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# Lecture # 09. Inputs and Production Functions

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Lecture # 09Inputs and Production Functions

Lecturer: Martin Paredes

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1. The Production FunctionMarginal and Average Products

Isoquants

Marginal Rate of Technical Substitution

2. Returns to Scale

3. Some Special Functional Forms

4. Technological Progress

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1. Inputs or factors of production areproductive resources that firms use to

manufacture goods and services.

Example: labor, land, capital

equipment…

2. The firm’s output is the amount of goods

and services produced by the firm.

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3. Production transforms a set of inputsinto a set of outputs

4. Technology determines the quantity of

output that is feasible to attain for a given

set of inputs.

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5. The production function tells us themaximum possible output that can be

attained by the firm for any given

quantity of inputs.

Q = F(L,K,T,M,…)

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6. A technically efficient firm is attaining themaximum possible output from its inputs

(using whatever technology is appropriate)

7. The firm’s production set is the set of all

feasible points, including:

The production function (efficient point)

The inefficient points below the production

function

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Example: The Production Function andTechnical Efficiency

Q

C

L

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Example: The Production Function andTechnical Efficiency

Q

D

C

L

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Example: The Production Function andTechnical Efficiency

Q

D

Production Function

Q = f(L)

C

L

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Example: The Production Function andTechnical Efficiency

Q

D

C

•A

•B

Production Function

Q = f(L)

L

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Example: The Production Function andTechnical Efficiency

Q

D

C

•A

•B

Production Function

Q = f(L)

Production Set

L

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Notes:The variables in the production function are

flows (amount of input per unit of time), not

stocks (the absolute quantity of the input).

Capital refers to physical capital (goods that

are themselves produced goods) and not

financial capital (money required to start or

maintain production).

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Utility FunctionSatisfaction from

purchases

Production Function

Output from inputs

2.

Derived from

preferences

Derived from

technologies

3.

Ordinal

Cardinal

1.

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4.Utility Function

Marginal Utility

5. Indifference Curves

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Marginal Rate of

Substitution

Production Function

Marginal Product

Isoquants

Marginal Rate of

Technical Substitution

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## 15.

Definition: The marginal product of an input is thechange in output that results from a small

change in an input

E.g.:

MPL = Q

L

MPK = Q

K

It assumes the levels of all other inputs are held

constant.

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Example: Suppose Q = K0.5L0.5Then:

MPL = Q = 0.5 K0.5

L

L0.5

MPK = Q = 0.5 L0.5

K

K0.5

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## 17.

Definition: The average product of an input isequal to the total output to be produced

divided by the quantity of the input that is

used in its production

E.g.:

APL = Q

L

APK = Q

K

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## 18.

Example: Suppose Q = K0.5L0.5Then:

APL = Q = K0.5L0.5 = K0.5

L

L

L0.5

APK = Q = K0.5L0.5 = L0.5

K

K

K0.5

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Definition: The law of diminishing marginalreturns states that the marginal product

(eventually) declines as the quantity used of a

single input increases.

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QExample: Total and Marginal Product

Q= F(L,K0)

L

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Example: Total and Marginal ProductQ

Q= F(L,K0)

Increasing

marginal

returns

Diminishing

marginal

returns

MPL maximized

L

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QExample: Total and Marginal Product

Q= F(L,K0)

MPL = 0 when

TP maximized

Increasing

total returns

Diminishing

total returns

L

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QExample: Total and Marginal Product

L

MPL maximized

TPL maximized where

MPL is zero. TPL falls

where MPL is negative;

TPL rises where MPL is

positive.

MPL

L

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There is a systematic relationship betweenaverage product and marginal product.

This relationship holds for any comparison

between any marginal magnitude with the

average magnitude.

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1. When marginal product is greater than averageproduct, average product is increasing.

E.g., if MPL > APL , APL increases in L.

2. When marginal product is less than average

product, average product is decreasing.

E.g., if MPL < APL, APL decreases in L.

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APLMPL

Example: Average and Marginal Products

MPL maximized

APL maximized

L

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QExample: Total, Average and Marginal Products

L

MPL maximized

APL

MPL

APL maximized

L

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Definition: An isoquant is a representation of allthe combinations of inputs (labor and capital)

that allow that firm to produce a given quantity

of output.

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KExample: Isoquants

Slope=dK/dL

0

Q = 10

L

L

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KExample: Isoquants

All combinations of (L,K) along the

isoquant produce 20 units of output.

Q = 20

Slope=dK/dL

0

Q = 10

L

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Example: Suppose Q = K0.5L0.5For Q = 20

=> 20 = K0.5L0.5

=> 400 = KL

=> K = 400/L

For Q = Q0

=> K = (Q0)2 /L

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## 32.

Definition: The marginal rate of technicalsubstitution measures the rate at which the

firm can substitute a little more of an input for a

little less of another input, in order to produce the

same output as before.

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Alternative Definition : It is the negative of the slopeof the isoquant:

MRTSL,K = — dK (for a constant level of

dL

output)

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We can express the MRTS as a ratio of themarginal products of the inputs in that basket

Using differentials, along a particular isoquant:

MPL . dL + MPK . dK = dQ = 0

Solving:

MPL = _ dK = MRTSL,K

MPK

dL

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Notes:If we have diminishing marginal returns, we also

have a diminishing marginal rate of technical

substitution.

In other words, the marginal rate of technical

substitution of labour for capital diminishes as

the quantity of labour increases along an

isoquant.

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Notes:If both marginal products are positive, the slope

of the isoquant is negative

For many production functions, marginal

products eventually become negative. Then:

MRTS < 0

We reach an uneconomic region of production

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## 37.

KExample: The Economic and the

Uneconomic Regions of Production

Isoquants

Q = 20

Q = 10

0

L

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KExample: The Economic and the

Uneconomic Regions of Production

A

B

Q = 20

Q = 10

0

L

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## 39.

KExample: The Economic and the

Uneconomic Regions of Production

A

B

Q = 20

Q = 10

MPL < 0

0

L

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## 40.

KExample: The Economic and the

Uneconomic Regions of Production

MPK < 0

A

B

Q = 20

Q = 10

MPL < 0

0

L

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## 41.

KExample: The Economic and the

Uneconomic Regions of Production

MPK < 0

Uneconomic Region

A

B

Q = 20

Q = 10

MPL < 0

0

L

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## 42.

KExample: The Economic and the

Uneconomic Regions of Production

MPK < 0

Uneconomic Region

A

B

Q = 20

Q = 10

Economic Region

MPL < 0

0

L

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