CHAPTER 8
THE DISCOVERY OF
PRODUCTION
AND ITS TECHNOLOGY
DISCOVERING PRODUCTION
• Primitive society
• Fruit and land
• Accidental discovery: jam
• Opportunity cost
• Cost of engaging in any activity
• Opportunity forgone - particular activity
• Normal profit
• Just sufficient to recover opportunity cost
• Extra-normal profit
• Return above normal profit
2
PRODUCTION FUNCTION AND
TECHNOLOGY
• Technology
• Set of technological constraints
• On production
• Combine inputs into outputs
3
PRODUCTION FUNCTION AND
TECHNOLOGY
• No free lunch assumption
• Production process
• Need inputs to produce outputs
• Non reversibility assumption
• Cannot run a production process in reverse
• Free disposability assumption
• Combination of inputs
• Certain output
• Or strictly less output
4
PRODUCTION FUNCTION AND
TECHNOLOGY
• Additivity assumption
• Produce output x
• One combination of inputs
• Produce output y
• Another combination of inputs
• Feasible: produce x+y
5
PRODUCTION FUNCTION AND
TECHNOLOGY
• Divisibility assumption
• Feasible input combination y
• Then, λy – feasible input combination
• 0≤ λ ≤ 1
6
PRODUCTION FUNCTION AND
TECHNOLOGY
• Convexity assumption
• Production activity: y
• Output: z
• Particular amounts of inputs
• Production activity: w
• Output: z
• Different amounts of inputs
• Produce: at least z
• Mix activities y (λ time) and w(1- λ time)
7
PRODUCTION FUNCTION AND
TECHNOLOGY
• Production function
• Maximum amount of output
• Given a certain level of inputs
• Output=f (input1, input2)
• Marginal product of input1
• the increase in output as a result of a marginal increase in
input1 holding input2 constant
• diminishing
8
ISOQUANT
• Isoquant
• Set of bundles
• Given production function
• Produce same output
• Most efficiently
ISOQUANT
Capital
III
II200
I100
0
Labor
All combinations of inputs along the same isoquant yield the same
output.
10
ISOQUANT
• Isoquants
• Never cross each other
• Farther from the origin greater outputs
• Slope
• Marginal rate of technical substitution
11
MARGINAL RATE OF TECHNICAL
SUBSTITUTION
Capital (x2)
α
3
2
7
β
4
0
9
11
Labor (x1)
The absolute value of the isoquant’s slope measures the rate at which one
input can be substituted for the other while keeping the output level constant.
12
MARGINAL RATE OF TECHNICAL
SUBSTITUTION
• Marginal rate of technical substitution (MRTS)
• Rate of substitution
• One input for another
• Constant output
13
THE PRODUCTION FUNCTION
Output (y)
The level of output
is a function of the
levels of capital and
labor used.
4
Labor (x1)
Capital (x2)
W
y1
W
y2
14
MARGINAL RATE OF TECHNICAL
SUBSTITUTION
• Marginal product of input x2 at point α
(change in output)
y


(change in the use of input x 2 given x1 ) x2
• MRTS of x2 for x1 at point α
y
Marginal product of x1 x1


Marginal product of x 2 y
x2
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DESCRIBING TECHNOLOGIES
• Returns to scale – ratio of
• Change in output
• Proportionate change in all inputs
• Constant returns to scale
• All inputs - increase by λ
• Output - increases by λ
16
DESCRIBING TECHNOLOGIES
• Increasing returns to scale
• All inputs - increase by λ
• Output - increases by more than λ
• Decreasing returns to scale
• All inputs - increase by λ
• Output - increases by less than λ
• Elasticity of substitution
• Substitute one input for another
• Given level of output
17
RETURNS TO SCALE
Capital (x2)
Capital (x2)
(a)
Capital (x2)
(b)
(c)
p2
D
4
C
2
1
0
A
3
B p1
8
4
6
12
Labor (x1)
2
1
0
A
B p1
10
4
12
6
Labor (x1)
Constant returns to scale. Increasing returns to scale.
Doubling the levels of
Doubling the levels of both
labor (from 3 to 6) and
inputs more than doubles
capital (from 2 to 4) also the output level
doubles the level of output
(from 4 to 8)
2
1
0
A
B p1
6
4
6
12
Labor (x1)
Decreasing returns to
scale. Doubling the levels
of both inputs less than
doubles the output level
18
TIME CONSTRAINTS
• Immediate run
• Period of time
• Cannot vary inputs
• Fixed factor of production
• Cannot be adjusted
• Given period of time
• Variable factor of production
• Can be adjusted
19
TIME CONSTRAINTS
• Short run
• Time period
• At least one factor of production – fixed
• Long run
• Time period
• All factors of production – variable
20
TIME CONSTRAINTS
• Long-run production function
• All inputs – variable
• Short-run production function
• Some inputs – variable
• Capital – fixed
• Labor – variable
21
FIGURE 8.5
C
With the level of
capital fixed at x2,
the output level is a
function solely of
the level of labor.
Short-run production function
B
Labor (x1)
Capital (x2)
x2
0
22
TIME CONSTRAINTS
• Total product curve
• Amount of output
• Add more and more units of variable input
• Hold one input constant
• Output – as we add more variable input
• First: increase at increasing rate
• After a point: Increase at decreasing rate
• Later: decrease
23
FIGURE 8.6
Output
Short-run production function inDlabor-output space
8 14
G
8
E
1
12
1
2
0
A
1
+1
+1
2
10
15 16
30
Labor
The level of the fixed input, capital, is suppressed.
24
TIME CONSTRAINTS
• Decreasing returns to factor
• Rate of output growth: decreasing
• Increase one input
• Other inputs – constant
• Marginal product curve
• Marginal product
• Factor of production
25
FIGURE 8.7
Marginal
product
Marginal product
e
d
1
2
0
1
10
30
Labor (x1)
The slope of the short-run production function measures the change
in the output level resulting from the introduction of 1 additional unit
of the variable input - labor.
26
THE PRODUCTION FUNCTION
• Cobb-Douglas production function
Q=AKαLβ
•
•
•
•
•
A – positive constant
0<α<1; 0<β<1
K – amount of capital
L – amount of labor
Q – output
27
THE PRODUCTION FUNCTION
• Returns to scale = (α+β)
• For λ K and λL:
Q’= A(λK)α(λL)β =λ α+β Q
• If α+β=1
• Linearly homogeneous
• Constant returns to scale Q=AKαL1-α
• If α+β>1
• Increasing returns to scale
• If α+β<1
• Decreasing returns to scale
28
THE PRODUCTION FUNCTION
• MRTS: dQ=0
dK  1    K


dL    L
• Elasticity of substitution
 KL
d ln( K / L)

1
d ln( MRTS )
29
THE PRODUCTION FUNCTION
• Q=AKαLβ; α+β=1
Q
K
Define : q  ; k 
L
L
Q
Q

Average products : APL   Ak ; APK   Ak 1
L
K
Q

Marginal products : MPL 
 (1  ) Ak ;
L
Q
MPK 
 Ak 1
K
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THE PRODUCTION FUNCTION
• Q=AKαLβ; α+β=1
• Share of capital in output: K∙MPK/Q=α
• Share of labor in output: L∙MPL/Q=1-α
• Elasticity of output
Q Q
QK 
 
K K
Q Q
QL 
  1 
L L
31