Frank C

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Frank C

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icroeconomics

Microeconom

ics

Exercise 2.6

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank Cowell Frank Cowell

March 2007 March 2007

Frank C

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Frank C

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icroeconomics

Microeconom

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Ex 2.6(1): Question

purposepurpose: examine properties of an important class of production function: examine properties of an important class of production function methodmethod: find MRTS and examine properties in relation to key parameter: find MRTS and examine properties in relation to key parameter

Frank C

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Frank C

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icroeconomics

Microeconom

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Ex 2.6(1): Find the MRTS

Differentiate the production functionDifferentiate the production function

Then the marginal product of input Then the marginal product of input ii is is

From the standard definition the MRTS isFrom the standard definition the MRTS is

Frank C

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Frank C

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icroeconomics

Microeconom

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Ex 2.6(1): Find the elasticity

Invert the MRTS formulaInvert the MRTS formula

Define two convenient changes of variable Define two convenient changes of variable := log (:= log (zz11//zz22) ) := log (:= log (11((zz)/)/22((zz))))

Then elasticity of substitution is just Then elasticity of substitution is just d d/d/d Written out in full:Written out in full:

Frank C

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Frank C

owell: M

icroeconomics

Microeconom

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Ex 2.6(2): Question and solution

The case The case → → ∞ yields ∞ yields = 0 = 0

The case The case → 0 yields → 0 yields = 1= 1

The case The case → 1 yields → 1 yields = ∞= ∞

Frank C

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Frank C

owell: M

icroeconomics

Microeconom

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Ex 2.6(3): Question

methodmethod: : Sketch isoquants for each of the special casesSketch isoquants for each of the special cases

Frank C

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Frank C

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icroeconomics

Microeconom

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Ex 2.6(3): → ∞

z2

z1

LeontiefLeontief

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Frank C

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icroeconomics

Microeconom

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Ex 2.6(3): → 0

z2

z1

Cobb-DouglasCobb-Douglas

Frank C

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Frank C

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icroeconomics

Microeconom

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Ex 2.6(3): → 1

z2

z1

LinearLinear

Frank C

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Frank C

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icroeconomics

Microeconom

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Ex 2.6: Things to remember

Many standard simple forms of isoquant map Many standard simple forms of isoquant map belong to same familybelong to same family

Parameter of production function is monotonic Parameter of production function is monotonic transform of elasticity of substitution.transform of elasticity of substitution.