MYOPIC OPTIMIZING, ECONOMIC GROWTH AND THE GOLDEN RULE

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1 12 Hong Kong Economic Papers, No. 10, October 1976 MYOPIC OPTIMIZING, ECONOMIC GROWTH AND THE GOLDEN RULE RICHARD H. DAY AND YIU-KWAN FAN* I Descriptive theories of economic growth involve the analysis of the time path of a model economy for various initial conditions and growth parameters. Normative theories study the existence and properties of intertemporally optimal growth trajectories using some kind of social objective function. Few examples of either class allow for the adjustment of preference, knowledge, technology and behaviour to new experience and changing environment. Most theories in the literature assume either that all generations within the time horizon never respond to changes by revising their behaviour or that the decision-makers of the present generation have perfect foresight and all uncertainties are absent. However, the growth path of art economy can be perceived better as a locus of consumption-investment choices of successive generations. Each generation solves its own problems of allocating income to consumption and to additions to capital stock which will be handed on to the next generation. Based on each generation's own preference, knowledge, taste and technology, the choice is myopic, but flexible. Each generation neither pretends that it has perfect foresight in making decisions for the unborn generations, nor restricts its descendants to a predetermined consumption pattern; yet it endows its heirs with the wherewithal to make decisions in its own right. In reality knowledge and preference evolve, foresight is limited, decisions are fraught with uncertainty, the environment is ever-changing, and each generation weighs its own consumption against greater wealth for its immediate descendants, based only on the information available at the time.* * Professor of Economics, University of Southern California and Lecturer in Economics, University of Hong Kong respectively. Our work was supported in part by the National Science Foundation. We have benefited from the helpful comments of Donald J. Harris and Fu-Sen Chen. 1 ``Any one generation determines only the size and composition of the capital stock it hands on to the next generation.. The significance of the capital stock that is handed on is, therefore, that it determines, insofar as is possible at the time, the range of choices among alternative consumption paths that will be open to the later generations. T. C. Koopmans [6, P. 11]. 2 ``The problem (of economic growth) takes on some of the aspects of the ascent of a mountain wrapped in fog. Rather than searching for a largely invisible optimal path, one may have to look for a good rule for choosing the next stretch of the path with the help of all information available at the time. T. C. Koopmans [6, p. 12]. ``Man is placed in the absurd position of having to optimize without knowing if his behaviour is optimal, of choosing without knowing

2 13 This idea of myopic optimization is the fundamental notion underlying Leontiefs graphical model [7] and Day's mathematical model of economic growth [3]. In the Leontief-Day (L-D) model the economy is assumed to possess a sequence of utility functions, the first member of which is only known by the economy's decision-makers at a given time. Hence optimizing over all generations is not carried out. Each period's decision is obtained by optimizing the objective function of that period. The entire growth path extending through generations is the locus of temporary equilibria obtained by joining these optimal `stretches'. It may or may not be optimal in retrospect according to some intertemporal criterion. In stark contrast to this approach is the optimal growth theory whose recent variants include among many others the works of Cass [1; 2], Koopmans [5], Samuelson [10] and Mirrlees [8]. In these models an entire sequence of utility functions is known in advance; the discounted sum of its members is maximized over an infinite or finite planning period. The conditions for the existence and stability of the optimal path are explored and analyzed. If it exists the optimal path represents the best possible track for the economy to follow during the planning period if the prescribed objective is to be achieved. But this entails complete certainty in production and consumption possibilities, and a preference function which is stationary throughout the entire time horizon under consideration. Intertemporal optimality is specific to the objective function; yet some criterion for comparison is desirable. The concept of the golden rule path, that path which maintains the maximum permanent flow of consumption per capita, has become the bench mark against which alternatives are compared.3 Various authors have shown that the optimal path is closely related to the golden rule path and that under certain conditions they converge.4 Of course the concept of intertemporal optimality is difficult even to define in a world as uncertain as the Leontief-Day decision-makers perceive. But this raises the question, under what conditions would myopic behaviour be intertemporally optimal in the golden rule sense? In this paper we provide an answer to this question. Indeed, we show that if the usual technological assumptions of the neoclassical world hold and assuming each generation is just as patient as the last, then there exists a golden rule level of patience that will achieve maximal perpetual consumption flow for all future generations. We also show that the associated golden rule path of capital accumulation is stable within bounds that depend on patience, the population growth rate and the rate of decrease in the marginal productivity of capital. As a by-product of the analysis we show that the Leontief-Day model subsumes the Solow-Swan model as a special case. II We begin our analysis by incorporating a changing labour force into the L-D model. Assume that the production relations of an economy can be described by 3 See [9]. 4 They include, for example, Cass [1], Samuelson [10], Koopmans [2] and von Weizsacker [11].

3 14 a twice-differentiable production function Yt = Ft(Kt, Ltt). (1) The functional form of (1) can change over time to allow for technological progress and changes in production possibilities. But for all t, the production function must be homogeneous of degree one, with positive but diminishing marginal productivity for each factor. Equation (1) can be rewritten in terms of the capital-labour ratio as yt = ft(kt) with f't(kt) > 0, f''t(kt) < 0. (2) The flow of homogeneous output is either consumed or added to the capital stock. Let Ct be total consumption. Then Yt = ct + Kt. Assuming for convenience zero depreciation of capital stock, output will be at a new level in the next period: yt+1 = Ft+lWt + Kt, L l t t+1 ). (3) However, since at time t, Ft+1 is not `known by the economy, it uses as a forecast of (3) the myopic approximation: c yttl = Yt + nkt. Ft,K& $1 (31 where Ft K is the marginal productivity > of capital perceived at time t. With population growing at rate nt at time t, the forecasting equation (3 ) can be rewritten in per capita terms as where ct 3 Ct -. Lt G+1 = y+ ([I + f&)1 f&) - f&)q } t As already noted, the economy at every t is assumed to possess a complete preference preordering over per capita present consumption ct and forecasted per capita perpetual income that could flow to future generations from the next period onward, yt+1 It weighs a unit of per capita present consumption against a higher level of per capita future perpetual income made possible by enlarging the capital stock. This preference preordering is represented by a continuous quasi-concave utility function: Ut = /J&t $+1>. (5)

4 This function completely preorders current choices but places only a partial preordering on the space of all possible consumption paths (ct, ct+1,...). As we shall see, it is nonetheless sufficient to determine a course for the economy. At the beginning of every period, the economy picks a point on its linearly approximated production possibility frontier to determine how output is to be divided between present consumption and capital accumulation according to the social utility function (5). Adding the non-negativity constraints ct, Ct+1 > 0 = (6) the sequential maximization of (5) subject to (4) and (6) period after period constitutes the solution to a recursive programming model because of the accounting identity: 15 or $= l 1 + nt-l Kytel - c;-~) + ktall where Ct*-l and ct*-1 are optimal aggregate and per capita consumption of the previous period respectively. The solutions to the problem for t-l become the parameters of the problem for t. Since ut is quasi-concave and continuous and ft is finite and non-negative for kt >0, the problem always possesses a solution. In order to derive moreinteresting results and facilitate closer comparison, let us assume that the utility function takes the form mf. U& Cttl) = ct t yttl mt > 0. (8) The parameter mt can be referred to as the `impatience parameter'. The higher the value of mt, the larger is the share of ct in per capita current output yt. Maximizing (8) subject to (4) and (6) and solving for C*t we have mt [l + f-&)1 f&i Ct* = ~ i l+m, f;(q) 1 = $(kt, mt). (9) The term f't(kt) can be considered as the approximate opportunity cost of each unit of present per capita consumption in terms of future output flow foregone; [1 + f't(kt)] ft(kt) is the expected output of the next period onward if nothing is consumed at all this period.

5 76 Therefore the decision-maker decides how much to consume this period by weighing the `benefits of non-consumption against the `cost of present consumption with his time-preference. The more impatient he is (hence mt is large), the higher is the level of present consumption; the higher the opportunity cost, the lower is c*t. Note also that c*t is not a function of population growth. Given c* t, ct+1 ctt1 = is obtained by substitution using (4): 1 + f;(kt) (1 + n,)tl + mt) f&)> (10) an equation that says that planned output of the next period is directly related to the productivity of capital, but inversely proportional to the population growth rate and the impatience parameter. From this two-period optimization problem we derive the history of capital accumulation using (7 ) and (9): 1 1 mt k,,l=- i [ -- 1 t nt 1 t mt (1 + rr+)f&) 1 f&> + $1. (11) This is the fundamental growth equation for this myopic optimizing model. The myopically optimal savind ratio can be derived from the consumption function. Let st be this ratio in period t. Then s*=l--= f&l C* t ( --I l I+% [I- G 1 = st(ktp mth (12) t Substituting (12) into (11), the fundamental growth equation can be written as kt+l = - st 1 t1 nt ' (k t, mjf&> + $1. The dynamic properties of this equation clearly depend on the historical evolution of technology ft, preferences u or impatience mt and the population growth rate nt. To proceed with the analysis therefore we assume (though the economy does not) that these are all constant, i.e., we assume that ft = f, mt = m and nt = n for all t. Let k be a stationary state for the process (11) and (13). We find that nk = s(k,m)f(k) (14) where s(km) is the myopically optimal savings rate in equilibrium. It is a routine

6 17 calculation to show that this equilibrium is stable if -1 < & b (km)f(k) + s(kmv W + 11 k~; 5 1 III Now let us compare our model with the basic one-sector neoclassical growth model. We use the Solow-Swan model, recasting it in discrete time for this purpose. The production function is assumed constant, Ft = F for all t. Instead of the variable saving ratio (12) derived from the myopic utility maximization problem we have the constant saving ratio s. Assuming a constant population growth rate n and with the savings-investment identity, we obtain in per capita terms the Solow-Swan process of capital accumulation: kttl = & b f&l + $1. (16) A comparison with (11) or (13) shows that it is subsumed as a special case by the L-D process. At a stationary state s f(k) = nk, a special case of (14). Stability requires + (17) -1 < * [s f (G) + l] < 1 (18) the Solow-Swan analogue of (15). The left-hand inequality holds by hypothesis. The right-hand inequality holds if the product of the propensity to save and the marginal product of stationary-state capital stock does not exceed the growth rate of the labour force, i.e., if s f (k> < n. (19) The golden rule of accumulation attains its full intuitive appeal when we associate time periods with generations. It specifies the conditions under which the perpetual level of consumption per generation is maximized. The well-known rule for the Solow-Swan world - the same in discrete time as in continuous time - states that nk* f (k*)k* f&*1 (20) or that n = f'(k*) when k* is the golden rule capital stock. Moreover (19) and (20)

7 18 together imply that the golden rule path, for aggregate capital stock, K(t) = (1 + n)tk*l(o) is stable so long as 0 < s < 1. There is a tendency to return to the path from small departures away from it. For the Leontief-Day world the golden rule holds in equilibrium if s(k*,m)f(k*) = f'(k*)k* (21) which combined with (14) shows that n = f'( *) that from (21) and (12) that as in the Solow-Swan case. Observe s(k*, m) = y&p - +)I = $y*. This implies that there exists an impatience parameter m* _ f (k*)[f(k*) - f (k*)k*l > o [f (k*)] 2k* t f(k*) such that s(k*, m*) is the saving ratio which inequilibrium sustains the golden rule. In order for the golden rule equilibrium to be stable in the L-D world, (15) has to be satisfied with n = f'(k*). This requires that technology, population growth and impatience must satisfy the condition n3 + n2 t mf(k*)f (k*) (1 t m)n2 I < 1, or, in a somewhat more comprehensible form (2 t m)(l t n)n2 - < f(k*)f (k*) < n2( 1 + n). m With f''< 0, the right-hand inequality always holds, and so long as the curvature of the production function is sufficiently small, the left-hand inequality will also be satisfied in theneighbourhood of k*. IV It would be ideal if we could predict the future correctly so that we could decide now the optimal path for future generations. But, given the changeability of our world and of human behaviour it is necessary to plan with incomplete information. Myopic optimizing would therefore seem to be a relevant concept for

8 19 describing the development process. Moreover, it is perhaps even desirable to retain flexibility in our plans so that our descendants will not be forced into a straightjacket made for them in our ignorance. It should be evident that intertemporal optimizing using information about the future that will eventually prove to be misleading can be sub-optimal or antioptimal. The contrasting possibility that myopic optimizing can generate trajectories that are not too `bad in a world of changing tastes and technology is one suggested by the fact - demonstrated here - that it can approach optimality in a world of constant tastes and technology. Though the practice of myopic optimizing and flexible planning under ignorance is widespread if not universal, the systematic study of such methods has received scanty attention by economic theorists. Leontief's assertion that ``further pursuit of these speculative arguments must clearly yield diminishing returns [7, p. 111] seems to us wholly unwarranted. Indeed Koopmans call for further analysis of the topic [6, p ] seems to us much the better advice. While our results make only a modest contribution of little practical importance they may at least serve as a useful pedagogical introduction to the subject.

9 20 REFERENCES 1. Cass, D.,``Optimum Growth in an Aggregative Model of Capital Accumulation, Review of Economic Studies, Vol. 32, July 1965, pp ``Optimum Growth in an Aggregative Model of Capital Accumulation, A Turnpike Theorem, Econometrica, Vol. 34, October 1966, pp Day, R.H., ``Flexible Utility and Myopic Expectations in Economic Growth, Oxford Economic Papers, Vol. 21, November 1969, pp ``Rational Choice and Economic Behaviour, Theory and Decision, Vol. 1, No. 3, 1971, pp Koopmans, T.C., ``On the Concept of Optimal Growth, in The Econometic Approach to Development Planning, Rand McNally, Chicago, 1965, pp ``Objectives, Constraints, and Outcomes in Optimal Growth Models, Econometrica, Vol. 35, January 1967, pp Leontief, W., ``Theoretical Note on Time-Preference, Productivity of Capital, Stagnation and Economic Growth, American Economic Review, Vol. 48, March 1958, pp Mirrlees, J., ``Optimal Growth When Technology is Changing, Review of Economic Studies, Vol. 34, January 1967, pp Phelps, E., Golden Rules of Economic Growth, W. W. Norton, New York, Samuelson, P.A., A., Catenary Turnpike Theorem Involving Consumption and the Golden Rules, American Economic Review, Vol. 55, June 1965, pp von Weizsacker, C.C., ``Existence of Optimal Programs of Accumulation for an Infinite Time Horizon, Review of Economic Studies, Vol. 32, April 1965, pp