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8 Oct 2015
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Frank Cowell: Lecture Examples
Example
1
x2
x1
• indiff curve u = 1• indiff curve u = 2• indiff curve u = 3
• From the equation• Equation of IC is
• Transformed utility function
8 Oct 2015
Frank Cowell: Lecture Examples 28 Oct 2015
Frank Cowell: Lecture Examples
Examplex2
x1
• Indifference curve (as before)• does not touch either axis
• Constraint set for given u• Cost minimisation must have interior solution
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Frank Cowell: Lecture Examples
• Lagrangian for cost minimisation
• For a minimum:
• Evaluate first-order conditions
Examplex2
x1
x*
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Frank Cowell: Lecture Examples
• 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
Example
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Frank Cowell: Lecture Examples
• From first-order conditions:
• Rearrange to get cost-min inputs:
• By definition minimised cost is:
• So cost function is
Example
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Frank Cowell: Lecture Examples
Example
x2
x1
x*
• Lagrangean for utility maximisation
• Evaluate first-order conditions
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Frank Cowell: Lecture Examples
Example
x2
x1
x*
• Optimal demands are
• So at the optimum
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Frank Cowell: Lecture Examples 98 Oct 2015
Frank Cowell: Lecture Examples
• Results from cost minimisation:
• Differentiate to get compensated demand:
• Results from utility maximisation:
Example
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Frank Cowell: Lecture Examples
• 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:
Example
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Frank Cowell: Lecture Examples
• 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:
Example
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Frank Cowell: Lecture Examples 138 Oct 2015
Frank Cowell: Lecture Examples
• 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:
Example
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