
© 19992003
Douglas A. Ruby
Revised: 01/17/2003
Production Relationships
Microeconomic Theory

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 =
P_{x}(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 isoprofit 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):
The combination of L_{0}, X_{0} corresponds to a level of profits
of p_{0}. Likewise the combination of
L_{2} (greater costs) and X_{2} (more revenue) also corresponds to
this same level of profits (p_{0})  revenue and
costs increase by the same amount. However, the combination of
L_{1} and X_{1} correspond to a greater level of profits relative to
the combination of L_{0}, X_{0} (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
X_{2} is impossible with a level of labor input of L_{1}.
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 isoprofit line 'profit_{1}. This point is known as a
producer optimum. The condition for this optimum is
formally defined as:
slope of an isoprofit line = slope of the production function
or
(w/P) = MP_{labor}
Changes to this producer optimum occur when there is a
change in factor prices {w, r, n, a}, output price 'P_{x}, fixed inputs
(K, M, R), or the level of technology 'f(.)' Some of these changes are modeled below:
Original Position
Productivity Increase
Wage Increase
In the case of an increase in the wage rate, we find that the slope of
any isoprofit 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:
 IsoProfit Line
 Marginal Productivity of Labor
 Marginal Rate of Transformation
 Producer Optimum
 Production Function
 Profit Maximization
 Real Wage
