1. Example indiff curve u = 1 indiff curve u = 2 indiff curve u = 3
From the equation
Equation of IC is x2
Check against the example in lecture 1
Transformed utility function x1
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3. Example Indifference curve (as before) does not touch either axis
Constraint set for given u
Cost minimisation must have interior solution x2 x1
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4. Example x* Lagrangian for cost minimisation For a minimum:
Evaluate first-order conditions x2 x* x1
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5. 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
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6. Example From first-order conditions: Rearrange to get cost-min inputs:
By definition minimised cost is:
So cost function is
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7. Example x* Lagrangean for utility maximisation
Evaluate first-order conditions x2 x* x1
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8. Example
Optimal demands are
So at the optimum x2 x* x1
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10. Example Results from cost minimisation:
Differentiate to get compensated demand:
Results from utility maximisation:
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11. 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:
Features of demand functions
Homogeneous of degree zero
Satisfy the “adding-up” constraint
Symmetric substitution effects
Negative own-price substitution effects
Income effects could be positive or negative:
in fact they are nearly always a pain
8 Oct 2015

12. Example in fact they are nearly always a pain
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:
Features of demand functions
Homogeneous of degree zero
Satisfy the “adding-up” constraint
Symmetric substitution effects
Negative own-price substitution effects
Income effects could be positive or negative:
in fact they are nearly always a pain
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14. 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:
8 Oct 2015