Exercise 2.6

1
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 ? ?8 yields s 0
• The case b ? 0 yields s 1
• The case b ? 1 yields s 8

6
Ex 2.6(3) Question

• method
• Sketch isoquants for each of the special cases

7
Ex 2.6(3) b ? ?8
z2
Leontief
z1
8
Ex 2.6(3) b ? 0
z2
Cobb-Douglas
z1
9
Ex 2.6(3) b ? 1
z2
Linear
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.