Growth

Mirrlees himself plays down his role in the 1962 paper with Kaldor, A New Model of Economic Growth’: ‘In the end he [Kaldor] generously made me a co-author’ (Nobelprize.org 1996). However, it stands as a milestone in his career: it became a much-cited paper which was part of standard courses on economic growth for two decades after it was written. It also put the name of ‘Mirrlees’ into the international limelight for the first time.

There are several economic ideas in the paper. These reflect the interests of Kaldor which were very much part of the Cambridge environment of the time. What interests us in this chapter is the way that Mirrlees uses his mathematics to bring these ideas to life. He developed a dynamical system of differential equations. The use of continuous time was a natural continuation of his interest in the Ramsey model. The Kaldor—Mirrlees model consists of some of the then standard Cambridge assumptions: Investment comes from savings which are a constant proportion of profits, and there is a vintage model of investment which drives productivity. Mirrlees takes these and develops a mathematical system of equations: he then jumps around differentiating integrals, integrating differential equations searching for a solution to the system of equations in ‘steady growth’. Existence and uniqueness are established in an appendix. The model also includes the ‘Golden Age’ assumption that expectations are fulfilled. For me, one of the interesting features of the paper is Section 15: Mirrlees ran numerical simulations of the model which were partly calibrated. Numerical simulation is very easy now due to software. In 1962, this would have been a much more laborious matter! It is in fact hard to believe that this model was written in 1962. The methods used by Mirrlees would at that time have been largely unknown to all but a few economists, mostly in the USA. The article itself had a distinguished life: probably due to the advanced nature of the mathematics, it was more cited than read. However, the citations died out after the neoclassical growth model became the orthodoxy in the 1970s.

The interest in development and growth continued. The issue of optimal growth in a Ramsey model with technology followed. It is illuminating to look at the opening paragraph of Mirrlees (1967: 95):

Ramsey’s famous paper created the subject of optimum economic growth. The chief deficiency in the subject as he left it, despite some illuminating comments, was the unrealism of his assumptions. I do not intend to make the shallow and pointless complaint that his assumptions are not exactly right. The point is that we have not, up to the present, known how to calculate optimum policies for models that might be a useful approximation to economies we know. It must now be assumed that technological change is an important influence on growth: models that neglect it are useful for developing analytic tools, but give no guidance for applications.

One can see here that Mirrlees views technological change as important for growth (and development, an issue to which we will come). He wanted to extend the Ramsey model to include this feature and that is exactly what he did. However, it also displays his belief that there is a need to analyse models that ‘approximate’ real economies.

Again, for me the most interesting thing about the paper is its great care and attention to detail. The paper was at the forefront of mathematical economics. Following Ramsey, economists tend to look at infinite-horizon models. For obvious reasons, the mathematics developed for physics and engineering only looks at finite-horizon models. Optimization in a finite horizon has both an initial condition and a terminal condition. These are usually enough to pick out a particular path as the optimal one. However, in infinite-horizon models the terminal condition is absent. Economics of course went on to develop the appropriate transversality condition: the condition that the value of the shadow price vector tends to zero as time tends to infinity. However, in the mid-1960s when the paper was written all this had yet to be sorted out. Mirrlees ends up with a saddlepath in his Figure 1, although he does not use the term ‘saddlepath’. He rules out the ‘explosive’ paths on economic grounds:

Paths reaching the z-axis cannot represent the optimum development, for x becomes negative, and they therefore do not even represent feasible developments of the economy ... Similarly if one.. .followed a path lying below RB [i.e. the saddlepath], there would be a lower rate of consumption at every subsequent date, which is obviously inferior. These paths we have rejected accumulate capital unnecessarily: with their associated prices, they fulfil all the competitive conditions, but they are not efficient (ibid.: 99—100).

The mathematical analysis looks at special cases (the production function is Cobb-Douglas or CES) and also looks at the solution when there are increasing returns and non-constant growth rates of the population and/or technology. The analysis is both innovative and exhaustive.

What did the young Mirrlees see as important in his work? Again, we can let him speak for himself:

Is it worth while to calculate optimum policies, and discover the features of optimum growth paths? We can, I think, make three kinds of claim for the usefulness of such a model as is discussed in this paper. First, it may be claimed that it helps us to judge the performance of actual economies; second, that it provides a method of estimating the optimum rate of investment for a government that determines the aggregate distribution of output between consumption and investment by fiscal and monetary means; third, that it provides a means of estimating accounting prices for the use of an economic administration that takes detailed economic decisions, and for its guidance in establishing a system of taxation (ibid.: 123).

The world view Mirrlees shared at the time was one of economic planning (‘administration’ and ‘decisions’) and a guide to taxation—the subject of which he was soon to move on to.