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 8 Oct 2015
8 Oct 2015
Example Indifference curve (as before) does not touch either axis Constraint set for given u Cost minimisation must have interior solution x2 x1 8 Oct 2015
Example x* Lagrangian for cost minimisation For a minimum: Evaluate first-order conditions x2 x* x1 8 Oct 2015
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
Example From first-order conditions: Rearrange to get cost-min inputs: By definition minimised cost is: So cost function is 8 Oct 2015
Example x* Lagrangean for utility maximisation Evaluate first-order conditions x2 x* x1 8 Oct 2015
Example Optimal demands are So at the optimum x2 x* x1 8 Oct 2015
8 Oct 2015
Example Results from cost minimisation: Differentiate to get compensated demand: Results from utility maximisation: 8 Oct 2015
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
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 8 Oct 2015
8 Oct 2015
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