Economic and Game Theory
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Any function with continuous second-order derivatives satisfy this property: their cross second-order derivatives are equal. Such functions may be called "smooth functions". The Cobb-Douglas production function is one of such functions. In Economics, 99% of the functions we work with are smooth, hence they satisfy this property (the only exceptions appear in very technical papers, so we don't have to worry that much about these exceptions. The only important exception though is the fixed proportions production function).
The marginal product of labor (PML) is indeed defined as the change in output with respect to a unit change in labor (L) with K held constant. But the MPL is itself another function of the same two variables: K and L. Therefore we can differentiate it again, but this time with respect to K, holding L constant. There's no problem if we do that, because the function we are differentiating is another function. What we get is the derivative of the MPL with respect to K (in symbols: dMPL/dK).
What the statement you mentioned says is: dMPL/dK equals dMPK/dL. In other words: The cross second-oder derivatives of the Cobb-Douglas function are symmetric, that is, they satisfy the property I mentioned in the begining (in symbols: d^2f/dKdL equals d^2f/dLdK). It just expresses in economic terms a mathematical property satisfied by the Cobb-Douglas function.
Take a look at any book on calculus, specifically, the chapters on derivatives of functions of many variables and Taylor approximations. You'll certainly find this property there. The proof is not simple, but you can try to verify this property by writing down any function of two variables and calculating its second-order cross derivatives. You'll see that they are equal. Try these: f(X,Y)=log(X)+loy(Y) and g(X,Y)=XY+X^2.
I don't quite understand. As the marginal product of labour is defined as the change in total product with respect to a unit change in labour with K held constant, how would an increase in L has the same impact on the marginal product of capital then? Thank you! [Manage messages]