Production

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Lecture # 09
Inputs and Production Functions
Lecturer: Martin Paredes
1. The Production Function
 Marginal 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 are
productive 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 inputs
into 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 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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6. A technically efficient firm is attaining the
maximum 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 and
Technical Efficiency
Q
C
•
L
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Example: The Production Function and
Technical Efficiency
Q
D
•
C
•
L
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Example: The Production Function and
Technical Efficiency
Q
D
•
Production Function
Q = f(L)
C
•
L
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Example: The Production Function and
Technical Efficiency
Q
D
C
•
•A
•
•B
Production Function
Q = f(L)
L
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Example: The Production Function and
Technical 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 Function
Satisfaction 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
6.
Marginal Rate of
Substitution
Production Function
Marginal Product
Isoquants
Marginal Rate of
Technical Substitution
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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
L
MPK = Q
K
 It assumes the levels of all other inputs are held
constant.
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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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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
L
APK = Q
K
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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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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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Q
Example: Total and Marginal Product
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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Q
Example: Total and Marginal Product
Q= F(L,K0)
MPL = 0 when
TP maximized
Increasing
total returns
Diminishing
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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 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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1. When marginal product is greater than average
product, 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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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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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
Slope=dK/dL
0
Q = 10
L
L
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K
Example: 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 = K0.5L0.5
 For Q = 20
=> 20 = K0.5L0.5
=> 400 = KL
=> K = 400/L
 For Q = Q0
=> K = (Q0)2 /L
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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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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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 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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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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K
Example: The Economic and the
Uneconomic Regions of Production
Isoquants
Q = 20
Q = 10
0
L
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K
Example: The Economic and the
Uneconomic Regions of Production
•
A
B
•
Q = 20
Q = 10
0
L
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K
Example: The Economic and the
Uneconomic Regions of Production
•
A
B
•
Q = 20
Q = 10
MPL < 0
0
L
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K
Example: The Economic and the
Uneconomic Regions of Production
MPK < 0
•
A
B
•
Q = 20
Q = 10
MPL < 0
0
L
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K
Example: 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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K
Example: 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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