By A. Harnack
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Additional info for An Intro to the Study of the Elements of the Diff and Int Calculus
Any extremal y,*(t), (4 = 1 , . . ,n), must by definition yield the extreme path value relative to all the neighboring paths. One type of neighboring path arises from varying only one of the functions y J ( t ) at a time, say, y,(t), while all the other y J ( t )functions are kept fixed. Then the functional depends only on the variation in y,(t) as if we are dealing with the case of a single state variable. 18)must still hold as a necessary condition, provided we change the y symbol to y , to reflect the new problem.
Hence, there exists a tradeoff between current profit and future profit. At any time t, with a given capital stock y, a specific investment decision-say, a decision to select the investment rate yl,--will result in the current profit F(t, y, y',). As to the effect of the investment decision on future profits, it enters through the intermediary of capital. The rate of capital accumulation is y',; if we can convert that into a value measure, then we can add it to the current profit F(t, y, y',) to see the overall (current as well as future) profit implication of the investment decision.
Multiplying botb sides by (1 + yt2), rearranging, and factoring out y', we yf in h r m s of c as follows: yt2 = c2/(1 - c2). Equivalently, rn " ,, 45 Special case IV: F = F ( t , y) In this special case, the argument y' is missing from the F function. Since we now have Fy,= 0, the Euler equation reduces simply to F, = 0, or, more explicitly, , The fact that the derivative y' does not appear in this equation means that the Euler equation is not a differential equation. The problem is degenerate.