chapter inputs and production functions - fep.up · pdf fileintroduction to inputs and...

6.4S U B S T I T U TA B I L I T Y A M O N G I N P U T S

6.1I N T R O D U C T I O N T O I N P U T S A N D P R O D U C T I O N F U N C T I O N S

APPLICATION 6.1 Competition Breeds Efficiency

6I N P U T S A N DP R O D U C T I O N

F U N C T I O N S

C H A P T E R

6.2P R O D U C T I O N F U N C T I O N S W I T H A S I N G L E I N P U T

6.3P R O D U C T I O N F U N C T I O N S W I T H M O R E T H A N O N E I N P U T

APPLICATION 6.2 High-Tech Workers versus Low-Tech Workers

APPLICATION 6.3 Elasticities of Substitution in German Industries

APPLICATION 6.4 Measuring Productivity

6.5R E T U R N S T O S C A L E APPLICATION 6.5 Returns to Scale in Electric Power

Generation

APPLICATION 6.6 Returns to Scale in Oil Pipelines

6.6T E C H N O L O G I C A L P R O G R E S S APPLICATION 6.7 Technological Progress in the U.K.

AppendixT H E E L A S T I C I T Y O F S U B S T I T U T I O N F O R A C O B B – D O U G L A S P R O D U C T I O N F U N C T I O N

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Can They Make It Better and Cheaper?The production of semiconductor chips—thin, glasslike wafers that are used to store information in

digital equipment—is costly, complex, and delicate.1 Production involves hundreds of steps and

takes place in facilities called fabs, expensive factories that can cost more than $1 billion to con-

struct. To avoid contaminating chips, fabs must be 1000 times cleaner than a hospital operating

room. Because the manufacturing process is so expensive and because a typical fab is obsolete in

3 to 5 years (and thus depreciates about $1 million a day), semiconductor producers must get their

manufacturing processes right. This necessitates careful facilities and operations planning.

An important trend in recent years in semiconductor manufacturing has been the substitution of

robots for humans to perform certain repetitive processing tasks. Despite protective clothing,

footwear, and headgear used by workers, robots are cleaner than human workers and thus offer the

prospect of higher chip yields (the fraction of good chips per total chips produced). Cleanliness is

vital because an invisible speck of dust can ruin a $20,000 wafer of chips. Since robots are not

cheap, semiconductor manufacturers face an important trade-off: Are the cost savings from better

chip yields and less labor worth the investments in state-of-the-art robotics? Some chip manufac-

turers have decided that they are, others that they are not.

This chapter lays the foundations for studying this type of economic trade-off.

C H A P T E R P R E V I E W In this chapter, you will

• Study production functions, which relate the quantity

of a firm’s output to the quantities of inputs the firm

employs.

• Learn about production functions with a single input,

using this analysis to develop the concepts of average and

marginal product of labor.

• Use these concepts to study production functions with

more than one input.

• Analyze substitutability among inputs and develop the

concept of elasticity of substitution.

• Examine some specific types of production functions.

• Learn about returns to scale—how increases in input

quantities affect the quantity of output.

• Study the notion of technological progress, whereby

firms increase output without increasing input or, equiva-

lently, maintain output while decreasing input.

1This example draws from John Teresko, “Robot Renaissance,” IndustryWeek (September 16, 1996), pp. 38–41.

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Production of goods and services involves transforming resources—such as laborpower, raw materials, and the services provided by facilities and machines—into fini-shed products. Semiconductor producers, for example, combine the labor servicesprovided by their employees and the capital services provided by fabs, robots, andprocessing equipment with raw materials, such as silicon, to produce finished chips.The productive resources, such as labor and capital equipment, that a firm uses tomanufacture goods and services are called inputs or factors of production, and theamount of goods and services produced is the firm’s output.

As our semiconductor example suggests, real firms can often choose one of severalcombinations of inputs to produce a given volume of output. A semiconductor firmcan produce a given number of chips using workers and no robots or using fewerworkers and many robots. The production function is a mathematical representationof the various technological recipes from which a firm can choose to configure its pro-duction process. In particular, the production function tells us the maximum quantityof output the firm can produce given the quantities of the inputs that it might employ.We will write the production function this way:

Q = f (L, K ) (6.1)

where Q is the quantity of output, L is the quantity of labor used, and K is the quantityof capital employed. This expression tells us that the maximum quantity of output thefirm can get depends on the quantities of labor and capital it employs. We could havelisted more categories of inputs, but many of the important trade-offs that real firmsface involve choices between labor and capital (e.g., robots and workers for semicon-ductor firms). Moreover, we can develop the main ideas of production theory usingjust these two categories of inputs.

The production function in equation (6.1) is analogous to the utility function inconsumer theory. Just as the utility function depends on exogenous consumer tastes,the production function depends on exogenous technological conditions. Over time,these technological conditions may change, an occurrence known as technologicalprogress, and the production function may then shift. We discuss technologicalprogress in Section 6.6. Until then, we will view the firm’s production function as fixedand unchangeable.

The production function in equation (6.1) tells us the maximum output a firmcould get from a given combination of labor and capital. Of course, inefficient man-agement could reduce output from what is technologically possible. Figure 6.1 de-picts this possibility by showing the production function for a single input, labor:Q = f (L) . Points on or below the production function make up the firm’sproduction set, the set of technically feasible combinations of inputs and outputs.Points such as A and B in the production set are technically inefficient (i.e., at thesepoints the firm gets less output from its labor than it could). Points such as C andD, on the boundary of the production set, are technically efficient. At these points,the firm produces as much output as it possibly can given the amount of labor itemploys.

If we invert the production function, we get a function L = g(Q), which tells usthe minimum amount of labor L required to produce a given amount of output Q. Thisfunction is the labor requirements function. If, for example, Q = √

L is the produc-tion function, then L = Q2 is the labor requirements function; thus, to produce anoutput of 7 units, a firm will need at least 72 = 49 units of labor.

6 . 1 I N T R O D U C T I O N T O I N P U T S A N D P R O D U C T I O N F U N C T I O N S 185

6.1I N T R O D U C T I O NT O I N P U T SA N DP R O D U C T I O NF U N C T I O N S

inputs Resources, such aslabor, capital equipment, andraw materials, that arecombined to producefinished goods.

factors of productionResources that are used toproduce a good.

output The amount of agood or service produced bya firm.

production function Amathematical representationthat shows the maximumquantity of output a firm canproduce given the quantitiesof inputs that it mightemploy.

production set The set oftechnically feasible combina-tions of inputs and outputs.

technically inefficientThe set of points in theproduction set at which thefirm is getting less outputfrom its labor than it could.

technically efficient Theset of points in the produc-tion set at which the firm isproducing as much output asit possibly can given theamount of labor it employs.

labor requirementsfunction A function thatindicates the minimumamount of labor required toproduce a given amount ofoutput.


186 C H A P T E R 6 I N P U T S A N D P R O D U C T I O N F U N C T I O N S

L, units of labor per year

Q, u

nits

of o

utpu

t per

yea

r

Technically inefficient

Technically efficient

B

A

C

D Q = f (L)

FIGURE 6.1 Technical Efficiency andInefficiencyAt points C and D the firm is technically efficient.It is producing as much output as it can with theproduction function Q � f (L) given the quantityof labor it employs. At points A and B the firm istechnically inefficient. It is not getting as muchoutput as it could with its labor.

firm faces competition from other firms. Caves andBarton found that manufacturing firms in industries thatdid not face much competition from foreign importstended to be more technically inefficient than firms inindustries that were subject to significant import com-petition. They also found that firms in industries withhigh levels of concentration (sales concentrated in rela-tively few firms) tended to be more technically ineffi-cient than firms in industries containing a large numberof smaller competitors. These findings suggest that thepressure of competition—whether from imports orother firms in the industry—tends to induce firms tosearch for ways to get as much output as they can fromtheir existing combinations of inputs, thus moving themcloser to the boundaries of their production sets.

A P P L I C A T I O N 6.1

Competition Breeds Efficiency

Using data from the U.S. Census of Manufacturing (agovernment survey taken every 5 years to track manu-facturing activity in the United States), Richard Cavesand David Barton studied the extent of technical ineffi-ciency among U.S. manufacturers.2 For the typical man-ufacturer in Caves and Barton’s study, the ratio of actualoutput to the maximum output that would be attain-able given the firm’s labor and capital employment was63 percent. (In the notation used in the text, we wouldsay that Q/f (L, K ) � 0.63 for the typical firm.) This find-ing implies that the typical U.S. manufacturer was tech-nically inefficient.

According to Caves and Barton, an important deter-minant of technical efficiency is the extent to which a

2Richard Caves and David Barton, Efficiency in U.S. Manufacturing Industries (Cambridge, MA: MIT Press,1990).

Because the production function tells us the maximum attainable output from agiven combination of inputs, we will sometimes write Q ≤ f (L, K) to emphasize thatthe firm could, in theory, produce a quantity of output that is less than the maximumlevel attainable given the quantities of inputs it employs.


The business press is full of discussions of productivity, which broadly refers to theamount of output a firm can get from the resources it employs. We can use the pro-duction function to illustrate a number of important ways in which the productivity ofinputs can be characterized. To illustrate these concepts most clearly, we will start ourstudy of production functions with the simple case in which the quantity of output de-pends on a single input, labor.

T O TA L P R O D U C T F U N C T I O N SSingle-input production functions are sometimes called total product functions.Table 6.1 shows a total product function for a semiconductor producer. It shows thequantity of semiconductors Q the firm can produce in a year when it employs variousquantities L of labor within a fab of a given size with a given set of machines.

Figure 6.2 shows a graph of the total product function in Table 6.1. This graph hasfour noteworthy properties. First, when L = 0, Q = 0. That is, no semiconductors canbe produced without using some labor. Second, between L = 0 and L = 12, output riseswith additional labor at an increasing rate (i.e., the total product function is convex).Over this range, we have increasing marginal returns to labor. When there are in-creasing marginal returns to labor, an increase in the quantity of labor increases totaloutput at an increasing rate. Increasing marginal returns are usually thought to occur be-cause of the gains from specialization of labor. In a plant with a small work force, work-ers may have to perform multiple tasks. For example, a worker might be responsible formoving raw materials within the plant, operating the machines, and inspecting the

6 . 2 P R O D U C T I O N F U N C T I O N S W I T H A S I N G L E I N P U T 187

6.2P R O D U C T I O NF U N C T I O N SW I T H A S I N G L EI N P U T

total product function Aproduction function. A totalproduct function with a sin-gle input shows how totaloutput depends on the levelof the input.

increasing marginal returnsto labor The region alongthe total product functionwhere output rises with addi-tional labor at an increasingrate.

TABLE 6.1 TotalProduct Function

L* Q

0 06 30

12 9618 16224 19230 150

*L is expressed in thousandsof man-hours per day, and Qis expressed in thousands ofsemiconductor chips per day.

L, thousands of man-hours per day

Q, t

hous

ands

of c

hips

per

day

Diminishingtotalreturns

Increasingmarginalreturns

Diminishingmarginalreturns

36301860

50

100

150

200

2412

Totalproductfunction

FIGURE 6.2 Total Product FunctionThe total product function shows the relationship between the quantity of labor (L) and thequantity of output (Q). Here the function has three regions: a region of increasing marginalreturns (L < 12); a region of diminishing marginal returns (12 < L < 24); and a region of diminish-ing total returns (L > 24).


finished goods once they are produced. As additional workers are added, workers canspecialize—some will be responsible only for moving raw materials in the plant; otherswill be responsible only for operating the machines; still others will specialize in inspec-tion and quality control. Specialization enhances the marginal productivity of workersbecause it allows them to concentrate on the tasks at which they are most productive.

Third, between L = 12 and L = 24, output rises with additional labor but at adecreasing rate (i.e., the total product function is concave). Over this range we havediminishing marginal returns to labor. When there are diminishing marginal re-turns to labor, an increase in the quantity of labor still increases total output but at adecreasing rate. Diminishing marginal returns set in when the firm exhausts its abilityto increase labor productivity through the specialization of workers.

Finally, when the quantity of labor exceeds L = 24, an increase in the quantity oflabor results in a decrease in total output. In this region, we have diminishing totalreturns to labor. When there are diminishing total returns to labor, an increase in thequantity of labor decreases total output. Diminishing total returns occur because ofthe fixed size of the fabricating plant: if the quantity of labor used becomes too large,workers don’t have enough space to work effectively. Also, as the number of workersemployed in the plant grows, their efforts become increasingly difficult to coordinate.3

M A R G I N A L A N D A V E R A G E P R O D U C TWe are now ready to characterize the productivity of the firm’s labor input. There aretwo related, but distinct, notions of productivity that we can derive from the produc-tion function. The first is the average product of labor, which we write as APL. Theaverage product of labor is the average amount of output per unit of labor.4 This isusually what commentators mean when they write about, say, the productivity of U.S.workers as compared to their foreign counterparts. Mathematically, the average prod-uct of labor is equal to:

APL = total productquantity of labor

= QL

Table 6.2 and Figure 6.3 show the average product of labor for the total product func-tion in Table 6.1. They show that the average product varies with the amount of labor


diminishing marginalreturns to labor The re-gion along the total productfunction in which outputrises with additional laborbut at a decreasing rate.

diminishing total returns tolabor The region along thetotal product function whereoutput decreases withadditional labor.

average product of laborThe average amount of out-put per unit of labor.

TABLE 6.2 Average Product of Labor

L Q APL �

6 30 512 96 818 162 924 192 830 150 5

Q

L

3We could also have diminishing total returns to other inputs, such as materials. For example, adding fer-tilizer to an unfertilized field will increase crop yields. But too much fertilizer will burn out the crop, andoutput will be zero.4The average product of labor is also sometimes called the average physical product of labor and is then writ-ten as APPL.


the firm uses. In our example, APL increases for quantities of labor less than L = 18and falls thereafter.

Figure 6.4 shows the graphs of the total product and average product curvessimultaneously. The average product of labor at any arbitrary quantity L0 correspondsto the slope of a ray drawn from the origin to the point along the total product func-tion corresponding to L0. For example, the height of the total product function atpoint A is Q0, and the amount of labor is L0. The slope of the line segment connectingthe origin to point A is Q0/L0, which is the average product APL0 per the equation dis-played above. At L = 18, the slope of a ray from the origin attains its maximal value,indicating that APL reaches its peak at this quantity of labor.

The other notion of productivity is the marginal product of labor, which we writeas MPL. The marginal product of labor is the rate at which total output changes as thefirm changes its quantity of labor:

MPL = change in total productchange in quantity of labor

= �Q�L

The marginal product of labor is analogous to the concept of marginal utility fromconsumer theory, and just as we could represent that curve graphically, we can alsorepresent the marginal product curve graphically, as shown in Figure 6.3. Marginalproduct, like average product, is not a single number but varies with the quantity oflabor. In the region of increasing marginal returns, where 0 ≤ L < 12, the marginalproduct function is increasing. When diminishing marginal returns set in, at L > 12,the marginal product function starts decreasing. When diminishing total returns set

6 . 2 P R O D U C T I O N F U N C T I O N S W I T H A S I N G L E I N P U T 189


MPL

APL

AP L,

MP L,

chi

ps p

er m

an-h

our

36301860

5

−5

−10

10

2412

APL is increasingso MPL > APL

APL is decreasingso MPL < APL

A

marginal product of laborThe rate at which total out-put changes as the quantityof labor the firm uses ischanged.

FIGURE 6.3 Average and Marginal Product FunctionsAPL is the average product function. MPL is the marginal product function. The marginal productfunction rises in the region of increasing marginal returns (L < 12) and falls in the region of dimin-ishing marginal returns (12 < L < 24). It becomes negative in the region of diminishing total returns(L > 24). At point A, where APL is at a maximum, APL � MPL.


in, at L > 24, the marginal product function cuts through the horizontal axis and be-comes negative. As shown in the upper panel in Figure 6.4, the marginal product cor-responding to any particular amount of labor L1 is the slope of the line that is tangentto the total product function at L1 (line BC in the figure). Since the slopes of these tan-gent lines vary as we move along the production function, the marginal product oflabor must also vary.

In most production processes, as the quantity of one input (e.g., labor) increases,with the quantities of other inputs (e.g., capital and land) held constant, a point will bereached beyond which the marginal product of that input decreases. This phenome-non, which reflects the experience of real-world firms, seems so pervasive that econo-mists call it the law of diminishing marginal returns.


law of diminishing marginalreturns Principle that asthe usage of one input in-creases, the quantities ofother inputs being held fixed,a point will be reached be-yond which the marginalproduct of the variable inputwill decrease.


MPL

APL

Q0

AP L,

MP L

, chi

ps p

er m

an-h

our)

18L0 L1

L0 L1

0 24


Totalproductfunction

Q, t

hous

ands

of c

hips

per

day

18

A

C

B

0 24

Average product atL0 equals slope ofray 0A

Marginal product atL1 equals slope ofline BC

FIGURE 6.4 Relationshipamong Total, Average, andMarginal Product FunctionsThe marginal product of labor atany point equals the slope of thetotal product curve at that point.The average product at any pointis equal to the slope of the rayfrom the origin to the totalproduct curve at that point.


R E L AT I O N S H I P B E T W E E N M A R G I N A LA N D A V E R A G E P R O D U C TAs with other average and marginal concepts you will study in this book (e.g., averagecost versus marginal cost), there is a systematic relationship between average productand marginal product. Figure 6.3 illustrates this relationship:

• When average product is increasing in labor, marginal product is greater thanaverage product. That is, if APL increases in L, then MPL > APL .

• When average product is decreasing in labor, marginal product is less than averageproduct. That is, if APL decreases in L, then MPL < APL .

• When average product neither increases nor decreases in labor because we are at apoint at which APL is at a maximum (point A in Figure 6.3), then marginalproduct is equal to average product.

The relationship between marginal product and average product is the same as therelationship between the marginal of anything and the average of anything. To illus-trate this point, suppose that the average height of students in your class is 160 cm.Now Mike Margin joins the class, and the average height rises to 161 cm. What do weknow about Mike’s height? Since the average height is increasing, the “marginalheight” (Mike Margin’s height) must be above the average. If the average height hadfallen to 159 cm, it would have been because his height was below the average. Finally,if the average height had remained the same when Mike joined the class, his heightwould have had to exactly equal the average height in the class.

The relationship between average and marginal height in your class is the same asthe relationship between average and marginal product shown in Figure 6.3. It is also therelationship between average and marginal cost that we will study in Chapter 8 andthe relationship between average and marginal revenue that we will see in Chapter 11.

6 . 3 P R O D U C T I O N F U N C T I O N S W I T H M O R E T H A N O N E I N P U T 191

6.3P R O D U C T I O NF U N C T I O N SW I T H M O R ET H A N O N EI N P U T

The single-input production function is useful for developing key concepts, such asmarginal and average product, and building intuition about the relationships betweenthese concepts. However, to study the trade-offs facing real firms, such as semicon-ductor companies thinking about substituting robots for humans, we need to studymultiple-input production functions. In this section, we will see how to describe amultiple-input production function graphically, and we will study a way to character-ize how easily a firm can substitute among the inputs within its production function.

T O TA L P R O D U C T A N D M A R G I N A L P R O D U C TW I T H T W O I N P U T STo illustrate a production function with more than one input, let’s consider a situationin which the production of output requires two inputs: labor and capital. This mightbroadly illustrate the technological possibilities facing a semiconductor manufacturercontemplating the use of robots (capital) or humans (labor).

Table 6.3 shows a production function (or, equivalently, the total product function)for semiconductors, where the quantity of output Q depends on the quantity of labor Land the quantity of capital K employed by the semiconductor firm. Figure 6.5 shows thisproduction function as a three-dimensional graph. The graph in Figure 6.5 is called a


total product hill, a three-dimensional graph that shows the relationship between thequantity of output and the quantity of the two inputs employed by the firm.5

The height of the hill at any point is equal to the quantity of output Q the firmproduces from the quantities of inputs it employs. We could move along the hill inany direction, but it is easiest to imagine moving in either of two directions. Startingfrom any combination of labor and capital, we could move eastward by increasing the


5In Figure 6.5, we show the “skeleton,” or frame, of the total product hill, so that we can draw variouslines underneath it. Figure 6.6 shows the same total product hill as a solid surface.

TABLE 6.3 Production Function for Semiconductors*

K**

0 6 12 18 24 30

0 0 0 0 0 0 06 0 5 15 25 30 23

L**12 0 15 48 81 96 7518 0 25 81 137 162 12724 0 30 96 162 192 15030 0 23 75 127 150 117

*Numbers in table equal the output that can be produced with various combinations of labor and capital.

**L is expressed in thousands of man-hours per day; K is expressed in thousands of machine-hours per day;and Q is expressed in thousands of semiconductor chips per day.

0 66

12

12

East

North

thousandsof machine-

hoursper day

Q (thousands ofsemiconductor chipsper day) = height ofhill at any point

thousandsof man-hours

per day

A

C

B

K

L,30

30

24

24

18

18

FIGURE 6.5 Total Product HillThe height of the hill at any point is equal to the quantity of output Q attainable from the quantities oflabor L and capital K corresponding to that point.

total product hill A three-dimensional graph of a pro-duction function.


quantity of labor or we could move northward by increasing the quantity of capital. Aswe move either eastward or northward, we move to different elevations along the totalproduct hill, where each elevation corresponds to the particular quantity of output.

Let’s now see what happens when we fix the quantity of capital at a particular level,say K = 24, and increase the quantity of labor. The outlined column in Table 6.3shows that when we do this, the quantity of output initially increases but then beginsto decrease (when L > 24). In fact, notice that the values of Q in Table 6.3 are identi-cal to the values of Q for the total product function in Table 6.1. This shows that thetotal product function for labor can be derived from a two-input production functionby holding the quantity of capital fixed at a particular level (in this case, at K = 24) andvarying the quantity of labor.

We can make the same point with Figure 6.5. Let’s fix the quantity of capital atK = 24 and move eastward up the total product hill by changing the quantity of labor.As we do so, we trace out the path ABC, with point C being at the peak of the hill. Thispath has the same shape as the total product function in Figure 6.2, just as the K = 24column in Table 6.3 corresponds exactly to Table 6.1.

Just as the concept of total product extends directly to the multiple input case, sotoo does the concept of marginal product. The marginal product of an input is the rateat which output changes as the firm changes the quantity of one of its inputs, holdingthe quantities of all other inputs constant. The marginal product of labor is given by:

MPL = change in quantity of output Qchange in quantity of labor L

∣∣∣∣

K is held constant

= �Q�L

∣∣∣∣

K is held constant(6.2)

Similarly, the marginal product of capital is given by:

MPK = change in quantity of output Qchange in quantity of capital K

∣∣∣∣

L is held constant

= �Q�K

∣∣∣∣

L is held constant(6.3)

The marginal product tells us how the steepness of the total product hill variesas we change the quantity of an input, holding the quantities of all other inputsfixed. The marginal product at any particular point on the total product hill is thesteepness of the hill at that point in the direction of the changing input. For example,in Figure 6.5, the marginal product of labor at point B—that is, when the quantity oflabor is 18 and the quantity of capital is 24—describes the steepness of the total prod-uct hill at point B in an eastward direction.

I S O Q U A N T STo illustrate economic trade-offs, it helps to reduce the three-dimensional graph of theproduction function (the total product hill) to two dimensions. Just as we used a con-tour plot of indifference curves to represent utility functions in consumer theory, wecan also use a contour plot to represent the production function. However, instead ofcalling the contour lines indifference curves, we call them isoquants. Isoquant means“same quantity”: any combination of labor and capital along a given isoquant allowsthe firm to produce the same quantity of output.


isoquant A curve thatshows all of the combina-tions of labor and capitalthat can produce a givenlevel of output.


To illustrate, let’s consider the production function described in Table 6.4 (thesame function as in Table 6.3). From this table we see that two different combinationsof labor and capital—(L = 6, K = 18) and (L = 18, K = 6)—result in an output ofQ = 25 units (where each “unit” of output represents a thousand semiconductors).Thus, each of these input combinations is on the Q = 25 isoquant.

The same isoquant is shown in Figure 6.6 (equivalent to Figure 6.5), illustratingthe total product hill for the production function in Table 6.4. Suppose that youstarted walking along the total product hill from point A with the goal of maintaininga constant elevation (i.e., a constant quantity of output). Line segment ABCDE is thepath you should follow. At each input combination along this path, the height of thetotal product hill is Q = 25 (i.e., each of these input combinations is on the Q = 25isoquant).


0 6

All combinations of L and K along path ABCDE produce25 units of output, whereeach "unit" represents athousand semiconductorchips.

612

12

East

NorthE

D

K

L,30

30

24

24

18

18

BA

C

thousandsof machine-

hoursper day

thousandsof man-hours

per day

FIGURE 6.6 Isoquants and the Total Product HillIf we start at point A and walk along the hill so that our elevation remains unchanged at 25 units of out-put, then we will trace out the path ABCDE. This is the 25-unit isoquant for this production function.

TABLE 6.4 Production Function for Semiconductors*

K**

0 6 12 18 24 30

0 0 0 0 0 0 06 0 5 15 25 30 23

L**12 0 15 48 81 96 7518 0 25 81 137 162 12724 0 30 96 162 192 15030 0 23 75 127 150 117

*Numbers in table equal the output that can be produced with various combinations of labor and capital.

**L is expressed in thousands of man-hours per day; K is expressed in thousands of machine-hours per day;and Q is expressed in thousands of semiconductor chips per day.


From this example, we can see that an isoquant is like a line on a topographical map,such as the one of Mt. Hood, in Oregon, in Figure 6.7. A line on this topographical mapshows points in geographic space at which the elevation of the land is constant. The totalproduct hill in Figure 6.6 is analogous to the three-dimensional map of Mt. Hood inpanel (a) of Figure 6.7, and the isoquants of the total product hill (see Figure 6.8) areanalogous to the lines on the topographical map of Mt. Hood in panel (b) of Figure 6.7.


TheChimney

Newton ClarkGlacier

EliotGlacier Cooper

Spur

CoalmanGlacier

ReidGlacier

IlluminationRock

Mississippi

CoeGlacier

PulpitRock

HotRocks

SteelCliff

White RiverGlacier

Mt. Hood Wilderness Area

8250

6000

6250

6500

7750

7750

8750

8500

8250

8000

8500

7500

7250

9500

9750

950092

50900085

008750

8250

11000

1075

0

1050

01025

01000

08250

9250

9000

8000

8000

72507500

6750

6500

7750

7000

75007250

7000

7500

7750

Newton ClarkGlacier

Newton ClarkGlacier

SandyGlacier

ZigzagGlacier

GlisanGlacier

EliotGlacier

TheChimney

Newton ClarkGlacier

EliotGlacier Cooper

Spur

CoalmanGlacier

ReidGlacier

IlluminationRock

Mississippi

CoeGlacier

PulpitRock

HotRocks

SteelCliff

White RiverGlacier

Mt. Hood Wilderness Area

(b)

(a)

FIGURE 6.7 Three-Dimensional and Topographic Map for Mt. HoodPanel (a) is a three-dimensional map of Mt. Hood. The product hill in Figure 6.6 is analogous to thiskind of map. Panel (b) shows a topographic map of Mt. Hood. A graph of isoquants (as in Figure 6.8) isanalogous to this topographic map.Source: www.delorme.com.


Figure 6.8 shows isoquants for the production function in Table 6.4 and Figure 6.6.The fact that the isoquants are downward sloping in Figure 6.8 illustrates an importanteconomic trade-off: A firm can substitute capital for labor and keep its output un-changed. If we apply this idea to a semiconductor firm, it tells us that the firm couldproduce a given quantity of semiconductors using lots of workers and a small number ofrobots or using fewer workers and more robots. Such substitution is always possiblewhenever both labor and capital (e.g., robots) have positive marginal products.

Any production function has an infinite number of isoquants, each one corre-sponding to a particular level of output. In Figure 6.8, isoquant Q1 corresponds to25 units of output. Notice that points B and D along this isoquant correspond to thehighlighted input combinations in Table 6.4. When both inputs have positive marginalproducts, using more of each input increases the amount of output attainable. Hence,isoquants Q2 and Q3, to the northeast of Q1 in Figure 6.8, correspond to larger andlarger quantities of output.

An isoquant can also be represented algebraically, in the form of an equation, as wellas graphically (like the isoquants in Figure 6.8). For a production function like the oneswe have been considering, where quantity of output Q depends on two inputs (quantityof labor L and quantity of capital K ) , the equation of an isoquant would express K interms of L. Learning-By-Doing Exercise 6.1 shows how to derive such an equation.

L E A R N I N G - B Y - D O I N G E X E R C I S E 6.1Deriving the Equation of an Isoquant

Problem

(a) Consider the production function whose equation is given by the formulaQ = √

K L. What is the equation of the isoquant corresponding to Q = 20?

E

S

D


60 18


Q1 = 25

Q2 > 25

Q3 > Q2

B

DK

, tho

usan

ds o

f mac

hine

-hou

rs p

er d

ay

18

6

FIGURE 6.8 Isoquants forthe Production Function inTable 6.4 and Figure 6.6Every input combination oflabor and capital along the Q1 � 25 isoquant (in particular,combinations B and D) producesthe same output, 25,000 semi-conductor chips per day. As wemove to the northeast, theisoquants correspond toprogressively higher outputs.


(b) For the same production function, what is the general equation of an isoquant, corre-sponding to any level of output Q?

Solution

(a) The Q = 20 isoquant represents all of the combinations of labor and capital that allowthe firm to produce 20 units of output. For this isoquant, the production function satisfiesthe following equation:

20 =√

K L (6.4)

To find the equation of the 20-unit isoquant, we solve this equation for K in terms of L. Theeasiest way to do this is to square each side of equation (6.4) and then solve for K in termsof L. Doing this yields K = 400/L. This is the equation of the 20-unit isoquant.

(b) In the general case, we begin with the production function itself: Q = √K L. To find

the general equation of an isoquant, we again square each side and solve for K in terms ofL. Doing this yields K = Q2/L. (If you substitute Q = 20 into this equation, you get theequation of the 20-unit isoquant that we solved for above.)

Similar Problems: 6.7 and 6.8

E C O N O M I C A N D U N E C O N O M I C R E G I O N SO F P R O D U C T I O NThe isoquants in Figure 6.8 are downward sloping: In the range of values of labor andcapital shown in the graph, as we increase the amount of labor we use, we can hold out-put constant by reducing the amount of capital. But now look at Figure 6.9, which showsthe same isoquants when we expand the scale of Figure 6.8 to include quantities of laborand capital greater than 24,000 man-hours and machine-hours per day. The isoquantsnow have upward-sloping and backward-bending regions. What does this mean?

The upward-sloping and backward-bending regions correspond to a situation inwhich one input has a negative marginal product, or what we earlier called diminish-ing total returns. For example, the upward-sloping region in Figure 6.9 occurs becausethere are diminishing total returns to labor (MPL < 0), while the backward-bendingregion arises because of diminishing total returns to capital (MPK < 0). If we havediminishing total returns to labor, then as we increase the quantity of labor, holdingthe quantity of capital fixed, total output goes down. Thus, to keep output constant(remember, this is what we do when we move along an isoquant), we must also increasethe amount of capital to compensate for the diminished total returns to labor.

A firm that wants to minimize its production costs should never operate in a re-gion of upward-sloping or backward-bending isoquants. For example, a semiconduc-tor producer should not operate at a point such as A in Figure 6.9 where there arediminishing total returns to labor. The reason is that it could produce the same out-put but at a lower cost by producing at a point such as E. By producing in the rangewhere the marginal product of labor is negative, the firm would be wasting money byspending it on unproductive labor. For this reason, we refer to the range in which iso-quants slope upward or bend backward as the uneconomic region of production.By contrast, the economic region of production is the region of downward-slopingisoquants. From now on, we will show only the economic region of production in ourgraphs.


uneconomic region of pro-duction The region ofupward-sloping or backward-bending isoquants. In the un-economic region, at least oneinput has a negative marginalproduct.

economic region ofproduction The regionwhere the isoquants aredownward sloping.


M A R G I N A L R AT E O F T E C H N I C A L S U B S T I T U T I O NA semiconductor firm that is contemplating investments in sophisticated roboticswould naturally be interested in the extent to which it can replace humans with robots.That is, the firm will need to consider the question: How many robots will it need toinvest in to replace the labor power of one worker? Answering this question will becrucial in determining whether an investment in robotics would be worthwhile.

The “steepness” of an isoquant determines the rate at which the firm can substi-tute between labor and capital in its production process. The marginal rate of tech-nical substitution of labor for capital, denoted by MRTSL,K, measures how steep anisoquant is. The MRTSL,K tells us the following:

• The rate at which the quantity of capital can be decreased for every one unitincrease in the quantity of labor, holding the quantity of output constant, or

• The rate at which the quantity of capital must be increased for every one unitdecrease in the quantity of labor, holding the quantity of output constant.

The marginal rate of technical substitution is analogous to the marginal rate ofsubstitution from consumer theory. Just as the marginal rate of substitution of good Xfor good Y is the negative of the slope of an indifference curve drawn with X on the


60 18 24

Economicregion

Uneconomic regionIsoquants arebackward bending(MPK < 0)

Isoquants areupward sloping(MPL < 0)


Q = 25

B

D

E A

K, t

hous

ands

of m

achi

ne-h

ours

per

day

18

24

6

FIGURE 6.9 Economic and Uneconomic Regions of ProductionThe backward-bending and upward-sloping regions of the isoquants make up the uneconomic region ofproduction. In this region, the marginal product of one of the inputs is negative. A cost-minimizing firmwould never produce in the uneconomic region.

marginal rate of technicalsubstitution of labor forcapital The rate at whichthe quantity of capital can bereduced for every one unitincrease in the quantity oflabor, holding the quantity ofoutput constant.


horizontal axis and Y on the vertical axis, the marginal rate of technical substitution oflabor for capital is the negative of the slope of an isoquant drawn with L on the hori-zontal axis and K on the vertical axis. The slope of an isoquant at a particular point isthe slope of the line that is tangent to the isoquant at that point, as Figure 6.10 shows.The negative of the slope of the tangent line is the MRTSL,K at that point.

Figure 6.10 illustrates the MRTSL,K along the Q = 1000 unit isoquant for a par-ticular production function. At point A, the slope of the line tangent to the isoquant is−2.5. Thus, MRTSL, K = 2.5 at point A, which means that, starting from this point,we can substitute 1.0 man-hour of labor for 2.5 machine-hours of capital, and outputwill remain unchanged at 1000 units. At point B, the slope of the isoquant is −0.4.Thus, MRTSL, K = 0.4 at point B, which means that, starting from this point, we cansubstitute 1.0 man-hour of labor for 0.4 machine-hours of capital without changingoutput.

As we move down along the isoquant in Figure 6.10, the slope of the isoquant in-creases (i.e., becomes less negative), which means that the MRTSL,K gets smaller andsmaller. This property is known as diminishing marginal rate of technical substi-tution. When a production function exhibits diminishing marginal rate of technicalsubstitution (i.e., when the MRTSL,K along an isoquant decreases as the quantity oflabor L increases), the isoquants are convex to the origin (i.e., bowed in toward theorigin).

We can show that there is a precise connection between MRTSL,K and the mar-ginal products of labor (MPL) and capital (MPK). Note that when we change the quan-tity of labor by �L units and the quantity of capital by �K units of capital, the changein output that results from this substitution would be as follows:

�Q = change in output from change in quantity of capital+ change in output from change in quantity of labor


50200

L, man-hours per day

A

K, m

achi

ne-h

ours

per

day

50

B20

Q = 1000

MRTSL,K at A = 2.5

MRTSL,K at B = 0.4

Slope of tangent line A = –2.5Slope of tangent line B = –0.4

FIGURE 6.10 MarginalRate of Technical Substitu-tion of Labor for Capital(MRTSL,K) along anIsoquantAt point A, the MRTSL,K is2.5. Thus, the firm can holdoutput constant by replac-ing 2.5 machine-hours ofcapital services with an ad-ditional man-hour of labor.At point B, the MRTSL,K is0.4. Here, the firm can holdoutput constant by replac-ing 0.4 machine-hours ofcapital with an additionalman-hour of labor.

diminishing marginal rateof technical substitutionA feature of a productionfunction in which themarginal rate of technicalsubstitution of labor forcapital diminishes as thequantity of labor increasesalong an isoquant.


From equations (6.2) and (6.3), we know,

change in output from change in quantity of capital = (�K )(MPK )change in output from change in quantity of labor = (�L)(MPL)

Thus, �Q = (�K )(MPK ) + (�L)(MPL). Along a given isoquant, output is unchanged(i.e., �Q = 0). So, 0 = (�K )(MPK ) + (�L)(MPL), or −(�K )(MPK ) = (�L)(MPL),which can be rearranged to

−�K�L

= MPL

MPK

But −�K/�L is the negative of the slope of the isoquant, which is equal to theMRTSL,K. Thus,

MPL

MPK= MRTSL, K (6.5)

This shows that the marginal rate of technical substitution of labor for capital is equalto the ratio of the marginal product of labor (MPL) to the marginal product of capital(MPK). (This is analogous to the relationship between marginal rate of substitutionand marginal utility that we saw in consumer theory.)

To illustrate why this relationship is significant, consider semiconductor produc-tion. Suppose that, at the existing input combination, an additional unit of labor wouldincrease output by 10 units, while an additional unit of capital (robots) would increaseoutput by just 2 units (i.e., MPL = 10, while MPK = 2). Thus, at our current inputcombination, labor has a much higher marginal productivity than capital. Equation (6.5)tells us that the MRTSL, K = 10/2 = 5, which means that the firm can substitute 1 unitof labor for 5 units of capital without affecting output. Clearly, a semiconductor firmwould want to know the marginal productivity of both inputs before making an in-vestment decision involving the mix between robots and human workers.

L E A R N I N G - B Y - D O I N G E X E R C I S E 6.2Relating the Marginal Rate of Technical Substitutionto Marginal Products

Problem At first glance, you might think that when a production functionhas a diminishing marginal rate of technical substitution of labor for capital,

it must also have diminishing marginal products of capital and labor. Show that this is nottrue, using the production function Q = K L, with the corresponding marginal productsMPK = L and MPL = K .

Solution First, note that MRTSL, K = MPL/MPK = K/L, which diminishes as L in-creases and K falls as we move along an isoquant. So the marginal rate of technical substitu-tion of labor for capital is diminishing. However, the marginal product of capital MPK is con-stant (not diminishing) as K increases (remember, the amount of labor is held fixed when wemeasure MPK). Similarly, the marginal product of labor is constant (again, because theamount of capital is held fixed when we measure MPL). This exercise demonstrates that it ispossible to have a diminishing marginal rate of technical substitution even though both ofthe marginal products are constant. The distinction is that in analyzing MRTSL,K, we movealong an isoquant, while in analyzing MPL and MPK, total output can change.

Similar Problem: 6.10

E

S

D



A semiconductor manufacturer considering the choice between robots and workerswould want to know how easily it can substitute between these inputs. The answer tothis question will determine, in part, a firm’s ability to shift from one mode of produc-tion (e.g., a high ratio of labor to capital) to another (e.g., a low ratio of labor to capi-tal) as the relative prices of labor and capital change. In this section, we explore how todescribe the ease or difficulty with which a firm can substitute between different inputs.

D E S C R I B I N G A F I R M ’ S I N P U T S U B S T I T U T I O NO P P O R T U N I T I E S G R A P H I C A L LYLet’s consider two possible production functions for the manufacture of semiconduc-tors. Figure 6.11(a) shows the 1-million-chip-per-month isoquant for the first produc-tion function, while Figure 6.11(b) shows the 1-million-chip-per-month isoquant forthe second production function.

These two production functions differ in terms of how easy it is for the firm tosubstitute between labor and capital. In Figure 6.11(a), suppose the firm operates atpoint A, with 100 man-hours of labor and 50 machine-hours of capital. At this point,it is hard for the firm to substitute labor for capital. Even if the firm quadruples its useof labor, from 100 to 400 man-hours per month, it can reduce its quantity of capital byonly a small amount—from 50 to 45 machine-hours—to keep monthly output at

6 . 4 S U B S T I T U TA B I L I T Y A M O N G I N P U T S 201

low-tech labor is about 6. That is, once the firm has de-termined its stock of computers, 1 high-tech worker canbe substituted for 6 low-tech workers and output willremain unchanged. The reason that this MRTS is solarge is that once the firm has invested in the acquisi-tion of computer equipment, the marginal product ofhigh-tech, computer-literate workers is much higherthan the marginal product of low-tech workers withfewer computer skills.

Lichtenberg notes that his estimate of the MRTS oflow-tech and high-tech workers is consistent with theexperience of real firms. He notes, for example, thatwhen a large U.S. telecommunications company de-cided to automate and computerize its responses tocustomer service inquiries, it hired 9 new computerprogrammers and information systems workers. Thesenew workers displaced 75 low-tech service workerswho had handled customer inquiries under the oldsystem. For every additional high-tech worker the firmhired, it was able to replace more than 8 low-techworkers (75/9 ≈ 8.3).

Over the last 20 years computers have become aubiquitous part of the business landscape. As this hashappened, firms have changed the composition oftheir work force, replacing “low-tech” workers with“high-tech” workers with greater knowledge aboutand experience in using computers.

Using data on employment and computer usageover the period 1988–1991, Frank Lichtenberg has esti-mated the extent to which computer equipment andcomputer-oriented personnel have contributed to out-put in U.S. businesses.6 As part of this study, Lichtenbergestimated the marginal rate of technical substitution ofhigh-tech labor—computer and information systemspersonnel—for low-tech labor—workers employed inactivities other than information systems and technol-ogy. If we hold a typical U.S. firm’s output fixed, andalso assume that its stock of computer equipmentremains fixed, then the MRTS of high-tech labor for


High-Tech Workers versus Low-Tech Workers

6.4S U B S T I T U T -A B I L I T YA M O N GI N P U T S

6F. Lichtenberg, “The Output Contributions of Computer Equipment and Personnel: A Firm-LevelAnalysis,” Economics of Innovation and New Technology, 3 (3–4, 1995), pp. 201–217.


1 million chips. Figure 6.11(a) also indicates that the firm would face a similar diffi-culty in substituting capital for labor. A large increase in the number of machine-hours(i.e., moving up the isoquant from point A) would yield only a small decrease in thenumber of man-hours.

By contrast, with the production function illustrated in Figure 6.11(b), the firm’ssubstitution opportunities are more abundant. Starting from the input combinationat point A, the firm can reduce its employment of capital significantly—from 50 to20 machine-hours—if it increases the quantity of labor from 100 to 400 man-hoursper month. Similarly, it could achieve significant reductions in man-hours by increas-ing machine-hours. Of course, whether it would want to do either would depend onthe relative cost of labor versus capital (an issue we will study in the next chapter), butthe point is that the firm can potentially make substantial labor-for-capital (or capital-for-labor) substitutions. In contrast to Figure 6.11(a), the production function inFigure 6.11(b) gives the firm more opportunities to substitute between labor andcapital.

A semiconductor firm would probably want to know whether its opportunities tosubstitute labor for capital are limited or abundant. But what distinguishes one situa-tion from the other? Note that in Figure 6.11(a), the MRTSL,K changes dramatically aswe move through point A on the 1-million-unit isoquant. Just above point A on theisoquant, MRTSL,K is quite large, almost infinite, but just beyond point A, theMRTSL,K abruptly shifts and becomes practically equal to 0. By contrast, as we movealong the isoquants in Figure 6.11(b), the MRTSL,K changes gradually.


4001000

L, man-hours per month

A

K, m

achi

ne-h

ours

per

mon

th

50 B45

Q = 1million

(a) Production Function

with Limited Input

Substitution Opportunities

4001000


A

K, m

achi

ne-h

ours

per

mon

th

50

B20 Q = 1

million

(b) Production Function

with Abundant Input

Substitution Opportunities

FIGURE 6.11 Input Substitution Opportunities and the Shape of IsoquantsIn panel (a), start from point A and move along the isoquant Q � 1 million (i.e., holding output con-stant). If the firm increases one input significantly (either L or K), it will only be able to reduce the otherinput by a small amount. The firm is in a position where there is virtually no substitutability betweenlabor and capital. By contrast, in panel (b) the firm has abundant substitution opportunities—that is, asignificant increase in one input would allow the firm to reduce the other input by a significant amount,holding output constant.

besa44438_ch06.qxd 09/23/2004 04:19 PM 2

This suggests that the ease or difficulty with which a firm can substitute amonginputs depends on the curvature of its isoquants. Specifically,

• When the production function offers limited input substitution opportunities,the MRTSL,K changes substantially as we move along an isoquant. In this case,the isoquants are nearly L-shaped, as in Figure 6.11(a).

• When the production function offers abundant input substitution opportunities,the MRTSL,K changes gradually as we move along an isoquant. In this case, theisoquants are nearly straight lines, as in Figure 6.11(b).

E L A S T I C I T Y O F S U B S T I T U T I O NThe concept of elasticity of substitution is a numerical measure that can help usdescribe the firm’s input substitution opportunities based on the relationships we justderived in the previous section. Specifically, the elasticity of substitution measures howquickly the marginal rate of technical substitution of labor for capital changes as wemove along an isoquant. Figure 6.12 illustrates elasticity of substitution. As labor issubstituted for capital, the ratio of the quantity of capital to the quantity of labor,known as the capital–labor ratio, K/L, must fall. The marginal rate of substitution ofcapital for labor, MRTSL,K, also falls, as we saw in the previous section. The elasticity ofsubstitution, often denoted by σ , measures the percentage change in the capital–laborratio for each 1 percent change in MRTSL,K as we move along an isoquant:

σ = percentage change in capital–labor ratiopercentage change in MRTSL, K

= %�( K

L

)

%�MRTSL, K(6.6)

Figure 6.12 illustrates the elasticity of substitution. Suppose a firm moves from theinput combination at point A (L = 5 man-hours per month, K = 20 machine-hours


elasticity of substitutionA measure of how easy it isfor a firm to substitute laborfor capital. It is equal to thepercentage change in thecapital–labor ratio for everyone percent change in themarginal rate of technicalsubstitution of capital forlabor as we move along anisoquant.

capital–labor ratio Theratio of the quantity of capi-tal to the quantity of labor.

1050


A

K, m

achi

ne-h

ours

per

mon

th

20

B10

Q = 1 million

MRTSL,K at B = 1

K/L at B = slope of ray 0B = 1

K/L at A = slope of ray 0A = 4

MRTSL,K at A = 4

FIGURE 6.12 Elasticity of Substitu-tion of Labor for CapitalAs the firm moves from point A to pointB, the capital–labor ratio K/L changesfrom 4 to 1 (−75%), as does the MRTSL,K.Thus, the elasticity of substitution oflabor for capital over the interval A to Bequals 1.


per month) to the combination at point B (L = 10, K = 10). The capital–labor ratioK/L at A is equal to the slope of a ray from the origin to A (slope of ray 0A = 4); theMRTSL,K at A is equal to the negative of the slope of the isoquant at A (slope ofisoquant = −4; thus, MRTSL, K = 4). At B, the capital–labor ratio equals the slope ofray 0B, or 1; the MRTSL,K equals the negative of the slope of the isoquant at B, also 1.The percent change in the capital–labor ratio from A to B is −75 percent (from 4 downto 1), as is the percent change in the MRTSL,K between those points. Thus, the elasticityof substitution over this interval is 1 (−75%/−75% = 1).



Elasticities of Substitution in German Industries7

Using data on output and input quantities over theperiod 1970–1988, Claudia Kemfert has estimated theelasticity of substitution between capital and labor in anumber of manufacturing industries in Germany. Table6.5 shows the estimated elasticities.

The results in Table 6.5 show two things. First, thefact that the estimated elasticity of substitution is lessthan 1 in all industries tells us that, generally speaking,labor and capital inputs are not especially substitutablein these industries. Second, the ease of substitutabilityof capital for labor is higher in some industries than inothers. For example, in the production of iron (elasticityof substitution equal to 0.50), labor and capital canbe substituted to a much greater extent than they

TABLE 6.5 Elasticities of Substitution in GermanManufacturing Industries, 1970–1988

Elasticity ofIndustry Substitution

Chemicals 0.37Stone and earth 0.21Iron 0.50Motor vehicles 0.10Paper 0.35Food 0.66

L, units of labor per year L, units of labor per year

(a) Isoquants for German iron production (b) Isoquants for German motor vehicle production

K, u

nits

of

capi

tal p

er y

ear

K, u

nits

of

capi

tal p

er y

ear

FIGURE 6.13 Isoquants for Iron and Motor Vehicle Production in GermanyThe higher elasticity of substitution of labor for capital in the iron industry [panel (a)] implies that laborand capital inputs are more easily substitutable in this industry than they are in the production ofmotor vehicles [panel (b)].

can in the production of motor vehicles (elasticity ofsubstitution 0.10). Figure 6.13 shows this graphically.Isoquants in iron production would have the shape ofFigure 6.13(a), while the isoquants in vehicle productionwould have the shape of Figure 6.13(b).

7This example is based on “Estimated Substitution Elasticities of a Nested CES Production FunctionApproach for Germany,” Energy Economics 20 (1998), pp. 249–264.


In general, the elasticity of substitution can be any number greater than or equalto 0. What is the significance of the elasticity of substitution?

• If the elasticity of substitution is close to 0, there is little opportunity to substi-tute between inputs. We can see this from equation (6.6), where σ will be closeto 0 when the percentage change in MRTSL,K is large, as in Figure 6.11(a).

• If the elasticity of substitution is large, there is substantial opportunity tosubstitute between inputs. In equation (6.6), this corresponds to the fact that σwill be large if the percentage change in MRTSL,K is small, as illustrated inFigure 6.11(b).

S P E C I A L P R O D U C T I O N F U N C T I O N SThe relationship between the curvature of isoquants, input substitutability, and theelasticity of substitution is most apparent when we compare and contrast a number ofspecial production functions that are frequently used in microeconomic analysis. Inthis section, we will consider four special production functions: the linear productionfunction, the fixed-proportions production function, the Cobb–Douglas productionfunction, and the constant elasticity of substitution production function.

Linear Production Function (Perfect Substitutes)In some production processes, the marginal rate of technical substitution of one inputfor another may be constant. For example, a manufacturing process may require energyin the form of natural gas or fuel oil, and a given amount of natural gas can always be sub-stituted for each liter of fuel oil. In this case, the marginal rate of technical substitutionof natural gas for fuel oil is constant. Sometimes a firm may find that one type of equip-ment may be perfectly substituted for another type. For example, suppose that a firmneeds to store 200 gigabytes of company data and is choosing between two types ofcomputers for that purpose. One has a high-capacity hard drive that can store 20 giga-bytes of data, while the other has a low-capacity hard drive that can store 10 gigabytes ofdata. At one extreme, the firm could purchase 10 high-capacity computers and no low-capacity computers (point A in Figure 6.14). At the other extreme, it could purchase no


100

5

10

20

L, quantity of low-capacity computers

A

C

BH, q

uant

ity o

f hig

h-ca

paci

ty c

ompu

ters

Slope of isoquants = –

Q = 200 gigabytes

1/2, a constant

FIGURE 6.14 Isoquants for a Linear Produc-tion FunctionThe isoquants for a linear production functionare straight lines. The MRTSL,H at any point on anisoquant is thus a constant.


high-capacity computers and 20 low-capacity computers (point B in Figure 6.14). Or,in the middle, it could purchase 5 high-capacity computers and 10 low-capacity com-puters (point C in Figure 6.14), because (5 × 20) + (10 × 10) = 200.

In this example, the firm has a linear production function whose equation wouldbe Q = 20H + 10L, where H is the number of high-capacity computers the firm em-ploys, L is the number of low-capacity computers the firm employs, and Q is the total gi-gabytes of data the firm can store. A linear production function is a production functionwhose isoquants are straight lines. Thus, the slope of any isoquant is constant and themarginal rate of technical substitution does not change as we move along the isoquant.

Because MRTSL,H does not change as we move along an isoquant, �MRTSL, H = 0.Using equation (6.6), this means that the elasticity of substitution for a linear produc-tion function must be infinite (σ = ∞). In other words, the inputs in a linear produc-tion function are infinitely (perfectly) substitutable for each other. When we have alinear production function, we say that the inputs are perfect substitutes. In ourcomputer example, the fact that low-capacity and high-capacity computers are perfectsubstitutes means that in terms of data storage capabilities, two low-capacity com-puters are just as good as one high-capacity computer. Or, put another way, the firmcan perfectly replicate the productivity of one high-capacity computer by employingtwo low-capacity computers.

Fixed-Proportions Production Function (Perfect Complements)Figure 6.15 illustrates a dramatically different case: isoquants for the production ofwater, where the inputs are atoms of hydrogen (H ) and atoms of oxygen (O). Sinceeach molecule of water consists of two hydrogen atoms and one oxygen atom, theinputs must be combined in that fixed proportion. A production function where theinputs must be combined in fixed proportions is called a fixed-proportions produc-tion function, and the inputs in a fixed-proportions production function are calledperfect complements.8 Adding more hydrogen to a fixed number of oxygen atomsgives us no additional water molecules; neither does adding more oxygen to a fixednumber of hydrogen atoms. Thus, the quantity Q of water molecules that we get isgiven by:

Q = min(

H2

, O)

where the notation min means “take the minimum value of the two numbers in theparentheses.”

When inputs are combined in fixed proportions, the elasticity of substitution iszero (i.e., σ = 0), because the marginal rate of technical substitution along the isoquantof a fixed-proportions production function changes from ∞ to 0 when we pass throughthe corner of an isoquant (e.g., point A, B, or C). A firm facing a fixed-proportions pro-duction function has no flexibility in its ability to substitute among inputs. We can seethis in Figure 6.15: to produce a single molecule of water, there is only one sensibleinput combination—two atoms of hydrogen and one atom of oxygen.

We often observe production processes with fixed proportions. The production ofcertain chemicals requires the combination of other chemicals, and sometimes heat, infixed proportions. Every bicycle must always have two tires and one frame. An auto-mobile requires one engine, one chassis, and four tires, and these inputs cannot be sub-stituted for one another.


fixed-proportions produc-tion function A produc-tion function where theinputs must be combined ina constant ratio to oneanother.

perfect complements (inproduction) Inputs in afixed-proportions productionfunction.

8The fixed-proportions production function is also called the Leontief production function, after the econo-mist Wassily Leontief, who used it to model relationships between sectors in a national economy.

linear production functionA production function of theform Q � aL + bK, where aand b are positive constants.

perfect substitutes (inproduction) Inputs in aproduction function with aconstant marginal rate oftechnical substitution.


Cobb–Douglas Production FunctionFigure 6.16 illustrates isoquants for the Cobb–Douglas production function, whichis intermediate between a linear production function and a fixed-proportions produc-tion function. The Cobb–Douglas production function is given by the formulaQ = ALα K β , where A, α, and β are positive constants (in Figure 6.16, their values are100, 0.4, and 0.6, respectively). With the Cobb–Douglas production function, capitaland labor can be substituted for each other. Unlike a fixed-proportions production


2 40

1

2

3

6

H, quantity of hydrogen atoms

B

C

A

O, q

uant

ity o

f oxy

gen

atom

s

Isoquant for 3 molecules of water

Isoquant for 2 molecules of water

Isoquant for 1 molecule of water

FIGURE 6.15 Isoquants for a Fixed-Proportions Production FunctionTwo atoms of hydrogen (H) and oneatom of oxygen (O) are needed to makeone molecule of water. The isoquants forthis production function are L-shaped,which indicates that each additionalatom of oxygen produces no additionalwater unless two additional atoms ofhydrogen are also added.

10 20 30 40 500

10

20

30

40

50


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FIGURE 6.16 Isoquants for aCobb–Douglas ProductionFunctionThe isoquants for a Cobb–Douglasproduction function are nonlineardownward-sloping curves.

Cobb–Douglas productionfunction A productionfunction of the formQ � ALαKβ , where Q isthe quantity of output fromL units of labor and K units ofcapital and where A, α, andβ are positive constants.


function, capital and labor can be used in variable proportions. Unlike a linear pro-duction function, though, the rate at which labor can be substituted for capital is notconstant as you move along an isoquant. This suggests that the elasticity of substitu-tion for a Cobb–Douglas production function falls somewhere between 0 and ∞. Infact, it turns out that the elasticity of substitution along a Cobb–Douglas productionfunction is always equal to 1. (This result is derived in the appendix to this chapter.)


function coefficients (especially α in the equationQ � ALαKβ , which reflects the productivity ofworkers) were affected by workplace practicesadopted by firms. Their findings were mixed. Totalquality management, a highly touted managementpractice adopted by many firms in the late 1980s andearly 1990s, aimed at increasing product quality orreducing manufacturing defects, was not associatedwith enhanced worker productivity. By contrast, theadoption of benchmarking practices (i.e., setting targetsbased on the successes of other firms in, say, reducingdefect rates) and the involvement of workers in regulardecision-making meetings seemed to have a positiveimpact on productivity.


Measuring Productivity

Because the Cobb–Douglas production function isthought to be a plausible way of characterizing manyreal-world production processes, it is often used byeconomists to study issues related to input productivityor production costs. For example, Sandra Black and LisaLynch estimated Cobb–Douglas production functionsto study the impact of “high performance” workplacepractices (such as total quality management or em-ployee involvement in decision making) on workerproductivity in U.S. firms.9 Specifically, Black and Lynchused data from the late 1980s and early 1990s to ex-plore whether changes in Cobb–Douglas production

1 20

1

2

L, units of labor services per year

σ = 0σ = 0.1

σ = 1

σ = 5σ = ∞

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FIGURE 6.17 Isoquants for theCES Production FunctionThis figure depicts the Q � 1 iso-quant for five different CES produc-tion functions, each correspondingto a different value of the elasticityof substitution σ . At σ � 0, the iso-quant is that of a fixed-proportionsproduction function. At σ � 1, theisoquant is that of a Cobb–Douglasproduction function. At σ = ∞,the isoquant is that of a linearproduction function.

9S. E. Black and L. M. Lynch, “How to Compete: The Impact of Workplace Practices and InformationTechnology on Productivity,” Review of Economics and Statistics, 83(3), (August 2001), pp. 434–445.


Constant Elasticity of Substitution Production FunctionEach of the three production functions discussed above is a special case of a produc-tion function called the constant elasticity of substitution (CES) productionfunction, which is given by the equation:

Q = [

a Lσ−1σ + b K

σ−1σ

] σσ−1

where a, b, and σ are positive constants (σ is the elasticity of substitution). Figure 6.17shows that as σ varies between 0 and ∞, the shape of the isoquants of the CES pro-duction function changes from the L-shape of the fixed-proportions production func-tion to the curve of the Cobb–Douglas production function to the straight line of thelinear production function.

Table 6.6 summarizes the characteristics of these four specific production functions.

6 . 5 R E T U R N S T O S C A L E 209

constant elasticity of sub-stitution (CES) productionfunction A type of produc-tion function that includeslinear production functions,fixed-proportions productionfunctions, and Cobb–Douglasproduction functions as spe-cial cases.

TABLE 6.6 Characteristics of Production Functions

Elasticity ofProduction Function Substitution (σ) Other Characteristics

Linear production function σ = ∞ Inputs are perfect substitutesIsoquants are straight lines

Fixed-proportions production function σ = 0 Inputs are perfect complementsIsoquants are L-shaped

Cobb–Douglas production function σ = 1 Isoquants are curvesCES production function 0 ≤ σ ≤ ∞ Includes other three production

functions as special casesShape of isoquants varies

returns to scale Theconcept that tells us thepercentage by which outputwill increase when all inputsare increased by a givenpercentage.

6.5R E T U R N ST O S C A L E

In the previous section, we explored the extent to which inputs could be substitutedfor each other to produce a given level of output. In this section, we study how in-creases in all input quantities affect the quantity of output the firm can produce.

D E F I N I T I O N SWhen inputs have positive marginal products, a firm’s total output must increase whenthe quantities of all inputs are increased simultaneously—that is, when a firm’s scale ofoperations increases. Often, though, we might want to know by how much output willincrease when all inputs are increased by a given percentage amount. For example, byhow much would a semiconductor firm be able to increase its output if it doubled itsman-hours of labor and its machine-hours of robots? The concept of returns to scaletells us the percentage increase in output when a firm increases all of its input quanti-ties by a given percentage amount:

Returns to scale = %� (quantity of output)%� (quantity of all inputs)

Suppose that a firm uses two inputs, labor L and capital K, to produce output Q. Nowsuppose that all inputs are “scaled up” by the same proportionate amount λ, whereλ > 1 (i.e., the quantity of labor increases from L to λL, and the quantity of capital


increases from K to λK ).10 Let φ represent the resulting proportionate increase in thequantity of output Q (i.e., the quantity of output increases from Q to φQ). Then:

• If φ > λ, we have increasing returns to scale. In this case, a proportionateincrease in all input quantities results in a greater than proportionate increase inoutput.

• If φ = λ, we have constant returns to scale. In this case, a proportionate in-crease in all input quantities results in the same proportionate increase in output.

• If φ < λ, we have decreasing returns to scale. In this case, a proportionateincrease in all input quantities results in a less than proportionate increase inoutput.

Figure 6.18 illustrates these three cases.Why are returns to scale important? When a production process exhibits increas-

ing returns to scale, there are cost advantages from large-scale operation. In particular,a single large firm will be able to produce a given amount of output at a lower cost perunit than could two equal-size smaller firms, each producing exactly half as much out-put. For example, if two semiconductor firms can each produce 1 million chips at$0.10 per chip, one large semiconductor firm could produce 2 million chips for lessthan $0.10 per chip. This is because, with increasing returns to scale, the large firmneeds to employ less than twice as many units of labor and capital as the smaller firmsto produce twice as much output. When a large firm has such a cost advantage oversmaller firms, a market is most efficiently served by one large firm rather than severalsmaller firms. This cost advantage of large-scale operation has been the traditional jus-tification for allowing firms to operate as regulated monopolists in markets such aselectric power and oil pipeline transportation.


increasing returns to scaleA proportionate increase inall input quantities resultingin a greater than proportion-ate increase in output.

constant returns to scaleA proportionate increase inall input quantities simulta-neously that results in thesame percentage increase inoutput.

decreasing returns to scaleA proportionate increase inall input quantities resultingin a less than proportionateincrease in output.

1 20

1

2

L(a) Increasing Returns to Scale

Q = 3Q = 2Q = 1

K

1 20

1

2

L(b) Constant Returns to Scale

Q = 3

Q = 2Q = 1

K

1 20

1

2

L(c) Decreasing Returns to Scale

Q = 2

Q = 3

Q = 1

K

FIGURE 6.18 Increasing, Constant, and Decreasing Returns to ScaleIn panel (a), doubling the quantities of capital and labor more than doubles output. In panel (b), dou-bling the quantities of capital and labor exactly doubles output. In panel (c), doubling the quantities ofcapital and labor less than doubles output.

10Therefore, the percentage change in all input quantities is (λ − 1) × 100 percent.


L E A R N I N G - B Y - D O I N G E X E R C I S E 6.3Returns to Scale for a Cobb–Douglas Production Function

Problem Does a Cobb–Douglas production function, Q = ALα K β, exhibitincreasing, decreasing, or constant returns to scale?

Solution Let L1 and K1 denote the initial quantities of labor and capital, and let Q1denote the initial output, so Q1 = ALα

1 K β

1 . Now let’s increase all input quantities bythe same proportional amount λ, where λ > 1, and let Q2 denote the resulting volume ofoutput: Q2 = A(λL1)α(λK1)β = λα+β ALα

1 K β

1 = λα+β Q1 . From this, we can see that if:

• α + β > 1, then λα+β > λ, and so Q2 > λQ1 (increasing returns to scale).

• α + β = 1, then λα+β = λ, and so Q2 = λQ1 (constant returns to scale).

• α + β < 1, then λα+β < λ, and so Q2 < λQ1 (decreasing returns to scale).

This shows that the sum of the exponents α + β in the Cobb–Douglas production functiondetermines whether returns to scale are increasing, constant, or decreasing. For this rea-son, economists have paid considerable attention to estimating this sum when studyingproduction functions in specific industries.

Similar Problems: 6.14, 6.15, and 6.16

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6 . 5 R E T U R N S T O S C A L E 211

production function other than Cobb–Douglas) havefound that electricity generation in large plants is proba-bly now characterized by constant returns to scale.12

It is possible that both conclusions are correct. Ifgeneration was characterized by increasing returnsto scale in the 1950s and 1960s but constant returns toscale thereafter, we should expect to see a growth in thescale of generating units throughout the 1950s and 1960sfollowed by smaller growth in later years. This is exactlywhat we observe. The average capacity of all units in-stalled between 1960 and 1964 was 151.7 megawatts. Bythe period 1970–1974, the average capacity of new unitshad grown to 400.3 megawatts. Over the next 10 years,the average capacity of new units continued to grow,but more slowly: Of all units installed between 1980 and1982, the average capacity was 490.3 megawatts.13


Returns to Scale in Electric PowerGeneration

Returns to scale have been thoroughly studied in electricpower generation, where the pioneering work was doneby economist Marc Nerlove.11 Using data from 145 elec-tric utilities in the United States during the year 1955,Nerlove estimated the exponents of a Cobb–Douglasproduction function and found that their sum wasgreater than 1. As illustrated in Learning-By-Doing Exer-cise 6.3, this implies that electricity generation is subjectto increasing returns to scale. Other studies in this sameindustry using data from the 1950s and 1960s also foundevidence of increasing returns to scale. However, studiesusing more recent data (and functional forms for the

11 Marc Nerlove, “Returns to Scale in Electricity Supply,” Chapter 7 in Carl F. Christ, ed., Measurement inEconomics: Studies in Honor of Yehuda Grunfeld (Stanford, CA: Stanford University Press, 1963), pp. 167–198.12 See T. G. Cowing and V. K. Smith, “The Estimation of a Production Technology: A Survey ofEconometric Analyses of Steam Electric Generation,” Land Economics (May 1978), pp. 157–170, andL. R. Christensen and W. Greene, “Economies of Scale in U.S. Electric Power Generation,” Journal ofPolitical Economy (August 1976), pp. 655–676.13These data come from Table 5.3 (p. 50) in P. L. Joskow and R. Schmalensee, Markets for Power:An Analysis of Electric Utility Deregulation (Cambridge, MA: MIT Press, 1983).



pipeline, a company must trade off the benefits of alarger throughput against the expense of generatingthat throughput.

Unlike electric power generation, in which returnsto scale have been estimated using statistical methods,returns to scale for oil pipelines can be deduced usingengineering principles. Applying such principles yields aCobb–Douglas production function14: Q � AH0.37K 1.73,where H denotes hydraulic horsepower, K denotes thesize of the pipe, Q denotes throughput, and A is a con-stant that depends on a variety of factors, including thelength of the pipeline, the variation in the terrain overwhich the pipeline travels, and the viscosity of the oil.Because the exponents of H and K add up to a numbergreater than 1, this production function exhibits in-creasing returns to scale. That is, if we double the diam-eter of the pipe and double the horsepower used topump the oil, the throughput of oil more than doubles.This implies that there are significant cost advantagesto building pipelines with large pipes and many power-ful pumping stations.


Returns to Scale in Oil Pipelines

Another industry in which returns to scale have beenextensively studied is the transportation of oil throughpipelines. The product of an oil pipeline is the volumeof oil transported through the pipeline during a givenperiod—usually called throughput and often measuredin barrels of oil per day. The throughput of an oilpipeline of a given length depends primarily on twofactors: pipe size (i.e., its diameter) and the amount ofpower (hydraulic horsepower) applied to the oil as ittravels through the pipeline. For a fixed amount ofhorsepower, a larger oil pipeline will result in a greaterthroughput. For a pipe of a given size, the more horse-power, the more throughput. In planning a pipeline, acompany controls both of these factors. It can in-crease the horsepower by increasing the number ofpumping stations it installs along the pipeline. And, ofcourse, it can choose the diameter of the pipelineitself. Pumping stations are costly, and pipe is moreexpensive as its diameter increases, so in planning a

R E T U R N S T O S C A L E V E R S U S D I M I N I S H I N GM A R G I N A L R E T U R N SIt is important to understand the distinction between the concepts of returns to scaleand marginal returns (see Section 6.2). Returns to scale pertains to the impact of anincrease in all input quantities simultaneously, while marginal returns (i.e., marginalproduct) pertains to the impact of an increase in the quantity of a single input, such aslabor, holding the quantities of all other inputs fixed.

Figure 6.19 illustrates this distinction. If we double the quantity of labor, from 10to 20 units per year, holding the quantity of capital fixed at 10 units per year, we movefrom point A to point B, and output goes up from 100 to 140 units per year. If we thenincrease the quantity of labor from 20 to 30, we move from B to C, and output goes upto 170. In this case, we have diminishing marginal returns to labor: The increase inoutput brought about by a 10-unit increase in the quantity of labor goes down as weemploy more and more labor.

By contrast, if we double the quantity of both labor and capital from 10 to 20 unitsper year, we move from A to D, and output doubles from 100 to 200. If we triple thequantity of labor and capital from 10 to 30, we move from A to E, and output triplesfrom 100 to 300. For the production function in Figure 6.19 we have constant returnsto scale but diminishing marginal returns to labor.

14This is an approximation of a formula presented in L. Cockenboo, Crude Oil Pipe Lines and Competition inthe Oil Industry (Cambridge, MA: Harvard University Press, 1955).


So far, we have treated the firm’s production function as fixed over time. But asknowledge in the economy evolves and as firms acquire know-how through experienceand investment in research and development, a firm’s production function will change.The notion of technological progress captures the idea that production functionscan shift over time. In particular, technological progress refers to a situation in whicha firm can achieve more output from a given combination of inputs, or equivalently,the same amount of output from lesser quantities of inputs.

We can classify technological progress into three categories: neutral technologicalprogress, labor-saving technological progress, and capital-saving technologicalprogress.15 Figure 6.20 illustrates neutral technological progress. In this case, anisoquant corresponding to a given level of output (100 units in the figure) shifts inward(indicating that lesser amounts of labor and capital are needed to produce a given out-put), but the shift leaves MRTSL,K, the marginal rate of technical substitution of laborfor capital, unchanged along any ray (e.g., 0A) from the origin. Under neutral techno-logical progress, each isoquant corresponds to a higher level of output than before, butthe isoquants themselves retain the same shape.

Figure 6.21 illustrates labor-saving technological progress. In this case, too, theisoquant corresponding to a given level of output shifts inward, but now along any rayfrom the origin, the isoquant becomes flatter, indicating that the MRTSL,K is less thanit was before. You should recall from Section 6.3 that MRTSL, K = MPL/MPK , so thefact that the MRTSL,K decreases implies that under this form of technological progressthe marginal product of capital increases more rapidly than the marginal product oflabor. This form of technological progress arises when technical advances in capitalequipment, robotics, or computers increase the marginal productivity of capital rela-tive to the marginal productivity of labor.

6 . 6 T E C H N O L O G I C A L P R O G R E S S 213

10 30200

10

20

30


Q = 200Q = 170

Q = 300

Q = 140Q = 100

A B C

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FIGURE 6.19 Diminishing Marginal Returnsversus Returns to ScaleThis production function exhibits constant returnsto scale but diminishing marginal returns to labor.

6.6TECHNOLOGICALP R O G R E S S

labor-saving technologicalprogress Technologicalprogress that causes themarginal product of capitalto increase relative to themarginal product of labor.

neutral technologicalprogress Technologicalprogress that decreases theamounts of labor and capitalneeded to produce a givenoutput, without affecting themarginal rate of technicalsubstitution of labor forcapital.

technological progressA change in a productionprocess that enables a firm toachieve more output from agiven combination of inputsor, equivalently, the sameamount of output from lessinputs.

15J. R. Hicks, The Theory of Wages (London: Macmillan, 1932).


Figure 6.22 depicts capital-saving technological progress. Here, as an isoquantshifts inward, MRTSL,K increases, indicating that the marginal product of labor in-creases more rapidly than the marginal product of capital. This form of technologicalprogress arises if, for example, the educational or skill level of the firm’s actual (andpotential) work force rises, increasing the marginal productivity of labor relative to themarginal product of capital.


0L, units of labor per year

Q = 100 isoquant beforetechnological progress

Q = 100 isoquant aftertechnological progress

A

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FIGURE 6.20 Neutral Technological Progress (MRTSL,K Remains the Same)Under neutral technological progress, an isoquant corresponding to any particular level of output shiftsinward, but the MRTSL,K (the negative of the slope of a line tangent to the isoquant) along any ray fromthe origin, such as 0A, remains the same.



Q = 100 isoquant aftertechnological progress

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FIGURE 6.21 Labor-SavingTechnological Progress(MRTSL,K Decreases)Under labor-saving technolog-ical progress, an isoquant cor-responding to any particularlevel of output shifts inward,but the MRTSL,K (the negativeof the slope of a line tangentto the isoquant) along any rayfrom the origin, such as 0A,goes down.

capital-saving technologicalprogress Technologicalprogress that causes the mar-ginal product of labor to in-crease relative to the mar-ginal product of capital.


L E A R N I N G - B Y - D O I N G E X E R C I S E 6.4Technological Progress

A firm’s production function is initially Q = √K L, with MPK =

0.5(√

L/√

K ) and MPL = 0.5(√

K/√

L) . Over time, the production functionchanges to Q = L

√K , with MPK = 0.5(L/

√K ) and MPL = √

K .

Problem

a) Verify that this change represents technological progress.

b) Show whether this change is labor-saving, capital-saving, or neutral.

Solution

a) With any positive amounts of K and L, more Q can be produced with the final produc-tion function. So there is technological progress.

b) With the initial production function, MRTSL, K = MPL/MPK = K/L . With the finalproduction function, MRTSL, K = MPL/MPK = (2K )/L . For any ratio of capital to labor(i.e., along any ray from the origin), MRTSL,K is higher with the second production func-tion. Thus, the technological progress is capital saving.

Similar Problems: 6.18, 6.19, and 6.20

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6 . 6 T E C H N O L O G I C A L P R O G R E S S 215



Q = 100isoquant aftertechnologicalprogress

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FIGURE 6.22 Capital-SavingTechnological Progress (MRTSL,K

Increases)Under capital-saving technologicalprogress, an isoquant correspond-ing to any particular level of out-put shifts inward, but the MRTSL,K

(the negative of the slope of a linetangent to the isoquant) along anyray from the origin, such as 0A,goes up.


CH A P T E R S U M M A R Y


rate of labor productivity declined considerably afterthe oil crisis of 1973. Since 1981, however, labor pro-ductivity in these industries has improved. This coin-cides with a period in which firms in these industriesaggressively reduced the sizes of their work forcesthrough layoffs.

Suer found evidence of significant technologicalprogress in the manufacturing industries he studied.This technological progress was not neutral. Instead, itresulted in an increase in the marginal product of capi-tal relative to the marginal product of labor (i.e., it waslabor-saving technological progress). The steep reduc-tion in labor employment in the industries that Suerstudied is a sign that this type of technologicalprogress was occurring.


Technological Progress in the U.K.

Banu Suer has estimated the magnitude and nature oftechnological progress in a number of manufacturingindustries in the United Kingdom.16 Among the indus-tries included in this study were general chemicals,pharmaceuticals, paint, soap, detergents, syntheticrubber, synthetic plastics and resins, dyes and pig-ments, and fertilizers. In the United Kingdom, theseindustries have traditionally been characterized byhigh profit margins, above-average rates of spendingon research and development as compared to U.K.manufacturing firms generally, and higher-than-averagerates of patent activity. These industries have gener-ally had a good productivity record, but their growth

16Banu Suer, “Total Factor Productivity Growth and Characteristics of the Production Technology in theUK Chemicals and Allied Industries,” Applied Economics, 27 (1995), pp. 277–285.

• The production function tells us the maximum quan-tity of output a firm can get as a function of the quanti-ties of various inputs that it might employ.

• Single-input production functions are total productfunctions. A total product function typically has threeregions: a region of increasing marginal returns, a regionof diminishing marginal returns, and a region of dimin-ishing total returns.

• The average product of labor is the average amountof output per unit of labor. The marginal product oflabor is the rate at which total output changes as thequantity of labor a firm uses changes.

• The law of diminishing marginal returns says that asthe usage of one input (e.g., labor) increases—the quan-tities of other inputs, such as capital or land, being heldfixed—then at some point the marginal product of thatinput will decrease.

• Isoquants depict multiple-input production func-tions in a two-dimensional graph. An isoquant shows allcombinations of labor and capital that produce the samequantity of output.

• For some production functions, the isoquants havean upward-sloping and backward-bending region. Thisregion is called the uneconomic region of production.

Here, one of the inputs has a negative marginal product.The economic region of production is the region ofdownward-sloping isoquants.

• The marginal rate of technical substitution of laborfor capital tells us the rate at which the quantity of capitalcan be reduced for every one unit increase in the quantityof labor, holding the quantity of output constant. Mathe-matically, the marginal rate of technical substitution oflabor for capital is equal to the ratio of the marginal prod-uct of labor to the marginal product of capital.

• Isoquants that are bowed in toward the origin exhibitdiminishing marginal rate of technical substitution.When the marginal rate of technical substitution oflabor for capital diminishes, fewer and fewer units ofcapital can be sacrificed as each additional unit of labor isadded along an isoquant.

• The elasticity of substitution measures the percent-age rate of change of K/L for each 1 percent change inMRTSL,K.

• Three important special production functions are thelinear production function (perfect substitutes), the fixed-proportions production function (perfect complements),and the Cobb–Douglas production function. Each ofthese is a member of a class of production functions knownas constant elasticity of substitution production functions.


• Returns to scale tell us the percentage by which out-put will increase when all inputs are increased by a givenpercentage. If a given percentage increase in the quanti-ties of all inputs increases output by more than that per-centage, we have increasing returns to scale. If a givenpercentage increase in the quantities of all inputs in-creases output by less than that percentage, we have de-creasing returns to scale. If a given percentage increasein the quantities of all inputs increases output by thesame percentage, we have constant returns to scale.

P R O B L E M S 217

• Technological progress refers to a situation in whicha firm can achieve more output from a given combina-tion of inputs, or equivalently, the same amount of out-put from smaller quantities of inputs. Technologicalprogress can be neutral, labor saving, or capital saving,depending on whether the marginal rate of technicalsubstitution remains the same, decreases, or increases fora given capital-to-labor ratio.

RE V I E W Q U E S T I O N S

1. We said that the production function tells us themaximum output that a firm can produce with its quanti-ties of inputs. Why do we include the word maximum inthis definition?

2. Suppose a total product function has the “traditionalshape” shown in Figure 6.2. Sketch the shape of the cor-responding labor requirements function (with quantityof output on the horizontal axis and quantity of labor onthe vertical axis).

3. What is the difference between average product andmarginal product? Can you sketch a total product func-tion such that the average and marginal product functionscoincide with each other?

4. What is the difference between diminishing totalreturns to an input and diminishing marginal returns to aninput? Can a total product function exhibit diminishingmarginal returns but not diminishing total returns?

5. Why must an isoquant be downward sloping whenboth labor and capital have positive marginal products?

6. Could the isoquants corresponding to two differentlevels of output ever cross?

7. Why would a firm that seeks to minimize its expen-ditures on inputs not want to operate on the uneconomicportion of an isoquant?

8. What is the elasticity of substitution? What does ittell us?

9. Suppose the production of electricity requires justtwo inputs, capital and labor, and that the productionfunction is Cobb–Douglas. Now consider the isoquantscorresponding to three different levels of output:Q = 100,000 kilowatt-hours, Q = 200,000 kilowatt-hours, and Q = 400,000 kilowatt-hours. Sketch theseisoquants under three different assumptions aboutreturns to scale: constant returns to scale, increasingreturns to scale, and decreasing returns to scale.

PR O B L E M S

6.1. A firm uses the inputs of fertilizer, labor, and hot-houses to produce roses. Suppose that when the quantityof labor and hothouses is fixed, the relationship betweenthe quantity of fertilizer and the number of roses pro-duced is given by the following table:

Tons of Number of Tons of Number ofFertilizer Roses Fertilizer Roses

Per Month Per Month Per Month Per Month

0 0 5 25001 500 6 26002 1000 7 25003 1700 8 20004 2200

a) What is the average product of fertilizer when 4 tonsare used?b) What is the marginal product of the sixth ton offertilizer?c) Does this total product function exhibit diminishingmarginal returns? If so, over what quantities of fertilizerdo they occur?d) Does this total product function exhibit diminishingtotal returns? If so, over what quantities of fertilizer dothey occur?

6.2. A firm is required to produce 100 units of outputusing quantities of labor and capital (L, K ) = (7, 6). Foreach of the following production functions, state whether


it is possible to produce the required output with the giveninput combination. If it is possible, state whether theinput combination is technically efficient or inefficient.a) Q = 7L + 8Kb) Q = 20

√K L

c) Q = min(16L, 20K )d) Q = 2(K L + L + 1)

6.3. For the production function Q = 6L2 − L3, fill inthe following table and state how much the firm shouldproduce so that:a) average product is maximizedb) marginal product is maximizedc) total product is maximizedd) average product is zero

L Q

0123456

6.4. Suppose that the production function for floppydisks is given by Q = K L2 − L3, where Q is the numberof disks produced per year, K is machine-hours of capital,and L is man-hours of labor.a) Suppose K = 600. Find the total product functionand graph it over the range L = 0 to L = 500. Thensketch the graphs of the average and marginal productfunctions. At what level of labor L does the averageproduct curve appear to reach its maximum? At whatlevel does the marginal product curve appear to reachits maximum?b) Replicate the analysis in (a) for the case in whichK = 1200.c) When either K = 600 or K = 1200, does the totalproduct function have a region of increasing marginalreturns?

6.5. Are the following statements correct or incorrect?a) If average product is increasing, marginal productmust be less than average product.b) If marginal product is negative, average productmust be negative.c) If average product is positive, total product must berising.


d) If total product is increasing, marginal product mustalso be increasing.

6.6. Economists sometimes “prove” the law of dimin-ishing marginal returns with the following exercise: Sup-pose that production of steel requires two inputs, laborand capital, and suppose that the production function ischaracterized by constant returns to scale. Then, if therewere increasing marginal returns to labor, you or I couldproduce all the steel in the world in a backyard blastfurnace. Using numerical arguments based on theproduction function shown in the following table, showthat this (logically absurd) conclusion is correct. The factthat it is correct shows that marginal returns to laborcannot be everywhere increasing when the productionfunction exhibits constant returns to scale.

L K Q

0 100 01 100 12 100 44 100 168 100 64

16 100 25632 100 1024

6.7. Suppose the production function is given by theequation Q = L

√K . Graph the isoquants correspond-

ing to Q = 10, Q = 20, and Q = 50. Do these iso-quants exhibit diminishing marginal rate of technicalsubstitution?

6.8. Consider again the production function for floppydisks: Q = K L2 − L3.a) Sketch a graph of the isoquants for this productionfunction.b) Does this production function have an uneconomicregion? Why or why not?

6.9. Suppose the production function is given by thefollowing equation (where a and b are positive con-stants): Q = a L + b K . What is the marginal rate oftechnical substitution of labor for capital (MRTSL,K) atany point along an isoquant?

6.10. You might think that when a production functionhas a diminishing marginal rate of technical substitutionof labor for capital, it cannot have increasing marginalproducts of capital and labor. Show that this is not true,using the production function Q = K 2L2, with the cor-responding marginal products MPK = 2K L2 andMPL = 2K 2L.

6.11. Suppose that a firm’s production function isgiven by Q = K L + K , with MPK = L + 1 and


MPL = K . At point A, the firm uses K = 3 units ofcapital and L = 5 units of labor. At point B, alongthe same isoquant, the firm would only use 1 unit ofcapital.a) Calculate how much labor is required at point B.b) Calculate the elasticity of substitution between Aand B. Does this production function exhibit a higheror lower elasticity of substitution than a Cobb–Douglasfunction over this range of inputs?

6.12. Two points, A and B, are on an isoquant drawnwith labor on the horizontal axis and capital on the ver-tical axis. The capital–labor ratio at B is twice that at A,and the elasticity of substitution as we move from A toB is 2. What is the ratio of the MRTSL,K at A versus thatat B?

6.13. Let B be the number of bicycles produced from Fbicycle frames and T tires. Every bicycle needs exactlytwo tires and one frame.a) Draw the isoquants for bicycle production.b) Write a mathematical expression for the productionfunction for bicycles.

6.14. What can you say about the returns to scale ofthe linear production function Q = a K + b L, where aand b are positive constants?

6.15. What can you say about the returns to scale ofthe Leontief production function Q = min(a K , b L) ,where a and b are positive constants?

6.16. A firm produces a quantity Q of breakfast cerealusing labor L and material M with the production func-tion Q = 50

√ML + M + L. The marginal product

functions for this production function are

MPL = 25

√

ML

+ 1

MPM = 25

√

LM

+ 1

a) Are the returns to scale increasing, constant, ordecreasing for this production function?

A P P E N D I X : A C O B B – D O U G L A S P R O D U C T I O N F U N C T I O N 219

b) Is the marginal product of labor ever diminishingfor this production function? If so, when? Is it evernegative, and if so, when?

6.17. Consider a CES production function given byQ = (K 0.5 + L0.5)2 .a) What is the elasticity of substitution for thisproduction function?b) Does this production function exhibit increasing,decreasing, or constant returns to scale?c) Suppose that the production function took the formQ = (100 + K 0.5 + L0.5)2 . Does this productionfunction exhibit increasing, decreasing, or constantreturns to scale?

6.18. Suppose a firm’s production function initiallytook the form Q = 500(L + 3K ) . However, as a result ofa manufacturing innovation, its production function isnow Q = 1000(0.5L + 10K ) .a) Show that the innovation has resulted in technologi-cal progress in the sense defined in the text.b) Is the technological progress neutral, labor saving,or capital saving?

6.19. A firm’s production function is initially Q =√K L, with MPK = 0.5(

√L/

√K ) and MPL =

0.5(√

K/√

L) . Over time, the production functionchanges to Q = K L, with MPK = L and MPL = K .a) Verify that this change represents technologicalprogress.b) Is this change labor saving, capital saving, or neutral?

6.20. A firm’s production function is initially Q =√K L, with MPK = 0.5(

√L/

√K ) and MPL =

0.5(√

K/√

L) . Over time, the production functionchanges to Q = K

√L, with MPK = √

L and MPL =0.5(K/

√L).

a) Verify that this change represents technologicalprogress.b) Is this change labor saving, capital saving, orneutral?

A P P E N D I X : The Elasticity of Substitution for a Cobb–Douglas Production Function

In this appendix we derive the elasticity of substitution for a Cobb–Douglas produc-tion function, f (L, K ) = ALα K β . The marginal product of labor and capital arefound by taking the partial derivatives of the production function with respect to labor



and capital, respectively (for a discussion of partial derivatives, see the MathematicalAppendix in this book):

MPL = ∂ f∂L

= αALα−1K β

MPK = ∂ f∂K

= β ALα K β−1

Now, recall that, in general,

MRTSL, K = MPL

MPK

Thus, for this Cobb–Douglas production function,

MRTSL, K = αALα−1K β

β ALα K β−1

= αKβL

Rearranging terms yields

KL

= β

αMRTSL, K (A6.1)

Therefore, �(K/L) = (β/α)�MRTSL, K or:

�( K

L

)

�MRTSL, K= β

α(A6.2)

Also, from (A6.1),

MRTSL, K( K

L

) = α

β(A6.3)

Now, using the definition of the elasticity of substitution in equation (6.6):

σ = %�( K

L

)

%�MRTSL, K= �

( KL

)

/ KL

(�MRTSL, KMRTSL, K

)

=(

�( K

L

)

�MRTSL, K

)(

MRTSL, KKL

)

(A6.4)

Substituting (A6.2) and (A6.3) into (A6.4) yields

σ = β

α× α

β= 1

That shows that the elasticity of substitution along a Cobb–Douglas production func-tion is equal to 1 for all values of K and L.


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