Intermediate*Microeconomics!(iv)!Risk,&Insurance&and&Monopolies&!!11! Choice!and!Markets!in!the!Presence!of!Risk!.......................................................!11—2!11.1! Introduction!to!risk!model!.................................................................................!11—2!11.1.1! Notation!.............................................................................................................!11—2!11.1.2! Risk!appetites!..................................................................................................!11—2!11.1.3! Example:!Ivan!and!his!dodgy!bonds!......................................................!11—3!11.1.4! Ivan’s!risk!aversion!graphically!..............................................................!11—3!11.1.5! Degree!of!risk!aversion!–!certainty!equivalent!.................................!11—4!11.1.6! Risk!premiums!................................................................................................!11—5!11.1.7! Other!risk!profiles!–!risk!neutral!and!risk!loving!............................!11—5!

11.2! Insurance!markets!.................................................................................................!11—7!11.2.1! Actuarially!fair!insurance!...........................................................................!11—7!

11.3! Modeling!consumption!of!insurance!.............................................................!11—8!11.3.1! The!model’s!framework!..............................................................................!11—8!11.3.2! The!choice!set!and!budget!line!................................................................!11—8!11.3.3! Preferences!......................................................................................................!11—9!11.3.4! Choice!..............................................................................................................!11—10!11.3.5! Choice!where!insurance!is!actuarially!unfair!.................................!11—11!

11.4! General!equilibrium!with!uncertainty!.......................................................!11—12!11.4.1! Uncertainty!without!aggregate!risk!....................................................!11—12!11.4.2! Choice!with!aggregate!risk!.....................................................................!11—13!

12! Monopolies!.....................................................................................................................!12—14!12.1! Elasticity!..................................................................................................................!12—14!12.2! Marginal!Revenue!...............................................................................................!12—15!12.3! Graphing!marginal!and!total!revenue!........................................................!12—16!12.4! Profit!maximization!...........................................................................................!12—17!12.4.1! Profit!maximization!in!the!market!......................................................!12—17!12.4.2! Profit!maximization:!a!monopoly!without!price!discrimination!.!12—17!12.4.3! Profit!maximization:!a!monopoly!and!first!degree!price!discrimination!...............................................................................................................!12—19!12.4.4! Profit!maximization:!a!monopoly!with!third!degree!price!discrimination!...............................................................................................................!12—20!12.4.5! Profit!maximization:!a!monopoly!and!secondWdegree!price!discrimination!...............................................................................................................!12—21!

! !

11 Choice(and(Markets(in(the(Presence(of(Risk(Risk,&expected&return,&expected&utility,&risk&appetite,&insurance&in&partial&equilibrium,&insurance&in&general&equilibrium.&

11.1 Introduction to risk model This!week,!we!will!be!extending!the!partial!and!general!equilibrium!models!to!a!situation!involving!risk.!!Instead!of!two!goods,!we!have!two!states! xG and! xB ,!the!good!and!bad!states!respectively.!!Thus,!instead!of!a!utility!function!of!u x1, x2( ) !we!have!a!utility!function!u xG, xB( ) .!!In!the!goods!model!we!could!have!all!of! x1 !or x2 ,!or!a!combination!of!both.!!In!this!model,!if!the!good!outcome!occurs,!the!bad!one!doesn’t,!and!vice!versa:!you!either!have! xG or! xB .!!We!are!focusing!on!state!independent!functions,!where!the!utility!of!one!outcome!is!different!from!the!utility!of!the!other:!i.e.!u xG( ) ≠ u xB( ) .!!One!of!the!outcomes!is!preferred,!or,!rather,!grants!the!agent!higher!utility.!!This!difference!may!be!insured!against!(which!we!will!explore!later!in!the!chapter).!

11.1.1 Notation(• Outcomes!

o Positive:! xG !o Negative:! xB !

• Probability!o Positive!!!Probability!of! xG = (1−δ) !o Negative!!!Probability!of! xB = δ !

• Expected!value!!!probability!weighted!estimate!of!earnings!o EV = (1−δ)• xG +δ • xB !

• Expected!utility!!!probability!weighted!estimate!of!utility!o AKA!von!NeumannWMorgenstern!expected!utility!function!o EU =U xB, xG( ) = δ •U xB( )+ (1−δ)•U(xG ) !

11.1.2 Risk(appetites((The!relationship!between!the!expected!value!and!expected!utility!defines!the!agent’s!appetite!for!risk.!!For!a!risk!averse!person,!the!utility!of!the!expected!value!of!a!gamble!is!higher!than!the!expected!utility!of!a!function,!i.e.:!

u EV( ) > EU !!

This!means!that!the!person!would!rather!have!the!expected!value!of!the!gamble!for!certain!than!face!the!risk.!!This!creates!a!concave!utility!function:!u(x) is!decreasing!in!income.!

11.1.3 Example:(Ivan(and(his(dodgy(bonds(Economic!circumstances:!

• Ivan!has!savings!worth!$100!• He!has!been!offered!to!invest!in!risky!bonds!

o (1−δ) = 0.75→ xG = 256 !o δ = 0.25→ xB =16 !

!Thus,!we!can!work!out!the!expected!value:!

!EV = 1−δ( )• xg +δ • xB

= 0.75•256+ 0.25•16=196

!

!What!about!his!risk!appetite?!Let!Ivan’s!utility!function!be!u x( ) = x .!!We!need!to!find!the!relationship!between!EU!and!u(EV).!

EU = u xG, xB( )= 1−δ( )•u xG( )+ δ( )u xB( )= 0.75( )• 256 + 0.25( ) 16=13

! ! u EV( ) = 196=14

!

!EU < u EV( ) ,!

Thus,!it!can!be!said!that!Ivan!is!risk!averse.!

11.1.4 Ivan’s(risk(aversion(graphically(!

!Here,!the!black!curve!represents!Ivan’s!utility!function.!!The!two!points!at!the!end!of!the!red!line!are!the!possible!outcomes!of!the!gamble.!!The!red!line!

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11

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15

16

X

u(x)

EV=196

EU

u(EV)

between!them!is!the!average!of!the!result,!with!expected!value!(the!risk!weighted!average)!delineated.!!The!u(x)!value!at!x=EV!intersecting!with!the!average!is!the!expected!utility.!!The!u(x)!value!at!x=EV!intersecting!with!the!utility!function!is!u(EV).!!Thus,!as!long!as!the!utility!function!is!concave!downwards,!the!agent!will!be!risk!averse,!preferring!the!weighted!average!to!a!risked!extreme.!

11.1.5 Degree(of(risk(aversion(–(certainty(equivalent(Above,!we!showed!that!Ivan!preferred,!at!the!expected!value!of!$196,!not!to!take!the!risk!than!to!take!the!risk.!!This!is!not!surprising,!since!he!is!being!paid!the!expected!winning!without!taking!the!risk,!and!he!is!risk!averse.!!However,!there!can!be!many!degrees!of!risk!aversion:!how!do!we!compare!different!risk!averse!preference!types?!!We!do!this!by!finding!the!certainty!equivalent:!the!minimum!amount!of!money!that!the!agent!would!have!to!be!paid!not!to!take!the!gamble.!!!Graphically,!it!can!be!found!by!finding!the!intersect!of!the!horizontal!line!originating!at!the!“average!between!results!line”!and!the!utility!function,!as!shown!below.!!

!At! xce ,!i.e.!at!$169,!Ivan!gets!the!same!utility!from!the!risk!with!the!potential!gain!as!from!the!monetary!payout.!!If!we!offered!him!$168,!he!would!prefer!to!take!the!gamble!than!to!accept!our!bribe!not!to.!!!!Solving!this!mathematically!–!u xce( ) = EU ,!we!get!the!same!result:!

u xce( ) = EU

xce = EU = 1−δ( )•u xG( )+δ •u xB( )

x12ce = .75( ) 256( )+ .25( ) 16( ) =13xce = 13( )2 =169

!

!

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0

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3

4

5

6

7

8

9

10

11

12

13

14

15

16

X

u(x)

EV=196

EU

u(EV)

xce=169

11.1.6 Risk(premiums(Since!Ivan!is!risk!averse,!it!is!obvious!that!he!would!be!willing!to!take!a!sum!of!money!smaller!than!the!expected!value!of!the!investment!in!order!to!avoid!the!risk!of!the!investment.!!This!difference!is!called!the!risk!premium:!the!greater!the!risk!premium,!the!more!risk!averse!the!individual!is.!!

RP = EV − xxe=196−169= 27

!

Graphically,!it!is!the!horizontal!distance!between!the!utility!function!and!the!average!line!originating!at! x = EV .!!

11.1.7 Other(risk(profiles(–(risk(neutral(and(risk(loving(

11.1.7.1 Risk(neutral(A!person!displaying!a!risk!neutral!profile!is!indifferent!between!taking!and!not!taking!risks.!

• u EV( ) = EU !• u x( ) !(The!utility!function)!is!linear!

o Therefore!marginal!utility!is!constant!in!x!(income)!• ! xce = EV∴RP = 0 !

11.1.7.2 Risk(loving(A!person!who!is!risk!loving!relishes!risk,!and!will!not!easily!forego!the!opportunity!for!a!flutter.!!They!would!prefer!to!take!the!gamble!then!to!get!the!expected!value!for!sure.!

• u EV( ) < EU !• u x( ) (The!utility!function)!is!convex!

o Marginal!utility!increases!in!income!(x)!• xCE > EV∴RP < 0 !

!! !

11.1.7.2.1 Example:!Lucy’s!choice!in!the!same!market?!Lucy!is!a!risk!lover,!with!a!utility!function!of!u x( ) = x2 .!!The!bond!she!is!offered!is!the!same!as!the!one!offered!to!Ivan.!

EV = 1−δ( ) xG + δ( ) xB= 0.75( ) 256( )+ 0.25( ) 16( )=196

u EV( ) = 196( )2

= 38, 416EU = 1−δ( )•u xG( )+ δ( )u xB( )

= 0.75( ) 256( )2 + 0.25( ) 16( )2

= 49,216u xCE( ) = EU

xCE2 = 49,216

xCE = 49216= 221.85 2.d.p( )

RP = EV − xCE=196− 221.85= −25.85

!

!!

!! !

170 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

260

0

50

100

150

200

250

X

u(x)

u(x)

EVxce

u(EV)

EU

11.2 Insurance markets Back!to!our!friend!Ivan,!we!see!that!he!is!extremely!risk!averse.!What!can!he!do!in!order!to,!thus,!minimize!his!risk?!He!can,!of!course,!take!out!some!dodgy!bond!insurance!!!But!how!can!we!put!this!into!our!model?!!Essentially,!an!insurance!contract!is!made!up!of!two!parts,!a!premium!(“p”)!that!is!paid!regardless!of!the!outcome;!and!the!benefit!(“b”)!that!is!paid!only!if!the!bad!outcome!ends!up!happening.!!Where!before! xG and! xB described!the!payout!in!the!good!and!bad!states!respectively,!this!is!no!longer!sufficient.!!

In!the!good!state,!Ivan!now!receives! xG − p ,!the!good!outcome!less!the!premium!paid!for!insurance.!!In!the!bad!state,!Ivan!receives! xB − p+ b ,!the!bad!outcome,!less!the!premium!paid!for!insurance,!but!then!with!the!addition!of!the!benefit!from!the!insurance!paid!out.!

11.2.1 Actuarially(fair(insurance(In!this!model,!we!assume!that!the!insurer!has!full!knowledge!about!the!probability!of!each!outcome!occurring.!!Further,!we!assume!that!there!is!no!correlation!between!events!(that!it!happens!one!year!does!not!alter!the!probability!of!it!occurring!in!the!next)!or!between!individuals.!!Thus,!in!every!year,!a!fraction!of!insurance!customers!(δ )!is!paid!a!benefit,!whilst!the!remainder!is!not!(1−δ ).!!If!this!market!is!in!perfect!competition,!there!will!be!zero!profit:!

!

π = 0 = 1−δ( ) p( )+ δ( ) p− b( )0 = p−δp+δp−δbδb = p

b = pδ

!

Thus,!if!the!benefit!is!the!premium!divided!by!the!risk!of!the!bad!state!occurring,!the!insurance!is!actuarially!fair.!! !

11.3 Modeling consumption of insurance We!can!model!the!income!that!Ivan!will!receive!for!either!result!on!a!twoWgoods!plane,!as!in!the!consumer!models!we!have,!before,!created.!

11.3.1 The(model’s(framework(!Here,!we!plot!a!( xG, xB )!plane,!with!each!axis!representing!payout!in!each!outcome.!!!

• First,!we!plot!the!endowment!point!( eG,eB )!• Second,!we!plot! xG = xB –!a!line!whereby!payout!is!identical!regardless!of!

the!outcome.!

!

11.3.2 The(choice(set(and(budget(line(

Let!us!look!at!the!concept!of!actuarial!fairness!we!discussed!above:!b = pδ.!

If!Ivan!spends!one!more!dollar!on!insurance,!his!benefit!will!increase!by!b = 1δ.!!

Thus,!his!payouts!in!the!good!and!bad!state!would!change!as!follows:!• In!the!good!state,!the!only!change!will!be!a!reduction!of!his!payout!by!one!

dollar:!the!dollar!he!spent!on!additional!insurance!!! xg = −1 !• In!the!bad!state,!there!will!be!an!increase!in!the!premium,!but!also!an!

increase!in!the!benefit!paid!by!the!insurance!company!!!

xB =1δ[benefit]−1[premium]= 1−δ

δ= γ !

Note,!gamma!is!only!equal!to!the!ratio!of!probabilities!where!the!insurance!is!actuarially!fair.!!This!helps!us!define!the!budget!line!of!the!insurance!purchaser:!

γ xG + xB = γeG + eBxB = γeG + eB −γ xG

!

!! !

3500 50 100 150 200 250 300

350

0

50

100

150

200

250

300

xg

xb

xg=xb

e(256,16)

In!this!example,!if!the!insurance!is!actuarially!fair!!!γ = 1−δδ

=.75.25

= 3 .!

This!creates!the!budget!line!as!follows:!xB = γeG + eB −γ xG → xB = 3( ) 256( )+ 16( )− 3( ) xGxB = −3xG + 784

!!!A!linear!function!

!

!

11.3.3 Preferences(Much!like!the!consumer!model,!to!find!the!consumer’s!choice,!we!need!to!find!their!indifference!curves.!!Here,!we!can!use!expected!utility:!

U xG, xB( ) = δ •u xB( )+ 1−δ( )•u xG( ) !Note,!the!“u()”!function!depends!on!the!participant’s!risk!appetite.!!In!this!example,!Ivan’s!utility!function!is!u = x .!Thus,!his!utility!function!is!!

U xG, xB( ) = .25( ) xB + .75( ) xG !

3500 50 100 150 200 250 300

350

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250

300

xg

xbxg=xb

e

Budget Line

11.3.4 Choice(As!in!the!consumer!model,!utility!is!maximized!where!the!slope!of!the!budget!line!is!equal!to!the!MRS!of!the!utility!function.!!Here,!the!slope!is!equal!to!γ =!W3!!!why!do!we!call!it!φ !here?!!What!about!the!MRS?!

!

U xG, xB( ) = 14!

"#$

%&xB

12 +

34!

"#$

%&x

12G

MRS = δ /δxGδ /δxB

=

1234x−12

G

1214xB−12

=3441xB12

xG12

= 3 xBxG

!

!

Equating!this!with!the!slope!can!give!us!the!optimum!consumption!point!of!insurance:!

!

MRS =Op.Cost

3 xBxG

= γ = 3

xG = xBxG = xB

!

!That!is,!the!agent!will!insure!so!that!they!have!the!same!payout!regardless!of!the!result.!!This%is%not%a%coincidence!%Whenever%we%have%actuarially%fair%insurance%and%an%agent%who%is%risk%averse,%they%will%always%choose%to%fully%insure.%!

!

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xg

xb

xg=xb

e

Budget Line

u(e)

u

Now!we!know!that!the!equilibrium!point!happens!when!the! xG = xB –!we!can!solve!this!by!equating!this!(the! 45 line)!with!the!budget!line!as!follows:!!

!

xB = −3xG + 784→ xB = xGxB = −3xB + 7844xB = 784xB = xG =196

!

11.3.5 Choice(where(insurance(is(actuarially(unfair(In!the!above!example!we!had!the!optimum!mix!of!a!risk!averse!agent!and!actuarially!fair!insurance.!!But!what!about!where!this!is!not!the!case,!where!the!price!of!insurance!gives!the!insurer!profits?!!

!b = p

δ→ b = 2p

δδb2= p

!

!Reconsider!after!homework.!!The!result!is,!the!budget!line!will!be!shallower!and!there!would!be!less!insurance!consumed!than!would!be!otherwise.!! !

11.4 General equilibrium with uncertainty In!the!above!model,!we!showed!partial!equilibrium:!where,!given!an!agent!and!an!insurer!(with!fixed!prices)!we!were!able!to!show!how!much!insurance!a!party!to!the!contract!would!purchase.!!Now,!we!sophisticate!the!model!by!removing!the!insurer!and!having,!instead,!two!parties!who!are!able!to!sell!insurance!to!each!other.!!We!do!this!to!see!whether,!in!the!insurance!market,!the!first!welfare!theorem!holds:!that!is,!that!the!market!efficiently!allocates!resources.!

11.4.1 Uncertainty(without(aggregate(risk(Let!us!model!an!example,!using!General!Equilibrium.!!!

• Janice!and!Brennan!live!on!opposite!sides!of!an!island.!!!• Every!year,!there!is!a!δ chance!of!drought!( xD ),!and!a! (1−δ) !chance!of!

rain!( xR ).!!!• Janice’s!plants!yield!more!fruit!in!the!rain!( eR

J > eDJ ),!and!Brennan’s!in!the!

drought!( eBR < eB

D ).!!• Janice!and!Brennan!are!risk!averse!

However,!on!this!Island,!there!is!no!aggregate!risk:!the!same!amount!of!fruit!grows!each!year,!the!presence!of!drought!or!rain!merely!changes!where!the!fruit!grows.!!Settings:!

• Endowment!o xR

J =12; xRB = 3→ 15∑ !

o xDJ = 3; xD

B =12→ 15∑ !• Utility!function!

o UJ xr, xd( ) = δ • xd + 1−δ( ) xr !

o UB xr, xd( ) = δ • xd + 1−δ( ) xr !

!Brennan's Bananas (Rain)

Bren

nan'

s Ba

nana

s (D

roug

ht)

J's Bananas (Rain)

Janice's Bananas (Drought)

Janice

Brennan

E

3

12

12

3

• Brennan!and!Janice!are!risk!averse:!both!want!to!fully!insure!as!long!as!contracts!are!fair!

• Competitive!equilibrium!implies!price!ratio!is!equal!to!both!MRS’s!

o PR*

PD* =MRSJ =MRSB !

• Both!Brennan!and!Janice!get!full!insurance!equilibrium!allocation!is!efficient:!First!Welfare!Theorem!holds!

11.4.2 Choice(with(aggregate(risk(Where!there!is!aggregate!risk!(the!total!output!for!one!event!is!greater!than!the!total!output!for!the!other)!then!total!insurance!is!not!possible.!!!Here,!however,!both!parties!will!try!to!insure!each!other,!and!will!sell!insurance!contracts!until!their!MRSs!are!equal.!!

!!!! !

Brennan's Bananas (Rain)

Bren

nan'

s Ba

nana

s (D

roug

ht)

Brennan

Janice

3

24

6

12e

E

12 Monopolies(Ever!week,!so!far,!we!have!discussed!markets!where!each!actor!is!sufficiently!small!such!that!they!cannot,!individually,!affect!the!market!equilibrium!price.!That!is,!producers!faced!perfect!elasticity:!if!they!raised!their!prices!even!by!a!fraction,!they!lost!all!of!their!customers.!!Where!they!don’t!face!perfect!elasticity,!they!have!some!market!power.!!Where!they!are!the!only!supplier,!they!have!a!monopoly:!a!monopolist!can!dictate!the!market!price.!!Monopolies!may!exist!where!there!are!no!substitutes!to!the!good!they!produce,!and!when!there!are!barriers!to!entry!preventing!other!producers!from!entering!the!market!(these!may!be!natural,!legal!or!otherwise).!!Will!monopolies!lead!to!an!efficient!allocation!of!resources,!or!will!it!lead!to!a!deadweight!loss?!!Is!this!changed!if!there!is!price!discrimination?!!Note:!in!the!below!analysis!we!assume!that!MWTP=Demand!Function!to!simplify!our!welfare!analyses.!

12.1 Elasticity Price!elasticity!is!the!responsiveness!of!demand!to!changes!in!price:!the!percentage!change!in!quantity!for!a!percentage!change!in!price.!!

! εD =%Δx%Δp

=Δx / xΔp / p

=ΔxΔp

px!

Thus,!even!where!there!is!a!linear!demand!curve!(the!same!change!in!price!always!leads!to!the!same!change!in!quantity)!there!will!be,!along!the!line,!different!elasticity’s,!since!elasticity!depends!on!the!starting!values!for!price!and!quantity.!!Given! x p( ) = 8− 2p ,!and!a!one!unit!decrease!in!price,!what!are!the!elasticities!at!the!following!points?!!

!

!

εDA =

ΔxΔp

px

=−2132

= −3

! ! !

εDB =

ΔxΔp

px

=−2124

= −1

! ! !

εDC =

ΔxΔp

px

=−2116

= −13

!

As!we!can!see,!even!where!the!function!is!linear,!we!have!changing!elasticity.!

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P

x(p)=800-2p

a

b

c

12.2 Marginal Revenue Marginal!revenue!is!the!change!in!revenue!for!a!given!change!in!x.!!!With!monopoly!conditions,!the!following!are!true:!

! TR = p x( )• x→MR = δTRx

= p x( )+ δpδx

• x !

not!dtr/dx?!!Total!revenue!is!equal!to!a!price!function!of!x!multiplied!by!“x”!purchased.!Therefore,!marginal!revenue!is!equal!to!the!change!in!total!revenue!for!a!change!in!x.!!This!is!the!same!as!the!price!function!added!to!the!derivative!with!respect!to!x!of!the!price!function!multiplied!by!the!x!value.!!It!is!helpful!to!express!this!in!terms!of!price!elasticity:!

!

MR = p x( )+ δpδx

• x

= p x( )+ δpδx

• x•p x( )p x( )

multiply by p x( )p x( )

=1

= p x( ) 1+ δpδx

xp x( )

!

"##

$

%&&

we factorise and remember that p x( )→ p

= p x( ) 1+ 1εD

!

"#

$

%&

!

!

12.3 Graphing marginal and total revenue Given!the!above!settings!( x p( ) = 8− 2p )!we!can!graph!the!function!of!x!alongside!the!marginal!revenue!curve.!!The!first!step!is!to!invert!x(p)!to!find!p(x):!! !

!x = 800− 2p

p x( ) = 400− x2!

Then,!we!find!total!revenue:!

!TR = p x( )• x

= 400x − x2

2

!

And!we!find!the!elasticity!function:!

!εD =

δxδp

px

= −2 px

!

From!total!revenue,!we!can!find!marginal!revenue:!

! MR =δTR x( )δx

= 400− x!

What’s!the!point!of!expressing!MR!in!terms!of!elasticity?!

!! !

100 1 2 3 4 5 6 7 8 9

0

1

2

3

4

X

P

x(p)

MR

12.4 Profit maximization We!will!analyze!profit!maximization!points!and!welfare!in!a!singleWprice!monopoly,!and!with!different!models!of!price!discrimination!that!can!allow!the!monopolist!to!extract!for!welfare!from!the!market.!

12.4.1 Profit(maximization(in(the(market(In!a!competitive!market,!market!demand!is!downward!sloping,!but!each!individual!firm!(being!small!on!the!scale!of!the!market)!faces!horizontal!demand:!perfect!elasticity.!!If!the!producer!sets!price!above!marginal!costs,!then!consumers!will!shift!to!another!producer!and!it!receives!no!revenue.!!Therefore,!to!maximize!profit,!it!must!set!prices!at!p=MC=MR.!

12.4.2 Profit(maximization:(a(monopoly(without(price(discrimination(A!monopoly!differs!here:!where!an!individual!firm!in!the!market!faces!a!horizontal!demand!curve!because!of!competition,!in!a!monopoly!where!there!is!no!competition,!the!individual!firm!faces!market!demand!that!is!downward!sloping.!!Therefore,!the!monopolist!is!able!to!set!price!above!MC!without!losing!all!his!customers.!!!!His!profit!maximization!problem!is!as!follows:!! max

xπ = TR x( )−TC x( ) = p x( )• x − c x( ) !

!

!

δπδx

= 0→MR x( ) =MC x( )

→ p x( ) 1+ 1εD

"

#$

%

&'=MC x( )

!

Thus,!the!monopolist!maximizes!his!profit!where!MR=MC.!!We!can!find!the!monopoly!markup!by!rearranging!terms:!

!

p x( ) 1+ 1εD

!

"#

$

%&=MC x( )

p x( )+p x( )εD

=MC x( )

p x( )−MC x( ) = −p x( )εD

p x( )−MC x( )p x( )

= −1εD

!

Thus,!the!price!elasticity!of!demand!is!a!good!measure!of!monopoly!power.!! !

12.4.2.1 Example(Given!the!above!settings!( x = 800− 2p ;MR = 400− x )!and!the!new!setting:!

MC x( ) = 32x !we!can!work!out!the!ideal!production!for!a!monopolist!to!maximize!

profit.!!Profit!is!maximized!where!MR=MC:!

!

MR =MC32x = 400− x

52x = 400

x =160

!

Now,!we!work!out!the!price!to!attract!that!level!of!demand:!

!

x = 800− 2p160 = 800− 2p2p = 800−160 = 640p = 320

!

Graphically,!we!work!this!out!by!finding!the!point!of!intersection!in!the!MR!and!MC!curves,!and!extending!that!to!the!demand!curve!above.!

!

12.4.2.2 Welfare(analysis(An!equilibrium!point!is!efficient!if!MSC=MSB.!!The!monopolist!produces!at!a!point!where!MSC<MSB,!an!inefficient!production!with!a!deadweight!loss!that!can!be!shown!graphically.!

!! !

100 1 2 3 4 5 6 7 8 9

0

1

2

3

4

X

P

x(p)

MR

MC

320

160

100 1 2 3 4 5 6 7 8 9

0

1

2

3

4

X

P

x(p)

MR

MC

Xm

Pm

PeDeadweight Loss

Xe

Consumer Surplus

ProducerSurplus

12.4.3 Profit(maximization:(a(monopoly(and(first(degree(price(discrimination(Where!there!is!firstWdegree!price!discrimination,!the!monopolist!is!aware!of!each!individual’s!MWTP,!and!is!able,!thus,!to!charge!him!or!her!exactly!what!he!or!she!is!willing!to!pay.!!Therefore,!the!price!is!no!longer!equal,!but!sits!on!the!demand!curve.!

• In!this!situation,!it!is!optimal!for!the!monopolist!to!produceP x( ) =MC x( ) .!• FirstWdegree!price!discrimination!gets!rid!of!the!deadweight!loss!incurred!

in!the!nonWdiscriminating!model,!but!it!shifts!all!of!the!benefit!to!the!producer.!

!

12.4.3.1 Introducing(two(part(tariffs(One!way!for!the!monopolist!to!extract!surplus!from!the!market!is!to!change!the!purchase!mechanism!from!a!price!to!a!twoWpart!tariff,!including!a!fixed!fee!(F)!and!a!perWunit!price!(p).!!The!total!amount!paid!is!thus!P x( ) = F + x• p .!!

!!

100 1 2 3 4 5 6 7 8 9

0

1

2

4

X

P

x(p)=P

MR

MC

X

ProducerSurplus

100 1 2 3 4 5 6 7 8 9

0

1

2

4

X

P

x(p)=P

MR

MC

X

Tariff

Per unit fee

12.4.4 Profit(maximization:(a(monopoly(with(third(degree(price(discrimination(Where!there!is!third!degree!price!discrimination,!the!monopolist!knows!the!MWTP!of!each!customer!but!is!not!able!to!charge!him!or!her!different!prices!perfectly.!!He!is,!however,!able!to!charge!different!amounts!based!on!observed!traits.!!Whilst!this!is!an!improvement!over!the!nonWdiscriminating!model!when!it!comes!to!deadweight!loss,!it!is!not!efficient:!there!is!still!usually!a!nonWzero!DWL.!!Let!us!create!an!example!where!there!are!two!types!of!consumer,!each!with!a!different!MWTP:!

• pL x( ) =110− 12x !

• pR x( ) = 60− 12x !

To!simplify!this!example,!we!will!set!the!MC!at!a!constant!10.!!Because!they!are!able!to!discriminate!based!on!observed!traits!(e.g.!being!a!student!or!a!pensioner),!we!just!create!two!graphs,!and!set!the!price!at!MR=MC!for!each!one.!!Why!is!the!MR!curve!here!not!parallel!to!the!P!curve!because!MC!is!static?!

!! !

X

P

x(p)=P

MR

MC

X

DWL

Producer Surplus

Consumer Surplus

P

L

X

P

x(p)=P

MR

MC

X

DWLProducer Surplus

Consumer SurplusP

R

12.4.5 Profit(maximization:(a(monopoly(and(secondRdegree(price(discrimination(SecondWdegree!price!discrimination!occurs!when!a!monopolist!is!not!able!to!identify!individual!consumers’!MWTP.!!Instead,!the!monopolist!must!tempt!different!classes!of!people!to!selfWidentify!with!different!pricing!mechanisms:!either!by!varying!quality!with!price!or!by!limiting!quantity!purchasable!at!a!single!price.!!Since!there!is!no!selfWidentification,!and!the!monopolist!must!tempt!consumers!into!paying!more,!there!will!be!a!DWL,!and!this!(alongside!the!monopolists!profits)!will!be!greater!than!in!first!or!third!degree!price!discrimination.!!To!figure!out!the!most!efficient!pricing!models,!let!us!create!a!funky!surplus!map.!!

We!have!the!same!settings!as!before:! p1 x( ) =110− 12x , p2 x( ) = 60− 1

2x !

! !

!Let!us!look!at!an!ideal!set!of!pricing!packages:!

• p1 90( ) = d + 900 !• p2 200( ) = d + f + g+ h( ) !

The!poorer!customer!an!only!afford!the!smaller!packet,!and!will,!thus!purchase!that:!he!gets!no!surplus!and!the!producer!receives!“d”!in!producer!surplus.!!Let!us!sketch!the!possible!choices!for!the!richer!customer.!!If!he!chooses!the!first!packet,!he!receives!“e”!as!a!surplus.!!If!he!chooses!the!second!packet,!he!gets!“e+g”!as!a!surplus,!leaving!the!producer!with!“d+f+h”!as!a!surplus.!!This!is!the!optimal!choice,!because!f>g.!

!!!The!poor!person!cannot!afford!the!second!option.!!The!riche!person!increases!his!welfare!by!purchasing!the!second!option!(albeit!only!by!a!fraction).!!Therefore,!the!monopolist!will!end!up!having!a!surplus!of!d+d+f+h.!!Shouldn’t!the!ideal!price!be!d+f+h!rather!than!d+f+g+h.!!If!it!is!the!second,!there!is!no!increase!in!welfare!and!the!rich!guy!will!only!buy!the!first!packet?!!!!

X

P

D1

MC

D2

D

E

F

G

H

10

100 20090

X

P

D1

MC

D2

D

E

F

G

H

10

100 20090

Poor guy packet 1

X

P

D1

MC

D2

D

E

F

G

H

10

100 20090

Rich guy packet 1

X

P

D1

MC

D2

D

E

F

G

H

10

100 20090

Rich guy packet 2