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Profit Maximizing Quantity of Variable Input -- Marginal Value Product and Marginal Input Cost

The previous page focused on the relationship between the level or quantity of variable input and the quantity of output (TPP). The discussion also addressed how managers can identify the range of input in which the business will logically want to produce (that is, Stage II wherein APP is declining but MPP is greater than zero).

This page discusses how the manager can further analyze the business to identify the specific level of variable input that will achieve the goal of profit maximization in the short run. The discussion requires that the focus shift from considering the quantity of physical units of variable input and output to considering the cost and revenue associated with the variable input and output.

Expressing the data in terms of dollars

The next step in the mental process is to convert the units of output into dollar amounts by multiplying the amount of output times the market price of the output. Axes on the graph are now dollar amount and quantity of variable input.

1.  Value of total product = total physical product x price of the output    VTP = TPP * P

The VTP curve illustrates the value of output at each level of variable input.

Graph 6

2.  Value of average product = average physical product x price of the output    VAP = APP * P = VTP/X

The VAP curve illustrates the value of output per unit of variable input at each level of variable input

3.  Marginal value product can be described several ways: 1) marginal physical product x price of output; 2) value of additional output resulting from the use of an additional unit of variable input; and 3) amount of increase in the firm's total revenue as a result of using one more unit of variable input. The two most critical factors are the price of the output (Py) and the amount of output the additional variable input will produce (MPPx)

MVP = MPPx * Py

= (VTP2 - VTP1)/(X2 - X1)

= (TPP2 - TPP1)/(X2 - X1) * Py

The MVP curve illustrates the value of output for each additional unit of variable input at each level of variable input; it is the value of the output that results from using one more unit of variable input.

In Stage II, the value of the output produced by using one more unit of variable input will be decreasing because the quantity of additional output (MPP) is declining in Stage II -- according to the law of diminishing marginal productivity.

Marginal input cost (MIC) is the cost of using an additional unit of variable input; restated, it is the change in total cost due to using additional units of variable input.

MIC = cost of one unit of X = (TC2 - TC1)/(X2 - X1)

TC = total cost; this concept is discussed more fully in another section.

The MIC will remain constant regardless of how much of the variable input is used. This is based on the assumption that no one business is large enough to influence the market price for the input regardless of how much of the input the business wants to use.

What level of output maximizes profit?

• As more variable input is used, the value of the product resulting from the additional input is declining (MVP). When the quantity of variable input reaches the level that the cost of the additional variable (MIC) exceeds the value of its additional output (MVP), the manager should decide to use no more of the variable input during this production period.
• Produce where the level of output has the value of last unit of output equal to the cost of the additional variable input needed to produce that last unit of output (MIC = MVP).  Businesses will continue to use a greater quantity of the variable input as long as the cost of the additional variable input (marginal input cost or MIC) is less than the additional revenue that the input will generate (marginal value product or MVP).
• Profit is not maximized where production is maximized because production is maximized where MVP is $0. Accordingly, MIC would have to be $0 (that is, free) to maximize profit where production is maximized. If there variable input has a cost, maximum profit is at some level of production less than maximum production.
• Note however, that this information does not reveal whether the business is earning a profit, that is, this information does not reveal whether total cost is less than or more than total revenue. Subsequent sections discuss how managers can analyze whether the business is earning a profit.

Graph 7

In summary

• MVP is the value of additional output resulting from the use of one more input of variable input.
• MIC is the cost of using one more input of variable input.
• Profit is maximized at the level of variable input where the MVP = MIC, that is, where the value of the additional output produced by using one more unit of variable input is equal to the cost of that last unit of variable input.

Next section addresses demand for variable input.

Last Updated October 22, 2009

Email: David.Saxowsky@ndsu.edu

This material is intended for educational purposes only. It is not a substitute for competent professional advice. Seek appropriate advice for answers to your specific questions.

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