Mall Boom or Mall Boom or BustBust

Kami ColdenKami Colden

Brad TeterBrad Teter

Devin WayneDevin Wayne

Jane ZieliekeJane Zielieke

Shan HuangShan Huang

Our PresentationOur Presentation

What is a mallWhat is a mall Discrete logistic growth modelDiscrete logistic growth model Assumptions we madeAssumptions we made Our modelOur model Our findingsOur findings Our conclusionOur conclusion

What is a Mall?What is a Mall?

A collection of independent retail stores, A collection of independent retail stores, services, and a parking area conceived, services, and a parking area conceived, constructed, and maintained by a constructed, and maintained by a management firm as a unit.management firm as a unit.

Shopping malls may also contain Shopping malls may also contain restaurants, banks, theatres, restaurants, banks, theatres, professional offices, service stations, professional offices, service stations, and other establishments.and other establishments.

ThunderbirdThunderbird

Located in Menomonie, Located in Menomonie, WIWI

London SquareLondon Square

Located in Eau Located in Eau Claire, WIClaire, WI Younker’sYounker’s

OakwoodOakwood Located in Eau Claire, WILocated in Eau Claire, WI 8 million visits per year 8 million visits per year 130 stores130 stores Key Attractions:Key Attractions:

Department StoresDepartment Stores Women's ApparelWomen's Apparel Housewares & HomeHousewares & Home Books & Books &

EntertainmentEntertainment Movie TheaterMovie Theater Food Court and Food Court and

RestaurantRestaurant

Mall of AmericaMall of America

Located in Bloomington, MNLocated in Bloomington, MN Currently the largest fully Currently the largest fully

enclosed retail and enclosed retail and entertainment complex in entertainment complex in the United States.the United States.

More than 520 storesMore than 520 stores 600,000 to 900,000 weekly 600,000 to 900,000 weekly

visits depending on seasonvisits depending on season Nearly $1.5 billion annually Nearly $1.5 billion annually

incomeincome

Discrete Logistic Discrete Logistic Growth ModelGrowth Model

Population ModelPopulation Model

X(n) = population of the mall at year nX(n) = population of the mall at year n

r = the intrinsic growth rate of the storesr = the intrinsic growth rate of the stores

The difference between the current and The difference between the current and previous year is represented by the previous year is represented by the equation: equation:

X(n + 1) – X(n) = rX(n)X(n + 1) – X(n) = rX(n)

Population Model (cont.)Population Model (cont.)

The population for the next year The population for the next year would be represented by the would be represented by the equation:equation:X(n+1) = RX(n) where R = r + 1X(n+1) = RX(n) where R = r + 1

Our model assumes that the growth Our model assumes that the growth rate is dependant on the rate is dependant on the population. So, growth rate would population. So, growth rate would be represented by r(x).be represented by r(x).

Carrying CapacityCarrying Capacity

The carrying capacity of the store The carrying capacity of the store population would be the maximum population would be the maximum number of stores possible given number of stores possible given current space restrictions. The current space restrictions. The carrying capacity is represented by carrying capacity is represented by a constant K.a constant K.

Ockham’s RazorOckham’s Razor

If there are several possible explanations If there are several possible explanations for some observation, and no significant for some observation, and no significant evidence to judge the validity of those evidence to judge the validity of those hypotheses, you should always use the hypotheses, you should always use the simplest explanation possible.simplest explanation possible.

Also known as the principle of parsimony Also known as the principle of parsimony – scientists should make no more – scientists should make no more assumptions or assume no more causes assumptions or assume no more causes than are absolutely necessary to explain than are absolutely necessary to explain their observations.their observations.

By Ockham’s RazorBy Ockham’s Razor

Growth rate would be linear (of the Growth rate would be linear (of the form r(x) = mx + b)form r(x) = mx + b)

r(0) = r(0) = (an intrinsic growth rate (an intrinsic growth rate without regard to restrictions like without regard to restrictions like space)space)

By Ockham’s Razor (cont.)By Ockham’s Razor (cont.)

r(K)= 0 (no growth)r(K)= 0 (no growth)

r(x) = -(r(x) = -( /K)x + /K)x + r(X(n)) = -(r(X(n)) = -( /K)x + /K)x +

(0, )

(K, 0)

Basic Logistic Population Basic Logistic Population ModelModel

X(n+1) – X(n) = [-(X(n+1) – X(n) = [-( /K)x + /K)x + ]X(n)]X(n)

X(n+1) = [-(X(n+1) = [-( /K)x + /K)x + ]X(n) + X(n)]X(n) + X(n)

X(n+1) = X(n)[1+ X(n+1) = X(n)[1+ (1-X(n)/K)](1-X(n)/K)]

Steady StateSteady State

A steady state is a point where an A steady state is a point where an system “likes” to remain once reached.system “likes” to remain once reached.

The fundamental equation X(n+1) = The fundamental equation X(n+1) = f(X(n)) is a 1f(X(n)) is a 1stst order recurrence equation. order recurrence equation.

To find the steady states of our model To find the steady states of our model solve the following equation for X:solve the following equation for X:

X[1+ X[1+ (1-X(n)/K)] = X(1-X(n)/K)] = X

X = 0 , X= KX = 0 , X= K

Steady State (cont.)Steady State (cont.)

Essentially, once the mall reaches Essentially, once the mall reaches capacity it has will most likely capacity it has will most likely remain full.remain full.

Conversely, once a mall becomes Conversely, once a mall becomes vacant it is highly unlikely that any vacant it is highly unlikely that any stores will be attracted to the stores will be attracted to the location.location.

StabilityStability

Stability is the tendency to Stability is the tendency to approach a steady state.approach a steady state.

To determine stability, find the To determine stability, find the derivative of f(x) = X[1+ derivative of f(x) = X[1+ (1-X(n)/K)](1-X(n)/K)] Which is: f’(x) = 1 + Which is: f’(x) = 1 + - (2 - (2 /K)X/K)X

Stable if |f’(x)| < 1Stable if |f’(x)| < 1

Stability (cont.)Stability (cont.)

Findings:Findings:

If the intrinsic growth rate is out of range, If the intrinsic growth rate is out of range, we find chaotic behavior in the model.we find chaotic behavior in the model.

f’(0)f’(0) f’(K)f’(K)

f’(0) = 1 + f’(0) = 1 + f’(K) = 1 + f’(K) = 1 + - 2 - 2 |1 + |1 + | < 1| < 1 = |1 – f| < 1= |1 – f| < 1

(0 is an unstable(0 is an unstable -1 < 1 - -1 < 1 - < 1 < 1

fixed point)fixed point) 0 < 0 < < 2 < 2

Assumptions

Assumptions We MadeAssumptions We Made

The mall is a fixed size and The mall is a fixed size and locationlocation

In our model we will be considering In our model we will be considering customers, stores, and mall customers, stores, and mall management.management.

Assumptions (cont.)Assumptions (cont.)

Mall management rationally and Mall management rationally and intentionally controls what they intentionally controls what they charge for rent in an effort to get a charge for rent in an effort to get a maximum profit for the mall.maximum profit for the mall.

Stores pass rent off to the Stores pass rent off to the customer within the prices of the customer within the prices of the products they sell.products they sell.

Assumptions (cont.)Assumptions (cont.)

SymbiosisSymbiosis Population of customers and stores are Population of customers and stores are

positively associated.positively associated. If one increases or decreases the other If one increases or decreases the other

follows until they reach capacity.follows until they reach capacity. Finite Carrying CapacityFinite Carrying Capacity

There is a maximum number of There is a maximum number of customers and stores a mall can have.customers and stores a mall can have.

Laws of economicsLaws of economics

Supply is positively associated with the price.Supply is positively associated with the price. Demand is negatively associated with the price.Demand is negatively associated with the price.

Demand Curve

Supply Curve

Equilibrium PointP

rice

(do

llar

s)

Quantity

Opportunistic RentOpportunistic Rent

Year n-1Year n-1 stores make a profitstores make a profit

Year nYear n mall management increases the rent mall management increases the rent

to maximize their profitto maximize their profit stores pass off the increase of rent to stores pass off the increase of rent to

the customers by increasing pricesthe customers by increasing prices

Opportunistic Rent (cont.)Opportunistic Rent (cont.)

Year n+1Year n+1 A noticeable loss in customers will be A noticeable loss in customers will be

observed and store will lose profitobserved and store will lose profit Year n+2Year n+2

stores will leave if not making a profitstores will leave if not making a profit mall management will have to decrease the mall management will have to decrease the

rent to keep stores or get new stores to move rent to keep stores or get new stores to move inin

This cycle will continue until mall This cycle will continue until mall management and the stores both reach management and the stores both reach an agreeable opportunistic rent.an agreeable opportunistic rent.

Misc. Factors Not Misc. Factors Not ConsideredConsidered

Niche effectiveness Niche effectiveness (different types of (different types of stores)stores)

Price elasticity Price elasticity (insensitivity to price (insensitivity to price change)change)

Economies of scale Economies of scale (more variety)(more variety)

Population of surrounding areaPopulation of surrounding area Attractiveness of the mallAttractiveness of the mall

Our Model

Formulating the Mall Formulating the Mall ModelModel

Let X(n) be the population of mall Let X(n) be the population of mall customers at year ncustomers at year n

Let Y(n) be the number of stores in Let Y(n) be the number of stores in the mall at year nthe mall at year n

Let K be the mall carrying capacity Let K be the mall carrying capacity of storesof stores

The CustomersThe Customers

Population of customers is Population of customers is proportional to the number of proportional to the number of stores in the mall:stores in the mall:

X(n + 1) = A * Y(n)X(n + 1) = A * Y(n)

where A is a multiple of the storeswhere A is a multiple of the stores

that are openthat are open Then A * K will be the customer Then A * K will be the customer

carrying capacity of the mallcarrying capacity of the mall

The StoresThe Stores

The store model based on the The store model based on the discrete logistic growth model isdiscrete logistic growth model is

Y(n + 1) = Y(n)[1 + Y(n + 1) = Y(n)[1 + (1 – Y(n) / K)](1 – Y(n) / K)] Where Where is the intrinsic growth rate is the intrinsic growth rate

(the rate at which the stores fill the (the rate at which the stores fill the mall)mall)

Minimum Operating CostsMinimum Operating Costs

ElectricityElectricity InsuranceInsurance Snow removalSnow removal Etc.Etc.

The Greed Factor (Opportunistic The Greed Factor (Opportunistic Rent)Rent)

Incorporating the greed factor into the Incorporating the greed factor into the customer modelcustomer model

X(n + 1) = A*Y(n) - R(X(n), Y(n))X(n + 1) = A*Y(n) - R(X(n), Y(n)) Where R(X, Y) represents the customers Where R(X, Y) represents the customers

attrition due to the greed factorattrition due to the greed factor Let R(X(n), Y(n)) = Let R(X(n), Y(n)) = (n)X(n) + (n)X(n) + (n)Y(n)(n)Y(n) For some positive sequences of {For some positive sequences of {(n)}, (n)},

(n)}(n)}

Building the Mall ModelBuilding the Mall Model

The CustomersThe CustomersX(n + 1) = A * Y(n) - X(n + 1) = A * Y(n) - (n)X(n) - (n)X(n) - (n)Y(n)(n)Y(n)

Where - Where - (n)X(n) - (n)X(n) - (n)Y(n) is customer attrition(n)Y(n) is customer attritionfrom last years price increasefrom last years price increase

The StoresThe StoresY(n + 2) = Y(n +1)[1 + Y(n + 2) = Y(n +1)[1 + (1 - Y(n) / K)] - B((1 - Y(n) / K)] - B((n)X(n) + (n)X(n) +

(n)Y(n)(n)Y(n)

Where the B is a constant multiplied by theWhere the B is a constant multiplied by thecustomer attrition in year ncustomer attrition in year n

Behold the Mall ModelBehold the Mall Model

Customers:Customers:

X(n + 1) = A * Y(n) - X(n + 1) = A * Y(n) - (n)X(n) - (n)X(n) - (n)Y(n) (n)Y(n)

Stores:Stores:

Y(n + 1) = y(n) )[1 + Y(n + 1) = y(n) )[1 + (1 - Y(n) / K)] (1 - Y(n) / K)] - B(- B((n - 1)X(n - 1) - (n - 1)X(n - 1) - (n - 1)Y(n - 1))(n - 1)Y(n - 1))

Mall Management & Mall Management & MoneyMoney

A large greed factor will produce A large greed factor will produce millions right away = no profits in millions right away = no profits in years to comeyears to come

Why?Why? Stores have moved or gone out of Stores have moved or gone out of

business, since increase in rent business, since increase in rent was passed onto customers, whom was passed onto customers, whom have gone elsewhere to find lower have gone elsewhere to find lower pricesprices

Mall Viability Mall Viability

The key to mall viability is a The key to mall viability is a function of the mall managements function of the mall managements long term profits long term profits

ΣΣ2424n=0n=0(((n)X(n) + (n)X(n) + (n)Y(n))(n)Y(n))

Want Want has high as possible has high as possible without driving stores out and new without driving stores out and new stores from moving in due to high rentstores from moving in due to high rent

Want to find sequences of {Want to find sequences of {(n)}, (n)}, {{(n)} which will maximize this sum(n)} which will maximize this sum

Our Model at WorkOur Model at Work

Many thanks toMany thanks to

Manager at Ben FranklinManager at Ben Franklin Marketing personal at Marketing personal at

Oakwood MallOakwood Mall www.britannica.com www.oakwoodmall.com www.mallofamerica.com And of course, Mr. And of course, Mr.

DeckelmanDeckelman