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Analysis of Production Function: MRTS and Elasticity of Substitution

Explore properties of a production function class by finding MRTS and elasticity, with key parameter examination and variable changes for elasticity calculation. Differentiate, invert formula, and sketch isoquants for special cases.

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Analysis of Production Function: MRTS and Elasticity of Substitution

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  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 → ∞ 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 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.

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