Menu pricing and matching of preferences for sushis

Menu pri ing and mat hing of preferen es for sushisAurélien Poissonnier(aurelien.poissonnier�insee.fr)Insee-Crest-LMA14 November 2013A. Poissonnier (Insee) Sushi menu Nov.13 1 / 37

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WarningAll omments are wel ome an be linked to several literatures

◮ produ t di�erentiation◮ prin ipal multi-agent◮ pri e dis riminationThe story behind it

A. Poissonnier (Insee) Sushi menu Nov.13 2 / 37

Plan1 General set-up2 Let's simplify �rst3 Generalization4 Risk Neutral Restaurant5 Risk Averse Restaurant6 Real menusA. Poissonnier (Insee) Sushi menu Nov.13 3 / 37

General set-upIn this model, �rms sellprepa ked multiprodu t o�erswhi h di�er by their relative omposition (relative quantity of ea hsushi)by their omposing elements (sushis)by their size (number of sushis)Consumerslike di�erent goods di�erentlymay be more or less hungry an always hose something else (pizza)A. Poissonnier (Insee) Sushi menu Nov.13 4 / 37

What is the demand for sushi menus depending on householdspreferen es ?What is the optimal pri ing poli y of the restaurants ?A. Poissonnier (Insee) Sushi menu Nov.13 5 / 37


Graphi ally: Salop 1979| 1 sushi-1sashimi

θ

onsumer's preferen esmenu o�ereddislike P

M

Figure : Consumers' preferen e with respe t to tuna sushis and sashimisUtility from eating menu M: U(M,P) = P×M = cos(θ)A. Poissonnier (Insee) Sushi menu Nov.13 7 / 37

Pri e dis rimination with bundling (Stole, 2007, se t.7)Sushi menus = mixed bundlingUse market ompetition and in�nite potential of di�erentiation todisregard market segmentation omponent pri ing is hardly a spe ial ase


A prin ipal multi-agent problemMirrlees 1976: labour market ex. of one prin ipal in ompetition withothers, intera ting with a ontinuum of agentsAt this pri e, I'd rather have a pizza...The prin ipalperfe tly monitors the agents' e�ort (pri e PM )but is unsure about his parti ipationtrade-o� between high parti ipation-low pri e and highpri e-low parti ipationThe agents ompare a given menu with everything elseutility from menu must be higher than a ba k-up solution (othermenu, pizza...)This ba k-up is modelled by a reservation utility or renoun ement ostU(M,P) > R = ρPMA. Poissonnier (Insee) Sushi menu Nov.13 9 / 37

θ

UtilityρPM

θmax

cos(θ)Figure : Consumers' utility as a fun tion of the angle between his preferen esand the menuwith θmax = acos(ρPM )A. Poissonnier (Insee) Sushi menu Nov.13 10 / 37

Expe ted pro�tAssume onsumers' preferen es uniformly distributed.Potential lients pass by and hoosePreferen es unknown to the restaurant and only partially revealed by hoi e: if θ < θ(max,M) they hose the menu at pri e PM .Demand fun tion:P(M) =

θ(max,M)

π=

acos(ρPM )

π(1)

CM the ost for produ ing menu M, the expe ted pro�t of therestaurant on menu M is proportional to:E(Profit) ≡ (PM − CM )P(M) = (PM − CM )

acos(ρPM )

π(2)A. Poissonnier (Insee) Sushi menu Nov.13 11 / 37


Ve tors and norms= ompositions and sizesS menu size, PM pri e, CM ost.M is a normalized ve tor for menu omposition (ea h dire tion is amenu item)SM: "two tuna sushis, 6 alifornia makis, 3 hi ken yakitoris, 1 misosoup and a bowl of ri e"P the relative preferen e for ea h item and A appetite.Utility

UA,P (S,M) = fA(S) ∗P×M = fA(S) ∗ cos(θ) (3)A eptan e angleθ(max,M) = acos

(

ρPM

fA(S)

) (4)A. Poissonnier (Insee) Sushi menu Nov.13 13 / 37

Multi menuA multinomial random variable: for ea h menu, out ome PM − CMhappens with probability P(M), and out ome no pro�t happening whenthe onsumer does not �nd any menu to his taste.E(Profit) ≡

∑

M

(PM − CM )P(M) =∑

M

(PM − CM )θ(max,M)

π(5)

=∑

M

(PM − CM )1

πacos

(

ρPM

fA(SM )

) (6)NB: as in Salop 1979, I assume that menu di�erentiation is given. I assume that asimilar result to E onomides 1989 holds, restaurants have an in entive to in reasemenu di�erentiation so that a preferen e draw is not within two menus a eptan e'sangleA. Poissonnier (Insee) Sushi menu Nov.13 14 / 37


Restaurant's programPri es will be independent, onsider one menu :MaxPM

(PM − CM )acos(BPM )

π(7)with B = ρ

fA(SM )Optimal pri e P̂M is the solution to∀M

∂E(Profit)

∂PM

= 0 =1

π

acos (BPM )−B(PM − CM )√

1− (BPM )2

(8)⇔ acos (BPM )

√

1− (BPM )2 = B(PM − CM ) (9)A. Poissonnier (Insee) Sushi menu Nov.13 16 / 37

Properties of the optimal pri es1 The optimal pri es are independent from one another2 They are also independent from the relative preferen es of the onsumer3 Pri es are in reasing in the produ tion ost but the mark-up isde reasing in the produ tion ost4 Pri es are de reasing in the onsumer's reservation utility per e (ρ)5 Pri es are in reasing but on ave fun tions of menu sizesA. Poissonnier (Insee) Sushi menu Nov.13 17 / 37


Expe ted pro�t and varian eE(Profit) ≡ (PM − CM )

acos(BPM )

π(10)

V(Profit) ≡ (PM − CM )2acos(BPM )

π

(

1−acos(BPM )

π

) (11)The restaurant solves the following optimization program:MaxPM

E(Profit)−κ

2V(Profit) (12)with κ the absolute risk aversion of the restaurantA. Poissonnier (Insee) Sushi menu Nov.13 19 / 37

CM 1PM

0

E(Profit) & V(Profit)

V(Profit)E(Profit)E(Profit)− 2V(Profit)E(Profit)− V(Profit)

Elligible Pri esProfit < 0 No Mat hP̂MP̃M

Figure : Expe ted pro�t and its varian e as fun tions of PM (with CM = 0.2and B = 1)A. Poissonnier (Insee) Sushi menu Nov.13 20 / 37

Properties of the varian e1 The varian e of the restaurant's pro�t is a hump shape fun tion onCM ≤ PM ≤ 1

B2 The varian e of the restaurant's pro�t is maximum for P̃M ≥ P̂MProperties of the optimal pri e in a risk averse restaurant1 To redu e the risk on its pro�t, the restaurant will hoose a pri elower than P̂M , the optimal pri e if it was risk neutral2 The pri e is in reasing in the produ tion ostA. Poissonnier (Insee) Sushi menu Nov.13 21 / 37

Multi (2) menuThe restaurant will solve the following problem:Max

P1,P2

E1 + E2 −κ

2(V1 + V2 − E1E2) (13)


0.0

0.2

0.4

0.6

0.8

1.0

0.00.2

0.40.6

0.81.0

0.00

0.05

0.10

0.15

0.20

0.25

P1

P2

E(Profit)

0.00

0.05

0.10

0.15

0.20

Figure : Shape of the Expe ted pro�t of the restaurant (under theassumptions that B1 = B2 and C1

B1

= 0.4,C2

B2

= 0.2)A. Poissonnier (Insee) Sushi menu Nov.13 23 / 37

0.0

0.2

0.4

0.6

0.8

1.0

0.00.2

0.40.6

0.81.0

0.00

0.02

0.04

0.06

0.08

0.10

P_1

P_2

V(Profit)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Figure : Shape of the Varian e of pro�t of the restaurant (under theassumptions that B1 = B2 and C1

B1

= 0.4,C2

B2

= 0.2)A. Poissonnier (Insee) Sushi menu Nov.13 24 / 37

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

c2

Pric

es

risk neutral P1risk neutral P2risk adverse P1risk adverse P2

2 4 6 8 10 12 14

0.4

0.5

0.6

0.7

0.8

absolute risk aversionP

rices

risk neutral Prisk adverse PFigure : Optimal pri es in a risk neutral or risk averse restaurant as fun tionsof the ost of menu 2 (C2) or the absolute risk aversion (κ)A. Poissonnier (Insee) Sushi menu Nov.13 25 / 37

0.0 0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

0.25

c2

Exp

ecte

d P

rofit

risk neutral profitrisk adverse profit

2 4 6 8 10 12 14

0.04

0.06

0.08

0.10

0.12

0.14

0.16

absolute risk aversionE

xpec

ted

Pro

fit

risk neutral profitrisk adverse profitFigure : Expe ted pro�t in a risk neutral or risk averse restaurant asfun tions of the ost of menu 2 (C2) or the absolute risk aversion (κ)A. Poissonnier (Insee) Sushi menu Nov.13 26 / 37

0.0 0.2 0.4 0.6 0.8 1.0

1.00

1.01

1.02

1.03

1.04

1.05

Relative Preferences

Rel

ativ

e P

rices

risk neutral P1/P2risk adverse P1/P2, k= 1risk adverse P1/P2, k= 3risk adverse P1/P2, k= 5risk adverse P1/P2, k= 7risk adverse P1/P2, k= 9risk adverse P1/P2, k= 11risk adverse P1/P2, k= 13risk adverse P1/P2, k= 15risk adverse P1/P2, k= 17risk adverse P1/P2, k= 19

Figure : Relative pri es as a fun tion of the relative preferen e for the twomenus (2 ompared to 1)(under the assumptions that B1 = B2 = 1 andC1,2 = 0.4)A. Poissonnier (Insee) Sushi menu Nov.13 27 / 37

1 2 3 4 5

6

8

10

12

14

Size of Menu 2

Pric

e

1

1.33

1.67

2

2.33

(<−) risk neutral (Menu 1)(<−) risk neutral (Menu 2)(−>) risk adverse (Menu 1)(−>) risk adverse (Menu 2)

1 2 3 4 5

3

4

5

6

Size of Menu 2P

rice

/ Siz

e

0.5

0.67

0.83

1

(<−) risk neutral (Menu 1)(<−) risk neutral (Menu 2)(−>) risk adverse (Menu 1)(−>) risk adverse (Menu 2)

Figure : Pri es of both menus as fun tions of the size of menu 2 (under theassumptions that κ = 6 and C1,2 = 0.4)A. Poissonnier (Insee) Sushi menu Nov.13 28 / 37


Sushishop.eu

Figure : Relatives pri es of sushishop's menus in 8 ountriesA. Poissonnier (Insee) Sushi menu Nov.13 30 / 37

Pri e on avity as a fun tion of size (on fren h menu)PM = α+ β1Nsushi + β2Nmaki + γ

(

∑

item

NitemPitem

)

+ δkid + δlunch + ε

∑

itemNitemPitem is the pri e of the platter if ea h item wasbought separatelyδkid, δlunch are a dummies for kids menu and lun h boxNsushi and Nmaki are introdu ed to apture a size e�e t beyondthe linear in rease in the produ tion ostα and δlunch are not signi� ant.γ = 93%: 7% dis ount with respe t to pur hasing omponentsseparately.Kids menus: 2e dis ount.negative size e�e ts: adding a sushi (resp. maki) ⇒ -14 ents(resp. -6 ents) but not signi� antA. Poissonnier (Insee) Sushi menu Nov.13 31 / 37

19 restaurants in Paris left bank and southern suburbsPM = α+ β ‖ Menu ‖1 +γ

(

∑

item

NitemPitem

)

+ δlunch + ε

PM = α+ β ‖ Menu ‖2 +γ

(

∑

item

NitemPitem

)

+ δlunch + ε

PM = α+ β ‖ Menu ‖1 +β2 ‖ Menu ‖21 +γ · · ·

PM = α+ β ‖ Menu ‖2 +β2 ‖ Menu ‖22 +γ · · ·Models 1 & 2 tested for the 19 restaurants: no signi� ant and negativesize e�e t.Models 3 & 4: pri es almost never on ave fun tions of sizeThree restaurants (asiat, hongjiu and sushi love) I �nd β2 < 0 andsigni� ant using the quadrati norm (resp. −5 ents per item, −2 entsper item and −4 ents per item), but small with respe t to a sushiwhi h osts separately more than 1.5e.A. Poissonnier (Insee) Sushi menu Nov.13 32 / 37

Some heaper itemsI estimate the following models:

PM = α+

(

∑

item

γitemNitem

)

+ δlunch + ε (14)For ea h restaurant, the model explains a large share of the pri es'varian e and the RMSE is around 1e.A. Poissonnier (Insee) Sushi menu Nov.13 33 / 37

model residual

−4 −2 0 2 4 6 8

asiat

directsushi

ginza

hongjiu

idsushi

mitaka

ninjasushi

ohsushi

sayorisushi

sushidaguerre

sushijuliette

sushiking

sushikyo

sushilove

sushiscene

tokyo

xinja

yasami

yoksushi

Figure:Residualsofthepri ingmodelsin19restaurantsA.Poissonnier(Insee)

SushimenuNov.1334/37

Salmon and TunaH0: the realtive pri e for sushi and sashimis re�e ts the relativepri e of tuna and salmon.reje ted in 9/13 asestuna sushis (reps. sashimis) are free in 5 (resp. 6) restaurantstuna (luxury �sh) an be preferred to salmon and restaurantsredu e its pri e to in rease the volume (instead of mark-up)Temaki (also nem, gyoza, tempura)4 to 5 e a pie e bought separatelyless than 3 e in a menu an be risk averse restaurant a ommodating for preferen esA. Poissonnier (Insee) Sushi menu Nov.13 35 / 37

Deserts and drinksMenus for two sometimes in lude drinks or deserts.They are largely overpri ed (+5-10 e)Not in single menus...Symmetri ally to rebate for most preferred menus, there is a ost tovery spe i� preferen esRi eIn menus, the pri e of a bowl of ri e is either non signi� ant or negative(in one restaurant only it is positive)It is the heapest way to in rease menu sizeCan be interpreted as small rebate for hungry ostumersA. Poissonnier (Insee) Sushi menu Nov.13 36 / 37

Residualnonlinearitiesmodel residual

−6 −4 −2 0 2 4 6 8

asiat

directsushi

ginza

hongjiu

idsushi

mitaka

ninjasushi

ohsushi

sayorisushi

sushidaguerre

sushijuliette

sushiking

sushikyo

sushilove

sushiscene

tokyo

xinja

yasami

yoksushi

menu residual price

+/−

std residualresidual price of selected m

enusm

ean residual of selected menus

Figure:Residualsofthepri ingmodelsin19restaurantsandidenti� ationofmenuswithoutraw�sh

A.Poissonnier(Insee)Sushimenu

Nov.1337/37

Economy & Finance

Menu pricing and matching of preferences for sushis