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# Bs Ch07 2 2006

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### Bs Ch07 2 2006

1. 1. Elasticity of Substitution <ul><li>How easy is it to substitute one input for another??? </li></ul><ul><li>Production functions may also be classified in terms of elasticity of substitution </li></ul><ul><ul><li>Shape of a single isoquant… </li></ul></ul><ul><li>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: </li></ul>
2. 2. Note Throughout <ul><li>Book uses ξ for substitution elasticity </li></ul><ul><li>I use σ </li></ul><ul><li>They are the same: ξ = σ </li></ul><ul><li>It just seems to me that σ is used more often in the literature…. </li></ul>
3. 3. Elasticity of Substitution <ul><li>Movement from A to B results in </li></ul><ul><ul><li>L becomes bigger, K becomes smaller </li></ul></ul><ul><ul><li>capital/labor ratio (K/L) decreasing </li></ul></ul><ul><ul><li>MRTS = -dK/dL = MP L /MP K </li></ul></ul><ul><ul><li>=> MRTS KL decreases </li></ul></ul><ul><li>Along a strictly convex isoquant, K/L and MRTS move in same direction </li></ul><ul><ul><li>Elasticity of substitution is positive </li></ul></ul><ul><li>Relative magnitude of this change is measured by elasticity of substitution </li></ul><ul><ul><li>If it is high, MRTS will not change much relative to K/L and the isoquant will be less curved (less strictly convex) </li></ul></ul><ul><ul><li>A low elasticity of substitution gives rather sharply curved isoquants </li></ul></ul>MRTS A MRTS B
4. 4. Elasticity of Substitution: Perfect-Substitute <ul><li> =  , a perfect-substitute technology </li></ul><ul><ul><li>Analogous to perfect substitutes in consumer theory </li></ul></ul><ul><ul><li>A production function representing this technology exhibits constant returns to scale </li></ul></ul><ul><ul><ul><li>ƒ(  K,  L) = a  K + b  L =  (aK + bL) =  ƒ(K, L) </li></ul></ul></ul><ul><ul><ul><li>All isoquants for this production function are parallel straight lines with slopes = -b/a </li></ul></ul></ul>
5. 5. Elasticity of substitution for perfect-substitute technologies σ = ∞
6. 6. Elasticity of Substitution: Leontief <ul><li> = 0, a fixed-proportions (or Leontief ) technology </li></ul><ul><ul><li>Analogous to perfect complements in consumer theory </li></ul></ul><ul><ul><li>Characterized by zero substitution </li></ul></ul><ul><li>A production technology that exhibits fixed proportions is </li></ul><ul><li>This production function also exhibits constant returns to scale </li></ul>
7. 7. Elasticity of substitution for fixed-proportions technologies <ul><li>Capital and labor must always be used in a fixed ratio </li></ul><ul><li>Marginal products are constant and zero </li></ul><ul><ul><li>Violates Monotonicity Axiom and Law of Diminishing Marginal Returns </li></ul></ul><ul><li>Isoquants for this technology are right angles => Kinked </li></ul><ul><ul><li>At kink, MRTS is not unique—can take on an infinite number of positive values </li></ul></ul><ul><ul><ul><li>K/L is a constant, d(K/L) = 0, which results in  = 0 </li></ul></ul></ul>σ = 0
8. 8. Elasticity of Substitution; Cobb-Douglas <ul><li> = 1, Cobb-Douglas technology </li></ul><ul><ul><li>Isoquants are strictly convex </li></ul></ul><ul><ul><ul><li>Assumes diminishing MRTS </li></ul></ul></ul><ul><li>An example of a Cobb-Douglas production function is </li></ul><ul><ul><li>q = ƒ(K, L) = aK b L d </li></ul></ul><ul><ul><ul><li>a, b, and d are all positive constants </li></ul></ul></ul><ul><li>Useful in many applications because it is linear in logs </li></ul>
9. 9. Isoquants for a Cobb-Douglas production function σ = 1
10. 10. Constant Elasticity of Substitution (CES) <ul><li> = some positive constant </li></ul><ul><li>Constant elasticity of substitution (CES) production function can be specified </li></ul><ul><ul><li>q =  [  K - ρ + (1 -  )L - ρ ] -1/ ρ </li></ul></ul><ul><ul><li> > 0, 0 ≤  ≤1, ρ ≥ -1 </li></ul></ul><ul><ul><ul><li> is efficiency parameter </li></ul></ul></ul><ul><ul><ul><li> is a distribution parameter </li></ul></ul></ul><ul><ul><ul><li> is substitution parameter </li></ul></ul></ul><ul><li>Elasticity of substitution is </li></ul><ul><ul><li> = 1/(1 +  ) </li></ul></ul><ul><ul><ul><li>Useful in empirical studies </li></ul></ul></ul>
11. 11. Investigating Production <ul><li>Spreadsheets available to assess Cobb-Douglas and Constant Elasticity of Substitution Production Functions. </li></ul><ul><li>On Website </li></ul><ul><li>I suggest reviewing them. </li></ul>
12. 12. Technical Progress/ Technological Change L K q 0 q 1 L 1 K 1 K 0 L 0 Technical Progress shifts the isoquant inward The same output can be produced with less/fewer inputs
13. 13. How to Measure Technical Progress? <ul><li>If q = A(t)f[K(t), L(t)], </li></ul><ul><ul><li>The term A(t) represents factors that influence output given levels of capital and labor. </li></ul></ul><ul><ul><li>Proxy for technical progress </li></ul></ul>
14. 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. 15. Technical Progress Continued Rate of Growth of Output is: <ul><li>Rate of Growth of Output is equal to </li></ul><ul><li>Rate of growth of autonomous technological change </li></ul><ul><li>Plus rate of growth of capital times e K (output elasticity of capital) </li></ul><ul><li>Plus rate of growth of labor times e L (output elasticity of labor) </li></ul>
16. 16. Historically Data from Robert Solow’s study of technological progress in the US, 1909 - 1949
17. 17. Annual Productivity Growth in Agriculture (1965 – 1994) (Nin et al., 2003) 2.50 1.19 0.96 W. Europe 1.55 0.63 0.67 E. Europe 0.98 0.52 0.53 South Amer. -0.53 1.32 0.36 Asia -0.32 -0.01 -0.26 Sub-Saharan Africa 0.20 0.01 0.05 ME/N. Africa Crops Livestock Agriculture Region
18. 18. How much can the world produce? <ul><li>DICE Model (W. Nordhaus – see Nordhaus and Boyer, 2000). </li></ul><ul><ul><li>D ynamic I ntegrated Model of C limate and the E conomy. </li></ul></ul><ul><li>Production: </li></ul><ul><ul><li>Q(t) = A(t)*(K(t) 0.30 L(t) 0.70 ) </li></ul></ul><ul><ul><ul><li>A(0) = 0.018 </li></ul></ul></ul><ul><ul><ul><li>K(t) = \$73.6 trillion </li></ul></ul></ul><ul><ul><ul><li>L(t) = 6,484 million (world population) </li></ul></ul></ul><ul><ul><ul><li>Q(t) denominated in \$ trillion/year </li></ul></ul></ul>
19. 19. Additional Assumptions <ul><li>A(t) increases at 0.37% per year. </li></ul><ul><ul><li>Global average increase in productivity. </li></ul></ul><ul><ul><li>Compare to alternative: 0.19% per year. </li></ul></ul>