1. Frank Cowell: Microeconomics Exercise 4.9 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

2. Frank Cowell: Microeconomics Ex 4.9(1) Question purpose: to analyse “short-run” constraints on the consumer purpose: to analyse “short-run” constraints on the consumer method: build model up step-by-step through the question parts. Start with simple Lagrangean maximisation method: build model up step-by-step through the question parts. Start with simple Lagrangean maximisation

3. Frank Cowell: Microeconomics Ex 4.9(1): Checking the U-function Given the utility function The indifference curves must look like this: They do not touch the axes… So it is clear that we cannot have a corner solution x2x2 x1x1

4. Frank Cowell: Microeconomics Ex 4.9(1): Setting up the problem From the question, the budget constraint is So the Lagrangean for the problem is We know that we must have an internal (tangency) solution So, differentiating, the first-order conditions are …plus the (binding) budget constraint

5. Frank Cowell: Microeconomics Ex 4.9(1): Ordinary demand functions From the FOCs we get Using this and the budget constraint we find = n/y. Using the value of in the FOCs we have   the ordinary demand functions for i=1,2,…,n… Take logs of the demand functions and differentiate to get the elasticities:

6. Frank Cowell: Microeconomics Ex 4.9(1): Solution functions The indirect utility function is just maximised utility expressed in terms of p and y    = V(p, y) = U(x * ) Evaluating this from x * we get: This gives a implicit relationship between  and y. Rearrange to get the cost (expenditure) function:

7. Frank Cowell: Microeconomics Ex 4.9(1): Compensated demand Take the cost function n[p 1 p 2 p 3 …p n e  ] 1/n Differentiate with respect to p 1 : This is the compensated demand function for good 1 Take logs and differentiate to get compensated elasticities:

8. Frank Cowell: Microeconomics Ex 4.9(2) Question purpose: introduce a single side-constraint purpose: introduce a single side-constraint method: show that modified model is closely related to original one. Reuse the original solution method: show that modified model is closely related to original one. Reuse the original solution

9. Frank Cowell: Microeconomics Ex 4.9(2): Modified problem x n is now fixed at A n   a contract with a high cancellation penalty? Define y' := y – p n A n Problem is equivalent to   max x 1 x 2 x 3 …x n  1 A n   subject to adjusted budget constraint: Apply results from part 1 to modified problem Ordinary demand is now: Compensated demand is:

10. Frank Cowell: Microeconomics Ex 4.9(2): Elasticities (ordinary ) Some results are just as before Own price: Cross-price (j<n) But something new for the nth (precommitted) good: This is just a pure income effect:   the person is precommitted to an amount A n   if the price goes up this reduces the income available to spend on other goods

11. Frank Cowell: Microeconomics Ex 4.9(2): Elasticities (compensated) Some results are essentially as before Own price: Cross-price (j<n) Note: the own-price effect is less elastic (closer to 0) Also for the nth (precommitted) good:

12. Frank Cowell: Microeconomics Ex 4.9(3) Question purpose: introduce many side-constraints purpose: introduce many side-constraints method: show that modified model is just a generalised version of that solved in part 2 method: show that modified model is just a generalised version of that solved in part 2

13. Frank Cowell: Microeconomics Ex 4.9(3): Further modified problem Given that for k = n – r,…,n we have x k fixed at A k The problem is equivalent to max x 1 x 2 x 3 …x m A´   where m := n – r – 1, A´ :=   subject to the adjusted budget constraint:   where Again apply results from previous parts Ordinary demand is now: Compensated demand is:

14. Frank Cowell: Microeconomics Ex 4.9(3): Elasticities (ordinary) Again, some results are just as before Own price: Cross-price (j < n − r) And now for all the precommitted goods: Interpretation of this income effect is just as in part 2

15. Frank Cowell: Microeconomics Ex 4.9(3): Elasticities (compensated) Results follow from part 2, replacing n  1 by m: Own price: Cross-price The smaller is m the less elastic is the own-price effect Also for all precommitted goods:

16. Frank Cowell: Microeconomics Ex 4.9: Points to remember The problem works just like the short-run for the firm The problem works just like the short-run for the firm The problem with one side-constraint follows just by replacing one variable by a constant The problem with one side-constraint follows just by replacing one variable by a constant The problem with many side constraints follows in a similar manner The problem with many side constraints follows in a similar manner Effect of adding more precommitment constraints: Effect of adding more precommitment constraints:  the smaller is the number m (i.e. the larger is r)…  …the less elastic is good 1 to its own price The result is similar to a rationing model The result is similar to a rationing model  but we cannot determine for which commodities the side-constraint is binding  this is arbitrarily given in the question