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 MPk = limt-> inf (f(K+t, )-f(K, ))/t or the partial derivative of f with respect to K]
Marginal Value Product is output price times MPK or P MPK. 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 = MVPk.
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.