Frank Cowell: Microeconomics
March 2007
Exercise 9.1
MICROECONOMICS
Principles and Analysis
Frank Cowell
Ex 9.1(1): Question
Frank Cowell: Microeconomics
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purpose: Analyse consumption externality and efficiency
method: Solve for equilibrium prices and allocation using standard
GE. Then examine source of inefficiency
Ex 9.1(1): incomes and demands
Frank Cowell: Microeconomics
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The term x1a is irrelevant to b-people's behaviour
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Incomes are
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so we could jump straight to demand functions…
…skip the Lagrangean step
We know that their demands will be given by
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Skip
Lagrangean
ya = 300 p1
ya = 200 p2
Both types have Cobb-Douglas utility functions
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they cannot do anything about it…
…although it affects their utility
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x1*a = ½ ya / p1 , x2*a = ½ ya / p2
x1*b = ½ yb / p1 , x2*b = ½ yb / p2
Ex 9.1(1): Lagrangean method
Frank Cowell: Microeconomics
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Lagrangean for either type can be written
 kx1h x2h + n[yh  p1x1h  p2 x2h ]
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FOC for an interior maximum
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kx2h  np1 = 0
kx1h  np2 = 0
yh  p1x1h  p2 x2h = 0
Substitute from FOC1, FOC2 into FOC3 to find n
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where n is a Lagrange multiplier
k is a constant (k =1 for type a , k =1/ x1a for type b)
yh  p1[np2 /k]  p2[np1 /k] = 0
n = ½kyh /p1p2
Substitute this value of n back into FOC2, FOC1 to get the demands:
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x1*h = ½ yh / p1
x2*h = ½ yh / p2
Ex 9.1(1): Equilibrium price ratio
Frank Cowell: Microeconomics
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Total demand for commodity 1 is
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There are 300N units of commodity 1
So the excess demand function for commodity 1 is
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E1 = [150 + 100/r ] N  300 N
= [100/r  150] N
To find equilibrium sufficient to put E1 = 0
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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
if E1 = 0 then E2 = 0 also
by Walras' Law
Clearly E1 = 0 exactly where r = ⅔
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the equilibrium price ratio
Ex 9.1(1): Equilibrium allocation
Frank Cowell: Microeconomics
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Take the equilibrium price ratio r = ⅔
Then, using the demand functions we find
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x1*a
x2*a
x1*b
x2*b
= ½ ⋅ 300
= ½ ⋅ 300r
= ½ ⋅ 200 / r
= ½ ⋅ 200
= 150
= 100
= 150
= 100
This is the equilibrium allocation
Ex 9.1(2): Question
Frank Cowell: Microeconomics
method:
 Verify that CE allocation is inefficient by finding a
perturbation that will produce a Pareto improvement
Ex 9.1(2): Source of inefficeincy
Frank Cowell: Microeconomics
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It is likely that the a-people are consuming “too much” of
good 1
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So try changing the allocation
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there is a negative externality
in the CE this is ignored
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
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given that, in equilibrium, r = MRS,
required adjustment is Dx2a = −rDx1a >0
Frank Cowell: Microeconomics
Ex 9.1(2): Pareto-improving
adjustment
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b-people's consumptions move in the opposite direction
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(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
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So it is possible to make a Pareto-improving perturbation
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move away from the CE
in such a way that some people's utility is increased
no-one else's utility decreases
Ex 9.1(3): Question and answer
Frank Cowell: Microeconomics
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Can this be done by just tweaking prices?
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This will not work
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increase relative price of commodity 1 for the a-people…
…relative to that facing the b-people?
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
Ex 9.1: Points to remember
Frank Cowell: Microeconomics
Be careful to model what is under each
agent’s control
 Use common-sense to spot Pareto
improvements
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