5 production analysis

CHAPTER 6 Production Theoryand Estimation

CHAPTER OUTLINE

6-1 The Organization of Production and the Production Function • The Organization ofProduction • The Production Function

6-2 The Production Function with One Variable Input • Total, Average, and Marginal Product• The Law of Diminishing Returns and Stages of Production

6-3 Optimal Use of the Variable Input

6-4 The Production Function with Two Variable Inputs • Production Isoquants • EconomicRegion of Production • Marginal Rate ofTechnical Substitution • Perfect Substitutes andComplementary Inputs

6-5 Optimal Combination of Inputs • Isocost Lines • Optimal Input Combination forMinimizing Costs or Maximizing Output • Profit Maximization • Effect of Change in InputPrices • Case Study 6-1: Substitutability between Gasoline Consumption and Driving Tlme

6-6 Returns to Scale • Case Study 6-2: Returns to Scale in U.s. Manufacturing Industries• Case Study 6-3: Bajaj Auto Limited-Smaller Is Better

6-7 Empirical Production Functions • Case Study 6-4: Output Elasticities in U.s.Manufacturing Industries

6-8 The Innovation Process • Meaning and Importance of Innovations • Case Study 6-5: HowDo Firms Get New Technology? • The Open Innovation Model

6-9 Innovation and Global Competitiveness • Innovations and the InternationalCompetitiveness of U.S. Firms • Case Study 6-6: How Xerox Lost and Regained InternationalCompetitiveness and Became a Leader in Information Technology • Case Study 6-7: The NewU.s. Digital Factory • Case Study 6-8: The Euro and the International Competitiveness ofEuropean Firms • The New Computer-Aided Production Revolution and the InternationalCompetitiveness of U.S. Firms

Summary • Problems

215

216 PARTTHREE Production and Cost Analysis

Appendix to Chapter 6: Production Analysis with Calculus • Constrained OutputMaximization • Constrained Cost Minimization • Profit Maximization • Appendix Problems

Supplementary Readings • Internet Site Addresses

KEY TERMS (in the order of their appearance)

ProductionInputsFixed inputsVariable inputsShort runLong runProduction functionTotal product (TP)Marginal product (MP)Average product (AP)Output elasticityLaw of diminishing returnsStages I. II. III of production

Marginal revenue product(MRP)

Marginal resource cost(MRC)

IsoquantRidge linesMarginal rate oftechnical

substitution (MRTS)Isocost lineExpansion pathConstant returns to scaleIncreasing returns to scaleDecreasing returns to scale

Cobb-Douglas productionfunction

Product innovationProcess innovationClosed innovation modelOpen innovation modelProduct cycle modelJust-in-time production

systemCompetitive benchmarkingComputer-aided design (CAD)Computer-aided manufacturing

(CAM)

InChapter 1, we defined the firm as an organization that combines and organizes labor,capital, and land or raw materials for the purpose of producing goods and services forsale. The aim of the firm is to maximize total profits or achieve some other related aim,

such as maximizing sales or growth. The basic production decision facing the firm is howmuch of the commodity or service to produce and how much labor, capital, and other re-sources or inputs to use to produce that output most efficiently. To answer these questions,the firm requires engineering or technological data on production possibilities (the produc-tionfunction) as well as economic data on input and output prices. This chapter providesthe framework for understanding the economics of production of the firm and derives a setof conditions for efficient production.

The chapter begins with a discussion of the production function, which summarizesthe engineering and technological production possibilities open to the firm. This generaldiscussion is extended to the specific case where there is a single variable input orresource (Section 6-2) and examines how much of the variable input the firm shouldemploy to maximize profits (Section 6-3). We then go on to examine the productionfunction when there are two variable inputs (Section 6-4) and to develop the conditionsfor their efficient combination in production (Section 6-5). In Section 6-6, we discussreturns to scale where all resources or inputs are variable. Section 6-7 discusses theempirical estimation of production functions. Section 6-8 deals with the innovationprocess. Finally, Section 6-9 deals with technological progress and innovations andtheir importance for the domestic and global competitiveness of firms. In the ChapterAppendix (which is optional), we use simple calculus to examine the conditions for

CHAPTER6 Production Theory and Estimation 217

maximizing output, minimizing costs, or maximizing profits. The nine case studies pre-sented throughout the chapter highlight the importance of production theory to the firmand its great relevance in managerial economics.

6-1 THE ORGANIZATION OF PRODUCTIONAND THE PRODUCTION FUNCTION

In this section, we examine first the organization of production and classify inputs into var-ious broad categories, and then we define the meaning and usefulness of the productionfunction in analyzing the firm's production activity.

The Organization of ProductionProduction refers to the transformation of inputs or resources into outputs of goods andservices. For example, IBM hires workers to use machinery, parts, and raw materials in fac-tories to produce personal computers. The output of a firm can either be a final commodity(such as a personal computer) or an intermediate product, such as semiconductors (whichare used in the production of computers and other goods). The output can also be a servicerather than a good. Examples of services are education, medicine, banking, communica-tion, transportation, and many others. Note that production refers to all of the activitiesinvolved in the production of goods and services, from borrowing to setting up or expand-ing production facilities, to hiring workers, purchasing raw materials, running quality con-trol, cost accounting, and so on, rather than referring merely to the physical transformationof inputs into outputs of goods and services.

Inputs are the resources used in the production of goods and services. As a convenientway to organize the discussion, inputs are classified into labor (including entrepreneurialtalent), capital, and land or natural resources. Each of these broad categories, however, in-cludes a great variety of the basic input. For example, labor includes bus drivers, assembly-line workers, accountants, lawyers, doctors, scientists, and many others. Inputs are alsoclassified as fixed or variable. Fixed inputs are those that cannot be readily changed dur-ing the time period under consideration, except perhaps at very great expense. Examples offixed inputs are the firm's plant and specialized equipment (it takes several years for IBMto build a new factory to produce computer chips to go into its computers). On the otherhand, variable inputs are those that can be varied easily and on very short notice. Exam-ples of variable inputs are most raw materials and unskilled labor.

The time period during which at least one input is fixed is called the short run,while the time period when all inputs are variable is called the long run. The length ofthe long run (i.e., the time period required for all inputs to be variable) depends on theindustry. For some, such as the setting up or expansion of a dry-cleaning business, thelong run may be only a few months or weeks. For others, such as the construction of anew electricity-generating plant, it may be many years. In the short run, a firm can in-crease output only by using more of the variable inputs (say, labor and raw materials) to-gether with the fixed inputs (plant and equipment). In the long run, the same increase inoutput could likely be obtained more efficiently by also expanding the firm's production


facilities (plant and equipment). Thus, we say that the firm operates in the short run andplans increases or reductions in its scale of operation in the long run. In the long run,technology usually improves, so that more output can be obtained from a given quantityof inputs, or the same output from less inputs.

The Production FunctionJust as demand theory centers on the concept of the demand function, production theory re-volves around the concept of the production function. A production function is an equa-tion, table, or graph showing the maximum output of a commodity that a firm can produceper period of time with each set of inputs. Both inputs and outputs are measured in physi-cal rather than in monetary units. Technology is assumed to remain constant during the pe-riod of the analysis.

For simplicity we assume here that a firm produces only one type of output (commod-ity or service) with two inputs, labor (L) and capital (K). Thus, the general equation of thissimple production function is

Q=f~L, K) [6-1]

Equation 6-1 reads: The quantity of output is a function of, or depends on, the quantity oflabor and capital used in production. Output refers to the number of units of the commod-ity (say, automobiles) produced, labor refers to the number of workers employed, andcapital refers to the amount of the equipment used in production. We assume that all unitsof Land K are homogeneous or identical. An explicit production function would indicateprecisely the quantity of output that the firm would produce with each particular set of in-puts of labor and capital. Although our discussion will be in terms of a single output pro-duced with only two inputs, the principles that we will develop are general and apply tocases in which the firm uses more than two inputs and produces more than one output (theusual situation).

Table 6-1 gives a hypothetical production function which shows the outputs (the Q's)that the firm can produce with various combinations of labor (L) and capital (K). The tableshows that by using 1 unit of labor (1L) and 1 unit of capital (1K), the firm would produce3 units of output (3Q). With 2L and lK, output is 8Q; with 3L and lK, output is 12Q; with3L and 2K, output is 28Q; with 4L and 2K, output is 30Q, and so on. Note also that labor and

_ Production Function with Two Inputs

Capital (K) 6 10 24 31 36 40 395 12 28 36 40 42 404 12 28 36 40 40 36 Output(Q)

3 10 23 33 36 36 33r 2 7 18 28 30 30 28K 1 3 8 12 14 14 12

2 3 4 5 6L~ Labor (L)

CHAPTER 6 Production Theory and Estimation 219

Output (0)

54

CillJ/lill r.t;

FIGURE 6-1 Discrete Production Surface The height of the bars refers to the maximum output (Q)that can be produced with each combination of labor (L) and capital (K) shown on the axes. Thus, thetops of all the bars form the production surface for the firm.

capital can be substituted for each other in production. For example, 12Q can be producedeither with 3L and lK or with lL and 4K. I Input prices will determine which of these twocombinations of labor and capital is cheaper. The output that the firm will want to produceis the one that maximizes its total profits. These questions will be examined and answeredlater in the chapter.

The production relationships given in Table 6-1 are shown graphically in Figure 6-1,which is three-dimensional. In Figure 6-1, the height of the bars refers to the maximumoutput that can be produced with each combination of labor and capital shown on the axes.Thus, the tops of all the bars form the production surface for the firm.

If we assume that inputs and outputs are continuously or infinitesimally divisible(rather than being measured in discrete units), we would have an infinite number of out-puts, each resulting from one of the infinite number of combinations of labor and capitalthat could be used in production. This is shown in Figure 6-2, in which the axes forming the

I 12Q could also be produced with lK and 6L instead of lK and 3L (see the last entry in the first row of thetable), but the firm would certainly not want to use this combination of labor and capital. Similarly, l2Q couldbe produced with lL and either 4K or 5K, but the firm would not want to use the latter input combination.


a

II

II

II~------------------------I

II

~ __ --"ZICIIIIIIII

------~DI

II

II

I/

II

II

I II I

~II B

II

I

.•..:::I0-.•..:::Io

o L

Labor

FIGURE 6-2 Continuous Production Surface The horizontal and inclined axes measure,respectively, the labor and capital inputs, while the vertical axis measures the height of the surface orthe maximum level of output resulting from each input combination-all assumed to be continuouslydivisible. The output generated by holding capital constant at Kl and increasing labor from zero to L2units is given by the height of cross section K1AB (with base parallel to the labor axis).

base of the figure measure the labor and capital inputs, while the height of the surface givesthe (maximum) level of output resulting from each input combination, all assumed to becontinuously divisible.

For example, by keeping the quantity of capital used at KI in Figure 6-2 and increas-ing the quantity of labor used from zero to ~ units (so that we are in the short run), the firmgenerate the output shown by the height of cross section KIAB (with base parallel to thelabor axis). On the other hand, by increasing the amount of labor used from zero to ~ unitsbut keeping capital constant at K2 rather than KI (so that we are still in the short run), thefirm generates the output shown by the top of cross section K2CD. If instead the firm keptlabor constant at LI and increased the quantity of capital used from zero to K2 units, thefirm's output would be the one shown by the top of cross section LIEF (with base parallelto the capital axis). With labor constant at~, on the other hand, the firm's output generatedby changing the quantity of capital used from zero to K2 units would be the one shown bythe height of cross section L2CD.

6-2 THE PRODUCTION FUNCTION WITH ONE VARIABLE INPUT

In this section, we present the theory of production when only one input is variable. Thus,we are in the short run. We begin by defining the total, the average, and the marginal prod-uct of the variable input and deriving from these the output elasticity of the variable input.


Total, Marginal, and Avetage Product of labor, and Output Elasticltv

(1) (2) (3) (4) (5)

Labor Output Marginal Average Output(number of or Product Product Elasticity

workers) Total Product of Labor of Labor of Labor

0 0

1 3 3 3 1

2 8 5 4 1.25

3 12 4 4 1

4 14 2 3.5 0.57

5 14 0 2.8 0

6 12 -2 2 -I

We will then examine the law of diminishing returns and the meaning and importance ofthe stages of production. These concepts will be used in Section 6-3 to determine the opti-mal use of the variable input for the firm to maximize profits.

Total, Average, and Marginal ProductBy holding the quantity of one input constant and changing the quantity used of the otherinput, we can derive the total product (TP) of the variable input. For example, by holdingcapital constant at 1 unit (i.e., with K = 1) and increasing the units of labor used from zeroto 6 units, we generate the total product of labor given by the last row in Table 6-1, whichis reproduced in column 2 of Table 6-2. Note that when no labor is used, total output orproduct is zero. With one unit of labor (lL), total product (TP) is 3. With 2L, TP = 8. With3L, TP = 12, and so on.2

From the total product schedule we can derive the marginal and average productschedules of the variable input. The marginal product (MP) of labor (MPL) is thechange in total product or extra output per unit change in labor used, while the averageproduct (AP) of labor (APL) equals total product divided by the quantity of labor used.That is,3

MPL

= !l.TPs:

APL

= TPL

[6-2]

[6-3]

Column 3 in Table 6-2 gives the marginal product of labor (MPL). Since labor increases by1 unit at a time in column 1, the MPL in column 3 is obtained by subtracting successive

2 The reason for the decline in TP when 6L is used will be discussed shortly.3 In terms of calculus, MPL = dTPldL.


quantities of TP in column 2. For example, TP increases from 0 to 3 units when the firstunit of labor is used. Thus, MPL = 3. For an increase in labor from lL to 2L, TP rises from3 to 8 units, so that MPL = 5, and so on. Column 4 of Table 6-2 gives the APL. This equalsTP (column 2) divided by L (column 1). Thus, with 1 unit of labor (IL), APL = 3. With 2L,APL = 4, and so on.

Column 5 in Table 6-2 gives the production or output elasticity of labor (EL). Thismeasures the percentage change in output divided by the percentage change in the quantityof labor used. That is,

E - %~QL- %M [6-4]

By rewriting Equation 6-4 in a more explicit form and rearranging, we get

E _ ~Q/Q _ ~Q/M _ MPL

L - MIL - QIL - APL [6-5]

That is, the output elasticity of labor is equal to the ratio of MPL to APL.4 For example,for the first unit of labor, EL = != 1. This means that from OL to lL (and with K = 1),TP or output grows proportionately to the growth in the labor input. For the second unitof labor, EL = 1.25 (i.e., TP or output grows more than proportionately to the increase in L),and so on.

Plotting the total, marginal, and average product of labor of Table 6-2 gives the corre-sponding product curves shown in Figure 6-3. Note that TP grows to 14 units with 4L,remains at 14 units with 5L, and then declines to 12 units with 6L (see the top panel ofFigure 6-3). The reason for this is that with the addition of the sixth worker, workers beginto get in each other's way and total product declines. In the bottom panel, we see that APL

rises to 4 units and then declines. Since the marginal product of labor refers to the changein total product per unit change in labor used, each value of the MPL is plotted halfway be-tween the quantities of labor used. Thus, the MPL of 3 units of output that results by goingfrom OL to lL is plotted at O.5L. The Mh of 5 that results from increasing labor from lL to2L is plotted at 1.5L, and so on. The MPL curve rises to 5 units of output at 1.5L and thendeclines. Past 4.5L, the Mh becomes negative.

Had the firm kept capital fixed at K = 2, increasing the amount of labor used fromOL to 6L would have given the TP shown by the second row from the bottom in Table 6-1. This would correspond to the cross section at K = 2 in Figure 6-1. From this TP func-tion and curve, we could then derive the MPL and APL functions as done in Table 6-2 andshown in two-dimensional space in Figure 6-3. While the actual values of TP, MPL, andAPL for K = 2 would differ from the corresponding ones with K = 1, the shape of thecurves would generally be the same (see Problem 2).

4 In terms of calculus,

aQ LEL =-.-aL Q

0 E14

12I /C ~ TP

t::I.."eCL

8•..0.•..::ICL.•..::I0Iii.•..0f-

3

0L

0 1 2 3 4 5 6Quantity of labor


t5::I.."e 4

CL

01 3Cl'" •.. 2

~ .~ APL

••• 0

I01.0

"> '"'" -.." .•.. 0I: 0

'"Iii -14 - J ~ .....••••... v LaborI: -2 I.~

'"::E

FIGURE 6-3 Total, Marginal, and Average Product of Labor Curves The top panel shows the totalproduct of labor curve. TP is highest between 4L and 5L. The bottom panel shows the marginal and theaverage product of labor curves. The MPL is plotted halfway between successive units of labor used. TheMPL curve rises up to lo5L and then declines, and it becomes negative past 4.5L. The APL is highestbetween 2L and 3L.

The law of Diminishing Returns and Stages of ProductionIn order to show graphically the relationship between the total product, on the one hand,and the marginal and average products of labor, on the other, we assume that labor time iscontinuously divisible (i.e., it can be hired for any part of a day). Then the TP, MPL, andAPL become smooth curves as indicated in Figure 6-4. The MPL at a particular point on theTP curve is given by the slope of the TP curve at that point. From Figure 6-4, we see thatthe slope of the TP curve rises up to point G (the point of inflection on the TP curve), is zeroat point J, and is negative thereafter. Thus, the MPL rises up to point G', is zero at point J',and is negative afterward.


TPJ

14

Labor

12TP

8

3

o IIii

MP I I__ L ----Stage I of labor_i_Stage II of labor_I •• ---Stage IIIof labor----+-

APL Ii G' i I

I H' iI i

o 1 2 5 63 4

5432

Labor

-2

o

FIGURE 6-4 Total, Marginal, and Average Product Curves, and Stages of Production With labortime continuously divisible, we have smooth TP, MP, and AP curves. The MPL (given by the slope of thetangent to the TP curve) rises up to point G', becomes zero atr, and is negative thereafter. The APL (givenby the slope of the ray from the origin to a point on the TP curve) rises up to point H' and declinesthereafter (but remains positive as long as TP is positive). Stage I of production for labor corresponds tothe rising portion of the APL• Stage II covers the range from maximum APL to where MPL is zero. Stage IIIoccurs when MPL is negative.

The APL is given by the slope of a ray from the origin to the TP curve. From Figure 6-4,we see that the slope of the TP curve rises up to point H and falls thereafter but remainspositive as long as TP is positive. Thus, the APL rises up to point H' and falls afterward.Note that at point H the slope of a ray from the origin to the TP curve (or APL) is equalto the slope of the TP curve (or Mh). Thus, Ah = MPL at point H' (the highest point onthe APL curve). Note also that APL rises as long as MPL is above it and falls when MPL isbelow it.

From Figure 6-4, we can also see that up to point G, the TP curve increases at an in-creasing rate so that the MPL rises. Labor is used so sc-arcely with the 1 unit of capital that


the MPL rises as more labor is used. Past point G, however, the TP curve rises at adecreasing rate so that the MPL declines. The declining portion of the MPL curve is areflection of the law of diminishing returns. This postulates that as we use more andmore units of the variable input with a given amount of the fixed input, after a point, weget diminishing returns (marginal product) from the variable input. In Figure 6-4, the lawof diminishing returns begins to operate after 1.5L is used (after point G' in the bottompanel of Figure 6-4). Note that diminishing returns is not a theorem that can be proved ordisproved with logic but is a physical law which has been found to be always empiricallytrue. It states that after a point, we will invariably get diminishing returns from the vari-able input. That is, as the firm uses more and more units of the variable input with thesame amount of the fixed input, each additional unit of the variable input has less and lessof the fixed input to work with and, after a point, the marginal product of the variable inputdeclines.

The relationship between the MPL and APL curves in the bottom panel of Fig-ure 6-4 can be used to define three stages of production for labor (the variable input). Therange from the origin to the point where the APL is maximum (point H' at 2.5L) is stage Iof production for labor. Stage II of production for labor extends from the point where theAPL is maximum to the point where the MPL is zero (i.e., from point H' at 2.5L to point J'at 4.5L). The range over which the MPL is negative (i.e., past point J' or with more than4.5L) is stage III of production for labor. The rational producer would not operate in stage IIIof labor, even if labor time were free, because MPL is negative. This means that a greateroutput or TP could be produced by using less labor! Similarly, he or she will not producein stage I for labor because (as shown in more advanced texts) this corresponds to stage IIIof capital (where the MP of capital is negativej' Thus, the rational producer will operatein stage II where the MP of both factors is positive but declining. The precise point withinstage II at which the rational producer operates will depend on the prices of inputs and out-put. This is examined in the next section.

6-3 I OPTIMAL USE OF THE VARIABLE INPUT

How much labor (the variable input in our previous discussion) should the firm use in orderto maximize profits? The answer is that the firm should employ an additional unit of laboras long as the extra revenue generated from the sale of the output produced exceeds the extracost of hiring the unit of labor (i.e., until the extra revenue equals the extra cost). For exam-ple, if an additional unit of labor generates $30 in extra revenue and costs an extra $20 tohire, it pays for the firm to hire this unit of labor. By doing so, the firm adds $30 to its rev-enues and $20 to its costs, so that its total profits increase. It does not pay, however, for thefirm to hire an additional unit of labor if the extra revenue it generates falls short of the extracost incurred. This is an example or application of the general optimization principle exam-ined in Chapter 2.

5 Stage [ of labor corresponds to stage III of capital only under constant returns to scale. This is the case whereoutput changes in the same proportion as the change in all inputs (see Section 6-6).


Marginal Revenue Product and Marginal Resource Cost of Labor

(1) (2) (3) (4) = (2) x (3) (5)

Units Marginal Marginal Marginalof Marginal Revenue Revenue Resource

Labor Product =p Product Cost=w2.5 4 $10 $40 $203.0 3 10 30 203.5 2 10 20 204.0 1 10 10 204.5 0 10 0 20

The extra revenue generated by the use of an additional unit of labor is called the mar-ginal revenue product of labor (MRPd. This equals the marginal product of labor (MPL)

times the marginal revenue (MR) from the sale of the extra output produced. That is,

MRPL = (MPd(MR) [6-6]

On the other hand, the extra cost of hiring an additional unit of labor or marginal resourcecost of labor (MRCd is equal to the increase in the total cost to the firm resulting from hir-ing the additional unit of labor. That is,

MRCL

= !1TCsi. [6-7]

Thus, a firm should continue to hire labor as long as MRPL > MRCL and until MRPL =MRCL. We can examine the optimal use of labor (and profit maximization) by thefirm facing the short-run production function discussed in Section 6-2 with the aid ofTable 6-3.

Column 2 in Table 6-3 gives the marginal product of labor read off from the MPL

curve in stage n of production in the bottom panels of Figures 6-3 and 6-4. The frac-tional units of labor are based on the assumption that the firm can hire labor for the fullor for half a day. Only the MPL in stage II is given in column 2 because the firm wouldnever produce in stage III of labor (where the MPL is negative) or in stage I of labor(which corresponds to the stage of negative marginal product for capital). Column 3gives the marginal revenue of $10 from the sale of each additional unit of the commod-ity produced, on the assumption that the firm is small and can sell the additional units ofthe commodity at the given market price (P) of $10. Column 4 gives the marginal rev-enue product of labor obtained by multiplying the MPL in column 2 by the MR = P ofthe commodity in column 3. Note that the MRPL declines because the MPL declines.Column 5 gives the marginal resource cost of labor on the assumption that the firm issmall and can hire additional units of labor at the constant market wage rate (w) of $20for each half-day of work.

From Table 6-3, we see that the firm should hire 3.5 units of labor because that iswhere MRPL = MRCL = $20. At less than 3.5L, MRPL > MRCL, and the firm would beadding more to its total revenues than to its total cost by hiring more labor. For example,


$

20 I "" MRCL = W = $20

40

30

10

oo 2.5 3.0 3.5 4.0 4.5 L

Units of labor used

FIGURE 6·5 Optimal Use of Labor It pays for the firm to hire more labor as long as the marginalrevenue product of labor (MRPLl exceeds the marginal resource cost of hiring labor (MRCL), and untilMRPL = MRCL• With MRCL = W= $20, the optimal amount of labor for the firm to use is 3.5 units. At 3.5L,

MRPL = MRCL = $20, and the firm maximizes total profits.

with 3L, MRPL = $30, and this exceeds the MRCL of $20. By hiring more labor, the firmwould increase its total profits. On the other hand, if the firm used more than 3.5L, MRPL <MRCL, and the firm would be adding more to its total costs than to its total revenue, andits total profits would be lower. For example, with 4L, MRPL = $10, while MRCL = $20.The firm could then increase its total profits by hiring less labor. Only with 3.5L willMRPL = MRCL = $20, and the firm's profits are maximized. Thus, the optimal use of laboris 3.5 units. Note that the marginal revenue product of labor (MRPd schedule in column 4of Table 6-3 represents the firm's demand schedule for labor. It gives the amount of labordemanded by the firm at various wage rates. For example, if the wage rate per day (w)were $40, the firm would hire 2.5 units of labor because that would be where MRPL =MRCL = W = $40. If w = $30, the firm would demand 3 units of labor. If w = $20, the firmwould demand 3.5L, and with w = $10, the firm would demand 4L. This is shown inFigure 6-5, where di. = MRPL represents the firm's demand curve for labor. The figureshows that if the wage rate per day (w) were constant at $20, the firm would demand 3.5L,as indicated above.

This discussion is applicable to any variable input, not just labor. To be noted is thatthe MRP of the variable input can be found not only by multiplying the marginal productof the input by the marginal revenue from the sale of the output produced but it can also beobtained from the change in the total revenue that would result per unit change in the vari-able input used (see Problem 5).


6-4 THE PRODUCTION FUNCTION WITH TWO VARIABLE INPUTS

We now examine the production function when there are two variable inputs. This can berepresented graphically by isoquants. In this section we define isoquants and discuss theircharacteristics. Isoquants will then be used in Section 6-5 to develop the conditions for theefficient combination of inputs in production.

Production IsoquantsAn isoquant shows the various combinations of two inputs (say, labor and capital) that thefirm can use to produce a specific level of output. A higher isoquant refers to a larger out-put, while a lower isoquant refers to a smaller output. Isoquants can be derived from Table6-4, which repeats the production function of Table 6-1 with lines connecting all thelabor-capital combinations that can be used to produce a specific level of output. Forexample, the table shows that 12 units of output (that is, 12Q) can be produced with 1 unitof capital (that is, 1K) and 3 units of labor (that is, 3L) or with lK and 6L. 6 The output of12Q can also be produced with IL and 4K, and IL and 5K. These are shown by the lowestisoquant in Figure 6-6. The isoquant is smooth on the assumption that labor and capital arecontinuously divisible. Table 6-4 also shows that 28Q can be produced with 2K and 3L, 2Kand 6L, 2L and 4K, and 2L and 5K (the second isoquant marked 28Q in Figure 6-6). Thetable also shows the various combinations of Land K that can be used to produce 36Q and40Q (shown by the top two isoquants in the figure). Note that to produce a greater output,more labor, more capital, or more of both labor and capital are required.

Economic Region of ProductionWhile the isoquants in Figure 6-6 (repeated in Figure 6-7) have positively sloped portions,these portions are irrelevant. That is, the firm would not operate on the positively slopedportion of an isoquant because it could produce the same level of output with less capitaland less labor. For example, the firm would not produce 36Q at point U in Figure 6-7 with

. : Production Function with Two Variable Inputs

Output (Q)

Capital (K) 6

5

43

i 2K 1

6 Since inputs are not free, a firm would produce 12Q with lK and 3L rather than with IK and 6L.


Capital

6

:] M

N

3

2

1

0

0 1

400

360

280

120

T5

6Labor

2 3 4 5

FIGURE 6-6 Isoquants An isoquant shows the various combinations of two inputs that can beused to produce a specific level of output (Q). From Table 6·4, we can see that 12Q can be producedwith 1L and SK (point M), 1L and 4K (point N), 2L and l.SK (point R), 3L and i« (point 5), or 6L and lK(point T). Higher isoquants refer to higher levels of output.

Capital

6

5

4

3

2

1

0

0 1 2 3 4 5 6Labor

FIGURE 6-7 The Relevant Portion of Isoquants The economic region of production is given by thenegatively sloped segment of isoquants between ridge lines OVI and OZI. The firm will not produce inthe positively sloped portion of the isoquants because it could produce the same level of output withboth less labor and less capital.

230 PART THREE Production and Cost Analysis

6L and 4K because it could produce 36Q by using the smaller quantities of labor and capi-tal indicated by point V on the same isoquant. Similarly, the firm would not produce 36Qat point W with 4L and 6K because it could produce 36Q at point Z with less Land K. Sinceinputs are not free, the firm would not want to produce in the positively sloped range ofisoquants.

Ridge lines separate the relevant (i.e., negatively sloped) from the irrelevant (or posi-tively sloped) portions of the isoquants. In Figure 6-7, ridge line OVI joins points on the var-ious isoquants where the isoquants have zero slope. The isoquants are negatively sloped tothe left of this ridge line and positively sloped to the right. This means that starting, for ex-ample, at point Von the isoquant for 36Q, if the firm used more labor, it would also haveto use more capital to remain on the same isoquant (compare point U to point V). Startingfrom point V, if the firm used more labor with the same amount of capital, the level of out-put would fall (i.e., the firm would fall back to a lower isoquant; see the dashed horizontalline in the figure). The same is true at all other points on ridge line OVI. Therefore, the MPL

must be negative to the right of this ridge line. This corresponds to stage IIIof productionfor labor.

Ridge line OZI joins points where the isoquants have infinite slope. The isoquants arenegatively sloped to the right of this ridge line and positively sloped to the left. This meansthat starting, for example, at point Z on the isoquant for 36Q, if the firm used more capi-tal, it would also have to use more labor to remain on the same isoquant (compare point Wwith point Z). Starting at point Z, if the firm used more capital with the same quantity oflabor, the level of output would fall (i.e., the firm would fall back to a lower isoquant; seethe dashed vertical line in the figure). The same is true at all other points on ridge line OZI.Therefore, the MPK must be negative to the left of or above this ridge line. This corre-sponds to stage IIIof production for capital.

Thus, we conclude that the negatively sloped portion of the isoquants within the ridgelines represents the relevant economic region of production. This refers to stage II of pro-duction for labor and capital, where the MPL and the MPK are both positive but declining.Producers will never want to operate outside this region. As a result, in the rest of the chap-ter we will draw only the negatively sloped portions of isoquants.

Marginal Rate of Technical SubstitutionWe saw in the previous section that isoquants are negatively sloped in the economicallyrelevant range. This means that if the firm wants to reduce the quantity of capital that ituses in production, it must increase the quantity of labor in order to remain on the sameisoquant (i.e., produce the same level of output). For example, the movement from pointN to point R on isoquant 12Q in Figure 6-8 indicates that the firm can give up 2.5K byadding 1L. Thus, the slope of isoquant l2Q between points Nand R is -2.5K/1 L. Betweenpoints Rand S, the slope of isoquant ] 2Q is -~, and so on.

The absolute value: of the slope of the isoquant is called the marginal rate of techni-cal substitution (MRTS). For a movement down along an isoquant, the marginal rate oftechnical substitution of labor for capital is given by -!1.K/M. We multiply !1.K/M by -I inorder to express the MRTS as a positive number. Thus, the MRTS between points Nand Ron the isoquant for 12Q is 2.:. Similarly, the MRTS between points Rand S is~. The MRTSat any point on an isoquant is given by the absolute slope of the isoquant at that point. Thus,


Capital

N "-- 400

2

- 360

4

3.5

3

-2.5K IIL__

1L

280

1 120

oo 1 2 3 3.5 4

labor

FIGURE 6·8 The Slope of Isoquants The absolute value of the slope of an isoquant is called themarginal rate of technical substitution (MRTS). Between points Nand Ron isoquant 12Q. MRTS = 2.5.Between points Rand S. MRTS =~. The MRTS at any point on an isoquant is given by the absolute slope ofthe tangent to the isoquant at that point. Thus. at point R. MRTS = 1.

the MRTS at point R is 1 (the absolute slope of the tangent to the isoquant at point R; seeFigure 6-8).7

The MRTS of labor for capital is also equal to MPJMPK. We can prove this by remem-bering that all points on an isoquant refer to the same level of output. Thus, for a movementdown a given isoquant, the gain in output resulting from the use of more labor must be equalto the loss in output resulting from the use ofless capital. Specifically, the increase in the quan-tity of labor used (M) times the marginal product of labor (MPL) must equal the reduction inthe amount of capital used (-b.K) times the marginal product of capital (MPK). That is,

(M)(MPL) = -(IlK)(MPK) [6-8]

7 The MRTS at a particular point on an isoquant when labor and capital are continuously divisible (so that theisoquant is a smooth curve) can be obtained by taking tbe total differential of the production function Q =f(L, K), setting this total differential equal to zero (since output does not change along a given isoquant), andsolving for dKldL (the slope of the isoquant). That is,

so that

iJQ iJQdQ = -·elL+-·dK = 0iJL iJK

elK = (-) iJQ I iJL.u. iJQI iJK

Since iJQliJL= MPL and iJQliJK= MPK,

elK = (_) MPL = MRTSdL MPK


so that

MPL = -M( = MRTSMh st. [6-9]

Thus, MRTS is equal to the absolute slope of the isoquant and to the ratio of the marginalproductivities.

Within the economically relevant range, isoquants are not only negatively sloped butalso convex to the origin (see Figure 6-8). The reason for this is that as the firm movesdown an isoquant and uses more labor and less capital, the MPL declines and the MPK in-creases (since the firm is in stage II of production for both labor and capital). With the MPL

declining and the MPK rising as we move down along an isoquant, the MPdMPK = MRTSwill fall (thus, the isoquant is convex to the origin).

Perfect Substitutes and Complementary InputsThe shape of an isoquant reflects the degree to which one input can be substituted for an-other in production. On one hand, the smaller the curvature of an isoquant, the greater is thedegree of substitutability of inputs in production. On the other hand, the greater the curva-ture of an isoquant, the smaller is the degree of substitutability.

At one extreme are isoquants that are straight lines, as shown in the left panel of Fig-ure 6-9. In this case, labor and capital are perfect substitutes. That is, the rate at which laborcan be substituted for capital in production (i.e., the absolute slope of the isoquant orMRTS) is constant. This means that labor can be substituted for capital (or vice versa) at theconstant rate given by the absolute slope of the isoquant. For example, in the left panel ofFigure 6-9, 2L can be substituted for lK regardless of the point of production on the iso-quant. In fact, point A on the labor axis shows that the level of output indicated by the middle

Capital Capital

2

66

4 4

2/

/o

Laboro

o 2 4 6 8 10 12 o 2 4 6Labor

FIGURE6-9 Perfect Substitutes and Complementary Inputs When an isoquant is a straight line (so thatits absolute slope or MRTS is constant), inputs are perfect substitutes. In the left panel, 2L can be substitutedfor lKregardless of the point of production on the isoquant. With the right-angled isoquants in the rightpanel, efficient production can take place only with 2K/1L. Thus, labor and capital are perfect complements.Using only more labor or only more capital does not increase output (that is, MPL =MPK= 0).


isoquant can be produced with labor alone (i.e., without any capital). Similarly, point B onthe capital axis indicates that the same level of output can be produced with capital only(i.e., without any labor). Examples of near-perfect input substitutability are oil and gas usedto operate some heating furnaces, energy and time in a drying process, and fish meal andsoybeans used to provide protein in a feed mix.8

At the other extreme of the spectrum of input substitutability in production are iso-quants that are at a right angle, as in the right panel of Figure 6-9. In this case, labor andcapital are perfect complements. That is, labor and capital must be used in the fixed pro-portion of 2KI1 L. In this case there is zero substitutability between labor and capital inproduction. For example, starting at point C on the middle isoquant in the right panel ofFigure 6-9, output remains unchanged if only the quantity of labor used is increased(that is, MPL = 0 along the horizontal portion of the isoquant). Similarly, output remainsunchanged if only the quantity of capital is increased (that is, MPK = 0 along the verticalportion of the isoquant). Output can be increased only by increasing the quantity of bothlabor and capital used in the proportion of 2K/IL. Examples of perfect complementaryinputs are certain chemical processes that require basic elements (chemicals) to be com-bined in a specified fixed proportion, engine and body for automobiles, two wheels anda frame for bicycles, and so on. In these cases, inputs can be used only in the fixedproportion specified (i.e., there is no possibility of substituting one input for another inproduction).

Although perfect substitutability and perfect complementarity of inputs in productionare possible, in most cases isoquants exhibit some curvature (i.e., inputs are imperfectsubstitutes), as shown in Figure 6-8. This means that in the usual production situation,labor can be substituted for capital to some degree. The smaller the degree of curvature ofthe isoquant, the more easily inputs can be substituted for each other in production. In ad-dition, when the isoquant has some curvature, the ability to substitute labor for capital (orvice versa) diminishes as more and more labor is substituted for capital. This is indicatedby the declining absolute slope of the isoquant or MRTS as we move down along an iso-quant (see Figure 6-8). The ability to substitute one input for another in production is ex-tremely important in keeping production costs down when the price of an input increasesrelative to the price of another.

~ OPTIMAL COMBINATION OF INPUTS

As we have seen in the previous section, an isoquant shows the various combinations oflabor and capital that a firm can use to produce a given level of output. In this section weexamine isocosts. An isocost line shows the various combinations of inputs that a firm canpurchase or hire at a given cost. By the use of isocosts and isoquants, we will then deter-mine the optimal input combination for the firm to maximize profits. In this section, wewill also examine input substitution in production as a result of a change in input prices.

8 The MPL also declines because less capital is used. On the other hand, the MPK increases because more laboris also used.


Capital14 A'

Capital

10 A 10

8 8

6

4 4

2

0 8 08*

0 2 4 6 8 10 labor 0 4 8 10 14 20 labor

FIGURE 6-10 Isocost Lines With total cost of e= $100 and W= r= $10, we have isocost line AB in the leftpanel, with vertical intercept of elr= $100/$10 = 10K and slope of-w/r= -$10/$10 = -1. With C = $140 andW= r= $10, we have isocost line A' a' in the right panel. With C' = $80 and W= r= $10, the isocost line isAna" in the right panel. On the other hand, with e = $100 and r= $10 but W= $5, we have isocost lineAB* inthe right panel, with vertical intercept of 10K and slope of -!.

Isocost LinesSuppose that a firm uses only labor and capital in production. The total costs or expendi-tures of the firm can then be represented by

C=wL+rK [6-10]

where C is total costs, w is the wage rate of labor, L is the quantity of labor used,r is the rental price of capital, and K is the quantity of capital used. Thus, Equa-tion 6- 10 postulates that the total costs of the firm (C) equals the sum of its expenditures onlabor (wL) and capital (rK). Equation 6-10 is the general equation of the firm's isocost lineor equal-co t line. It shows the various combinations of labor and capital that the firm canhire or rent at a given total cost. For example, if C = $100, w = $10, and r = $10, the firmcould either hire 10L or rent 10K, or any combination of Land K shown on isocost line ABin the left panel of Figure 6-10. For each unit of capital the firm gives up, it can hire one ad-ditional unit of labor. Thus, the slope of the isocost line is -1.

By subtracting wL from both sides of Equation 6-10 and then dividing by r, we get thegeneral equation of the isocost line in the following more useful form:

C wK=---L

r r[6-11 ]

where Clr is the vertical intercept of the isocost line and -wlr is its slope. Thus, for C =$100 and w = r= $10, the vertical intercept is Clr = $1001$10 = 10K, and the slope is -wlr =-$10/$10 = -1 (see isocost line AB in the left panel of Figure 6-10).

A different total cost by the firm would define a different but parallel isocost line, whiledifferent relative input prices would define an isocost line with a different slope. For example,


Capital

A'14

5

140

11

10

8

2

o B'

o 2 3 5 8 10 12 14Labor

FIGURE 6-11 Optimal Input Combination The optimal input combination is given by points D, E,and Fat which isoquants 8Q, 10Q, and 14Q are tangent to isocosts A"8", A8, and A'B', respectively. Byjoining the origin with points D, E, and F, we get the expansion path of the firm. At the optimal inputcombinations (tangency points), the absolute slope of the isoquants (MRTS = MPL!MPK) equals theabsolute slope of the isocost lines (wi r), so that MPl [w = MPKI r.

an increase in total expenditures to C = $140 with unchanged w = r = $10 would give isocostline A'B' in the right panel of Figure 6-10, with vertical intercept C'Ir = $1401$10 = 14Kand slope of -wlr=-$101$10=-I. If total expenditures declined to C' = $80 with unchangedw= r=$1 0, the isocost line would beAIB", with vertical intercept of C'lr= $801$10 = 8K andslope of -wi r = -$10/$10 = -I. On the other hand, with e = $100 and r = $10 but w = $5,we would have isocost lineAB*, with vertical interceptofClr=$l 001$10= 10K and slope of-wlr = -$5/$10 = -!.

Optimal Input Combination for Minimizing Costsor Maximizing OutputThe optimal combination of inputs needed for a firm to minimize the cost of producing agiven level of output or maximize the output for a given cost outlay is given at the tangencypoint of an isoquant and an isocost. For example, Figure 6-11 shows that the lowest cost ofproducing 10 units of output (i.e., to reach isoquant lOQ) is given by point E, where isoquantIOQ is tangent to isocost line AB. The firm uses 5L at a cost of $50 and 5K at a cost of $50also, for a total cost of $100 (shown by isocost AB). The output of IOQ can also be regardedas the maximum output that can be produced with an expenditure of $100 (i.e., with isocostlineAB).


Note that the finncould also produce 10Q atpoint G(with 3Land 11K) or at pointH (with12L and 2K) at a co t of $140 (isocost A'B'). But this would not represent the least-cost inputcombination required to produce 10Q. In fact, with an expenditure of$140 (i.e., with isocostA'B'), the firm could reach isoquant 14Q at point F (see Figure 6-11). Similarly, the firm couldproduce 8Q efficiently at point D on isocost A"B" at a cost of $80 or inefficiently at points Jand M on isocost AB at a cost of $100. Thus, the optimal input combination needed to mini-mize the cost of producing a given level of output or the maximum output that the firm can pro-duce at a given cost outlay is given at the tangency of an isoquant and an isocost.

Joining points of tangency of isoquants and isocosts (i.e., joining points of optimalinput combination ) gives the expansion path of the firm. For example, line ODEF inFigure 6-11 is the expansion path for the firm. It hows that the minimum cost of reachingisoquants (i.e., producing) 8Q, lOQ, and 14Q are $80, $100, and $140, given by pointsD, E, and F, respectively. It also shows that with total costs of $80, $100, and $140 (i.e.,with isocosts A"B", AB, and A'B') the maximum outputs that the firm can produce are 8Q,lOQ, and 14Q, respectively. To be noted is that not all expansion paths are straight linesthrough the origin as in Figure 6-11, though this is the case for most production functionsestimated empirically (see Section 6-7).

With optimal input combinations (i.e., at the points of tangency of isoquants and iso-cost lines), the (absolute) slope of the isoquant or marginal rate of technical substitution oflabor for capital i equal to the (absolute) slope of the isocost line or ratio of input prices.That is,

MRIS = wr

[6-12]

Since the MRTS = MPJMPK, we can rewrite the condition for the optimal combination ofinputs as

MPL W

MPK r

Cross-multiplying, we get

MPL MPK=w r

[6-13 ]

[6-14 ]

Equation 6-14 indicates that to minimize production costs (or to maximize output fora given cost outlay), the extra output or marginal product per dollar spent on labor must beequal to the marginal product per dollar spent on capital. If MPL = 5, MPK = 4, and w = r,the firm would not be maximizing output or minimizing costs since it is getting more extraoutput for a dollar spent on labor than on capital. To maximize output or minimize costs,the firm would have to hire more labor and rent less capital. As the firm does this, the MPL

declines and the MP K increases (since the firm is in stage II of production for Land K). Theprocess would have to continue until condition 6-14 holds. Ifw were higher than r, the MPL

would have to be proportionately higher than the MPK for condition 6-14 to hold. The samegeneral condition for the optimal input combination would have to hold regardless of thenumber of inputs. That is, the MP per dollar spent on each input would have to be the samefor all inputs.


Profit MaximizationIn order to maximize profits, a firm should employ each input until the marginal revenueproduct of the input equals the marginal resource cost of hiring the input. With constantinput prices, this means that the firm should hire each input until the marginal revenueproduct of the input equals the input price. This is a simple extension of the profit maxi-mization condition discussed in Section 6-3 for the case of a single variable input. Withlabor and capital as the variable inputs, the firm will maximize profits by hiring labor andcapital until the marginal revenue product of labor (MRPd equals the wage rate (w) anduntil the marginal revenue product of capital (MRPK) equals the rental price of capital (r).That is, in order to maximize profits, a firm should hire labor and capital until

MRPL=w

MRPK=r

[6-15]

[6-16]

Hiring labor and capital so that Equations 6-15 and 6-16 hold implies that condition 6-14for optimal input combination will also be satisfied. To see this, remember from Section 6-3and Equation 6-6 that the marginal revenue product (MRP) of an input equals the marginalproduct of the input (MP) times the marginal revenue (MR) generated from the sale of the out-put. Thus, we can rewrite Equations 6-15 and 6-16 as

(MPL)(MR) = w

(MPK)(MR) = r

[6-17]

[6-18]

Dividing Equation 6-17 by Equation 6-18 gives

MPL =~MPJ(. r

[6-19]

Cross-multiplying in Equation 6-19 gives the condition for the optimal combination of in-puts given by Equation 6-14:

MPL = MPKw r

[6-14 ]

Note that there is an optimal input combination for each level of output (see points D,E, and F in Figure 6-11), but only at one of these outputs (the one where MRP of each inputequals the input price) will the firm maximize profits. That is, to maximize profits, the firmmust produce the profit-maximizing level of output with the optimal input combination. Byhiring inputs so that Equations 6-15 and 6-16 are satisfied, however, both conditions aremet at the same time. That is, the firm will be producing the best or profit-maximizing levelof output with the optimal input combination.

Effect of Change in Input PricesStarting from an optimal input combination, if the price of an input declines, the firmwill substitute the cheaper input for other inputs in production in order to reach anew optimal input combination. For example, Figure 6-12 shows that with C = $100 and


Capital

10

7

5

3

oo 5

8*

8 20 Labor10 14

FIGURE 6-12 Input Substitution in Production With C= $100 and w= r= $10, the optimal inputcombination to produce 10Q is 5K and 5L (point E, where isoquant 10Q is tangent to isocost A8). Atpoint E, K/L = 1. If r remains at $10 but wfalls to $5, the firm can reach isoquant 10Q with C= $70.The optimal combination of Land K is then given by point Rwhere isocost A*8' is tangent to isoquant10Q,andK/L=~.

If the price of gasoline were $2 per gallon and ifthe individual could have earned $4 per hour byworking rather than driving, the optimal combina-tion of the inputs of gasoline and travel time wouldbe at point B where the isoquant is tangent to iso-cost DE. At point B, the absolute slope of isocostDE is ~ (the ratio of the price of 1 gallon of gasolineto the cost of 1 hour of driving time). The minimumcost for the trip is then $48 (6 hours of driving timeat $4 per hour plus 12 gallons of gasoline at $2 pergallon). If the price of gasoline fell from $2 to $1per gallon, the optimal combination of driving timeand gasoline would be given instead by point Cwhere the isoquant is tangent to isocost line FG. Atpoint C the absolute slope of isocost FG is ± andreflects the new relative price of gasoline ($1 pergallon) and traveling time ($4 per hour). The mini-mum cost of the trip would then be $35 (5 hours of

CASE STUDY 6-1Substitutability between Gasoline Consumptionand Driving Time

Higher highway peed reduces driving time butincrease gasoline consumption by reducing gasmileage. The trade-off between gasoline consump-tion and traveling time for a trip of 360 miles can berepresented by the isoquant shown in Figure 6-13.The isoquant shows that at 50 miles per hour, the360 miles can be covered in 7.2 hours (that is, 7 hoursand 12 minutes) with 10 gallons of gasoline, at36 miles per gallon (point A). At 60 miles per hour,the 360 miles can be covered in 6 hours with 12 gallonsof gasoline, at 30 miles per gallon (point B). Drivingat 60 miles per hour saves I hour and 12 minutesof travel time but increases gasoline consumptionby 2 gallons. At 72 miles per hour, the trip will take 5hours and 15 gallons of gasoline, at 24 miles per gal-lon (point C). Driving at 72 miles per hour saves an-other I hour of travel time but increases gasolineconsumption by another 3 gallons.


Driving time(hours)

I0

7.2

6

5

:L '" ~ : •• miles

E

I I I I0 8 10 12 15

Gasoline (gallons)

FIGURE 6·13 The Optimal Combination of Gasoline and Driving Time At the gasoline price of $2per gallon and foregone earnings of $4 per hour for driving, the minimum cost of a 360-mile trip isgiven by point 8, at which the isoquant is tangent to isocost DE. The optimal driving speed is 60 milesper hour so that the trip would take 6 hours and 12 gallons of gasoline at a total cost of $48. Iftheprice of gasoline falls to $1, the minimum cost of the trip is given by point C,at which the isoquant istangent to isocost FG. The optimal driving speed is 72 miles per hour so that the trip would take5 hours and 15 gallons of gasoline at a total cost of $35.

traveling time at $4 per hour plus 15 gallons ofgasoline at $1 per gallon).

Thus, the repeal of the 55-miles-an-hour law in1987 (which had been imposed in 1974 at the start ofthe petroleum crisis to save gasoline) was a rationalresponse to the decline in gasoline prices (in relationto other prices) and the increase in real wages that

had taken place since 1981. Opposition to the repealof the 55-miles-an-hour law came primarily fromthose who believe that lower speed limits save lives.Relatively low gasoline prices from 1986 to 2003did, however, bring to a halt progress in energyefficiency.

Source: Charles A. Lave, "Speeding, Coordination, and the 55-MPH Limit," American Economic Review (December1985), pp. 1159-1164; "Death Rate on U.S. Roads Reported at a Record Low," The New York Times (October 27,1998), p. 16; "Looking for Ways to Save Gasoline," The Wall Street Journal (July 12,2001), p. AI; "Senate KillsEffort to Raise Cars' Fuel Efficiency," The Wall Street Journal (March 14, 2002), p. A2; "Fuel Economy Below 80'sLevels, E.P.A. Says," The New York Times (October 30, 2002), p. C4; "How Much Is Your Time Worth?" The New YorkTimes (February 26, 2003), p. DI; "At $2 a Gallon, Gas Is Still Worth Guzzling," The New York Times (May 16,2004), sec. 4, p. 14; and "New Gas-Mileage Standards May Vary by Vehicle Weight, The Wall Street Journal (July 10,2005), p. A2.


w = r = $10, the optimal input combination is 5K and 5L given by point E, where iso-quant 10Q is tangent to isocost AB (as in Figure 6-11). At point E, the capital-labor ratio(that is, KIL) is 1.

If r remains at $10 but w falls to $5, the isocost line becomes AB*, and the firm canreach isoquant 14Q with C = $100 (point N in Figure 6-12). The firm can now reach iso-quant lOQ with C = $70. This is given by isocost A *B', which is parallel to AB* (that is,-wlr = -~ for both) and is tangent to isoquant lOQ at point R. At point R, the firm uses3K at a cost of $30 and 8L at a cost of $40, for a total cost of $70. At point R, KIL = ~(as compared with KIL = 1 at point E before the reduction in w). Thus, with a reductionin w (and constant r), a lower C is required to produce a given level of output. To mini-mize the production cost of producing lOQ, the firm will have to substitute L for K inproduction, so that KIL declines.

The ease with which the firm can substitute L for K in production depends on the shapeof the isoquant. As we have seen in Section 6-4, the flatter the isoquant, the easier it is tosubstitute L for K in production. On the other hand, if the isoquant is at a right angle (as inthe right panel of Figure 6-9), no input substitution is possible (that is, MRTS = 0). Then,KIL will be constant regardless of input prices. Case Study 6-1 examines the trade-offbetween consumption and traveling time on the nation's highway.

j 6-6 RETURNS TO SCALE

Returns to scale refers to the degree by which output changes as a result of a given changein the quantity of all inputs used in production. There are three types of returns to scale:constant, increasing, and decreasing. If the quantity of all inputs used in production is in-creased by a given proportion, we have constant returns to scale if output increases in thesame proportion; increasing returns to scale if output increases by a greater proportion;and decreasing returns to scale if output increases by a smaller proportion. That is, sup-pose that starting with the general production function

Q=f(L, K) [6-1]

we multiply Land K by h, and Q increases by 1, as indicated in Equation 6-20:

AQ =f(hL, hK) [6-20]

We have constant, increasing, or decreasing returns to scale, respectively, depending onwhether A = h, A> h, or A < h.

For example, when all inputs are doubled, we have constant, increasing, or decreasingreturns to scale, respectively, if output doubles, more than doubles, or less than doubles. Thisis shown in Figure 6-14. In all three panels of Figure 6-14, we start with the firm using 3Land 3K and producing 100Q (point A). By doubling inputs to 6L and 6K, the left panel showsthat output also doubles to 200Q (point B), so that we have constant returns to scale. Thecenter panel shows that output triples to 300Q (point C), so that we have increasing returnsto scale, while the right panel shows that output only increases to 150Q (point D), so that wehave decreasing returns to scale. Here h (the increase in Land K) is 100 percent, while (theincrease in Q) is 100 percent in the left panel, 200 percent in the middle panel, and 50 per-cent in the right panel.


Capital Capital Capital

61-----V. I6 62000 3000'tzf:± 3 3

0

i 1000

0 0o 3 6 Labor 0 3 6 0 3 6

FIGURE 6-14 Constant, Increasing, and Decreasing Returns to Scale In all three panels of this figurewe start with the firm using 3L and 3K and producing 100Q (point A). By doubling inputs to 6L and 6K,the left panel shows that output also doubles to 200Q (point B), so that we have constant returns toscale; the center panel shows that output triples to 300Q (point C), so that we have increasing returnsto scale; while the right panel shows that output only increases to 150Q (point D), so that we havedecreasing returns to scale.

CASE STUDY 6-2Returns to Scale in U.S. Manufacturing Industries

Table 6-5 reports the estimated returns to scale in18 manufacturing industries in the United States in1957. A value of I refers to constant returns to scale,a value greater than J refers to increasing returns toscale, and a value less than 1 refers to decreasing re-turns to scale. The table shows that for a doubling of(i.e., with a 100 percent increase in) all inputs, output

would rise by III percent in the furniture industry,by 109 percent in chemicals, but only by 95 percentin petroleum (the last entry in the second column ofthe table). Although only the textile industry seemsto face constant returns to scale exactly, most otherindustries are very close to it.

Estimated Returns to Scale in U.S. Manufacturing

Industry Returns to Scale Industry

Furniture 1.11 Stone, clay, etc.Chemicals 1.09 Fabricated metalsPrinting 1.08 Electrical machineryFood, beverages 1.07 Transport equipmentRubber, plastics 1.06 Nonelectrical machineryInstruments 1.04 TextilesLumber 1.04 Paper and pulpApparel 1.04 Primary metalsLeather 1.04 Petroleum

Returns to Scale

1.03

1.03

1.03

1.02

1.02

1.00

0.98

0.96

0.95

Source: J. Moroney,"Cobb-Douglas Production Functionsand Returns to Scale in U.S. ManufacturingIndustry,"Western Economic Journal (December 1967), pp. 39-51.

1500

Labor

By doing this the company was able to improveits output/employee/year from 67.7 in 1997 to 101 in2002 and then to 132 in 2004, an increase of almost50 percent in 7 years. The obvious goal to increasethe productivity was made possible, says Mr MadhurBajaj, vice-chairman. Productivity has gone up sig-nificantly in the last few years, from the level of 67.7vehicles per employee/year to the level of 132 vehi-cles per employee/year in the year ended 2004. Con-sequently, the company's R&D was focused on up-grading current products, developing new scooters,and cost reduction mainly in the entry-level seg-ments. The company is setting up a new facility atthe Akurdi complex near Pune for design, analysis,prototyping, and testing.

In terms of growth, the company witnessed asharp increase of 55.4 percent in its sales in the year2002 as compared to market growth of 40 percentduring the previous year. Bajaj Auto was also able tooffset its high raw material cost due to increased pro-ductivity of its employees, which also led to reduc-tion in labor cost as percentage of net sales from 5.7percent to 5 percent. In 2002, Bajaj had a workforceof 13,000 employees with a network of 422 dealersand over 1,200 authorized service centers.

CASE STUDY 6-3Bajaj Auto limited-Small Is Better

Bajaj Auto Limited was established in 1945 as atrading company and obtained its production licensein 1959. It is one of India's largest two-wheelermanufacturers and was a dominant player until theearly 1990s. The two-wheeler industry thrives in de-veloping countries, especially in densely populatedcountries like India. But with income levels rising,people are opting for entry-level motorcycles ratherthan scooters.

The two-wheeler industry grew by 11.6 per-cent-from 5.05 million units in 2002-03 to 5.64million units in 2003-04. However, due to globalcompetition, Bajaj's market share declined from 49.3percent in 1994 to 38.9 percent in 1999. In order toovercome this problem, Bajaj Auto has decided thatsmaller is better and reduced significantly its headcount, first in 1998 and then in 2001 by more than3,000 persons. In 2000-01, the company cut down itsstaff strength from 17,213 to 13,819. Bajaj has of-fered four VRS schemes till date for its staff and twosuch for its workers, and the attempt compacted itsemployee strength by 9,384 (from 21,373 in 1997 to11,531 in 2004), resulting in a reduction of Rs 70.6crore per annum in its wage bill.

Production and Employee Size of Bajaj Auto Ltd

Source: Company Annual Report 2004-05, India Infoline 2004, The Hindu Business Line, October 2003.

Year1997

1998

1999

2000

2001

2002

2003

2004

Production1,439,174

1,354,482

1,381,765

1,432,471

1,212,748

1,356,463

1,457,066

1,516,876

242

No. of Employees21,273

18.589

18,585

17,213

13,81913,482

12,338

11,531

Output/Employee/Year67.7

72.9

74.3

83.2

87.8

100.6

118.1131.5


Increasing returns to scale arise because as the scale of operation increases, a greaterdivision of labor and specialization can take place and more specialized and productive ma-chinery can be used. Decreasing returns to scale, on the other hand, arise primarily becauseas the scale of operation increases, it becomes ever more difficult to manage the firm ef-fectively and coordinate the various operations and divisions of the firm. In the real world,the forces for increasing and decreasing returns to scale often operate side by side, with theformer usually overwhelming the latter at small levels of output and the reverse occurringat very large levels of output.

For example, in Table 6-1, we saw that with lL and lK, we have 3Q. With 2L and 2K,we have 18Q. Thus, we have increasing returns to scale over this range of outputs. However,doubling the amount of labor and capital used from 3L and 3K to 6L and 6K only increasesoutput from 33 to 39 units, so that we have decreasing returns to scale over this larger rangeof outputs. In the real world, most industries seem to operate near the range of constant re-turns to scale where the forces of increasing and decreasing returns to scale are more or lessin balance (see Case Study 6-2). But, as Case Study 6-3 shows, Bajaj Auto Ltd facesdecreasing returns to scale because of excess manpower and thus reduced its employeesover a time period.

I 6-7 I EMPIRICAL PRODUCTION FUNCTIONS

The production function most commonly used in empirical estimation is the power func-tion of the form

Q=AKaLb [6- 21]

where Q, K, and L refer, respectively, to the quantities of output, capital, and labor, and A,a, and b are the parameters to be estimated empirically. Equation 6-21 is often referredto as the Cobb-Douglas production function in honor of Charles W. Cobb and Paul H.Douglas, who introduced it in the 1920s.9

The Cobb-Douglas production function has several useful properties. First, the mar-ginal product of capital and the marginal product of labor depend on both the quantity ofcapital and the quantity of labor used in production, as is often the case in the real world. 10

Second, the exponents of K and L (i.e., a and b) represent, respectively, the output elastic-ity of labor and capital (EK and EL), and the sum of the exponents (i.e., a + b) measures the

9 See C. W. Cobb and P. H. Douglas, "A Theory of Production," American Economic Review (March 1928),pp.139-165.10 The equation for the marginal product of capital is

oQ QMPK = - = aAKa - IU = a·-oK K

Similarly, the equation for the marginal product of labor is

MPL = oQ = bAKaU-1 = b.goL L

The MPK and MPL are positive and diminishing throughout (i.e., the Cobb-Douglas production function ex-hibits only stage II of production for capital and labor).


returns to scale. If a + b = 1, we have constant returns to scale; if a + b > I, we have in-creasing returns to scale; and if a + b < 1, we have decreasing returns to scale. II Third, theCobb-Douglas production function can be estimated by regression analysis by transform-ing it into

In Q = In A + a In K + b In L [6-22]

which is linear in the logarithms. Finally, the Cobb-Douglas production function can eas-ily be extended to deal with more than two inputs (say, capital, labor, and natural resourcesor capital, production labor, and nonproduction labor).

The Cobb-Douglas production function can be estimated either from data for a singlefirm, industry, or nation over time (i.e., using time-series analysis), or for a number offirms, industries, or nations at one point in time (i.e., using cross-sectional data). In either

II The output elasticity of capital is

Similarly,

and EK + EL = a + b = returns to scale.

have slightly increasing returns to scale in furnitureproduction.

The values of a, b, and c reported in Table 6-7were estimated by regression analysis with cross-sectional data for each of 18 industries for the year1957, using a Cobb-Douglas production function ex-tended to three inputs and transformed into naturallogarithms. All estimated coefficients, with the ex-ception of c for the rubber and plastics industry, werepositive, as expected. Of the 54 estimated coeffi-cients, 39 were statistically significant at the 5 per-cent level and all 18 regressions explained morethan 95 percent of the variations in output (i.e., R2 ex-ceeded 0.95). However, only the first four industrieslisted in the table and fabricated metals have returnsto scale that are statistically different from 1 at the5 percent level of significance. In all the otherindustries, constant or near-constant returns to scaleseemed to prevail.

CASE STUDY 6-4Output Elasticities in U.S. Manufacturing Industries

Table 6-7 reports the estimated output elasticities ofcapital (a), production workers (b), and nonproduc-tion workers (c) for the same 18 manufacturing in-dustries examined in Case Study 6-2. The value ofa = 0.205 for furniture means that a I percent in-crease in the quantity of capital used (holding thenumber of production and nonproduction workersconstant) results in a 0.205 percent increase in thequantity of furniture produced. The value of b =0.802 means that a 1 percent increase in the num-ber of production workers used (while holding thestock of capital and the number of nonproductionworkers used constant) increases Q by 0.802 per-cent. Finally, the value of c = 0.102 means that a1 percent increase in the number of nonproductionworkers used (together with a constant amount ofthe other inputs) results in Q increasing by 0.102percent. Increasing all three inputs at the same timeby 1 percent leads to Q rising by a + b + c = 0.205 +0.802 + 0.102 = 1.11 percent. This means that we


Output Elasticities of Capital (0), Production Workers (b), andNonproduction Workers (c) in U.S. Manufacturing Industries

Output Elasticity of

Production Nonproduction ReturnsCapital Workers Workers to Scale

Industry (0) (b) (c) (o+b+c)

Furniture 0.205 0.802 0.102 1.110

Chemicals 0.200 0.553 0.336 1.089

Printing 0.459 0.045 0.574 1.078

Food and beverages 0.555 0.439 0.076 1.070

Rubber and plastics 0.48] 1.033 -0.458 1.056

Instruments 0.205 0.819 0.020 1.044

Lumber 0.392 0.504 0.145 1.041

Apparel 0.128 0.437 0.477 1.041

Leather 0.076 0.441 0.523 1.040

Stone, clay, etc. 0.632 0.032 0.366 1.030

Fabricated metals 0.151 0.512 0.364 1.027

Electrical machinery 0.368 0.429 0.229 1.026

Transport equipment 0.234 0.749 0.041 1.024

Nonelectrical machinery 0.404 0.228 0.389 1.021

Textiles 0.121 0.549 0.334 1.004

Paper and pulp 0.420 0.367 0.197 0.984

Primary metals 0.371 0.077 0.509 0.957

Petroleum 0.308 0.546 0.093 0.947

Source: J. Moroney, "Cobb-Douglas Production Functions and Returns to Scale in U.S. ManufacturingIndustry," Western Economic Journal (December 1967), pp. 39-51.

case, the researcher faces three potential difficulties:

1. If the firm produces a number of different products, output may have to be mea-sured in monetary rather than in physical units, and this will require deflating thevalue of output by the price index in time-series analysis or adjusting for price dif-ferences for firms and industries located in different regions in cross-sectionalanalysis.

2. Only the capital consumed in the production of the output should be counted, ideally.Since machinery and equipment are of different types and ages (vintages) and pro-ductivities, however, the total stock of capital in existence has to be used instead.

3. In time-series analysis a time trend is also usually inch.sed to take into consider-ation technological changes over time, while in cross-sectional analysis we mustascertain that all firms of industries use the same technology (the best available).

Case Study 6-4 gives the estimated output elasticities of capital, production workers, andnonproduction workers of the U.S. manufacturing industries examined in Case Study 6-2.


6-8 THE INNOVATION PROCESS

In this section, we examine the meaning and importance of technological progress and theprocess by which firms get new technology under the closed versus the open innovationmodel.

Meaning and Importance of InnovationsThe introduction of innovations is the single most important determinant of a firm's long-term competitiveness at home and abroad. Innovations are basically of two types-productinnovation (which refers to the introduction of new or improved products) and processinnovation (the introduction of new or improved production processes). Contrary topopular belief, most innovations are incremental and involve more or less contin-uous small improvements in products or processes rather than a major technologicalbreakthrough. 12Furthermore, most innovations involve the commercial use of ideas thathave been around for years. For example, it took a quarter of a century before firms (pri-marily Japanese ones) were able to perfect the flat video screen (invented in the mid-1960sby George Heilmeier of RCA) and introduce them commercially in portable PCs.

Innovations can be examined with isoquants. A new or improved product requires anew isoquant map showing the various combinations of inputs to produce each level ofoutput of the new or improved product. On the other hand, a process innovation can beshown by a shift toward the origin of product isoquants, showing that each level of out-put can be produced with fewer inputs after the innovation than before.

Unless a firm aggressively and continuously improves the product or the productionprocess, it will inevitably be overtaken by other more innovative firms. This is how theXerox Corporation, the inventor of the copier in 1959, lost its competitive edge to Japanesecompetitors in the 1970s before it shook off its complacency and learned again how tocompete during the 1980s. To be successful in today's world, firms must adopt a globalcompetitive strategy, which means that they must continually scout the world for new prod-uct ideas and processe . It i also crucial for firms to have a presence, first through exportsand then by local production, in the world's major markets. Larger sales mean both im-proving economie of scale in production and distribution, and being able to spend more onresearch and development to stay ahead of the competition.

The introduction of innovations is also stimulated by strong domestic rivalry and geo-graphic concentration-the former because it forces firms to constantly innovate or losemarket share (and even risk being driven entirely out of the market), the latter because itleads to the rapid spread of new ideas and the development of specialized machinery andother inputs for the industry. It is sharp domestic rivalry and great geographic concentrationthat made Japanese firms in many high-tech industries such fierce competitors in worldmarkets during the 1980s.13

i2 See "Less Glamour, More Profits," The Economist (April 24, 2004), p. II.nSee Michael E. Porter, "The Competitive Advantage of Nations," Harvard Business Review (March-April 1990),pp. 75-93; Walter B. Wriston, "The State of American Management," Harvard Business Review (January-February 1990), pp. 78-83; and "Competition: How American Industry Stacks Up Now," Fortune (April 18,1994), pp. 52-M.


CASE STUDY 6-5How Do Firms Get New Technology?

Table 6-8 provides the results of a survey of 650 ex-ecutives in 130 industries on the methods that U.S.firms use to acquire new technology on process andproduct innovations, arranged from the most impor-tant to the least important. We see that the most im-portant method of acquiring product and processinnovations is by independent research and develop-ment (R&D) by the firm. The other methods arrangedin order of decreasing importance are: licensingtechnology by the firms that originally developedthe technology; publications or technical meetings;reverse engineering (i.e., taking the competitiveproduct apart and devising a method of producing a

similar product); hiring employees of innovatingfirms; patent disclosures (i.e., from the detailed infor-mation available from the patent office, which can beused to develop a similar technology or product insuch a way as not to infringe on the patent); or infor-mation from conversations with employees of inno-vating firms (who may inadvertently provide secretinformation in the course of general conversations).For product innovations, reverse engineering be-comes more important than licensing, and hiring em-ployees from innovating firms is more important thanpublications or technical meetings. At least, this wasthe case in the past.

_ Methods of Acquiring New Technology

Rank

Process ProductMethod of Acquisition Innovation Innovation

Independent R&D I I

Licensing 2 3

Publications or technical meetings 3 5

Reverse engineering 4 2

Hiring employees of innovating firm 5 4

Patent disclosures 6 6

Conversations with employeesof innovating firm 7 7

Source: R. E. Levin, "Appropriability, R&D Spending, and Technological Performance," American Economic Review(May 1988), pp. 424-428; P. F. Drucker, "The Discipline of Innovation." Harvard Business Review (November-December1998); "GE Goes Back to the Future," The Wall Street Journal (May 7, 2002), p. B I; and "The Way to the Future,"Business Week (October 11,2004), pp. 92-97.

The risk in introducing innovations is usually high. For example, 8 out of lO newproducts fail shortly after their introduction. Even the most carefully introduced innova-tions can fail, as evidenced by the failure of RJR Nabisco Inc.'s "smokeless cigarette"and Coca-Cola's change of its 99-year-old recipe in 1985. In general, the introduction ofa new product or concept (such as McDonald's hamburgers and Sony Walkmans) is morelikely to succeed than changing an existing product (such as launching a new soup,

5 production analysis

Documents

wall street journal

harvard business review

western economic journal

american economic review

open innovation model

positively sloped portion

optimal driving speed

economically relevant range