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 MRTS d L K d MRTS MRTS d L K L K d MRTS L K MRTS
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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
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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
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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
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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
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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
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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
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Investigating Production
Spreadsheets available to assess Cobb-Douglas and Constant Elasticity of Substitution Production Functions.
On Website I suggest reviewing them.
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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
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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
),(
),(),(
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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:
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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)
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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
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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
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How much can the world produce?
DICE Model (W. Nordhaus – see Nordhaus and Boyer, 2000). Dynamic Integrated Model of Climate and the