## Digression Optimal Control Theory

In recent decades neoclassical growth theory has leaned heavily on a branch of mathematics known as optimal control theory. The idea that economic growth proceeds along an optimal path was first introduced by Frank Ramsey in 1928 to test Pigou's idea that people tend to save too little and under-invest due to myopia (short-sightedness) about the future (Ramsey 1928). Ramsey's model postulated a single homogeneous capital good, and assumed that future growth follows an optimal path determined by maximizing the time integral of 'utility' L. It assumed diminishing returns for both utility and capital productivity.

Utility in his model was a function of consumption C, defined as total output Y minus savings/investment. Evidently total output Y is equal to consumption C plus capital accumulation during the year. The latter is equal to new investment (equated with savings sY where s is the savings rate) minus depreciation.

Here SK is the annual depreciation of capital K. Rearranging terms we get,

Output Y is assumed to be a function of capital stock K, so output per capita y is a function of capital per capita k. We want to maximize the integral over utility L from the present to a distant future time tx, where the integrand is a function of k:

It is also usual (though Ramsey himself did not resort to this device) to introduce a discount function exp (—gt) in the integral. This supposedly reflects the myopia or time preference mentioned above. For instance, one might choose a utility function L of the form:

where h and g are parameters. Thus the utility L becomes a function of k, its time derivative k and time t. Several mathematical conditions also apply.

The condition for a minimum (actually any extreme value) of the integral is that the so-called Euler-Lagrange equation must be satisfied at all points within the range of integration, namely:

Lagrange also introduced a method of introducing constraints with undetermined multipliers. These multipliers later evolved into so-called co-state variables. The Euler-Lagrange differential equation determines k as a function of x. (This is the central result in the calculus of variations.) It is important to emphasize that the Euler-Lagrange equation is quite general: it determines the functional form of extremum of any line integral over a function L of some variable (such as k), the time derivative of that variable, and time itself. The next step, due to Hamilton, was to introduce a 'conjugate' variable, defined by

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