Salop.E2.80.99s Circle Model Location model

1 salop’s circle model

1.1 example of second good

1.1.1 assumptions
1.1.2 analysis

salop’s circle model

one of famous variations of hotelling’s location model salop’s circle model. similar previous spatial representations, circle model examines consumer preference regards geographic location. however, salop introduces 2 significant factors: 1) firms located around circle no end-points, , 2) allows consumer choose second, heterogeneous good.

an example of second good
assumptions

assume consumers equidistant 1 around circle. model occur 1 time period, in 1 product purchased. consumer have choice of purchasing variations of product (a differentiated product) or product b (an outside good; undifferentiated product).

there 3 firms located equidistant around circle. each firm offers variation of product a, , outside firm offers good, product b.

analysis

in example, consumer wants purchase ideal variation of product a. willing purchase product, given within constraint of utility, transportation/distance costs, , price.

the utility



u



{\displaystyle u\,}

particular product @ distance



d



{\displaystyle d\,}

represented in following equation:

u
(
d
,

d

1


)
=
u
−
r

|

d
−

d

1



|




{\displaystyle u(d,d_{1})=u-r|d-d_{1}|\,}

where



u



{\displaystyle u\,}

utility superior brand,



r



{\displaystyle r\,}

denotes rate @ inferior brand lowers utility superior brand,



d



{\displaystyle d\,}

location of superior brand, ,




d

1





{\displaystyle d_{1}\,}

location of consumer. distance between brand , consumer thereby given in




|

d
−

d

1



|




{\displaystyle |d-d_{1}|\,}

.

the consumer’s primary goal maximize consumer surplus, i.e. purchase product best satisfies combination of price , quality. although consumer may receive more pleasure superior brand, inferior brand may maximize surplus



c
s



{\displaystyle cs\,}

given by:

u
(
d
,

d

1


)
−
p
=
c
s



{\displaystyle u(d,d_{1})-p=cs\,}

, difference between utility of product @ location



d



{\displaystyle d\,}

, price



p



{\displaystyle p\,}

.

now suppose consumer has option purchase outside, undifferentiated product b. consumer surplus gained product b denoted




u

∗





{\displaystyle u^{*}\,}

.

therefore, given amount of money, consumer purchase superior variation of product on product b long as

u
(
d
,

d

1


)
−
p
≥

u

∗





{\displaystyle u(d,d_{1})-p\geq u^{*}\,}

, consumer surplus superior variation of product greater consumer surplus gained product b.

alternatively, consumer purchases superior variation of product long as

u
−

u

∗


−
r

|

d
−

d

1



|

−
p
≥
0



{\displaystyle u-u^{*}-r|d-d_{1}|-p\geq 0\,}

, difference between surplus of superior variation of product , surplus gained product b positive.