©2005, Southwestern

Slides by Pamela L. Hall

Western Washington University

Production Technology

Chapter 7

2

Introduction Aim in this chapter

Investigate purely technical relationship of combining inputs to produce outputs

• Presents a physical constraint on society’s ability to satisfy wants

Classify factors going into production process Derive a production function that establishes a relationship between

production factors and a firm’s output Discuss Law of Diminishing Marginal Returns and stages of production Develop concept of isoquants

• When two production factors are allowed to vary

Can substitute one factor for another Measure of this ability is elasticity of substitution

Effect of proportional changes in all inputs is called returns to scale Can classify production functions in terms of their elasticity of substitution

and returns to scale attributes

3

Factors of Production For economic modeling, factors of production are

generally classified as Capital

• Durable manmade inputs Are themselves produced goods

Labor • Time or service individuals put into production

Land • All natural resources (for example, water, oil, and climate)

Classification allows us to conceptualize simple cases first Then extend analysis to higher dimensions that are more general

(realistic)

4

Factors of Production Time also enters into production process

Economists generally divide time into three periods, based on ability to vary inputs• Market period

All inputs are fixed

• Short-run period Some inputs are fixed and some are variable

• Long-run period All inputs are variable

In terms of actual time, market-period, short-run, and long-run intervals can vary considerably from one firm to another, Depends on nature of a particular firm

Division of time into three periods is a simplification With intertemporal substitution among stages

More general models incorporating numerous time stages are less restrictive in their assumptions Called dynamic models

5

Production Functions Firms are interested in turning inputs into outputs with the

objective of maximizing profit Formalized by a production function

q = ƒ(K, L, M)• Where q is output of a particular commodity

• K is capital

• L is labor

• M is land or natural resources

For any possible combination of inputs, production function records maximum level of output that can be produced from that combination

In market period all inputs are fixed, so level of output cannot be varied

6

Production Functions Denote K°, L°, and M° as the fixed level of capital, labor, and land Production from these fixed inputs is fixed at q°, so

q° = ƒ(K°, L°, M°)

If capital and labor could be varied with only land fixed, then a short-run production function would be q = ƒ(K, L, M°) Now possible to vary output by changing either K or L

• Or both K and L

In long run, all inputs could be varied, so only restriction on output is technology Production function represents set of technically efficient production

processes• Yields highest level of output for a given set of inputs

7

Production Functions Generally, technical aspects of production do impose

restrictions on profit Assumptions (axioms) concerning these aspects are required for

developing economic models

Two axioms generally underlie a production function Monotonicity

• Implies that if a firm can produce q with a certain level of inputs Should be able to produce at least q if there exists more of every input Assumes free disposal of inputs

• Implies that all marginal products of the variable inputs are positive at their profit-maximizing level

Strict convexity• Analogous to Strict Convexity Axiom in consumer theory

8

Variations in One Input (Short Run) Marginal Product

Marginal product (MP) of variable input Change in output, Δq, resulting from a unit change of the variable

input• Holding all other inputs constant

If capital is variable input, then marginal product of capital is

Alternatively, if labor is variable input, then marginal product of labor is

MP is analogous to concept of marginal utility except that MP is a cardinal number measured on the ratio scale Not an ordinal number

9

Variations in One Input (Short Run) Marginal Product

Distances between any levels of MP are of a known size measured in physical quantities Bushels, crates, pounds, etc.

Consider following cubic production function with labor as variable input q = 6L2 – ⅓L3

Marginal product of labor is • MPL = 12L – L2

Graph of production function and MPL is provided in Figure 7.1

10

Figure 7.1 Stages of production and MPL and APL

11

Variations in One Input (Short Run) Marginal Product

At first, for low levels of labor, total product (TP) is increasing at an increasing rate Slope of TP or MPL is rising

At point of inflection, slope is at its maximum MPL is also at a maximum

To right of maximum MPL, TP is still increasing But at a decreasing rate MPL is positive, but falling

At maximum TP, slope of TP curve is zero Corresponding to MPL = 0

When TP is falling, MPL is negative According to Monotonicity Axiom, given free disposal, a firm will not

operate in negative range of MPL Generally assumed that MPL ≥ 0

12

Average Product According to U.S. Department of Labor, output per hour of labor for nonfarm business

increased at an annual rate of 2.1% from 1991 to 2000 Measure of productivity is measured in physical quantities Called average product (AP) of an input

Defined, for labor, as APL = q/L

In general, average product (AP) is output (TP) divided by input In Figure 7.1, APL at first increases, reaches a maximum, then declines

Productivity of labor, as measured by APL, changes as additional workers are employed Results from short-run condition that all other inputs remain fixed At first, with a relatively small number of workers for a large amount of other inputs

• Adding an additional worker increases productivity of all workers APL increases

However, a point is reached where labor is no longer relatively limited compared with fixed inputs

• An additional worker will result in APL declining

13

Average Product

Graphically, we can determine APL from TP curve by considering a line (cord) through origin Slope of a cord through origin is TP divided by labor

• Since APL is defined as TP divided by labor Slope of a cord through origin is APL at a level of labor where cord

intersects TP

As number of workers increases, at first cord shifts upward and slope of the cord increases

• Resulting in increased APL

Can continue to shift cord upward and it will continue to intersect TP curve until it finally is tangent to TP curve

• At this point, APL is at its maximum

14

Law of Diminishing Marginal Returns and Stages of Production

A firm’s costs will depend on Prices it pays for inputs Technology of combining inputs into output

In short run firm can change its output by adding variable inputs to fixed inputs Output may at first increase at an increasing rate However, given a constant amount of fixed inputs, output will at some point

increase at a decreasing rate• Occurs because at first variable input is limited compared with fixed input

As additional workers are added, productivity remains very high Output, or TP, increases at an increasing rate However, as more of variable input is added, it is no longer as limited

Eventually, TP will still be increasing, but at a decreasing rate MPL will still be positive, but declining

Called Law of Diminishing Marginal Returns (or just diminishing returns)

15

Law of Diminishing Marginal Returns and Stages of Production

As indicated in Figure 7.1, diminishing marginal returns starts at point A MPL is at a maximum

To the left of point A there are increasing returns and at point A constant returns exist

Between points A and B, where MPL is declining, diminishing marginal returns exist

To the right of point B, marginal productivity is both diminishing and negative (MPL < 0), which violates Monotonicity Axiom

TP curve will at some point increase only at a decreasing rate (concave) due to Law of Diminishing Marginal Returns

Some production functions may not exhibit increasing returns at first In fact no firm with a profit-maximizing objective will operate in area of increasing

returns or negative returns• Production functions generally will only be concave

With diminishing marginal returns throughout production process Depicted in Figure 7.2

16

Figure 7.2 Production function with diminishing marginal returns throughout

17

Law of Diminishing Marginal Returns and Stages of Production

In Figure 7.2 MPL and APL decline throughout Cobb-Douglas production function can also only

exhibit diminishing marginal returns throughout production process Can characterize production where all marginal products

are positive Useful for representing firms’ technology constraints

• Given that profit-maximizing firms will only operate in area of diminishing marginal returns where all marginal products are positive

Illustrated in Figure 7.3 for a variable level of labor

18

Figure 7.3 Cobb- Douglas production function with labor as the only variable input

19

Relationship of Marginal Product to Average Product

In area of diminishing marginal returns, marginal product can intersect with average product

As indicated in Figure 7.1, this intersection of MPL and APL occurs where APL is at a maximum

If addition to total, marginal unit, is greater (less) [equal to] than overall average Average will rise (fall) [neither rise nor fall]

Taking derivative of average results in relationship between marginal product and average product

Marginal product is average product plus an adjustment factor (APL/L)L

If slope of APL is zero (rising) [falling] Adjustment factor is zero (> 0) [< 0]

• MPL = APL (MPL > APL) [MPL < APL]

20

Output Elasticity Another important relation between an average and marginal

product is output elasticity Measures how responsive output is to a change in an input

For example, output elasticity of labor, denoted L, is defined as proportionate rate of change in q with respect to L

Given production function q = ƒ(K, L) Output elasticity of labor is

L = (ln q)/ (ln L) = (q/L)(L/q) = MPL/APL

When MPL > APL, L > 1; when 0 < MPL < APL, 0 < L < 1; and when MPL < 0, L < 1 Illustrated in Figure 7.1

21

Table 7.1 Estimated output elasticities for milk

22

Stages of Production

Firm must determine profit-maximizing amount of an available input it should employ Use technology of production to determine at what

stage of production to add a variable input, say, labor Exact profit-maximizing level of labor within this

stage depends on • Cost of labor

• Price received for the firm’s output

Specifically, we divide short-run production function into three stages of production

23

Stages of Production Stage I includes area of increasing returns and extends up to point

where average product reaches a maximum Illustrated in Figure 7.1 Includes a portion of marginal product curve that is declining

• Marginal product is greater than average product, so average product is rising As long as average product is rising, firm will add variable inputs Fixed inputs are present in uneconomically large proportion relative to variable input Variable input is limited relative to fixed inputs

• Rational profit-maximizing producer would never operate in Stage I of production Firm would not produce in short run

Would produce by using fewer units of fixed inputs in long run Fixed inputs become variable Reduction of fixed inputs would result in entire set of product curves shifting

leftward Results in Stage I ending at a lower level of output Illustrated in Figure 7.4

24

Figure 7.4 Shifts in stages of production with a reduction in the level of fixed inputs

25

Stages of Production Rational producer will also not operate in Stage III of production

Range of negative marginal product for variable input In Stage III, TP is actually declining as more of the variable input is added

Figures 7.1, 7.2, and 7.4 illustrate Stage III Additional units of the variable input Stage III actually cause a decline in total

output Even if units of variable input were free, a rational producer would not employ them

beyond the point of zero marginal product In Stage III, variable input is combined with fixed input in uneconomically large

proportions Indeed, point of zero MP, for variable input, is called intensive margin

Point of maximum AP of variable input is called extensive margin• A firm will operate between extensive and intensive margins

Stage II of production Both AP and MP of variable input are positive but declining Output elasticity is between 0 and 1 In contrast, output elasticity for variable input is < 0 in Stage III and > 1 in Stage I

26

Two Variable Inputs Assumed a different combination of, say two, inputs will produce same level of

output For example, in manufacturing microwave ovens, greater use of plastics may be

substituted for a reduction in metal use Indifference curves represent a consumer’s preferences for different

combinations of two goods with utility remaining constant In production theory isoquants represent different input combinations that may be

used to produce a specified level of output• Iso means equal and quant stands for quantity

An isoquant is a locus of points representing same level of output or equal quantity

• For movements along an isoquant Level of output remains constant Input ratio changes continuously

Isoquants are the same concept as indifference mapping Equal utility along same indifference curve replaced by equal output level along

same isoquant Figure 7.5 represents a possible production function for two inputs

27

Figure 7.5 Isoquant map for two variable inputs, capital, K, and labor, L

28

Marginal Rate of Technical Substitution (MRTS)

In Figure 7.5, isoquants are drawn with a negative slope Based on assumption that substituting one input for another can result

in output not changing

A measure for this substitution is marginal rate of technical substitution (MRTS) Defined as negative of slope of an isoquant

Measures how easy it is to substitute one input for another holding output constant

• Similar to concept of MRS in consumer theory

MRTS measures reduction in one input per unit increase in the other that is just sufficient to maintain a constant level of output

29

Convex and Negatively Sloping Isoquants

Can establish underlying assumptions of negatively sloped and convex-to-the-origin isoquant by developing relationship between MRTS and MPs MRTS (K for L) = MPL ÷ MPK

• Take total derivative of production function, q = ƒ(K, L) dq = MPLdL + MPKdK

Along an isoquant dq = 0, output is constant• Thus MPLdL = -MPKdK

Solving for the negative of the slope of the isoquant yields

Along an isoquant, gain in output from increasing L slightly is exactly balanced by loss in output from a suitable decrease in K

For isoquants to be negatively sloped, both MPL and MPK must be positive Ridgelines trace out boundary in isoquant map where marginal products are positive

• See Figure 7.6

Ridgelines are isoclines (equal slopes) where MRTS is either zero or undefined for different levels of output

30

Figure 7.6 Ridgelines in the isoquant map

31

Convex and Negatively Sloping Isoquants

MRTS results in isoquants drawn strictly convex to origin Result is analogous to relationship between MRS and strictly convex indifference

curves For high ratios of K to L MRTS is large

• Indicating that a great deal of capital can be given up if one more unit of labor becomes available

Assumption of strictly convex isoquants is related to Law of Diminishing Marginal Returns

Given MRTS(K for L) = MPL/MPK Movement from A to B in Figure 7.6 results in an increase in labor

• Corresponding decrease in MPL

• Decrease in capital with a corresponding increase in MPK

A firm will always operate in Stage II of production Characterized by diminishing marginal returns Stage II of production, for both the variable inputs, is represented by strictly convex

isoquants In Figure 7.6, a rational producer will only operate somewhere between points D and

C

32

Stages of Production in the Isoquant Map

Can illustrate stages of production in isoquant map by fixing one of the inputs

A situation where capital is fixed at some level is indicated by horizontal line at A in Figure 7.7 In short run, firm must operate somewhere on this line At Stage I, labor input is small relative to fixed level of capital

• Marginal product of capital and MRTS are negative

• Isoquants have positive slopes

• At point B, MRTS is undefined, MPK is zero, and APL equals MPL

This is demarcation between Stages I and II of production

In Stage II of production, all isoquants are strictly convex and have negative slopes

At point C, marginal product of labor is zero• Corresponds to line of demarcation between Stages II and III

33

Figure 7.7 Stages of production in the isoquant map

34

Classifying Production Functions

Production functions represent tangible (measurable) productive processes Economists pay more attention to actual form of

these functions than to form of utility functions• Resulted in classification of production functions in

terms of returns to scale and substitution possibilities Empirical estimates of actual production functions

For some production processes it may be extremely difficult if not impossible to substitute one input for another

35

Returns to Scale Measure how output responds to increases or decreases in all inputs

together Long-run concept since all inputs can vary For example, if all inputs are doubled, returns to scale determine

whether output will double, less than double, or more than double In many cases, it is difficult to change some inputs at will and increase

inputs proportionally Firms do attempt to control as much of environmental conditions as feasible

• Examples in agriculture include greenhouses or pesticides

Assuming it is possible to proportionally change all inputs, a production function can exhibit constant, decreasing, or increasing returns to scale across different output ranges However, it is generally assumed, for simplicity, production functions only

exhibit either constant, decreasing, or increasing returns to scale

36

Returns to Scale Specifically, given production function

q= ƒ(K, L)

A explicit definition of constant returns to scale is ƒ(K, L) = ƒ (K, L) = q, for any > 0

If all inputs are multiplied by some positive constant , output is multiplied by that constant also

If production function is homogeneous Constant returns to scale production function is homogeneous

of degree 1 or linear homogeneous in all inputs• Isoquants are radial blowups and equally spaced as output expands

(Figure 7.8)

37

Figure 7.8 Returns to scale

38

Returns to Scale

Decreasing returns to scale exists if output is increased proportionally less than all inputs ƒ(K, L) < ƒ(K, L) = q

Increasing returns to scale exists if output increases more than proportional increase in inputs ƒ(K, L) > ƒ(K, L) = q

39

Determinants of Returns to Scale Adam Smith established that returns to scale is result of two forces

Division of labor • An increase in all inputs increases division of labor and results in increased

efficiency Production might more than double

Managerial difficulties• Result in decreased efficiency

Production might not double

Early 20th century concept of assembly-line mass production is based on division of labor Each worker has a specialized task to perform for each product being

assembled• Worker becomes very skilled at this task

Increases productivity Example: Henry Ford experienced increasing returns to scale in automobile

manufacturing

40

Determinants of Returns to Scale One cause of managerial difficulties in mass

production is required stockpiling of parts and supplies Inventory control must be maintained, where an accounting

of parts is required• Results in a significant amount of inputs allocated to storage and

accounting of inventories Results in decreasing returns to scale

Just-in-time delivery systems are helping to mitigate these factors

• One problem with just-in-time production Increased vulnerability of firms to supply disruptions

Without a stockpile of parts, such disruptions could shut down production fairly quickly

41

Determinants of Returns to Scale

Postindustrial manufacturing is shifting away from mass production of a standardized product and evolving toward mass customization Called agile manufacturing

• Results in increasing returns to scale

42

Determinants of Returns to Scale

As a firm increases in size by increasing all inputs, another possible cause of decreasing returns to scale is Allocation of inputs for environmental and local

service projects• As a firm employs more inputs and increases output, it

becomes increasingly more exposed to public concerns associated with its production practices

To enhance and maintain goodwill within its community, firm will allocate additional inputs for environmental and local service projects

Contributes to decreasing returns to scale

43

Returns to Scale and Stages of Production

Determine relationship between returns to scale and stages of production by assuming a linear homogeneous production function (homogeneous of degree 1) Implies a constant returns to scale production function

Applying Euler’s Theorem to production function q = ƒ(K, L) we obtain q = L(MPL) + K(MPK)

• Dividing by L gives APL = MPL + (K/L)MPK

• Solving for MPK yields MPK = (L/K)(APL – MPL)

44

Returns to Scale and Stages of Production

Assuming constant returns to scale, we define stages of production as Stage I

• MPL > APL > 0, MPK < 0

Stage II • APL > MPL > 0, APK > MPK > 0

Stage III • MPL < 0, MPK > APK > 0

Stages I and III are symmetric for a constant returns to scale production function Given Monotonicity Axiom, only relevant region for production

is Stage II

45

Elasticity of Substitution A firm may compensate for a decrease in use of one input by an

increase in use of another Heinrich von Thunen collected evidence from his farm in Germany that

suggested ability of one input to compensate for another was significant Postulated principle of substitutability

• Possible to produce a constant output level with a variety of input combinations

Principle of substitutability is not an economic law • There are production functions for which inputs are not substitutable

• However, for those functions where inputs are substitutable Degree that inputs can be substituted for one another is an important technical

relationship for producers

Production functions may also be classified in terms of elasticity of substitution Measures how easy it is to substitute one input for another Determines shape of a single isoquant

46

Elasticity of Substitution In Figure 7.9 consider a movement from A to B

Results in capital/labor ratio (K/L) decreasing

Profit-maximizing firm is interested in determining a measure of ease in which it can substitute K for L If MRTS does not change at all for changes in K/L, the two inputs are

perfect substitutes If MRTS changes rapidly for small changes in K/L, substitution is

difficult If there is an infinite change in the MRTS for small changes in K/L

(called fixed proportions), substitution is not possible• A scale-free measure of this responsiveness is elasticity of substitution

47

Figure 7.9 Capital/labor ratio° and MRTS, K°/L° > K'/L'

48

Elasticity of Substitution Defined as percentage change in K/L divided by percentage change in

MRTS

Along a strictly convex isoquant, K/L and MRTS move in same direction Elasticity of substitution is positive

In Figure 7.9, a movement from A to B results in both K/L and MRTS declining Relative magnitude of this change is measured by elasticity of substitution

• If it is high, MRTS will not change much relative to K/L and the isoquant will be less curved (less strictly convex)

• A low elasticity of substitution gives rather sharply curved isoquants

Possible for the elasticity of substitution to vary for movements along an isoquant and as the scale of production changes However, frequently elasticity of substitution is assumed constant

49

Elasticity of Substitution: Perfect-Substitute

= , a perfect-substitute technology Analogous to perfect substitutes in consumer

theory A production function representing this

technology exhibits constant returns to scale• ƒ(K, L) = aK + bL = (aK + bL) = ƒ(K, L)

• All isoquants for this production function are parallel straight lines with slopes = -b/a

See Figure 7.10

50

Figure 7.10 Elasticity of substitution for perfect-substitute technologies

51

Elasticity of Substitution: Leontief

= 0, a fixed-proportions (or Leontief ) technology Analogous to perfect complements in consumer theory Characterized by zero substitution

A production technology that exhibits fixed proportions is

This production function also exhibits constant returns to scale

52

Elasticity of Substitution: Leontief

Figure 7.11 illustrates a fixed proportions function Capital and labor must always be used in a fixed

ratio Marginal products are constant and zero

Violates Monotonicity Axiom and Law of Diminishing Marginal Returns

Isoquants for this technology are right angles Are not smooth curves, but are kinked

• At kink, MRTS is not unique—can take on an infinite number of positive values

K/L is a constant, d(K/L) = 0, which results in = 0

53

Figure 7.11 Elasticity of substitution for fixed-proportions technologies

54

Elasticity of Substitution; Cobb-Douglas

= 1, Cobb-Douglas technology Isoquants are strictly convex

• Assumes diminishing MRTS (Figure 7.12)

An example of a Cobb-Douglas production function is q = ƒ(K, L) = aKbLd

• a, b, and d are all positive constants

Useful in many applications because it is linear in logs

55

Figure 7.12 Isoquants for a Cobb-Douglas production function

56

Elasticity of Substitution; Cobb-Douglas

= some positive constant Constant elasticity of substitution (CES) production

function can be specified q = [K- + (1 - )L-]-1/

> 0, 0 ≤ ≤1, ≥ -1 is efficiency parameter is a distribution parameter

is substitution parameter

Elasticity of substitution is = 1/(1 + )

• Useful in empirical studies