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.