The contribution of Carlo Casarosa on the forerunners of the life cycle hypothesis by Franco Modigliani and Richard Brumberg

Abstract

The Italian economist Carlo Casarosa published a study in 2002 claiming that there existed two contributions—one by Roy Harrod (1948, Lecture Two) and one by James Duesenberry (1949, Chapter III, Sect. 9)—that could be considered forerunners of Modigliani and Brumberg’s pioneering works about the Life Cycle Hypothesis (LCH). Those contributions, according to the Italian scholar, had been, until that moment, scarcely known by the scientific community. Particularly, he pointed out that, while Modigliani in 1970 eventually recognized Harrod’s contribution to the LCH, he never mentioned Duesenberry’s work. After presenting and commenting on the content of Casarosa’s investigation, in this paper we present the results of an extended historiographical analysis that we have carried out on the early literature about the LCH. Rather interestingly, our analysis shows that several economists, especially from Cambridge (UK), had acknowledged the role of Harrod (together with the one of Frank Ramsey) since early 1950s, while Duesenberry’s contribution to the LCH was completely ignored.

1 Introduction

Franco Modigliani and his brilliant graduate student Richard Brumberg developed the Life-Cycle-Hypothesis (LCH henceforth) in two pioneering articles, written between 1952 and 1953: the first one, Utility Analysis and the Consumption Function: An Interpretation of Cross-Section Data, concerned with the microeconomic implications of the LCH, was completed in 1952 and then published in 1954 in the book Post-Keynesian Economics edited by Kenneth Kurihara. The second one, Utility analysis, and aggregate consumption functions: an attempt at integration, developing the macroeconomic implications of the model, was completed in 1953 and only published in 1980 in the second volume of The Collected Papers of Franco Modigliani, edited by Andrew Abel. The delay in the publication of the second manuscript, which however had been circulating within the academy since the early 1950s and referred to as “maybe the most famous unpublished paper in post-war economics” (Phelps 1968, p. 499), was “the direct result of the deep personal loss Modigliani felt with the untimely death of Brumberg shortly after the manuscript was completed” (Merton 1987, p. 147 1n).

Soon after the publication of the first work in 1954 (and even before that year, as explained by Fisher 1987, p. 177 and Farrell 1959, p. 678), the LCH had been the object of vigorous and persistent scrutiny both from a theoretical and empirical point of view, being, in particular, in competition with another emerging ‘new theory’ on consumption behavior: Milton Friedman’s 1957Permanent Income Theory (PIT henceforth). Hence, Modigliani spent a lot of energy in responding to these criticisms, presenting theoretical extensions and empirical tests through the involvement of numerous co-authors.¹ After such a long list of articles, rather surprisingly, in 1970, in a study published in a Festschrift for Roy Harrod, for the first time Modigliani recognized that through his Lecture Two of Towards a Dynamic Economics published in 1948, Harrod had been a pioneering contributor of the LCH model.² Since then, Modigliani acknowledged the contribution of the Oxford economist on “hump-saving” on several other occasions (e.g., Modigliani 1975, p. 5; 1986, p. 300).

Although there exist a significant number of works about Franco Modigliani’s life and intellectual activity,³ to the best of our knowledge, Carlo Casarosa is the only scholar providing a detailed and clear assessment of Harrod ‘s contribution to the LCH in a paper published in 2002 and written in Italian.⁴ Moreover, rather surprisingly, Casarosa is also the only scholar to unveil Duesenberry’s contribution to the LCH, contained in Chapter III, Sect. 9 of his volume Income, Saving and the Theory of Consumer Behavior, published in 1949 and mostly known for the chapters about the relative-income theory of consumption.⁵

Given this premise, the aim of the present work is two-fold: first, to present Casarosa’s valuable analysis to the non-Italian reader. Second, to extend his work in documenting the pioneering contributions of Harrod and Duesenberry to the LCH. In fact, Casarosa’s main focus was to clearly pinpoint the common traits and the differences between the mentioned contributions, but he did not (and could not, given the length of his article) carry out a complete historiographical analysis on whether and to what extent such forerunners of the LCH had been already recognized or mentioned by previous works.

Therefore, after presenting a commented summary of Casarosa’s study, we will present the results of our historiographical research on the early literature about the LCH. It will be shown that, since the publication of the first article by Modigliani and Brumberg in 1954, there had been scholars, especially from Cambridge (UK), already pointing to the works of Harrod and, also, to that of Frank Ramsey.⁶ Rather interestingly, we show that both Richard Brumberg (in an article posthumously published in 1956) and William Hamburger, a Ph.D. student of Franco Modigliani (in two papers published in 1954 and 1955), had mentioned the contributions by Harrod, Ramsey, and Duesenberry, although, as for the latter, not with a clear reference to his work on the LCH. Hence, anticipating our conclusions, we can say that our historiographical scrutiny corroborates Casarosa’s thesis concerning the long oblivion of Harrod’s Lecture Two and, in particular, of Duesenberry’s pioneering insight into the LCH.

The work is organized as follows: in Sect. 2 we recall the main features and implications of the LCH, developed by Modigliani and Brumberg; in Sect. 3 we present and discuss Casarosa’s analysis on Harrod’s and Duesenberry’s contributions as forerunners of the LCH. Section 4 provides some new historiographical evidence on the early studies on the precursors of the LCH. Conclusions end the work.

2 Modigliani-Brumberg’s life cycle hypothesis in Casarosa’s work

Casarosa’s article, La relazione tra tasso di risparmio e tasso di crescita nella teoria del ciclo di vita: da Harrod a Modigliani-Brumberg (The Relation Between Aggregate Saving Rate and Rate of Growth in the Life Cycle Theory: From Harrod to Modigliani-Brumberg), was published in the Italian journal of history of economic thought, Il Pensiero economico italiano, in 2002 from page 51 to 89 (the references of this paragraph are from that source). The Italian economist lays down his findings in strictly chronological order: first, he presents the analysis of the theory of saving contained in Lecture Two of Harrod’s book Towards a Dynamic Economics, published in 1948 (pp. 52–61). Second, he develops the analysis of Duesenberry’s contribution to the LCH, contained in Sect. 9, Chapter III of his 1949 book Income, Saving and the Theory of Consumer Behavior (pp. 61–66). Finally, after presenting the version of Modigliani-Brumberg (M-B henceforth) of the LCH (pp. 66–81), recapitulates his findings (pp. 81–86) and concludes (pp. 86–89). To summarize Casarosa’s analysis, we recall the main features of the LCH as put forward by M-B, which we will use as a benchmark in the assessment of both Harrod’s and Dusenberry’s theories of saving.

2.1 “The ‘ray of light’ of Modigliani and Brumberg, in the car, on the road to…Urbana” ⁷

Introducing the M-B works, Casarosa recalls that.

in the United States, many macroeconomists tried to empirically verify the hypotheses made by Keynes regarding the form of the aggregate function of consumption (and saving) and, more generally, tried to explain the data on consumption and saving that came from various sources. In this perspective, an apparent contradiction immediately emerged: on the one hand, the data from national accounts and household budgets seemed to confirm the Keynesian hypothesis according to which, both at the individual and at the aggregate level, the propensity to save is an increasing function of income; on the other hand, the Kuznets data showed that in the period 1886–1929 the aggregate propensity to save had remained substantially constant, despite the significant increase in per capita income and therefore seemed to invalidate that same hypothesis.

Modigliani is one of the first to commit himself to solving the puzzle and with an article, published in 1949, formulates a hypothesis, which is then baptized the Duesenberry-Modigliani hypothesis, as also Duesenberry, working independently and on the basis of partially different arguments, comes to the same conclusions at the same time.

(Casarosa 2002, p. 61).⁸

Quoting from Modigliani’s autobiographic book Adventures of an economist Casarosa points out that Modigliani, after his contribution on the relative income hypothesis of 1949, had no longer written on saving, being engaged in a large-scale project on the subject of firms’ behavior planning under uncertainty (Expectations and Business fluctuations) at University of Illinois since November 1948 (Modigliani 2001, pp. 51 ff.). The occasion to return on the subject of saving occurred in 1952 when Kenneth Kurihara asked him to contribute with an article on saving to a collective volume on Post-Keynesian economics and he, admittedly, had not yet clear what to write, until, in a now-legendary episode, he and Richard Brumberg, a first-year graduate student that he had met at University of Illinois in 1952 and involved in the writing of the article, during a conversation driving back to Urbana “were rewarded by a ray of light”: they had the intuition that an individual’s saving is meant to accumulate resources to pursue a stable average consumption over the life span, and, in particular, after retirement (Modigliani 2001, p. 59), which is the fundamental motive for saving at the basis of the LCH. On the basis of this exciting intuition, M-B started to write down the two papers of the 1950s.

Casarosa, before introducing the LCH, anticipates his thesis as follows:

The fundamental importance of retirement savings (the “ray of light” on the road to … Urbana) was perceived by both Harrod and Duesenberry; the model of the individual’s life cycle had been outlined, albeit in an embryonic way, by Harrod; the conclusion that in a stationary economy saving and propensity to save are zero and that, in a growing economy with constant per capita income, the aggregate propensity to save is an increasing function of the population growth rate, had been reached by both Harrod and Duesenberry. Finally, Harrod took for granted the existence of a positive relationship between the aggregate propensity to save and the growth rate of per capita income, and Duesenberry also expressed himself in the same sense, even though he ended up stating that in reality the propensity to save was positively correlated to the difference between the growth rate of per capita income and the interest rate.

(Casarosa 2002, p. 68).

2.2 The LCH model

Casarosa recalls that M-B in their 1954 seminal paper explored the microeconomic features of an economy in which agents behave in a life-cycle manner.⁹ Given that the model is well known (Modigliani 1966, 1975, 1986), we summarize its main features into some Propositions, as also Casarosa did, in order to compare them with the findings of Duesenberry and Harrod.

First, Modigliani’s LCH extends Irving Fisher’s two-period model (see Fisher 1907, 1930)¹⁰ to a multiperiod framework in which individuals work when young and then retire when old, maintaining the assumption that agents maximize a lifetime utility function with respect to their own lifetime consumption. Under the assumption of homothetic preferences and other simplifying assumptions, the solution to such a problem, exploiting the individual’s lifetime budget constraint, yields that consumption in any period, $c_{t}$, is a function of the expected (discounted) lifetime wealth $W_{t}$ (and of the interest rate)¹¹: $c_{t} = c_{t} (W_{t} ,r)$.

In words, individuals select an optimal lifetime consumption path so that they can maintain consumption relatively constant throughout their life in the face of time-varying resources, and in particular, in the face of retirement period. Consequently, the present value of consumption is equal to the present value of expected lifetime wealth (human capital plus starting value of assets). Moreover, consumption in each period turns out to be a function of expected lifetime wealth and, given that income is only a small fraction of $W_{t}$, it follows that consumption in each period is basically independent of current income; as a consequence, any unexpected positive (negative) shock in income will be accumulated through saving (decumulated through dissaving), in order to increase (decrease) lifetime resources and thus, proportionally increase (decrease) lifetime consumption.

Casarosa, then, recalls that building on this microeconomic behavior, M-B draw some major implications for the aggregate economy in their second contribution (M-B 1980), which he summarizes, as for the so-called “stripped-down version” (see Modigliani 1986, p. 300), through the following Propositions (text within square brackets is added by the authors), where $Y,W$ are aggregate income and aggregate wealth, respectively:

Proposition 1

In a stationary economy, aggregate saving [S] is zero and, therefore, the aggregate propensity to save is also zero [while the aggregate wealth-income ratio, $w \equiv \frac{W}{Y}$, is equal to half the length of retirement].

(Casarosa 2002, p. 71).

Casarosa provides the proof of the first proposition both formally and by using the famous graph of “hump wealth” firstly presented in Modigliani (1966). Moreover, he presents numerical proof obtained by simulating an Overlapping Generations Model (OLG) economy in a stationary state, composed of 4 cohorts (see Table 1, p. 73). Further, he explains the economic rationale of this finding on the grounds that, given the assumptions of the simplified model, in a stationary economy the positive savings of working cohorts are exactly equal and thus offset by the negative savings (decumulation) of the retirees.

As for the case of a steadily growing economy either because of steadily growing population at a constant rate n or because of steadily growing per-capita income (so that the expression for the growth rate of the economy, $\rho$, is: $\rho = \left( {1 + n} \right)\left( {1 + g} \right) - 1 \approx n + g$), Casarosa, quite interestingly, reports two tables (Table 2, p. 75 and Table 3, p. 77), in which both the longitudinal path of consumption and saving of different cohorts and the aggregate variables are reported as functions of the growth rates of population and income per-capita, respectively. Moreover, he summarizes the findings for an economy with steadily growing population as follows:

Proposition 2

If per capita income remains constant, but the population increases at a constant rate, n, the aggregate propensity to save is positive, and an increasing function of the population growth rate and, for n which tends to infinity, it tends to a value equal to the propensity to save of the active generations [i.e., it is a concave function and does not depend on the level of income].

(Casarosa 2002, p. 74).

As for the economic rationale of this result, given that the Italian author’s explanation faithfully follows the line of exposition that Modigliani has provided on several occasions (e.g., Modigliani 1966, 1975, 1986) for the sake of brevity, we report here Modigliani’s own words:

when the source of growth in population, the mechanism behind positive saving may be labeled the Neisser effect (see his 1944 article): younger households in their accumulation phase account for a larger share of population, and retired dis-savers for a smaller share, than in the stationary society”.

(Modigliani 1986, p. 301).

Turning to the case of an economy displaying a steady growth of per-capita income and constant population, Casarosa writes:

Proposition 3

If the population remains constant, but per capita income increases at a constant rate, g, the aggregate propensity to save is positive and an increasing function of the growth rate of per capita income, with a maximum value equal to the average of the maximum values towards which the propensities to save of active generations tend, as g tends to infinity [i.e., it is a concave function and does not depend on the level of income].

(Casarosa 2002, p. 78).

Again, in Modigliani’s words:

When the growth is due to productivity, the mechanism at work may be called the Bentzel (1959) effect (who independently called attention to it). Productivity growth implies that younger cohorts have larger lifetime resources than older ones, and, therefore, their savings are larger than the dissaving of the poorer, retired cohorts.

(Modigliani 1986, p. 302).

Casarosa recalls that Modigliani worked in a framework in which agents plan their consumption “as though they did not anticipate the future growth of income”, that is, using static expectations rather than the “rational expectations”, that would be fathered by Muth in his famous work some years later (1961).¹²

Casarosa does not provide a proposition for the results on the wealth-income ratio, although he presents the numerical results and their economic interpretation. Again, for the sake of exposition, we summarize M-B’s findings by adding the following Proposition:

Proposition 4

The wealth-income ratio is a decreasing function of the rate of growth ρ, thus being largest at zero growth and equal to one-half times the length of retirement period.

The economic rationale underlying the shape of the relation between w and ρ can be better appreciated, again, by discussing two cases separately: the case with constant rate of growth of population and constant per-capita income, and the case with stationary population and steadily growing productivity, respectively. As for the former case, continuing to follow Modigliani’s argument, since the whole wealth is expected to be mainly concentrated among the older households, and the demographic weight of such cohorts is reduced by an increase in the growth of population, it follows that the ratio of wealth to income is expected to fall as the rate of population growth increases (Modigliani 1966, p. 166). As for the latter case, Casarosa explains that “since patrimonial wealth is concentrated in the hands of the older generations, its amount is proportional to the income received in the past by these generations and, therefore, its relative value with respect to current income is the smaller the greater the proportional gap between current income and income from previous periods” (Casarosa 2002, p. 79).

One comment is worth doing: as shown by Modigliani (1966), the relation between the aggregate saving rate and economic growth can be obtained from the steady-state equation $s \equiv \frac{S}{Y} = \frac{{{\Delta }W}}{{\text{W}}}\frac{W}{Y} = \rho w\left( \rho \right)$, where $S,Y,s,$ are aggregate saving, aggregate income, aggregate saving rate, and the last expression $w\left( \rho \right)$ recognizes that the aggregate wealth-income ratio, $w$, is a function of the rate of growth of the economy. Given that $w$ depends on life-cycle consumption and earnings, it follows that $s$ is independent of the level of current aggregate income. Moreover, it is worth recalling that a first-order expansion of the saving rate around $\rho = 0$ provides the following approximation for the aggregate saving rate: $s \approx \frac{{\left( {L - N} \right)}}{2}\rho$, with L the length of adult life and N the length of the working period, which Modigliani (1986) summarizes as follows: “the main parameter that controls the wealth-income ratio and the saving rate for given growth is the prevailing length of retirement” (Modigliani 1986, p. 301).

Casarosa (2002) dedicates a section to explain how the theoretical model of M-B was able to reconcile the apparently contradictory empirical evidence concerning consumption (and saving) behavior that emerged from budget data, on one hand, and longitudinal data, on the other hand (pp. 80–81). In particular, he explains how the presence of wealth in the consumption function, due to the life-cycle behavior, is responsible for the anticyclical (procyclical) movements of the average propensity to consume (to save) in the short run, and of its constancy in the long run (ibidem).

As a final comment, it is worth noticing that later contributions by Modigliani, alone and with other authors,¹³ have addressed the issue of family composition, social security, and bequests motives, just to mention a few, which either corroborated or weakened some of the main propositions of the “stripped-down version”. The collection of new data, increasingly available at the international level, facilitated the empirical tests of the LCH and of its extended versions. In any case, such later extensions showed that the original framework was sufficiently general to include several other aspects that may affect consumption and saving decisions, both at the micro and at the macro level.

3 The forerunners of the LCH

We now turn to present how Casarosa gives an account of the contributions that, in apparently independent ways,¹⁴ Harrod and Duesenberry supplied to the theory of saving and, in particular, to the LCH.

3.1 Harrod’s theory of saving

As already mentioned, Harrod develops his pioneering contribution to the LCH in Lecture Two of his book Towards a dynamic economics of 1948 as a part of a more general issue, which Casarosa summarizes as follows:

• (a) determination of the conditions that allow the economic system to grow at the natural rate, that is, at a rate consistent with the dynamics of the population and technical progress;

• (b) examination of the stability of the natural growth path (i.e., examination of the economic cycle);

• (c) identification of economic policies capable of ensuring that the economic system moves along or converges towards the path of natural growth.

(Casarosa 2002, p. 52).

The Italian economist underlines how, since the introductory part of his book, Harrod explicitly declares to adopt a framework in which the dynamics of both population and technological progress are taken as exogenous.¹⁵ Concerning the latter, Harrod introduces a type of technological progress which he defines as “neutral”, in the sense that, for a given interest rate, it does not affect the capital income ratio (or capital coefficient) (Harrod 1948, p. 22). We note that, by doing this, Harrod is aware of being in “flat contrast’ (ibidem, 1948, p. 20) with the classical view, according to which the achievement of the long-run equilibrium would have been driven by the Malthusian population adjustment mechanism and by the decreasing returns from land. Under these assumptions, Harrod further develops on his 1939 article and argues that an economic system is able to maintain the natural rate of growth, with a constant interest rate, only if the aggregate propensity to save (s) is equal to $s = g \cdot \frac{K}{Y}$, where g is the natural rate of growth and K/Y is the ratio between capital and aggregate output.¹⁶ Then, the author wonders if this level of propensity to save is plausible and, to answer this question, analyses the supply of savings, to which he dedicates the entire second chapter of his book. After this premise, Casarosa gives an account of Harrod’s tripartition of different types of savings: for personal use, for inheritance, and for business reasons – which he refers to as corporate saving. Of these three types, in turn, Harrod focuses mainly on the first, considering the others of minor importance and, at any rate, somehow related to personal savings (Casarosa 2002, p. 53).

The Italian economist shows how Harrod, on the basis of the Fisherian approach and his assumption on time preference, reaches the conclusion that the individual propensity to save (for personal use):

• (a) is an increasing function of the per-capita income (mainly due to the effect that the latter exerts on the time preference T);

• (b) is an increasing function of the real interest rate (on the basis of calculations that he affirms to have carried out given “a wide range of values within the bounds of probability” for parameters T and e; see Harrod 1948, p. 49);

• (c) is a function of the expected rate of increase of income. This relation has an ambiguous sign in that, on one hand, higher expected income implies higher future consumption and, thus, the need for higher saving during the accumulation phase (in the spirit of M-B), on the other hand, the expected increase of income can allow individuals to pursue higher consumption since the early stages of life (in the spirit of Friedman’s PIT).

Interestingly enough, we notice that Harrod addresses the case in which “a representative man foresaw the prospect of a rising income in his lifetime, not merely owing to his own relative advancement which he might expect in any case, but owing to the general advance of the community” (Harrod 1948, p. 54). This is clearly an anticipation of the “rational expectation” case, which would be introduced more than ten years later by Muth (1961) and more than 20 years later in macroeconomic models by Lucas. However, Harrod concludes that in these circumstances, although it could be argued that “this should reduce his need to accumulate in order to cover his own future contingencies […] it has no effect whatever on the rate of growth in the size of humps [saving for retirement, N. o. A.]” (ibidem).

Harrod, then, states that, given that such accumulation process occurs mainly in the working period for the life-cycle motive, there is decumulation in the retirement period. For this reason, Harrod uses the term “humped-saving” to indicate the component of personal saving that generates the “hump wealth” in a household’s lifetime.¹⁷

3.1.1 Aggregate saving

After presenting the individual saving behavior as it emerges from the Fisherian approach in his LCH and the longitudinal path of wealth, Harrod turns to consider the aggregate level, starting from the case of a stationary economy. Casarosa points out that the English economist correctly argues that:

In a society in which population and the state of technology are stationary, saving [for retirement N. o. A.] should be zero. Members of each generation will save for themselves, but the older members of the population will be simultaneously dis-saving an equal amount.

(Harrod 1948, p. 45).

Casarosa comments that, as a matter of fact, this is the content of Proposition 1 emerging in M-B and recalled in the previous section.

When addressing the case of growing population and constant per-capita income, Casarosa gives an account of Harrod’s conclusion (comments within square brackets are our own):

If population increases, while technology is stationary, it appears that the hump sector of capital accumulation [aggregate saving] is likely to increase at the same rate as the population. The hump sector is the sum of all the capitals [saving] intended to be dissipated by individuals now living; in a stationary population, this dissipation [dissaving from retirees] would be exactly balanced by the hump-savings [positive saving] of the younger people. In a steadily growing population, the number of humps [number of cohorts with positive saving, hence the positive saving of the workers] is being increased, and therefore the size of the sums of all humps [aggregate saving] is growing at the same rate as the population itself. Thus hump-accumulations will increase [aggregate saving is positive and grows] at such a rate that if the other sectors of accumulation increased at the same rate, the demand for new capital [investment] would be precisely met [by the supply of saving] at a constant rate of interest.

(Harrod 1948, pp. 51–52).

Thus, according to Harrod, when population grows at a constant rate, aggregate saving is positive and increases at the same rate as population (and of aggregate income as well), so that the aggregate propensity to save is positive and constant. Even though Harrod does not explicitly conclude that the saving rate is an increasing function of the population growth rate, Casarosa argues that, at all evidence, Harrod correctly anticipated the “Neisser effect” (Casarosa 2002, p. 83) and the conclusion that, in presence of a steadily growing population and constant per-capita income, the saving rate is independent of the aggregate current income.¹⁸

However, according to Casarosa, Harrod achieves the wrong conclusion that between the aggregate propensity to save and the rate of growth of population there exists a proportionality relation, on the grounds that “the aggregate propensity to save would be equal to the product of the natural rate of growth and the [constant, N. o. A.] capital-output ratio” (Casarosa 2002, p. 57). In fact, as shown by M-B (1980) and Modigliani (1966), in a pure LCH model such a relation is monotonic but concave, given that the wealth-income ratio is a decreasing and concave function of the rate of growth. Although Harrod does not explicitly state nor comment on the proportionality result, we agree with Casarosa, given that Harrod seems to retain that the capital coefficient (K/Y) is not affected by the rate of growth of population.

When discussing the case of constant population and increasing per-capita income, Harrod’s argument becomes more complex. Harrod states in fact that, other things equal, “hump-saving is likely to increase in proportion to income”, (Harrod 1948, p. 53), probably because, as argued by Casarosa, “the propensity to save of younger cohorts increases with respect to that pursued by current retirees during their working life” (Casarosa 2002, p. 58), and thus, Harrod would point out “an increasing and proportional relationship between the rate of growth of per-capita income and aggregate saving” (ibidem, p. 57). But then the Oxford Economist goes on arguing that individual preferences are likely to change over time, in particular, the time preference T is likely to increase, in such a way that “this would involve the aggregate of hump-saving increasing more rapidly than income” (Harrod 1948, p. 53). Consequently, “so far as hump accumulations [aggregate saving, N. o. A.] are concerned, […] a rise of income per head is likely to cause them to grow at a greater rate than income” (ibidem, pp. 54–55). Through this argument, Harrod concludes that in the presence of a constant rate of technological progress, the average propensity to save in general is not constant and, instead, it increases through time, thus never reaching a stable or economically meaningful steady state, at a constant rate of interest.

As for corporate saving, Harrod argues that it may be expected to vary with requirements and consequently to become positive as population is increasing and in harmony with the rate of growth of population (ibidem, p. 52). As for saving for posterity, Casarosa underlines that the Oxford economist seems to conclude that “the higher the population growth rate, the greater the probability that the aggregate propensity to save for hereditary purposes increases less (and even much less) than in proportion to the population growth rate” (Casarosa 2002, p. 59), the reason being that the higher the number of children, the higher the sacrifice, in terms of current consumption, entailed in transferring constant per-capita units of purchasing power to descendants.

Hence, comparing the case of a stationary population with that of an increasing one, we note that, according to Harrod (1948), “an accumulation requiring a falling rate of interest is much more likely in the former case” (p. 53. The references of this paragraph are from the same source), because “in that case, no new capital at all is required with interest constant [because both net investment and hump saving are zero, N. o. A.] while some increase in average estates passing at death is almost certain so that with interest unchanged we could confidently expect redundancy of saving. In the other case, it is not clear whether such a redundancy is even probable”. In the case of increasing income per head, Harrod states again that corporate saving is “likely to respond positively to the extra-requirements due to technical advance”, and, although “nothing very definite can be said” about saving for hereditary purposes, it is reasonable to suppose that, other things being equal, the “average size of legacies might increase at the same rate as average income per head”. However, given the effect that increasing per-capita income exerts on time preference, it may be also possible that “average amount passing at death might rise more rapidly than income per head”.

3.1.2 The economy’s instability and policy implications

After his analysis on Harrod’s contribution, Casarosa concludes that Harrod, although on a mainly intuitive basis, clearly built up an LCH, that is, a framework “of a man saving during early and middle life in order to buy an annuity on retirement” (Harrod 1948, p. 49). Moreover, Casarosa underlines that Harrod correctly anticipated the conclusion that in a stationary economy, with constant population and per-capita income, retirement savings are zero and the aggregate propensity to save is also zero. In the event that productivity is constant, but the population grows continuously, aggregate retirement savings are positive and the aggregate propensity to save is also positive and constant; however, as already mentioned, Casarosa notices that Harrod gets to the wrong conclusion that such a relation is proportional and not concave, as later clarified by M-B.

The relationship between the aggregate propensity to save and productivity growth is barely sketched, although the author seems to argue that it is like the one concerning the saving rate and population growth (although without explaining why),¹⁹ under constant individuals’ time preference, while it tends to increase through time given the positive effect that increasing per-capita income exerts on saving through the change in time preference. Casarosa comments that this conclusion on saving redundancy at a constant interest rate (which, we recall, can also emerge in an economy with stationary population and positive saving for posterity), represents a further confirmation of the Keynesian concern about over-saving as a condition preventing the achievement of full employment, although obtained in a dynamic framework (Casarosa 2002, p. 60; see also the comments of Robinson 1949, p. 76).

Casarosa mentions how the analysis of the properties of the aggregate saving function and of the aggregate saving rate developed in Lecture Two allows Harrod to answer, in the following chapters, the questions that are at the core of his book, that is, whether an economy with steadily growing population and “neutral” technological progress that occurs at a constant pace, too, can grow at the constant (natural) rate, given a constant interest rate. In fact, as recalled by Casarosa (Casarosa 2002, p. 60), on one hand, Harrod has shown that there is no presumption that thriftiness tends to adapt itself to the rate of capital accumulation required to sustain a steady expansion of production with rate of interest constant; moreover, we add, he doubts that, under laissez fair, financial markets can be able to manage the systematic fall in the interest rate required to guarantee a balanced growth (Harrod 1948, pp. 58–62).²⁰ Hence, in the next chapters, Harrod proposes and discusses some public policies aimed at ensuring balanced growth: in particular, the reduction of the interest rate accompanied by a tax cut and the parallel increase of the monetary financing of public expenditure.

3.2 Duesenberry’s relative income theory and his insights into the LCH

We now turn to document how Modigliani’s LCH shares elements of strong proximity with Duesenberry’s model of saving. We will follow the lines traced by Casarosa (2002), who—as already recalled—firstly documented how Duesenberry, in his 1949 book Income, Saving and the Theory of Consumer Behavior, clearly anticipates some of the main conclusions concerning the LCH model.²¹

Duesenberry pursued the ambitious objective of unveiling not only the short-run determinants of the aggregate propensity to save but also those influencing its long-run behavior, in such a way to explain the empirical evidence emerging from both budget and time-series data. Following Keynes’s argument, Duesenberry believes that consuming and saving decisions are heavily influenced by social norms, although without abandoning the reference to the Fisherian approach.²²

3.2.1 The Dusenberry-Modigliani hypothesis

Duesenberry believed that consumption is a “social” rather than an individual decision-making process and doubted that it is fully based on mere rational calculations oriented towards the future. Therefore, rather than assuming directly, as in the theory of absolute income, that the poorest families naturally have a greater propensity to consume (and therefore lower propensity to save), he derived this property from the hypothesis that utility functions are socially determined or that they are interconnected between households. On the basis of this assumption, therefore, the poorest families have a higher-than-average propensity to consume as they try to achieve a standard of living more similar to, or less distant than that of, other households.

Another implication of what Duesenberry calls the demonstration effect (Duesenberry 1949, p. 27) is that, if the distribution of income remains constant, even in the presence of economic growth, the average propensity to consume of households within a cross-section should be correlated negatively to their income, while it should remain constant at the aggregate level. The demonstration effect introduced by Duesenberry, therefore, represents a possible explanation of the “empirical puzzle” provided by Kuznets (1942, 1946).

A second property of the utility functions of households, based on the relative income theory, is that they are linked intertemporally within the same consumption unit. This means that households not only compare their current consumption level with that of other families but also with the levels that they have experienced in the past because spending and saving decisions leave a “memory” of the life associated with them (ibidem, pp. 27–32).²³ This assumption would explain also why the average propensity to consume is counter-cyclical along the business cycle, i.e., increases recessions and decreases in the expansionary phases of the economy.

The theory of relative income, although promising, was surpassed by subsequent studies, in particular by the s. c. “wealth theories” (see Martini and Spataro 2021). However, Duesenberry’s contribution, after years of relative oblivion, has experienced some new popularity in modern neo-Keynesian and neoclassical macroeconomic theory.²⁴

As mentioned, Casarosa gives an account of the Duesenberry-Modigliani hypothesis by pointing out that, while both economists in their contributions published in 1949, shared the same explanation about the determinants of the short-run (procyclical) behavior of the aggregate saving rate—on the grounds that individuals tend to pursue a constant rate of consumption in reaction to changes of their income (see Casarosa 2002, p. 62) and, in particular, through the “ratchet effect” in Duesenberry’s view—they depart from each other as for the explanation of the long-run constancy of saving rate. Modigliani, at that time (1949), believed that it was “the very nature of the capitalistic development that neutralized the tendency of the average propensity to consume to fall, through the continuous creation of new goods and the continuous improvement of the quality of existing goods” (ibidem). As for Duesenberry, he is more prone to believe that the causes of the long-run constancy of the aggregate saving rate are strictly connected with the very nature of saving decisions. Hence, in the third Chapter, entitled A Reformulation of the Theory of Saving, he presents his theory of saving decisions.

3.2.2 The determinants of the aggregate propensity to save in Duesenberry (1949)

Duesenberry, besides mentioning the social nature of individual preferences, anchors his theory on individual’s saving decision to the Fisherian approach of intertemporal choices. In particular, in Sect. 9 of Chapter III, entitled The Role of Population Growth, the American economist discusses the motives for saving which can affect the shape of the aggregate saving function.

Casarosa (2002, p. 63) summarizes the ideas of Duesenberry by recalling that, according to the latter, albeit the reasons for saving can be different, both in nature and in relative importance, “no one will deny that a very large part of saving is done either for retirement or protection of dependents. Saving for contingencies and future purchases of durable goods, or education of children, falls into the same class” (Duesenberry 1949, p. 41). And then Duesenberry adds:

To the extent that saving for retirement is important, the age distribution of the population and the rate of growth of income will be important in the determination of the rate of saving. These variables are important because they determine the size of negative savings by retired persons relative to the size of positive savings by persons preparing for retirement.

(Duesenberry 1949, pp. 41–42).

Then, Duesenberry analyzes aggregate saving of a stationary economy and argues that:

Indeed, if all saving was done for retirement, a community with stable population and income would have zero aggregate savings. Once the age distribution had reached equilibrium, the dissavings of retired persons would just balance off the positive savings of young people.

(Duesenberry 1949, p. 42).

Casarosa (2002) argues that probably, Duesenberry considers this proposition completely obvious and, therefore, not worthy of explanation (p. 64). Then, the American economist comments on the case of economic growth, starting from the case of population growth and constant per-capita income.

A growing society with stable income per capita will have positive savings even if every individual liquidates all his savings before he dies. By comparison with a stable population, a growing one will have larger proportion of young people and a smaller proportion of old people. The savings of the young people will more than balance off the dissavings of the older ones. The aggregate savings ratio will therefore vary with the age distribution of the population even with constant income per capita.

(Duesenberry 1949, p. 42).

Casarosa (2002) comments that “Duesenberry does not specify the quomodo of the variation, but from the whole of the reasoning it is evident that, in his opinion, the greater the growth rate of the population, the greater the ‘weight’ of young people and, therefore, the greater the aggregate propensity to save” (p. 64). In other words, Duesenberry clearly identifies the role of retirement saving as a source of aggregate wealth and the s. c. “Neisser effect” (although without mentioning Neisser), according to which the aggregate saving rate is an increasing function of the population growth rate.

Then Dusenberry turns to discuss the case of constant population and increasing per-capita income and states that “even a stable population would have positive net savings if it had a rising income per capita. But the connection between the rate of increase of income and the rate of saving is not a simple one” (Duesenberry 1949, p. 42. The references of this paragraph are from the same source). He starts considering the case in which “the value of assets is simply the accumulated amount of past savings” (which he interprets as stemming from the case of zero interest rate). Under these circumstances, Duesenberry, correctly, argues that “the rate of saving would be directly connected with the rate of growth of income”. The reason for this situation lies in the fact that “the bank accounts of dissavers would depend on a weighted average of past incomes”, while “the positive savings of those still earning would depend on current income, which by assumption, is higher than the past income”. So that, he can conclude that if “each individual planned to die with zero assets the positive savings would have to be greater than the negative ones”.

Casarosa (2002) comments that “Duesenberry's reasoning is certainly correct, because young people, obtaining higher incomes than those received by the elderly during their working life, plan a higher level of consumption for the retirement period than that of the current elderly and therefore, they realize a flow of savings greater than that achieved in the past by the latter” (p. 64). However, he goes on noticing that.

by pursuing this path, Duesenberry could easily have shown that the aggregate propensity to save is an increasing function of the growth rate of per capita income. In reality, similarly to what he did for the population growth rate, he limits himself to stating that ‘the rate of increase of income [per capita, added by Casarosa] will thus be one of the important determinants of the rate of net saving’ (Duesenberry 1949, p. 43). However, from the context, it seems clear that, if the interest rate on assets is zero, the aggregate propensity to save is an increasing function of the growth rate of per capita income.

(Casarosa 2002, p. 65).

In our own words, Duesenberry had correctly grasped and convincingly explained the Bentzel effect, although without discussing, as both Harrod and M-B did, the issue of individuals' expectations on future income, nor whether the case of zero interest rate is compatible with the condition of steady growth.

After presenting the analysis of the zero-interest-rate, which however he considers “certainly unrealistic”, Duesenberry addresses the situation in which assets are not simply the sum of past savings but, rather, “are equal to the present value of expected future income from property” (ibidem, p. 43) and here his explanations become, however, less clear. He argues that, in this case, the growth rate of assets is not necessarily dependent on the amount of savings invested, but, rather, on the extent to which dis-savers (asset owners) “share in the results of the innovation process” (ibidem), which is the source of per-capita growth income. If, for example,

Old people hold a cross-section of all types of assets, the amounts they have for liquidation will increase in proportion to income. In this case, their dissaving ought to rise as fast as the positive saving of young people. Consequently, a stable population with rising income could have no net saving (if all savings were for retirement purposes).

(Duesenberry 1949, p. 43).

Casarosa interprets Duesenberry’s line of reasoning as the case in which the rate of interest is equal to the rate of growth of per-capita income²⁵ and then he points out that:

Duesenberry’s reasoning, however, is flawed by the fact that he seems to exclude interest accrued on retirement wealth from current income. Indeed, if we include interest in current income it is quite obvious that Duesenberry’s argument for the null interest rate hypothesis also holds for the hypothesis of interest rate equal to the growth rate of per capita income and, in fact, for any value of the interest rate; this is because, whatever the value of the interest rate, young people receive incomes (including interest) higher than those previously received by the elderly and, therefore, realize savings flows higher than those realized at the time by the latter and, consequently, higher than the negative savings they make during the retirement period.

(Casarosa 2002, p. 65).²⁶

In fact, Duesenberry seems to abandon the previous line of reasoning according to which saving and dissaving are proportional to lifetime resources of different cohorts and concludes that the saving rate will be proportional to the per-capita income growth rate only if the share of dis-savers who own stocks (rather than fixed-income assets) is sufficiently low, which can depend, among other things, on the distribution of income within the population. For these reasons, Duesenberry (1949, p. 45) concludes that the aggregate propensity to save a) is independent of the absolute level of aggregate income and b) is an increasing function of the population growth rate, while, c) it can be an increasing function of the per-capita income growth rate, provided that the rate of economy’s rate of growth is higher than the interest rate.²⁷ Notice that Duesenberry does not even mention the properties of the wealth-income ratio.

4 The forerunners of the LCH in the early literature

To the best of our knowledge, no other author before Carlo Casarosa has clearly disentangled and discussed the role of M-B, on one hand, and Harrod and Duesenberry, on the other hand, in building up the LCH. Given that the Italian economist did not (and could not, given his 39-page-long analysis) carry out a complete historical and bibliographic analysis on whether and to what extent such forerunners of the LCH had been already recognized or mentioned by previous works, in this Section we make one step further by providing a historical review of the major works citing the forerunners of the LCH.

As already mentioned, Modigliani never acknowledged Duesenberry’s nor Ramsey’s insights into the LCH and he cited for the first time Harrod’s 1948 contribution in 1970, in a chapter of a book published in honor of the Oxford economist. Since then, Modigliani would recall Harrod’s role as a forerunner of the LCH on several other occasions (e. g, Modigliani 1975, p. 5; 1986, p. 300).²⁸ Given that we could not find any work discussing the contribution of Duesenberry to the LCH, in the next subsections we will focus on the analysis of the works that discussed Harrod’s and Ramsey’s insights into the LCH.

4.1 Lecture Two of Harrod’s “Towards a Dynamic Economics” before the M-B-LCH

Since the publication of Harrod’s book, several authors have provided reviews, comments, and critiques to the innovative framework of the English economist. Most of them were concerned with the third chapter, containing the three fundamental equations of Harrod and his “proof” of the instability of the long-run equilibrium (see Hansen 1949, 1952, Hicks 1949, Walker 1949, Atkinson 1949, Alexander 1950). The most relevant early works with respect to Lecture Two, which we are aware of, are the comments contained in Higgins (1948, pp. 177–181), Graaff (1950), and, to a lesser extent, Robinson (1949, pp. 72–76).

Commenting on the results of Lecture Two, Higgins argues that they are of little help in that they do not provide clear testable implications, being based on parameters (such as T and e) which cannot be observed. While Higgins’ latter critique was probably legitime for that time, recent econometric literature, since the pioneering work of Hall (1978), has been largely involved in the estimation of such parameters (i.e., individual discount rate and the intertemporal elasticity of substitution, exactly focusing on the Keynes-Ramsey equation used by Harrod). Higgins also argues that Harrod’s assumption of a decreasing time preference (increasing T), pointing to an increasing propensity to save, is at odds with the empirical evidence provided by Harrod himself on page iv (ibidem, p. 179) and, in any case, with the more general evidence that the average propensity to save did not display any long-run change (ibidem, p. 180). Finally, he casts serious doubts about the stability of the financial markets in presence of Harrod’s proposed progressive reductions in the interest rate (ibidem).

Going to Graaff, in his 1950 paper entitled Mr. Harrod on Hump Saving, he clarifies the meaning of the equations laid out by Harrod on page 42 as stemming from an intertemporal utility maximization problem, and corrected them as follows:

$$C_{t} = C_{r} \left( {1 - e\left\{ {1 - \frac{1}{{T^{r - 1} R^{r - 1} }}} \right\}} \right)\;\;\;\;\;\;\;\;\;\;\left[ {{\text{G}}{.1}} \right]$$

$$\left( {C_{0} + ... + C_{n} } \right) - \left( {Y_{0} + ... + Y_{n} } \right) = \mathop \sum \limits_{1}^{n - 1} \left( {Y_{r} - C_{r} } \right)\left( {R^{n - k} - 1} \right)\;\;\;\;\left[ {{\text{G}}{.2}} \right]$$

specifying that $e$ is the “average elasticity (over the relevant range) of the curve giving the marginal utility of consumption. (It is not, as Mr. Harrod seems inclined to say, the average elasticity of the income utility curve)” (Graaff 1950, p. 82).²⁹ Moreover, Graaff traces back the origins of Harrod’s theory in Ramsey (1928). In Graaff’s words, Ramsey “reasoned that a community with a stationary population can be divided up into a number of classes, the men in each class being identical in all respects, but having their birthdays spread out evenly over the course of a lifetime” (Graaff 1950, p. 85). It is worth noting that, while Ramsey’s growth model would be fully acknowledged by the neoclassical literature on economic growth of the 1960s, his (brief) contribution to conceive the overlapping generation model would be broadly neglected until recent years (see Duarte 2009b and Attanasio 2015 for a deeper discussion). Duarte (2009b) points out that “Harrod ([1936] 1965) initially saw Ramsey’s contribution as a possible foundation for his exogenous saving function and that Harrod (1960, 280–81) stated that the ideas he developed in Lecture Two of his 1948 book were “much influenced” by Ramsey. However, over time he broke away from Ramsey to the point of repudiating his utilitarian analysis in 1960 (Young 1989, 183)” (Duarte, 2009b, p. 172).³⁰

As for Joan Robinson, in her 1949 review of Towards a Dynamic Economics she follows the same argument used by Hansen (1946) in questioning the positive effect of population growth on aggregate saving, on the grounds that “the more rapid the rate of growth of population the larger is likely to be the average size of a family, and the smaller the margin above subsistence from a given individual income” (p. 74), thus anticipating the debate on the effects of taking family composition in the LCH properly into account (Fisher 1956 and Leff 1969). While praising Harrod for “directing attention to long-term problems” (p. 72), Robinson criticizes him for omitting the role of wealth on the rate of saving (p. 74) and, more generally, the issue of distribution of income and wealth upon thriftiness, commenting that “Mr. Harrod dismisses the whole question of distribution with some hints about the political instability of an egalitarian society” (p. 83). Rather interestingly, Robinson also notices Harrod’s logical shift of emphasis from investment to thriftiness when the “General Theory is transposed from short-period to long-period terms” (p. 72). In other words, according to the Cambridge economist, while in the General theory saving was somehow taken as fixed and the main question was whether investment tends to reach the full employment level, when considering the long-run, as Harrod does, the issue is rather to unveil the “influences which determine thriftiness and to inquire whether there are any cross-connections between capital requirement [investment, N. o. A.] and thriftiness which tend to keep them in harmony” (ibidem). In fact, the emphasis on saving will be at the heart of both Harrod’s analysis and later contributions to LCH and, more in general, to economic growth.

We conclude by noticing that differently from Modigliani, Harrod does not seem to have fully recognized the mechanics behind his pioneering OLG model and its policy implications. For example, when discussing the effects of the falling interest rate on aggregate savings, Harrod writes that “we need only have regard to the total effect of a falling rate upon hump-saving, the distinction between the near and more distant effect disappearing” (Harrod 1948, p. 50). On one hand, he so fails to recognize that different generations will be affected by interest changes to different extents. On the other hand, the idea that a once and for all reduction of the interest rate will not suffice to reduce the mismatch between aggregate saving and capital requirements seems, although on a different basis, somehow in line with M-B 1980 argument on the need for a continuous price deflation (rather than a once for all price fall) that, operating through the “Pigou effect” on wealth on different generations, would be necessary to systematically reduce saving so as to restore full employment (however, M-B doubt the real effectiveness of such an instrument; see M-B. 1980, p. 151- 158).³¹

4.2 Harrod’s 1948 theory of saving after the publication of M-B-LCH

After the publication of M-B (1954), several other authors have mentioned Harrod’s insight into the LCH. M. R. Fisher, a well-known economist from Cambridge (UK), in his 1956 article carried out an assessment of the empirical implications of both LCH and Milton Friedman’s Permanent Income Theory (PIT) and proposed some theoretical extension to the LCH, to take into account family composition. In this work, the author refers to Harrod’s work as a forerunner of the LCH (M. R. Fisher 1956, p. 218).³² Some years later, another economist from Cambridge (UK), Michael James Farrell, wrote about the LCH-PIT debate (Farrell 1959), citing Harrod (1948) as a forerunner of the LCH. Referring to both LCH and PIT, Farrell writes that:

the dates are misleading, as the theories were circulated in mimeograph and widely discussed as early as 1953. They had been partly anticipated by Harrod (1948) and Vickrey (1947), but these writers were interested primarily in other problems. The present paper owes a great deal to all three writers [i.e., Modigliani, Brumberg and Friedman, N. o. A.], and particularly to personal discussions with them during 1953–54.

(Farrell 1959, p. 678, 1n).

The same author, in 1970, insists that:

the concept of “hump-saving”, like so much of modern economics, seems to have originated with Sir Roy Harrod (1948, "Lecture Two: The Supply of Saving"). He comments that "in a society in which population and the state of technology are stationary, [aggregate hump-saving N. o. A.] should be zero" and goes on to speculate about the effects of population growth and increasing per capita income. It was, however, left to the now famous (but still unpublished) paper by M-B to make any calculations of the magnitude of these effects.

(Farrell 1970, p. 873).

Finally, Edmund Phelps (1968), dealing with the effect of population growth on the aggregate saving rate within the “life cycle” models, cites Cassel (1932), Ramsey (1928), Fisher (1930), and Harrod (1948), besides M-B (1980), Tobin (1967) and Diamond (1965) (pp. 449–450) as those who gave a contribution to the building of the LCH.³³

Among authors involved in a historical reconstruction of Modigliani’s contribution to economics and LCH, Mayer underlines that it was Harrod, with his 1948 book Towards a Dynamic Economics, who can be considered the anticipator of the life cycle approach (Mayer 1972, p. 25).³⁴ The same author mentions a few lines of Duesenberry’s contribution to the theory of (retirement) saving, albeit without further discussing the point (ibidem, pp. 32–33).

4.3 The predecessors of the LCH in the writings of William Hamburger and Richard Brumberg

Mayer also dedicates a paragraph on William Hamburger’s contributions (1951, 1954, 1955), stating that, although the theoretical model is not fully developed, these works can be considered de facto the first attempts to formulate an empirically testable version of a “wealth theory” of consumption (not of saving). William Hamburger had been a Ph.D. student at University of Chicago under the supervision of Milton Friedman, O. H. Brownlee and Franco Modigliani (Hamburger 1955, p. 1, 1n.) and, after completing his unpublished Ph.D. dissertation thesis in 1951 (cited in M-B 1954, p. 393, 10n and, more extensively in M-B 1980, pp. 178),³⁵ published two articles in 1954 and 1955. For this reason, we can consider the latter a “direct source” of the origins of the LCH, together with other co-authors of Modigliani, comprising Richard Brumberg. However, since both Hamburger’s and Brumberg’s Ph.D. dissertations were not published, we could infer their content from indirect sources only, and thus in the present paper we restrict our analysis to their published works.

In his 1955 paper on Econometrica, entitled The Relation of Consumption to Wealth and the Wage Rate, Hamburger writes that “this paper was presented at the December, 1953, joint meeting of the Econometric Society and the American Economic Association. It is based on the statistical portion of my dissertation, ‘Consumption and Wealth’, completed in 1951 at the University of Chicago” (p.1, 1n).³⁶ Indeed, the 1955 published article most likely was an advanced version of the paper mentioned by M-B (1954, 1980), with the addition of the data covering years 1951–1952. In fact, the theoretical part is embryonic and the author seems to point to Friedman’s PIT (no mention to the life-cycle is provided). However, besides citing M-B (1954, 1980) and Brumberg’s 1953 unpublished dissertation at the Department of Economics of Carnegie Institute of Technology (p. 12, 9n; although with a mistake, in that Brumberg completed his thesis at Johns Hopkins University), Hamburger, cites also his article, The Determinants of Aggregate Consumption, published in the 1954–1955 issue of The Review of Economic Studies (p. 2, 2n).

In this article, Hamburger provides a rather informal theory on the determinants of consumption, with particular emphasis on lifetime resources and tastes, again prefiguring a PIT model, rather than an LCH. In the footnotes of the introductive section of his article, Hamburger, after acknowledging Milton Friedman’s helpful comments (p. 24, 1n), lists a series of references concerning contributions that “related consumption to the theory of choice” (p. 23) including, among the theoretical works: Ramsey (1928), Duesenberry (1949, pp. 6–46) and, as for empirical contributions, Tobin (1947), Hamburger (1951) Brumberg (1953) (this time correctly reporting author’s affiliation at Johns Hopkins University) and both M-B (1954, 1980). Although the citation of Duesenberry includes the pages of Sect. 9 of Chapter III, in the text there is no reference to the LCH nor to its implications at the aggregate level. Moreover, while citing Ramsey (1928), Hamburger does not cite Harrod (1948).

Finally, in the issue of March 1956 of The Economic Journal, a posthumous empirical article of Richard Brumberg was published with the title An approximation to the aggregate saving function. In the introductory footnote, Richard Stone and Carl Christ, editors of the article, specify that the paper, a “reasonable approximation to consumption levels during the war years”, was left in draft form due to the premature death of Richard Brumberg in August 1954 and that they limited their intervention to minor editorial changes (Brumberg 1956, p. 66 1n). In the main text, Brumberg recalls that “the extensive literature on the theory of the determinants of personal saving contains one thread of thought that has become increasingly explicit in recent years”, that is, “connections between an individual's rate of saving at a moment in time and the total resources that he expects to command during his life” (ibidem). The contributions that he cites, besides M-B (1954), are Ramsey (1928), Harrod (1948), Lecture Two: The Supply of Saving, Graaff (1950), Tobin (1947), Hamburger (1951) Friedman (1957) “to be published”. Citation of Duesenberry (1949) (ibidem, p. 70) is only concerned with the “relative income” hypothesis, not with the LCH. We do not know whether these references belong to the “minor editorial changes” operated by Stone and Christ to Brumberg’s draft nor whether they were already present in Brumberg's unpublished dissertation of 1953. However, it is possible that Brumberg came to know about Harrod’s and Ramsey’s contributions while being in Cambridge, where also Graaff, Malcolm R. Fisher, and Farrell worked. Future studies on this issue will help to clarify the point. In any case, Modigliani will cite both Hamburger (1955) and Brumberg (1956) in later works (for example, in Ando and Modigliani 1963, while in his 1966 article he only cites Brumberg’s work).

5 Conclusions

This work presents and extends Carlo Casarosa’s 2002 contribution on the predecessors of the Life Cycle Hypothesis (LCH) on consumption and saving by Franco Modigliani and Richard Brumberg (M-B): following Casarosa’s 2002 work, the paper has shown that the LCH model was somehow anticipated not only by Sir Roy Harrod, as acknowledged by Modigliani in 1970, but also by Duesneberry (1949). Hence, by presenting Casarosa’s contribution to the non-Italian reader, we have clarified that Harrod’s (1948) and Duesenberry’s (1949) theories, though traditionally included in different groups of theories by such authors as Mayer (1972), share a relevant feature: they both anticipate, although in unformalized way, the main macroeconomic implications of the LCH developed by Modigliani and his coauthors some years later, which is, in fact, a “wealth theory” of consumption and saving.

While these works seem to have been written independently from each other, all of them have been inspired by the rich debate of the 1940s on the theoretical and empirical shortcomings of the Keynesian view on consumption and saving behavior. Moreover and somehow ironically, they both resorted to the “out-of-fashion” Fisher’s marginalist approach of intertemporal choices to provide a theoretical solution, similarly to the Permanent Income Theory of Milton Friedman, though Harrod drew also on the insights provided by Ramsey in his 1928 article on optimal savings.

According to Casarosa, both Harrod and Duesenberry “formulate immediately, albeit, on an exclusively intuitive basis, the first two fundamental propositions of life cycle theory: in stationary economy aggregate savings are zero, while, in an economy in which per capita income remains constant but population grows steadily, the aggregate propensity to save is a positive and increasing function of the population growth rate” (Casarosa 2002, p. 82). Moreover, Harrod and Duesenberry, albeit for different reasons, fail to correctly and fully explain the relationship between aggregate propensity to save and rate of growth of per-capita income (in fact, also Modigliani had to condition his results on the restrictive assumption of static expectations). Moreover, while according to Duesenberry and M-B the aggregate saving rate is independent of the level of income, in Harrod’s view the two variables are correlated, at least in the case of increasing per-capita income (ibidem, p. 65). Furthermore, Casarosa has pointed out that neither Harrod nor Duesenberry provides an analysis of the wealth-income ratio, although Harrod considered it a constant. As known, the latter assumption, which brings about a proportional relation between the saving rate and the rate of growth of the economy, represents one of the causes for the economy’s instability in Harrod’s model. On the other hand, the Italian economist adds that M-B, besides providing and correcting the propositions of their predecessors on the basis of a theoretically grounded model, explicitly unveil the retirement motive as a source of a considerably high share of the existing wealth in a society, putting under the spotlight the role of the length of the retirement period as a key parameter in shaping the wealth/income ratio and in determining the relation between the aggregate saving rate and the rate of growth of the economy (ibidem, 86).

Our study on Casarosa’s contribution allows us to offer some other observations concerning the forerunners of the LCH.

First, while Harrod did not directly carry out any empirical tests on his theoretical model, both Duesenberry and M-B did. However, differently from Modigliani, Duesenberry seems to have failed to draw all the implications from his theory of saving and, specifically, the consequence that, in presence of an LCH-OLG model, aggregate consumption (or saving) is a function of both aggregate disposable income and aggregate wealth.³⁷

Another relevant difference between the three authors who are the focus of our work is that the LCH of Modigliani and Brumberg produced (or the authors were able to draw) particularly relevant lessons as for policy implications, the main being that, when dynamic economic models are considered, characterized by finite time horizon and heterogeneous individuals as in the case of the OLG-LCH framework, both stock and flow variables must be taken into consideration. Furthermore, the impact and the long-run effects, which may be quite different (Modigliani 1961, p. 730) should be accounted for, because they may affect different generations in different ways, as in the case of the “Burden of national debt”—the so-called “displacement effect of capital” (Barnett and Solow 2000, p. 236)—discussed by Modigliani in his 1961 paper, or even the case of price deflation discussed in M-B (1980, pp. 151 ff.) In fact, the consumption function of the LCH model was set at the basis of the consumption module of the Fed-MIT-Penn macro-econometric model for the American economy, a significant project in which Modigliani and Ando had been involved since mid-1960s (see Szenberg and Ramrattan 2008, Chapter 6).

In conclusion, while from the very outset of the debate on the LCH originated by M-B writings, several authors, especially from Cambridge (UK), had acknowledged the contributions of both Harrod and Ramsey (including two graduate students of Modigliani such as William Hamburger and Richard Brumberg), Duesenberry’s insight into the LCH had been instead completely neglected until Casarosa had unveiled it. In this respect, however, it is worth remembering that, in his correspondence with Strotz in 1953, Modigliani mentioned James Duesenberry among colleagues who had already read the paper (see Rancan 2020, p. 134, 3n).³⁸ This fact documents that Modigliani had already asked Duesenberry to read the paper in which he and his brilliant graduate student had formalized and corrected the pioneering insights that Duesenberry had formulated on the LCH on an intuitive basis. As known, however, the paper would have been published only almost thirty years later, due to the difficulties that Modigliani underwent after Brumberg’s premature death (ibidem, pp. 119 ff.).

The main goal of the paper was to document Casarosa’s valuable contribution, also to the non-Italian reader, in clearly disentangling the role of Harrod, Duesenberry, and Modigliani-Brumberg in pioneering the LCH, and at the same time, we wanted to expose his work to rigorous scrutiny based on historiographical evidence, from which, as it is evident, it has been confirmed and corroborated.

Anyway, we are aware that, even if we have (hopefully) reached our goal, the work is not over yet and we are in fact already researching to provide new and more complete results on other potential forerunners of the LCH. In doing this, again quoting Casarosa, we do not aim “to undermine at all the fundamental importance of the contributions of Modigliani-Brumberg and of the subsequent ones of Modigliani and his collaborators” (p. 68); rather, we aim to tribute a fair acknowledgment to each of “the founding fathers of this theoretical vein, so relevant to modern macroeconomics and, to some extent (see the overlapping generation model), to economic theory in general” (ibidem).