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 and when a mismatch among inputs occurs.

The Stages of Production

Stage I exists where MP_L > AP_L 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 (AP_L). If the margin is greater than the average, the margin is "pulling" the average up.

Stage II exists where AP_L > MP_L > 0. In this stage, increasing the amount of labor used leads to additional output although output per worker (AP_L) 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:

MP_L = w / P

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 / AP_L

thus as AP_L ↑, 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/MP_L)

as MP_L↓,MC ↑, and vice-versa.

If we accept that the firm will only operate in Stage II where

AP_L > MP_L

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 in the table below).

The Shut-down point

We can rearrange our conditions for Profit Maximization:

MP_L = w/P

P = w/MP_L

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:

If:

Scenario:	A	B	C
Price =	$15.00	$10.00	$7.00
Output (X) =	12 units	11 units	10 units
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 (scenario A), 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 (scenario B), 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 (scenario C), 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

MC < AVC

MP_L > AP_L

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.