## A Producer Optimum

A producer optimum represents a solution to a problem facing all business firms -- maximizing the profits from the production and sales of goods and services subject to the constraint of market prices, technology and market size. This problem can be described as follows:

max p = Px(X) - [wL + rK + nM + aR]
s.t
X = f(L,K,M,R)
.

In this optimization problem, the profit equation represents the objective function and the production function represents the constraint. The firm must determine the appropriate input- output combination as defined by this constraint in the attempt to maximize profits.

The objective function can be rewritten in the form of 'X = f(L)' as follows:

X = [(p + FC)/P] + (w/P)L

where FC represents the fixed costs of production (rK + nM + aR). This expression is known as an iso-profit line with the term in the brackets being the intercept that represents a given level of profits and the term (w/P)-- also known as the real wage rate, represents the slope of this line. Any point on a particular line represents i given level of profits. For example, in the diagram below (click on the next button several times):

Figure 1, A Producer Optimum The combination of L0, X0 corresponds to a level of profits of p0. Likewise the combination of L2 (greater costs) and X2 (more revenue) also corresponds to this same level of profits (p0) -- revenue and costs increase by the same amount. However, the combination of L1 and X1 correspond to a greater level of profits relative to the combination of L0, X0 (revenue increases more than costs).

By adding the production function to the above diagram, we find that the input-output combinations as defined by points 'a', 'b', and 'c' are all within the limits of available technology. Point 'd' however, is unattainable -- a level of output of X2 is impossible with a level of labor input of L1.

At point 'b', we find that we achieve the greatest level of profits possible with this existing level of technology. At this point the production function is just tangent to iso-profit line 'profit1. This point is known as a producer optimum. The condition for this optimum is formally defined as:

slope of an iso-profit line = slope of the production function

or

(w/P) = MPlabor

This optimum implies that the last worker hired is paid a real wage (the purchasing power of wages) exactly equal to her/his marginal productivity. We will see in other readings that this relationship can be rearranged as:

w = MPlabor x P

where 'w' represents the nominal wage rate (labor supply in perfectly competitive labor markets) and MPlabor x P represents (derived) demand for labor by the firm.

We can also rearrange as follows:

P = w/MPlabor = MC

In this expression, we find that a producer optimum is consistent with profit maximizing behavior by the firm in the short run operating in a competitive market (as a price-taker). In this case, the competitive price P represent s the marginal revenue from selling one more unit of the product. On the right-hand side, w/MPL can be shown to represent the marginal cost of producing one more unit.

Changes to this producer optimum occur when there is a change in factor prices {w, r, n, a}, output price 'Px, fixed inputs (K, M, R), or the level of technology 'f(.)' Some of these changes are modeled below:

Figure 2, Changes in Productivity or Wages Original Position Productivity Increase Wage Increase

In the case of an increase in the wage rate, we find that the slope of any iso-profit line becomes steeper and thus tangent to the production function at some point to the left of the original. At this new producer optimum, we find that the firm will react by hiring less labor now that this input is more expensive, and as a consequence reduces the level of output produced. In this example, revenue falls, and the costs of production increase (less labor but at a higher wage rate). The profits of the firm will be reduced.

An increase in labor productivity (either due to better technology or the availability of more capital will have the opposite effect. The firm will hire more labor (if possible at existing wage rates), produce more output for sale and (assuming that output prices remain the same) achieve a greater level of profits.

Concepts for Review:
• Iso-Profit Line
• Marginal Productivity of Labor
• Marginal Rate of Transformation
• Producer Optimum
• Production Function
• Profit Maximization
• Real Wage