lecture # 09 inputs and production functions lecturer: martin paredes
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Lecture # 09Lecture # 09
Inputs and Production Inputs and Production FunctionsFunctions
Lecturer: Martin ParedesLecturer: Martin Paredes
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1. The Production Function Marginal and Average Products Isoquants Marginal Rate of Technical Substitution
2. Returns to Scale3. Some Special Functional Forms4. 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
L
Q
•C
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Example: The Production Function and Technical Efficiency
L
Q
••
C
D
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Example: The Production Function and Technical Efficiency
Q = f(L)
L
Q
••
C
D
Production Function
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Example: The Production Function and Technical Efficiency
Q = f(L)
L
Q
•
•••
C
D
A
B
Production Function
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Example: The Production Function and Technical Efficiency
Q = f(L)
L
Q
•
•••
C
D
A
B
Production Set
Production Function
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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 Production Function
1.
Satisfaction from purchases
Output from inputs
2.
Derived from preferences
Derived from technologies
3.
Ordinal Cardinal
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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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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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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 APK = Q L 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
L
Q= F(L,K0)
Example: Total and Marginal Product
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Q
L
MPL maximized
Q= F(L,K0)
Example: Total and Marginal Product
Increasing marginal returns
Diminishing marginal returns
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Q
L
MPL = 0 whenTP maximized
Q= F(L,K0)
Example: Total and Marginal Product
Diminishing total returns
Increasing total returns
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Example: Total and Marginal Product
L
MPL
Q
LMPL maximized
TPL maximized whereMPL is zero. TPL fallswhere MPL is negative;TPL rises where MPL ispositive.
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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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Example: Average and Marginal Products
L
APL
MPL
MPL maximized
APL maximized
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Example: Total, Average and Marginal Products
L
APL
MPL
Q
LMPL maximized
APL maximized
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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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Example: Isoquants
L
K
Q = 10
0
Slope=dK/dL
L
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L
Q = 10
Q = 20
All combinations of (L,K) along theisoquant produce 20 units of output.
0
Slope=dK/dL
K
Example: Isoquants
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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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Example: The Economic and the Uneconomic Regions of Production
L
K
Q = 10
Q = 20
0
Isoquants
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Example: The Economic and the Uneconomic Regions of Production
L
K
Q = 10
Q = 20
0
• •BA
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Example: The Economic and the Uneconomic Regions of Production
L
K
Q = 10
Q = 20
0
MPL < 0
• •BA
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Example: The Economic and the Uneconomic Regions of Production
L
K
Q = 10
Q = 20
0
MPK < 0
MPL < 0
• •BA
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Example: The Economic and the Uneconomic Regions of Production
L
K
Q = 10
Q = 20
0
MPK < 0
MPL < 0
• •BA
Uneconomic Region
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Example: The Economic and the Uneconomic Regions of Production
L
K
Q = 10
Q = 20
0
MPK < 0
MPL < 0
• •BA
Uneconomic Region
Economic Region
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