Production and Cost Fixed Cost

¶ … Production and Cost

Fixed Cost describes the investment a firm has to sink before any variable cost can be employed, and which they have to pay regardless of how many units they produce, or if they produce any at all. This is the physical plant; cost of borrowing capital; land rental; license cost, or any number of costs that the firm must pay in order to employ the first variable-cost input, using the fixed cost to produce output, or go out of business. These are costs that do not change regardless of how much output is produced. Variable cost is the cost of inputs that use the fixed investment to produce output. Fixed investment, in land or capital, will not produce output until a unit of variable cost is applied to it, typically labor and raw materials (Ben-Akiva, 2008 p. 4); energy; any inputs the producer has the choice to employ or not, and how much, or trade for substitutes. We can bundle them all together and call them 'variable cost' (Henderson and Quandt, 1958 p.55).

Whether the producer can choose how much or how to apply or substitute a factor of production defines whether it is fixed or not, which also defines the long and short run:

The short run is defined as the period in which the producer cannot change at least one input, i.e. that cost is fixed. The long run is the period over which the producer can change all factors of production, or as Alfred Marshall defined it "any factor of production can be varied" (quoted in Reynolds, 1971 p. 22). We will consider some of the assumptions buried in this apparently simple definition below.

Table 1. Costs, averaged per unit of Variable Inputs

FC = 100

Output

AFC

MC

AVC

ATC

0

undefined

0

1

100/1=100

100/1=100

(100+100)/1=200

2

100/2=50

(100+100)/2=100

(FC+2x100)/2=150

3

100/3=33.3

(200+100)/3=100

(FC+3x100)/3=133

4

100/4=25

(300+150)/4=113

(FC+450)/4=138

5

100/5=20

(450+200)/5=130

(100+450+200)/5=150

Presented in Table I are units of output; Average Fixed Cost (AFC), where Fixed Cost (FC) is 100 regardless of how many units of output are produced; the Marginal Cost per unit of output; Average Variable Cost with increasing costs to scale (see below); and Average Total Cost (ATC), fixed plus variable costs over number of units of output.

Diminishing marginal returns is one of many assumptions included in models that take place under perfect competition. What this means is that there is a limit to output from fixed investment where adding one more unit of variable inputs will ultimately not produce another unit of output. Laymen call this crowding, or better yet gridlock: Adding a 999th worker at your local McDonalds would probably not get the burgers out faster, or even one more burger made than adding the 998th worker. We'll revisit this briefly but the effect this has on the short-run cost curves, is to cause ATC and AVC to start to rise, after they initially fall and flatten out, where they would fall indefinitely were diminishing marginal productivity relaxed, or not yet achieved. The firm wants to minimize total cost, in order to maximize revenue per unit, which is where MR intersects MC. Let's develop the model and watch this play out.

Fixed cost in this example is 100. Regardless of how many units of output are produced, fixed cost remains the same. But the more units of output (Y) we produce, the lower is the Average Fixed Cost per unit. Therefore the AFC curve is down sloping (Table 2). This model uses only one variable input, which is entirely possible, to enhance clarity and avoid marginal rate of technical substitution between inputs (the graph should intercept the origin, because output of zero units still costs 100$). The fact that FC is 100 for any amount of Y indicates this is a short-run story, or else the FC entries would not all be the same (Petroff, 1989 n. pag.). We'll get to that. Some authors use infinity symbol for AFC where no inputs have been employed (Reynolds, 1971 p. 18) but I call that "undefined" (~) because you can't technically divide by zero. Other authors leave it empty or don't use tables at all (Henderson and Quandt, 1958 p. 55-67).

FC

Y (Output)

FC

AFC

0

undefined

1

2

50

3

33

4

25

5

20

Table 2: Fixed Cost,

Average Fixed Cost

Y

MC

AVC

1

2

3

4

5

Table 3: Marginal Cost (MC) and Average Variable Cost (AVC) from Table 1.

Short run Average Variable Cost (AVC) in this model experiences increasing costs, which is given here to demonstrate AVC. This does not always have to be the case but usually is eventually for a wide variety of reasons (Braff, 1969 p. 50) we'll return to. AVC in Table 3 is derived from the calculations in the AVC column in Table 1. The reason AVC starts to rise is because MC increases: We could relax that assumption. It often takes many units for this to happen, and this is not the only possible outcome (Robinson, 1959 p. 48).

When we sum AFC and AVC from Table 1 we get ATC, Average Total Cost (Table 4). This is total cost, i.e. summed cost of variable cost plus original fixed cost, divided by the number of units of output. ATC increases once variable inputs are applied, then decreases, then starts to increase again, because of increasing marginal costs we saw in Table 1 (and 3).

Y

AFC

AVC

ATC

0

0

1

2

50

3

33

4

25

5

20

Table 4: Average Fixed, Variable

and Total Cost

FC = 100

Table 5: Putting them all together

"~" = "undefined"

Y

AFC

MC

AVC

ATC

0

0

1

2

50

3

33

4

25

5

20

Now we can see and in fact already know from above that AFC falls as more inputs are employed, because AFC = fixed cost divided by number of units of output, where the denominator is increasing; AVC is flat and then begins to rise as marginal costs start increasing, and short-run average total cost falls at first as AFC falls with increasing production (denominator in AVC = (the sum of MC)/# of units Y), but then begins to rise because return from adding more inputs start to fall (in this model). ATC is the sum of average fixed and average variable cost; ATC must therefore always be above both AVC and ATC curves. Again, the firm wants to minimize cost and maximize revenue, and will produce the number of units of Y where the slope of ATC = zero.

This is a simplification in many ways. Rarely does production take a single variable input, although it can (Reynolds, 1971 p. 16); producing the primary output often generates a derivative secondary product (Henderson, 1958 p. 67); we are considering a single firm under perfect competition rather than a monopoly or oligopoly, i.e. The firm is a price taker and there is unrestricted entry, which very rarely happens to the degree we discuss it in theory (Knight, 1967 p. 81); and while we'd need to introduce a complex production function to describe more inputs, we can consider multiple variable inputs all at once and call it marginal cost for ease of presentation. Robinson sequentially eases these constraints and describes the different relationships the cost curves may take in the quasi- and very long run (1959 p. 48-49). Keynes discussed increasing marginal cost driven by and affecting wages as employment approaches full employment (1973, p. 289). Bober explores the limitations to conditions where diminishing returns is applicable, and offers alternative interpretations (2001 p. 22-23) for a principle that has become so central to the theory of production and cost as to be "dignified by being alluded to as a law, since in theory and practice it is inviolable" (Braff, 1969 p. 51).