Download The Firm: Optimisation

Example
x2
•
indiff curve u = 1
•
indiff curve u = 2
•
indiff curve u = 3
•
From the equation
•
Equation of IC is
•
Transformed utility function
x1
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Frank Cowell: Lecture Examples
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8 Oct 2015
Frank Cowell: Lecture Examples
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Example
x2
• Indifference curve (as before)
• does not touch either axis
• Constraint set for given u
• Cost minimisation must have interior solution
x1
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Frank Cowell: Lecture Examples
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Example
• Lagrangian for cost minimisation
x2
• For a minimum:
• Evaluate first-order conditions

x*
x1
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Frank Cowell: Lecture Examples
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Example
• First-order conditions for cost-min:
• Rearrange the first two of these:
• Substitute back into the third FOC:
• Rearrange to get the optimised Lagrange multiplier
8 Oct 2015
Frank Cowell: Lecture Examples
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Example
• From first-order conditions:
• Rearrange to get cost-min inputs:
• By definition minimised cost is:
• So cost function is
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Frank Cowell: Lecture Examples
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Example
• Lagrangean for utility maximisation
• Evaluate first-order conditions
x2

x*
x1
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Frank Cowell: Lecture Examples
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Example
• Optimal demands are
x2
• So at the optimum

x*
x1
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Frank Cowell: Lecture Examples
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8 Oct 2015
Frank Cowell: Lecture Examples
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Example
• Results from cost minimisation:
• Differentiate to get compensated demand:
• Results from utility maximisation:
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Frank Cowell: Lecture Examples
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Example
• Ordinary and compensated demand for good 1:
• Response to changes in y and p1:
• Use cost function to write last term in y rather than u:
• Slutsky equation:
• In this case:
8 Oct 2015
Frank Cowell: Lecture Examples
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Example
• Take a case where income is endogenous:
• Ordinary demand for good 1:
• Response to changes in y and p1:
• Modified Slutsky equation:
• In this case:
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Frank Cowell: Lecture Examples
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8 Oct 2015
Frank Cowell: Lecture Examples
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Example
• Cost function:
• Indirect utility function:
• If p1 falls to tp1 (where t < 1) then utility rises from u to u′:
• So CV of change is:
• And the EV is:
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Frank Cowell: Lecture Examples
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