Stability Analysis Of John Hicks

General Equilibrium Models and Stability Analysis Of John Hicks

Hicks began lecturing on the general equilibrium theories of Walras (1934b) and Pareto in 1930 and had written many articles and his Theory of Wages from that general perspective but it was not until the publication of Value and Capital (1939a) that his ideas were formed into a coherent framework. The book was finished at Cambridge but it was basically an LSE book. Both its title and contents have led many people to suggest that it is a work of bridge-building between micro- and macro-economics-in particular, laying the micro foundations of macro theory. However, Hicks disputes this and sees it instead as an attempt to link together static neo-classical theory and the more dynamic theories of capital accumulation. Consequently it may be helpful if we first sketch a plan of this classic work. The analysis proceeds as follows.

(i) First the Hicks-Allen contribution to demand theory is outlined and the stability of equilibrium is analysed; first in relation to two goods and then in a multiple exchange system. At this stage Hicks introduces the concepts of 'perfect' and 'imperfect' stability and shows that the instability of exchange arises only from asymmetrical income effects.

(ii) The firm, production and the technical elasticities of substi­tution are then introduced and a system in which both exchange and production respond to price incentives is analysed and the properties required for the stability of the whole system are then established along the lines of exchange stability.

(iii) The dynamic part of the book begins with an outline of the method of analysis to be employed. The Hicksian 'week' is introduced and defined as that period during which prices are constant. It was essentially a dynamic concept and it provided the framework for a discussion of equilibrium over time-what Hicks termed temporary equilibrium. This rep­resented an heroic attempt to break out of the straitjacket of static theory by specifically analysing the effects of price expectations: i.e. during the 'week' both current prices and expected prices were allowed to influence production and consumption plans.

(iv) The difficult problems of capital, interest, money and speculation are then preliminarily analysed. At this stage the relationship between long and short rates is introduced and the expectational theory of the term structure of interest rates is formulated for the first time; the loanable funds versus liquidity preference debate is cleared up along general equilibrium lines; the close substitutability between money and other assets is established; the ex ante/ex post distinction between investment and saving is cleared up along with the concept of income; and the elasticity of expectations is introduced for the first time.

(v) The scene is then set for an analysis of the working of the dynamic system. The effect of price expectations on produc­tion plans and thereby their influence on the temporary equilibrium of the whole system is then formalised. The stability of the system is shown to be dependent on certain stabilisers-in particular inelastic expectations, normal prices, rigid money wages and contracts.

(vi) Finally, the implications of the analysis for real-world problems such as capital accumulation and the trade cycle are tackled.

The number of novel concepts introduced in this work is remarkable, not to mention older concepts which are rigorously restated and misconceptions cleared up. Thus economists were provided with a far more comprehensive overview of the entire system of production and exchange than had ever been presented before. However, the significance of many of these concepts only became fully appreciated later and this can be illustrated by analysing two of the most important in terms of subsequent developments - Hicks's stability analysis and the concept of temporary equilibrium.

Stability analysis

Hicksian stability conditions depend on the negative slope of the excess demand function -i.e. stability requires that a rise in price makes supply greater than demand. This ensures the stability of equilibrium even in the case where the supply curve is downward sloping as long as it is steeper than the demand curve - this represents stability in the Walrasian sense rather than in the sense of Marshall. In the case of multiple exchange Hicks defined perfect stability as occurring when a movement away from equilibrium sets up forces tending to restore equilibrium both when other prices are given and when other prices are adjusted so as to preserve equilibrium in the other markets. Imperfect stability occurs when equilibrium is restored only after all other price repercussions are allowed for.

However, the analysis soon came under attack. Specifically, in the case of multiple exchange the Hicksian stability condition raises problems because it takes no account of convergence, i.e. no dynamic adjustment mechanism is incorporated into the system. Samuelson was to point out that for stability - in the sense of convergence towards the equilibrium - one needs the further condition that the rate of change of price in each market is proportional to the excess demand in that market. Otherwise the system might move away from equilibrium once a disturbance occurs (e.g. the cobweb model). Samuelson insisted that an understanding of the dynamics of change was crucial even in comparative static analysis: 'One interested only in fruitful statics must study dynamics' (Samuelson, 1947).

Basically Hicks had defined perfect stability in terms of the cross elasticities of demand as expressed by the derivatives of the excess demand functions - the impact on the excess demand for the ith good of a change in the price of the jth good. However, Samuelson pointed out that stability actually required an analysis of whether the excess demands got smaller over time, i.e. whether deviations from equilibrium were gradually eliminated. This involved an analysis of the dynamics of the system. Specifically, what was needed was the time derivative of price (its rate of change) with respect to the volume of excess demand in the system. True dynamic stability requires that the roots of this equation (the characteristic equation of the dynamic adjustment process) have negative real parts and Samuelson insisted that this requirement is not equivalent to the Hicks conditions.

Samuelson's criticisms provided the foundations of dynamic analysis and this was unquestionably an important advance. However, although Samuelson's foundations were completely general it was soon pointed out that in practice this generality took the analysis further away from a discussion, of real economic problems. Samuelson's necessary and sufficient conditions, for example, were based entirely on the mathematics of the excess demand functions not on the economics. This is in contrast to Hicks's stability conditions which were specifically derived from his economic model. Later Metzler and Morishima were to prove that the Samuelson and Hicks conditions were equivalent in most cases and that, moreover, the Hicks conditions were necessary if useful applications of Samuelson's correspondence principle were to be achieved within a Walrasian price system. Thus although Hicks's method could be criticised, the stability conditions themselves have stood the test of time.. As Mundell puts it: 'The stability analysis introduced by Hicks has been one of the most successful failures in economic theory' (in Woolfe, 1968, Chap. 18).

In fact, Hicks'showed characteristic foresight in refusing to accept Samuelson's propositions as an unequivocal step forward. Thus in the 1946 second edition of Value and Capital he criticised Samuelson's dynamic analysis for being too mechanical.' T reduced the purely mechanical part of my dynamic theory to the simplest terms . . . But in so doing I did leave myself free to make some progress with the less mechanical parts - expectations and so on. [Thus], I should be sorry to abandon it altogether in favour of pure concentration on mechanism.' The progress with the less mechanical parts refers to the temporary equilibrium concept to which we now turn.