# Optimal Savings

**DOI:**https://doi.org/10.1057/978-1-349-95189-5_1329

## Abstract

How much should a nation save or, to put it differently, what is the optimal rate of growth? This question is at the heart of the extensive literature on ‘optimum savings’ which developed as a complement to the literature on descriptive growth models in the 1950s and 1960s. Let it be noted that the reasonableness of the question presupposes a utilitarian welfare-theoretic outlook, which locates a source of ‘market failure’ in the intertemporal context stemming from what A.C. Pigou (1928) had described as a defective telescopic faculty. While Böhm-Bawerk, Fisher and other economists had noted the fact the individuals show a preference for advancing the timing of future satisfaction, they refrained from making any normative statement. Instead they constructed theories of interest which utilized this crucial behavioural characteristic on the part of individual economic agents. Pigou, however, read into the fact that individuals discount future satisfaction at a positive rate, that is display impatience, ‘a far reaching economic disharmony’ (1928, p. 26). This made him seriously question the ‘optimality’ of the rate of savings thrown up by an otherwise fully competitive market even under conditions of full employment. Pigou’s ideas on this question received support from the Cambridge philosopher-mathematician Frank P. Ramsey, who took the next most important step of determining a rule for determining the optimum rate of savings based on the logic of intertemporal utility maximization, one of the early exercises in economics using the technique of classical calculus of variations. Ramsey was relatively precise in laying down the normative postulates underlying his enquiry, ingenious in deriving the characteristics of the optimal path and not so much concerned with demonstrating that an optimal solution will always exist even on the premises laid out by him.

How much should a nation save or, to put it differently, what is the optimal rate of growth? This question is at the heart of the extensive literature on ‘optimum savings’ which developed as a complement to the literature on descriptive growth models in the 1950s and 1960s. Let it be noted that the reasonableness of the question presupposes a utilitarian welfare-theoretic outlook, which locates a source of ‘market failure’ in the intertemporal context stemming from what A.C. Pigou (1928) had described as a defective telescopic faculty. While Böhm-Bawerk, Fisher and other economists had noted the fact the individuals show a preference for advancing the timing of future satisfaction, they refrained from making any normative statement. Instead they constructed theories of interest which utilized this crucial behavioural characteristic on the part of individual economic agents. Pigou, however, read into the fact that individuals discount future satisfaction at a positive rate, that is display impatience, ‘a far reaching economic disharmony’ (1928, p. 26). This made him seriously question the ‘optimality’ of the rate of savings thrown up by an otherwise fully competitive market even under conditions of full employment. Pigou’s ideas on this question received support from the Cambridge philosopher-mathematician Frank P. Ramsey, who took the next most important step of determining a rule for determining the optimum rate of savings based on the logic of intertemporal utility maximization, one of the early exercises in economics using the technique of classical calculus of variations. Ramsey was relatively precise in laying down the normative postulates underlying his enquiry, ingenious in deriving the characteristics of the optimal path and not so much concerned with demonstrating that an optimal solution will always exist even on the premises laid out by him.

Ramsey’s paper was much appreciated by John Maynard Keynes who, in his obituary note on Ramsey’s untimely death, which appeared in the *Economic Journal*, called it ‘one of the most remarkable contributions to mathematical economics ever made’ (1930).

Despite Keynes, Ramsey’s paper received very little attention for nearly three decades, partly, because of the ‘Great Depression’ where ‘excessive savings’ in the sense of too high a propensity to save appeared to many economists including Keynes himself, to be the problem, and the emergence of a new welfare economics, which found the cardinal approach towards utility embedded in Ramsey’s formulation of the problem extremely questionable, if not unacceptable.

During the late 1950s, however, attention was redirected to the question which Ramsey had posed, especially by those who were particularly concerned with problems of development planning in relation to low income countries. Experience of sustained full employment in the advanced capitalist countries, obvious inadequacy of the stock of capital in the poorer countries from the point of view of generating employment at an adequate level of remuneration, and a back-door entry of ‘cardinal utility’ via the von Neumann–Morgenstern axioms, although applicable only to risky prospects, made the intellectual environment more receptive to the class of issues that Ramsey had dealt with in his 1928 paper.

Discussion was initiated by Tinbergen (1960), Goodwin (1961) and Chakravarty (1962a) from a development theoretic point of view the motivation was to help planners arrive at optimal growth paths for labour surplus economies based on explicit parametric forms of utility and production functions.

These exercises showed that even in relatively simple cases, optimal paths do not always exist for an open-ended future. The special nature of the assumptions made by Ramsey became more evident through extensive investigations initiated by Koopmans (1960) on the axiomatization of intertemporal utility functions which were in some suitable sense continuous. While Koopmans was concerned with complete and continuous preference orderings, a different approach was taken by Von Weizsäcker (1965) and dealt with a partial order on the programme space defined by the principle of ‘overtaking’. A consumption path *C*_{t} is said to overtake an alternative path *C*_{t} if there exists a time *T** such that \( {\int}_0^Tu\left({c}_t\right)\mathrm{d}t>{\int}_0^TU\left({c}_t^{\ast}\right)\mathrm{d}t \) for all *T* ⩾ *T**. The overtaking criterion, being a partial order, allows for non-comparable paths but as subsequent discussion showed this may not matter under certain economically relevant conditions, thereby providing an extension of the Ramsey criterion which deals with improper integrals of the form \( {\int}_0^{\infty }u\left(c(t)\right) \mathrm{d}t \) with concave *U*(*c*(*t*)) functions.

While Ramsey dealt with a stationary population, during the 1960s characterization of ‘optimal growth paths’ in the ‘overtaking sense’ was extended to situations involving exogenously growing population. A reasonably complete analysis was given by Cass (1965) and Koopmans (1965) for the one good case with continuous time and twice continuously differentiable production and utility functions.

A multisectoral generalization of the original Ramsey model was carried out by Samuelson and Solow (1956) in the mid-1950s. During the 1960s, Gale (1967) and others derived multisectoral generalizations for situations involving growing populations, using once again the ‘overtaking’ criterion.

Aggregative models involving exogenous technical change were carried out by Mirrlees (1967), Inagaki (1970) and several others. These authors used explicit ‘time discounting’ and obtained for certain special case lower bounds which a constant rate of time discounting must obey.

- A.
Ramsey characterized the social welfare function or more accurately, the welfare function over time as the integral of deviations of current utility levels from a postulated finite upper bound on instantaneous utility levels, denoting it by ‘Bliss’ ‘B’, assumed zero time discounting concave utility functions, a stationary population and no technical progress. He distinguished between two types of ‘bliss’, one due to capital saturation and the other due to utility saturation. In compact mathematical notation. Ramsey’s problem was to minimize an expression \( {\int}_0^{\infty}\left(B-U\Big(c(t)\right) \mathrm{d}t \) subject to \( c(t)+{k}^{\prime }(t)=f\left(l,k\right) \) where

*c*(*t*) stands for consumption at time*t, k*(*t*) denoted the stock of capital at ‘*t*’ and ‘*l*’ for a given labour force. \( {k}^{\prime}\left(t\equiv dk/ dt\right) \) represents the rate of capital formation, measured on a ‘net’ basis. This is a standard problem in the calculus of variations excepting for the choice of an infinite time horizon, as can be seen through substitution. The integral is of the form \( {\int}_0^{\infty}\left(k,{k}^{\prime}\right) \mathrm{d}t \). Using the Euler necessary condition for a minimum value of the functional one can write down implicitly the optimal path for savings over time, provided it exists. In general, the path will not belong to a class of paths characterized by a constant saving ratio over time. Concavity of utility function*u*(*c*) and diminishing returns to capital assure that the second order conditions are also satisfied. Ramsey, however, succeeded in deducing through an elegant transformation of the independent variable, namely, time, a very remarkable rule which optimal paths must necessarily satisfy. Keynes provided an intuitive explanation for the same rule. The ‘Keynes–Ramsey’ rule states that the optimal rate of capital accumulation at any given instant of time multiplied by the marginal utility of optimal consumption at that point of time must equal the excess of the bliss level of utility over the utility of the current optimal level of consumption. The remarkable thing about the Keynes–Ramsey rule is that it is ‘altogether independent of the production function except in so far as this determines bliss, the maximum rate of utility obtainable’ (Ramsey 1928).

In the presence of time discounting, the integrand becomes *F*(*k, k', t*). With this modification, the Euler differential equation which in general constitutes a second order nonlinear differential equation does not necessarily possess a first integral and hence, the optimum growth path does not lend itself to a simple characterization in terms of decision rule which is formally independent of time (‘*t*’).

Ramsey assumes a stationary population, although he allowed for utility maximizing choice on the part of current labour. In the context of the discussion that took place in the early 1960s, population was generally assumed to be growing at an exogenously given rate. Thus *L*(*t*) was put at *L*_{0}e^{nt}. With this modification, the Ramsey concept of ‘bliss’ has to be altered.

Assuming a constant returns to scale production function and expressing all relevant variables on a per worker basis, one can derive the relationship \( c(t)+{k}^{\prime }(t)=f\left(k(t)\right)- nk \). Assuming that we are considering only steady growth paths, we have *k′*(*t*) = 0 and a time independent expression \( c=f(k)- nk\ c \). The expression *c* is maximized for *k* such that *f ′*(*k*) = *n*. Under some mild restriction on *f*(*k*) this expression can be solved for a finite value of ‘*k*’ and the corresponding consumption level *ĉ. ĉ* can be interpreted as the highest level of sustainable consumption per worker over time, that is the best among the steady states for a given technology. Instead of Ramsey’s expression *B*, we can now write \( {\int}_0^{\infty}\left(u\left(\widehat{c}\right)-u(c)\right) \mathrm{d}t \) as the integral, and minimize this modified functional subject to the production conditions given earlier. *u*(*ĉ*) is generally referred to in the literature as the utility of consumption attached to the ‘golden rule of accumulation’.

Koopmans (1965) and Cass (1965) demonstrated that if \( k(0)<\widehat{k} \) the optimal paths will approach \( \widehat{k} \) from below over time whereas if \( k(0)<\widehat{k} \) it will approach it from above.

Cass included time discounting (*p* > 0), and obtained the ‘modified golden rule’ for which *f'*(*k*) = *n* + *p* and deduced the optimal growth path.

For the case of *p* = 0, the Keynes–Ramsey rule is restored again for the same reason as noted in the Ramsey case. However, when population is growing, it is not clear whether one should use the instantaneous utility function *u*(*c*) where ‘*c*’ is consumption per worker (or per capita, if the participation rate is constant) or a different social welfare functional altogether. Thus, Arrow and Kurz (1970) have argued in favour of maximizing an expression \( {\int}_0^{\infty }{\mathrm{e}}^{- Pt}u\left[c(t)\right]P(t),\mathrm{d}t \) where *P*(*t*) stands for population at time ‘*t*’ on the ground that ‘if more people benefit, so much the better’ (Arrow and Kurz 1970, p. 12). It is clear that in this case, a *P* > 0 is essential if an optimal solution is to exist at all.

The extension of the model to many sector cases was first attempted by Samuelson and Solow (1956). They showed that the Fisher arbitrage rule regarding prices over time could be extended to an *n-*good case as a necessary property of all optimal paths, no matter which specific utility function is used as it depends only on the question of intertemporal efficiency. An analogue of a ‘golden rule’ was obtained in situations involving no joint production, a single consumer good and relevant convexity condition.

*n-*dimensional cases involving discounted utility functions can throw up pathologies which are not present in simpler cases, especially if joint production is allowed (Samuelson and Liviatan 1969).

- B.
Revival of discussion on the optimum rate of savings in the early 1960s was motivated by policy considerations. Dissatisfaction with a politically determined rate of savings or with the market solution, especially when the capital market was considered to be subject to considerable imperfection, led economists to look more closely into the character of growth paths based on an ethically explicit criterion function over time. As time is open-ended, the discussion veered towards problems posed by an infinite planning horizon. With the discovery that optimal paths may not exist with otherwise well behaved production and utility functions, economists devoted a great deal of attention to possible modifications of the Ramsey–Pigou valuation premises to get around the non-existence problem. Koopmans, in particular, felt the need for introducing an assumption relating to time discounting to get over the problem of non-existence and so did Arrow and Kurz.

Some authors tried to explore the sensitivity of finite horizon optimal growth paths to terminal conditions, which in the nature of the case, has to be arbitrary. The idea behind these exercises was to offer an alternative to the procedure of discounting which equally violated the postulate of ethical neutrality between generations (Chakravarty 1962b), the aim being to examine whether optimal paths will prove insensitive at least in their initial phase to terminal conditions, provided the horizon was sufficiently long. Brock (1971) subsequently generalized this type of analysis quite considerably.

Based on these discussions, Hammond and Mirrlees (1973) proposed a category of growth profiles which seemed to avoid the Scylla of ‘time discounting’ and the Charybdis of a given ‘terminal capital stock’, by suggesting a category of paths called ‘agreeable paths’ with the property that if an optimum path exists over an infinite time horizon, it is agreeable.

Furthermore, ‘an agreeable path exists if and only if a perpetually feasible) locally optimal path exists’. It is then the maximal locally optimal path’ (Hammond and Mirrlees 1973). Hammond subsequently extended the analysis to a multisectoral context (1976).

Agreeable paths possess an operational appeal to planners and therefore need to be pursued in greater depth. Among areas of current interest, one can also mention models which relax the assumption of additive separability, which does not seem to be sufficiently strongly grounded in ethical intuition, as well as the assumption of ‘stationarity’ in the sense defined first by Koopmans (1960).

Despite the existence of several unsolved problems, literature on ‘optimal savings’ has been of interest to economic theorists for having explored with considerable thoroughness the ‘open-endedness’ of the future from a national decision-theoretic point of view and for providing a convenient parametric method of generating optimal growth paths in a precise sense of the term with associated dual prices, which can be used for social benefit–cost analysis. It has also posed a philosophical issue of broader interest as to whether one can adopt ethical principles that are independent of environmental consideration in the broad sense of the term (i.e., population growth and/or technological progress).

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