**MVP = w**

MVP stands for marginal value product (not most valuable player.

Write y = f(K,…) for the production function.

The (discrete approximation to the) marginal product of input
K (capital) is MPK = f(K+1,…) - f(K,…). In words, the
amount of additional output caused by one unit more of input.
[The exact definition is MP_{k} = lim_{t-> inf}
(f(K+t,…)-f(K,…))/t or the partial derivative of f with
respect to K]

Marginal Value Product is output price times MP_{K} or
P MP_{K}. It is the number of additional dollars a firm
receives from the additional output it sells that was caused by
increasing K by one unit.

Let w be the cost of a unit of K. w is the price of an input.

**A Profit Maximizing Firm employs an input, K, until w = MVP**_{k}**.**

The picture above shows three possible choices of the input k. At k*, MVP is greater than w. Thus the firm would make more money by using more k: Another unit of k would only cost w which is less than the amount of money the firm would receive for its additional output, MVP(k*,..). Similarly, the firm should use less than k***, since using a unit less would only result in foregone revenue of MVP(k***,…) which is less than the cost of buying another unit of k, w. That leaves only k**, where MVP = w.