. © 1999 - 2004
Douglas A. Ruby


A Producer Optimum

Microeconomics
Tutorial: A Producer Optimum

The optimization problem facing the producer (business firm) may be stated as:
max p = PxX - [wL + rK +nM + aR]
s.t.
X = f(L;K,M,R) -- Labor is the only variable input
  • Drag the green triangle along the horizontal axis to represent different levels of Labor input (and corresponding changes in output). Watch the numeric changes in Profits and try to find the input/output combination which maximizes profits. Once you have identified this combination, Click on the 'Iso-profit Lines' button to visually understand this optimum choice.
  • You can click on the number boxes to make changes to the 'Wage Rate' or 'Fixed Costs' and repeat the above step.
  • Press 'Reset' to start over.
Experiment with different values for the wage rate (the price of labor) and fixed costs, identifying different profit-maximizing combinations of input and output.


A. Simulate changes to the Labor input/Output combinations by dragging the green triangle along the horizontal axis. Observe changes in the level of Profits. How might you explain these change in profit levels? At what input/output combination are profits maximized? What is the level of profits?

B. How does a changing the wage rate to $8.00 affect the profit-maximizing level of labor input and profits?

C. Press 'Reset'. Click on the 'Fixed Cost' number box and change this value to $500.00. How does this increase in 'Fixed Costs' affect the profit-maximizing level of labor input? How about the level of profits?




Answers...
A. Given our condition for a Producer Optimum:
MPL = W/P

ΔQ/ΔL = W/P

PΔQ = WΔL

ΔRevenue = ΔCosts
For levels of labor input below 100 units, an increase in the amount of labor (and corresponding levels of output),
MPL > W/P or PΔQ > WΔL. -- Profits will increase with increasing amounts of labor input.
Beyond 100 units of labor,
MPL < W/P or PΔQ < WΔL. -- Profits decline with increasing amounts of labor input.
The profit-maximizing level of labor input is 100 units resulting in 100 units of output. This result is consistent with the condition for a Producer Optimum:
MPL = 0.5010L-0.50 = [w/P] = [$10.00/20.00]

    = 5L-0.50 = 1/2
The level of profits is $1,000.

B. A decrease in the wage rate will increase the profit-maximizing level of labor input. As Wages , [W/P] . At 100 units of labor input, MPL > [W/P] (Press the Iso-Profit Lines button). By using more labor input, the MPL (due to diminishing marginal productivity)such that the optimal amount of labor is 157 units and profits are equal to $1250.00.

C. An increase in fixed costs does not change the profit maximizing level of labor input. The profit maximizing condition is based on the relationship between changes in revenue (PΔQ) and changes in variable costs (WΔL). Profits are reduced by the change in fixed costs for all input/output combinations.