Production Function, Slides of Technology

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1
Production Functions
[See Chap 9]
2
Production Function
The firm’s production function for a
particular good (q) shows the maximum
amount of the good that can be produced
using alternative combinations of inputs.
q= f(z1,… , zN)
Examples (with N=2):
z1= capital, z2= labor.
z1= skilled labor, z2= unskilled labor
z1= capital, z2= land.
3
Marginal Product
The marginal product is the additional output
that can be produced by employing one more
unit of the input while holding other inputs
constant:
1
1
11
ofproduct marginal f
z
q
MPz =
==
pf3
pf4
pf5
pf8
pf9
pfa

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1

Production Functions

[See Chap 9]

2

Production Function

  • The firm’s production function for a

particular good ( q ) shows the maximum

amount of the good that can be produced

using alternative combinations of inputs.

q = f ( z 1 , … , zN )

  • Examples (with N=2):
    • z 1 = capital, z 2 = labor.
    • z 1 = skilled labor, z 2 = unskilled labor
    • z 1 = capital, z 2 = land.

3

Marginal Product

  • The marginal product is the additional output that can be produced by employing one more unit of the input while holding other inputs constant:

1 1

marginal productof (^11) z f q z MP = ∂

4

Average Product

  • Input productivity can be measured by

average product

1

2 1

(^1) z

f z z z

AP = q = (^1 , )

z 2

5

ISOQUANTS

6

Isoquants

  • An isoquant shows the combinations of

z 1 and z 2 that can produce a given level

of output ( q 0 )

f ( z 1 , z 2 ) = q 0

  • Like indifference curves for technology.

10

MRTS and Marginal Products

  • Take the total differential of the production

function:

2 1 2 1 2

dz MP dz MP d z z

f dz z

f dq (^21) 1

  • Along an isoquant dq = 0, so

2

1 q q MP

MP dz

MRTS =− dz = (^1) = (^0)

2

11

PROPERTIES OF TECHNOLOGY

12

1. Monotonicity

  • A production function is monotone if

f(z 1 ,z 2 ) is strictly increasing in both inputs.

  • This implies that
    • isoquants are thin
    • isoquants do not cross
    • isoquants are downward sloping.

i i

f z

f

13

2. Quasi-Concavity

  • Suppose z=(z 1 ,z 2 ) and z’=(z 1 ’,z 2 ’) are two

input bundles.

  • f(.) is quasi-concave in z if whenever

f(z)≥f(z’) then

f(tz+(1-t)z’)≥f(z’) 1>t>0.

  • Implications
    • Isoquants are convex.
    • MRTS decreases in z 1 , as move along isoquant.

14

3. Concavity

  • Suppose z=(z 1 ,z 2 ) and z’=(z 1 ’,z 2 ’) are two input bundles.
  • f(.) is concave in z if f(tz+(1-t)z’) ≥ tf(z) + (1-t)f(z’) 1>t>0.
  • Implies quasi-concavity (convex isoquants).
  • Implies diminishing marginal productivity:
  • Implies constant or decreasing returns to scale.

2 11 0 1

2 1

(^1) = ≤ ∂ =∂ ∂

∂ (^) f z

f z

MP^22 = 22 ≤ 0 ∂ =∂ ∂

∂ (^) f z

f z

MP 2 2

2

15

4. Returns to Scale

  • How does output respond to increases

in all inputs together?

  • suppose that all inputs are doubled, would output double?
  • The effect of a proportional change in all

inputs on output is called the returns to

scale

19

Perfect Substitutes

  • Suppose that the production function is

q = f ( z 1 , z 2 ) = az 1 + bz 2

  • Isoquants are straight lines.
    • MRTS is constant, since MP 1 =a and MP 2 =b.
  • Production function exhibits constant

returns to scale

f ( tz 1 , tz 2 ) = atz 1 + btz 2 = t ( az 1 + bz 2 ) = tf ( z 1 , z 2 )

20

Perfect Substitutes

z 1

z 2

q 1 q^2 q^3

Capital and labor are perfect substitutes

slope = -a/b

21

Perfect Complements

  • Suppose that the production function is q = min ( az 1 , bz 2 ) a , b > 0
  • Capital and labor must always be used

in a fixed ratio

  • the firm will always operate along a ray where z 1 / z 2 is constant

22

Perfect Complements

z 1

z 2

q 1

q 2

q 3

No substitution between labor and capital

is possible

q 3 /a

q 3 /b

23

Cobb-Douglas

  • Suppose that the production function is q = f ( z 1 , z 2 ) = z 1 az 2 b^ a,b > 0
  • Returns to scale f ( tz 1 , tz 2 ) = ( tz 1 ) a ( tz 1 ) b^ = ta + b^ z 1 az 1 b^ = ta + bf ( z 1 , z 1 )
  • if a + b = 1 ⇒ constant returns to scale
  • if a + b > 1 ⇒ increasing returns to scale
  • if a + b < 1 ⇒ decreasing returns to scale

24

Generalized Subs/Comps

  • Generalized perfect substitutes

q = f ( z 1 , z 2 ) = ( az 1 + b z 2 )γ

  • Generalized perfect complements

q = f ( z 1 , z 2 ) = (min(a z 1 ,b z 2 ))γ

  • Constant returns if γ=1.
  • Increasing returns if γ>1.
  • Decreasing returns if γ<1.

28

Technical Progress

  • Suppose that the production function is q = A ( t ) f ( z 1 , z 2 )

where A ( t ) represents all factors that

affect the production of q other than z 1

and z 2

  • Changes in A over time represent technical progress
  • We would imagine that dA / dt > 0