Production

ECO61 Microeconomic AnalysisUdayan Roy

Fall 2008

What is a firm?

• A firm is an organization that converts inputs such as labor, materials, energy, and capital into outputs, the goods and services that it sells.– Sole proprietorships are firms owned and run by a

single individual.– Partnerships are businesses jointly owned and

controlled by two or more people.– Corporations are owned by shareholders in

proportion to the numbers of shares of stock they hold.

What Owners Want?

• Main assumption: firm’s owners try to maximize profit

• Profit () is the difference between revenues, R, and costs, C:

= R – C

What are the categories of inputs?• Capital (K) - long-lived inputs.

– land, buildings (factories, stores), and equipment (machines, trucks)

• Labor (L) - human services – managers, skilled workers (architects, economists,

engineers, plumbers), and less-skilled workers (custodians, construction laborers, assembly-line workers)

• Materials (M) - raw goods (oil, water, wheat) and processed products (aluminum, plastic, paper, steel)

How firms combine the inputs?

• Production function is the relationship between the quantities of inputs used and the maximum quantity of output that can be produced, given current knowledge about technology and organization

Production Function

• Formally,q = f(L, K)

– where q units of output are produced using L units of labor services and K units of capital (the number of conveyor belts).

ProductionFunction

q = f(L, K)

Output q

Inputs(L, K)

The production function may simply be a table of numbers

The production function may be an algebraic formula

}12,24{min18

1224

15 4.06.0

KLQ

KLQ

KLQ

Just plug in numbers for L and K to get Q.

Marginal Product of Labor• Marginal product of labor (MPL ) - the change in

total output, Q, resulting from using an extra unit of labor, L, holding other factors constant:

L

QMPL

Average Product of Labor• Average product of labor (APL ) - the ratio of

output, Q, to the number of workers, L, used to produce that output:

L

QAPL

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 inputs– E.g., by substituting K for L

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Isoquants

• An isoquant identifies all input combinations (bundles) that efficiently produce a given level of output– Note the close similarity to indifference curves– Can think of isoquants as contour lines for the

“hill” created by the production function

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Family of Isoquants

K, U

nits

of c

apita

l per

day

e

b

a

d

fc

63210 L, Workers per day

6

3

2

1

q = 14

q = 24

q = 35

The production function above yields the isoquants on the left.

Properties of Isoquants

• Isoquants are thin• Do not slope upward• Two isoquants do not cross• Higher-output isoquants lie farther from the

origin

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Figure 7.10: Properties of Isoquants

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Figure 7.10: Properties of Isoquants

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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

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Marginal Rate of Technical Substitution

• Marginal Rate of Technical Substitution for labor with capital (MRTSLK): the amount of capital needed to replace labor while keeping output unchanged, per unit of replaced labor– Let K be the amount of capital that can replace L units

of labor in a way such that total output ― Q = F(L,K) ― is unchanged.

– Then, MRTSLK = - K / L, and– K / L is the slope of the isoquant at the pre-change

inputs bundle. – Therefore, MRTSLK = - slope of the isoquant

Marginal Rate of Technical Substitution

• marginal rate of technical substitution (MRTS) - the number of extra units of one input needed to replace one unit of another input that enables a firm to keep the amount of output it produces constant

L

KMRTS

laborin increase

capitalin increase

Slope of Isoquant!

How the Marginal Rate of Technical Substitution Varies Along an Isoquant

K,

Uni

ts o

f ca

pita

l per

day

L, Workers per day

45

7

10

16a

b

cd

e

q = 10

K = –6

L = 1

0 1

1

1

1

2 3

–3

–2

–1

4 5 6 7 8 9 10

MRTS in a Printing and Pu blishing U.S. Firm

Substitutability of Inputs and Marginal Products.

• Along an isoquant output doesn’t change (q = 0), or:

(MPL x ΔL) + (MPK x ΔK) = 0.

– or

Increase in q per extra unit of labor

Extra units of labor

Increase in q per extra unit of capital

Extra units of capital

isoquant of slope

0

MRTSMP

MP

L

K

LMP

MPK

LMPKMP

K

L

K

L

LK

Figure 7.12: MRTS

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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

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Figure 7.13: Declining MRTS

• We often assume that MRTSLK decreases as we increase L and decrease K

• Why is this a reasonable assumption?

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Extreme Production Technologies

• Two inputs are perfect substitutes if their functions are identical– Firm is able to exchange one for another at a fixed rate– Each isoquant is a straight line, constant MRTS

• Two inputs are perfect complements when– They must be used in fixed proportions– Isoquants are L-shaped

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Substitutability of Inputs

Substitutability of Inputs

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

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Returns to Scale

• Constant returns to scale (CRS) - property of a production function whereby when all inputs are increased by a certain percentage, output increases by that same percentage.

f(2L, 2K) = 2f(L, K).

Returns to Scale (cont).

• Increasing returns to scale (IRS) - property of a production function whereby output rises more than in proportion to an equal increase in all inputs

f(2L, 2K) > 2f(L, K).

Returns to Scale (cont).

• Decreasing returns to scale (DRS) - property of a production function whereby output increases less than in proportion to an equal percentage increase in all inputs

f(2L, 2K) < 2f(L, K).

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 inputs– Its production function shifts upward at each

combination of inputs– May be either general change in productivity of

specifically linked to use of one input• Productivity improvement that leaves MRTS

unchanged is factor-neutral

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The Cobb-Douglas Production Function

• It one is the most popular estimated functions.

q = ALK

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

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Figure: 7.16: Cobb-Douglas Production Function

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