Chapter 7

Technology and Production

McGraw-Hill/Irwin Copyright © 2008 by The McGraw-Hill Companies, Inc. All Rights Reserved.

Main Topics

Production technologiesProduction with one variable inputProduction with two variable inputsReturns to scaleProductivity differences and

technological change

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Production Technologies

Firms produce products or services, outputs they can sell profitably

A firm’s production technology summarizes all its production methods for producing its output

Different production methods can use the same amounts of inputs but produce different amounts of output

A production method is efficient if there is no other way for the firm to produce more output using the same amounts of inputs

7-3

Production Technologies:An Example

Firm producing garden benchesAssembles benches from pre-cut kitsHired labor is only input that can be varied

One worker produces 33 benches in a weekTwo workers can produce different numbers of

benches in a week, depending on how they divide up the assembly tasksEach work alone, produce total of 66 benchesHelp each other, produce more

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Production Technologies: An Example

Table 7.1: Inputs and Output for Various Methods of Producing Garden Benches

Production Method

Number of Assembly Workers

Benches Produced Per

WeekEfficient?

A 1 33 Yes

B 2 66 No

C 2 70 No

D 2 74 Yes

E 4 125 No

F 4 132 Yes

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Production Possibilities Set

A production possibilities set contains all combinations of inputs and outputs that are possible given the firm’s technologyOutput on vertical axis, input on horizontal axis

A firm’s efficient production frontier shows the input-output combinations from all of its efficient production methodsCorresponds to the highest point in the production

possibilities set on the vertical line at a given input level

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Figure 7.2: Production Possibility Set for Garden Benches

LLLLFQ 25102 23 7-7

Production Function

Mathematically, describe efficient production frontier with a production functionOutput=F(Inputs)

Example: Q=F(L)=10LQ is quantity of output, L is quantity of laborSubstitute different amounts of L to see how output

changes as the firm hires different amounts of laborAmount of output never falls when the amount

of input increases Production function shows output produced for

efficient production methods

7-8

Short and Long-Run Production

An input is fixed if it cannot be adjusted over any given time period; it is variable if it can be

Short run: a period of time over which one or more inputs is fixed

Long run: a period over time over which all inputs are variable

Length of long run depends on the production process being consideredAuto manufacturer may need years to build a new

production facility but software firm may need only a month or two to rent and move into a new space

7-9

Average and Marginal Products

Average product of labor is the amount of output that is produced per worker:

Marginal product of labor measures how much extra output is produced when the firm changes the amount of labor it uses by just a little bit:

L

LF

L

QAPL

L

LLFLF

L

QMPL

7-10

Diminishing Marginal Returns

Law of diminishing marginal returns: eventually the marginal product for an input decreases as its use increases, holding all other inputs fixed

Table 7.3: Marginal Product of Producing Garden Benches

Number of Workers

Benches Produced Per Week

MPL

0 0 --

1 33 33

2 74 41

3 111 37

4 132 21

7-11

Relationship Between AP and MP

Compare MP to AP to see whether AP rises or falls as more of an input is added

MPL shows how much output the marginal worker addsIf he is more productive than average, he brings the

average upIf he is less productive than average, he drives the

average downRelationship between a firm’s AP and MP:

When the MP of an input is (larger/smaller/the same as) the AP, the marginal units (raise/lower/do not affect) the AP

7-12

AP and MP Curves

When labor is finely divisible, AP and MP are graphed as curves

For any point on a short run production function:AP is the slope of the straight line

connecting the point to the originMP equals the slope of the line tangent to

the production function at that point

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Figure 7.4: Marginal Product of Labor

7-14

Figure 7.6: Average and Marginal Product Curves

AP curve slopes upward when it is below MP

AP slopes downward when it is above MP

AP is flat where the two curve cross

7-15

Production with Two Variable Inputs

Most production processes use many variable inputs: labor, capital, materials, and land

Capital inputs include assets such as physical plant, machinery, and vehicles

Consider a firm that uses two inputs in the long run:Labor (L) and capital (K)Each of these inputs is homogeneousFirm’s production function is Q = F(L,K)

7-16

Production with Two Variable Inputs

When a firm has more than one variable input it can produce a given amount of output with many different combinations of inputsE.g., by substituting K for L

Productive Inputs Principle: Increasing the amounts of all inputs strictly increases the amount of output the firm can produce

7-17

Sample Problem 1 (7.7):

Suppose that a firm’s production function is Q = F(L) = L3 – 200L2 + 10,000L. Its marginal product of labor is MPL = 3L2 – 400L +10,000. At what amount of of labor input are the firm’s average and marginal product of labor equal? Confirm that the average and marginal product curves satisfy the relationship discussed in the text.

Isoquants

An isoquant identifies all input combinations that efficiently produce a given level of outputNote the close parallel to indifference curvesCan think of isoquants as contour lines for

the “hill” created by the production functionFirm’s family of isoquants consists of

the isoquants for all of its possible output levels

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Figure 7.8: Isoquant Example

7-20

Properties of Isoquants

Isoquants are thinDo not slope upwardThe boundary between input

combinations that produce more and less than a given amount of output

Isoquants from the same technology do not cross

Higher-level isoquants lie farther from the origin

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Figure 7.10: Properties of Isoquants

7-22

Figure 7.10: Properties of Isoquants

7-23

Substitution Between Inputs

Rate that one input can be substituted for another is an important factor for managers in choosing best mix of inputs

Shape of isoquant captures information about input substitution Points on an isoquant have same output but different input mix Rate of substitution for labor with capital is equal to negative

the slope Marginal Rate of Technical Substitution for input X

with input Y: the rate as which a firm must replace units of X with units of Y to keep output unchanged starting at a given input combination

7-24

Figure 7.12: MRTS

7-25

MRTS and Marginal Product

Recall the relationship between MRS and marginal utility

Parallel relationship exists between MRTS and marginal product

The more productive labor is relative to capital, the more capital we must add to make up for any reduction in labor; the larger the MRTS

K

LLK MP

MPMRTS

7-26

Figure 7.13: Declining MRTS

Often assume declining MRTS

Here MRTS declines as we move along the isoquant, increasing input X and decreasing input Y

7-27

Extreme Production Technologies

Two inputs are perfect substitutes if their functions are identicalFirm is able to exchange one for another at a fixed

rateEach isoquant is a straight line, constant MRTS

Two inputs are perfect complements whenThey must be used in fixed proportionsIsoquants are L-shaped

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Figure 7.14: Perfect Substitutes

7-29

Figure 7.15: Fixed Proportions

7-30

Cobb-Douglas Production Function

Common production function in economic analysis

Introduced by mathematician Charles Cobb and economist (U.S. Senator) Paul Douglas

General form:

Where A, , and are parameters that take specific values for a given firm

KALKLFQ ,

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Cobb-Douglas Production Function

A shows firm’s general productivity level and affect relative productivities of labor

and capital

Substitution between inputs:

1

1

KALMP

KALMP

K

L

KALKLFQ ,

L

KMRTSLK

7-32

Figure: 7.16: Cobb-Douglas Production Function

7-33

Sample Problem 2 (7.8):

Suppose that John, April, and Tristan have two production plants for producing orange juice. They have a total of 850 crates of oranges and the marginal product of oranges in plant 1 is

and in plant 2 is

What is the best assignment of oranges between the two plants?

11 000,1 OMPO 2

2 2200,1 OMPO

Sample Problem 3:

Suppose a XYZ Inc. operates to production plants which have Cobb-Douglas production functions. The MRTS for each plant is:

If both plants face the same labor and capital costs, and α=1/3 and β=2/3 in plant one and α=2/3 and β=1/3, which plant is more labor intensive.

L

KMRTSLK

Returns to Scale

Types of Returns to ScaleProportional change in

ALL inputs yields…What happens when all

inputs are doubled?

ConstantSame proportional change in

outputOutput doubles

IncreasingGreater than proportional

change in outputOutput more than

doubles

DecreasingLess than proportional

change in outputOutput less than doubles

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Figure 7.17: Returns to Scale

7-37

Productivity Differences and Technological Change

A firm is more productive or has higher productivity when it can produce more output use the same amount of inputsIts production function shifts upward at each

combination of inputsMay be either general change in productivity

of specifically linked to use of one inputProductivity improvement that leaves

MRTS unchanged is factor-neutral

7-38

Sample Problem 4:

Find the returns to scale for the following production functions:Q = L1/2K1/3M1/3

Q = L1/2 + K1/4

Q = L1/2*(L+K)1/2