X - Essential Microeconomics

Essential Microeconomics
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2.2 BUDGET-CONSTRAINED CHOICE WITH TWO COMMODITIES
Continuity of demand
2
Income effects
6
Quasi-linear, Cobb-Douglas and CES preferences
9
Expenditure function
14
Substitution effects and Compensated demand
17
Elasticity of substitution
19
Determinants of demand elasticity
25
© John Riley
October 4, 2014
Essential Microeconomics
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Consider the choice problem of a consumer with income I facing a price vector p.
Max{U ( x) | p  x  I , x 
n
}
x
(2.2-1)
We assume that the local non-satiation assumption holds and that the utility function is continuous and
strictly quasi-concave on
2
.
The local non-satiation assumption ensures that the consumer spends all
his or her income and strict quasi-concavity ensures that there is a unique1 solution x0  x( p, I ) .
__________
1
Suppose x and x are both solutions. Then U ( x )  U ( x ) and because U is strictly quasi-concave it follows
0
1
0


1
1
0
that for any convex combination x , U ( x )  U ( x ) . Finally, because p  x  I and p  x  I , then
0
p  x  I . Thus x  is feasible and strictly preferred.
© John Riley
October 4, 2014
Essential Microeconomics
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Given uniqueness and continuity of preferences, it is intuitively plausible that the implied demand
function x( p, I ) must be continuous. This follows from the Theorem of the Maximum (See EM
Appendix C.)
Proposition 2.2-1: Theorem of the Maximum (I)
Consider the maximization problem
Max{ f ( x, ) | x  X ( ),   A} where X ( )  {x 
x
n
,
hi ( x, )  0, i  1,..., m} .
If f and X ( ) are continuous 1 and, for all  , there is a unique solution x ( ) , then x ( ) is continuous.
For the most general propositions about consumers (and firms as well) continuity is all that we need.
However, to simplify modeling, it is convenient to assume some degree of differentiability as well.
Then the necessary conditions for a maximum are restrictions on the gradient vector of the maximand
and constraint functions.
_______
1
The mapping from a parameter to a set is called a correspondence. For a formal definition of a continuous
correspondence, see EM Appendix C.
© John Riley
October 4, 2014
Essential Microeconomics
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Assumptions
Differentiability:
The utility function is continuously differentiable on
Strictly increasing preferences:
For all x 
n
,
2
.
U
( x) 0 .
x
U
 , j  1,..., n .
x j 0 x
j
Whenever we wish to avoid corner solutions we also assume that lim
Forming the Lagrangian for the maximization problem (2.2-1),
L  U   ( I  p  x) .
FOC
L U *

( x )   p j  0, j  1,..., n , with equality if x j  0
x j x j
Note that because the marginal utility of at least one commodity is strictly positive, the shadow price
(or marginal utility of income) must be strictly positive.
© John Riley
October 4, 2014
Essential Microeconomics
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Henceforth we will focus primarily on the simplest 2 commodity case. If both commodities are
consumed, the FOC can be rewritten as follows.
U U
x1 x2

.
p1
p2
(2.2-2)
One extra dollar spent on commodity 1 yields
spent on commodity 1 is
1
additional units thus the marginal value per dollar
p1
1 U
. Spending on commodity 1 is increased until this is equal to the
p1 x1
marginal value of spending an additional dollar on commodity 2. We can also rewrite the FOC as
follows:
U
x
p
MRS ( x* )  1  1 .
U p2
x2
That is, the slope of the indifference curve must equal
the slope of the budget line in Figure 2.2-1.
© John Riley
Figure 2.2-1: Budget constrained choice
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Essential Microeconomics
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Income Effects
Figure 2.2-2 shows the path of the consumer’s choice
x*  x( p, I ) as his income increases. As shown, this
“Income Expansion Path” is initially positively sloped
i.e.
x1
x
and 2 are both positive.
I
I
Figure 2.2-2: Income Expansion Path
In this case the commodities are said to be normal in the
neighborhood of the optimum.
However, as depicted, for higher incomes consumption of commodity 1 declines as income increases.
In this case commodity 1 is “inferior” in the neighborhood of the optimum.
To facilitate comparisons across commodities, it is helpful to consider the proportional effects on
demand as income changes, in other words the income elasticity of demand
 (x j , I ) 
© John Riley
I x j
.
x j I
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Essential Microeconomics
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In Figure 2.2-2 the slope of the Income Expansion path at x*  x( p, I ) is steeper than the line
joining x* and the origin, that is
dx2
dx1
IEP
x2
x2*

I

 .
x1 x1*
I
Rearranging this inequality,
 ( x2 , I ) 
I x2
I x1

  ( x1 , I ) .
*
*
x2 I
x1 I
Figure 2.2-2: Income Expansion Path
Appealing to the following lemma, the income elasticities
weighted by their expenditure shares must sum to 1. Thus, in Figure 2.2-2, the income elasticity of
demand for commodity 2 exceeds 1 and for commodity 1 is less than 1.
Lemma 2.2-2: Income Elasticities Weighted by Expenditure Shares Sum to 1
k1 ( x1* , I )  k2 ( x2* , I )  1
where k j 
© John Riley
p j x*j
I
is the expenditure share for commodity j.
October 4, 2014
Essential Microeconomics
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Proof: To establish this proposition, we substitute the consumer’s choice into the budget constraint and
differentiate by I.
x1*
x2*
p1
 p2
 1.
I
I
Rearranging the left-hand side,
p1 x1* I x1*
p2 x2* I x2*
(
) *
(
) *
 k1 ( x1* , I )  k2 ( x2* , I )  1 .
I x1 I
I x2 I
Q.E.D.
© John Riley
October 4, 2014
Essential Microeconomics
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We now examine income elasticities for three commonly used utility functions.
Example 1: Quasi-Linear Preferences
Definition: Preferences are quasi-linear if they can be represented by the utility function
U ( x)  v( x1 )   x2 .
For such a utility function the marginal rate of substitution (MRS) at x* is
dx2
dx1
U U ( x* )
U
x1 v( x1* )


.
U

x2
Income
Expansion Path
The MRS is independent of commodity 2 which means that
the indifference curves are vertically parallel.
As depicted in Figure 2.2-3, it follows that the income
Figure 2-2-3: Quasi-linear preferences
expansion path is first horizontal and then vertical.
Over the range in which both commodities are consumed, it follows that the income elasticity of
commodity 1 is zero. Given Lemma 2.2-2, the income elasticity of commodity 2 is the inverse of the
expenditure share.
© John Riley
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October 4, 2014
Essential Microeconomics
Example 2: Cobb-Douglas Preferences1
Differentiating by x1 ,
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U ( x)  x11 x22 , 1, 2  0
U
U
U  2U
 1 x11 1 x2 2  1 . Similarly,

.
x1
x1
x2
x2
At the maximum the FOC must be satisfied, hence
U U
x1 x2

.
p1
p2
Substituting and then dividing by U ,
1
p1 x1

2
p2 x2

hence p j x j 

U
,
j
.

____
1
Since u( x)  ln U ( x)  1 ln x1   2 ln x2 , U ( x) is strictly quasi-concave and so the FOC are sufficient.
© John Riley
October 4, 2014
Essential Microeconomics
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We then solve for  by substituting back into the budget constraint.
p1 x1  p2 x2 
  2
1   2
.
 I , hence   1
I

Demand for commodity j is therefore x j ( p, I ) 
j
I
.
p j 1   2
Finally, substituting back into the utility function, maximized utility is
U ( x( p, I ))  (
© John Riley
1
p1
)1 (
2
p2
) 2 (
I
)1  2 .
1   2
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October 4, 2014
Essential Microeconomics
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1 1
Example 3: CES Preferences U ( x)  (1 x1

1 1
  2 x2

)
1
1 1
, 1 , 2 ,  0,   1
From the definition of U,
1 1
U ( x)
1 1
 1x1
1 1
  2 x2  .
Differentiating by x j ,
1
(1   )U
1
 1
U  jU 
U
 1
1
.

 (1   ) j x j . Hence
1
x j
x j
x j
We follow the same steps as for Example 2.
Substituting into the FOC and then dividing by
1
1
p1 x1

2
1
p2 x2


1
.
1

U ,
(2.2-3)
U
 j p j1
j 
) and so p j x j 
Hence x j  (
.
pj

© John Riley
October 4, 2014
Essential Microeconomics
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We then solve for   by substituting back into the budget constraint
p1 x1  p2 x2 
1

(1 p11   2 p21 )  I , hence
1


1 p11
I
.
  2 p21
Demand for commodity j is therefore
 j p j1
I
x j ( p, I ) 
(
).
p j 1 p11   2 p21
(2.2-4)
Substituting x( p, I ) into the utility function and collecting terms,
U ( x( p, I )) 
I
1
 1
 1 1
(1 p1   2 p2 )
.
Note that the demand functions given by (2.2-4) reduce to the Cobb-Douglas
demand functions if   1.
█
1 1
Note: For   1 , maximizing U ( x) is equivalent to minimizing v( x)  1 x1
11
equivalent to maximizing the concave function u ( x)  1 x1
11
  2 x2
1 1
  2 x2
which, in turn, is
. Thus the necessary conditions are
sufficient.
© John Riley
October 4, 2014
Essential Microeconomics
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Dual optimization problem: expenditure minimization
Given the very weak assumption of local non-satiation,
for any budget constrained utility maximization problem
there is a “dual” optimization problem.
As we shall see, this dual problem is very useful in
understanding the determinants of demand.
Suppose that x* is a solution to a consumer’s maximization
problem. That is,
x*  arg Max{U ( x) | x  0, p  x  I } .
x
Such a consumption bundle is depicted in Figure 2.2-4.
© John Riley
Figure 2.2-4: Expenditure Minimization
October 4, 2014
Essential Microeconomics
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Consider any consumption bundle xˆ such that p  xˆ  I . If  is sufficiently small the neighborhood
N ( xˆ,  ) lies in the budget set. If the local non-satiation property holds, then there exists some xˆˆ in this
neighborhood that is strictly preferred to xˆ . Then xˆ cannot be optimal. Hence
p  x  I  U ( x)  U ( x* ) .
Equivalently,
U ( x)  U ( x* )  p  x  p  x* .
Thus, x* is expenditure minimizing, among all consumption bundles that are preferred to x* . We
summarize this result as follows.
Lemma 2.2-3: Duality Lemma
If the local non-satiation assumption holds and x*  arg Max{U ( x) | x  0, p  x  I } ,
x
then U ( x)  U ( x* )  p  x  p  x* and so x*  arg Min{ p  x | x  0, U ( x)  U ( x* )}.
x
© John Riley
October 4, 2014
Essential Microeconomics
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For any level of utility, U and price vector p we define the expenditure function M ( p,U ) to be the
minimum expenditure needed to achieve the utility level U .
Definition: Expenditure Function
M ( p,U )  Min{ p  x | U ( x)  U }
x
While it is not difficult to solve for the expenditure function,
it is often more convenient to solve first for the
indirect utility function
V ( p, I )  Max{U ( x) | p  x  I } .
x
Given local non-satiation, the consumer spends his entire
Figure 2.2-5: Maximized utility as a function of income
income.
Moreover the higher his income the greater is his utility.
Thus maximized utility V ( p, I ) is a strictly increasing function of income. This is depicted in Figure
2.2-5. The expenditure function is then the inverse mapping from U to I.
© John Riley
October 4, 2014
Essential Microeconomics
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Compensated demand
Let xc ( p,U ) be the solution to the dual problem, that is xc ( p,U ) solves
M ( p,U )  Min{ p  x | U ( x)  U }
x
This is known as the consumer’s compensated demand. Consider the effect on compensated demand of
an increase in the price of commodity 1. This is depicted in Figure 2.2-6 for price vectors p 0 and p1 .
As the price of commodity 1 rises, the consumer is compensated so that he is just able to maintain the
utility level U 0 .
Figure 2.2-6: Substitution effect
© John Riley
October 4, 2014
Essential Microeconomics
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The following useful property of the expenditure function is an immediate implication of the Envelope
Theorem. 1
M
 x c ( p ,U 0 )
p
Informally, if the price of commodity j rises and the consumer maintains his consumption plan, his
extra expenditure is x*j . This is the direct effect. The indirect effect associated with adjusting to the
change in the price is of second order.
____
1
See Exercise 2.2-2. Converting the expenditure minimization problem to a maximization problem,
so

L   p  x   (U ( x)  U )
M L

 x j .
p j p j
© John Riley
October 4, 2014
Essential Microeconomics
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Substitution Effect
The effect on demand of a compensated price change is called the substitution effect. As Figure 2.2-6
illustrates, the size of this effect depends critically on the curvature of the indifference curve. In the
x2c ( p,U 0 )
left diagram, as the price ratio changes, the consumption ratio c
changes a lot. That is, the
0
x1 ( p,U )
substitution effect is large. In the right diagram a price change has a small effect on the consumption
ratio so the substitution effect is small. As we shall see, the elasticity of the consumption ratio with
respect to the price ratio is a very useful measure of price sensitivity.
x2c p1
Definition: Elasticity of substitution    ( c , ) .
x1 p2
Example: CES utility function
From equation (2.2-3),
x2c  2 p1
  .
c
x1 1 p2
Taking the logarithm,
x2c
p
ln( c )   ln( 1 )  constant.
p2
x1
© John Riley
October 4, 2014
Essential Microeconomics
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d
As is readily confirmed,  ( y, x)  x ln y . Hence,
dx
x2c p1
( c , )  .
x1 p2
Thus for the CES utility function, the parameter  is the elasticity of substitution.
█
Quick review of elasticity
For any function y  f ( x) let y be the change in y when the change in x is x . A measure of the
sensitivity of y with respect to x is the percentage change in y divided by the percentage change in x.
100
y
x x y
/ 100

y
x
y x
An advantage of this measure is that it is independent of the units in which the two variable are
measured. To see this suppose we rescale units so that xˆ  x and yˆ  y . Then
xˆ yˆ x y

.
yˆ xˆ y x
© John Riley
October 4, 2014
Essential Microeconomics
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(Point) Elasticity
We define the point elasticity to the limit as x  0
y x dy
 ( y, x)  x lim

.
x0
y
x
y dx
Then from the above observation
Note that
 (  y ,  x )   ( y, x)
d
1 dy
ln y 
. Therefore the point elasticity can be rewritten as follows:
dx
y dx
 ( y, x)  x dy  x d
y dx
dx
ln y .
The following lemma is an immediate implication.
Lemma 2.2-4:
x2c p1
   ( c , )   ( x2c , p1 )   ( x1c , p1 )
x1 p2
We appeal to this lemma in proving the following result.
© John Riley
October 4, 2014
Essential Microeconomics
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Proposition 2.2-5: Elasticity of Substitution and Compensated Own Price Elasticity

( x2c , p1 )

k1
where k1 
p1 x1
px
and
 ( x1c , p1 )  (1  k1 )
Proof: To demonstrate equivalence, first note that around the indifference curve as p1 rises we have
U x1c U x2c

 0.
x1 p1 x2 p1
Also, from the first order condition, the marginal utility of each commodity is proportional to its price.
Hence
x1c
x2c
p1
 p2
 0.
p1
p1
(2.2-5)
Dividing by x1c and rearranging this equation,
p1 x1c
p2 x2c
p2 x2c p1 x2c
k2 p1 x2c
 c
 (
) c

, where k j  p j xcj / p  xc .
c
c
c
k1 x2 p1
x1 p1
x1 p1
p1 x1 x2 p1
© John Riley
October 4, 2014
Essential Microeconomics
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Therefore
 ( x1c , p1 )   k2  ( x2c , p1 ) .
k1
(2.2-6)
From Lemma 2.2-4,
   ( x2c , p1 )   ( x1c , p1 ) .
From (2.2-6),
 ( x1c , p1 )   k2  ( x2c , p1 ) .
k1
Substituting for the second term on the right hand side
   ( x2c , p1 ) 
k2
 ( x2c , p1 )  1  ( x2c , p1 ) .
k1
k1
Substituting this expression into (2.2-6),
 ( x1c , p1 )  k2  (1  k1 ) .
Q.E.D.
© John Riley
October 4, 2014
Essential Microeconomics
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Remark: Note that the compensated own price elasticity,
 ( x1c , p1 )  (1  k1 ) , is bounded from
below by the elasticity of substitution. Moreover, if the expenditure share is small, the elasticity of
substitution is a good approximation for the compensated own price elasticity.
Decomposition of price effects
To understand the impact of a price
C
change it proves helpful to decompose this
B
B
A
A
into two parts: a compensated price effect
and an income effect. Consider the figure.
The left hand diagram illustrates the effect
of an increase in the price of commodity 1.
O
O
Figure 2.2-7: Decomposition of the price effect into income and substitution effects
Suppose next that the individual is fully
compensated as the price rises. In the right hand diagram in Figure 2.2-7 the consumer moves along his
indifference curve from A to C substituting commodity 2 for commodity 1. This is the substitution
© John Riley
October 4, 2014
Essential Microeconomics
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effect of the price increase. In the second step, the extra compensation is taken away and the budget
line is pulled in towards the origin. The consumer then moves from C to B. Note that if, as depicted,
commodity 1 is a normal good, both the substitution and income effects are negative. That is, the
bigger (i.e. the more negative) the substitution effect and the bigger the income effect, the greater will
be the total effect on demand for commodity 1.
Slutsky Equation
We now consider the decomposition of the price effect in mathematical terms. If M ( p,U ) is
minimized total expenditure at utility level U , and x1 ( p, I ) is the consumer’s demand for commodity
1, the compensated demand is x1c  x1 ( p, M ( p,U )) . Differentiating by p1 , the slope of the
compensated demand curve is
x1c x1 x1 M


p1 p1 I p1
Yet we have seen that
© John Riley
M
 x1 . Substituting into the above expression and rearranging,
p1
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Essential Microeconomics
x j
p1

x cj
p1
total price
compensated
effect
price effect
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 x1
x j
I
.
Slutsky equation
income
effect
In particular, the Slutsky decomposition of the "own price effect" is as follows:
x1 x1c
x

 x1 1 .
p1 p1
I
(2.2-7)
Determinants of Demand Price Elasticity
Using the Slutsky equation, we can develop insights into the determinants of demand elasticity.
Converting (2.2-7) into elasticity form,
p1 x1 p1 x1c p1 x1 I x1
.


x1 p1 x1 p1
I x1 I
where x1c  x1c ( p,U ) is the compensated demand for commodity 1. Hence
 ( x1, p1 )   ( x1c , p1)  k1 ( x1, I ) .
From Proposition 2.2-5,
© John Riley
(2.2-8)
 ( x1c , p1 )  (1  k1 ) .
October 4, 2014
Essential Microeconomics
Substituting for
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 ( x1c , p1 ) in equation (2.2-8) we have the following proposition.
Proposition 2.2-6: Decomposition of Own Price Elasticity
 ( x1, p1 )  (1  k1)  k1 ( x1, I )
It follows that the absolute value of the own price elasticity must lie between the income elasticity and
the elasticity of substitution. Holding the expenditure share constant, the higher the elasticity of
substitution or the income elasticity, the more negative is the own price elasticity. Moreover, the higher
the expenditure share of commodity 1, the greater the weight on the income elasticity. Intuitively, a
higher share means that a price rise requires a bigger change in income for the individual to be
compensated. Thus the income effect on the change in demand is greater.
© John Riley
October 4, 2014

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