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:

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 = MP L /MP K => MRTS KL 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 MRTS A MRTS B

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) = a  K + b  L =  (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) = aK b L d 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 q0q0 q1q1 L1L1 K1K1 K0K0 L0L0 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

14 Technical Progress Continued Divide result on previous page by q and adjust = output elasticity wrt capital = e K = output elasticity wrt labor = e L Some identities:

15 Technical Progress Continued Rate of Growth of Output is: Rate of Growth of Output is equal to Rate of growth of autonomous technological change Plus rate of growth of capital times e K (output elasticity of capital) Plus rate of growth of labor times e L (output elasticity of labor)

16 Historically Data from Robert Solow’s study of technological progress in the US,

17 Annual Productivity Growth in Agriculture (1965 – 1994) (Nin et al., 2003) RegionAgricultureLivestockCrops ME/N. Africa Sub-Saharan Africa Asia South Amer E. Europe W. Europe

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) 0.30 L(t) 0.70 ) A(0) = 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.