Consumer
demand
The individuals maximize their utility (5.1) subject to the budget constraint
y{^) ~
/o
PU)^{J)^J
^^^ ^ti^ non-negativity constraints
c{j) >
0. The individual
demand function
c{p{j)^ 6)
is implicitly given by the first order conditions
v'{c{p{3),0))
=
\(0)p{j)
iip{j)<v'{0)/\{e)
(5.3)
c{p{3),e)
=
0
iip{j)>v'{Q)/\{9)
where A(^) denotes the Lagrangian multiplier (i.e. the marginal utility of income
of an agent with relative income
0).
The above first order conditions yield the
demand for each good
j
G [0,
A^]
by an agent with endowment level
6.
We note
that, in general, nonnegativity constraints may become binding because we have
assumed that
v'{{))
is finite. If '^'(0) <
X{9)p{j),
consumer
0
does not want to
consume good
j
because the price
p{j)
is too high.
As long as agent
6
is consuming good j , the individual reaction to a price
change can be found by differentiating equation (5.3) (we suppress the j-index)
dc/dp
= \/v"{c)
— ^v'[c)/v"{c).
The price elasticity of the individual demand
curve varies with across consumers with different endowment levels
'^ ^~
dp
c{0)
c{0)v^\c{0))'
Pricing decisions of firms.
The market for each good is monopolistic. There is a mass of
N
monopolists who
are unique suppliers for their respective product and who set prices to maximize
profits. The market demand curve that monopolistic producer for good
j
faces,
it given by horizontal aggregation of individual demand curves for this product.
Obviously, the market demand function
x{p{j))
depends on the price
p{j)
and on
the distribution of income,
F{6)
{p{j))=^
[
c{p{j),e)dF{0).
Jeivd)
HPU)
The slope of the market demand function can then be written as (for ease of
notation we will suppress the j'-index and those arguments of the c(-)- and
x{-)-

124
5. Markups and Exclusion
functions that are not directly relevant)^
dx
1
f~^ v'{c{0))
dp
pL^^.v'\c{e))'^^^^^'
where individuals with endowment 6* < ^ do not consume this good as in that
case
v'{0) < X{0)p.
Monopolist
j
chooses the price that maximizes profits taking the prices of all
other firms as given. Formally, monopolist
j
solves the problem
w
max
p{j) x{p{j))
x{p{j)).
P{j)
CL
The solution to this problem is given by the familiar Lerner index (again, for ease
of notation, we suppress the j-index)
p
6{p)
Je{^) x{p)
c{e,p)v'\c{e,p))
Formula (5.4) tells us that the monopoly price depends on a weighted average of
individual price elasticities of demand
[—v'{c)/ {cv"{c))]^
taking the consumption
shares
c{0)/x
as weights.
5.2.2
Restrictions on Preferences and Distribution
It is obvious that, without putting more structure both on preferences and dis-
tribution, not much can be said about the general equilibrium of this model. The
main problem is that we cannot explicitly solve for the monopoly price which
makes the model intractable. Hence we proceed by putting more structure on
both the subutility function
v{-)
as well as the distribution function
F{-).
We
make the following assumptions:
(i)
Inequality
only consists of two groups. There is a group of poor households,
indexed by P, of size /?, and a group of rich households, indexed by i?, of size
1
—
p.
The relative income of the poor household is denoted by
6p
whereas the

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- Fall '18
- Supply And Demand, Asymmetric Equilibria