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© 2000-2003
Douglas A. Ruby
Revised:01/18/2003


Production Relationships

A Producer Optimum

Profit Maximization

Microeconomic Theory
The Costs of Production
in the Short Run
A prelude to understanding the costs of production in the short run is a discussion of the stages of production. These stages represent different relationships between the quantities of the variable factor input used (typically labor) and the quantities of the fixed factors of production available.

I. The Stages of Production:
Stage I exists where MPL > APL that is, where using more labor (the variable factor of production) leads to more output (X) and more effective use of the fixed factors or production. This is evidenced by increases in Average Productivity (APL). If the margin is greater than the average, the margin is "pulling" the average up.

Stage II exists where APL > MPL > 0. In this stage, increasing the amount of labor used leads to additional output although output per worker (APL) is declining. If the margin is less than the average, then the margin is pulling the average down.

Stage III is where the Marginal Productivity of Labor is negative -- additional labor input results in less output (negative returns). In this stage of production, there is too much of the variable input relative to the amounts of fixed factors of production available.

Thus in Stage I there is too much of the fixed factors of production relative to the variable factors of production and the firm should increase production. In Stage III the opposite is true (too much of the variable factor relative to the fixed factors) and the firm shoud reduce the level of production (by using less of the variable factor -- labor). Stage II represents a balance between the fixed and variable factors of production and the firm should produce in this range. The exact amount of labor to be used would be determined by the condition for a producer optimum:

MPL = w/P
II. Production Costs
A production function describes the underlying technology that governs the conversion of inputs into the desired output. By simply pre-multiplying the quantity of each factor of production by its associated factor price, this production technology can be either modeled by or govern related cost relationships. This relationship is known as the dual relationship between production and costs.

These costs can be described as follows:

Total Costs: TC = Variable Costs (VC) + {Fixed Costs (FC)} or
VC = wL and
FC = {rK + nM + aR}
In per-unit terms:
Average Variable Costs:
AVC = VC / X
= wL / X
= w(L/X)
= w / APL
 thus as APL , AVC , and vice-versa.

Average Fixed Costs:
AFC = FC / X

 and as X , AFC ,

Average Total Costs:
ATC = TC/X or AVC + AFC
and Marginal Costs:
MC = DTotal Costs/DX
= DVariable Costs/DX
= D(wL)/DX
= w(DL/DX) = w/(DX/DL)

MC = (w/MPL)

 as MPL , MC , and vice-versa.

If we accept that the firm will only operate in Stage II where APL > MPL then given the dual nature of production and costs we have:
MC > AVC
and additionally:
as X , MC
This segment of Marginal Costs represents the Supply curve for the firm (shown in red below).
Table 1 -- The Costs of Production
Production Function: Xx = 18L2 - L3     -- Wage Rate = 25.00
Labor
Input		 Output
[X]		 Marginal
Product		 Fixed
Costs		 Variable
Costs		 Total
Costs		 Average
Variable Costs		 Average
Total Costs		 Marginal
Costs
0 0 NA 100.00 0.00 50.00 NA NA NA

III. The Shut-down point
We can rearrange our conditions for Profit Maximization:
MPL = w/P
as:
P = w/MPL
with the right-hand side term being Marginal Costs:
P = MC
The profit-maximizing firm will produce a level of output where market price just covers the marginal cost of production for that level of output. Now suppose that we have the following subset of data:

Output (X) FC VC TC AVC ATC MC
10 50 92 142 9.20 14.90 7.00
11 50 100 150 9.10 14.20 8.00
12 50 115 165 9.60 13.80 15.00
13 50 140 190 10.80 14.60 25.00

If:
Price = 15.00 10.00 7.00
Output (X) = 12 11 10
Revenue = 180.00 110.00 70.00
Total Costs = 165.00 150.00 142.00
Profit = +15.00 -32.00 -72.00

At a market price of 15, the profit-maximizing firm will produce a level of output equal to 12 units and earn (abnormal) profits of 15.

At a market price of 10.00, the profit-maximizing firm (now loss minimizing) will produce 11 units of output. Even though there are losses of (-32.00), it is still to the advantage of the firm to continue to operate. If the firm were to shut-down, it would still be responsible for its fixed costs of 50.00. In this case, as long as:

ATC > P > AVC
Revenue still covers all of the Variable Costs and makes a contribution against the Fixed Costs of production.

At a lower market price of 7.00, the firm would choose to produce 10 units of output. However, the market price does not even cover the per-unit (average) variable costs and thus total losses exceed the fixed costs of the firm. In this case where:

P < AVC
it is better for the firm to cease operation. Also note that when this is the case,
P < AVC and P = MC
so,
MC < AVC
or
MPL > APL
and the firm is trying to operate in Stage I of production.

In summary, we can define the relevant supply decisions by the firm, in the short run, as being where:

P, MC > AVC
and given that this is consistent with Stage II or production:
as X , MC ,
As market price (P) increases, the profit maximizing firm will offer more output (X) to the market.
Concepts for Review:
  • Average Fixed Cost (AFC)
  • Average Productivity (AP)
  • Average Total Cost (ATC)
  • Average Variable Cost (AVC)
  • Costs (of Production)
  • Fixed Factors of Production
  • Marginal Costs (MC)
  • Marginal Productivity (MP)
  • [A] Producer Optimum
  • Profit Maximization
  • [Sales] Revenue
  • [the] Short Run
  • Stage I (of Production)
  • Stage II (of Production)
  • Stage III (of Production)
  • Supply Curve (for the firm)
  • Total Costs (TC)
  • Variable Costs (VC)
  • Variable Factor of Production