The Theory of Production started with an analysis of the physical relationship between
inputs and outputs in the productive process. We used the Theory of Production to develop
general rules by which any good could be produced by efficiently utilizing inputs.
Efficiency required that all potential methods of production be organized so that reducing
the amount of any input would result in a diminished amount of output. This quality was
termed technical efficiency. It formed the basis for productivity curves and
isoquants.
You should remember that the short run is a situation when at least one input is fixed. For example, if the production function has the form Q=f(K,L), where K is fixed capital in the short run, then the only way to change output is by changing the variable factor, labor (L).
Figure 8.1 shows a short run production curve which relates total output to the amount of labor used. We note that short run production is dependent on the amount of capital being used by making the total product of labor curve (TPL) a function of a particular amount of capital, K*.
The problem we face is to relate this short run production curve to a short run cost curve. To do this, we first note that total cost is given by TC=wL+rK, where w is the price per unit of labor and r is the price per unit of capital. Since K is fixed, at K*, the short run fixed cost (FC) is rK*. Short run variable cost (VC) is the cost associated with labor, wL. As the amount of labor changes variable cost changes, as does output.
To derive the short run variable cost curve note that labor is measured along the horizontal axis in figure 8.1. If this axis is scaled by w (multiply labor by w) the TPL(K*) curve will be scrunched if w is less than 1 and stretched if w exceeds 1. But the basic shape of the curve is not changed. The horizontal axis now measures short run variable cost, while the vertical axis still measures output. The orientation of the graph shows what output would be with each level of variable cost.
.
To get to more standard orientation, we need simply rotate this graph 90 degrees,
and then flip it symmetrically over the vertical axis, which now measures w times labor,
that is variable cost. We rename the vertical axis cost, and the curve VC (for variable
cost) to reflect the new meaning, and get figure 8.3b. Notice that VC is a function of K*,
as was TPL.
The variable cost curves shown in figure 8.3b tells what the total variable cost
will be for any level of output. It is derived, as we just showed, directly from the short
run production curve. To find short run total cost we need to add the fixed cost
component. Fixed cost is rK*, which by definition does not change in the short run. Thus,
the fixed cost curve is a horizontal line at rK*. Total cost is fixed cost plus variable
cost, TC=FC+VC, so it is the vertical sum of each cost at each output, as shown in figure
8.5. For each output FC is the same, and VC+FC equals TC. Thus, the vertical distance
between the TC and VC curves is always equal to FC.
The inflection points of the variable and total cost curves have specific meanings. In the production curve (refer back to figure 8.1) at L1 the marginal product of labor, Q/L, reaches its maximum; it is the point at which diminishing marginal productivity of labor sets in. L3 was the labor usage beyond which the marginal product of labor became negative; using more labor actually lowered total output.
These points are reflected in the cost curves illustrated in figure 8.6. Up to an output of Q1 the slopes of both the variable cost and total cost curves were getting flatter. The slope of the variable cost curve, DVC/DQ, is called the marginal cost, denoted MC. It tells how variable cost will change as output changes. And since in the short run the only thing that can change total cost is variable cost, MC also equals DTC/DQ. (Notice that TC=FC+VC. As output varies TC changes only because VC does. FC is constant. Thus, TC=VC.) At Q1, then, marginal cost begins to increase. And at Q3, where negative MPL first occurred, MC, the slope of the tangent to VC and TC, becomes infinity.
We can use some simple math to more directly relate marginal cost and the marginal product of labor. By definition, MPL=DQ/DL, which can be rearranged to Q=MPLxL. Marginal cost is given by MC=DTC/DQ=dMC/dQ. In the short run, with fixed input prices, the only way total cost can change is by changing the amount of labor used. That is, TC=wL. Substituting for TC and Q gives the relationship MC=w/MPL.
When a firm produces a good it often wants to know the cost per unit of output. Economists term this cost average total cost (ATC), where ATC is simple TC/Q. Average total cost is often simply called average cost. By decomposing total cost into variable and fixed costs, we can derive the family of average cost curves. That is, note that TC=VC+FC, and write ATC=(VC+FC)/Q=(VC/Q)+(FC/Q). The first term is average variable cost (AVC), and is the variable cost per unit of output. Average fixed cost (AFC), the second term, is the fixed cost per unit of output. So ATC=AVC+AFC.
Average variable cost has an inverse relationship to the average product of labor,
similar to the relationship between marginal cost and marginal product of labor. APL=Q/L,
while AVC=wL/Q. Notice, by substitution, AVC=w/APL. So if the APL is increasing, the AVC
must be decreasing, and vice versa.
The isoquant-isocost analysis offers an alternative approach for looking at short run costs and productivity. Figure 8.7 shows how. Suppose a firm is originally producing Q0 in an economically efficient manner. You should recall that this means the isoquant for Q0 is tangent to the isocost at that point. It is the least cost method for producing Q0, and thus the average cost of producing Q0 is lower using the combination (L0,K0) than any other point on that isoquant.
But suppose the firm needs to decrease its output, say to Ql. In the short run, the only way to do so is by lowering labor, to Ll. Total cost falls to TCl, which is less than TC0. Similarly, if the firm needs to increase its output to Qh, it can only do it by increasing labor to Lh. In both cases, the firm's only option is to produce inefficiently in the short run.
The different points that a firm will use at various levels of production is called the expansion path. In the short run, because capital is fixed, the expansion path is horizontal at K0. As with cost curves, there is a different short run expansion path associated with each level of capital, but they are all horizontal.
One thing we cannot tell from this graph is what has happened to average cost.
Depending on the production returns to scale, TCh/Qh may be less
than, equal to, or greater than TC0/Q0. The same uncertainty applies
to TCl/Ql.
.
As recession and new loan defaults gripped North Carolina in 1989 and 1990, two manufacturers of pre-fabricated homes saw conspicuous decreases in the demand for their products (WSJ, 4/4/91). Sales decreased, prices fell, and the firms found themselves on the verge of bankruptcy. Manufactured Homes, Inc., tried to rely on staying power. It kept its operations fundamentally intact and found itself essentially insolvent by early 1991. Yet the other firm, Oakwood Homes Corporation, survived the downturn to record a 93 percent increase in earnings during fiscal year 1990 by radically altering its operations. The different results of these similar firms turned on the ability to adjust overall production measures quickly. It provides an illustration of long run versus short run costs.
You should recall that in the long run all factors of production are variable. This enables firms to move to the lowest cost method of production for any given output, unlike the short run where a firm may find itself to produce at an inefficient point. In figure 8.7 the short run expansion path showed that only if output was Qo was it least cost. With capital fixed at Ko, producing outputs of Qh or Ql forced the firm to be inefficient. It could lower the cost of Qh by increasing capital and decreasing labor, while Ql could be produced for less by using more labor than Ll and less capital than Ko. Of course, if capital was at a lower level, say Kl in figure 8.8, the short run expansion path (the dotted horizontal line) shows that Ql could be produced in a least cost manner, but the cost of producing both Qo and Qh is higher than if Ko capital were available.
.
Figure 8.8 indicates that for every given level of capital there exists a set of short run
cost curves, each of which shows the efficient combination of inputs for a particular
output, but which also indicates the cost of producing other outputs with that capital.
We translate the short run costs implied by the isoquant mapping in figure 8.8 to cost curves in figure 8.10. Each level of capital traces out an average cost curve. For the low output, Ql, the lowest average cost is achieved on the cost curve generated by Kl amount of capital, hence the label AC(Kl). Similarly, the lowest average costs for Qo and Qh are achieved if Ko and Kh levels of capital are used, respectively.
More generically, figure 8.11 gives a family of short run cost curves, each dependent
on the amount of capital available. The curves shown (SAC1 to SAC4)
are for four specific levels of capital. Other levels of capital would give different
short run cost curves. In fact, with infinitesimal changes in the amount of capital we
could generate a continuous set of short run average cost curves. The lower bound of all
these short run average cost curves, called the envelope of the curves, gives the long
run average cost curve (LAC), shown by the dotted line in figure 8.11.
An alternative way of finding the long run average cost curve is to use the expansion path that comes from the efficient points of production, as illustrated on the isoquant-isocost graph. The long run expansion path is the locus of input combinations a firm would use in producing different levels of output in the long run, that is, when all inputs are variable. Since firms would want to produce each output at the lowest cost, this would just be the collection of efficient points of production, as illustrated in figure 8.15. By dividing the indicated (lowest possible) cost for each output level by the output, the LAC curve can be mapped.
The shape of the expansion path tells how efficient technology will adjust as output
increases. Figure 8.15 shows a technology that would require a decreasing capital to labor
ratio as output increases if average cost is to be kept to a minimum. If the expansion
path curved upward, then the efficient ratio of capital to labor would increase as output
increased. When the efficient technology requires a constant capital labor ratio, for
example when production is with a fixed proportion production function, the long run
expansion path will be a straight line, the slope of which equals the efficient
capital-labor ratio.
Earlier we explored the technical relationship between output and the scale of operations by Returns to Scale. You should recall that returns to scale relates the change in output to a proportionate change in the amount of inputs used. If output increases more than proportionately, there are increasing returns to scale. Proportionate or less than proportionate changes in output indicate constant returns to scale and decreasing returns to scale, respectively.
A concept closely related to returns to scale is economies of scale, which is the cost equivalent analysis relating changes in output and average cost. In a fundamental way, economies of scale is simply an analysis of the long run average cost curve.
When the long run expansion path is a straight line, as shown in figure 8.16, relating returns to scale and economies of scale is easy. Look at the expansion path as output increases from Q1 to Q2. Point 2 lies exactly twice as far from the origin as point 1, indicating that the scale has doubled. Thus, the isocost curve that passes through point 2 is for twice the cost as the isocost curve passing through point 1, as indicated by using 2C1 as the numerator of the intercepts for the line through point 2.
We can now see how, with the linear expansion path, returns to scale relate inversely to economies of scale. The cost of moving from Q1 to Q2 have doubled, so what happens to average cost will depend on how output has changed. For an output level of Q1, average cost, AC1 is just C1 divided by Q1, that is, AC1=C1/Q1. Similarly, average cost at Q2, indicated by AC2, is given by the formula AC2=2C1/Q2. If Q2=2Q1, which means there are constant returns to scale, then AC2=2C1/2Q1=C1/Q1=AC1. When long run average cost is constant over a range of output we term this constant economies of scale. Similarly, if Q2<2Q1, meaning there are decreasing returns to scale, AC2 exceeds AC1, termed diseconomies of scale. Finally if Q2>2Q1, there are increasing returns to scale and average cost is decreasing. Such deceases in average cost are known as economies of scale. Because these are scale changes, so capital is not fixed, we know this is a long run change, not short run.
Keep in mind that the straight line expansion path means that the economically efficient ratio of inputs, that is the capital to labor ratio that minimizes the cost of producing any output, is constant. If the efficient ratio of capital to labor is changing, so the expansion path is curved, our analysis is only slightly more complicated.
Figure 8.17 replicates the isoquant-isocost graph of figure 8.16, but allows the expansion path to curve. In this case, efficient production requires that an increasing amount of labor per unit of capital be used as output gets large. Except for this change, the analysis of economies of scale is the same. Again the isocost lines were drawn so the cost has doubled, with wL2+rK2, the total cost of producing Q2, being twice wL1+rK1, the total cost of producing Q1. So if Q2>2Q1, there are increasing returns to scale and decreasing average cost. Corresponding results hold for decreasing and constant returns to scale.
The difference is that we no longer have a true scale change. As drawn, with the
expansion path curving downward, labor has increased at a greater rate than capital. A
true change of scale would be along a straight ray from the origin through point 1 and to
the outer isocost, intersecting it at point 3. Labor would increase less, to L3
instead of L2, and capital more, to K3 not K2. But
production at point 3 would not be economically efficient in the long run. When expansion
paths are curved, we lose the clear inverse relationship between returns to scale and
economies of scale.