Chapter 2: Demand Theory - faculty.ses.wsu.edufaculty.ses.wsu.edu/Espinola/Demand...
Transcript of Chapter 2: Demand Theory - faculty.ses.wsu.edufaculty.ses.wsu.edu/Espinola/Demand...
Utility Maximization Problem
โข Consumer maximizes his utility level by selecting a bundle ๐ฅ (where ๐ฅ can be a vector) subject to his budget constraint:
max๐ฅโฅ0
๐ข(๐ฅ)
s. t. ๐ โ ๐ฅ โค ๐ค
โข Weierstrass Theorem: for optimization problems defined on the reals, if the objective function is continuous and constraints define a closed and bounded set, then the solution to such optimization problem exists.
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Utility Maximization Problem
โข Existence: if ๐ โซ 0 and ๐ค > 0 (i.e., if ๐ต๐,๐ค is closed and bounded), and if ๐ข(โ) is continuous, then there exists at least one solution to the UMP.โ If, in addition, preferences are strictly convex, then the
solution to the UMP is unique.
โข We denote the solution of the UMP as the argmax of the UMP (the argument, ๐ฅ, that solves the optimization problem), and we denote it as ๐ฅ(๐, ๐ค). โ ๐ฅ(๐, ๐ค) is the Walrasian demand correspondence, which
specifies a demand of every good in โ+๐ฟ for every possible
price vector, ๐, and every possible wealth level, ๐ค.
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Utility Maximization Problem
โข Walrasian demand ๐ฅ(๐, ๐ค)at bundle A is optimal, as the consumer reaches a utility level of ๐ข2 by exhausting all his wealth.
โข Bundles B and C are not optimal, despite exhausting the consumerโs wealth. They yield a lower utility level ๐ข1, where ๐ข1 < ๐ข2.
โข Bundle D is unaffordable and, hence, it cannot be the argmax of the UMP given a wealth level of ๐ค.
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Properties of Walrasian Demand
โข If the utility function is continuous and preferences satisfy LNS over the consumption set ๐ = โ+
๐ฟ , then the Walrasian demand ๐ฅ(๐, ๐ค) satisfies:
1) Homogeneity of degree zero:
๐ฅ ๐, ๐ค = ๐ฅ(๐ผ๐, ๐ผ๐ค) for all ๐, ๐ค, and for all ๐ผ > 0
That is, the budget set is unchanged!
๐ฅ โ โ+๐ฟ : ๐ โ ๐ฅ โค ๐ค = ๐ฅ โ โ+
๐ฟ : ๐ผ๐ โ ๐ฅ โค ๐ผ๐ค
Note that we donโt need any assumption on the preference relation to show this. We only rely on the budget set being affected.
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x1
x2
wp2
ฮฑ wฮฑ p2
=
wp1
ฮฑ wฮฑ p1
=
x (p,w), unaffected
Properties of Walrasian Demand
Advanced Microeconomic Theory 6
โ Note that the preference relation can be linear, and homog(0) would still hold.
2) Walrasโ Law:
๐ โ ๐ฅ โค ๐ค for all ๐ฅ = ๐ฅ(๐, ๐ค)
It follows from LNS: if the consumer selects a Walrasian demand ๐ฅ โ ๐ฅ(๐, ๐ค), where ๐ โ ๐ฅ < ๐ค, then it means we can still find other bundle ๐ฆ, which is ฮตโclose to ๐ฅ, where consumer can improve his utility level.
If the bundle the consumer chooses lies on the budget line, i.e., ๐ โ ๐ฅโฒ = ๐ค, we could then identify bundles that are strictly preferred to ๐ฅโฒ, but these bundles would be unaffordable to the consumer.
Properties of Walrasian Demand
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x
x2
x1
Properties of Walrasian Demand
8
โ For ๐ฅ โ ๐ฅ(๐, ๐ค), there is a bundle ๐ฆ, ฮตโclose to ๐ฅ, such that ๐ฆ โป ๐ฅ. Then, ๐ฅ โ ๐ฅ(๐, ๐ค).
Properties of Walrasian Demand
3) Convexity/Uniqueness:
a) If the preferences are convex, then the Walrasian demand correspondence ๐ฅ(๐, ๐ค)defines a convex set, i.e., a continuum of bundles are utility maximizing.
b) If the preferences are strictly convex, then the Walrasian demand correspondence ๐ฅ(๐, ๐ค)contains a single element.
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Properties of Walrasian Demand
Convex preferences Strictly convex preferences
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( , )x x p w
Unique ( , )x p w
UMP: Necessary Condition
max๐ฅโฅ0
๐ข ๐ฅ s. t. ๐ โ ๐ฅ โค ๐ค
โข We solve it using Kuhn-Tucker conditions over the Lagrangian ๐ฟ = ๐ข ๐ฅ + ๐(๐ค โ ๐ โ ๐ฅ),
๐๐ฟ
๐๐ฅ๐=
๐๐ข(๐ฅโ)
๐๐ฅ๐โ ๐๐๐ โค 0 for all ๐, = 0 if ๐ฅ๐
โ > 0
๐๐ฟ
๐๐= ๐ค โ ๐ โ ๐ฅโ = 0
โข That is in the interior optimum, ๐๐ข(๐ฅโ)
๐๐ฅ๐= ๐๐๐ for every
good ๐, which implies๐๐ข(๐ฅโ)
๐๐ฅ๐๐๐ข(๐ฅโ)
๐๐ฅ๐
=๐๐
๐๐โ ๐๐ ๐ ๐,๐ =
๐๐
๐๐โ
๐๐ข(๐ฅโ)
๐๐ฅ๐
๐๐=
๐๐ข(๐ฅโ)
๐๐ฅ๐
๐๐
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UMP: Sufficient Condition
โข When are Kuhn-Tucker (necessary) conditions, also sufficient?
โ That is, when can we guarantee that ๐ฅ(๐, ๐ค) is the max of the UMP and not the min?
โข Answer: if ๐ข(โ) is quasiconcave and monotone, and has Hessian ๐ป๐ข(๐ฅ) โ 0 for ๐ฅ โ โ+
๐ฟ , then the Kuhn-Tucker FOCs are also sufficient.
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UMP: Sufficient Condition
โข Kuhn-Tucker conditions are sufficient for a max if:
1) ๐ข(๐ฅ) is quasiconcave, i.e., convex upper contour set (UCS).
2) ๐ข(๐ฅ) is monotone.3) ๐ป๐ข(๐ฅ) โ 0 for ๐ฅ โ โ+
๐ฟ .โ If ๐ป๐ข ๐ฅ = 0 for some ๐ฅ,
then we would be at the โtop of the mountainโ (i.e., blissing point), which violates both LNS and monotonicity.
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x1
x2
wp2
wp1
x * โ x (p,w)
slope = - p2
p1
Ind. Curve
UMP: Violations of Sufficient Condition
1) ๐(โ) is non-monotone:
โ The consumer chooses bundle A (at a corner) since it yields the highest utility level given his budget constraint.
โ At point A, however, the tangency condition
๐๐ ๐ 1,2 =๐1
๐2does not
hold.
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UMP: Violations of Sufficient Condition
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x2
x1
A
B
C
Budget line
โ The upper contour sets (UCS) are not convex.
โ ๐๐ ๐ 1,2 =๐1
๐2is not a
sufficient condition for a max.
โ A point of tangency (C) gives a lower utility level than a point of non-tangency (B).
โ True maximum is at point A.
2) ๐(โ) is not quasiconcave:
UMP: Corner Solution
โข Analyzing differential changes in ๐ฅ๐ and ๐ฅ๐, that keep individualโs utility unchanged, ๐๐ข = 0,
๐๐ข(๐ฅ)
๐๐ฅ๐๐๐ฅ๐ +
๐๐ข(๐ฅ)
๐๐ฅ๐๐๐ฅ๐ = 0 (total diff.)
โข Rearranging,
๐๐ฅ๐
๐๐ฅ๐= โ
๐๐ข ๐ฅ๐๐ฅ๐
๐๐ข ๐ฅ๐๐ฅ๐
= โ๐๐ ๐๐,๐
โข Corner Solution: ๐๐ ๐๐,๐ >๐๐
๐๐, or alternatively,
๐๐ข ๐ฅโ
๐๐ฅ๐
๐๐>
๐๐ข ๐ฅโ
๐๐ฅ๐
๐๐, i.e.,
the consumer prefers to consume more of good ๐.
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UMP: Corner Solution
โข In the FOCs, this implies:
a)๐๐ข(๐ฅโ)
๐๐ฅ๐โค ๐๐๐ for the goods whose consumption is
zero, ๐ฅ๐โ= 0, and
b)๐๐ข(๐ฅโ)
๐๐ฅ๐= ๐๐๐ for the good whose consumption is
positive, ๐ฅ๐โ> 0.
โข Intuition: the marginal utility per dollar spent on good ๐ is still larger than that on good ๐.
๐๐ข(๐ฅโ)
๐๐ฅ๐
๐๐= ๐ โฅ
๐๐ข(๐ฅโ)
๐๐ฅ๐
๐๐
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UMP: Corner Solution
โข Consumer seeks to consume good 1 alone.
โข At the corner solution, the indifference curve is steeper than the budget line, i.e.,
๐๐ ๐1,2 >๐1
๐2or
๐๐1
๐1>
๐๐2
๐2
โข Intuitively, the consumer would like to consume more of good 1, even after spending his entire wealth on good 1 alone.
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x1
x2
wp2
wp1
slope of the B.L. = - p2
p1
slope of the I.C. = MRS1,2
UMP: Lagrange Multiplier
โข ๐ is referred to as the โmarginal values of relaxing the constraintโ in the UMP (a.k.a.โshadow price of wealthโ).
โข If we provide more wealth to the consumer, he is capable of reaching a higher indifference curve and, as a consequence, obtaining a higher utility level.
โ We want to measure the change in utility resulting from a marginal increase in wealth.
Advanced Microeconomic Theory 19
x1
x2
w 0
p2
{x +: u(x) = u 1}2
{x +: u(x) = u 0}2
w 1
p2
w 0
p1
w 1
p1
UMP: Lagrange Multiplier
โข Analyze the change in utility from change in wealth using chain rule (since ๐ข ๐ฅ does not contain wealth),
๐ป๐ข ๐ฅ(๐, ๐ค) โ ๐ท๐ค๐ฅ(๐, ๐ค)
โข Substituting ๐ป๐ข ๐ฅ(๐, ๐ค) = ๐๐ (in interior solutions),
๐๐ โ ๐ท๐ค๐ฅ(๐, ๐ค)
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UMP: Lagrange Multiplier
โข From Walrasโ Law, ๐ โ ๐ฅ ๐, ๐ค = ๐ค, the change in expenditure from an increase in wealth is given by
๐ท๐ค ๐ โ ๐ฅ ๐, ๐ค = ๐ โ ๐ท๐ค๐ฅ ๐, ๐ค = 1
โข Hence, ๐ป๐ข ๐ฅ(๐, ๐ค) โ ๐ท๐ค๐ฅ ๐, ๐ค = ๐ ๐ โ ๐ท๐ค๐ฅ ๐, ๐ค
1
= ๐
โข Intuition: If ๐ = 5, then a $1 increase in wealth implies an increase in 5 units of utility.
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Walrasian Demand
โข We found the Walrasian demand function, ๐ฅ ๐, ๐ค , as the solution to the UMP.
โข Properties of ๐ฅ ๐, ๐ค :1) Homog(0), for any ๐, ๐ค, ๐ผ > 0,
๐ฅ ๐ผ๐, ๐ผ๐ค = ๐ฅ ๐, ๐ค(Budget set is unaffected by changes in ๐ and ๐ค of the same size)
2) Walrasโ Law: for every ๐ โซ 0, ๐ค > 0 we have ๐ โ ๐ฅ = ๐ค for all ๐ฅ โ ๐ฅ(๐, ๐ค)
(This is because of LNS)
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Walrasian Demand: Wealth Effects
โข Normal vs. Inferior goods
๐๐ฅ ๐,๐ค
๐๐ค
><
0 normalinferior
โข Examples of inferior goods:
โ Two-buck chuck (a really cheap wine)
โ Walmart during the economic crisis
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Walrasian Demand: Wealth Effects
โข An increase in the wealth level produces an outward shift in the budget line.
โข ๐ฅ2 is normal as ๐๐ฅ2 ๐,๐ค
๐๐ค> 0, while
๐ฅ1 is inferior as ๐๐ฅ ๐,๐ค
๐๐ค< 0.
โข Wealth expansion path: โ connects the optimal consumption
bundle for different levels of wealth
โ indicates how the consumption of a good changes as a consequence of changes in the wealth level
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Walrasian Demand: Wealth Effects
โข Engel curve depicts the consumption of a particular good in the horizontal axis and wealth on the vertical axis.
โข The slope of the Engel curve is:
โ positive if the good is normal
โ negative if the good is inferior
โข Engel curve can be positively slopped for low wealth levels and become negatively slopped afterwards.
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x1
Wealth, w
ลต
Engel curve for a normal good (up to income ลต), but inferior
good for all w > ลต
Engel curve of a normal good, for all w.
Walrasian Demand: Price Effects
โข Own price effect:๐๐ฅ๐ ๐, ๐ค
๐๐๐
<>
0UsualGiffen
โข Cross-price effect:๐๐ฅ๐ ๐, ๐ค
๐๐๐
><
0Substitutes
Complements
โ Examples of Substitutes: two brands of mineral water, such as Aquafine vs. Poland Springs.
โ Examples of Complements: cars and gasoline.
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Indirect Utility Function
โข The Walrasian demand function, ๐ฅ ๐, ๐ค , is the solution to the UMP (i.e., argmax).
โข What would be the utility function evaluated at the solution of the UMP, i.e., ๐ฅ ๐, ๐ค ?
โ This is the indirect utility function (i.e., the highest utility level), ๐ฃ ๐, ๐ค โ โ, associated with the UMP.
โ It is the โvalue functionโ of this optimization problem.
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Properties of Indirect Utility Function
โข If the utility function is continuous and preferences satisfy LNS over the consumption set ๐ = โ+
๐ฟ , then the indirect utility function ๐ฃ ๐, ๐ค satisfies:
1) Homogenous of degree zero: Increasing ๐ and ๐คby a common factor ๐ผ > 0 does not modify the consumerโs optimal consumption bundle,๐ฅ(๐, ๐ค), nor his maximal utility level, measured by ๐ฃ ๐, ๐ค .
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Properties of Indirect Utility Function
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x1
x2
wp2
ฮฑ wฮฑ p2
=
wp1
ฮฑ wฮฑ p1
=
x (p,w) = x (ฮฑp,ฮฑ w)
v (p,w) = v (ฮฑp,ฮฑ w)
Bp,w = Bฮฑp,ฮฑw
Properties of Indirect Utility Function
2) Strictly increasing in ๐ค:
๐ฃ ๐, ๐คโฒ > ๐ฃ(๐, ๐ค) for ๐คโฒ > ๐ค.
3) non-increasing (i.e., weakly decreasing) in ๐๐
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Properties of Indirect Utility Function
33
W
P1
w2
w1
4) Quasiconvex: The set ๐, ๐ค : ๐ฃ(๐, ๐ค) โค าง๐ฃ is convex for any าง๐ฃ.
- Interpretation I: If (๐1, ๐ค1) โฟโ (๐2, ๐ค2), then (๐1, ๐ค1) โฟโ (๐๐1 +(1 โ ๐)๐2, ๐๐ค1 + (1 โ ๐)๐ค2); i.e., if ๐ด โฟโ ๐ต, then ๐ด โฟโ ๐ถ.
Properties of Indirect Utility Function
- Interpretation II: ๐ฃ(๐, ๐ค) is quasiconvex if the set of (๐, ๐ค) pairs for which ๐ฃ ๐, ๐ค < ๐ฃ(๐โ, ๐คโ) is convex.
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W
P1
w*
p*
v(p,w) < v(p*,w
*)
v(p,w) = v1
Properties of Indirect Utility Function
- Interpretation III: Using ๐ฅ1 and ๐ฅ2 in the axis, perform following steps:
1) When ๐ต๐,๐ค, then ๐ฅ ๐, ๐ค
2) When ๐ต๐โฒ,๐คโฒ, then ๐ฅ ๐โฒ, ๐คโฒ
3) Both ๐ฅ ๐, ๐ค and ๐ฅ ๐โฒ, ๐คโฒ induce an indirect utility of ๐ฃ ๐, ๐ค = ๐ฃ ๐โฒ, ๐คโฒ = เดค๐ข
4) Construct a linear combination of prices and wealth:
เต ๐โฒโฒ = ๐ผ๐ + (1 โ ๐ผ)๐โฒ
๐คโฒโฒ = ๐ผ๐ค + (1 โ ๐ผ)๐คโฒ ๐ต๐โฒโฒ,๐คโฒโฒ
5) Any solution to the UMP given ๐ต๐โฒโฒ,๐คโฒโฒ must lie on a lower indifference curve (i.e., lower utility)
๐ฃ ๐โฒโฒ, ๐คโฒโฒ โค เดค๐ข
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Properties of Indirect Utility Function
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x1
x2
Bp,wBpโ,wโBpโโ,wโโ
x (p,w)
x (pโ,wโ )
Ind. Curve {x โโ2:u(x) = ลซ}
WARP and Walrasian Demand
โข Relation between Walrasian demand ๐ฅ ๐, ๐คand WARP
โ How does the WARP restrict the set of optimal consumption bundles that the individual decision-maker can select when solving the UMP?
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WARP and Walrasian Demand
โข Take two different consumption bundles ๐ฅ ๐, ๐ค and ๐ฅ ๐โฒ, ๐คโฒ , both being affordable ๐, ๐ค , i.e.,
๐ โ ๐ฅ ๐, ๐ค โค ๐ค and ๐ โ ๐ฅ ๐โฒ, ๐คโฒ โค ๐ค
โข When prices and wealth are ๐, ๐ค , the consumer chooses ๐ฅ ๐, ๐ค despite ๐ฅ ๐โฒ, ๐คโฒ is also affordable.
โข Then he โrevealsโ a preference for ๐ฅ ๐, ๐ค over ๐ฅ ๐โฒ, ๐คโฒwhen both are affordable.
โข Hence, we should expect him to choose ๐ฅ ๐, ๐ค over ๐ฅ ๐โฒ, ๐คโฒ when both are affordable. (Consistency)
โข Therefore, bundle ๐ฅ ๐, ๐ค must not be affordable at ๐โฒ, ๐คโฒ because the consumer chooses ๐ฅ ๐โฒ, ๐คโฒ . That is,
๐โฒ โ ๐ฅ ๐, ๐ค > ๐คโฒ.
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WARP and Walrasian Demand
โข In summary, Walrasian demand satisfies WARP, if, for two different consumption bundles, ๐ฅ ๐, ๐ค โ ๐ฅ ๐โฒ, ๐คโฒ ,
๐ โ ๐ฅ ๐โฒ, ๐คโฒ โค ๐ค โ ๐โฒ โ ๐ฅ ๐, ๐ค > ๐คโฒ
โข Intuition: if ๐ฅ ๐โฒ, ๐คโฒ is affordable under budget set ๐ต๐,๐ค, then ๐ฅ ๐, ๐ค cannot be affordable
under ๐ต๐โฒ,๐คโฒ.
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Checking for WARP
โข A systematic procedure to check if Walrasian demand satisfies WARP:โ Step 1: Check if bundles ๐ฅ ๐, ๐ค and ๐ฅ ๐โฒ, ๐คโฒ are
both affordable under ๐ต๐,๐ค. That is, graphically ๐ฅ ๐, ๐ค and ๐ฅ ๐โฒ, ๐คโฒ have to lie on
or below budget line ๐ต๐,๐ค.
If step 1 is satisfied, then move to step 2.
Otherwise, the premise of WARP does not hold, which does not allow us to continue checking if WARP is violated or not. In this case, we can only say that โWARP is not violatedโ.
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Checking for WARP
- Step 2: Check if bundles ๐ฅ ๐, ๐ค is affordable under ๐ต๐โฒ,๐คโฒ.
That is, graphically ๐ฅ ๐, ๐ค must lie on or below budge line ๐ต๐โฒ,๐คโฒ.
If step 2 is satisfied, then this Walrasian demand violates WARP.
Otherwise, the Walrasian demand satisfies WARP.
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Checking for WARP: Example 1
โข First, ๐ฅ ๐, ๐ค and ๐ฅ ๐โฒ, ๐คโฒ are both affordable under ๐ต๐,๐ค.
โข Second, ๐ฅ ๐, ๐ค is not affordable under ๐ต๐โฒ,๐คโฒ.
โข Hence, WARP is satisfied!
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x1
x2
w p2
wp1
Bp,w
Bp ,w
x (p,w)
x (p ,w )
Checking for WARP: Example 2
โข The demand ๐ฅ ๐โฒ, ๐คโฒunder final prices and wealth is not affordable under initial prices and wealth, i.e., ๐ โ ๐ฅ ๐โฒ, ๐คโฒ > ๐ค.โ The premise of WARP
does not hold.
โ Violation of Step 1!
โ WARP is not violated.
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Checking for WARP: Example 3
โข The demand ๐ฅ ๐โฒ, ๐คโฒunder final prices and wealth is not affordable under initial prices and wealth, i.e., ๐ โ ๐ฅ ๐โฒ, ๐คโฒ > ๐ค.โ The premise of WARP
does not hold.
โ Violation of Step 1!
โ WARP is not violated.
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Checking for WARP: Example 4
โข The demand ๐ฅ ๐โฒ, ๐คโฒunder final prices and wealth is not affordable under initial prices and wealth, i.e., ๐ โ ๐ฅ ๐โฒ, ๐คโฒ > ๐ค.โ The premise of WARP
does not hold.
โ Violation of Step 1!
โ WARP is not violated.
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Checking for WARP: Example 5
46
x1
x2
w p2
wp1
Bp,w
Bp ,w x (p,w)x (p ,w )
โข First, ๐ฅ ๐, ๐ค and ๐ฅ ๐โฒ, ๐คโฒ are both affordable under ๐ต๐,๐ค.
โข Second, ๐ฅ ๐, ๐ค is affordable under ๐ต๐โฒ,๐คโฒ, i.e., ๐โฒ โ ๐ฅ ๐, ๐ค < ๐คโฒ
โข Hence, WARP is NOTsatisfied!
Implications of WARP
โข How do price changes affect the WARP predictions?โข Assume a reduction in ๐1
โ the consumerโs budget lines rotates (uncompensated price change)
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Implications of WARP
โข Adjust the consumerโs wealth level so that he can consume his initial demand ๐ฅ ๐, ๐ค at the new prices.
โ shift the final budget line inwards until the point at which we reach the initial consumption bundle ๐ฅ ๐, ๐ค (compensated price change)
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Implications of WARP
โข What is the wealth adjustment?๐ค = ๐ โ ๐ฅ ๐, ๐ค under ๐, ๐ค
๐คโฒ = ๐โฒ โ ๐ฅ ๐, ๐ค under ๐โฒ, ๐คโฒ
โข Then,โ๐ค = โ๐ โ ๐ฅ ๐, ๐ค
where โ๐ค = ๐คโฒ โ ๐ค and โ๐ = ๐โฒ โ ๐.
โข This is the Slutsky wealth compensation: โ the increase (decrease) in wealth, measured by โ๐ค, that we
must provide to the consumer so that he can afford the same consumption bundle as before the price change, i.e., ๐ฅ ๐, ๐ค .
49
Implications of WARP: Law of Demand
โข Suppose that the Walrasian demand ๐ฅ ๐, ๐ค satisfies homog(0) and Walrasโ Law. Then, ๐ฅ ๐, ๐ค satisfies WARP iff:
โ๐ โ โ๐ฅ โค 0
whereโ โ๐ = ๐โฒ โ ๐ and โ๐ฅ = ๐ฅ ๐โฒ, ๐คโฒ โ ๐ฅ ๐, ๐ค
โ ๐คโฒ is the wealth level that allows the consumer to buy the initial demand at the new prices, ๐คโฒ = ๐โฒ โ ๐ฅ ๐, ๐ค
โข This is the Law of Demand: quantity demanded and price move in different directions.
50
Implications of WARP: Law of Demand
โข Does WARP restrict behavior when we apply Slutsky wealth compensations? โ Yes!
โข What if we were not applying the Slutsky wealth compensation, would WARP impose any restriction on allowable locations for ๐ฅ ๐โฒ, ๐คโฒ ?โ No!
51
Implications of WARP: Law of Demand
โข Can ๐ฅ ๐โฒ, ๐คโฒ lie on segment A?1) ๐ฅ ๐, ๐ค and ๐ฅ ๐โฒ, ๐คโฒ are both affordable under
๐ต๐,๐ค.
2) ๐ฅ ๐, ๐ค is affordable under ๐ต๐โฒ,๐คโฒ. โ WARP is violated if ๐ฅ ๐โฒ, ๐คโฒ lies on segment A
โข Can ๐ฅ ๐โฒ, ๐คโฒ lie on segment B?1) ๐ฅ ๐, ๐ค is affordable under ๐ต๐,๐ค, but ๐ฅ ๐โฒ, ๐คโฒ is
not. โ the premise of WARP does not holdโ WARP is not violated if ๐ฅ ๐โฒ, ๐คโฒ lies on segment B
53
Implications of WARP: Law of Demand
โข What did we learn from this figure?1) We started from ๐ป๐1, and compensated the wealth of
this individual (reducing it from ๐ค to ๐คโฒ) so that he could afford his initial bundle ๐ฅ ๐, ๐ค under the new prices. From this wealth compensation, we obtained budget line ๐ต๐โฒ,๐คโฒ.
2) From WARP, we know that ๐ฅ ๐โฒ, ๐คโฒ must contain more of good 1. That is, graphically, segment B lies to the right-hand side of
๐ฅ ๐, ๐ค .
3) Then, a price reduction, ๐ป๐1, when followed by an appropriate wealth compensation leads to an increase in the quantity demanded of this good, โ๐ฅ1. This is the compensated law of demand (CLD).
54
Implications of WARP: Law of Demand
โข Practice problem:โ Can you repeat this analysis but for an increase in
the price of good 1? First, pivot budget line ๐ต๐,๐ค inwards to obtain ๐ต๐โฒ,๐ค.
Then, increase the wealth level (wealth compensation) to obtain ๐ต๐โฒ,๐คโฒ.
Identify the two segments in budget line ๐ต๐โฒ,๐คโฒ, one to the right-hand side of ๐ฅ ๐, ๐ค and other to the left.
In which segment of budget line ๐ต๐โฒ,๐คโฒ can the Walrasian demand ๐ฅ(๐โฒ, ๐คโฒ) lie?
Advanced Microeconomic Theory 55
Implications of WARP: Law of Demand
โข Is WARP satisfied under the uncompensated law of demand (ULD)?โ ๐ฅ ๐, ๐ค is affordable under
budget line ๐ต๐,๐ค, but ๐ฅ(๐โฒ, ๐ค) is not. Hence, the premise of WARP is
not satisfied. As a result, WARP is not violated.
โ But, is this result imply something about whether ULD must hold? No! Although WARP is not violated,
ULD is: a decrease in the price of good 1 yields a decrease in the quantity demanded.
Advanced Microeconomic Theory 56
Implications of WARP: Law of Demand
โข Distinction between the uncompensated and the compensated law of demand:โ quantity demanded and price can move in the same
direction, when wealth is left uncompensated, i.e.,
โ๐1 โ โ๐ฅ1 > 0as โ๐1 โ โ๐ฅ1
โข Hence, WARP is not sufficient to yield law of demand for price changes that are uncompensated, i.e.,
WARP โ ULD, butWARP โ CLD
Advanced Microeconomic Theory 57
Implications of WARP: Slutsky Matrix
โข Let us focus now on the case in which ๐ฅ ๐, ๐ค is differentiable.
โข First, note that the law of demand is ๐๐ โ ๐๐ฅ โค 0(equivalent to โ๐ โ โ๐ฅ โค 0).
โข Totally differentiating ๐ฅ ๐, ๐ค ,๐๐ฅ = ๐ท๐๐ฅ ๐, ๐ค ๐๐ + ๐ท๐ค๐ฅ ๐, ๐ค ๐๐ค
โข And since the consumerโs wealth is compensated, ๐๐ค = ๐ฅ(๐, ๐ค) โ ๐๐ (this is the differential analog of โ๐ค = โ๐ โ ๐ฅ(๐, ๐ค)).โ Recall that โ๐ค = โ๐ โ ๐ฅ(๐, ๐ค) was obtained from the
Slutsky wealth compensation.
Advanced Microeconomic Theory 58
Implications of WARP: Slutsky Matrix
โข Substituting,๐๐ฅ = ๐ท๐๐ฅ ๐, ๐ค ๐๐ + ๐ท๐ค๐ฅ ๐, ๐ค ๐ฅ(๐, ๐ค) โ ๐๐
๐๐ค
โข or equivalently,
๐๐ฅ = ๐ท๐๐ฅ ๐, ๐ค + ๐ท๐ค๐ฅ ๐, ๐ค ๐ฅ(๐, ๐ค)๐ ๐๐
Advanced Microeconomic Theory 59
Implications of WARP: Slutsky Matrix
โข Hence the law of demand, ๐๐ โ ๐๐ฅ โค 0, can be expressed as
๐๐ โ ๐ท๐๐ฅ ๐, ๐ค + ๐ท๐ค๐ฅ ๐, ๐ค ๐ฅ(๐, ๐ค)๐ ๐๐
๐๐ฅ
โค 0
where the term in brackets is the Slutsky (or substitution) matrix
๐ ๐, ๐ค =๐ 11 ๐, ๐ค โฏ ๐ 1๐ฟ ๐, ๐ค
โฎ โฑ โฎ๐ ๐ฟ1 ๐, ๐ค โฏ ๐ ๐ฟ๐ฟ ๐, ๐ค
where
๐ ๐๐ ๐, ๐ค =๐๐ฅ๐(๐, ๐ค)
๐๐๐+
๐๐ฅ๐ ๐, ๐ค
๐๐ค๐ฅ๐(๐, ๐ค)
Advanced Microeconomic Theory 60
Implications of WARP: Slutsky Matrix
โข Proposition: If ๐ฅ ๐, ๐ค is differentiable, satisfies WL, homog(0), and WARP, then ๐ ๐, ๐ค is negative semi-definite,
๐ฃ โ ๐ ๐, ๐ค ๐ฃ โค 0 for any ๐ฃ โ โ๐ฟ
โข Implications:โ ๐ ๐๐ ๐, ๐ค : substitution effect of good ๐ with respect to
its own price is non-positive (own-price effect)โ Negative semi-definiteness does not imply that
๐ ๐, ๐ค is symmetric (except when ๐ฟ = 2). Usual confusion: โthen ๐ ๐, ๐ค is not symmetricโ, NO!
Advanced Microeconomic Theory 61
Implications of WARP: Slutsky Matrix
โข Proposition: If preferences satisfy LNS and strict convexity, and they are represented with a continuous utility function, then the Walrasian demand ๐ฅ ๐, ๐คgenerates a Slutsky matrix, ๐ ๐, ๐ค , which is symmetric.
โข The above assumptions are really common. โ Hence, the Slutsky matrix will then be symmetric.
โข However, the above assumptions are not satisfied in the case of preferences over perfect substitutes (i.e., preferences are convex, but not strictly convex).
Advanced Microeconomic Theory 62
Implications of WARP: Slutsky Matrix
โข Non-positive substitution effect, ๐ ๐๐ โค 0:
๐ ๐๐ ๐, ๐คsubstitution effect (โ)
=๐๐ฅ๐(๐, ๐ค)
๐๐๐
Total effect:โ usual good
+ Giffen good
+๐๐ฅ๐ ๐, ๐ค
๐๐ค๐ฅ๐(๐, ๐ค)
Income effect:+ normal good
โ inferior good
โข Substitution Effect = Total Effect + Income Effect
โ Total Effect = Substitution Effect - Income Effect
Advanced Microeconomic Theory 63
Implications of WARP: Slutsky Equation
๐ ๐๐ ๐, ๐คsubstitution effect (โ)
=๐๐ฅ๐(๐, ๐ค)
๐๐๐
Total effect
+๐๐ฅ๐ ๐, ๐ค
๐๐ค๐ฅ๐(๐, ๐ค)
Income effect
โข Total Effect: measures how the quantity demanded is affected by a change in the price of good ๐, when we leave the wealth uncompensated.
โข Income Effect: measures the change in the quantity demanded as a result of the wealth adjustment.
โข Substitution Effect: measures how the quantity demanded is affected by a change in the price of good ๐, after the wealth adjustment.โ That is, the substitution effect only captures the change in demand due to variation
in the price ratio, but abstracts from the larger (smaller) purchasing power that the consumer experiences after a decrease (increase, respectively) in prices.
Advanced Microeconomic Theory 64
โข Reduction in the price of ๐ฅ1. โ It enlarges consumerโs set of feasible
bundles. โ He can reach an indifference curve
further away from the origin.
โข The Walrasian demand curve indicates that a decrease in the price of ๐ฅ1 leads to an increase in the quantity demanded.โ This induces a negatively sloped
Walrasian demand curve (so the good is โnormalโ).
โข The increase in the quantity demanded of ๐ฅ1 as a result of a decrease in its price represents the total effect (TE).
Advanced Microeconomic Theory 65
Implications of WARP: Slutsky EquationX2
X1
I2
I1
p1
X1
p1
A
B
a
b
โข Reduction in the price of ๐ฅ1.โ Disentangle the total effect into the
substitution and income effectsโ Slutsky wealth compensation?
โข Reduce the consumerโs wealth so that he can afford the same consumption bundle as the one before the price change (i.e., A).โ Shift the budget line after the price change
inwards until it โcrossesโ through the initial bundle A.
โ โConstant purchasing powerโ demand curve (CPP curve) results from applying the Slutsky wealth compensation.
โ The quantity demanded for ๐ฅ1 increases from ๐ฅ1
0 to ๐ฅ13.
โข When we do not hold the consumerโs purchasing power constant, we observe relatively large increase in the quantity demanded for ๐ฅ1 (i.e., from ๐ฅ1
0 to ๐ฅ12 ).
Advanced Microeconomic Theory 66
Implications of WARP: Slutsky Equationx2
x1
I2
I1
p1
x1
p1
I3
CPP demand
A
B
C
a
bc
โข Reduction in the price of ๐ฅ1.โ Hicksian wealth compensation (i.e.,
โconstant utilityโ demand curve)?
โข The consumerโs wealth level is adjusted so that he can still reach his initial utility level (i.e., the same indifference curve ๐ผ1as before the price change).โ A more significant wealth reduction than
when we apply the Slutsky wealth compensation.
โ The Hicksian demand curve reflects that, for a given decrease in p1, the consumer slightly increases his consumption of good one.
โข In summary, a given decrease in ๐1produces: โ A small increase in the Hicksian demand for
the good, i.e., from ๐ฅ10 to ๐ฅ1
1. โ A larger increase in the CPP demand for the
good, i.e., from ๐ฅ10 to ๐ฅ1
3. โ A substantial increase in the Walrasian
demand for the product, i.e., from ๐ฅ10 to ๐ฅ1
2.Advanced Microeconomic Theory 67
Implications of WARP: Slutsky Equationx2
x1
I2
I1
p1
x1
p1
I3
CPP demand
Hicksian demand
โข A decrease in price of ๐ฅ1 leads the consumer to increase his consumption of this good, โ๐ฅ1, but:
โ The โ๐ฅ1 which is solely due to the price effect (either measured by the Hicksian demand curve or the CPP demand curve) is smaller than the โ๐ฅ1 measured by the Walrasian demand, ๐ฅ ๐, ๐ค , which also captures wealth effects.
โ The wealth compensation (a reduction in the consumerโs wealth in this case) that maintains the original utility level (as required by the Hicksian demand) is larger than the wealth compensation that maintains his purchasing power unaltered (as required by the Slutsky wealth compensation, in the CPP curve).
Advanced Microeconomic Theory 68
Implications of WARP: Slutsky Equation
Substitution and Income Effects: Normal Goods
โข Decrease in the price of the good in the horizontal axis (i.e., food).
โข The substitution effect (SE) moves in the opposite direction as the price change.
โ A reduction in the price of food implies a positive substitution effect.
โข The income effect (IE) is positive (thus it reinforces the SE).
โ The good is normal.
Advanced Microeconomic Theory 69
Clothing
FoodSE IE
TE
A
B
C
I2
I1
BL2
BL1
BLd
Substitution and Income Effects: Inferior Goods
โข Decrease in the price of the good in the horizontal axis (i.e., food).
โข The SE still moves in the opposite direction as the price change.
โข The income effect (IE) is now negative (which partially offsets the increase in the quantity demanded associated with the SE). โ The good is inferior.
โข Note: the SE is larger than the IE.
Advanced Microeconomic Theory 70
I2
I1
BL1 BLd
Substitution and Income Effects: Giffen Goods
โข Decrease in the price of the good in the horizontal axis (i.e., food).
โข The SE still moves in the opposite direction as the price change.
โข The income effect (IE) is still negative but now completely offsets the increase in the quantity demanded associated with the SE. โ The good is Giffen good.
โข Note: the SE is less than the IE.
71
I2
I1BL1
BL2
Substitution and Income Effects
SE IE TE
Normal Good + + +Inferior Good + - +Giffen Good + - -
Advanced Microeconomic Theory 72
โข NG: Demand curve is negatively sloped (as usual)
โข GG: Demand curve is positively sloped
Substitution and Income Effects
โข Summary:1) SE is negative (since โ ๐1 โ โ ๐ฅ1)
SE < 0 does not imply โ ๐ฅ1
2) If good is inferior, IE < 0. Then,
TE = เธSEโ เธ
โ เธIEโ
+
โ if IE><
SE , then TE(โ)TE(+)
For a price decrease, this impliesTE(โ)TE(+)
โโ ๐ฅ1
โ ๐ฅ1
Giffen goodNonโGiffen good
3) Hence,a) A good can be inferior, but not necessarily be Giffenb) But all Giffen goods must be inferior.
Advanced Microeconomic Theory 73
Expenditure Minimization Problem
โข Expenditure minimization problem (EMP):min๐ฅโฅ0
๐ โ ๐ฅ
s.t. ๐ข(๐ฅ) โฅ ๐ข
โข Alternative to utility maximization problem
Advanced Microeconomic Theory 75
Expenditure Minimization Problem
โข Consumer seeks a utility level associated with a particular indifference curve, while spending as little as possible.
โข Bundles strictly above ๐ฅโ cannot be a solution to the EMP:โ They reach the utility level ๐ขโ But, they do not minimize total
expenditure
โข Bundles on the budget line strictly below ๐ฅโ cannot be the solution to the EMP problem:โ They are cheaper than ๐ฅโ
โ But, they do not reach the utility level ๐ข
Advanced Microeconomic Theory 76
X2
X1
*x
*UCS( )x
u
Expenditure Minimization Problem
โข Lagrangian๐ฟ = ๐ โ ๐ฅ + ๐ ๐ข โ ๐ข(๐ฅ)
โข FOCs (necessary conditions)๐๐ฟ
๐๐ฅ๐= ๐๐ โ ๐
๐๐ข(๐ฅโ)
๐๐ฅ๐โค 0
๐๐ฟ
๐๐= ๐ข โ ๐ข ๐ฅโ = 0
Advanced Microeconomic Theory 77
[ = 0 for interior solutions ]
Expenditure Minimization Problem
โข For interior solutions,
๐๐ = ๐๐๐ข(๐ฅโ)
๐๐ฅ๐or
1
๐=
๐๐ข(๐ฅโ)
๐๐ฅ๐
๐๐
for any good ๐. This implies, ๐๐ข(๐ฅโ)
๐๐ฅ๐
๐๐=
๐๐ข(๐ฅโ)
๐๐ฅ๐
๐๐or
๐๐
๐๐=
๐๐ข(๐ฅโ)
๐๐ฅ๐๐๐ข(๐ฅโ)
๐๐ฅ๐
โข The consumer allocates his consumption across goods until the point in which the marginal utility per dollar spent on each good is equal across all goods (i.e., same โbang for the buckโ).
โข That is, the slope of indifference curve is equal to the slope of the budget line.
Advanced Microeconomic Theory 78
EMP: Hicksian Demand
โข The bundle ๐ฅโ โ argmin ๐ โ ๐ฅ (the argument that solves the EMP) is the Hicksian demand, which depends on ๐ and ๐ข,
๐ฅโ โ โ(๐, ๐ข)
โข Recall that if such bundle ๐ฅโ is unique, we denote it as ๐ฅโ = โ(๐, ๐ข).
Advanced Microeconomic Theory 79
Properties of Hicksian Demand
โข Suppose that ๐ข(โ) is a continuous function, satisfying LNS defined on ๐ = โ+
๐ฟ . Then for ๐ โซ0, โ(๐, ๐ข) satisfies:1) Homog(0) in ๐, i.e., โ ๐, ๐ข = โ(๐ผ๐, ๐ข) for any ๐, ๐ข,
and ๐ผ > 0. If ๐ฅโ โ โ(๐, ๐ข) is a solution to the problem
min๐ฅโฅ0
๐ โ ๐ฅ
then it is also a solution to the problemmin๐ฅโฅ0
๐ผ๐ โ ๐ฅ
Intuition: a common change in all prices does not alter the slope of the consumerโs budget line.
Advanced Microeconomic Theory 80
Properties of Hicksian Demand
โข ๐ฅโ is a solution to the EMP when the price vector is ๐ = (๐1, ๐2).
โข Increase all prices by factor ๐ผ
๐โฒ = (๐1โฒ , ๐2
โฒ ) = (๐ผ๐1, ๐ผ๐2)
โข Downward (parallel) shift in the budget line, i.e., the slope of the budget line is unchanged.
โข But I have to reach utility level ๐ข to satisfy the constraint of the EMP?
โข Spend more to buy bundle ๐ฅโ(๐ฅ1
โ, ๐ฅ2โ), i.e.,
๐1โฒ ๐ฅ1
โ + ๐2โฒ ๐ฅ2
โ > ๐1๐ฅ1โ + ๐2๐ฅ2
โ
โข Hence, โ ๐, ๐ข = โ(๐ผ๐, ๐ข)
Advanced Microeconomic Theory 81
x2
x1
Properties of Hicksian Demand
2) No excess utility:
for any optimal consumption bundle ๐ฅ โ โ ๐, ๐ข , utility level satisfies ๐ข ๐ฅ = ๐ข.
Advanced Microeconomic Theory 82
x1
x2
wp2
wp1
x โ
x โ h (p,u)
uu 1
Properties of Hicksian Demand
โข Intuition: Suppose there exists a bundle ๐ฅ โ โ ๐, ๐ข for which the consumer obtains a utility level ๐ข ๐ฅ = ๐ข1 > ๐ข, which is higher than the utility level ๐ข he must reach when solving EMP.
โข But we can then find another bundle ๐ฅโฒ = ๐ฅ๐ผ, very close to ๐ฅ (๐ผ โ 1), for which ๐ข(๐ฅโฒ) > ๐ข.
โข Bundle ๐ฅโฒ:โ is cheaper than ๐ฅโ exceeds the minimal utility level ๐ข that the consumer must
reach in his EMP
โข For a given utility level ๐ข that you have to reach in the EMP, bundle โ ๐, ๐ข does not exceed ๐ข, since otherwise you could find a cheaper bundle that exactly reaches ๐ข.
Advanced Microeconomic Theory 83
Properties of Hicksian Demand
3) Convexity:
If the preference relation is convex, then โ ๐, ๐ข is a convex set.
Advanced Microeconomic Theory 84
Properties of Hicksian Demand
4) Uniqueness:
If the preference relation is strictly convex, then โ ๐, ๐ขcontains a single element.
Advanced Microeconomic Theory 85
Properties of Hicksian Demand
โข Compensated Law of Demand: for any change in prices ๐ and ๐โฒ,
(๐โฒโ๐) โ โ ๐โฒ, ๐ข โ โ ๐, ๐ข โค 0
โ Implication: for every good ๐,(๐๐
โฒ โ ๐๐) โ โ๐ ๐โฒ, ๐ข โ โ๐ ๐, ๐ข โค 0
โ This is definitely true for compensated demand.
โ But, not necessarily for Walrasian demand (which is uncompensated):
โข Recall the figures on Giffen goods, where a decrease in ๐๐ in fact decreases ๐ฅ๐ ๐, ๐ข when wealth was uncompensated.
Advanced Microeconomic Theory 86
The Expenditure Function
โข Plugging the result from the EMP, โ ๐, ๐ข , into the objective function, ๐ โ ๐ฅ, we obtain the value function of this optimization problem,
๐ โ โ ๐, ๐ข = ๐(๐, ๐ข)
where ๐(๐, ๐ข) represents the minimal expenditure that the consumer needs to incur in order to reach utility level ๐ข when prices are ๐.
Advanced Microeconomic Theory 87
Properties of Expenditure Function
โข Suppose that ๐ข(โ) is a continuous function, satisfying LNS defined on ๐ = โ+
๐ฟ . Then for ๐ โซ0, ๐(๐, ๐ข) satisfies:1) Homog(1) in ๐, i.e.,
๐ ๐ผ๐, ๐ข = ๐ผ ๐ โ ๐ฅโ
๐(๐,๐ข)
= ๐ผ โ ๐(๐, ๐ข)
for any ๐, ๐ข, and ๐ผ > 0.
We know that the optimal bundle is not changed when all prices change, since the optimal consumption bundle in โ(๐, ๐ข) satisfies homogeneity of degree zero.
Such change just makes it more or less expensive to buy the same bundle.
Advanced Microeconomic Theory 88
Properties of Expenditure Function
2) Strictly increasing in ๐: For a given prices, reaching a higher utility requires higher expenditure:
๐1๐ฅ1โฒ + ๐2๐ฅ2
โฒ > ๐1๐ฅ1 + ๐2๐ฅ2
where (๐ฅ1, ๐ฅ2) = โ(๐, ๐ข)and (๐ฅ1
โฒ , ๐ฅ2โฒ ) = โ(๐, ๐ขโฒ).
Then,๐ ๐, ๐ขโฒ > ๐ ๐, ๐ข
Advanced Microeconomic Theory 89
x2
x1
x2
x1
Properties of Expenditure Function
3) Non-decreasing in ๐๐ for any good ๐:Higher prices mean higher expenditure to reach a given utility level. โข Let ๐โฒ = (๐1, ๐2, โฆ , ๐๐
โฒ , โฆ , ๐๐ฟ) and ๐ =(๐1, ๐2, โฆ , ๐๐, โฆ , ๐๐ฟ), where ๐๐
โฒ > ๐๐.โข Let ๐ฅโฒ = โ ๐โฒ, ๐ข and ๐ฅ = โ ๐, ๐ข from EMP under
prices ๐โฒ and ๐, respectively.โข Then, ๐โฒ โ ๐ฅโฒ = ๐(๐โฒ, ๐ข) and ๐ โ ๐ฅ = ๐(๐, ๐ข).
๐ ๐โฒ, ๐ข = ๐โฒ โ ๐ฅโฒ โฅ ๐ โ ๐ฅโฒ โฅ ๐ โ ๐ฅ = ๐(๐, ๐ข)
โ 1st inequality due to ๐โฒ โฅ ๐โ 2nd inequality: at prices ๐, bundle ๐ฅ minimizes EMP.
Advanced Microeconomic Theory 90
Properties of Expenditure Function
4) Concave in ๐:
Let ๐ฅโฒ โ โ ๐โฒ, ๐ข โ ๐โฒ๐ฅโฒ โค ๐โฒ๐ฅโ๐ฅ โ ๐ฅโฒ, e.g., ๐โฒ๐ฅโฒ โค ๐โฒ าง๐ฅand ๐ฅโฒโฒ โ โ ๐โฒโฒ, ๐ข โ ๐โฒโฒ๐ฅโฒโฒ โค ๐โฒโฒ๐ฅโ๐ฅ โ ๐ฅโฒโฒ, e.g., ๐โฒโฒ๐ฅโฒโฒ โค ๐โฒโฒ าง๐ฅ
This implies๐ผ๐โฒ๐ฅโฒ + 1 โ ๐ผ ๐โฒโฒ๐ฅโฒโฒ
โค ๐ผ๐โฒ าง๐ฅ + 1 โ ๐ผ ๐โฒโฒ าง๐ฅ
๐ผ เธ๐โฒ๐ฅโฒ
๐(๐โฒ,๐ข)
+ 1 โ ๐ผ ๐โฒโฒ๐ฅโฒโฒ
๐(๐โฒโฒ,๐ข)
โค [๐ผ๐โฒ + 1 โ ๐ผ ๐โฒโฒ
าง๐
] าง๐ฅ
๐ผ๐(๐โฒ, ๐ข) + 1 โ ๐ผ ๐(๐โฒโฒ, ๐ข) โค ๐( าง๐, ๐ข)concavity
Advanced Microeconomic Theory 91
x2
x1
x
x
Relationship between the Expenditure and Hicksian Demand
โข Letโs assume that ๐ข(โ) is a continuous function, representing preferences that satisfy LNS and are strictly convex and defined on ๐ = โ+
๐ฟ . For all ๐ and ๐ข,
๐๐ ๐,๐ข
๐๐๐= โ๐ ๐, ๐ข for every good ๐
This identity is โShepardโs lemmaโ: if we want to find โ๐ ๐, ๐ข and we know ๐ ๐, ๐ข , we just have to differentiate ๐ ๐, ๐ข with respect to prices.
โข Proof: three different approaches1) the support function2) first-order conditions 3) the envelop theorem (See Appendix 2.2)
Advanced Microeconomic Theory 93
Relationship between the Expenditure and Hicksian Demand
โข The relationship between the Hicksian demand and the expenditure function can be further developed by taking first order conditions again. That is,
๐2๐(๐, ๐ข)
๐๐๐2 =
๐โ๐(๐, ๐ข)
๐๐๐
or๐ท๐
2๐ ๐, ๐ข = ๐ท๐โ(๐, ๐ข)
โข Since ๐ท๐โ(๐, ๐ข) provides the Slutsky matrix, ๐(๐, ๐ค), then
๐ ๐, ๐ค = ๐ท๐2๐ ๐, ๐ข
where the Slutsky matrix can be obtained from the observable Walrasian demand.
Advanced Microeconomic Theory 94
Relationship between the Expenditure and Hicksian Demand
โข There are three other important properties of ๐ท๐โ(๐, ๐ข), where ๐ท๐โ(๐, ๐ข) is ๐ฟ ร ๐ฟ derivative matrix of the hicksian demand, โ ๐, ๐ข :1) ๐ท๐โ(๐, ๐ข) is negative semidefinite
Hence, ๐ ๐, ๐ค is negative semidefinite.
2) ๐ท๐โ(๐, ๐ข) is a symmetric matrix Hence, ๐ ๐, ๐ค is symmetric.
3) ๐ท๐โ ๐, ๐ข ๐ = 0, which implies ๐ ๐, ๐ค ๐ = 0.
Not all goods can be net substitutes ๐โ๐(๐,๐ข)
๐๐๐> 0 or net
complements ๐โ๐(๐,๐ข)
๐๐๐< 0 . Otherwise, the multiplication
of this vector of derivatives times the (positive) price vector ๐ โซ 0 would yield a non-zero result.
Advanced Microeconomic Theory 95
Relationship between Hicksian and Walrasian Demand
โข When income effects are positive (normal goods), then the Walrasian demand ๐ฅ(๐, ๐ค) is above the Hicksian demand โ(๐, ๐ข) .โ The Hicksian demand
is steeper than the Walrasian demand.
Advanced Microeconomic Theory 96
x1
x2
wp2
wp1
x1
p1
wp1
p1
p1
x*1 x1
1 x21
h (p,ลซ)
x (p,w)
p1
I1
I3
I2x1
x 3x 2
x*
SE IE
Relationship between Hicksian and Walrasian Demand
โข When income effects are negative (inferior goods), then the Walrasian demand ๐ฅ(๐, ๐ค) is below the Hicksian demand โ(๐, ๐ข) .โ The Hicksian demand
is flatter than the Walrasian demand.
Advanced Microeconomic Theory 97
x1
x2
wp2
wp1
x1
p1
wp1
p1
p1
x*1 x1
1x21
h (p,ลซ)
x (p,w)
p1
I1
I2
x1
x 2x*
SE
IE
a
bc
Relationship between Hicksian and Walrasian Demand
โข We can formally relate the Hicksian and Walrasian demand as follows:โ Consider ๐ข(โ) is a continuous function, representing
preferences that satisfy LNS and are strictly convex and defined on ๐ = โ+
๐ฟ .โ Consider a consumer facing ( าง๐, เดฅ๐ค) and attaining utility
level เดค๐ข.โ Note that เดฅ๐ค = ๐( าง๐, เดค๐ข). In addition, we know that for any
(๐, ๐ข), โ๐ ๐, ๐ข = ๐ฅ๐(๐, ๐ ๐, ๐ข๐ค
). Differentiating this
expression with respect to ๐๐, and evaluating it at ( าง๐, เดค๐ข), we get:
๐โ๐( าง๐, เดค๐ข)
๐๐๐=
๐๐ฅ๐( าง๐, ๐( าง๐, เดค๐ข))
๐๐๐+
๐๐ฅ๐( าง๐, ๐( าง๐, เดค๐ข))
๐๐ค
๐๐( าง๐, เดค๐ข)
๐๐๐Advanced Microeconomic Theory 98
Relationship between Hicksian and Walrasian Demand
โข Using the fact that ๐๐( าง๐,เดฅ๐ข)
๐๐๐= โ๐( าง๐, เดค๐ข),
๐โ๐( าง๐,เดฅ๐ข)
๐๐๐=
๐๐ฅ๐( าง๐,๐( าง๐,เดฅ๐ข))
๐๐๐+
๐๐ฅ๐( าง๐,๐( าง๐,เดฅ๐ข))
๐๐คโ๐( าง๐, เดค๐ข)
โข Finally, since เดฅ๐ค = ๐( าง๐, เดค๐ข) and โ๐ าง๐, เดค๐ข =
๐ฅ๐ าง๐, ๐ าง๐, เดค๐ข = ๐ฅ๐ าง๐, เดฅ๐ค , then
๐โ๐( าง๐,เดฅ๐ข)
๐๐๐=
๐๐ฅ๐( าง๐, เดฅ๐ค)
๐๐๐+
๐๐ฅ๐( าง๐, เดฅ๐ค)
๐๐ค๐ฅ๐( าง๐, เดฅ๐ค)
Advanced Microeconomic Theory 99
Relationship between Hicksian and Walrasian Demand
โข But this coincides with ๐ ๐๐ ๐, ๐ค that we discussed in the Slutsky equation.
โ Hence, we have ๐ท๐โ ๐, ๐ข
๐ฟร๐ฟ
= ๐ ๐, ๐ค .
โ Or, more compactly, ๐๐ธ = ๐๐ธ + ๐ผ๐ธ.
Advanced Microeconomic Theory 100
Relationship between Walrasian Demand and Indirect Utility Function
โข Letโs assume that ๐ข(โ) is a continuous function, representing preferences that satisfy LNS and are strictly convex and defined on ๐ = โ+
๐ฟ . Support also that ๐ฃ(๐, ๐ค) is differentiable at any (๐, ๐ค) โซ 0. Then,
โ
๐๐ฃ ๐,๐ค
๐๐๐๐๐ฃ ๐,๐ค
๐๐ค
= ๐ฅ๐(๐, ๐ค) for every good ๐
โข This is Royโs identity.
โข Powerful result, since in many cases it is easier to compute the derivatives of ๐ฃ ๐, ๐ค than solving the UMP with the system of FOCs.
Advanced Microeconomic Theory 101
Summary of Relationships
โข The Walrasian demand, ๐ฅ ๐, ๐ค , is the solution of the UMP.โ Its value function is the indirect utility function, ๐ฃ ๐, ๐ค .
โข The Hicksian demand, โ(๐, ๐ข), is the solution of the EMP. โ Its value function is the expenditure function, ๐(๐, ๐ข).
โข Relationship between the value functions of the UMP and the EMP:โ ๐ ๐, ๐ฃ ๐, ๐ค = ๐ค, i.e., the minimal expenditure needed in
order to reach a utility level equal to the maximal utility that the individual reaches at his UMP, ๐ข = ๐ฃ ๐, ๐ค , must be ๐ค.
โ ๐ฃ ๐, ๐(๐, ๐ข) = ๐ข, i.e., the indirect utility that can be reached when the consumer is endowed with a wealth level w equal to the minimal expenditure he optimally uses in the EMP, i.e., ๐ค =๐(๐, ๐ข), is exactly ๐ข.
Advanced Microeconomic Theory 102
Summary of Relationships
โข Relationship between the argmax of the UMP (the Walrasian demand) and the argmin of the EMP (the Hicksian demand):โ ๐ฅ ๐, ๐(๐, ๐ข) = โ ๐, ๐ข , i.e., the (uncompensated)
Walrasian demand of a consumer endowed with an adjusted wealth level ๐ค (equal to the expenditure he optimally uses in the EMP), ๐ค = ๐ ๐, ๐ข , coincides with his Hicksian demand, โ ๐, ๐ข .
โ โ ๐, ๐ฃ(๐, ๐ค) = ๐ฅ ๐, ๐ค , i.e., the (compensated) Hicksian demand of a consumer reaching the maximum utility of the UMP, ๐ข = ๐ฃ (๐, ๐ค), coincides with his Walrasian demand, ๐ฅ ๐, ๐ค .
Advanced Microeconomic Theory 104
Summary of Relationships
โข Finally, we can also use:โ The Slutsky equation:
๐โ๐(๐, ๐ข)
๐๐๐=
๐๐ฅ๐(๐, ๐ค)
๐๐๐+
๐๐ฅ๐(๐, ๐ค)
๐๐ค๐ฅ๐(๐, ๐ค)
to relate the derivatives of the Hicksian and the Walrasian demand.โ Shepardโs lemma:
๐๐ ๐, ๐ข
๐๐๐= โ๐ ๐, ๐ข
to obtain the Hicksian demand from the expenditure function.โ Royโs identity:
โ
๐๐ฃ ๐, ๐ค๐๐๐
๐๐ฃ ๐, ๐ค๐๐ค
= ๐ฅ๐(๐, ๐ค)
to obtain the Walrasian demand from the indirect utility function.
Advanced Microeconomic Theory 106
Summary of Relationships
Advanced Microeconomic Theory 107
x(p,w)
v(p,w) e(p,u)
h(p,u)
The UMP The EMP
e(p,v(p,w))=w
v(p,e(p,w))=u
(1) (1)
(2)
3(a)
3(b)
4(a) Slutsky equation
(Using derivatives)