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Advanced Microeconomics

Consumer Choice

Ronald Wendner

Department of EconomicsUniversity of Graz, Austria

Course # 320.911

Consumer Choice

� Commodities

� Consumption set

� Competitive budgets

� Demand functions & comparative statics

� WARP and the law of demand

R. Wendner (U Graz, Austria) Microeconomics 2 / 28

Commodities

Commodities

Decision problem of consumer: choice of goods/services (=commodities)

L commodities, indexed by 1, 2, ...,L

commodity vector x = (x1, x2, ..., xL) ∈ RL (commodity space)

definition of commodity: time, location, state of nature


Consumption set

The consumption set

X ⊂ RL feasible x

Examples


Consumption set

for the time being X = RL+ = {x ∈ RL | xl ≥ 0, l = 1, ...,L}

Queries.

Give an example of an X with an institutional constraint.What are important properties of X as defined above?


Budget set

Walrasian (competitive) budget sets

Constraints on choice: restrictions on X ; prices p, wealth w

p = (p1, p2, ..., pL) ∈ RL++ = {p ∈ RL | pl > 0, l = 1, ...,L}

price taking assumption

w ∈ R+

Walrasian (competitive) budget set: Bp,w = {x ∈ RL+ | p · x ≤ w} ⊂ X

Budget hyperplane {x ∈ RL | p · x = w} (“budget line” for L = 2)


Budget set

X convex ⇒ Bp,w convex


Budget set

Query. Vector p is orthogonal to any vector on budget hyperplane. Why?


Demand & comparative statics

Demand functions and comparative statics

Definition (HD0)f (y1, ..., yN ) is HD0 if f (y1, ..., yN )=f (λy1, ..., λyN ), λ ∈ R++

x(p,w) Walrasian demand correspondence: %-max x in Bp,w

x(p,w) is HD0: x(p,w)=x(λp, λw), λ ∈ R++

HD0 implies that we can normalize prices w/o loss of generality

pl = 1 for one lw = 1∑L

l=1 pl = 1

Bp,w is HD0 in (p,w), i.e.: Bλp,λw = Bp,w



Definition (Walras law, version 1)For p � 0 and w > 0, p · x = w for all x(p,w).

Query. Suppose L = 2, and consider the following Walrasian demandfunctions:

x1(p,w) = α

p1w

x2(p,w) = (1− α)βp2

w

Does this Walrasian demand function satisfy HD0 and Walras law whenβ ∈ (0, 1)?



Comparative statics

x(p,w) =

x1(p,w)...

xL(p,w)

∈ RL+

Wealth (Income) effects

Dwx(p,w) =

∂ x1(p,w)∂ w...

∂ xL(p,w)∂ w

∈ RL

Price effects

Dpx(p,w) =

∂ x1(p,w)∂ p1

· · · ∂ x1(p,w)∂ pL

.... . .

...∂ xL(p,w)∂ p1

· · · ∂ xL(p,w)∂ pL

with dimension L × L



Wealth (income) effects

normal goods: ∂ xl/∂ w ≥ 0 ; otherwise inferior goods

Engel functions: x(p̄,w)

image in RL+: Ep̄ = {x(p̄,w) |w > 0}: wealth (income) expansion path



Price effects

regular goods: ∂ xl/∂ pl ≤ 0 ; otherwise Giffen goods

Offer curve in RL+: Op̄−k ,w̄ = {x(p̄−k , w̄) | pk > 0}

where p−k = (p1, p2, ...pk−1, pk+1, ...pL)



� Which restrictions are imposed on comparative staticsif x(p,w) is HD0 and Walras law holds?

Restrictions by HD0

xl(λp, λw)− xl(p,w) = 0 , l = 1, ...,L (1)

differentiate (1) w.r.t. λ and set λ = 1 then

L∑k=1

∂ xl(p,w)∂ pk

pk + ∂ xl(p,w)∂ w w = 0 , l = 1, ...,L

Dpx(p,w)p + Dwx(p,w)w = 0

I p- and w- effects sum up to zero



Aggregation (elasticity-formulation of the above)

εlk(p,w) = ∂ xl(p,w)∂ pk

pk

xl(p,w) , εlw(p,w) = ∂ xl(p,w)∂ w

wxl(p,w)

L∑k=1

εlk(p,w) + εlw(p,w) = 0



� Restrictions by Walras law

L∑l=1

plxl(p,w) = w ⇔ p · x(p,w) = w (2)

Cournot aggregation: differentiate (2) w.r.t. pk

L∑l=1

pl∂ xl(p,w)∂ pk

+ xk(p,w) = 0 , k = 1, ...,L

p ·Dpx(p,w) + x(p,w)′ = 0

Engel aggregation: differentiate (2) w.r.t. w

L∑l=1

pl∂ xl(p,w)∂ w = 1 , p ·Dwx(p,w) = 1



Definition (Budget share spent on l)bl(p,w) = plxl(p,w)/w

Elasticity representation of the Cournot aggregation

L∑l=1

bl(p,w)εlk(p,w) + bk(p,w) = 0

Elasticity representation of the Engel aggregation

L∑l=1

bl(p,w)εlw(p,w) = 1

I PS2


WARP & Law of demand

WARP and the Compensated Law of demand

Definitionx(p,w) satisfies WARP if for any (p,w) and (p′,w′) the following holds: Ifp · x(p′,w′) ≤ w and (p,w) 6= (p′,w′), then p′ · x(p,w) > w′



Query. Do (c), (d), (e) violate the WARP?

I WARP poses restriction on demand behavior



� Compensated price effects

∆pl →

(i) change in relative price pl/pk ;

(ii) change in real wealth w/pl (we want to get rid of this effect)

price- and wealth effects to be disentangled

Slutsky wealth compensation: (p,w)→ (p′,w′)

w = p · x(p,w)w′ = p′ · x(p,w)

}∆w = ∆p · x(p,w)



Compensated price change when p1 ↓

Query. How do budget lines look like with an uncompensated price change?



PropositionLet x(p,w) satisfy HD0 and Walras law. Then, for any compensated price changefrom (p,w) to (p′,w′):

WARP ⇔ (p′ − p) · [x(p′,w′)− x(p,w)] ≤ 0

in short notation ∆p ·∆x ≤ 0

p and x move in opposing directions for compensated price changes

Most simple case: only pl changes: ∆p = (0, 0, ...,∆pl , ...0)

∆p ·∆x = ∆pl ∆xl ≤ 0

pl ↑⇒ xl ↓ (compensated)



� Uncompensated-, compensated ∆p and WARP

right: uncompensated ∆p with p1 ↓ and x1 ↓



Proposition 1: to check for WARP, only compensated changes needed to bechecked

∆p ·∆x ≤ 0

Suppose x(p,w) is differentiable

dp · dx ≤ 0 (*)

dw = x(p,w) · dp

WARP implicates on total differential of x = x(p,w)



dx = Dpx(p,w)dp + Dwx(p,w)dw

= Dpx(p,w)dp + Dwx(p,w)[x(p,w) · dp]

= [Dpx(p,w) + Dwx(p,w)x ′(p,w)]dp

Query. Determine the dimensions of all expressions above. Are thecomformibility conditions satisfied?

From WARP (*) it follows:

dp · [Dpx(p,w) + Dwx(p,w)x ′(p,w)]︸︷︷︸S(p,w)

dp ≤ 0

I S(p,w) Slutsky substitution matrix



Wealth (income) compensated substitution effects

S(p,w) =

s11(p,w) · · · s1L(p,w)...

. . ....

sL1(p,w) · · · sLL(p,w)

slk(p,w) = ∂ xl(p,w)∂ pk

+ ∂ xl(p,w)∂ w xk(p,w)

slk (income) compensated substitution effects



PropositionLet x(p,w) satisfy WL, HD0, and WARP. Then, for all dp ∈ RL

dp · S(p,w)dp ≤ 0 .

Thus: S(p,w) is negative semidefinite

I sll ≤ 0 (non-positive own-price effects)

I Giffen good (∂ xl/∂ pl > 0) ⇒ ∂ xl(p,w)/∂ w < 0 (inferior)



� A few results

WARP ⇔ Compensated law of demand

WARP ⇒ S(p,w) negative semidefinite (NSD)

NSD ⇒ sll(p,w) ≤ 0 (nonpositive own-price effects)


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