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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.
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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.