EC202: Worked Example #3.8 Frank Cowell April 2004 This presentation covers exactly the material set out in the file WorkedExamples.pdf, but with the addition of a few graphics and comments To start the presentation select Slideshow\View Show or click on icon below left. Mouse click or [Enter] to advance through slide show

WX3.8: Consumption and rationing Part 1: standard demand functions

WX3.8: Utility maximisation Maximise x 1 x 2 x 3 x 4 subject to The Lagrangean is Differentiating, the FOC is which implies Using the budget constraint we get = 4/M. So we have Ordinary demand function

WX3.8: Derive related functions Substitute optimal demands in utility function Indirect utility function Rearrange to get M as a function of u Cost function Differentiate: Compensated demand function

WX3.8: Ordinary elasticity Take the ordinary demand function Take logs and differentiate with respect to log p 1 and with respect to log p j

WX3.8: Compensated elasticity Take the compensated demand function Again take logs and differentiate with respect to log p 1 and with respect to log p j

WX3.8: Consumption and rationing Part 2: introduce a side constraint

WX3.8: Modify the problem x 4 is now fixed at A 4 Define M' := M – p 4 A 4 Problem is equivalent to maximising x 1 x 2 x 3 A 4 subject to budget with adjusted income M'. Use results from part 1 applied to 3-good economy

WX3.8: Ordinary elasticity again Take the ordinary demand function Take logs and differentiate with respect to log p 1 and with respect to log p j Take logs and differentiate with respect to log p 4 An income effect

WX3.8: Compensated elasticity again Take the compensated demand function Again take logs and differentiate Closer to 0 than before

WX3.8: Consumption and rationing Part 3: more constraints

WX3.8: More constraints Reapply the method with one fewer commodity: Differentiate again: Closer to 0 than before

WX3.8: Consumption and rationing Part 4: Interpretation  Model illustrates the comparative statics of someone who is subject to a quota ration.  But not rich enough to determine which commodities are consumed at a conventional equilibrium and which will be constrained by the ration.  Parts 2 and 3 show clearly how the compensated demand becomes “steeper” the more external constraints are imposed

WX3.8: Points to remember Modify the problem where appropriate to get a more tractable equivalent. Re-use the solution to one part of the problem to build the next.