consumer choice 1md3405/choice_ma_consumer_2_17.pdf · hicksian demand and the expenditure function...
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Consumer Choice 1
Mark Dean
GR5211 - Microeconomic Analysis 1
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The Indirect Utility Function
• Imagine that the consumer can choose to live in two differentcountries
• In country 1 they would face prices p1 and have income w1• In country 2 they would face prices p2 and have income w2
• Which country would they prefer to live in?• i.e. what are there preferences over budget sets?
• which we can denote by �∗
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The Indirect Utility Function
• Here is one possibility• Figure out one of the best items in budget set 1 (i.e.x(p1,w1))
• Figure out one of the best items in budget set 2 (i.e.x(p2,w2))
• The consumer prefers budget set 1 to budget set 2 if theformer is preferred to the latter
• i.e. we can define �∗on the set of budget sets by(p1, x1) � ∗(p2, x2)
if and only if x1 � x2
for x1 ∈ x(p1, x1) and x2 ∈ x(p2, x2)
• Can you think of reasons why this might not be the rightmodel?• Temptation• Uncertainty• Regret
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The Indirect Utility Function
• If � can be represented by a utility function we can define theindirect utility function
v(p,w) = u(x(p,w))
• v now represents the preferences �∗on the space of budgetsets
• Proof?
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Properties of the Indirect Utility Function
• Property 1:
v(αp, αw) = v(p,w) for α > 0
• Follows from the fact that x(αp, αw) = x(p,w)
• Property 2: v(p,w) is non increasing in p and increasing inw
• Assuming non satiation
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Properties of the Indirect Utility Function
• Property 3: v is quasiconvex: i.e. the set
{(p,w)|v(p,w) ≤ v}
is convex for all v
• Proof left as an exercise
• Property 4: If � is continuous then �∗ is continuous• Follows from the Theorem of the Maximum
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The Story of The Turtle
• From Ariel Rubinstein
• The furthest a turtle can travel in 1 day is 1 km• The shortest length of time it takes for a turtle to travel 1kmis 1 day
• No, we didn’t know what he was on about either• But bear with me...
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The Story of The Turtle
• Is this always true?• No! Requires two assumptions
1 The turtle can travel a strictly positive distance in any positiveperiod of time
2 The turtle cannot jump a positive distance in zero time
• So much for zoology, what has this got to do with economics?
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Expenditure Minimization
• It is going to be very useful to define Expenditureminimization problem• This is the dual of the utility maximization problem
• Prime problem (utility maximization)
choose x ∈ RN+
in order to maximize u(x)
subject toN
∑i=1pixi ≤ w
• Dual problem (cost minimization)
choose x ∈ RN+
in order to minimizeN
∑i=1pixi
subject to u(x) ≥ u
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Expenditure Minimization
• Are these problems ‘the same’?• In general, no
• Like the teleporting turtle
• However, if we rule out teleportation (and laziness) then theywill be the same.
• What assumptions allow us to do that?
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Duality
TheoremIf u is monotonic and continuous then x∗ is a solution to the primeproblem with prices p and wealth w it is a solution to the dualproblem with prices p and utility v(p,w)
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Duality
Proof.
• Assume not, then there exists a bundle x such that
u(x) ≥ v(p,w) = u(x∗)
with
∑ pi xi < ∑ pix∗i = w
• But this means, by monotonicity, that there exists an ε > 0such that, for
x ′ =
x1 + εx2 + ε...
xN + ε
∑ pix ′i < w
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Duality
Proof.
• By monotonicity, we know that u(x ′) > u(x) ≥ u(x∗), and sox∗ is not a solution to the prime problem
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Duality
TheoremIf u is monotonic and continuous then x∗ is a solution to the dualproblem with prices p and utility u∗ it is a solution to the primeproblem with prices p and wealth ∑ pix∗i
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Duality
Proof.
• Assume not, then there exists a bundle x such that
∑ pi xi ≤∑ pix∗i
withu(x) > u(x∗) ≥ u∗
• By continuity, there exists an ε > 0 such that, for allx ′ ∈ B(x , ε), u(x ′) > u(x∗)
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Duality
Proof.
• In particular, there is an ε > 0 such that
x ′ =
x1 − εx2 − ε...
xN − ε
and u(x ′) > u(x∗) ≥ u∗
• But ∑ pix ′i < ∑ pi xi ≤ ∑ pix∗i , so x∗ is not a solution to the
prime problem.
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Hicksian Demand and the Expenditure Function
• The dual problem allows us to define two new objects• The Hicksian demand function
h(p, u) = arg minx∈X ∑ pixi
subject to u(x) ≥ u
• This is the demand for each good when prices are p and theconsumer must achieve utility u
• Note difference from Walrasian demand
• The expenditure function
e(p, u) = minx∈X ∑ pixi
subject to u(x) ≥ u
• This is the amount of money necessary to achieve utility uwhen prices are p
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Properties of the Hicksian Demand Function
• Assume that we are dealing with continuous, non-satiatedpreferences
• Fact 1: h is homogenous of degree zero in prices - i.e.h(αp, u) = h(p, u) for α > 0
• Follows from the fact that increasing all prices by α does notchange the tangency conditions
• i.e. the slope of the ’budget line’remains the same
• Fact 2: No excess utility - i.e. u(h(p, u)) = u• Follows from continuity (why?)
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Properties of the Hicksian Demand Function
• Fact 3: If preferences are convex then h is a convex set. Ifpreferences are strictly convex then h is unique• Proof: say that x and y are both in h(p, u). Then
∑ pi xi = ∑ pi yi = e(p, u)
• Implies that for any α ∈ (0, 1) and z = αx + (1− α)y
∑ pi zi = ∑ pi (αxi + (1− α)yi )
= α ∑ pi xi + (1− α)∑ pi yi= e(p, u)
• Also, as preferences are convex, z � x , and sou(z) ≥ u(x) = u
• If preferences are strictly convex, then z � x• But, by continuity, exists ε > 0 such that z ′ � x allz ′ ∈ B(x , ε)
• Implies that there is a z ′ such that u(z ′) > u and∑ pi zi < ∑ pi xi
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Properties of the Expenditure Function
• Again, assume that we are dealing with continuous,non-satiated preferences
• Fact 1: e(αp, u) = αe(p, u)
• Follows from the fact that h(αp, u) = h(p, u)
• Fact 2: e is strictly increasing in u and non-decreasing in p• Strictly increasing due to continuity and non-satiation• Only non-decreasing because may already be buying 0 of somegood
• Fact 3: e is continuous in p and u• Logic follows from the theorem of the maximum (though can’tbe applied directly)
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Properties of the Expenditure Function
• Fact 4: e is concave in p• Proof: fix a u. we need to show that
e(p′′, u) ≥ αe(p, u) + (1− α)e(p′, u)
wherep′′ = αp + (1− α)p′
• Let x ′′ ∈ h(p′′, u), then
e(p′′, u) = ∑ p′′i x′′i
= ∑(αpi + (1− α)p′i
)x ′′i
= α ∑ pi x′′i + (1− α)∑ p′i x
′′i
≥ α ∑ pi xi + (1− α)∑ p′i x′i
= αe(p, u) + (1− α)e ′(p, u)
where x ∈ h(p, u) and x ′ ∈ h(p′, u)
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Properties of the Expenditure Function
• This is quite an important and intuitive property• Implies that if we look at how expenditure changes as afunction of one price it looks like this ...
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Properties of the Expenditure Function
• Think of a price increase from p1 to p2• If the consumer couldn’t change their allocation thenexpenditure would go from e1 to e3
• This is an upper bound on the true increase in expenditure.
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Comparative Statics
• We will now put the above machinery to work to learn aboutthe relationship between the various measures we haveintroduced
• This will also allow us to say something about thecomparative statics of these functions - for example howdemand changes with price
• Before doing so, it will be worth reviewing a very usefulmathematical result
• The Envelope Theorem• See Mas-Colell section M.L
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The Envelope Theorem
• Consider a constrained optimization problem
choose x
in order to maximize f (x : q)
subject to
g1(x : q) = 0...
gN (x : q) = 0
• Where q are some parameters of the problem (for exampleprices)
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The Envelope Theorem
• Assume the problem is well behaved, and let
• x(q) be (a) solution to the problem if the parameters are q• v(q) = f (x(q) : q)
• Key question: how does v alter with q• i.e. how does the value that can be achieved vary with theparameters?
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The Envelope Theorem
• Say that both x and q are single valued• And say that there are no constraints• Chain rule gives
∂v∂q=
∂f∂q+
∂f∂x
∂x∂q
• But note that if we are at a maximum
∂f∂x= 0
• and so∂v∂q=
∂f∂q
• Only the direct effect of the change in parametersmatters
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The Envelope Theorem
• This result generalizes
Theorem (The Envelope Theorem)In the above decision problem
∂v(q)∂q
=∂f (x(q) : q)
∂q−∑
nλn
∂gn(x(q) : q)∂q
where λn is the Lagrange multiplier on the nth constraint
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Hicksian Demand and The Expenditure Function
• We can now apply the envelope theorem to get someinteresting results relating the various functions that we havedefined
• First, the relationship between the expenditure function andHicksian demand
Theorem (Shephard’s Lemma)Say preferences are continuous, locally non satiated and strictlyconvex then
hl (p, u) =∂e(p, u)
∂pl
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Hicksian Demand and The Expenditure Function
Proof.EMP is
minN
∑i=1pixi
subject to u(x) ≥ u
Applying the envelope theorem directly gives the result
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Hicksian Demand and The Expenditure Function
CorollaryAssume h is continuously differentiable, and let
Dph(p, u) =
∂h1∂p1
· · · ∂h1∂pM
......
∂hM∂p1
· · · ∂hM∂pM
Then
1 DPh(p, u) = D2pe(p, u)
2 DPh(p, u) is negative semi definite
3 DPh(p, u) is symmetric
4 DPh(p, u)p = 0
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Hicksian Demand and The Expenditure Function
Proof.
1 Follows directly from previous claim
2 Follows from (1) and the fact that e is concave
3 Follows from (1) and the fact that matrices of secondderivatives are symmetric
4 Follows from the homogeneity of degree zero of h, so
h(αp, u)− h(p, u) = 0
Differentiating with respect to α gives the desired result
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Walrasian Demand and The Indirect Utility Function
Theorem (Roy’s Identity )Say preferences are continuous, locally non satiated and strictlyconvex then
xl (p,w) = −∂v (p,w )
∂pl∂v (p,w )
∂w
Proof.Applying the envelope theorem tells us that
∂v(p,w)∂pl
= −λxl (p,w)
also
λ =∂v(p,w)
∂w
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Walrasian and Hicksian Demand
• Perhaps more usefully we can relate Hicksian and WalrasianDemand
Theorem (The Slusky Equation)Let preferences be continuous, strictly convex and locallynon-satiated and u = v(p,w)
∂hl (p, u)∂pk
=∂xl (p,w)
∂pk+
∂xl (p,w)∂w
xk (pw ,w)
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Walrasian and Hicksian Demand
• Proof.By duality, we know
hl (p, u) = xl (p, e(p, u))
Differentiating both sides with respect to pk gives
∂hl (p, u)∂pk
=∂xl (p,w)
∂pk+
∂xl (p,w)∂w
∂e(p, u)∂pk
but we know that
∂e(p, u)∂pk
= hk (p, u) = xk (p, e(p, u)) = xk (p,w)
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Walrasian and Hicksian Demand
• Why is this useful?• Define the Slutsky Matrix by
Sl ,,k =∂xl (p,w)
∂pk+
∂xl (p,w)∂w
xk (p,w)
• The above theorem tells us that
S = DPh(p, u)
• And so S must be negatively semi definite, symmetric andS .p = 0
• Also note that S is observable (if you know the demandfunction)
• It turns out this result is if and only if: Demand isrationalizable if and only if the resulting Slutsky Matrix hasthe above properties
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Walrasian and Hicksian Demand
• It also helps us understand how demand changes as respondto own prices.
• We now need one more theorem
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Law of Compensated Demand
Theorem (The Law of Compensated Demand)Assume preferences are continuous, locally non satiated andstrictly convex, then for any p′, p′′
(p′′ − p′)(h(p′′, u)− h(p′, u) ≤ 0
Proof.As h minimizes expenditure we have
p′′h(p′′, u) ≤ p′′h(p′, u)
andp′h(p′′, u) ≥ p′h(p′, u)
Subtracting the two inequalities gives the result
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Law of Compensated Demand
• An immediate corollary is that the compensated priceelasticity of demand is non positive• An increase in the price of good l reduces the Hicksiandemand for good l
• Back to the Slutsky equation we l = k we have∂hl (p, u)
∂pl− ∂xl (p,w)
∂wxl (p,w) =
∂xl (p,w)∂pl
• Does ∂xl (p,w )∂pl
have to be negative?• No! Giffen Goods
• But this can only happen if the income effect∂xl (p,w)
∂wxl (p,w)
• Overwhelms the substitution effect∂hl (p, u)
∂pl