I couldn't find a general, fully algebraic solution for the Cobb-Douglas "optimal consumption bundle" problem for n goods online. For sure, this is an easy question which appears in almost every microeconomic theory textbook out there in some form, but seemingly always with more variables determined. In case anyone needs it, I wrote it up. This is an interesting problem in basic multivariable calculus (which is why I wrote it up) because it's an example of a case in which one can maximize a function with an infinite or unknown number of input variables.

Optimal consumption bundles in an n-good Cobb-Douglas model

So there are some interesting points to this solution. (You made a typo on line five btw).

ReplyDeleteFirstly, there is an easy way to find the solution. Firstly, note that the maximum of f is also a local maximum of f^x, where x is any positive real number. Thus, the important part of the maximisation condition is the direction in alpha, beta, gamma space, and a condition like alpha+beta+gamma = constant, gives you all possible directions. Next, imagine that you are matching contours. You are looking for a maximum of f which corresponds to a tangent of g(a, b, c) at some point (a, b, c), and because the form of a plane means that bringing in a new dimension does not change the tangent vectors in the original directions, the solution for each dimension is independent of the parameters that determine b,c.....etc, thus the one dimensional solution, a=d/A is clearly prototypical, and the general solution must be of the form (alpha^{x}*d/A. Finally, by substituting into g, and comparing to our remarks on the symmetries of f, it is `obviously' linear in alpha.

This approach is very powerful when the form of g is such that adding new dimensions will not change the tangent vectors of the old dimensions, and allows you to build general solutions out of trivial ones. Formally, you can see this because you can rescale g to d=1, so if you imagine that there are only two dimensions, a,b, and you add a third, then for any point c, the two dimensions are like just changing d, which we can rescale to one anyway, and preserve the tangent vector in a, b.

Secondly, a good exam question extension is to ask a student to find the maximum of

a+b+c

subject to d=A*exp(a/alpha)+B*exp(b/beta)....

this works because the total derivative of log f is 1/f df, and since f is known to be non zero, any point where df=0 is preserved (i.e. local maximums along a direction g) so this is just a rewritten version of the original problem.