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Chapter 3
Demand Theory
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The Budget Constraint
 Attainable
consumption bundles
are bundles that the consumer can
afford to buy.
 Attainable consumption bundles
satisfy the following inequality known
as the budget constraint.
p1x1 + p2x2 ≤ M
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Figure 3.1 Attainable consumption bundles
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Opportunity Cost, Real Income and
Relative Prices

Rewriting the budget constraint by solving for X2
gives:
x2 = M/p2 – (p1/p2)x1
Where:
M/p2 is real income
P1/P2 is the relative price
The relative price shows that the
opportunity cost of good 1 is P1/P2 units of
good 2. P1/P2 is the absolute value of the slope of
the budget line.
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Endowments Rather Than Money
Sometimes an endowment of goods is
assumed rather than cash.
 Sally owns apples x10 and eggs x20.
 Her budget constraint is:

p1x1 + p2x2 ≤ p1x10 + p2x20
 Solving for x2:
x2 = (p1x10 + p2x20)/p2 – (p1/p2)/x1
As before, the budget constraint depends upon relative
prices and real income (the endowment).
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Figure 3.2 The budget line with endowments
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The Choice Problem
 The
non-satiation assumption implies
that utility maximizing consumption lies
on the budget line.
 The consumer choice problem is:
maximize U(x1, x2) by choice of x1 & x2,
subject to constraint p1x1 + p2x2 = M
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The Choice Problem
 In
principle we refer to the solution
(x1*, x2*) as endogenous variables, as
these variables are determined within the
model.
 The actual values of X1* and X2* depend
on the exogenous variables in the
model, (p1, p2 and M) and on the specific
form of the utility function.
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Figure 3.3 Non-satiation and the utilitymaximizing consumption bundle
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Demand Functions
X1* = D1(p1, p2, M)
X2* = D2(p1, p2, M)
 These
demand function equations
simply say that the choice of X1* and
X2* depend upon the prices of all
items in the consumption bundle and
the budget devoted to that bundle.
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Anna’s optimal choice when both
goods are perfect substitutes
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Graphic Analysis of Utility Maximization
 Assume
indifference curves are
smooth and strictly convex.
 Interior solutions are where
quantities of both goods are positive.
 Corner solution is one where the
quantity of one good is positive and
the quantity of the other is zero.
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Interior Solution

1.
2.
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An interior solution is described
by:
P1x1* + P2x2* Ξ M, the optimal
bundle lies on the budget line.
MRS(X1*, X2*) Ξ P1/P2 , the slope of
the indifference curve equals the
slope of the budget line at the
optimal bundle.
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Figure 3.5 The utility-maximizing
consumption bundle
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Figure 3.6 Essential goods
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Corner Solutions
A corner solution graphically lies not in
the interior between the two axis, but at a
corner where the budget line intersects
one of the two axes.
 For example, if at the point where the
budget line intersects the X2 axis, the
budget line is steeper than the
indifference curve, only good 2 will be
purchased.

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Figure 3.7 Inessential goods
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Excise Tax Versus Lump-Sum Tax
 Given
a choice between a lump sum
tax and an excise tax that raises the
same revenue, the consumer will
choose the lump sum tax (see Figure
3.8).
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Figure 3.8 Excise versus lump-sum taxes
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Figure 3.9 Cash transfer versus
in-kind transfers
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Figure 3.10 Optimal
consumption with endowments
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Figure 3.11 Normal and inferior goods
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Figure 3.12 Engel curves
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Figure 3.13 The consumption response to a
change in the price of another good
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Consumption Response to a
Change in Price
 The
price-consumption path
connects the utility maximizing
bundles that arise from a change in
the price of p1 or p2.
 Note that when p1 changes, M and p2
are assumed to be constant.
Likewise if p2 were to change, M and
p1 are assumed to be constant.
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Figure 3.14 The price-consumption
path and the demand function
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Elasticity
 Elasticity
is a measure of
responsiveness of the quantity
demanded for one good to a change
in one of the exogenous variables:
price or income.
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Figure 3.15 The need for a unit-free
measure of responsiveness
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Own-Price Elasticity

Own-price elasticity (Ell) relates to how
much the one good changes when its own
price changes.
 Ell=
% change in x1 / % change in P1
Ell  (x1 / x1) /( p1 / p1)
OR
E11  (x1 / p1)( p1 / x1)
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Elasticity
 If
we allow changes in the exogenous
variables to approach zero we obtain
marginal or point elasticity.
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Elasticity
Arc elasticity measures discrete changes
in x1 when there is a discrete change in
p1,p2 or M).
 By allowing changes in the exogenous
variables to approach zero gives marginal
or point elasticity.
 Price elasticity of demand for a good is
the elasticity of quantity consumed per
capita with respect to the price of the
good.

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Income Elasticity
 The
income elasticity of demand is
the elasticity of quantity consumed
per capita with respect to income per
capita.
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Income Elasticity Formula
E1m  (x1 / M )( M / x1)
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Cross Price Elasticity
 The
cross price elasticity of
demand for good 1 with respect to
the price of good 2, is the elasticity
of per capita consumption of good 1
with respect to p2.
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Cross Price Elasticity Formula
E 12  (x1 / p 2)( p 2 / x1)
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