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### Determination of Optimal Employment of an Input

 To see how the economic productivity of an input, as defined by its marginal revenue product, is related to the factor for productive purposes, one need only consider the simple question. If the price of input X in the production system is \$100, how many units of X would a firm use? If the marginal revenue products exceed the related cost, that level of such input number could be employed.

The relationship between resource productivity as measured by the marginal revenue product and optimal employment or factor use can be maximization. If marginal revenue exceeds marginal cost, profits most increase. In the context of production decisions, this means that if the marginal revenue product of an input, i.e., the marginal revenue generated by its employment in a production system exceeds its marginal cost, then profits are increased as input employment increases.

Similarly, when the marginal revenue product is less than the cost of the factor, marginal profit is negative, so the firm would reduce employment of that factor.

This concept of optimal resource employment can be clarified by examining a very simple production system in which a single variable input labor (L), is used to product a single product, Q. Profit maximization requires that production be at a level such that marginal revenue equals marginal cost. Since the only available factor in the system is input L, the marginal cost of production can be expressed as:

MC = ∆C/∆Q = PL/MPL …………….. (i)

That is dividing PL, the price of a marginal unit of L, by MPL, the number of units output gained by the employment of an added unit of L, provides a measure of the marginal cost of producing each additional unit of the product.

Since MR must equal MC at the profit maximizing output level, MR can be substituted for MC in equation (i), resulting in the expression,

MR = PL/MPL ……………………. (ii)

Equation (ii) must hold for profit maximization since it was demonstrated immediately above that the right-handed side of equation (ii) is just another expression for MC. Solving equation (ii) for PL results in PL = MR x MPL which is defined as the marginal revenue product of L.

PL = MRPL …………………….. (iii)

Equation (iii) states the general result that a profit maximizing firm will always employs an input up to the point where its MRP is equal to its cost. If the MRP exceeds the cost of the input profits are increased by employing additional units of the factor.
Similarly, when the resources’ price is greater than its MRP, profit is increased by using less of the factor.

Only at the level of usage where MRP = P are profits maximized than the costs incurred (PL). Only at L where PL = MRPL, will total profits be maximized. If PL were higher the quantity of L demanded would be reduced. Similarly, if PL were lower the quantity of L purchased would be greater.