1. Frank Cowell: Microeconomics Exercise 2.6 MICROECONOMICS Principles and Analysis Frank Cowell March 2007

2. Frank Cowell: Microeconomics Ex 2.6(1): Question purpose: examine properties of an important class of production function purpose: examine properties of an important class of production function method: find MRTS and examine properties in relation to key parameter method: find MRTS and examine properties in relation to key parameter

3. Frank Cowell: Microeconomics Ex 2.6(1): Find the MRTS Differentiate the production function Differentiate the production function Then the marginal product of input i is Then the marginal product of input i is From the standard definition the MRTS is From the standard definition the MRTS is

4. Frank Cowell: Microeconomics Ex 2.6(1): Find the elasticity Invert the MRTS formula Invert the MRTS formula Define two convenient changes of variable Define two convenient changes of variable   := log (z 1 /z 2 )   := log (  1 (z)/  2 (z)) Then elasticity of substitution is just  d  /d  Then elasticity of substitution is just  d  /d  Written out in full: Written out in full:

5. Frank Cowell: Microeconomics Ex 2.6(2): Question and solution The case  →  ∞ yields  = 0 The case  →  ∞ yields  = 0 The case  → 0 yields  = 1 The case  → 0 yields  = 1 The case  → 1 yields  = ∞ The case  → 1 yields  = ∞

6. Frank Cowell: Microeconomics Ex 2.6(3): Question method: Sketch isoquants for each of the special cases Sketch isoquants for each of the special cases

7. Frank Cowell: Microeconomics Ex 2.6(3):  →  ∞ z2z2 z1z1 Leontief

8. Frank Cowell: Microeconomics Ex 2.6(3):  → 0 z2z2 z1z1 Cobb-Douglas

9. Frank Cowell: Microeconomics Ex 2.6(3):  → 1 z2z2 z1z1 Linear

10. Frank Cowell: Microeconomics Ex 2.6: Things to remember Many standard simple forms of isoquant map belong to same family Many standard simple forms of isoquant map belong to same family Parameter of production function is monotonic transform of elasticity of substitution. Parameter of production function is monotonic transform of elasticity of substitution.