intermediate micro lecture 9 - georgetown...

Consumer’s surplus CV/EV Examples

Consumer’s surplus

Intermediate Micro

Lecture 9

Chapter 14 of Varian


Welfare analysis

I Last few lectures: how does ∆pi affect demand?I Today: how does ∆pi (or other change) affect consumers’s

well-being?I Bad news: multiple methodsI Good news: Agree on good/badI Good news: Similar magnitude


Consumer’s surplus: a micro principles review

I Start with discretegoods model

I Reservation price: rj ,price (per unit) atwhich consumer isindifferent betweenbuying j and j − 1units of the good

I Gives $-value ofconsumption of 1 unit



Suppose price is p̄ (purpleline)

I Choice: x = 2I

Gross consumer surplus:$-valuation of totalbenefit from chosenx .

I Area of 2 left-mostbars

I r1 + r2



Suppose price is p̄ (purpleline)

I Choice: x = 2I Consumer’s surplus:

$-valuation of benefitfrom chosen x net offoregone consumptionof other goods.

I Area above theprice line

I r1 + r2 − 2p̄


Non-discrete goods

I Can find CS when xis not discrete good

I Use demand function:xi (pi , p−i ,m)

CS =

∫ ∞p̄

x(p, p−i ,m)dp


Non-discrete goods

I Or, useinverse demand functionpi (xi , p−im)

I pi so thatxi (pi , p−i ,m) = xi

CS =

∫ x∗

0p(x , p−i ,m)−p̄dx


Welfare analysis

I CS alone not veryinteresting

I ∆CS with policychange is informative

I $-value (-cost) ofprice change

I Example: ↑ p, from p̄to p′


Multiple consumers

I Units for CS are $s

I We can sum CSsacross manyindividuals

I Can also add inproducer surpluses

I Note PS is income forhouseholds


Great, let’s stop here

Issues with CS

1. It requires taking integralsI It can be done!

2. How well do ”Purchase costs” represent utility?I Income effectI Quasilinear utility - perfectlyI Other utility functions - well, maybe not


Using the indifference curve

Other ways to measure impact of price increase

1. Compensating variation: Increase in m needed, after priceincrease to restore utility to pre-change level

I Price change and ↑ m happenI How much to compensate for price change?

2. Equivalent variation: Decrease in m sufficient, before pricechange, to bring utility to post-change level

I ↓ m happens instead of price changeI What cost is equivalent to effect of price change?

CV and EV can also be used for price decreases


Compensating variation

I p1 ↑ from p to p′

I p2 = 1I CV is ∆m so that

budget lineI with slope −p′I tangent to old

indifference curve

I ∆xmax2 = ∆m

p2= ∆m


Equivalent variation

I p1 ↑ from p to p′

I p2 = 1I EV is −∆m so that

budget lineI with slope −pI tangent to new

indifference curve

I ∆xmax2 = ∆m

p2= ∆m


What we’re doing

I How far has theindifference curvemoved?

I Like, in $ terms

I Use budget lines tomeasure


CV vs EV

I CV 6= EV (Almost always)

I CV: in post-change $s

I EV in pre-change $s

I They don’t have samebuying power

For ↑ pI CV ≥ EV , usually >

I Equal with quasilinearutility

I = ∆CS , too

I Equal if CV, or EV,= 0


Example: Cobb-Douglas utility


u(x , y) = x2y

m = 300, py = 1

px changes from p = 5 to p′ = 4Find the CV and EV


Example: Quasilinear utility


u(x , y) = 4√x + y

m = 40, py = 1

px changes from p = 0.8 to p′ = 1Find the CV and EV


Quasilinear utility - a general result

Compensating variation

v(x∗′) + [m + CV − p′x∗′] = v(x∗) + [m − px∗]

CV = [v(x∗)− px∗] − [v(x∗′)− p′x∗′]CV = [v(x∗)− v(x∗′)] − [px∗ − p′x∗′]CV = [∆utility from x ] − [∆expenditure on x ]


Quasilinear utility - a general result

Equivalent variation

v(x∗′) + [m − p′x∗′] = v(x∗) + [m − EV − px∗]

EV = [v(x∗)− px∗] − [v(x∗′)− p′x∗′]EV = [v(x∗)− v(x∗′)] − [px∗ − p′x∗′]EV = [∆utility from x ] − [∆expenditure on x ]

So, CV = EV


Quasilinear utility and CS

I FOC for x : p = v ′(x)

I Loss of direct utilitywhen x falls from x∗

to x∗′ is∫x∗′

x∗v ′(x)dx

I = v(x∗)− v(x∗′)

I Blue area

For ↑ p


Quasilinear utility and CS

I Reduced spendingfrom ↓ x goes to y

I Green area

I Increased spendingfrom ↑ p comes fromy

I Red area

I Total effect:∆CS = CV = EV

I Only for quasilinearutility

−∆CS = v(x∗)− v(x∗′)− p[x∗ − x∗′] +x∗′[p′ − p]blue area −green box +red box

intermediate micro lecture 9 - georgetown...

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