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icroeconomics

Microeconom

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Exercise 4.9

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank CowellFrank Cowell

November 2006 November 2006

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Ex 4.9(1) Question

purposepurpose: to analyse “short-run” constraints on the consumer: to analyse “short-run” constraints on the consumer methodmethod: build model up step-by-step through the question : build model up step-by-step through the question

parts. Start with simple Lagrangean maximisationparts. Start with simple Lagrangean maximisation

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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

x2

x1

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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

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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:

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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:

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Ex 4.9(1): Compensated demand

Take the cost function n[p1p2p3…pne]1/n

Differentiate with respect to p1:

This is the compensated demand function for good 1 Take logs and differentiate to get compensated

elasticities:

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Ex 4.9(2) Question

purposepurpose: introduce a single side-constraint: introduce a single side-constraint methodmethod: show that modified model is closely related to : show that modified model is closely related to

original one. Reuse the original solutionoriginal one. Reuse the original solution

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Ex 4.9(2): Modified problem 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:

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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 An

if the price goes up this reduces the income available to spend on other goods

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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:

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Ex 4.9(3) Question

purposepurpose: introduce many side-constraints: introduce many side-constraints methodmethod: show that modified model is just a : show that modified model is just a

generalised version of that solved in part 2generalised version of that solved in part 2

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Ex 4.9(3): Further modified problem 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:

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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

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Ex 4.9(3): Elasticities (compensated)

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:

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Ex 4.9: Points to remember

The problem works just like the short-run for the firmThe problem works just like the short-run for the firm The problem with one side-constraint follows just by The problem with one side-constraint follows just by

replacing one variable by a constantreplacing one variable by a constant The problem with many side constraints follows in a similar The problem with many side constraints follows in a similar

mannermanner Effect of adding more precommitment constraints:Effect of adding more precommitment constraints:

the smaller is the number the smaller is the number m m (i.e. the larger is (i.e. the larger is rr)… )… ……the less elastic is good 1 to its own pricethe less elastic is good 1 to its own price

The result is similar to a rationing modelThe result is similar to a rationing model but we cannot determine for which commodities the side-constraint but we cannot determine for which commodities the side-constraint

is bindingis binding this is arbitrarily given in the questionthis is arbitrarily given in the question