Using methods of optimal control theory the problem is usually formulated as that of finding the time-paths of saving and investment which will give the preferred time-path of consumption, given the production function for the economy, the rate of growth of the labour force and the initial values of all the relevant variables. The second type of model was originated by the physicist John von Neumann games, theory of), and studies a model in which there are many commodities or sectors. The production of each commodity at one point in time requires inputs of other commodities produced at the preceding point in time. The question then arises as to whether it is possible to find a time-path of outputs of commodities such that they are all growing at the same rate and, if so, to investigate the properties of such a balanced growth path. The question of optimality is concerned with finding the maximal such growth path, i.e. that along which the common growth rate is the largest possible. <=$> turnpike theorems.

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