Lecture # 09. Inputs and Production Functions презентация

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

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

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.

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,…)

Definitions
A technically efficient firm is attaining the maximum possible output from its inputs (using whatever technology is appropriate)
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

Example: The Production Function and Technical Efficiency
L
Q
•
C

Example: The Production Function and Technical Efficiency
L
Q
•
•
C
D

Example: The Production Function and Technical Efficiency
Q = f(L)
L
Q
•
•
C
D
Production Function

Example: The Production Function and Technical Efficiency
Q = f(L)
L
Q
•
•
•
•
C
D
A
B
Production Function

Example: The Production Function and Technical Efficiency
Q = f(L)
L
Q
•
•
•
•
C
D
A
B
Production Set
Production Function

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).

Comparison between
production function
and utility function

Comparison between
production function
and utility function

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.

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
Marginal Product

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

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
Average Product

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.

Q
L
Q= F(L,K0)
Example: Total and Marginal Product

Q
L
MPL maximized
Q= F(L,K0)
Example: Total and Marginal Product
Increasing marginal returns
Diminishing marginal returns

Q
L
MPL = 0 when
TP maximized
Q= F(L,K0)
Example: Total and Marginal Product
Diminishing total returns
Increasing total returns

Example: Total and Marginal Product
L
MPL
Q
L
MPL maximized
TPL maximized where
MPL is zero. TPL falls
where MPL is negative;
TPL rises where MPL is
positive.

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.

Marginal and Average Products
When marginal product is greater than average product, average product is increasing.
E.g., if MPL > APL , APL increases in L.
When marginal product is less than average product, average product is decreasing.
E.g., if MPL < APL, APL decreases in L.

Example: Average and Marginal Products
L
APL
MPL
MPL maximized
APL maximized

Example: Total, Average and Marginal Products
L
APL
MPL
Q
L
MPL maximized
APL maximized

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.

Example: Isoquants
L
K
Q = 10
0
Slope=dK/dL
L

L
Q = 10
Q = 20
All combinations of (L,K) along the
isoquant produce 20 units of output.
0
Slope=dK/dL
K
Example: Isoquants

Isoquants
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

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.
Marginal Rate Of Technical Substitution

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)

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

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.

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 < 0
We reach an uneconomic region of production

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

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

Example: The Economic and the Uneconomic Regions of Production
L
K
Q = 10
Q = 20
0
MPL < 0
•
•
B
A

Example: The Economic and the Uneconomic Regions of Production
L
K
Q = 10
Q = 20
0
MPK < 0
MPL < 0
•
•
B
A

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