Demand Theory Utility Maximization Problem

• Consumer maximizes his utility level by selecting a bundle 푥 (where 푥 can be a vector) subject to his budget constraint: max 푢(푥) 푥≥0 s. t. 푝 ∙ 푥 ≤ 푤 • 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.

2 Utility Maximization Problem

• Existence: if 푝 ≫ 0 and 푤 > 0 (i.e., if 퐵푝,푤 is closed and bounded), and if 푢(∙) is continuous, then there exists at least one solution to the UMP. – If, in addition, preferences are strictly convex, then the solution to the UMP is unique. • We denote the solution of the UMP as the argmax of the UMP (the argument, 푥, that solves the optimization problem), and we denote it as 푥(푝, 푤). – 푥(푝, 푤) is the Walrasian demand correspondence, which 퐿 specifies a demand of every good in ℝ+ for every possible vector, 푝, and every possible wealth level, 푤.

3 Utility Maximization Problem

• Walrasian demand 푥(푝, 푤) at bundle A is optimal, as the consumer reaches a utility level of 푢2 by exhausting all his wealth. • Bundles B and C are not optimal, despite exhausting the consumer’s wealth. They yield a lower utility level 푢1, where 푢1 < 푢2. • Bundle D is unaffordable and, hence, it cannot be the argmax of the UMP given a wealth level of 푤.

4 Properties of Walrasian Demand

• If the utility function is continuous and preferences 퐿 satisfy LNS over the consumption set 푋 = ℝ+, then the Walrasian demand 푥(푝, 푤) satisfies: 1) Homogeneity of degree zero: 푥 푝, 푤 = 푥(훼푝, 훼푤) for all 푝, 푤, and for all 훼 > 0 That is, the budget set is unchanged! 퐿 퐿 푥 ∈ ℝ+: 푝 ∙ 푥 ≤ 푤 = 푥 ∈ ℝ+: 훼푝 ∙ 푥 ≤ 훼푤 Note that we don’t need any assumption on the preference relation to show this. We only rely on the budget set being affected.

5 Properties of Walrasian Demand

x2 – Note that the preference relation can be linear, and α w w homog(0) would α p = p 2 2 still hold. x (p,w), unaffected

x α w = w 1 α p1 p1

Advanced Microeconomic Theory 6 Properties of Walrasian Demand

2) Walras’ Law: 푝 ∙ 푥 ≤ 푤 for all 푥 = 푥(푝, 푤) It follows from LNS: if the consumer selects a Walrasian demand 푥 ∈ 푥(푝, 푤), where 푝 ∙ 푥 < 푤, then it means we can still find other bundle 푦, which is ε–close to 푥, where consumer can improve his utility level. If the bundle the consumer chooses lies on the budget line, i.e., 푝 ∙ 푥′ = 푤, we could then identify bundles that are strictly preferred to 푥′, but these bundles would be unaffordable to the consumer.

7 Properties of Walrasian Demand

x2 – For 푥 ∈ 푥(푝, 푤), there is a bundle 푦, ε–close to 푥, such that 푦 ≻ 푥. Then, 푥 ∉ 푥(푝, 푤).

x

x1

8 Properties of Walrasian Demand

3) Convexity/Uniqueness: a) If the preferences are convex, then the Walrasian demand correspondence 푥(푝, 푤) 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 푥(푝, 푤) contains a single element.

9 Properties of Walrasian Demand

Convex preferences Strictly convex preferences

Unique x ( p , w ) x x(,) p w

10 UMP: Necessary Condition

max 푢 푥 s. t. 푝 ∙ 푥 ≤ 푤 푥≥0 • We solve it using Kuhn-Tucker conditions over the Lagrangian 퐿 = 푢 푥 + 휆(푤 − 푝 ∙ 푥), ∗ 휕퐿 휕푢(푥 ) ∗ = − 휆푝푘 ≤ 0 for all 푘, = 0 if 푥푘 > 0 휕푥푘 휕푥푘 휕퐿 = 푤 − 푝 ∙ 푥∗ = 0 휕휆 휕푢(푥∗) • That is in the interior optimum, = 휆푝푘 for every 휕푥푘 good 푘, which implies 휕푢(푥∗) 휕푢(푥∗) 휕푢(푥∗) 휕푥푙 푝푙 푝푙 휕푥푙 휕푥푘 휕푢(푥∗) = ⇔ 푀푅푆 푙,푘 = ⇔ = 푝푘 푝푘 푝푙 푝푘 휕푥푘 11 UMP: Sufficient Condition

• When are Kuhn-Tucker (necessary) conditions, also sufficient? – That is, when can we guarantee that 푥(푝, 푤) is the max of the UMP and not the min? • Answer: if 푢(∙) is quasiconcave and monotone, and 퐿 has Hessian 훻푢(푥) ≠ 0 for 푥 ∈ ℝ+, then the Kuhn- Tucker FOCs are also sufficient.

12 UMP: Sufficient Condition

• Kuhn-Tucker

conditions are x2 sufficient for a max if:

1) 푢(푥) is quasiconcave, w p2 i.e., convex upper contour set (UCS). x * ∈ x (p,w) 2) 푢(푥) is monotone. Ind. Curve 3) 훻푢(푥) ≠ 0 푥 ∈ ℝ퐿 for +. p1 slope = - – If 훻푢 푥 = 0 for some 푥, p2 then we would be at the

“top of the mountain” (i.e., w x1 blissing point), which p1 violates both LNS and monotonicity. 13 UMP: Violations of Sufficient Condition

1) 풖(∙) is non-monotone:

– The consumer chooses bundle A (at a corner) since it yields the highest utility level given his budget constraint.

– At point A, however, the tangency condition 푝1 푀푅푆 1,2 = does not 푝2 hold.

14 UMP: Violations of Sufficient Condition

2) 풖(∙) is not quasiconcave: – The upper contour sets (UCS) are not convex. x2 푝1 – 푀푅푆 1,2 = is not a 푝2 sufficient condition for a A max. C – A point of tangency (C) B gives a lower utility level Budget line

than a point of non- x1 tangency (B). – True maximum is at point A. 15 UMP: Corner Solution

• Analyzing differential changes in 푥푙 and 푥푙, that keep individual’s utility unchanged, 푑푢 = 0,

푑푢(푥) 푑푢(푥) 푑푥푙 + 푑푥푘 = 0 (total diff.) 푑푥푙 푑푥푘 • Rearranging, 푑푢 푥 푑푥푘 푑푥푙 = − = −푀푅푆푙,푘 푑푥푙 푑푢 푥 푑푥푘 푑푢 푥∗ 푑푢 푥∗ 푝푙 푑푥푙 푑푥푘 • Corner Solution: 푀푅푆푙,푘 > , or alternatively, > , i.e., 푝푘 푝푙 푝푘 the consumer prefers to consume more of good 푙.

16 UMP: Corner Solution

• In the FOCs, this implies: 휕푢(푥∗) a) ≤ 휆푝 for the whose consumption is 휕푥 푘 푘 ∗ zero, 푥푘= 0, and 휕푢(푥∗) b) = 휆푝 for the good whose consumption is 휕푥 푙 푙 ∗ positive, 푥푙 > 0. • Intuition: the marginal utility per dollar spent on good 푙 is still larger than that on good 푘. 휕푢(푥∗) 휕푢(푥∗) 휕푥푙 = 휆 ≥ 휕푥푘 푝푙 푝푘

17 UMP: Corner Solution

• Consumer seeks to consume good 1 alone. x2 • At the corner solution, the is steeper than the budget line, i.e., w p1 p2 slope of the B.L. = - 푝1 푀푈1 푀푈2 p2 푀푅푆1,2 > or > 푝2 푝1 푝2

• Intuitively, the consumer slope of the I.C. = MRS1,2 would like to consume more of good 1, even after spending his entire wealth on good 1 alone. w x1 p1

18 UMP: Lagrange Multiplier

• 휆 is referred to as the “marginal values of relaxing the x2 constraint” in the UMP (a.k.a. “shadow price of wealth”).

• If we provide more wealth to w 1 p2 the consumer, he is capable of w 0 reaching a higher indifference p2 2 1 curve and, as a consequence, {x +: u(x) = u } obtaining a higher utility level. {x 2: u(x) = u 0} – We want to measure the + 0 1 change in utility resulting from w w x1 p p a marginal increase in wealth. 1 1

Advanced Microeconomic Theory 19 UMP: Lagrange Multiplier

• Analyze the change in utility from change in wealth using chain rule (since 푢 푥 does not contain wealth), 훻푢 푥(푝, 푤) ∙ 퐷푤푥(푝, 푤) • Substituting 훻푢 푥(푝, 푤) = 휆푝 (in interior solutions), 휆푝 ∙ 퐷푤푥(푝, 푤)

20 UMP: Lagrange Multiplier

• From Walras’ Law, 푝 ∙ 푥 푝, 푤 = 푤, the change in expenditure from an increase in wealth is given by 퐷푤 푝 ∙ 푥 푝, 푤 = 푝 ∙ 퐷푤푥 푝, 푤 = 1 • Hence, 훻푢 푥(푝, 푤) ∙ 퐷푤푥 푝, 푤 = 휆 푝 ∙ 퐷푤푥 푝, 푤 = 휆 1 • Intuition: If 휆 = 5, then a $1 increase in wealth implies an increase in 5 units of utility.

21 Walrasian Demand

• We found the Walrasian demand function, 푥 푝, 푤 , as the solution to the UMP. • Properties of 푥 푝, 푤 : 1) Homog(0), for any 푝, 푤, 훼 > 0, 푥 훼푝, 훼푤 = 푥 푝, 푤 (Budget set is unaffected by changes in 푝 and 푤 of the same size) 2) Walras’ Law: for every 푝 ≫ 0, 푤 > 0 we have 푝 ∙ 푥 = 푤 for all 푥 ∈ 푥(푝, 푤) (This is because of LNS)

22 Walrasian Demand: Wealth Effects

• Normal vs. Inferior goods 휕푥 푝,푤 > 0 normal 휕푤 < inferior • Examples of inferior goods: – Two-buck chuck (a really cheap wine) – Walmart during the economic crisis

23 Walrasian Demand: Wealth Effects

• An increase in the wealth level produces an outward shift in the budget line.

휕푥 푝,푤 • 푥 is normal as 2 > 0, while 2 휕푤 휕푥 푝,푤 푥 is inferior as < 0. 1 휕푤 • Wealth expansion path: – connects the optimal consumption bundle for different levels of wealth – indicates how the consumption of a good changes as a consequence of changes in the wealth level

24 Walrasian Demand: Wealth Effects

• Engel curve depicts the consumption of a particular good in the horizontal axis and Wealth, w wealth on the vertical axis. Engel curve for a (up to income ŵ), but for all w > ŵ • The slope of the Engel curve is: – positive if the good is normal Engel curve of a normal – negative if the good is inferior good, for all w. ŵ • Engel curve can be positively slopped for low wealth levels and become negatively slopped x afterwards. 1

25 Walrasian Demand: Price Effects

• Own price effect: 휕푥푘 푝, 푤 < Usual 0 휕푝푘 > Giffen • Cross-price effect: 휕푥푘 푝, 푤 > Substitutes 0 휕푝푙 < Complements – Examples of Substitutes: two brands of mineral water, such as Aquafine vs. Poland Springs. – Examples of Complements: cars and gasoline.

26 Walrasian Demand: Price Effects

• Own price effect

Usual good Giffen good

27 Walrasian Demand: Price Effects

• Cross-price effect

Substitutes Complements

28 Indirect Utility Function

• The Walrasian demand function, 푥 푝, 푤 , is the solution to the UMP (i.e., argmax).

• What would be the utility function evaluated at the solution of the UMP, i.e., 푥 푝, 푤 ? – This is the indirect utility function (i.e., the highest utility level), 푣 푝, 푤 ∈ ℝ, associated with the UMP. – It is the “value function” of this optimization problem.

29 Properties of Indirect Utility Function

• If the utility function is continuous and preferences satisfy LNS over the consumption 퐿 set 푋 = ℝ+, then the indirect utility function 푣 푝, 푤 satisfies:

1) Homogenous of degree zero: Increasing 푝 and 푤 by a common factor 훼 > 0 does not modify the consumer’s optimal consumption bundle, 푥(푝, 푤), nor his maximal utility level, measured by 푣 푝, 푤 .

30 Properties of Indirect Utility Function

x2

v (p,w) = v (αp,α w)

α w = w α p2 p2 x (p,w) = x (αp,α w)

Bp,w = Bαp,αw

x α w = w 1 α p1 p1

31 Properties of Indirect Utility Function

2) Strictly increasing in 푤: 3) non-increasing (i.e., weakly decreasing) in 푝푘 푣 푝, 푤′ > 푣(푝, 푤) for 푤′ > 푤.

32 Properties of Indirect Utility Function

4) Quasiconvex: The set 푝, 푤 : 푣(푝, 푤) ≤ 푣ҧ is convex for any 푣ҧ. - Interpretation I: If (푝1, 푤1) ≿∗ (푝2, 푤2), then (푝1, 푤1) ≿∗ (휆푝1 + (1 − 휆)푝2, 휆푤1 + (1 − 휆)푤2); i.e., if 퐴 ≿∗ 퐵, then 퐴 ≿∗ 퐶.

W

w2

w1

P1

33 Properties of Indirect Utility Function

- Interpretation II: 푣(푝, 푤) is quasiconvex if the set of (푝, 푤) pairs for which 푣 푝, 푤 < 푣(푝∗, 푤∗) is convex.

W v(p,w) = v1

w*

v(p,w) < v(p*,w*)

p* P1

34 Properties of Indirect Utility Function

- Interpretation III: Using 푥1 and 푥2 in the axis, perform following steps: 1) When 퐵푝,푤, then 푥 푝, 푤 ′ ′ 2) When 퐵푝′,푤′, then 푥 푝 , 푤 3) Both 푥 푝, 푤 and 푥 푝′, 푤′ induce an indirect utility of 푣 푝, 푤 = 푣 푝′, 푤′ = 푢ത 4) Construct a linear combination of and wealth: 푝′′ = 훼푝 + (1 − 훼)푝′ ൠ 퐵 ′′ ′′ 푤′′ = 훼푤 + (1 − 훼)푤′ 푝 ,푤 5) Any solution to the UMP given 퐵푝′′,푤′′ must lie on a lower indifference curve (i.e., lower utility) 푣 푝′′, 푤′′ ≤ 푢ത

35 Properties of Indirect Utility Function

x2

x (p’,w’ )

Ind. Curve {x ∈ℝ2:u(x) = ū} x (p,w)

x1 Bp’’,w’’ Bp,w Bp’,w’

36 WARP and Walrasian Demand

• Relation between Walrasian demand 푥 푝, 푤 and WARP – How does the WARP restrict the set of optimal consumption bundles that the individual decision- maker can select when solving the UMP?

37 WARP and Walrasian Demand

• Take two different consumption bundles 푥 푝, 푤 and 푥 푝′, 푤′ , both being affordable 푝, 푤 , i.e., 푝 ∙ 푥 푝, 푤 ≤ 푤 and 푝 ∙ 푥 푝′, 푤′ ≤ 푤 • When prices and wealth are 푝, 푤 , the consumer chooses 푥 푝, 푤 despite 푥 푝′, 푤′ is also affordable. • Then he “reveals” a preference for 푥 푝, 푤 over 푥 푝′, 푤′ when both are affordable. • Hence, we should expect him to choose 푥 푝, 푤 over 푥 푝′, 푤′ when both are affordable. (Consistency) • Therefore, bundle 푥 푝, 푤 must not be affordable at 푝′, 푤′ because the consumer chooses 푥 푝′, 푤′ . That is, 푝′ ∙ 푥 푝, 푤 > 푤′.

38 WARP and Walrasian Demand

• In summary, Walrasian demand satisfies WARP, if, for two different consumption bundles, 푥 푝, 푤 ≠ 푥 푝′, 푤′ , 푝 ∙ 푥 푝′, 푤′ ≤ 푤 ⇒ 푝′ ∙ 푥 푝, 푤 > 푤′

• Intuition: if 푥 푝′, 푤′ is affordable under budget set 퐵푝,푤, then 푥 푝, 푤 cannot be affordable under 퐵푝′,푤′.

39 Checking for WARP

• A systematic procedure to check if Walrasian demand satisfies WARP: – Step 1: Check if bundles 푥 푝, 푤 and 푥 푝′, 푤′ are both affordable under 퐵푝,푤. . That is, graphically 푥 푝, 푤 and 푥 푝′, 푤′ have to lie on or below budget line 퐵푝,푤. . 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 “WARP is not violated”.

40 Checking for WARP

- Step 2: Check if bundles 푥 푝, 푤 is affordable under 퐵푝′,푤′. . That is, graphically 푥 푝, 푤 must lie on or below budge line 퐵푝′,푤′. . If step 2 is satisfied, then this Walrasian demand violates WARP. . Otherwise, the Walrasian demand satisfies WARP.

41 Checking for WARP: Example 1

• First, 푥 푝, 푤 and 푥 푝′, 푤′ are both x2 affordable under 퐵푝,푤.

• Second, 푥 푝, 푤 is not Bp,w affordable under w x (p,w) 퐵푝′,푤′. p 2

x (p ,w ) • Hence, WARP is Bp ,w satisfied! w x1 p1

42 Checking for WARP: Example 2

• The demand 푥 푝′, 푤′ under final prices and wealth is not affordable under initial prices and wealth, i.e., 푝 ∙ 푥 푝′, 푤′ > 푤. – The premise of WARP does not hold. – Violation of Step 1! – WARP is not violated.

43 Checking for WARP: Example 3

• The demand 푥 푝′, 푤′ under final prices and wealth is not affordable under initial prices and wealth, i.e., 푝 ∙ 푥 푝′, 푤′ > 푤. – The premise of WARP does not hold. – Violation of Step 1! – WARP is not violated.

44 Checking for WARP: Example 4

• The demand 푥 푝′, 푤′ under final prices and wealth is not affordable under initial prices and wealth, i.e., 푝 ∙ 푥 푝′, 푤′ > 푤. – The premise of WARP does not hold. – Violation of Step 1! – WARP is not violated.

45 Checking for WARP: Example 5

• First, 푥 푝, 푤 and 푥 푝′, 푤′ are both x2 affordable under 퐵푝,푤.

• Second, 푥 푝, 푤 is Bp,w 퐵 ′ ′ affordable under 푝 ,푤 , w ′ i.e., 푝 ∙ 푥 푝, 푤 < 푤′ p 2

x (p ,w ) • Hence, WARP is NOT x (p,w) Bp ,w satisfied! w x1 p1

46 Implications of WARP

• How do price changes affect the WARP predictions? • Assume a reduction in 푝1 – the consumer’s budget lines rotates (uncompensated price change)

47 Implications of WARP

• Adjust the consumer’s wealth level so that he can consume his initial demand 푥 푝, 푤 at the new prices. – shift the final budget line inwards until the point at which we reach the initial consumption bundle 푥 푝, 푤 (compensated price change)

48 Implications of WARP

• What is the wealth adjustment? 푤 = 푝 ∙ 푥 푝, 푤 under 푝, 푤 푤′ = 푝′ ∙ 푥 푝, 푤 under 푝′, 푤′

• Then, ∆푤 = ∆푝 ∙ 푥 푝, 푤 where ∆푤 = 푤′ − 푤 and ∆푝 = 푝′ − 푝.

• This is the Slutsky wealth compensation: – the increase (decrease) in wealth, measured by ∆푤, that we must provide to the consumer so that he can afford the same consumption bundle as before the price change, i.e., 푥 푝, 푤 .

49 Implications of WARP:

• Suppose that the Walrasian demand 푥 푝, 푤 satisfies homog(0) and Walras’ Law. Then, 푥 푝, 푤 satisfies WARP iff: ∆푝 ∙ ∆푥 ≤ 0 where – ∆푝 = 푝′ − 푝 and ∆푥 = 푥 푝′, 푤′ − 푥 푝, 푤 – 푤′ is the wealth level that allows the consumer to buy the initial demand at the new prices, 푤′ = 푝′ ∙ 푥 푝, 푤

• This is the Law of Demand: quantity demanded and price move in different directions.

50 Implications of WARP: Law of Demand

• Does WARP restrict behavior when we apply Slutsky wealth compensations? – Yes!

• What if we were not applying the Slutsky wealth compensation, would WARP impose any restriction on allowable locations for 푥 푝′, 푤′ ? – No!

51 Implications of WARP: Law of Demand

• See the discussions on the next slide

52 Implications of WARP: Law of Demand

• Can 푥 푝′, 푤′ lie on segment A? 1) 푥 푝, 푤 and 푥 푝′, 푤′ are both affordable under 퐵푝,푤. 2) 푥 푝, 푤 is affordable under 퐵푝′,푤′. ⇒ WARP is violated if 푥 푝′, 푤′ lies on segment A

• Can 푥 푝′, 푤′ lie on segment B? ′ ′ 1) 푥 푝, 푤 is affordable under 퐵푝,푤, but 푥 푝 , 푤 is not. ⇒ the premise of WARP does not hold ⇒ WARP is not violated if 푥 푝′, 푤′ lies on segment B

53 Implications of WARP: Law of Demand

• What did we learn from this figure? 1) We started from 훻푝1, and compensated the wealth of this individual (reducing it from 푤 to 푤′) so that he could afford his initial bundle 푥 푝, 푤 under the new prices. . From this wealth compensation, we obtained budget line 퐵푝′,푤′. 2) From WARP, we know that 푥 푝′, 푤′ must contain more of good 1. . That is, graphically, segment B lies to the right-hand side of 푥 푝, 푤 . 3) Then, a price reduction, 훻푝1, when followed by an appropriate wealth compensation leads to an increase in the quantity demanded of this good, ∆푥1. . This is the compensated law of demand (CLD).

54 Implications of WARP: Law of Demand

• Practice problem: – Can you repeat this analysis but for an increase in the price of good 1?

. First, pivot budget line 퐵푝,푤 inwards to obtain 퐵푝′,푤. . Then, increase the wealth level (wealth compensation) to obtain 퐵푝′,푤′.

. Identify the two segments in budget line 퐵푝′,푤′, one to the right-hand side of 푥 푝, 푤 and other to the left.

. In which segment of budget line 퐵푝′,푤′ can the Walrasian demand 푥(푝′, 푤′) lie?

Advanced Microeconomic Theory 55 Implications of WARP: Law of Demand

• Is WARP satisfied under the uncompensated law of demand (ULD)? – 푥 푝, 푤 is affordable under ′ budget line 퐵푝,푤, but 푥(푝 , 푤) is not. . Hence, the premise of WARP is not satisfied. As a result, WARP is not violated.

– 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

• Distinction between the uncompensated and the compensated law of demand: – quantity demanded and price can move in the same direction, when wealth is left uncompensated, i.e., ∆푝1 ∙ ∆푥1 > 0 as ∆푝1 ⇒ ∆푥1

• Hence, WARP is not sufficient to yield law of demand for price changes that are uncompensated, i.e., WARP ⇎ ULD, but WARP ⇔ CLD

Advanced Microeconomic Theory 57 Implications of WARP: Slutsky Matrix

• Let us focus now on the case in which 푥 푝, 푤 is differentiable. • First, note that the law of demand is 푑푝 ∙ 푑푥 ≤ 0 (equivalent to ∆푝 ∙ ∆푥 ≤ 0). • Totally differentiating 푥 푝, 푤 , 푑푥 = 퐷푝푥 푝, 푤 푑푝 + 퐷푤푥 푝, 푤 푑푤 • And since the consumer’s wealth is compensated, 푑푤 = 푥(푝, 푤) ∙ 푑푝 (this is the differential analog of ∆푤 = ∆푝 ∙ 푥(푝, 푤)). – Recall that ∆푤 = ∆푝 ∙ 푥(푝, 푤) was obtained from the Slutsky wealth compensation.

Advanced Microeconomic Theory 58 Implications of WARP: Slutsky Matrix

• Substituting, 푑푥 = 퐷푝푥 푝, 푤 푑푝 + 퐷푤푥 푝, 푤 푥(푝, 푤) ∙ 푑푝 푑푤 • or equivalently, 푇 푑푥 = 퐷푝푥 푝, 푤 + 퐷푤푥 푝, 푤 푥(푝, 푤) 푑푝

Advanced Microeconomic Theory 59 Implications of WARP: Slutsky Matrix

• Hence the law of demand, 푑푝 ∙ 푑푥 ≤ 0, can be expressed as 푇 푑푝 ∙ 퐷푝푥 푝, 푤 + 퐷푤푥 푝, 푤 푥(푝, 푤) 푑푝 ≤ 0 푑푥 where the term in brackets is the Slutsky (or substitution) matrix 푠11 푝, 푤 ⋯ 푠1퐿 푝, 푤 푆 푝, 푤 = ⋮ ⋱ ⋮ 푠퐿1 푝, 푤 ⋯ 푠퐿퐿 푝, 푤 where 휕푥푙(푝, 푤) 휕푥푙 푝, 푤 푠푙푘 푝, 푤 = + 푥푘(푝, 푤) 휕푝푘 휕푤

Advanced Microeconomic Theory 60 Implications of WARP: Slutsky Matrix

• Proposition: If 푥 푝, 푤 is differentiable, satisfies WL, homog(0), and WARP, then 푆 푝, 푤 is negative semi-definite, 푣 ∙ 푆 푝, 푤 푣 ≤ 0 for any 푣 ∈ ℝ퐿

• Implications: – 푠푙푘 푝, 푤 : substitution effect of good 푙 with respect to its own price is non-positive (own-price effect) – Negative semi-definiteness does not imply that 푆 푝, 푤 is symmetric (except when 퐿 = 2). . Usual confusion: “then 푆 푝, 푤 is not symmetric”, NO!

Advanced Microeconomic Theory 61 Implications of WARP: Slutsky Matrix

• Proposition: If preferences satisfy LNS and strict convexity, and they are represented with a continuous utility function, then the Walrasian demand 푥 푝, 푤 generates a Slutsky matrix, 푆 푝, 푤 , which is symmetric.

• The above assumptions are really common. – Hence, the Slutsky matrix will then be symmetric.

• 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

• Non-positive substitution effect, 푠푙푙 ≤ 0:

휕푥 (푝, 푤) 휕푥 푝, 푤 푠 푝, 푤 = 푙 + 푙 푥 (푝, 푤) 푙푙 휕푝 휕푤 푙 substitution effect (−) 푙 Total effect: Income effect: − usual good + normal good + Giffen good − inferior good

• Substitution Effect = Total Effect + Income Effect ⇒ Total Effect = Substitution Effect - Income Effect

Advanced Microeconomic Theory 63 Implications of WARP:

휕푥 (푝, 푤) 휕푥 푝, 푤 푠 푝, 푤 = 푙 + 푙 푥 (푝, 푤) 푙푙 휕푝 휕푤 푙 substitution effect (−) 푙 Total effect Income effect

• Total Effect: measures how the quantity demanded is affected by a change in the price of good 푙, when we leave the wealth uncompensated. • Income Effect: measures the change in the quantity demanded as a result of the wealth adjustment. • Substitution Effect: measures how the quantity demanded is affected by a change in the price of good 푙, after the wealth adjustment. – 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 Implications of WARP: Slutsky Equation

X2

• Reduction in the price of 푥1. – It enlarges consumer’s set of feasible bundles. A – He can reach an indifference curve further away from the origin. B I2 • The Walrasian indicates I1 that a decrease in the price of 푥1 leads to an increase in the quantity X1 demanded.

– This induces a negatively sloped p1 Walrasian demand curve (so the good

is “normal”). p1 a • The increase in the quantity demanded of 푥1 as a result of a decrease in its price represents the b total effect (TE).

X Advanced Microeconomic Theory 1 65 Implications of WARP: Slutsky Equation

x2

• Reduction in the price of 푥1. – Disentangle the total effect into the substitution and income effects A – Slutsky wealth compensation? C B • Reduce the consumer’s wealth so that he I2 can afford the same consumption bundle I3 as the one before the price change (i.e., A). I1

– Shift the budget line after the price change x1 inwards until it “crosses” through the initial bundle A.

– “Constant purchasing power” demand p curve (CPP curve) results from applying the 1 Slutsky wealth compensation. p1 a – The quantity demanded for 푥1 increases 0 3 from 푥1 to 푥1 .

• When we do not hold the consumer’s c b purchasing power constant, we observe relatively large increase in the quantity 0 2 CPP demand demanded for 푥1 (i.e., from 푥1 to 푥1 ).

x Advanced Microeconomic Theory 1 66 Implications of WARP: Slutsky Equation

x2

• Reduction in the price of 푥1. – Hicksian wealth compensation (i.e., “constant utility” demand curve)? • The consumer’s wealth level is adjusted

so that he can still reach his initial utility I2

level (i.e., the same indifference curve 퐼1 I3 as before the price change). I1 – A more significant wealth reduction than when we apply the Slutsky wealth x1 compensation. – The Hicksian demand curve reflects that, for p a given decrease in p1, the consumer slightly 1 increases his consumption of good one.

p1 • In summary, a given decrease in 푝1 produces: – A small increase in the Hicksian demand for 0 1 the good, i.e., from 푥1 to 푥1 . – A larger increase in the CPP demand for the 0 3 good, i.e., from 푥1 to 푥1 . CPP demand – A substantial increase in the Walrasian Hicksian demand demand for the product, i.e., from 푥0 to 푥2. 1 1 x Advanced Microeconomic Theory 1 67 Implications of WARP: Slutsky Equation

• A decrease in price of 푥1 leads the consumer to increase his consumption of this good, ∆푥1, but:

– The ∆푥1 which is solely due to the price effect (either measured by the Hicksian demand curve or the CPP demand curve) is smaller than the ∆푥1 measured by the Walrasian demand, 푥 푝, 푤 , which also captures wealth effects. – The wealth compensation (a reduction in the consumer’s 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 Substitution and Income Effects: Normal Goods • Decrease in the price of the good in the horizontal axis (i.e., food). Clothing • The substitution effect (SE)

moves in the opposite direction BL2 as the price change. BL1 – A reduction in the price of food implies a positive BL C substitution effect. d A

I2 • The income effect (IE) is B

positive (thus it reinforces the I1

SE). SE IE TE Food – The good is normal.

Advanced Microeconomic Theory 69 Substitution and Income Effects: Inferior Goods • Decrease in the price of the good in the horizontal axis (i.e., food). • The SE still moves in the opposite direction as the price change. I2 • The income effect (IE) is now negative (which partially offsets the increase in the quantity demanded associated I1 BL with the SE). 1 BLd – The good is inferior. • Note: the SE is larger than the IE. Advanced Microeconomic Theory 70 Substitution and Income Effects: Giffen Goods • Decrease in the price of the good in the horizontal axis (i.e., food). • The SE still moves in the opposite direction as the price change. I2 • The income effect (IE) is still negative but now completely offsets the increase in the

quantity demanded associated I1 BL2 BL with the SE. 1 – The good is Giffen good. • Note: the SE is less than the IE.

71 Substitution and Income Effects

SE IE TE Normal Good + + + Inferior Good + - + Giffen Good + - -

• NG: Demand curve is negatively sloped (as usual) • GG: Demand curve is positively sloped

Advanced Microeconomic Theory 72 Substitution and Income Effects

• Summary: 1) SE is negative (since ↓ 푝1 ⇒ ↑ 푥1) . SE < 0 does not imply ↓ 푥1 2) If good is inferior, IE < 0. Then, > TE(−) TE = SEณ − IEณ ⇒ if IE SE , then < TE(+) − ต− + For a price decrease, this implies TE(−) ↓ 푥 Giffen good ⇒ 1 TE(+) ↑ 푥1 Non−Giffen good 3) Hence, a) A good can be inferior, but not necessarily be Giffen b) But all Giffen goods must be inferior.

Advanced Microeconomic Theory 73 Expenditure Minimization Problem

Advanced Microeconomic Theory 74 Expenditure Minimization Problem

• Expenditure minimization problem (EMP): min 푝 ∙ 푥 푥≥0 s.t. 푢(푥) ≥ 푢

• Alternative to utility maximization problem

Advanced Microeconomic Theory 75 Expenditure Minimization Problem

• Consumer seeks a utility level associated with a particular indifference curve, while X2 spending as little as possible. • Bundles strictly above 푥∗ cannot be a solution to the EMP: UCS(x* ) – They reach the utility level 푢 – But, they do not minimize total expenditure u x* • Bundles on the budget line strictly below 푥∗ cannot be the solution to the EMP problem: – They are cheaper than 푥∗ X1 – But, they do not reach the utility level 푢

Advanced Microeconomic Theory 76 Expenditure Minimization Problem

• Lagrangian 퐿 = 푝 ∙ 푥 + 휇 푢 − 푢(푥)

• FOCs (necessary conditions) 휕퐿 휕푢(푥∗) = 푝푘 − 휇 ≤ 0 [ = 0 for interior 휕푥푘 휕푥푘 solutions ] 휕퐿 = 푢 − 푢 푥∗ = 0 휕휇

Advanced Microeconomic Theory 77 Expenditure Minimization Problem

• For interior solutions, 휕푢(푥∗) ∗ 휕푢(푥 ) 1 휕푥푘 푝푘 = 휇 or = 휕푥푘 휇 푝푘 for any good 푘. This implies, 휕푢(푥∗) 휕푢(푥∗) 휕푢(푥∗) 휕푥푘 휕푥푙 푝푘 휕푥푘 = or = 휕푢(푥∗) 푝푘 푝푙 푝푙 휕푥푙 • 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 “bang for the buck”). • That is, the slope of indifference curve is equal to the slope

of the budget line. Advanced Microeconomic Theory 78 EMP: Hicksian Demand

• The bundle 푥∗ ∈ argmin 푝 ∙ 푥 (the argument that solves the EMP) is the Hicksian demand, which depends on 푝 and 푢, 푥∗ ∈ ℎ(푝, 푢) • Recall that if such bundle 푥∗ is unique, we denote it as 푥∗ = ℎ(푝, 푢).

Advanced Microeconomic Theory 79 Properties of Hicksian Demand

• Suppose that 푢(∙) is a continuous function, 퐿 satisfying LNS defined on 푋 = ℝ+. Then for 푝 ≫ 0, ℎ(푝, 푢) satisfies: 1) Homog(0) in 푝, i.e., ℎ 푝, 푢 = ℎ(훼푝, 푢) for any 푝, 푢, and 훼 > 0. . If 푥∗ ∈ ℎ(푝, 푢) is a solution to the problem min 푝 ∙ 푥 푥≥0 then it is also a solution to the problem min 훼푝 ∙ 푥 푥≥0 . Intuition: a common change in all prices does not alter the slope of the consumer’s budget line.

Advanced Microeconomic Theory 80 Properties of Hicksian Demand

• 푥∗ is a solution to the EMP when the price vector is 푝 = (푝1, 푝2). x2 • Increase all prices by factor 훼 ′ ′ ′ 푝 = (푝1, 푝2) = (훼푝1, 훼푝2) • Downward (parallel) shift in the budget line, i.e., the slope of the budget line is unchanged. • But I have to reach utility level 푢 to satisfy the constraint of the EMP? • Spend more to buy bundle ∗ ∗ ∗ 푥 (푥1, 푥2), i.e., ′ ∗ ′ ∗ ∗ ∗ x 푝1푥1 + 푝2푥2 > 푝1푥1 + 푝2푥2 1 • Hence, ℎ 푝, 푢 = ℎ(훼푝, 푢)

Advanced Microeconomic Theory 81 Properties of Hicksian Demand

2) No excess utility: x for any optimal 2 consumption bundle 푥 ∈ ℎ 푝, 푢 , utility w level satisfies p2 푢 푥 = 푢. x ∈ h (p,u) x ’

u 1 u w x1 p1

Advanced Microeconomic Theory 82 Properties of Hicksian Demand

• Intuition: Suppose there exists a bundle 푥 ∈ ℎ 푝, 푢 for which the consumer obtains a utility level 푢 푥 = 푢1 > 푢, which is higher than the utility level 푢 he must reach when solving EMP. • But we can then find another bundle 푥′ = 푥훼, very close to 푥 (훼 → 1), for which 푢(푥′) > 푢. • Bundle 푥′: – is cheaper than 푥 – exceeds the minimal utility level 푢 that the consumer must reach in his EMP • For a given utility level 푢 that you have to reach in the EMP, bundle ℎ 푝, 푢 does not exceed 푢, since otherwise you could find a cheaper bundle that exactly reaches 푢. Advanced Microeconomic Theory 83 Properties of Hicksian Demand

3) Convexity: If the preference relation is convex, then ℎ 푝, 푢 is a convex set.

Advanced Microeconomic Theory 84 Properties of Hicksian Demand

4) Uniqueness: If the preference relation is strictly convex, then ℎ 푝, 푢 contains a single element.

Advanced Microeconomic Theory 85 Properties of Hicksian Demand

• Compensated Law of Demand: for any change in prices 푝 and 푝′, (푝′−푝) ∙ ℎ 푝′, 푢 − ℎ 푝, 푢 ≤ 0 – Implication: for every good 푘, ′ ′ (푝푘 − 푝푘) ∙ ℎ푘 푝 , 푢 − ℎ푘 푝, 푢 ≤ 0 – This is definitely true for compensated demand. – But, not necessarily for Walrasian demand (which is uncompensated):

• Recall the figures on Giffen goods, where a decrease in 푝푘 in fact decreases 푥푘 푝, 푢 when wealth was uncompensated.

Advanced Microeconomic Theory 86 The Expenditure Function

• Plugging the result from the EMP, ℎ 푝, 푢 , into the objective function, 푝 ∙ 푥, we obtain the value function of this optimization problem, 푝 ∙ ℎ 푝, 푢 = 푒(푝, 푢) where 푒(푝, 푢) represents the minimal expenditure that the consumer needs to incur in order to reach utility level 푢 when prices are 푝.

Advanced Microeconomic Theory 87 Properties of Expenditure Function

• Suppose that 푢(∙) is a continuous function, 퐿 satisfying LNS defined on 푋 = ℝ+. Then for 푝 ≫ 0, 푒(푝, 푢) satisfies: 1) Homog(1) in 푝, i.e., 푒 훼푝, 푢 = 훼 푝 ∙ 푥∗ = 훼 ∙ 푒(푝, 푢) 푒(푝,푢) for any 푝, 푢, and 훼 > 0. . We know that the optimal bundle is not changed when all prices change, since the optimal consumption bundle in ℎ(푝, 푢) 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 풖:

For a given prices, x reaching a higher utility 2 requires higher expenditure: ′ ′ 푝1푥1 + 푝2푥2 > 푝1푥1 + 푝2푥2

where (푥1, 푥2) = ℎ(푝, 푢) x2 ′ ′ ′ and (푥1, 푥2) = ℎ(푝, 푢 ).

Then, x1 x1 푒 푝, 푢′ > 푒 푝, 푢

Advanced Microeconomic Theory 89 Properties of Expenditure Function

3) Non-decreasing in 풑풌 for any good 풌: Higher prices mean higher expenditure to reach a given utility level. • Let 푝′ = (푝 , 푝 , … , 푝′ , … , 푝 ) and 푝 = 1 2 푘 퐿 ′ (푝1, 푝2, … , 푝푘, … , 푝퐿), where 푝푘 > 푝푘. • Let 푥′ = ℎ 푝′, 푢 and 푥 = ℎ 푝, 푢 from EMP under prices 푝′ and 푝, respectively. • Then, 푝′ ∙ 푥′ = 푒(푝′, 푢) and 푝 ∙ 푥 = 푒(푝, 푢). 푒 푝′, 푢 = 푝′ ∙ 푥′ ≥ 푝 ∙ 푥′ ≥ 푝 ∙ 푥 = 푒(푝, 푢)

– 1st inequality due to 푝′ ≥ 푝 – 2nd inequality: at prices 푝, bundle 푥 minimizes EMP. Advanced Microeconomic Theory 90 Properties of Expenditure Function

4) Concave in 풑:

′ ′ ′ ′ ′ Let 푥 ∈ ℎ 푝 , 푢 ⇒ 푝 푥 ≤ 푝 푥 x2 ∀푥 ≠ 푥′, e.g., 푝′푥′ ≤ 푝′푥ҧ and 푥′′ ∈ ℎ 푝′′, 푢 ⇒ 푝′′푥′′ ≤ 푝′′푥 ∀푥 ≠ 푥′′, e.g., 푝′′푥′′ ≤ 푝′′푥ҧ x This implies 훼푝′푥′ + 1 − 훼 푝′′푥′′ ≤ 훼푝′푥ҧ + 1 − 훼 푝′′푥ҧ x 푒(푝′,푢) 푒(푝′′,푢) 훼 푝ฑ′푥′ + 1 − 훼 푝′′푥′′ ≤ [훼푝′ + 1 − 훼 푝′′]푥ҧ x 푝ҧ 1 훼푒(푝′, 푢) + 1 − 훼 푒(푝′′, 푢) ≤ 푒(푝ҧ, 푢) concavity

Advanced Microeconomic Theory 91 Connections

Advanced Microeconomic Theory 92 Relationship between the Expenditure and Hicksian Demand • Let’s assume that 푢(∙) is a continuous function, representing preferences that satisfy LNS and are 퐿 strictly convex and defined on 푋 = ℝ+. For all 푝 and 푢, 휕푒 푝,푢 = ℎ푘 푝, 푢 for every good 푘 휕푝푘 This identity is “Shepard’s lemma”: if we want to find ℎ푘 푝, 푢 and we know 푒 푝, 푢 , we just have to differentiate 푒 푝, 푢 with respect to prices. • Proof: three different approaches 1) the support function 2) first-order conditions 3) the envelop theorem (See Appendix 2.2) Advanced Microeconomic Theory 93 Relationship between the Expenditure and Hicksian Demand • The relationship between the Hicksian demand and the expenditure function can be further developed by taking first order conditions again. That is, 2 휕 푒(푝, 푢) 휕ℎ푘(푝, 푢) 2 = 휕푝푘 휕푝푘 or 2 퐷푝푒 푝, 푢 = 퐷푝ℎ(푝, 푢)

• Since 퐷푝ℎ(푝, 푢) provides the Slutsky matrix, 푆(푝, 푤), then 2 푆 푝, 푤 = 퐷푝푒 푝, 푢 where the Slutsky matrix can be obtained from the observable Walrasian demand.

Advanced Microeconomic Theory 94 Relationship between the Expenditure and Hicksian Demand • There are three other important properties of 퐷푝ℎ(푝, 푢), where 퐷푝ℎ(푝, 푢) is 퐿 × 퐿 derivative matrix of the hicksian demand, ℎ 푝, 푢 : 1) 퐷푝ℎ(푝, 푢) is negative semidefinite . Hence, 푆 푝, 푤 is negative semidefinite. 2) 퐷푝ℎ(푝, 푢) is a symmetric matrix . Hence, 푆 푝, 푤 is symmetric. 3) 퐷푝ℎ 푝, 푢 푝 = 0, which implies 푆 푝, 푤 푝 = 0. 휕ℎ (푝,푢) . Not all goods can be net substitutes 푙 > 0 or net 휕푝 휕ℎ (푝,푢) 푘 complements 푙 < 0 . Otherwise, the multiplication 휕푝푘 of this vector of derivatives times the (positive) price vector 푝 ≫ 0 would yield a non-zero result.

Advanced Microeconomic Theory 95 Relationship between Hicksian and Walrasian Demand • When income effects x2 w are positive (normal p2 goods), then the x* x 2 x 3

Walrasian demand x1 I2 I3 푥(푝, 푤) is above the I1

w w x1 Hicksian demand p1 p 1 p ℎ(푝, 푢) . 1 SE IE

– The Hicksian demand p1 is steeper than the p1 p 1

Walrasian demand. x (p,w)

h (p,ū)

* 1 2 x1 x1 x1 x1 Advanced Microeconomic Theory 96 Relationship between Hicksian and Walrasian Demand

x2 • When income effects w are negative (inferior p2 x* 2 goods), then the x I2 Walrasian demand x1

푥(푝, 푤) I is below the 1 w w x p 1 Hicksian demand SE 1 p 1 p1 ℎ(푝, 푢) . IE a – The Hicksian demand p1 p1

is flatter than the p 1 c b

Walrasian demand. h (p,ū)

x (p,w)

* 2 1 x1 x1 x1 x1 Advanced Microeconomic Theory 97 Relationship between Hicksian and Walrasian Demand • We can formally relate the Hicksian and Walrasian demand as follows: – Consider 푢(∙) is a continuous function, representing preferences that satisfy LNS and are strictly convex and 퐿 defined on 푋 = ℝ+. – Consider a consumer facing (푝ҧ, 푤ഥ) and attaining utility level 푢ത. – Note that 푤ഥ = 푒(푝ҧ, 푢ത). In addition, we know that for any (푝, 푢), ℎ푙 푝, 푢 = 푥푙(푝, 푒 푝, 푢 ). Differentiating this 푤 expression with respect to 푝푘, and evaluating it at (푝ҧ, 푢ത), we get: 휕ℎ (푝ҧ, 푢ത) 휕푥 (푝ҧ, 푒(푝ҧ, 푢ത)) 휕푥 (푝ҧ, 푒(푝ҧ, 푢ത)) 휕푒(푝ҧ, 푢ത) 푙 = 푙 + 푙 휕푝푘 휕푝푘 휕푤 휕푝푘 Advanced Microeconomic Theory 98 Relationship between Hicksian and Walrasian Demand 휕푒(푝ҧ,푢ഥ) • Using the fact that = ℎ푘(푝ҧ, 푢ത), 휕푝푘 휕ℎ푙(푝ҧ,푢ഥ) 휕푥푙(푝ҧ,푒(푝ҧ,푢ഥ)) 휕푥푙(푝ҧ,푒(푝ҧ,푢ഥ)) = + ℎ푘(푝ҧ, 푢ത) 휕푝푘 휕푝푘 휕푤

• Finally, since 푤ഥ = 푒(푝ҧ, 푢ത) and ℎ푘 푝ҧ, 푢ത = 푥푘 푝ҧ, 푒 푝ҧ, 푢ത = 푥푘 푝ҧ, 푤ഥ , then

휕ℎ푙(푝ҧ,푢ഥ) 휕푥푙(푝ҧ,푤ഥ) 휕푥푙(푝ҧ,푤ഥ) = + 푥푘(푝ҧ, 푤ഥ) 휕푝푘 휕푝푘 휕푤

Advanced Microeconomic Theory 99 Relationship between Hicksian and Walrasian Demand

• But this coincides with 푠푙푘 푝, 푤 that we discussed in the Slutsky equation.

– Hence, we have 퐷푝ℎ 푝, 푢 = 푆 푝, 푤 . 퐿×퐿 – Or, more compactly, 푆퐸 = 푇퐸 + 퐼퐸.

Advanced Microeconomic Theory 100 Relationship between Walrasian Demand and Indirect Utility Function • Let’s assume that 푢(∙) is a continuous function, representing preferences that satisfy LNS and are strictly 퐿 convex and defined on 푋 = ℝ+. Support also that 푣(푝, 푤) is differentiable at any (푝, 푤) ≫ 0. Then,

휕푣 푝,푤 휕푝푘 − 휕푣 푝,푤 = 푥푘(푝, 푤) for every good 푘 휕푤 • This is Roy’s identity. • Powerful result, since in many cases it is easier to compute the derivatives of 푣 푝, 푤 than solving the UMP with the system of FOCs.

Advanced Microeconomic Theory 101 Summary of Relationships

• The Walrasian demand, 푥 푝, 푤 , is the solution of the UMP. – Its value function is the indirect utility function, 푣 푝, 푤 . • The Hicksian demand, ℎ(푝, 푢), is the solution of the EMP. – Its value function is the expenditure function, 푒(푝, 푢). • Relationship between the value functions of the UMP and the EMP: – 푒 푝, 푣 푝, 푤 = 푤, 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, 푢 = 푣 푝, 푤 , must be 푤. – 푣 푝, 푒(푝, 푢) = 푢, 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., 푤 = 푒(푝, 푢), is exactly 푢.

Advanced Microeconomic Theory 102 Summary of Relationships

Advanced Microeconomic Theory 103 Summary of Relationships

• Relationship between the argmax of the UMP (the Walrasian demand) and the argmin of the EMP (the Hicksian demand): – 푥 푝, 푒(푝, 푢) = ℎ 푝, 푢 , i.e., the (uncompensated) Walrasian demand of a consumer endowed with an adjusted wealth level 푤 (equal to the expenditure he optimally uses in the EMP), 푤 = 푒 푝, 푢 , coincides with his Hicksian demand, ℎ 푝, 푢 . – ℎ 푝, 푣(푝, 푤) = 푥 푝, 푤 , i.e., the (compensated) Hicksian demand of a consumer reaching the maximum utility of the UMP, 푢 = 푣 (푝, 푤), coincides with his Walrasian demand, 푥 푝, 푤 .

Advanced Microeconomic Theory 104 Summary of Relationships

Advanced Microeconomic Theory 105 Summary of Relationships

• Finally, we can also use: – The Slutsky equation: 휕ℎ푙(푝, 푢) 휕푥푙(푝, 푤) 휕푥푙(푝, 푤) = + 푥푘(푝, 푤) 휕푝푘 휕푝푘 휕푤 to relate the derivatives of the Hicksian and the Walrasian demand. – Shepard’s lemma: 휕푒 푝, 푢 = ℎ푘 푝, 푢 휕푝푘 to obtain the Hicksian demand from the expenditure function. – Roy’s identity: 휕푣 푝, 푤 휕푝 − 푘 = 푥 (푝, 푤) 휕푣 푝, 푤 푘 휕푤 to obtain the Walrasian demand from the indirect utility function.

Advanced Microeconomic Theory 106 Summary of Relationships

The UMP The EMP 4(a) Slutsky equation x(p,w) (Using derivatives) h(p,u)

3(a)

(1) (1)

3(b)

(2) v(p,w) e(p,u) e(p,v(p,w))=w v(p,e(p,w))=u

Advanced Microeconomic Theory 107 Summary of Relationships

The UMP The EMP 4(a) Slutsk equation x(p,w) (For derivatives) h(p,u)

3(a)

4(b) Shepard s (1) (1) Roy s lemma Identity 4(c) 3(b)

(2) v(p,w) e(p,u) e(p,v(p,w))=w v(p,e(p,w))=u

Advanced Microeconomic Theory 108