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

2. 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

3. 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

4. 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:

5. 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 = ∞

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

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

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

9. Ex 2.6(3): b → 1 z2
Linear
4isoqs z1

10. 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.