier was also run with different rates of growth of the labor 
 force, all other initial conditions identical. In the earlier years, the 
 lower rate of population growth (1% yearly) leads to higher per-worker 
 consumption because of the capital dilution effect than does 2% population 
 growth. But after tim or three decades (just how long depends on the 
 assumptions) the higher rate of population growth comes to have higher 
 per-worker consumption due to its faster rate of productivity growth. 
 And a present-value computation shows that the higher rate of population 
 growth yields better results up to very significant discount rates, and 
 at all discount rates where thereare meaningful differences between the 
 
-18- 
 
 various population-growth assumptions. These results may be seen in 
 Table 1. 
 
 V. SUMMARY AND CONCLUSIONS 
 
 Arrow's (and Kaldor's) representation of learning-by-doing and the 
 current state of technique as a function of capital rather than of cumu- 
 lated output — the latter being the empirical basis for the learning-by- 
 doing concept — leads to major confusion. An obvious though not-very- 
 important example is that a constant production level implies no progress 
 in such a formulation, though there is solid evidence that learning-by- 
 doing continues as cumulated output increases. 
 
 An important implication of the learning-by-doing phenomenon (as 
 with almost any model where technical progress is endogenous) is that 
 it leads to a different conclusion about population growth than does 
 standard growth theory, implying that the eventual consumption path is 
 higher with faster population growth. Arrow's model does not show this 
 clearly, Kaldor somehow reached the opposite conclusion, and standard 
 growth-theoretic texts (e.g., Solow, 1970; Wan, 1971; Brems, 1973; Dixit, 
 1975) — even the specialized summary of growth theory with respect to 
 population growth of Pitchford (1974) — do not mention this implication. 
 
 The most important implication of learnlng-by-doing for population 
 policy — with respect to immigration as well as natural increase — is that 
 a population of larger size implies faster growth of consumption per 
 head than does a smaller population size. Arrow's formulation does not 
 reveal this implication, and empirical work built on Arrow's model does 
 not reach this conclusion. The paper unravels an apparent paradox and 
 
-19- 
 
 shows that a corrected formulation of the learning-by-doing inodel does 
 indeed have this important implication, which is consistent with rele- 
 vant empirical data of several kinds. 
 
 M/D/:42 
 
-20- 
 
 REFERENCES 
 
 Armen A. Alchian, "Reliability of Progress Curves in Airframe Production," 
 Econometrica , Vol, 31, No. 4, October 1963, 679-93. 
 
 K. J. Arrow, "The Economic Implications of Learning by Doing," Review 
 of Economic Studies , XXIX (3), No. 80, June 1962, 155-73. 
 
 H. Asher, Cost Quantity Relationships in the Airframe Industry . (Santa 
 Monica, California: RAND Report No. R-291, July 1956). 
 
 N. Baloff, "The Learning Curve — Some Controversial Issues," Journal of 
 Industrial Economics , XIV, July 1966, 275-282. 
 
 H. Earkai and D. Levhari, "The Impact of Experience on Kibbutz Farming," 
 The Review of Economics and Statistics , LV, Feb. 1973, 56-63. 
 
 Kans Brems, Labor, Capital and Growth . (Lexington, Mass.: Lexington 
 Books, 1973). 
 
 Hollis Chenery and Moses Syrquin, Patterns of Development, 1950-1970 . 
 (New York: Oxford, 1975). 
 
 Colin Clark, Conditions of Economic Progress . 3rd ed. (New York: 
 Macmillan, 1957). 
 
 A. K. Dixit, The Theory of Equilibrium Growth . (New York: Oxford 
 University Press, 1976). 
 
 Walter A. Eltis, Growth and Distribution . CLondon: Macmillan, 1973). 
 
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Table 1 
 
 Present Values (600 Years) of Per-Capita Consumption Streams 
 With Two Different Initial Population Sizes 
 And Various Rates of Savings and Population Growth 
 
 b = .5 
 
 L = 1000 L = 2000 
 
 K = 1000 K = 2000 
 
 L 
 
 K 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0% 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 
 
 
 
 .653 
 
 .732 
 
 .315 
 
 .1497 
 
 a42 
 
 .111 
 
 .869 
 
 .897 
 
 .366 
 
 .221 
 
 .156 
 
 .120 
 
 
 
 E+3 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+3 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+2 
 
 
 
 4 
 
 ,190 
 
 .120 
 
 .403 
 
 .228 
 
 .157 
 
 .119 
 
 .495 
 
 .215 
 
 .566 
 
 .288 
 
 .187 
 
 .137 
 
 
 
 E+4 
 
 E+3 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+4 
 
 E+3 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+2 
 
 1 
 
 2 
 
 .103 
 
 .179 
 
 .383 
 
 .211 
 
 .146 
 
 .112 
 
 .198 
 
 .295 
 
 .498 
 
 .250 
 
 .166 
 
 .124 
 
 
 
 E+5 
 
 E+3 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+5 
 
 E+3 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+2 
 
 1 
 
 4 
 
 .191 
 
 .282 
 
 .470 
 
 .236 
 
 .858 
 
 .118 
 
 .372 
 
 .489 
 
 .644 
 
 .290 
 
 .183 
 
 .133 
 
 
 
 E+5 
 
 E+3 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+5 
 
 E+3 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+2 
 
 1 
 
 a 
 
 .353 
 
 .405 
 
 .616 
 
 .278 
 
 .176 
 
 .128 
 
 .692 
 
 .838 
 
 .897 
 
 .356 
 
 .211 
 
 .147 
 
 
 
 E+5 
 
 E+3 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+5 
 
 E+3 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+2 
 
 2 
 
 4 
 
 .913 
 
 .448 
 
 .879 
 
 .249 
 
 .155 
 
 .115 
 
 .182 
 
 .888 
 
 .148 
 
 .326 
 
 .185 
 
 .131 
 
 
 
 E+6 
 
 E+4 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+7 
 
 E+4 
 
 E+3 
 
 E+2 
 
 E+2 
 
 E+2 
 
 2 
 
 3 
 
 .173 
 
 .842 
 
 .141 
 
 .313 
 
 .178 
 
 .126 
 
 .346 
 
 .167 
 
 .249 
 
 .436 
 
 .221 
 
 .148 
 
 
 
 E+7 
 
 E+4 
 
 E+3 
 
 E+2 
 
 E+2 
 
 E+2 
 
 E+7 
 
 E+5 
 
 E+3 
 
 E+2 
 
 E+2 
 
 E+2 
 
^