Inputs and Production Functions

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Lecture 09 Inputs and Production
Functions Lecturer Martin Paredes
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Outline

• The Production Function
• Marginal and Average Products
• Isoquants
• Marginal Rate of Technical Substitution
• Returns to Scale
• Some Special Functional Forms
• Technological Progress

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Definitions

• Inputs or factors of production are productive
  resources that firms use to manufacture goods and
  services.
• Example labor, land, capital equipment
• The firms output is the amount of goods and
  services produced by the firm.

4
Definitions

• Production transforms a set of inputs into a set
  of outputs
• Technology determines the quantity of output that
  is feasible to attain for a given set of inputs.

5
Definitions

• The production function tells us the maximum
  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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Definitions

• A technically efficient firm is attaining the
  maximum possible output from its inputs (using
  whatever technology is appropriate)
• The firms 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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Q
Example The Production Function and
Technical Efficiency
C

L
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Q
Example The Production Function and
Technical Efficiency
D

C

L
9
Q
Example The Production Function and
Technical Efficiency
Production Function
Q f(L)
D

C

L
10
Q
Example The Production Function and
Technical Efficiency
Production Function
Q f(L)
D

C


B

A
L
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Q
Example The Production Function and
Technical Efficiency
Production Function
Q f(L)
D

C


B

A
Production Set
L
12

• 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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Comparison between production function and
utility function
Utility Function Production Function
1. Satisfaction from purchases Output from inputs
2. Derived from preferences Derived from technologies
3. Ordinal Cardinal
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Comparison between production function and
utility function
Utility Function Production Function
4. Marginal Utility Marginal Product
5. Indifference Curves Isoquants
6. Marginal Rate of Substitution Marginal Rate of Technical Substitution
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Marginal Product

• Definition The marginal product of an input is
  the change in output that results from a small
  change in an input
• E.g. MPL ?Q MPK ?Q
• ?L ?K
• It assumes the levels of all other inputs are
  held constant.

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Marginal Product
Example Suppose Q K0.5L0.5 Then MPL ?Q
0.5 K0.5 ?L L0.5 MPK ?Q 0.5
L0.5 ?K K0.5
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Average Product
Definition The average product of an input is
equal to the total output to be produced divided
by the quantity of the input that is used in its
production E.g. APL Q APK Q L
K
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Average Product
Example Suppose Q K0.5L0.5 Then 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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Law of Diminishing Marginal Returns
Definition The law of diminishing marginal
returns states that the marginal product
(eventually) declines as the quantity used of a
single input increases.
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Example Total and Marginal Product
Q
Q F(L,K0)
L
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Example Total and Marginal Product
Q
Q F(L,K0)
Increasing marginal returns
Diminishing marginal returns
MPL maximized
L
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Example Total and Marginal Product
Q
Q F(L,K0)
MPL 0 when TP maximized
Diminishing total returns
Increasing total returns
L
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Q
Example 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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Marginal and Average Products

• There is a systematic relationship between
  average product and marginal product.
• This relationship holds for any comparison
  between any marginal magnitude with the average
  magnitude.

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Marginal and Average Products

• When marginal product is greater than average
  product, average product is increasing.
• E.g., if MPL gt APL , APL increases in L.
• When marginal product is less than average
  product, average product is decreasing.
• E.g., if MPL lt APL, APL decreases in L.

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APL MPL
Example Average and Marginal Products
MPL maximized
APL maximized
L
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Q
Example Total, Average and Marginal Products
L
MPL maximized
APL MPL
APL maximized
L
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Isoquants
Definition An isoquant is a representation of
all the combinations of inputs (labor and
capital) that allow that firm to produce a given
quantity of output.
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K
Example Isoquants
Q 10
SlopedK/dL
L
L
0
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K
Example Isoquants
All combinations of (L,K) along the isoquant
produce 20 units of output.
Q 20
Q 10
SlopedK/dL
L
0
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Isoquants

• Example Suppose Q K0.5L0.5
• For Q 20 gt 20 K0.5L0.5
• gt 400 KL
• gt K 400/L
• For Q Q0 gt K (Q0)2 /L

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Marginal Rate Of Technical Substitution
Definition The marginal rate of technical
substitution 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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Marginal Rate Of Technical Substitution
Alternative Definition It is the negative of
the slope of the isoquant MRTSL,K dK
(for a constant level of dL output)
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Marginal Product and the Marginal Rate of
Technical Substitution

• We can express the MRTS as a ratio of the
  marginal 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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Marginal Product and the Marginal Rate of
Technical Substitution

• 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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Marginal Product and the Marginal Rate of
Technical Substitution

• 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 lt 0
• We reach an uneconomic region of production

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K
Example The Economic and the Uneconomic Regions
of Production
Isoquants
Q 20
Q 10
L
0
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K
Example The Economic and the Uneconomic Regions
of Production
Q 20
B


A
Q 10
L
0
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K
Example The Economic and the Uneconomic Regions
of Production
Q 20
B


A
Q 10
MPL lt 0
L
0
40
K
Example The Economic and the Uneconomic Regions
of Production
MPK lt 0
Q 20
B


A
Q 10
MPL lt 0
L
0
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K
Example The Economic and the Uneconomic Regions
of Production
MPK lt 0
Uneconomic Region
Q 20
B


A
Q 10
MPL lt 0
L
0
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K
Example The Economic and the Uneconomic Regions
of Production
MPK lt 0
Uneconomic Region
Q 20
B


A
Q 10
Economic Region
MPL lt 0
L
0