Advanced Microeconomics Comparative statics and duality theory
Harald Wiese
University of Leipzig
Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 62 Part B. Household theory and theory of the …rm
1 The household optimum 2 Comparative statics and duality theory 3 Production theory 4 Cost minimization and pro…t maximization
Harald Wiese (University of Leipzig) Advanced Microeconomics 2 / 62 Comparative statics and duality theory Overview
1 The duality approach 2 Shephard’slemma 3 The Hicksian law of demand 4 Slutsky equations 5 Compensating and equivalent variations
Harald Wiese (University of Leipzig) Advanced Microeconomics 3 / 62 Maximization and minimization problem
Maximization problem: Minimization problem: Find the bundle that Find the bundle that maximizes the utility for a minimizes the expenditure given budget line. needed to achieve a given utility level.
x2 indifference curve with utility level U
A
C B budget line with income level m
x1
Harald Wiese (University of Leipzig) Advanced Microeconomics 4 / 62 The expenditure function and the Hicksian demand function I
Expenditure function:
e : R` R R, ! (p, U¯ ) e (p, U¯ ) := min px 7! x with U (x ) U¯ The solution to the minimization problem is called the Hicksian demand function:
χ : R` R R` , ! + (p, U¯ ) χ (p, U¯ ) := arg min px 7! x with U (x ) U¯
Harald Wiese (University of Leipzig) Advanced Microeconomics 5 / 62 The expenditure function and the Hicksian demand function II
Problem Express e in terms of χ and V in terms of the household optima!
Lemma For any α > 0:
χ (αp, U¯ ) = χ (p, U¯ ) and e (αp, U¯ ) = αe (p, U¯ ) .
Obvious?!
Harald Wiese (University of Leipzig) Advanced Microeconomics 6 / 62 The expenditure function and the Hicksian demand function III
Problem Determine the expenditure function and the Hicksian demand function for a 1 a the Cobb-Douglas utility function U (x1, x2) = x1 x2 with 0 < a < 1! Hint: We know that the indirect utility function V is given by:
V (p, m) = U (x (p, m)) a 1 a m m = a (1 a) p1 p2 a 1 a a 1 a = m. p p 1 2
Harald Wiese (University of Leipzig) Advanced Microeconomics 7 / 62 Hicksian demand and the expenditure function
utility expenditure maximization minimization
objective function utility expenditure
parameters prices p, income m prices p, utility U¯ notation for x (p, m) χ (p, U¯ ) best bundle(s) name of Marshallian Hicksian demand function value of V (p, m) e (p, U¯ ) objective function = U (x (p, m)) = p χ (p, U¯ )
Harald Wiese (University of Leipzig) Advanced Microeconomics 8 / 62 Applying the Lagrange method (recipe)
` ¯ L (x, µ) = ∑ pg xg + µ [U U (x)] g =1 with: U – strictly quasi-concave and strictly monotonic utility function; prices p >> 0; µ > 0 – Lagrange multiplier - translates a utility surplus (in case of U (x) > U¯ ) into expenditure reduction Increasing consumption has a positive direct e¤ect on expenditure, but a negative indirect e¤ect via a utility surplus (scope for expenditure reduction) and µ
Harald Wiese (University of Leipzig) Advanced Microeconomics 9 / 62 Applying the Lagrange method (recipe)
Di¤erentiate L with respect to xg :
∂L (x1, x2, ..., µ) ∂U (x1, x2, ..., x`) ! ∂U (x1, x2, ..., x`) ! pg = pg µ = 0 or = ∂xg ∂xg ∂xg µ and hence, for two goods g and k
∂U (x1,x2,...,x`) x p ∂ g =! g or MRS =! MOC U (x ,x ,...,x ) ∂ 1 2 ` pk ∂xk
Note also: ∂L(x,µ) = U¯ U (x) =! 0 ∂µ
Harald Wiese (University of Leipzig) Advanced Microeconomics 10 / 62 Applying the Lagrange method Comparing the Lagrange multipliers
λ – the shadow price for utility maximization (translates additional ∂V income m into higher utility: λ = ∂m ); µ – the shadow price for expenditure minimization (translates ¯ additional utility U¯ into higher expenditure: = ∂e(p,U ) ). µ ∂U¯ We note without proof 1 µ = λ
Harald Wiese (University of Leipzig) Advanced Microeconomics 11 / 62 The duality theorem here, duality does work
x2 indifference curve with utility level U At the budget line through point C A (budget m), the household optimum C is at point B and B budget line with income level m the utility is U (B) = V (p, m) . The expenditure
x1 needed to obtain B’sutility level or A’sutility level is equal to m :
e (p, V (p, m)) = m. Harald Wiese (University of Leipzig) Advanced Microeconomics 12 / 62 The duality theorem here, duality does work
x2 indifference curve with utility level U The minimal expenditure for the A indi¤erence curve passing through A C (utility level U¯ ) is B budget line with income level m denoted by e (p, U¯ ) and achieved by bundle B.
x1 With income e (p, U¯ ) (at point C), the highest achievable utility is V (p, e (p, U¯ )) = U¯ . Harald Wiese (University of Leipzig) Advanced Microeconomics 13 / 62 The duality theorem Conditions for duality
Theorem ` Let U : R+ R be a continuous utility function that obeys local nonsatiation! and let p >> 0 be a price vector. ) If x (p, m) is the household optimum for m > 0:
χ (p, V (p, m)) = x (p, m)
e (p, V (p, m)) = m. If χ (p, U¯ ) is the expenditure-minimizing bundle for U¯ > U (0):
x (p, e (p, U¯ )) = χ (p, U¯ )
V (p, e (p, U¯ )) = U¯ .
Harald Wiese (University of Leipzig) Advanced Microeconomics 14 / 62 The duality theorem here, duality does not work
x2 At (p, m) , the household optimum is at the bliss point with V (p, m) = 9. A B 9 The expenditure 7 needed to obtain 6 this utility level is smaller than m :
x1 e (p, V (p, m)) < m. Here, local nonsatiation is violated.
Harald Wiese (University of Leipzig) Advanced Microeconomics 15 / 62 Main results
Theorem Consider a household with a continuous utility function U, Hicksian demand function χ and expenditure function e. Shephard’slemma: The price increase of good g by one small unit increases the expenditure necessary to uphold the utility level by χg . Roy’sidentity: A price increase of good g by one small unit decreases the budget available for the other goods by χg and indirect utility by ∂V the product of the marginal utility of income ∂m and χg . Hicksian law of demand: If the price of a good g increases, the Hicksian demand χg does not increase. ∂χ (p,U¯ ) ¯ The Hicksian cross demands are symmetric: g = ∂χk (p,U ) . ∂pk ∂pg Slutsky equations see next slide
Harald Wiese (University of Leipzig) Advanced Microeconomics 16 / 62 Main results
Theorem Money-budget Slutsky equation:
∂xg ∂χg ∂xg = χg . ∂pg ∂pg ∂m Endowment Slutsky equation:
endowment money ∂xg ∂χg ∂xg = + ωg χg . ∂pg ∂pg ∂m
Harald Wiese (University of Leipzig) Advanced Microeconomics 17 / 62 Shephard’slemma Overview
1 The duality approach 2 Shephard’slemma 3 The Hicksian law of demand 4 Slutsky equations 5 Compensating and equivalent variations
Harald Wiese (University of Leipzig) Advanced Microeconomics 18 / 62 Shephard’slemma
Assume a price increase for a good g by one small unit. To achieve the same utility level, expenditure must be increased by at most ∂e χg ∂pg Shephard’sLemma (see manuscript):
∂e = χg ∂pg
Harald Wiese (University of Leipzig) Advanced Microeconomics 19 / 62 Roy’sidentity
Duality equation: U¯ = V (p, e (p, U¯ )) .
Di¤erentiating with respect to pg :
∂V ∂V ∂e 0 = + ∂pg ∂m ∂pg ∂V ∂V = + χg (Shephard’slemma) ∂pg ∂m
Roy’sidentity: ∂V = ∂V χ . ∂pg ∂m g
Harald Wiese (University of Leipzig) Advanced Microeconomics 20 / 62 The Hicksian law of demand Overview
1 The duality approach 2 Shephard’slemma 3 The Hicksian law of demand 4 Slutsky equations 5 Compensating and equivalent variations
Harald Wiese (University of Leipzig) Advanced Microeconomics 21 / 62 Compensated (Hicksian) law of demand
Assume: ` p and p0 – price vectors from R ; ` ` χ (p, U¯ ) R+ and χ (p0, U¯ ) R+ – expenditure-minimizing bundles necessary2 to achieve a utility of2 at least U¯ .
p0 χ U¯ , p0 p0 χ (U¯ , p) . expenditure- expenditure- minimizing| {z } minimizing| {z } bundle at p0 bundle at p
... (see the manuscript) If the price of one good increases, the Hicksian demand for that good ) cannot increase: ∂χg 0. ∂pg
Harald Wiese (University of Leipzig) Advanced Microeconomics 22 / 62 Concavity and the Hesse matrix concavity
De…nition Let f : M R be a function on a convex domain M R`. ! ) f is concave if
f (kx + (1 k) y) kf (x) + (1 k) f (y) for all x, y M and for all k [0, 1] (for – convex). 2 2 f is strictly concave if
f (kx + (1 k) y) > kf (x) + (1 k) f (y) holds for all x, y M with x = y and for all k (0, 1) (for < – strictly convex). 2 6 2
Harald Wiese (University of Leipzig) Advanced Microeconomics 23 / 62 Hesse matrix
De…nition Let f : R` R be a function. ! The second-order partial derivative of f with respect to xi and xj (if it exists) is given by ∂ ∂f (x ) 2 ∂xi ∂ f (x) fij (x) := = . ∂xj ∂xi ∂xj If all the second-order partial derivatives exist, the Hesse matrix of f is given by f11 (x) f12 (x) f1` (x) f21 (x) f22 (x) Hf (x) = 0 1 . B f (x) f (x) f (x) C B `1 n2 `` C @ A Problem Determine the Hesse matrix for f (x, y) = x2y + y 2. Harald Wiese (University of Leipzig) Advanced Microeconomics 24 / 62 Hesse matrix
Believe me (or check in the script) that Lemma (diagonal entries) If a function f : R` R is concave (strictly concave), the diagonal entries of its Hesse matrix! are non-positive (negative).
Remember: For f : R R, f is concave i¤ f 00 (x) 0 for all x R (chapter on von Neumann-Morgenstern! utility) 2 Lemma (symmetry) If all the second-order partial derivatives of f : R` R exist and are continuous, then !
fij (x) = fji (x) for all i, j = 1, ..., `
Lemma (expenditure fct. concave) The expenditure function is concave. Harald Wiese (University of Leipzig) Advanced Microeconomics 25 / 62 Hesse matrix of expenditure function
∂2e(p,U¯ ) ∂2e(p,U¯ ) ∂2e(p,U¯ ) 2 (∂p1 ) ∂p1 ∂p2 ∂p1 ∂p` 2 2 0 ∂ e(p,U¯ ) ∂ e(p,U¯ ) 1 ∂p ∂p 2 He (p, U¯ ) = 2 1 (∂p2 ) . B C B 2 2 2 C B ∂ e(p,U¯ ) ∂ e(p,U¯ ) ∂ e(p,U¯ ) C 2 B ∂p ∂p1 ∂p ∂p2 C B ` ` (∂p`) C @ A
Harald Wiese (University of Leipzig) Advanced Microeconomics 26 / 62 Compensated (Hicksian) law of demand
By Shephard’slemma: ∂e (p, U¯ ) = χg (p, U¯ ) ∂pg Forming the derivative of the Hicksian demand, we …nd
∂e(p,U¯ ) 2 ∂χ (p, U¯ ) ∂ ∂p ∂ e (p, U¯ ) g = g = ∂pk ∂pk ∂pg ∂pk with two interesting conclusions: 1. g = k Hicksian law of demand (by lemma “expenditure fct. concave” and lemma “diagonal entries”): ∂χ (p, U¯ ) g 0 ∂pg
Harald Wiese (University of Leipzig) Advanced Microeconomics 27 / 62 Substitutes and complements (the Hicksian de…nition)
2. g = k : If6 the o¤-diagonal entries in the expenditure function’sHesse matrix are continuous, lemma “symmetry” implies ∂χ (p, U¯ ) ∂χ (p, U¯ ) g = k . ∂pk ∂pg
De…nition Goods g and k are substitutes if ∂χ (p, U¯ ) ∂χ (p, U¯ ) g = k 0; ∂pk ∂pg complements if
∂χ (p, U¯ ) ∂χ (p, U¯ ) g = k 0. ∂pk ∂pg
Harald Wiese (University of Leipzig) Advanced Microeconomics 28 / 62 Every good has at least one substitute.
For all α > 0 : χg (αp, U¯ ) = χg (p, U¯ ) . Di¤erentiating with the adding rule (chapter on preferences) yields
` ∂χg (αp, U¯ ) ∂χg (αp, U¯ ) ∂χg (p, U¯ ) ∑ pk = = = 0 k=1 ∂pk ∂α ∂α
) ` ∂χg (αp, U¯ ) ∂χg (αp, U¯ ) ∑ pk = pg 0. k=1, ∂pk ∂pg k =g 6 0 ) | {z } Lemma Assume ` 2 and p >> 0. Every good has at least one substitute.
Harald Wiese (University of Leipzig) Advanced Microeconomics 29 / 62 Slutsky equations Overview
1 The duality approach 2 Shephard’slemma 3 The Hicksian law of demand 4 Slutsky equations 5 Compensating and equivalent variations
Harald Wiese (University of Leipzig) Advanced Microeconomics 30 / 62 Three e¤ects of a price increase
∂χ (p, U¯ ) ∂x (p, m) 1 0 and 1 0 ∂p1 ! ∂p1 ?
1 Substitution e¤ect or opportunity-cost e¤ect: p1 " p1/p2 ) " x1 and x2 ) # " 2 Consumption-income e¤ect: p1 " overall consumption possibilities decrease ) x1 if 1 is a normal good ) # 3 Endowment-income e¤ect: p1 " value of endowment increases ) x1 if 1 is a normal good ) "
Harald Wiese (University of Leipzig) Advanced Microeconomics 31 / 62 Two di¤erent substitution e¤ects De…nitions
In response to a price change, there are two di¤erent ways to keep real income constant: Old-household-optimum substitution e¤ect Old-utility-level substitution e¤ect
x2 substitution x2 substitution budget line budget line (old bundle (old utility level purchasable) achievable)
new new budget old budget budget old budget line line line line
x1 x1
Harald Wiese (University of Leipzig) Advanced Microeconomics 32 / 62 Two di¤erent substitution e¤ects Bahncard 50
Example G
Two goods: train substitution budget line rides T and other (old utility level willingness achievable) goods G to pay
pT = 0.2 (per budget line with free kilometer), budget bahncard line pG = 1. without bahncard “Bahncard 50”: T pT reduced to 0.1. Willingness to pay for the “Bahncard 50”?
Harald Wiese (University of Leipzig) Advanced Microeconomics 33 / 62 Two di¤erent substitution e¤ects Tax and rebate
Example G tax•and•rebate You smoke 10 cigarettes budget line per day. The budget government is concerned deficit in terms of about your health. good G Quantity tax of 10 cents, but old budget line
rebate of 1 Euro Fags per day. Budget de…cit in terms of the other goods?
Harald Wiese (University of Leipzig) Advanced Microeconomics 34 / 62 The Slutsky equation for the money budget Derivation
Duality equation: χg (p, U¯ ) = xg (p, e (p, U¯ )) Di¤erentiate with respect to pk
∂χ ∂x ∂x ∂e g = g + g ∂pk ∂pk ∂m ∂pk ∂xg ∂xg = + χk (Shephard’sLemma) ∂pk ∂m The Slutsky equation (g = k):
∂xg ∂χg ∂xg = χg . ∂pg ∂pg ∂m 0 > 0 Hicksian|{z} for normal|{z} goods law of demand
Harald Wiese (University of Leipzig) Advanced Microeconomics 35 / 62 The Slutsky equation for the money budget Implications
The Slutsky equation:
∂xg ∂χg ∂xg = χg . ∂pg ∂pg ∂m 0 > 0 Hicksian|{z} for normal|{z} goods law of demand
g normal g ordinary ) g normal e¤ect of a price increase stronger on Marshallian demand than on Hicksian) demand g inferior income e¤ect may outweigh substitution e¤ect — > Gi¤en) good
Harald Wiese (University of Leipzig) Advanced Microeconomics 36 / 62 The Slutsky equation for the money budget
Assume (pˆg , U¯ ) .
By duality, χg (pˆg , U¯ ) = xg (pˆg , e (pˆg , U¯ )) . g normal Hicksian demand curves steeper than Marshallian demand curves)
pg Hicksian demand curves
cg (pg ,U )
Marshallian demand curve pg xg (pg ,e(pg ,U ))
cg (pg ,U ) xg , cg
= xg (pg ,e(pg ,U ))
Harald Wiese (University of Leipzig) Advanced Microeconomics 37 / 62 The Slutsky equation for the endowment budget Derivation
endowment ∂xg (p, ω) ∂pk money ∂xg (p, p ω) = ∂pk money money ∂xg ∂xg ∂ (p ω) = + ∂pk ∂m ∂pk money money ∂xg ∂xg = + ωk (de…nition of dot product) ∂pk ∂m money money ∂χg ∂xg ∂xg = χk + ωk (money-budget Slutsky equation) ∂pk ∂m ∂m money ∂χg ∂xg = + (ωk χk ) . ∂pk ∂m
Harald Wiese (University of Leipzig) Advanced Microeconomics 38 / 62 The Slutsky equation for the endowment budget Implications
The Slutsky equation:
endowment money ∂xg ∂χg ∂xg = + ωg χg . ∂pg ∂pg ∂m 0 > 0 < 0 for| net{z demander} |{z} for| a{z normal} good g
g normal and household net demander g ordinary ) g normal and household net supplier g may be non-ordinary )
Harald Wiese (University of Leipzig) Advanced Microeconomics 39 / 62 The Slutsky equation for the endowment budget Application: consumption today versus consumption tomorrow
The intertemporal budget equation in future value terms:
(1 + r) x1 + x2 = (1 + r) ω1 + ω2.
The Slutsky equation:
endowment money ∂x1 ∂χ1 ∂x1 = + (ω1 χ ). ∂ (1 + r) ∂ (1 + r) ∂m 1 0 > 0 > 0 |for lender{z } | {z } for normal| {z good} …rst-period consumption
Harald Wiese (University of Leipzig) Advanced Microeconomics 40 / 62 The Slutsky equation for the endowment budget Application: leisure versus consumption
The budget equation:
wxR + pxC = w24 + pωC .
The Slutsky equation:
money ∂xendowment ∂χ ∂x R = R + R (24 χ ) . ∂w ∂w ∂m R 0 0 > 0 by de…nition |{z} for| normal{z } | {z } good recreation
Thus, if the wage rate increases, it may well happen that the household works ...
Harald Wiese (University of Leipzig) Advanced Microeconomics 41 / 62 The Slutsky equation for the endowment budget Application: contingent consumption
The budget equation: γ γ x1 + x2 = (A D) + A 1 γ 1 γ with γK – payment to the insurance if K is to be paid to the insuree in case of damage D. The Slutsky equation for consumption in case of damage:
endowment money ∂x1 ∂χ1 ∂x1 γ = γ + (A D χ1) . ∂ 1 γ ∂ 1 γ ∂m 0 0 > 0 in case of for| normal{z } | {z } | {z } good consumption a nonnegative insurance in case of damage
Harald Wiese (University of Leipzig) Advanced Microeconomics 42 / 62 Compensating and equivalent variations Overview
1 The duality approach 2 Shephard’slemma 3 The Hicksian law of demand 4 Slutsky equations 5 Compensating and equivalent variations
Harald Wiese (University of Leipzig) Advanced Microeconomics 43 / 62 Compensating and equivalent variations
De…nition A variation is equivalent to an event, if both (the event or the variation) lead to the same indi¤erence curve EV (event) ; ! A variation is compensating if it restores the individual to its old indi¤erence curve (prior to the event) CV (event) . !
Harald Wiese (University of Leipzig) Advanced Microeconomics 44 / 62 Compensating and equivalent variations
Equivalent Compensating variation variation
in lieu of an event because of an event
monetary variation monetary variation is equivalent compensates for event (i.e., achieving the same utility) (i.e., holding utility constant)
Harald Wiese (University of Leipzig) Advanced Microeconomics 45 / 62 Compensating and equivalent variations The case of good air quality
Change of air quality:
income
m3 EV (a ® b) = CV (b ® a) a b m2 EV (b ® a) = CV (a ® b) m1 I 2
I 1
q1 q 2 air quality
Harald Wiese (University of Leipzig) Advanced Microeconomics 46 / 62 Compensating and equivalent variations
Compensation money if some amount of money is given to the individual: ! CV (degr.) – the compensation money for the degradation of the air quality. Willingness to pay if money is taken from the individual. EV (degr.) – the willingness! to pay for the prevention of the degradation. If the variation turns out to be negative, exchange EV for EV or EV for EV (similarly for CV ).
Harald Wiese (University of Leipzig) Advanced Microeconomics 47 / 62 Compensating or equivalent variation?
Example Consumer’scompensating variation: A consumer asks himself how much he is prepared to pay for a good. Consumer’sequivalent variation – the compensation payment for not getting the good. You go into a shop and ask for compensation for not taking (stealing?) the good. Producer’scompensating variation – the compensation money he gets for selling a good. Producer’sequivalent variation: The producer asks himself how much he would be willing to pay if the good were not taken away from him.
Harald Wiese (University of Leipzig) Advanced Microeconomics 48 / 62 Price changes
The willingness to pay for the price decrease of good g:
CV ph pl = EV pl ph . g ! g g ! g The compensation money for the price increase of good g:
EV ph pl = CV pl ph . g ! g g ! g CV ph pl < EV ph pl (for normal goods, see below); 1 ! 1 1 ! 1 cv and ev – if we are not sure whether a change is good or bad. Lemma Consider the event of a price change from pold to pnew . Then:
Uold : = V pold , m = V (pnew , m + cv) , CV = cv and j j Unew : = V (pnew , m) = V pold , m + ev , EV = ev . j j Harald Wiese (University of Leipzig) Advanced Microeconomics 49 / 62 Price changes Exercise
Problem Tell the sign of cv and ev for a price increase of all goods.
Harald Wiese (University of Leipzig) Advanced Microeconomics 50 / 62 Price changes
Price increase of good 1:
x2
new CV CV = p1 × CV1 2 C = p2 × CV 2
A O
x CV 1 1
Harald Wiese (University of Leipzig) Advanced Microeconomics 51 / 62 Price changes
Example a 1 a Cobb-Douglas utility function: u (x1, x2) = x1 x2 with (0 < a < 1) . h l h By a price decrease from p1 to p1 < p1 (for example, Bahncard 50)
a 1 a m m a h (1 a) p p2 1 utility at the old, high price a 1 a | m + cv{z ph pl } m + cv ph pl = a 1 ! 1 (1 a) 1 ! 1 . l p p1 ! 2 ! utility at the new, lower price and compensating variation
| {z l a } h l p1 cv p1 p1 = 1 h m < 0. ! p1 !
Harald Wiese (University of Leipzig) Advanced Microeconomics 52 / 62 Price changes Exercises
Problem Determine the equivalent variation for a price decrease in case of Cobb-Douglas utility preferences.
Problem Determine the compensating variation and the equivalent variation for the h l h price decrease from p1 to p1 < p1 and the quasi-linear utility function given by u (x1, x2) = ln x1 + x2 (x1 > 0)!
Assume m > 1! Hint: the household optimum is x (m, p) = p2 , m 1 . p2 p1 p2
Harald Wiese (University of Leipzig) Advanced Microeconomics 53 / 62 Applying duality Implicit de…nition of compensating variation
Implicit de…nition: Uold := V pold , m = V (pnew , m + cv) Duality equation e (p, V (p, m)) = m leads to e pold , V pold , m = m (1) e (pnew, V (pnew, m + cv)) = m + cv (2)
) cv = e (pnew , V (pnew , m + cv)) m (2) = e pnew , Uold e pold , Uold (1) and implicit de…nition The household is given, or is relieved of, the money necessary to uphold the old utility level.
Harald Wiese (University of Leipzig) Advanced Microeconomics 54 / 62 Applying duality Implicit de…nition of equivalent variation
Implicit de…nition: Unew := V (pnew , m) = V pold , m + ev Duality equation e (p, V (p, m)) = m leads to
e (pnew , V (pnew , m)) = m (1) e pold , V pold , m + ev = m + ev (2) )
ev = e pold , V pold , m + ev m (2) = e pold , Unew e (pnew , Unew ) (1) and implicit de…nition Assume pnew < pold . The equivalent variation is the amount of money necessary to increase the household’sincome from m = e (pnew , Unew ) to e pold , Unew .
Harald Wiese (University of Leipzig) Advanced Microeconomics 55 / 62 Variations for a price change and Hicksian demand Applying the fundamental theorem of calculus
ph h l g h cv pg pg = χ pg , V pg , m dpg l g ! pg Z by (if you want) cv ph pl = e pl , V ph , m e ph , V ph , m g ! g g g g g = e ph , V ph , m e pl , V ph , m g g g g h ph i h g = e pg , V p , m g l pg ph h g ∂e p, V pg , m = dpg (Fundamental Theorem) pl ∂p Z g g ph g h = χ pg , V pg , m dpg (Shephard’slemma). l g pg Z Harald Wiese (University of Leipzig) Advanced Microeconomics 56 / 62 Variations for a price change and Hicksian demand Applying the fundamental theorem of calculus
pg
h cg (pg ,V (pg ,m))
h pg
h l CV (pg ® pg ) l pg
cg
Harald Wiese (University of Leipzig) Advanced Microeconomics 57 / 62 Variations for a price change and Hicksian demand Comparisons
Theorem h l h Assume any good g and any price decrease from pg to pg < pg .
ph h l g h cv pg pg = χ pg , V pg , m dpg . l g ! pg Z If g is a normal good:
ph h l g l h CV pg pg xg (pg ) dpg CV pg pg . l ! pg ! Z (Hicksian) Marshallian (Hicksian) willingness to pay loss compensation | {z } willingness| {z to pay} | {z }
Harald Wiese (University of Leipzig) Advanced Microeconomics 58 / 62 Variations for a price change and Hicksian demand Comparisons for normal goods
pg h l loss compensation: c g (p g,V (p g, m)) c g (p g ,V (p g ,m)) l h Hicksian CV (pg ® p g ) demand = EV ph ® p l > 0 curves ( g g ) willingness to pay: h p h l g CV (p g ® p g ) l h = EV (p g ® p g )< 0
l Marshallian p g demand curve
xg (p g ,m)
xg , c g
Harald Wiese (University of Leipzig) Advanced Microeconomics 59 / 62 Variations for a price change and Hicksian demand Consumers’rent
De…nition proh The Hicksian consumer’srent at price pˆg < pg is given by
Hicks proh CR (pˆg ) : = CV p pˆg g ! proh pg proh = χg pg , V pg , m dpg . pˆg Z
Harald Wiese (University of Leipzig) Advanced Microeconomics 60 / 62 Further exercises I
Problem 1 Determine the expenditure functions and the Hicksian demand function for U (x1, x2) = min (x1, x2) and U (x1, x2) = 2x1 + x2. Can you con…rm the duality equations
χ (p, V (p, m)) = x (p, m) and x (p, e (p, U¯ )) = χ (p, U¯ )?
Harald Wiese (University of Leipzig) Advanced Microeconomics 61 / 62 Further exercises II
Problem 2 Derive the Hicksian demand functions and the expenditure functions of the two utility functions:
(a) U(x1, x2) = x1 x2, (b) U(x1, x2) = min (a x1, b x2) with a, b > 0. Problem 3 Verify Roy’sidentity for the utility function U(x1, x2) = x1 x2! Problem 4 Draw a …gure that shows the equivalent variation following a price increase.
Harald Wiese (University of Leipzig) Advanced Microeconomics 62 / 62