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IntroductionThe consumer problem
Duality
Advanced Microeconomics
Consumer theory: optimization and duality
Jan Hagemejer
October 23, 2012
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
DualityIntroduction
Introduction
The plan:
1 The utility maximization
2 The expenditure minimization
3 Duality of the consumer problem
4 Some examples
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
The consumer problem
In general, the consumer problem can be state as:
choose the best bundle that the consumer can a�ord
or: choose a bundle x � y for x and any y 's in the budget set B,given prices p and wealth w
or: if we have a utility function representing �, maximize utilitysubject to the budget constraint (given by p and w).
the correspondence between prices p, wealth w and the consumerchosen bundle is the demand correspondence.
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
The utility maximization problem (UMP)
We will asume that the consumer has a rational, continuous andlocally nonsatiated preference relation.
u(x) is a continuous utility function representing consumerpreferences
the consumption set is X = RL+
The utility maximization problem is de�ned as:
Maxx≥0 u(x)subject to p · x ≤ w
If u(x) is well behaved, then this problem has a solution x(p,w)which is the so-called Walrasian demand correspondence
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
The UMP (for interior solution)
The UMP is usually set up as a Kuhn-Tucker sort of problem.
Let us write down the Lagrange function:
L = u(x) + λ(w − p · x)
The �rst order conditions for an interior solution:
∂u(x)∂x1− λp1 ≤ 0
...∂u(x)∂xL− λpL ≤ 0
p · x − w ≤ 0
which gives (if interior solution)∂u(x)∂x
l∂u(x)∂x
k
= plpk
for all l , k
and hence: MRSlk = plpk
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
Marginal rate of substitution
Let's totally di�erentiate u = u(x) for a zero change in utility:
du = 0 =L∑
l=1
∂u(x)
∂xldxl
Lets assume that dxl 6= 0 and dxk 6= 0 and all other dxn = 0:
0 =∂u(x)
∂xldxl +
∂u(x)
∂xkdxk
Rearrange:
− dxl
dxk=
∂u(x)∂xl∂u(x)∂xk
= MRSlk
So at the optimal choice the ratio in which the consumer is willingto give away l for k is equal to the ratio of prices.
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
The UMP
The KT procedure says that either λ = 0 or the budget constraint isbinding (which is usually the case).
Also, we might have xl = 0 and it that case the relevant FOC is
satisi�ed with inequality, i.e. ∂u(x)∂xl
< λpL (corner solution)
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
The UMP without corner solutions
Example: Cobb-Douglas utility. The general case:
u(x) =L∏
i=1
xαi
i
with∑
i αi = 1.
The two good case:
u(x) = xα11xα22
with α1 + α2 = 1.
The UMP:max
x1,x2∈Xu(x) subject to p1x1 + p2x2 ≤ w
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
The Cobb Douglas continued
L = u(x) + λ0(w −L∑
l=1
plxl)
Why we do not need to care about corner solutions (xi = 0)? Why wouldthe only constraint be always binding (equality constraint)?
In the two good case:
L = xα11xα22
+ λ0(w −L∑
l=1
plxl)
Solution procedure.... (in class).
Solution of the problem (Walrasian demand function):
x1 = α1w
p1and x2 = α2
w
p2.
It also implies constant expenditure shares equal to αi .Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
The UMP with corner solutions
If we suspect there may be corner solutions xl = 0 for some l , then weneed to set up the full Kuhn-Tucker problem with:
The budget constraint∑l
l=1plxl − w with the Lagrange multiplier
λ0
L inequality constraints xl ≥ 0 with the L Lagrange multipliers λl
L = u(x) + λ0(w −L∑
l=1
plxl) +L∑
l=1
λl(xl)
Then for the inequality constraints it is either (λl = 0 and xl > 0) or(λl > 0 and xl = 0). You have to check all the combinations!
Example: For the utility function u(x) =∑L
l=1alxl , l = 2, �nd the
demand function.
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
The UMP with corner solutions
L = a1x1 + a2x2 + λ0(w −l∑
l=1
plxl) +l∑
l=1
λl(xl)
FOC's are:[x1] a1 − λ0p1 + λ1 = 0[x2] a2 − λ0p2 + λ2 = 0[λ0] p1x1 + p2x2 − w ≤ 0,[λ1] x1 ≥ 0, λ1 ≥ 0, λ1x1 = 0[λ2] x2 ≥ 0, λ2 ≥ 0, λ2x2 = 0
λ ≥ 0, λ0(w − p1x1 − p2x2) = 0,
Case 1: λ0 > 0, and all λl = 0, therefore all xl > 0 (interior solution)
from �rst 2 FOCs we have: a1p1
= λ0 and a2p2
= λ0 → a1/p1 = a2/p2 orp1p2
= a1a2
or better a1p1
= a2p2
(the expenditures on one unit of MU are
equal).
Only at that price ratio demand is a correspondence: p1x1 + p2x2 = w .All other cases are corner solutions.
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
The UMP with corner solutions
Case 2:
λ0 > 0, and λ1 > 0, λ2 = 0 therefore x1 = 0 and x2 > 0 (corner solution)
The FOC's become:
p2x2 = w and therefore x2 = wp2.
a2 − λ0p2 = 0 and a1 − λ0p1 + λ1 = 0⇒ a1p1
= λ0 − λ1p1< λ0 = a2
p2, so
a1p1< a2
p2
Case 3:
λ0 > 0, and λ1 = 0, λ2 > 0 therefore x1 > 0 and x2 = 0 (corner solution)
The FOC's become:
p1x1 = w and therefore x1 = wp1.
a1 − λ0p1 = 0 and a2 − λ0p2 + λ2 = 0⇒ a2p2
= λ0 − λ2p2< λ0 = a1
p1, so
a2p2< a1
p1
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
The demand correspondence
In our problem the �nal demand is:
x(p) =
x1 = w
p1, x2 = 0. if a1
p1> a2
p2
x1, x2 : p1x1 + p2x2 = w if p1p2
= a1a2
x2 = wp2, x1 = 0 if a1
p1< a2
p2
As long as the bang for the buck is equal, we have the interior solution,otherwise only corner solutions.
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
Walrasian demand correspondence
The Walrasian demand correspondence x(p,w) assigns a set of chosenconsumption bundles for each price-wealth pair (p,w)
It can be multi-valued. If single valued we call it a demand function
Under the conditions of continuity and representation of u(x) theWalrasian demand correspondence possesses the following properties:
1 Homogeneity of degree zero in (p,w)2 Walras law: p · x = w (the budget constraint is binding)3 Convexity/uniqueness: if � is convex, so that u(·) is quasiconcave,
then x(p,w) is a convex set.
if � is strictly convex, so that u(·) is strictly quasiconcave, then
x(p,w) has just one element.
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
Walrasian demand correspondence
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
Properties of Walrasian demand
Wealth e�ects given the vector of prices p on the demand for good
l , partial derivative: ∂xl (p,w)∂w .
In matrix notation:
Dwx(p,w) =
∂x1(p,w)∂w...
∂xl (p,w)∂w...
∂xL(p,w)∂w
∂xl (p,w)∂w > 0, good is normal, if all > 0 then demand is normal
∂xl (p,w)∂w < 0, good is inferior.
Demand as a function of wealth x(p̄,w), Engel function
Wealth expansion path: Ep = {x(p̄,w) : w > 0}Income elasticity of demand: εw = ∂x(p,w)
∂ww
x(p,w) , necessity <1,
luxury >1
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
Wealth e�ects
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
Price e�ects
We can measure the e�ects of prices on the demand for goods.
The price e�ect is de�ned as:∂xl (p,w)∂pl
and usually < 0. If > 0 then
so-called Gi�en good.
In matrix notation
Dpx(p,w) =
∂x1(p,w)∂p1
∂x1(p,w)∂pL
.... . .
∂xL(p,w)∂p1
. . . ∂xL(p,w)∂pL
In that context we can de�ne the own- and cross-price elasticity of
demand ∂xl (p,w)∂pl
plxl (p,w) and
∂xl (p,w)∂pk
plxk (p,w) where k 6= l
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
Demand for good as a function of own price
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
O�er curve
OC - a locus of points demanded in over all possible values of one of theprices (in R2).
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
The Gi�en good
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
The indirect utility function
Once we have the optimal choice, x(p,w) we can plug it back intothe utility function.
u(x(p,w)) = v(p,w) is the indirect utility function
it says what the level of utility is, given prices and wealth and utilitymaximization
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
What for?
for example we can �nd thelevels of prices that generatesame utility given wealth
or the relationship wealthand utility at �xed prices
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
Expenditure minimization
We can go back and rede�ne our problem.
Instead of UMP, let us think of the consumer that has a desired levelof utility.
He wants to obtain this level of utility at the lowest possibleexpenditure.
the analogy of the production level and the cost minimization isobvious
The problem is set up as follows:
minx≥0 px subject to u(x) ≥ u
The solution is h(p, u), the demand for goods given prices andutility, the so called Hicksian demand (contrast it to x(p,w)).
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
The expenditure function
Once we have the solution to the problem, we can calculate the actualexpenditure:
e(p, u) =L∑
l=1
plh(p, u)
It is the �cost� of generating/obtaining a level of utility u given the set ofprices.
Why is it useful?
given the prices it determines a one-to-one relationship betweenmoney/expenditure and utility
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
Expenditure minimization
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The UMPWalrasian demandThe EMP
Hicksian demand and e�ects of a price change
∆wHicks is the the �Hicksian compensation�.
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
Duality of the consumer problem
The two problems can be related to each other:
1 x(p,w) = h(p, v(p,w))
2 x(p, e(p, u)) = h(p, u)
3 e(p, v(p,w)) = w
4 v(p, e(p, u)) = u
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
Some more nice properties
To recover Hicksian demand from expenditure function
If u(·) is a continuous utility function. For all p and u the Hicksiandemand h(p, u) is the derivative vector of the expenditure functionwith respect to prices.
h(p, u) = ∇pe(p, u)
To recover Walrasian demand from indirect utility function (Roy'sidentity - check for assumptions):
x(p,w) = − 1
∇wv(p,w)∇pv(p,w)
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
The Slutsky Equation
Suppose that u(·) is a continuous utility function representing a locallynonsatiated and strictly convex preference relation � de�ned on theconsumption set X = RL
+. Then for all (p,w), and u = v(p,w), we have
∂hl(p, u)
∂pk=∂xl(p,w)
∂pk+∂xl(p,w)
∂wxk(p,w)
and we can also rewrite it as:
∂hl(p, u)
∂pk︸ ︷︷ ︸Substitution e�ect
− ∂xl(p,w)
∂wxk(p,w)︸ ︷︷ ︸
Income e�ect
=∂xl(p,w)
∂pk
Jan Hagemejer Advanced Microeconomics
IntroductionThe consumer problem
Duality
Recap
Jan Hagemejer Advanced Microeconomics
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