MICROECONOMICS Principles and Analysis Frank Cowell Exercise 2.6 MICROECONOMICS Principles and Analysis Frank Cowell March 2007

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

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

Ex 2.6(1): Find the elasticity Invert the MRTS formula Define two convenient changes of variable z := log (z1/z2) q := log (f1(z)/f2(z)) Then elasticity of substitution is just  dz/dq Written out in full:

Ex 2.6(2): Question and solution The case b → ∞ yields s = 0 The case b → 0 yields s = 1 The case b → 1 yields s = ∞

Ex 2.6(3): Question method: Sketch isoquants for each of the special cases

Ex 2.6(3): b → ∞ z2 Leontief 4isoqs z1

Ex 2.6(3): b → 0 z2 Cobb-Douglas 4isoqs z1

Ex 2.6(3): b → 1 z2 Linear 4isoqs z1

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