Transcript Ex 4.9
Frank Cowell: Microeconomics November 2006 Exercise 4.9 MICROECONOMICS Principles and Analysis Frank Cowell Ex 4.9(1) Question Frank Cowell: Microeconomics 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 Given the utility function The indifference curves must look like this: x2 x1 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 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 Ex 4.9(1): Ordinary demand functions Frank Cowell: Microeconomics From the FOCs we get Using this and the budget constraint we find l = n/y. Using the value of l 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: Ex 4.9(1): Solution functions Frank Cowell: Microeconomics The indirect utility function is just maximised utility expressed in terms of p and y 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 Take the cost function n[p1p2p3…pneu]1/n Differentiate with respect to p1: 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 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 xn is now fixed at An a contract with a high cancellation penalty? Define y' := y – pnAn Problem is equivalent to max x1x2x3…xn1An subject to adjusted budget constraint: Apply results from part 1 to modified problem Ordinary demand is now: Compensated demand is: Ex 4.9(2): Elasticities (ordinary ) Frank Cowell: Microeconomics 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 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 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: Ex 4.9(3) Question Frank Cowell: Microeconomics 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 Given that for k = n – r,…,n we have xk fixed at Ak The problem is equivalent to max x1x2x3…xmA´ 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: Ex 4.9(3): Elasticities (ordinary) Frank Cowell: Microeconomics 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 Ex 4.9(3): Elasticities (compensated) Frank Cowell: Microeconomics Results follow from part 2, replacing n1 by m: Own price: Cross-price The smaller is m the less elastic is the own-price effect Also for all precommitted goods: Ex 4.9: Points to remember Frank Cowell: Microeconomics 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: 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 but we cannot determine for which commodities the side-constraint is binding this is arbitrarily given in the question