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.
Analysis of Production Function: MRTS and Elasticity of Substitution
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Presentation Transcript
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 z1
Ex 2.6(3): b → 0 z2 Cobb-Douglas z1
Ex 2.6(3): b → 1 z2 Linear 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.
