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

2. Ex 4.9(1) Question
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

3. 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

4. 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. Ex 4.9(1): Ordinary demand functions
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:

6. Ex 4.9(1): Solution functions
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:

7. Ex 4.9(1): Compensated demand
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:

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

9. 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…xn1An
subject to adjusted budget constraint:
Apply results from part 1 to modified problem
Ordinary demand is now:
Compensated demand is:

10. 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

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

13. 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:

14. 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. 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. Ex 4.9: Points to remember
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