1

Elasticity of Substitution How easy is it to substitute one input for another???

Production functions may also be classified in terms of elasticity of substitution Shape of a single isoquant…

Elasticity of Substitution is a measure of the proportionate change in K/L (capital to labor ratio) relative to the proportionate change in MRTS along an isoquant:

)ln(

ln

)(%

%lim

0 MRTSdL

Kd

MRTSMRTSd

LK

LKd

MRTSL

K

MRTS

2

Note Throughout

Book uses ξ for substitution elasticity I use σ They are the same: ξ = σ It just seems to me that σ is used more often

in the literature….

3

Elasticity of Substitution

Movement from A to B results in L becomes bigger, K becomes

smaller capital/labor ratio (K/L) decreasing

MRTS = -dK/dL = MPL/MPK

=> MRTSKL decreases Along a strictly convex isoquant, K/L and

MRTS move in same direction Elasticity of substitution is positive

Relative magnitude of this change is measured by elasticity of substitution If it is high, MRTS will not change

much relative to K/L and the isoquant will be less curved (less strictly convex)

A low elasticity of substitution gives rather sharply curved isoquants

MRTSA

MRTSB

4

Elasticity of Substitution: Perfect-Substitute

= , a perfect-substitute technology Analogous to perfect substitutes in consumer

theory A production function representing this

technology exhibits constant returns to scale• ƒ(K, L) = aK + bL = (aK + bL) = ƒ(K, L)

• All isoquants for this production function are parallel straight lines with slopes = -b/a

5

Elasticity of substitution for perfect-substitute technologies

σ = ∞

6

Elasticity of Substitution: Leontief

= 0, a fixed-proportions (or Leontief ) technology Analogous to perfect complements in consumer theory Characterized by zero substitution

A production technology that exhibits fixed proportions is

This production function also exhibits constant returns to scale

7

Elasticity of substitution for fixed-proportions technologies

Capital and labor must always be used in a fixed ratio

Marginal products are constant and zero Violates Monotonicity Axiom and Law

of Diminishing Marginal Returns Isoquants for this technology are right

angles => Kinked At kink, MRTS is not unique—can take

on an infinite number of positive values

• K/L is a constant, d(K/L) = 0, which results in = 0

σ = 0

8

Elasticity of Substitution; Cobb-Douglas

= 1, Cobb-Douglas technology Isoquants are strictly convex

• Assumes diminishing MRTS

An example of a Cobb-Douglas production function is q = ƒ(K, L) = aKbLd

• a, b, and d are all positive constants

Useful in many applications because it is linear in logs

9

Isoquants for a Cobb-Douglas production function

σ = 1

10

Constant Elasticity of Substitution (CES)

= some positive constant Constant elasticity of substitution (CES) production

function can be specified q = [K-ρ + (1 - )L- ρ]-1/ρ

> 0, 0 ≤ ≤1, ρ ≥ -1 is efficiency parameter is a distribution parameter is substitution parameter

Elasticity of substitution is = 1/(1 + )

• Useful in empirical studies

11

Investigating Production

Spreadsheets available to assess Cobb-Douglas and Constant Elasticity of Substitution Production Functions.

On Website I suggest reviewing them.

12

Technical Progress/Technological Change

L

K

q0

q1

L1

K1

K0

L0

Technical Progress shifts the isoquant inward

The same output can be produced with less/fewer inputs

13

How to Measure Technical Progress?

If q = A(t)f[K(t), L(t)], The term A(t) represents factors that influence output

given levels of capital and labor. Proxy for technical progress

dt

dL

dL

df

dt

dK

dK

df

LKf

q

A

q

dt

dA

dt

dqdt

LKdfALKf

dt

dA

dt

dq

),(

),(),(

14

Technical Progress Continued

Divide result on previous page by q and adjust

Ldt

dL

LKf

L

dL

df

Kdt

dK

LKf

K

dK

df

Adt

dA

qdt

dq

dt

dL

dL

df

dt

dK

dK

df

LKfAdt

dA

qdt

dq

),(),(

),(

111

LdtdLKdt

dKAdtdAqdt

dq ;;;

f

K

dK

df= output elasticity wrt capital= eK f

L

dL

df= output elasticity wrt labor= eL

Some identities:

15

Technical Progress Continued

Rate of Growth of Output is:

L

Le

K

Ke

A

A

q

qLK

Rate of Growth of Output is equal to

• Rate of growth of autonomous technological change• Plus rate of growth of capital times eK (output elasticity of capital)• Plus rate of growth of labor times eL (output elasticity of labor)

16

Historically

00.1;65.0;75.1;35.0;75.2 L

Le

K

Ke

q

qLK

50.1)00.1(*65.0)75.1(*35.075.2

A

A

L

Le

K

Ke

q

q

A

ALK

Data from Robert Solow’s study of technological progress in the US, 1909 - 1949

17

Annual Productivity Growth in Agriculture (1965 – 1994) (Nin et al., 2003)

Region Agriculture Livestock Crops

ME/N. Africa 0.05 0.01 0.20

Sub-Saharan Africa

-0.26 -0.01 -0.32

Asia 0.36 1.32 -0.53

South Amer. 0.53 0.52 0.98

E. Europe 0.67 0.63 1.55

W. Europe 0.96 1.19 2.50

18

How much can the world produce?

DICE Model (W. Nordhaus – see Nordhaus and Boyer, 2000). Dynamic Integrated Model of Climate and the

Economy. Production:

Q(t) = A(t)*(K(t)0.30L(t)0.70)• A(0) = 0.018• K(t) = $73.6 trillion• L(t) = 6,484 million (world population)• Q(t) denominated in $ trillion/year

19

Additional Assumptions

A(t) increases at 0.37% per year. Global average increase in productivity. Compare to alternative: 0.19% per year.

0

10

20

30

40

50

60

70

80

90

100

2005 2015 2025 2035 2045 2055 2065 2075 2085 2095 2105

Year

Out

put

(Tril

lions

US

$)

Base

Slow Productivity Growth