No Such Thing as a Free Lunch: Principles of Economics (Part 35)

Economics is a strange science. Our subject deals with some of the most important as well as mundane issues that impinge on the human condition.

Dale T. Mortensen

 Short-Run Costs and Output Decisions

(Part B)

by

Charles Lamson

 Variable Costs

Total Variable Cost (TVC) Total variable cost (TVC) is the sum of those costs that vary with the level of output in the short run. To produce more output, a firm uses more inputs. The cost of additional output depends directly on the additional inputs that are required and how much they cost.

As you saw in Parts 30-33 of this analysis, input requirements are determined by technology. Firms generally have a number of production techniques available to them, and the option they choose is assumed to be the one that produces the desired level of output at the least cost. To find out which technology involves the least cost, a firm must compare the total variable costs of producing that level of output using different production techniques.

This is as true of small businesses as it is of large manufacturing firms. Suppose, for example, that you are a small farmer. A certain amount of work has to be done to plant and harvest your 120 acres. You might hire four farmhands and divide up the task, or you might buy several pieces of complex farm machinery (capital) and do the work single-handedly. Your final choice depends on a number of things. What machinery is available? What does it do? Will it work on small fields such as yours? How much will it cost to buy each piece of equipment? What wage will you have to pay farm hands? How many will you need to get the job done? If Machinery is expensive and labor is cheap, you will probably choose the labor-intensive technology. If farm labor is expensive and the local farm equipment dealer is going out of business, you might get a good deal on some machinery and choose the capital-intensive method.

Having compared the costs of alternative production techniques, The firm may be influenced in its choice by the current scale of its operation. Remember, in the short run a firm is locked into a fixed scale of operations.  A firm currently producing on a small scale may find that a labor-intensive technique is the least costly, whether or not labor is comparatively expensive. The same firm producing on a larger scale might find a capital-intensive technique less costly.

The total variable cost curve is a graph that shows the relationship between total variable cost and the level of a firm's output (q). At any given level of output, total variable cost depends on (1) the techniques of production that are available and (2) the prices of the inputs required by each technology. To continue to examine this relationship in more detail, let us look at some hypothetical production figures.

Table 2 presents an analysis that might lie behind three points on a typical firm's total variable cost curve. In this case, there are two production techniques available, A and B, one somewhat more capital intensive than the other. We will assume that the price of labor is $1 per unit and the price of capital is $2 per unit. For the purposes of this example, we focus on variable capital---that is, on capital that can be changed in the short run. In practice, some capital (such as buildings and large, specialized machines) is fixed in the short run. In one example, we will use K to denote variable capital. Remember, however, that the firm has other capital, capital that is fixed in the short run.

Analysis reveals that to produce one unit of output, the labor-intensive technique is least costly. Technique A requires 4 units of both capital and labor, which would cost a total of $12. Technique B requires 6 units of labor but only two units of capital for a total cost of only $10. To maximize profit, the firm would use technique B to produce one unit. The total variable cost of producing one unit of output would thus be $10.

The relatively labor-intensive technique B is also the best method of production for 2 units of output. By using B, the firm can produce 2 units for $18. If the firm decides to produce 3 units of output, however, technique A is the cheaper. By using the least-cost technology (A), the total variable cost of production is $24. The firm will use three units of capital at $2 each and 6 units of labor at $1 each. 

Figure 3 graphs the relationship between variable cost and output based on the data in Table 2, assuming the firm chooses, for each output, the least-cost technology.

The total variable cost curve embodies information about both factor, or input, prices and technology. It shows the cost of production using the best available technique at each output level given current factor prices.

Marginal Cost (MC) The most important of all cost concepts is that of marginal cost (MC), the increase in total cost that results from the production of one more unit of output. Let us say, for example, that a firm is producing 1000 units of output per period and decides to raise its rate of output to 1001. Producing the extra unit raises costs, and the increase---that is, the cost of producing the 1001st unit is the marginal cost. Focusing on the "margin" is one way of looking at variable costs: Marginal costs reflect the changes in variable costs because they vary when output changes. Fixed costs do not change when output changes.

Table 3 shows how marginal cost is derived from total variable cost by simple subtraction. The total variable cost of producing the first unit of output is $10. Raising production from 1 unit to 2 units increases total variable cost from $10 to $18; the difference is the marginal cost of the second unit, or $8. Raising output from 2 to 3 units increases total variable cost from $18 to $24. The marginal cost of the third unit, therefore, is $6.

It is important to think for a moment about the nature of marginal cost. Specifically, marginal cost is the cost of the added input, or resources, needed to produce one additional unit of output. Look back at Table 2, and think about the additional capital and labor needed to go from one unit to two units. Producing 1 unit of output with technique B requires 2 units of capital and 6 units of labor; producing 2 units of output using the same technique requires 4 units of capital and 10 units of labor. Plus, the second unit requires 2 additional units of capital and 4 additional units of labor. What then is the added, or marginal, cost of the second unit? Two units of capital cost $2 each ($4 total) and four units of labor cost $1 each (another $4), for a total marginal cost of $8, which is exactly the number we derived in Table 3.

While the easiest way to derive marginal cost is to look at total variable cost and subtract, do not lose sight of the fact that when a firm increases its output level, it hires or demands more inputs. Marginal cost measures the additional cost of inputs required to produce each successive unit of output.

The Shape of the Marginal Cost Curve in the Short Run The assumption of a fixed factor of production in the short run means that a firm is stuck at its current scale of operation (in our example, the size of the plant). As a firm tries to increase its output, it will eventually find itself trapped by that scale. Thus, our definition of the short run also implies that marginal cost eventually rises with output. The Firm can hire more labor and use more materials---that is, it can add variable inputs---but diminishing returns eventually set in.

Recall the sandwich shop, with one grill and too many workers trying to prepare sandwiches on it, from Part 33 of this analysis. With a fixed grill capacity, more laborers could make more sandwiches, but the marginal product of each successive cook declined as more people tried to use the grill. If each additional unit of labor adds less and less to total output, it follows that it requires more labor to produce each additional unit of output. Thus, each additional unit of output costs more to produce. In other words, diminishing returns, or decreasing marginal product, implies increasing marginal cost (Figure 4).

Recall too the accountant who helps people file their tax returns. He has an office in his home and works alone. His fixed factor of production is that there are only 24 hours in a day and he has only so much stamina. In the long run, he may decide to hire and train an associate, but in the meantime (the short-run) he has to decide how much to produce, and that decision is constrained by his current scale of operation. The biggest component of the accountant's cost is time. When he works, he gives up leisure and other things that he could do with his time. With more and more clients, he works later and later into the night. As he does so, he becomes less and less productive, and his hours become more and more valuable for sleep and relaxation. In other words, the marginal cost of doing each successive tax return rises.

To reiterate, in the short run, every firm is constrained by some fixed input that (1) leads to diminishing returns to variable inputs and (2) limits its capacity to produce. As a firm approaches that capacity, it becomes increasingly costly to produce successively higher levels of output. Marginal costs ultimately increase with output in the short run. 

Notice that up to 100 units, marginal cost decreases and the variable cost curve becomes flatter. The slope of the total variable cost curve is declining---that is, total variable cost increases, but at a decreasing rate. Beyond 100 units of output, marginal cost increases and the total variable cost curve gets steeper---total variable costs continue to increase, but at an increasing rate.

A more complete picture of the costs of a hypothetical firm appears in Table 4. Column 2 shows total variable costs---derived from information on input prices and technology. Column 3 derived marginal cost by simple subtraction. For example, raising output from 3 units to 4 units increases variable costs from $24 to $32, making the marginal cost of the fourth unit $8 ($32 - $24). The marginal cost of the 5th unit is $10, the difference between $32 (TVC) for 4 units and $42 (TVC) for five units.

Average Variable Cost (AVC) Average variable cost (AVC) is total variable cost divided by the number of units of output (q):

In Table 4 we calculate AVC in column 4 by dividing the numbers in column 2 (TVC) by the numbers in column 1 (q). For example, if total variable cost of producing 5 units of output is $42, then the average variable cost is $42 divided by 5, or $8.40.

Marginal cost is the cost of one additional unit. Average variable cost is the total variable cost divided by the total number of units produced.

Graphing Average Variable Costs and Marginal Costs The relationship between average variable cost and marginal cost can be Illustrated graphically. When marginal cost is below average variable cost declines toward it. When marginal cost is above average variable cost, average variable cost increases toward it.

If you follow this logic you will see that marginal cost intersects average variable cost at the lowest, or minimum, point of AVC.

An example using test scores should help you to understand the relationship between MC and AVC. Consider the following sequence of test scores: 95, 85, 92, 88. The average of these four is 90. Suppose you get an 80 on your 5th test. The score will drag down your average to 88. Now suppose that you get an 85 on your 6th test. The score is higher than 80 but it is still below your 88 average. As a result, your average continues to fall (from 88 to 87.5), even though your marginal test score rose. If instead of an 85 you got an 89---just one point over your average---you have turned your average around; it is now rising. 

*MAIN SOURCE: CASE & FAIR, 2004, PRINCIPLES OF ECONOMICS, 7TH ED., PP. 154-159*

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