1. MICROECONOMICS Principles and Analysis Frank Cowell
Exercise 9.1
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2007

2. Ex 9.1(1): Question
purpose: Analyse consumption externality and efficiency
method: Solve for equilibrium prices and allocation using standard GE. Then examine source of inefficiency

3. Ex 9.1(1): incomes and demands
The term x1a is irrelevant to b-people's behaviour
they cannot do anything about it…
…although it affects their utility
Incomes are
ya = 300 p1
ya = 200 p2
Both types have Cobb-Douglas utility functions
so we could jump straight to demand functions…
…skip the Lagrangean step
We know that their demands will be given by
x1*a = ½ ya / p1 , x2*a = ½ ya / p2
x1*b = ½ yb / p1 , x2*b = ½ yb / p2
Skip
Lagrangean

4. Ex 9.1(1): Lagrangean method
Lagrangean for either type can be written
kx1h x2h + n[yh  p1x1h  p2 x2h ]
where n is a Lagrange multiplier
k is a constant (k =1 for type a , k =1/ x1a for type b)
FOC for an interior maximum
kx2h  np1 = 0
kx1h  np2 = 0
yh  p1x1h  p2 x2h = 0
Substitute from FOC1, FOC2 into FOC3 to find n
yh  p1[np2 /k]  p2[np1 /k] = 0
n = ½kyh /p1p2
Substitute this value of n back into FOC2, FOC1 to get the demands:
x1*h = ½ yh / p1
x2*h = ½ yh / p2

5. Ex 9.1(1): Equilibrium price ratio
Total demand for commodity 1 is
N [ x1*a + x1*b ] = N [ ½ ⋅300 + ½ ⋅ 200/r ]
where N is the large unknown number of traders
and r := p1 / p2
only the price ratio matters in the solution
There are 300N units of commodity 1
So the excess demand function for commodity 1 is
E1 = [ /r ] N  300 N
= [100/r  150] N
To find equilibrium sufficient to put E1 = 0
if E1 = 0 then E2 = 0 also
by Walras' Law
Clearly E1 = 0 exactly where r = ⅔
the equilibrium price ratio

6. Ex 9.1(1): Equilibrium allocation
Take the equilibrium price ratio r = ⅔
Then, using the demand functions we find
x1*a = ½ ⋅ 300 = 150
x2*a = ½ ⋅ 300r = 100
x1*b = ½ ⋅ 200 / r = 150
x2*b = ½ ⋅ 200 = 100
This is the equilibrium allocation

7. Ex 9.1(2): Question method:
Verify that CE allocation is inefficient by finding a perturbation that will produce a Pareto improvement

8. Ex 9.1(2): Source of inefficeincy
It is likely that the a-people are consuming “too much” of good 1
there is a negative externality
in the CE this is ignored
So try changing the allocation
so that the a-people consume less of good 1
Dx1a < 0
but where the a-people's utility remains unchanged
The means that their consumption of good 2 must increase
given that, in equilibrium, r = MRS,
required adjustment is Dx2a = −rDx1a >0

9. Ex 9.1(2): Pareto-improving adjustment
b-people's consumptions move in the opposite direction
(there is a fixed total amount of each good)
Dx1b = −Dx1a > 0
Dx2b = −Dx2a < 0
Effect on their utility can be computed thus:
Dlog Ub = Dx1b / x1b + Dx2b / x2b − Dx1a /x1a
= [ − 1/150 + ⅔(1/100) − 1/150] Dx1a
= − Dx1a / 150 >0
So it is possible to make a Pareto-improving perturbation
move away from the CE
in such a way that some people's utility is increased
no-one else's utility decreases

10. Ex 9.1(3): Question and answer
Can this be done by just tweaking prices?
increase relative price of commodity 1 for the a-people…
…relative to that facing the b-people?
This will not work
a-people’s income is also determined by p1 …
…and their resulting consumption of commodity 1 is independent of price
A rationing scheme may work

11. Ex 9.1: Points to remember
Be careful to model what is under each agent’s control
Use common-sense to spot Pareto improvements