Frank Cowell Microeconomics November 2006 Exercise 4 9

Frank Cowell: Microeconomics November 2006 Exercise 4. 9 MICROECONOMICS Principles and Analysis Frank Cowell

Ex 4. 9(1) Question Frank Cowell: Microeconomics n n 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

Ex 4. 9(1): Checking the U-function Frank Cowell: Microeconomics n Given the utility function n The indifference curves must look like this: x 2 x 1 n n They do not touch the axes… So it is clear that we cannot have a corner solution

Ex 4. 9(1): Setting up the problem Frank Cowell: Microeconomics n n From the question, the budget constraint is So the Lagrangean for the problem is n We know that we must have an internal (tangency) solution So, differentiating, the first-order conditions are n …plus the (binding) budget constraint n

Ex 4. 9(1): Ordinary demand functions Frank Cowell: Microeconomics n From the FOCs we get n Using this and the budget constraint we find l = n/y. Using the value of l in the FOCs we have n u n the ordinary demand functions for i=1, 2, …, n… Take logs of the demand functions and differentiate to get the elasticities:

Ex 4. 9(1): Solution functions Frank Cowell: Microeconomics n n The indirect utility function is just maximised utility expressed in terms of p and y u u = V(p, y) = U(x*) Evaluating this from x* we get: This gives a implicit relationship between u and y. Rearrange to get the cost (expenditure) function:

Ex 4. 9(1): Compensated demand Frank Cowell: Microeconomics n n Take the cost function n[p 1 p 2 p 3…pneu]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:

Ex 4. 9(2) Question Frank Cowell: Microeconomics n n purpose: introduce a single side-constraint method: show that modified model is closely related to original one. Reuse the original solution

Ex 4. 9(2): Modified problem Frank Cowell: Microeconomics n xn is now fixed at An u n n a contract with a high cancellation penalty? Define y' : = y – pn. An Problem is equivalent to u u max x 1 x 2 x 3…xn 1 An subject to adjusted budget constraint: n Apply results from part 1 to modified problem n Ordinary demand is now: n Compensated demand is:

Ex 4. 9(2): Elasticities (ordinary ) Frank Cowell: Microeconomics n Some results are just as before n Own price: n Cross-price (j<n) n But something new for the nth (precommitted) good: n This is just a pure income effect: u u the person is precommitted to an amount An if the price goes up this reduces the income available to spend on other goods

Ex 4. 9(2): Elasticities (compensated) Frank Cowell: Microeconomics n Some results are essentially as before n Own price: n Cross-price (j<n) n Note: the own-price effect is less elastic (closer to 0) Also for the nth (precommitted) good: n

Ex 4. 9(3) Question Frank Cowell: Microeconomics n n purpose: introduce many side-constraints method: show that modified model is just a generalised version of that solved in part 2

Ex 4. 9(3): Further modified problem Frank Cowell: Microeconomics n n Given that for k = n – r, …, n we have xk fixed at Ak The problem is equivalent to max x 1 x 2 x 3…xm. A´ u u u where m : = n – r – 1, A´ : = subject to the adjusted budget constraint: where n Again apply results from previous parts Ordinary demand is now: n Compensated demand is: n

Ex 4. 9(3): Elasticities (ordinary) Frank Cowell: Microeconomics n Again, some results are just as before n Own price: n Cross-price (j < n − r) n And now for all the precommitted goods: n Interpretation of this income effect is just as in part 2

Ex 4. 9(3): Elasticities (compensated) Frank Cowell: Microeconomics n Results follow from part 2, replacing n 1 by m: n Own price: n Cross-price n The smaller is m the less elastic is the own-price effect Also for all precommitted goods: n

Ex 4. 9: Points to remember Frank Cowell: Microeconomics n n 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 many side constraints follows in a similar manner Effect of adding more precommitment constraints: u u n 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 u u but we cannot determine for which commodities the side-constraint is binding this is arbitrarily given in the question