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In the previous chapter, the state of balanced growth of all outputs with continued full employment of labour was described in terms of six sets of inequalities. Such a state of affairs has been called a Silvery Equilibrium, which is reduced to the familiar Golden Equilibrium in the particular case where no worker saves and no capitalist consumes at all. In this chapter, the existence is established of a balanced growth at the natural rate in one step but not two. The impossibility of perverse cases is demonstrated if various assumptions put forward in Ch. 6 are made. The new argument uses game theory as before. The different sections of the chapter look at the game‐theoretic approach, the preliminary production‐pricing game, the existence and uniqueness of the ‘fair’ wage rate, equivalence between the ‘game‐theoretic’ and ‘balanced growth’ solutions, groping for the equilibrium game, the equilibrium game and the Silvery Equilibrium, and characterization of the fair wage rate as the warranted and the greatest technically possible real wage rate.

The argument developed in the previous chapter was based on the tacit assumption that the production of goods by means of goods and labour is independent of the feeding of labour, and was described in terms of the matrices of input and output coefficients and the vector of labour‐input coefficients, all of which were considered as technologically given constants. However, it is difficult to obtain precise figures of these coefficients, especially for the labour‐input coefficients, without knowing how and at what level people are fed; the productivity of labour depends not only on technology in the narrowest sense but also on the workers’ state of health, their living and working conditions, etc. Further, in a slave economy, with the wages fixed at a subsistence level, the productivity of labour was low, and it is this that led to its replacement by a more productive system—the capitalist economy. It is not surprising that outputs of goods might increase, even though the allocation of available goods among industries and families became unfavourable for the former; in fact, the positive indirect effect on outputs of a transfer of goods from industries to families causing an improvement in the welfare of the workers might be so strong as to overcome the negative direct effect on outputs of the decrease in industrial inputs. Thus, the production of goods and the feeding of men should be treated as an inseparable process, and this is the approach taken in this chapter, which discusses: the production of goods and ‘men’ by means of goods and ‘men’; the properties of aggregate production processes; conditions for the Silvery Equilibrium; modification of the rule of competitive pricing such that Keynesian involuntary unemployment is allowed for; the existence of ‘quasi‐equilibrium’ processes and prices; the upper semicontinuity of quasi‐equilibrium prices; the equilibration of quasi‐equilibrium; efficiency and optimality in an economy with a flexible labour force; instantaneous efficiency and Pareto optimality of the Silvery Equilibrium; the possibility of optimum cyclical growth of production of goods and men; and the intertemporal efficiency and Pareto optimality of the Golden Equilibrium.

The problem of optimum savings has been discussed by Ramsey on the assumption of a constant population and later by a number of economists on the more general assumption that the labour force expands at a constant exogenously fixed rate; different rates of population growth lead to different solutions; i.e. the path of optimum capital accumulation is relative to the population growth. In contrast, Meade and others have been concerned with the problem of optimum population, assuming among other things that at any given time the economy is provided with a given rate of savings as well as a given stock of capital equipment to be used; it follows that the path of optimum population is relative to capital accumulation. It is evident that these two partial optimization procedures should be synthesized so as to give a genuine supreme path, which is optimum with respect to both capital and population. This final chapter generalizes the Ramsey–Meade problem in that direction and shows that two kinds of long‐run paths—efficient and optimum paths—will under some conditions converge to the Golden Growth path when the time horizon of the paths becomes infinite; the two long‐run tendencies that are derived may be regarded as extensions of those discussed in the chapters entitled First and Second Turnpike Theorems. The different sections of the chapter discuss: the generalized Ramsey–Meade problem; the finding that the Golden Equilibrium rate of growth is greater than the Silvery Equilibrium rate; the Average Final State Turnpike Theorem; the strong superadditivity of processes—a sufficient condition for strong convergence; the tendency towards the ‘top facet’ as the general rule; cyclic phenomena; the Average Consumption Turnpike Theorem and its proof; and aversion to fluctuation in consumption.

Chapter 10 was concerned with the Final State Turnpike Theorem on the assumptions that consumption of each good per worker is fixed throughout the planning period and that the authorities try to maximize the stocks of goods that they can bestow, at the horizon, upon the future citizens; this chapter looks at a Second Turnpike Theorem. The partial optimization for the sake of the future should more properly be superseded by a general mutual optimization, so that the benefits from the properties initially available are shared between the people living in the planning period and those after that; this would inevitably cause confrontation with one of the hardest problems of economics—the interpersonal and intertemporal comparisons of utilities. In this chapter, attempts to solve the crux of the problem are abandoned and the other extreme is addressed: the conditions are derived for Ramsey optimality as distinct from DOSSO efficiency, i.e. optimization is in favour of the people in the planning period, and the satisfaction of the future residents is pegged at a certain level, of which the present residents approve. Among all feasible programmes that leave, at the end of the planning period, necessary amounts of goods for the future residents, the question is whether the people living choose a single one that is most preferable from their own point of view, i.e. there is a switch over of ideology from abstinence for the future to satisfaction in the transient life. The different sections of the chapter include discussion of: two norms of optimum growth—the Golden Balanced Growth path and the Consumption Turnpike; the existence of the Consumption Turnpike; the Silvery Rule of Accumulation’ the singular case where there is no discrimination between the living and the coming people; the Consumption Turnpike Theorem—the cases of the subjective time‐preference factor not being greater than the growth factor of the population, and of the former being greater than the latter; and an example of a cyclic Ramsey‐optimum growth.

Investigates whether a series of the short‐run equilibria starting from an arbitrarily (or historically) given capital–labour endowment will eventually approach the state of the long‐run or ‘Silvery Equilibrium’. Taking a neoclassical approach to growth equilibrium, it is found that once a more general model of flexible population growth is used, the Silvery Equilibrium may be unstable even though the relative capital‐intensity condition is satisfied. Also, the stability depends not only upon the capital intensities of the two industries but also upon the relative steepness of the warranted‐rate‐of‐growth and the natural‐rate‐of‐growth curve and the flexibility of workers’ and capitalists’ consumption–savings decisions.

Among the assumptions underlying the analysis of stability of growth equilibrium already made in the book, the following two have played the most important roles in deriving the conclusions: first, prices and the wage rate are perfectly flexible so that the price of any good or any factor of production will go down to zero if excess supply of it cannot be eliminated (the Rule of Free Goods); second, unless the price of the capital service is zero, the existing stock of capital is fully utilized and investment is made according to the Acceleration Principle. However, as soon as a progression is made from the Walras‐type ‘flexprice’ model (where quantities are fixed in the short run, and prices adjust faster than quantities) to a ‘fixprice’ model (where prices are fixed in the short run, and quantities adjust faster than prices), either full employment of labour or full utilization of capital is no longer automatically established. Also, in the absence of full utilization of capital, it is evident that investment decisions do not obey the Acceleration Principle. The first four sections of this chapter look at price flexibility and full employment, the possibility of a Keynesian short‐run equilibrium with unemployment, the induction of centrifugal forces around the Silvery Equilibrium by the Harrodian investment function, and the necessity of unemployment in a ‘fixprice’ economy. The last section looks at the possibility of avoiding this last Iron Rule: that in fixprice economies where the Rule of Competitive Pricing does not work, a state of full employment cannot be kept unless the warranted rate of growth is equated with the natural rate of growth.

This chapter considers the dynamic method developed by Erik Lindahl in 1929–30. It was worked out at just the same time as Keynes's Treatise; but it was (at least initially) quite independent of the Treatise, nor did Keynes know anything appreciable about it until after he had written the General Theory.

This chapter discusses the Fixprice method. It compares the method to the Temporary Equilibrium method. It considers the inherent tendency of the Fixprice method to ‘go macro’. It addresses questions such as: What is the Fixprice theory of the working of a single market? In what ways does it differ from a Flexprice theory?

The conditions for the Golden Equilibrium have been established earlier in the book and this chapter turns to an examination of the economy for stability; it asks whether a Hicks–Malinvaud competitive equilibrium trajectory starting from the historically given initial point approaches nearer and nearer to the state of Golden Equilibrium when the order of the path gets larger. This problem, which amounts to asking whether an economy obeying the principle of competition can attain a Golden Age, is discussed repeatedly in this chapter and the following one. Convergence of this sort will be compared with another kind of convergence recently dealt with by many writers under the common heading of Turnpike Theorems, particular applications of which may occur in more or less planned economies but not in purely competitive economies. In this chapter, the simple case of ‘L‐shaped’ indifference curves is examined. The different sections of the chapter compare the Hicks–Malinvaud equilibrium trajectory (Hicks–Malinvaud equilibrium growth path) with the DOSSO‐efficient path, discuss the Final State Turnpike Theorem, offer a proof of the theorem by the jyoseki (a formula in the game of go), present a lemma by Gale, discuss the convergence to the Turnpike, discuss yosses (the final part of a game of go) of the proof and cyclic exceptions, and look at the tendency towards the Golden Equilibrium of a competitive economy with no planning authorities.

Earlier chapters in the book have: introduced consumer's choice into the conventional framework of economic growth originated by J. von Neumann, and assumed that consumers are classified into two broad groups of persons—the worker and the capitalist (Ch. 6 ); observed that a balanced growth equilibrium obtained when only the worker consumes and only the capitalist saves is distinguished as the ‘best’ one from all other possible states of balanced growth and is, therefore, referred to as the Golden Equilibrium (Ch. 10); and concentrated on a particular economy where the capitalist is thrifty enough to carry out no consumption of goods at all while the worker is well paid so that he/she can buy goods in the Golden Equilibrium amounts, thus enabling the establishment of convergence to the Turnpike (Ch. 10). The following question is then naturally asked: Is the Golden Equilibrium still stable when the assumption of rigid consumption is replaced by the more realistic one that the worker's demand for consumption goods depends on prices and the wage income? In association with the assumption of rigid consumption, Ch. 10 made another powerful assumption that there is no shortage at all in the supply of labour; an economy was considered where the labour force grows at a constant rate, which is exogenously determined, and it was found that a state of balanced growth is compatible with such a flexible demand schedule. Therefore, it is suspected that the flexible demand for consumption goods is an additional cause of the cyclic behaviour of the Hicks–Malinvaud competitive equilibrium path and the DOSSO‐efficient growth path. The different sections of this chapter look at the possibility that flexible demand for consumption goods may cause cycles, the possibility of a Hicks–Malinvaud path in this case (and a numerical example), the DOSSO efficiency in the case of flexible consumption, and a DOSSO zigzag.