Consumption goods, the things that provide individuals with utility, are the result of a process that combines raw materials and talent into a finished good. A common perception of production is that it involves physical inputs and a physical output. Oil is refined to produce gasoline. Steel, plastic and rubber are combined with labor to give us automobiles. Or water, seed, fertilizers, land and labor are used to produce agricultural crops. But production is much more. In the growing service sector of the economy, individual talent and time are the principle inputs. Lawyers, economists and other consultants "produce" advice and counsel. Restaurants provide meals that individuals do not need to cook themselves. At the same time, the individual enjoys sustenance, entertainment and time not spent cooking. Although there is no lasting physical output, a good has been produced which gave utility to the person doing the consuming.
The Theory of Production deals with this process from an abstract perspective often removed from the physical and mechanical phenomena usually thought of as production. It works to develop general rules by which any good, whether it is a physical product, a service, or a combination of the two, can be produced in a cost efficient manner.
Before moving on, a word about the "production of economics." Economists as a
group are efficient individuals. That means we try to conserve effort whenever possible.
Moreover, as a group we are fairly bright. If something works well in one use we look for
additional uses. Such is the case with production theory. You will find, with minor
modifications, that the theory of production is very similar to utility theory. There are
production analogues to utility (output), marginal utility (marginal products), utility
maximization (cost minimization), and just about every other concept we addressed in
utility theory. Mostly, we use the old tools, but give them new names. If you look for
these similarities, and use the relationships we developed for utility theory but apply
the new names, the theory of production should come easy.
During the first quarter of 1991 the U.S. automobile industry, like automobile manufacturers from around the world, faced diminished demand. Uncertainty about the economy, recessionary trends, and the residual effect of the Persian Gulf war left most automobile producers with a large inventory of cars, and the need to cut production and costs. A series of articles in The Wall Street Journal detailed the problems and choices facing car companies.
On February 22, 1991 The Journal reported that the Ford Motor Company would have only three of its 14 domestic assembly plants operating at full capacity. Nine plants would shut down completely, and the remaining two would run just one shift per day. Approximately 22,000 workers would be idled, and about 41,000 fewer vehicles than were previously planned would be produced. General Motors and Chrysler reported similar cuts. Overall, domestic automobile production was down 14.8 percent from the previous year.
Other articles showed how salaried workers, as well as production line workers, were being affected. Ford wanted to cut costs some $3 billion dollars, but found many cost cutting efforts that would effect line workers were prohibited by union contracts. Instead, Ford laid off white collar workers (March 1, 1991 in The Wall Street Journal). Similarly, Chrysler reduced its salaried employees by 3000 workers, and GM planned a reduction of 15,000 white collar workers (WSJ, 2/15/91). At the same time the automobile companies thought seriously about the long term prospects of keeping all of their manufacturing plants.
This discussion of cut-backs in the automobile industry engenders two concepts of primary importance in the Theory of Production. First is the idea that production is the procedure of taking inputs and combining them through some technological process to produce a final output, and that process is often complicated and convoluted. It may take many levels of labor and capital, along with specific skills, to produce the final good.
In the automobile industry, for example, producing one car requires raw materials like steel, plastic and rubber, and assembly-line labor. But more also is needed. The capital equipment (machinery and buildings) that is the assembly line, management at the plant and higher up, and marketing all go into producing the "good" that is an automobile. There are many types of inputs; labor, raw materials, physical capital equipment, and engineering, managerial and marketing know-how.
The second point is that the inputs can be combined in many different ways. It is often possible to substitute one input for another. The Journal reported that two of Ford's plants would run single shifts. Quite likely, the plants were tooled to produce different types of automobiles, so running a single plant with two shifts would not give the desired mix of styles. But to some degree, it is possible to substitute capital for labor, or labor for capital, and get the desired output. Alternatively, better or more intensive management (the use of white collar workers) may make it possible to curtail the number of assembly-line workers without cutting production.
Economists represent the (usually) complicated process of production with the production function. The production function is a mathematical or graphical representation that relates the quantity of a good that will be produced in a given period of time when inputs are combined using a technological process. Simply put, the quantity of output per period, Q, is a function of the amounts of inputs per period used. In functional form Q=f(x1,x2,...,xn) where the xi indicate the amounts of different inputs that are used in the production process. In the automobile example, x1 could be assembly-line labor, x2 could be capital equipment, x3 could be steel, and x4 could be managerial labor.
You probably realize already that the production function is similar to the utility function we used in chapter 4. However, there is a very significant difference. The utility function gives an ordinal measure of utility. But the production function is a cardinal measure. If one set of values for the xi result in Q=15, and another set of xi give Q=30, then the second output level is twice as large as the first. Thus, we can compare the magnitudes of different outputs as well as the relative ordering.
An important component of the production function is the listing of inputs. All necessary and potential inputs are arguments in the production function, just like the utility function listed all the possible consumer goods as arguments. As we did in utility theory, for the most part we will simplify the production model to a two input world, and call these inputs, generically, capital, denoted K, and labor, denoted L. This is commonly written Q=f(K,L).
In its indefinite form the production function specifies possible technological tradeoffs during the process which combines the inputs into a final good. Engineers tend to focus on the detailed relationships that may exist between inputs. Economists tend to be not so exacting. While both professions spend time trying to estimate production functions, economists will be more general in the specification of relationships, and also tend to aggregate inputs into broader classifications. One thing to keep in mind is that the function f(x1,...,xn) specifies all the technologically efficient input combinations. By technical efficiency we mean that if the value of one of the arguments is reduced, and the other arguments are held fixed, then output will decrease. In short, the production function requires that there be no extraneous or wasted inputs.
There are numerous acceptable forms for f(x1,...,xn). Later we will specify some characteristics that economists have generally agreed apply almost universally to the production process. However, the form of f(x1,...,xn) carries information about two very important characteristics of the production process. First, it tells us how easy it is to substitute one good for another.
At one extreme is an almost unlimited ability to substitute different inputs. Suppose, for example, a two input production process has the linear form Q=K+2L, where K is capital and L is labor. Initially let Q=20, K=10 and L=5. If we reduce K by two units but add 1 unit of L, output would remain the same. And, as long as we constrain K and L to be nonnegative (negative inputs really do not have a meaning), we can continually make this substitution. Capital and labor substitute at the ratio of 2 units of capital for each unit of labor in the production process. If instead Q=K+4L, the ratio for substituting capital for labor would be 4 to 1, but again would remain constant.
Linear production functions commonly arise when inputs are perfect substitutes, that is, the inputs are completely interchangeable. For example, if economic consulting output is classified as hours spent preparing analysis and advising clients, then C=M+F, where C is the amount of consulting produced, M the amount of male labor used, and F the amount of female labor used. In this case, male and female labor are perfect substitutes in the productive process, and tradeoff at the rate of one male hour for one female hour.
The other extreme allows no substitutability. We generally use the functional form Q=min(aK,bL) to indicate that output, Q, is the smaller of the terms in the parentheses. Thus, if aK<bL, Q=aK. Of course, if aK>bL, then Q=bL. This is called a fixed proportion production function, because efficiency requires that the inputs be used in the fixed proportion so that K/L=b/a. Fixed proportion production functions occur when it is not possible to substitute inputs. If, for example, the perfect martini requires 6 parts gin to 1 part vermouth, then you cannot produce one by using 5 parts gin to 2 parts vermouth. The proportions would be wrong, and you would not have the perfect martini.
Most production processes fall within the extremes of substitutability. Sometimes it is easy to substitute one input for another, and at other times substitution is difficult. Often it will depend on the amount of each input that is being used.
A common production function is the Cobb-Douglas function, which has the form Q=AKaLb, where A, a and b are all parameters. Suppose A equals 1, and a and b both equal 1/2. Then for an output of 100, we could have K and L both equal 100, or we could have K=25 and L=400, or K=16 and L=625. From the original point of (100,100) it takes 300 units of labor to substitute for 75 units of capital (to get to (25,400)), a ratio of 4 to 1. But to move to the next point (16, 625) requires that 225 more units of labor be used to substitute for just 9 units of capital, a ratio of 25 to 1. As the amount of capital gets smaller, the labor needed to substitute if we reduce capital by an additional unit is increasing. The amount of labor needed to replace a unit of capital if output is to remain constant is called the marginal rate of technical substitution (MRTS) of labor for capital. As the amount of an input gets small, the MRTS of the other input for the first input increases. Notice the similarity between the MRTS and the marginal rate of substitution from utility theory. We will talk about the MRTS more later in this chapter.
Although we often think of production processes as technologically fixed relationships
that require inputs to be used in fixed proportions, as we saw in the operating room
example in Box 1, usually it is possible to make incremental changes in the amount of
inputs used. It may be possible to do a heart procedure with one fewer nurse, but it will
take more operating room time. Steel manufacturers may add more capital in the form of
robots, or hire more workers. In an office, there are some tradeoffs possible between
secretarial and support staff and personal computers given to managers and analysts.
Small changes in the level of inputs will change the total output produced. The change in output that comes from a small change in the level of one input, holding the levels of all other inputs constant, is called a marginal product, denoted MP. Formally, let L1 and K1 indicate initial levels of two inputs, labor and capital. Then Q1=f(L1,K1) would be the initial level of output. If we let the utilization of L increase by a small amount, say D L, so L2=L1+D L then output changes to Q2=f(L2,K1). Note that neither K nor f changed, indicating "all else is constant." The marginal product of L (MPL) is (Q2-Q1)/(L2-L1). The numerator of this value is the change in total output, and the denominator is the change in the amount of L used. Thus, MPL=D Q/D L or in calculus, MPL=¶Q/¶L.
Technology determines the characteristics of the marginal productivity of an input. Often the marginal product of one input will depend on the amounts of other inputs that are also used. For example, when digging ditches, adding another laborer will give the largest increase in total product, and thus have a higher marginal product, if a shovel is available for the laborer. If instead she must share a shovel with another worker, her marginal product would be smaller.
Examples of how capital can augment productivity abound. In the late 1980s and early 1990s the Wm. Wrigley Jr. Co., makers of Wrigley's chewing gums, upgraded its equipment by installing superior machines for wrapping and other tasks. The result was an increase in labor productivity of 30 percent (The Wall Street Journal, 5/29/91, p.A6).
Eventually we expect marginal products to decrease. As more and more of an input is used in a productive process, holding the amounts of other inputs constant, the marginal product of the last unit added will get smaller. When marginal productivity is decreasing it is termed Diminishing Marginal Products. The intuition of diminishing marginal productivity stems from such various sources as specialization of inputs, differing abilities, and crowding effects.
Often, specialization of inputs will improve marginal productivity up to a point. Machines built for a particular job are more efficient at accomplishing that job than machines which have more general uses. So we see different types of shovels, hammers, and trucks. Labor is especially adept at specialization. One idea behind assembly lines is that workers can become particularly adept at one particular task, thus increasing their productivity. Of course, over-specialization is possible, leading to boredom and discontent that eventually leads to diminished productivity on the part of added labor.
Differing abilities and crowding also can lead to diminishing marginal productivity. If the most capable laborer is used first, then adding additional workers will require using less able individuals who cannot produce as much. The marginal product of labor falls as more workers are used. Crowding can cause inefficiencies that hurt productivity if too many workers are used in a single area.
Graphically, the relationship between marginal product and total product is given in figure 7.1. Holding constant the amount used of other inputs, as the quantity of labor, L, increases, the marginal product of labor initially increases because of specialization, the quality of labor, and other reasons discussed above. This continues up to L1. From L1 to L2 the marginal product of labor is positive but decreasing. The negative slope of the marginal product curve beyond L1 indicates that it is point at which diminishing marginal productivity of labor begins. At L2 the marginal product of labor becomes negative. Adding more workers beyond L2 will actually lower total output.
The graph of marginal product is derived from the total product curve. In fact, the marginal product is the slope of the total product curve. Thus, when marginal product is increasing, the total product curve is upward sloping and getting steeper. As diminishing marginal product of labor begins, at L1, there is a inflection point in the total product curve, which becomes flatter and eventually curves downward as MPL becomes negative.
..
.
.
There is one other productivity curve of interest, the average product of labor (APL)
curve. Figure 7.2 shows how APL is related to MPL. The average product of labor is the
total product divided by the amount of labor used. That is, APL=TP/L. It can be measured
as the slope of the line connecting the origin to the total product curve. In figure 7.2,
APL is increasing until L3 units of labor are used, at which point APL starts
falling. L3 is noted on figure 7.1 to show you this point. The average product
of labor reaches its highest point when it equals the marginal product of labor, as shown
in figure 7.2.
A numerical example will help understand the relationship between the amount of an input used and the measures of productivity. Table 7.1 gives some example values. For a given amount of capital the marginal product of labor first increases, until 5 units of labor are used, and then starts to decrease, for labor units 6 and above. But the APL continues to increase through the sixth unit of labor, and does not start to fall until the seventh unit of labor is utilized, at which point the MPL is less than the APL. Notice that the total product, the output, keeps increasing as long as the MPL is positive.
Table
7.1 An Example of Labor Productivity
|
Labor Units used |
Total Product (output) |
Average Product |
Marginal
Product |
|
0 |
0 |
0 |
- |
|
1 |
2 |
2 |
2 |
|
2 |
5 |
2.5 |
3 |
|
3 |
9 |
3 |
4 |
|
4 |
14 |
3.5 |
5 |
|
5 |
19 |
3.8 |
4 |
|
6 |
23 |
3.8 |
4 |
|
7 |
26 |
3.7 |
3 |
|
8 |
28 |
3.5 |
2 |
You might think that an increase in productivity is always a good thing. When measured as we did, as the incremental value of additional units of input, it probably is. But you should keep in mind that we have been measuring (total, marginal and average) productivity holding all else constant, especially the amount of other inputs. Failure to keep other things the same can cause confusing and misleading conclusions. Aggregate measures of productivity provide an example.
In the third quarter of 1990 the government reported that American workers' productivity climbed at the fastest rate in two years. Economists, however, saw this as a bad sign, because it came about from a drop in the number of hours worked. The productivity increase indicated that firms were making do with fewer workers, thus employment was being cut.
The government measure of productivity is output per hour worked. For the period from
July to September 1990 it was 1.6 percent higher than in the previous three months.
However, the number of hours worked fell 0.1 percent for the same period, and that was why
productivity had gone up so fast. (From the Spokesman Review, p.A11, 11/7/90)
Up to this point we have allowed labor to change but held capital fixed. In such a situation labor is called the variable factor of production, and capital is called the fixed factor. When at least one input is fixed the producer is said to be in the short run. Over a longer planning horizon, when all inputs are variable, the producer is said to be in the long run.
It is possible that capital is variable with labor fixed. Union contracts that prohibit layoffs set up such situations. Then labor is a fixed factor too. And if capital is allowed to vary we can find total, average and marginal product curves for labor, we can find them for capital. The marginal product of capital is the change in output from using another unit of capital, holding labor constant. It presumes using more capital with no more labor.
In the long run both labor and capital can vary. Thus, we need a way to represent the choices a producer faces when all inputs are variable. If the technology allows input substitution we must be concerned with different combinations of inputs giving the same output. We have already approached this problem peripherally, but now confront it directly.
Recall that the production function, Q=f(K,L), indicates what total product will be for given amounts of capital and labor. Thus, if we hold Q fixed, we can determine the different combinations of K and L that will give the same output. A graph of all these combinations is called an isoquant. It is the production equivalent to the indifference curve we studied in chapter 4. An isoquant shows all the possible combinations of capital and labor that produce the same total output.
Figure 7.3 shows a typical isoquant. All points on each curve represent the same specific quantity of output. Thus, Q'=f(K1,L1) and Q'=f(K2,L2). In fact, every point on the curve shows the same level of output, Q'. A curve further from the origin than Q', say as Q" as shown in figure 7.3, indicates a larger output. That is, Q" is larger than Q', signifying that any combination of inputs that lies on the curve labeled Q" gives more output than any combination of inputs on the curve Q'.
The shape of the isoquant indicates how easy or difficult it is to substitute inputs in
the production process. Earlier we noted that a linear isoquant shows a constant ability
to substitute inputs. Recall, for example, the production function Q=K+2L which we used
earlier in this chapter. Holding Q constant, we can rearrange this equation to K=Q-2L. For
each set level of output we can reduce capital by two units for every unit of L we use.
Figure 7.4 shows isoquants for this production function for Q=10 and Q=20.
. .
.
More commonly is the case where it is not possible to continually substitute inputs at a constant rate. This is the situation of a diminishing marginal rate of substitution. The extreme is the fixed proportion production function. Fixed proportion production functions have "L=shaped" isoquants, as shown in figure 7.5 for the function Q=min(2K,L) for Q=10 and Q=20. Notice that the corner occurs where 2K=L=Q.
When substitution is possible, but increasingly difficult the isoquant will have a convex shape (like the ones showed in figure 7.3). Of course, curves that are more convex (curves closer to being "L=shaped") correspond to a technology which allows only a limited ability to substitute goods. The slope of the isoquant gives us this measure. Look at figure 7.6. Since points 1, 2 and 3 all are on the isoquant labeled Q, all three combinations of capital and labor give the same output. If the firm moves from the input combination at point 1 to the combination indicated by point 2, the change in labor is L2-L1=D L2 and the change in capital is K2-K1=D K2. Notice, labor is being substituted for capital. How much capital each additional unit of labor will displace is given by the slope of the isoquant, D K/D L. Of course, the negative of D K/D L is just the marginal rate of technical substitution mentioned earlier.
Denote the MRTS we just found as MRTS(K for L). It tells us how much K can be replaced by each additional unit of labor without affecting the rate of output. This helps us see several important characteristics of the MRTS. First, since the inverse of D K/D L is D L/D K, the MRTS(L for K) is simply the inverse of the MRTS(K for L). Second, the convex shape of the isoquant shows that the MRTS(K for L) is decreasing as we move downward to the right on the curve. In figure 7.6 the move from point 2 to point 3 makes an equal addition of labor as the move from point 1 to point 2. The points are positioned so D L3=L3-L2 is the same size change as D L2. But the new labor replaces a smaller amount of capital. D K3=K3-K2 is smaller than D K2. The MRTS(K for L) has diminished. This means that the amount of capital that another unit of labor will compensate for gets smaller and smaller as production uses more labor and less capital.
We can use the fact that the MRTS is the negative of the slope of the isoquant to see that diminishing MRTS is consistent with diminishing marginal products. MPL is represented on an isoquant graph by a shift across curves, holding capital fixed, as in figure 7.7. Originally output was Q1 being produced with K1 of capital and L1 of labor (point 1). If labor increases to L2, output will increase to Q2 (point 2). MPL=(Q2-Q1)/(L2-L1). Capital is fixed at K1. By diminishing marginal products we assume that the MPL of labor has decreased as labor increased from L1 to L2.
To get back to the original output of Q1 we can decrease capital. As shown, if capital falls from K1 to K2 (a move from point 2 to point 3) output falls from Q2 to Q1. Holding labor fixed at L2 now, the MPK=(Q1-Q2)/(K2-K1). Both the numerator and the denominator are negative, so the MPK is positive. By diminishing marginal products, MPK should be greater at K2 then at K1. The move from point 1 to point 3 brought with it a smaller MPL, from more labor being used, and a larger MPK, from less capital being used.
We can find this result directly from the formula for marginal product. The definition of the marginal product of labor is MPL=DQ/DL. Rearranging this equation gives that DQ=MPLxDL. Similarly, from the marginal product of capital DQ=MPKxDK. As we move along a single isoquant we know that DQ=0. That is, the change in output from changing labor must exactly offset the change in output from changing capital. Thus, MPLxDL+MPKxDK=0 along an isoquant. Rearranging this equation gives that -DK/DL=MPL/MPK. Since the left hand side of this equation is the MRTS, we find that the MRTS=MPL/MPK.
The location and shape of the isoquant is dependent on the technology of production.
Figure 7.8 shows two representative isoquants for production using capital and labor. In
panel a, the location of the curve is skewed upward to the left, and the isoquant is
somewhat steep. This indicates that labor is more important in the productive process than
is capital. If only a small amount of labor is used, say L1, an output of Q'
can be reached only if a lot of capital (K1) is also used. The steepness
indicates that a small increment in labor will offset a large amount of capital. But in
panel b the roles of labor and capital in the production function are reversed. A small
amount of capital will replace a lot of labor, but it would take a lot of labor to
supplant a little capital. So the isoquant in panel a shows a production function that
weights labor more heavily, while the one in panel b gives a greater weight to capital.
The isoquant offers a menu of capital and labor combinations that will produce the same output. Firms must choose between these possibilities. A major concern of firms when making a choice of inputs is the cost of each possibility. In this section we explore the problem of choosing the actual input combination.
Assume that the firm must pay a given amount for each unit of labor and each unit of capital it uses. Call the cost per unit of labor the wage, w, and the cost per unit of capital its rental rate, r. Then any particular capital-labor combination has a cost C given by C=wL+rK. This is a linear function where C is the cost of using a particular level of L and K. For any given C we can use this equation to find the different blends of capital and labor that will have the same cost. Again, you should recognize that this relationship is very similar to a budget constraint. The different combinations of capital and labor that produce the same cost describe an isocost curve.
By rearranging the isocost equation to K=(C/r)-(w/r)L we can graph it in the same space
as the isoquant. Notice, this states K as a function of L, with a K-intercept of C/r and a
slope of -w/r. The L-intercept is C/w. An isocost line is illustrated in figure 7.9. Just
as with an isoquant, the farther the isocost line is from the origin, the higher the cost.
Thus, in figure 7.10, C" is larger than C'.
.
.
.
The point of introducing the isocost line is to find the cost minimizing input combination for any given output. By superimposing isocost lines on a given isoquant, we can find the least expensive blend of capital and labor that will give that output. It will be the point on the isoquant that also lies on the isocost line closest to the origin.
It turns out that the cost minimizing input combination is found at the tangent between the isocost line and the isoquant. In figure 7.11 the isocost line for C4 is the lowest one shown. But it is not possible to produce Q' at that cost. The firm must spend at least C3, for inputs L3 and K3, if it wishes to produce Q'. If the firm were at point 2, which lies on the isoquant for Q', it could lower its cost by substituting capital for labor. From point 1, the firm could cut costs by substituting labor for capital.
Tangent curves have the same slope. The slope of the isocost curve is -w/r, and the slope of the isoquant is
-MRTS=-MPL/MPK. Thus, the least cost method of producing any given output requires that the input combination be such that the MRTS=MPL/MPK=w/r. We can rearrange this to MPL/w=MPK/r, which more generally says that the additional output obtained for the last dollar spent on each input should be equal.
It is easiest to understand this rule by looking at some possible adjustments if the rule is violated. Suppose for some output the marginal product of labor is 10 and the marginal product of capital is 40. Moreover, suppose capital costs two dollars per unit and labor costs one dollar per unit. If labor is reduced by one unit, total product would fall by 10 units, but one dollar is now available to spend on capital. Only 1/4 unit of capital, costing fifty cents, is needed to restore output to its original level (since the MPK=40, 1/4 of a unit gives an additional output of 10 units) so total cost can be reduced by fifty cents. Of course, since labor has decreased, the marginal product of labor will increase (by diminishing marginal productivity), and an increase in capital lowers the marginal product of capital. The net change is the same output for fifty cents less in cost. The opportunity to lower cost remains until MPL/w=MPK/r.
This rule for cost minimization applies no matter what the output and no matter how many inputs there are to production. In a three input process, with capital, labor and raw materials, M, where the cost per unit of M is p, the cost minimization rule is
MPK/r=MPL/w=MPM/p. The marginal adjustments hold as well.
Cost minimization helps explain why different countries, and even distinct areas in the same country, may use assorted processes to produce the same good, especially when the same technology is available to all producers. Lower relative labor costs in one area or country may make one technology more attractive, while lower relative capital costs favor another technology elsewhere.
We see this in figure 7.12. An isoquant for a particular output level is shown. In one country (country a) labor is expensive relative to capital. This is shown by the solid isocost curve, which has the slope -wa/ra. The least cost method of production calls for the input combination given at point a, at a cost of Ca. It uses little labor and much capital. Country b faces relatively inexpensive labor (the dotted isocost with slope -wb/rb). In country b the input combination that should be used is shown by point b, with a cost of Cb. Country b should use more labor and less capital than country a.
Note that we cannot say if the relative costs are different between the two countries. We know that wa/ra>wb/rb and that Ca/ra>Cb/rb (and that Ca/wa<Cb/wb), but we do not know the relationship between Ca and Cb.
In many less developed countries capital is relatively expensive and labor is relatively cheap. Thus, these countries will find it more cost effective to use production processes that employ a relatively large amount of labor. There is more manual labor used in road building, farming and construction. But in the United States, where labor is relatively expensive, firms find it more cost efficient to utilize relatively more capital.
The importance of relative input prices comes clear when looking at particular
products. Both capital and labor are absolutely cheaper in Brazil than the United States,
yet Brazilian steel manufacturers use more capital than do U.S. steel producers, simply
because the relative price of capital is lower. Similarly, changes in relative
input prices can explain why firms adopt new technologies. When personal computers were
new and relatively expensive, firms kept a high level of labor for support staff. As the
price of personal computers dropped machinery replaced labor. Scanning devices in grocery
stores illustrate this idea. A single checker can now handle more customers, and at the
same time the scan updates inventory, so less labor is needed than before.
Efficiency in production requires that the last dollar spent on each input give the same marginal product. Because of short run constraints, firms do not continually adjust on the margin to changing input prices. Instead, firms monitor changes in the relative prices of inputs. Any adjustments in long run production takes into account the input prices.
Changes in relative input prices pivot the isocost curve. In figure 7.13 we see how the optimal input combination changes with the prices. Originally a firm paid a wage of w1 for each unit of labor it used and a rate of r for each unit of capital. The optimal combination of inputs to produce an output of Q' is L1 units of labor and K1 units of capital. Now suppose the wage rate increases to w2 per unit, while the cost per unit of capital remains at r. The slope of the isocost curve increases since w1/r<w2/r, and the most efficient way to produce Q' is to use L2 units of labor and K2 units of capital. If the firm keeps output at Q', in the long run it will substitute capital for labor.
We can see that besides using more capital, the firm will also find producing Q' more expensive. The intercept of the vertical (capital) axis is the total cost divided by the price per unit of capital. Remember that r has not changed, so any shift is due to different values for the numerator, which is just total cost. Thus, C2 clearly exceeds C1.
All this discussion has been predicated on output being held at Q'. As we shall see in
the next chapter, however, if the cost of production changes the firm may change the
amount it produces. Thus, while we can compare the efficient input mix for various
input prices, we cannot say conclusively what will happen to the amount of each input a
firm employs. If the firm holds output constant it will substitute away from the input
which has had a relative price increase, and towards the input which has become relatively
cheaper (from labor towards capital in the example here), but it is by no means certain
that output will remain the same.
Previous sections have shown two possible production decisions. The first was to change output by varying the amount of a single input. The other had a focus of input substitutability, and was the basis for establishing optimal input combinations for any given output. Implicit for the latter case was, although all inputs were variable, if one increased the other decreased. But what would happen if all inputs increased? We would, of course, expect output to go up. When all inputs are varied in the same proportion the firm is said to have changed its scale. The technical relationship between output and the scale is measured by Returns to Scale. It tells us how output will change when all the inputs are changed in the same proportion. Returns to scale is an inherently long run concept since all inputs must be variable. It is associated with the efficient size of a firm.
Proportionate changes in inputs can be represented easily using the isoquant mapping. Figure 7.14 depicts two rays extending out from the origin in the capital-labor space we used for drawing isoquants. Any two points along a single ray indicate a scale change, that is, proportionate changes in the inputs. For example, along ray R1 a move from point 1 to point 2 will double both capital and labor. Similarly, moving from point 3 to point 4 on ray R2 increases the scale by 50 percent. The different rays represent distinct capital-labor ratios. As pictured, ray R1 indicates a higher K/L ratio than does ray R2.
If we show the isoquants, as we do in figure 7.15, it is possible to measure the
returns to scale. For simplicity we have left off ray R2. The isoquant that
goes through point 1 is for an output level of Q1, and the isoquant through
point 2 is for output Q2. We can now define three types of returns to scale.
These definitions depend on the relationship between Q1 and Q2.
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A production process is said to demonstrate increasing returns to scale if the change in output is more than proportionate to the change in inputs. Recall that point 2 uses twice the inputs as point 1 (scale has been doubled). If Q2 exceeds twice Q1 then we have increasing returns to scale.
Alternatively, the change in output can be less than proportionate. As depicted in figure 7.15, Q2 would be less than 2Q1. In this case the production process demonstrates decreasing returns to scale. As you might expect, if the change in output is proportionate to the change in inputs, we have what is termed constant returns to scale. Such would be the case if Q2 just equaled twice Q1.
Different production functions show assorted returns to scale. Suppose the production function is linear, for example Q=2K+3L. Then there are constant returns to scale. Similarly, the fixed proportion production function, Q=min(aK,bL) has constant returns to scale. But Q=KL is an increasing returns to scale production function, while Q=(KL)0.3 demonstrates decreasing returns to scale.
The returns to scale of a production function are not necessarily constant. Often as
crowding or managerial inefficiencies set in a production process will go from increasing
returns to scale to constant and eventually decreasing returns to scale. Moreover, the
capital-labor ratio can be a primary determinant of the returns to scale of a production
process. In figure 7.16 ray R2 is again shown. Point 5 shows an alternative
capital-labor combination that produces output Q1. Doubling the amounts of
inputs used will move the scale to point 6. As drawn, doubling the scale of operations
from 5 to 6 has less of an effect on output than does doubling the scale from 1 to 2.
Thus, in this case, a lower capital-labor ratio lowers the returns to scale of the
production process. If Q2 is twice Q1, then a capital-labor ratio of
K1/L1 has constant returns to scale, but a ratio of K3/L3,
which is smaller, has decreasing returns to scale, because point 6 is on a lower isoquant
than point 2.
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You might think that if this were the case, a smart firm would use the higher capital to
labor ratio. After all, increasing returns to scale seems to be a desirable property. But
we must be cognizant of the costs of inputs. As we saw above, for any given output the
input combination which minimizes costs is where the MRTS=w/L. In figure 7.17 we add
isocost curves. If the input prices are w and r as shown, the optimal input combination
for producing Q1 occurs at point 5. This is a capital-labor ratio that shows
decreasing returns to scale. Additionally, with a production function that has varying
returns to scale depending on the input ratio, the optimal capital-labor ratio may change
with output. Thus, the least cost method of producing Q2 is at point 7, which
is not a scale change from point 5. Capital and labor changed in different
proportions.
A Mathematical Interpretation of Returns to Scale
Functional notation offers a way to clearly represent scale changes. Let Q1=f(K,L). A scale change can be represented by a positive constant, c, by which we multiply each of the input values. If c>1 we are increasing the scale of operations, and if 0<c<1, scale is being cut. To simplify the explanation, we will assume c>1. Let Q2=f(cK,cL). Then mathematically returns to scale may be defined as follows:
if Q2>cQ1 then we have increasing returns to scale;
if Q2=cQ1 then we have constant returns to scale:
and
if Q2<cQ1 then we have decreasing returns to scale.
As an example look at the Cobb-Douglas production function which has the form Q=AKaLb.
Let K'=cK and L'=cL. Then Q'=AK'aL'b. By substitution, Q'=A(cK)a(cL)b
which reduces to ca+bAKaLb or Q'=ca+bQ. If
a+b<1, then ca+b<c, so this is a case of decreasing returns to scale.
Alternatively, a+b>1 indicates increasing returns to scale, and a+b=1 indicates
constant returns to scale.