Franchise Contracting: The Effects of The Entrepreneur’s Timing

Option and Debt Financing

Volodymyr Babich and Christopher S. Tang∗

July 1, 2014

Abstract

We solve a sequential-moves game that involves three players: the franchisor, the entrepreneur,

and the banks. The franchisor chooses values of contract terms (a one-time franchise fee and

a royalty rate for on-going payments). The entrepreneur dynamically decides when to sign

this contract and open a store, applying for debt financing to cover the initial investment. In

response to the entrepreneurs application, banks competitively determine loan rates. We find

that the franchisor should use royalty cash flows and not franchise fee to extracts value from

the entrepreneur. This is a new explanation of an empirical evidence that franchise contracts

favor royalties over fees. To account for the possibility of the entrepreneurs bankruptcy and

bankruptcy costs, the franchisor should decrease royalty rate. However, despite lower rate, the

threshold for the entrepreneur to open the store is higher in the model with financing than in the

model without financing. This threshold is much higher than it would have been for the inte-

grated system, which in turn is higher than the static break-even-NPV threshold. If a franchisor

ignores financing considerations, she will suffer from having to wait longer for a store opening

and from a higher entrepreneur’s bankruptcy probability. We predict that the franchisor is the

main beneficiary of the entrepreneur’s greater initial wealth and that the franchisor will benefit

if she assumes a greater share of the store’s operating costs.

∗Babich: McDonough School of Business, Georgetown University, 37th and O Streets NW, Washington, DC 20057,E-mail: vob2@georgetown.edu; Tang: UCLA Anderson School, 110 Westwood Plaza, Los Angeles, CA, 90095, E-mail:chris.tang@anderson.ucla.edu.

1 Introduction

According to Blair and Lafontaine (2011) (§1.1), a franchise agreement is a contractual relationship

between two firms where one firm (the franchisee) pays the other firm (the franchisor) for the right

to sell the franchisor’s product and/or the right to use its trademarks and business format in

a given location for a specified period of time. Franchising contributes significantly to the US

economy. Based on the 2007 US Economic Census,1 franchises comprise 10.5% of all businesses in

295 industries covered by the Census. Franchises account for 9.7% of the annual payroll and 13.5% of

the total employment. The majority of franchises are in the Accommodation and Food sector (43%),

with the retail sector being second (27%). For this paper a good set of examples of franchises comes

from a list of top ten franchises for 2014 presented by The Entrepreneur magazine: Anytime Fitness,

Hampton Inn, Subway, Supercuts, and Jimmy Jones. To make it easier to differentiate between

franchisor and franchisee, we shall refer to the owner of the franchisee firm as the entrepreneur,

who we shall imagine to be a small-business owner and not a wealthy businessmen such as Richard

Branson or Elon Musk. We shall refer to franchise outlets as stores.

While franchising is a cornucopia of research questions, probably one of the most important

ones is how to choose the contract terms. A typical franchise contract comprises a franchise fee

and a royalty rate (Blair and Lafontaine 2011, §3). The franchise fee is a one-time payment

from the entrepreneur to the franchisor, paid at the time the contract is signed. The royalty

rate is a fraction of revenues or profits from the store paid by the entrepreneur to the franchisor.

Franchise fees and royalty rates are sometimes divided into multiple categories (e.g., royalty rates

are sometimes presented as advertising fees, Sen 1993, Bhattacharyya and Lafontaine 1995), but

their nature remains — fees are paid once at the contract signing and royalty rates represent on-

going payments. Numerous empirical studies of franchise contracts show that fees represent a small

percentage of the total value transferred to the franchisor. For example, Shane et al. (2006) report

that the mean fee is $17,690, the mean royalty rate is 5.27%, and the mean initial CAPX investment

is $122,970. As a comparison, Blair and Lafontaine (2011) calculate (footnote 68) that a royalty

rate 5% on revenues of $500,000 per year (relatively low number) over a 15-year contract adds up

to $375,000; and a franchise fee of $20,000 is just slightly above 5% of the total royalty payments.

Why the value from royalties should be so high relative to fees is somewhat a puzzle because

many economic models predict the opposite. Under perfect information, economists suggest that

the franchisor should simply sell the rights to the business outright to the entrepreneur. In fact, as

1http://www.census.gov/econ/census/pdf/franchises_snapshot.pdf

1

long as there is an agreement regarding the value of on-going cash flows, this value can be captured

by the one-time fee (Caves and Murphy 1976, Blair and Kaserman 1982). Moreover, in the presence

of the moral hazard problem with respect to the entrepreneur’s effort, it is optimal to set the royalty

rate to zero to make the entrepreneur the full residual-rights claimant of the store (Brickley 2002).

A number of explanations for this empirical puzzle has been proposed. For example, Blair

and Kaserman (1982) and Bhattacharyya and Lafontaine (1995) suggest that a positive royalty

rate serves to provide incentives to the franchisor in the presence of the moral hazard problem

with respect to the franchisor’s effort. Alternatively, Lafontaine (1992) postulates that if the

entrepreneur is risk-averse, the higher royalty rates serve to shift risk from store cash flows to

the franchisor. As yet another explanation, Mathewson and Winter (1985) and Kaufmann and

Lafontaine (1994) suggest that entrepreneurs’ financial constraints limit the size of the upfront

fees. Thus, franchisors must use royalty payments to extract franchisees’ surplus. Klein (1995)

criticizes the financial-constraints explanation, pointing out that entrepreneurs must have access

to efficient financial markets. Instead, he proposes that royalty payments are needed because there

are legal restrictions on contract termination, tied to large initial payments. Finally, the structure

of the contract payments could simply be mimicking the structure of the franchisor’s costs as the

franchisor provides services to the franchisee (Sen 1993).

An important contribution of our paper is to propose a new explanation to the empirical puzzle

of the dominance of royalties over fees. Specifically, we prove that the royalties should be used

instead of fees when the entrepreneur has the option to decide when to open the store. The second

important contribution of our paper is to refine the discussion of financing considerations and their

effects on franchise contracts, by accounting for limited entrepreneurs’ internal capital (as discussed

in Kaufmann and Lafontaine 1994), access to financial markets to acquire external financing (as

claimed by Klein 1995), and bankruptcies.

In this paper we present a model that captures two salient features: (1) the entrepreneur has

the option to decide on when to open the store and (2) the entrepreneur’s financing considerations.

These two features are essential in practice, yet they have received a scant attention in franchise

theory so far. We study how these two features affect franchise contracts, the timing of store

opening, the division of values between the franchisor and the franchisee, system coordination, and

the franchise system advantages over the company store system. We derive insights into franchise

contracting and into sharing financial burdens among franchisors and franchisees.

To appreciate the importance and relevance of the two key features of this paper, let’s review

an example of a franchising process, based on the description from Subway’s website, part of which

2

we reproduced in Table 1.

10 Quick and Easy Steps to the Grand Opening of Your Store:

1. Request a Franchise Brochure2. Submit Franchise Application3. Meet Local Development Agent4. Review Disclosure Document5. Conduct Local Research6. Secure Financing7. Sign Franchise Agreement8. Attend Training9. Secure a Location and Build Your Store

10. Celebrate Grand Opening

Table 1: Steps to Opening a Subway Franchise StoreSource: http://www.subway.com/subwayroot/Own_a_Franchise/nextsteps.aspx

There are three main takeaways from the franchising process in Table 1 that we capture in

our analysis. First, the contract terms are advertised publicly (on the website2), prior to any

entrepreneur contacting the franchisor. Specifically, at the time of writing of this paper, anyone

contemplating opening a Subway store is offered terms of $15,000 in franchise fees and then 12.5%

in on-going royalty payments (8% counted as royalty rate and 4.5% as advertising fee) on gross

sales minus sales tax. Here, similar to Lafontaine (1992) and Sen (1993), we combine royalty rates

and advertising fees.

Second, an entrepreneur, having read the contract terms, has an option to study this opportunity

and decide when and if to sign the contract. The entrepreneur does not have to decide immediately.

There is no ultimatum to accept the contract or walk away forever. Thus, franchise contracts differ

from some other forms of business contracts studied in the literature.

Third, financing is essential. Subway estimates capital requirements for opening a store in

the US to be between $116,200 and $262,8503. Typically, entrepreneurs have personal financial

resources, but those are not sufficient to cover the entire investment. Subway offers contact infor-

mation of several financial institutions, who can provide entrepreneurs with loans. Defaults on such

loans and bankruptcies of entrepreneurs are not uncommon. For example, according to the Small

Business Administration dataset,4 Blimpie’s franchisees defaulted on 48% of loans; Matco Tools,

Quizno’s Subs, Cold Stone Creamery, AAMCO Transmissions franchisees defaulted on 37% of loans

each; MAACO Auto Painting Center and Long John Silver’s franchisees defaulted on 25% of their

2http://www.subway.com/subwayroot/Own_a_Franchise/default.aspx3http://www.subway.com/subwayroot/own_a_franchise/PDFs/Capital_Req_US_Canada.pdf4Small Business Administration (SBA) loan failure rates for franchisees in a franchise brand https://opendata.

socrata.com/Banking-Finance-and-Insurance/SBA-Loan-failure-rate-for-franchisees-in-a-franchi/

yiht-4fbq

3

loans. In contrast, franchisees of Denny’s Restaurant, Jiffy Lube, and Burger King franchisees

defaulted on 10% of the loans, Dairy Queen and Subway Sandwich Shop franchisees defaulted on

9% of the loans; Little Caesar’s Pizza and Pizza Hut failed on 4% of loans. Concerns about loan

defaults and associated costs (Opler and Titman 1994) may cause banks to refuse financing to the

entrepreneurs or to offer financing at unattractive terms. To illustrate, Clifford (2010) reports that

following the 2008 financial crisis, lending to franchisees fell from $11.7 billion in 2008 to $7.5 billion

in 2009.

To analyze interactions between contract terms, dynamic entrepreneur’s response, and financ-

ing, we incorporate these three takeaways into our model. We solve a sequential-moves game that

involves three players: the franchisor, the entrepreneur, and the banks. The franchisor chooses

values of contract terms. The entrepreneur dynamically decides when to sign this contract and

open a store, applying for debt financing to cover the initial investment. In response to the en-

trepreneurs application, banks competitively determine loan rates. Thus, we solve an optimization

problem over the contract terms, which affects the solution of the optimal stopping time problem

of the entrepreneur (i.e., the threshold for opening a store), which is subject to financial market

equilibrium constraints. We value streams of cash flows to each player as contingent claims on

store profits. We also allow for the store profits process to be imperfectly observable prior to store

opening.

We find that in the optimal contract royalties should be positive and fees should be zero even

without financial constraints. In equilibrium, the store will be opened later than the store in the

integrated system (i.e., the critical threshold on store profits at which the decision to open the store

is made is higher in the franchise system), which in turn will be opened later than static positive-

NPV-rule would suggest. Interestingly, the addition of the entrepreneur’s need to raise external

financing lowers royalty rates, but the store opening is delayed further. Because of the intrinsic non-

linearities that come with external financing, the unobservability of the store profits prior to opening

affects the optimal decisions much more in the model with financing. We calculate the ramifications

if franchisor, mistakenly, ignores the entrepreneur’s need for capital. These ramifications include

higher royalty rates, longer waits for store openings, and higher rates of bankruptcies. We find

that the franchisor would benefit from either sharing more of the costs of running the store with

the franchisee or by offering initial financing (our results resonate with the empirical findings on

financing and franchise growth in Shane et al. 2006).

4

2 Models and Assumptions

We study three models, which are, in the increasing level of complexity: Model I—an integrated

system with neither franchise contracts nor financing concerns, Model II—a decentralized franchise

system, that considers franchise contracts but not financing concerns, and Model III—our main

model—a decentralized franchise system with franchise contracts and financing. In this section we

describe the main model (Model III), while highlighting, in passing, the differences with the other

two models.

There are three players in Model III: the franchisor (she), the entrepreneur (he), and the bank

(it). Figure 1 illustrates the setup. The protagonist of this paper is the franchisor and the focal point

of our analysis is the franchisor’s decisions. In Models III and II, the franchisor owns franchise rights,

and she “rents” them to an entrepreneur (possibly one of many in a pool of entrepreneurs) in return

for payments, specified by the contract. We shall discuss cash flows between the franchisor and the

entrepreneur shortly. The entrepreneur needs the franchise right to open the store. Investments in

the store require external financing in Model III, and the entrepreneur secures it from a bank (one

of many perfectly competitive banks).

Investment

& C

osts

Store P

rofit

Non-operating

Costs

Entrepreneur BanksFranchisor

Store

Figure 1: Model III — Decentralized franchise system with financing — Players: franchisor, entrepreneur,and bank (one of many). Store is the physical asset. Arcs indicate exchange of cash flows and rights.

In Models II and I all firms are assumed to have unlimited capital and financing sources are

not important. In Model I the franchisor and the entrepreneur are combined into a single decision

maker. We shall describe cash flows between the entrepreneur and the bank shortly. All three

players are risk-neutral (or uncertainty in store profit constitutes a diversifiable risk), and all players

use ρ as their continuously compounded discount rate.

A common feature in all models is S(t)—the store sales (in monetary units) net of operating

costs per unit of time (we discuss explicit sharable operating costs below). In the sequel, for brevity,

we shall refer to S(t) as store profit. All other important quantities are contingent on the value

S(t) and its evolution over time. The process for store profit is modeled according to an arithmetic

5

Wiener process with drift α and volatility σ:

dS(t) = αdt+ σdZ(t). (1)

Under process (1), the store profit, S(t), can be negative. This captures practice better (Radner

and Shepp 1996), making (1) preferable to a geometric Wiener process assumption typically used

in real options models (Trigeorgis 1996), as the geometric Wiener process does not allow negative

values and is more appropriate for modeling stock or commodity prices. However, if the drift α

is positive, the negative values of S(t) are unlikely. Other assumptions for the stochastic process

for S(t) (e.g., adding features such as jumps or mean-reversion) jeopardize analytical tractability,

while not adding substantial qualitative insights. A discussion of the consequences of alternative

assumptions for S(t) is presented in §5.

Before the store is open, the everyone observes a noisy signal S(t) of the true value S(t).

Information can come from sales of similar stores and is stipulated by the disclosure requirements

under the franchise law. (The Federal Trade Commission requires disclosure of contract information

of current and former franchisees5 and prospective franchisees may contact the existing ones for

information.) A possible interpretation is that S(t) represents an “industry average” projection

of the cash flows. At the investment time, the store-specific uncertainty η is resolved and S(t) =

S(t) + η, where E[η] = 0. The process for S(t) is given by (1) as well.

Figure 2 illustrates the timeline in Model III and Table 2 summarizes the timing of cash flows.

Non-operating

costs

Franchise fee,Store CAPX

Operating costs,Store profit,

Royalty payments

Bankruptcy costs

No

n-operating

store

Operating store

7/1/2014 - 7/8/2014 7/1/2014 - 7/8/2014

τo τB

Figure 2: Model III — Decentralized franchise system with financing — Timeline

Time Franchisor Entrepreneur Bank

t ∈ [0, τo) −fdowndtt = τo x −mo −Lt ∈ (τo, τB) [yS(t)− cF − fup]dt [(1− y)S(t)− cE − iL]dt iLdtt = τB −bF −bE −bB

Table 2: Cash flows over time in Model III — Decentralized franchise system with financing

5http://www.business.ftc.gov/documents/bus70-franchise-rule-compliance-guide, Rule 20

6

At time t = 0 (see Figure 2), the franchisor offers a contract to the entrepreneur. As discussed in

§1 the franchise contract comprises x (franchise fee), and y (royalty rate). According to the franchise

process in §1, we assume that the terms of the contract are set at time t = 0, in anticipation of all

subsequent events. This is a reasonable approximation of practice. If contracts change over time,

they do so at a much slower rate than other events in the franchise system. For instance, terms

could be changed at the end of the contract term, but in practice franchise contracts are signed for

multiple years (e.g., 7-Eleven-Japan franchise contracts are for 15 years6, Subway contracts are for

20 years) and are renewable. Considering the long time horizons of contracts, we approximate them

by infinity. In the paper, we accept the structure of the contract and contracting process as given

from practice. We also do not question why the practice of franchising exists in the first place.

Blair and Lafontaine (2011) review research of these and other questions related to franchising.

Having observed the contract, the entrepreneur may sign the contract and open the store right

away, or he may wait and until the store is “profitable enough.” We shall distinguish between the

time period when the store is not operating, followed by the period when the store is operating.

Define τo ≥ 0 as the time the entrepreneur accepts the franchisor’s contract and opens the store.

This time corresponds to the projected store profit level So as follows:

τodef= inf{t ≥ 0 : S(t) ≥ So}. (2)

During time t ∈ [0, τo), the store is not operating and the franchisor pays fdown per unit of time

(e.g., to maintain franchise rights, to cover lease on land where the store is to be opened). Neither

the entrepreneur nor the bank are actively participating and they do not experience any cash flows

(see Table 2).

At time t = τo, the contract is signed, the entrepreneur pays the franchise fee x to the franchisor,

and makes the capital investment CAPX into the store. Thus, the total capital needed by the

entrepreneur at time t = τo is x+CAPX. In Model III, we assume that the entrepreneur has initial

capital mo < CAPX. Therefore, as frequently happens in practice, the entrepreneur needs external

financing. For that the entrepreneur secures a bank loan for the amount Ldef= x+CAPX −mo (it

is suboptimal to borrow a larger amount).

Loan terms are fixed at time t = τo. Specifically, in return for the loan L, the entrepreneur

promises to pay iL per unit of time to the bank until bankruptcy time t = τB ≥ τo. Interest rate i

is determined from the perfectly competitive equilibrium in the banking industry.

6http://www.7andi.com/ir/pdf/annual/2012 11.pdf

7

During t ∈ (τo, τB) (see Figure 2), the store is operating, generating S(t) (given in (1)) in profit

per unit of time. Fraction yS(t) of these profit goes to the franchisor, and the remainder (1−y)S(t)

goes to the entrepreneur. Furthermore, there is an additional operating cost, not accounted for in

S(t). This cost is c = cF + cE per unit of time. This cost can include costs of materials, labor, and

effort. We shall assume that part cF of the cost is paid by the franchisor, while part cE of the cost is

paid by the entrepreneur. In addition, when the store is operating, the franchisor pays fup per unit

of time (e.g., for advertisement, logistics of supplying the store). This cost could contain cost fdown,

that is fup = fdown+extra costs. Finally, the entrepreneur pays loan interest iL per unit of time to

the bank, as long as he can, that is, as long as (1− y)S(t) > cE + iL. Otherwise, the entrepreneur

defaults on loan payments and goes into bankruptcy. These cash flows are summarized in Table 2.

The assumption of an exogenous bankruptcy threshold is consistent with the finance literature

on pricing corporate debt (seminal papers making the same assumption are Merton 1974, Black

and Cox 1976, and Longstaff and Schwartz 1995; other papers in the same camp include Geske

1977, Ho and Singer 1982, and Kim et al. 1993). An alternative assumption (e.g., Leland 1994)

endogenizes the choice of the bankruptcy threshold. However, this requires that the equity holders

must possess the power of controlling the timing of bankruptcy, have an ability to raise capital

to meet the liquidity needs meanwhile, and have power to renegotiate financial contracts with

the bond holders. These assumptions may be acceptable for large corporations, but not for small

entrepreneurs who open franchise stores for Subway or Quiznos. Neither modeling assumption

(exogenous v. endogenous threshold) is a true representation of reality. However, for small business

owners liquidity concerns are paramount. Thus, it is reasonable to assume an exogenous bankruptcy

threshold, tied to liquidity. We leave analysis of the endogenous bankruptcy threshold to future

research. Thus, define bankruptcy threshold (corresponding to (1− y)S(t) > cE + iL) as

SBdef= cE+iL

1−y (3)

and bankruptcy time is

τBdef= inf{t ≥ τo : S(t) ≤ SB}. (4)

At the time of bankruptcy, t = τB, all three players incur bankruptcy costs: bF , bE , and bB

(again, see Table 2). These costs can be reputation costs, costs of lost market share, legal costs, etc.

Time of bankruptcy τB is the end of the planning horizon. This is a reasonable assumption for a

significant subset of franchises. When the entrepreneur goes into bankruptcy the franchisor cannot

immediately replace him with another one. If the store profit is too low for one entrepreneur it is

too low for others as well. Furthermore, there is evidence that consumers react to bankruptcies

8

with lower demand (e.g., Opler and Titman, 1994). Particular franchise locations may be acquired

by competition or rented out by the land owner (e.g., a mall operator) to another business7. An

alternative assumption is that a default of the current entrepreneur is an option for the franchisor

to begin looking for another entrepreneur. This makes the franchisor’s problem recursive and more

challenging solve mathematically. With profit at the bankruptcy levels, the option to look for

another entrepreneur is unlikely to be very valuable.

Cash flows in the integrated system (Model I) and in the franchise without financing (Model

II) are given in Table 3. In these models, there is no bankruptcy and the planning horizon does

not end at τB. The value τo is determined endogenously in all three models. This value is different

in each of the models. It will be clear from the context which of the three τo values we are using.

Time Integrated (Model I) Franchisor (Model II) Entrepreneur (Model II)

t ∈ [0, τo) −fdowndt −fdowndtt = τo −CAPX x −CAPX − xt ∈ (τo,∞) [S(t)− c− fup]dt [yS(t)− cF − fup]dt [(1− y)S(t)− cE ]dt

Table 3: Cash flows over time for the integrated system (Model I) and for the franchise system withoutfinancial concerns (Model II)

To prepare for the upcoming analysis, we introduce the tool of contingent claim pricing through

a solution of differential equations. Readers familiar with this methodology should proceed to §3.

2.1 Preliminaries—Contingent claims valuation

To make this paper self-contained, we present a contingent claim valuation approach that will be

applied later (for more details, see Dixit and Pindyck 1994). Consider a security which pays its

holder C(t, S)dt per unit of time and that can be exchanged for a one-time payment of E(t, S).

The value of this security F (t, S) depends on on the value S(t) and, therefore, this security is a

contingent claim on S.

In the region where the option to exhange is not exercised, the holder of the security expects

rate of return ρ + λβ from it, where ρ is the risk-free rate, β is the volatility of F (related to the

volatility of S and correlation between S and the market), and λ is the market price of risk. If

investors in the economy are risk-neutral or the underlying asset is not correlated with the market,

then λβ part of the expected return equals zero and the holder of the security expects a return rate

of ρ. Thus, in equilibrium, the expected return from holding F , comprising capital gains and cash

7After KFC franchisee, Denny Wagstaff, filed for bankruptcy in 2011, his stores in California and Minnesota wereacquired by AFC Enterprises Inc., operator of Popeye’s Louisiana Kitchen restaurants. http://www.redding.com/

news/2012/nov/12/kentucky-fried-chicken-franchise-owner-closes-29/

9

flows equals the return that investors demand:

E[dF ] + Cdt = ρFdt. (5)

By Ito’s lemma,

dF = Ftdt+ FSdS +1

2σ2FSSdt =

(Ft + αFS +

1

2σ2FSS

)dt+ σFSdW. (6)

where sub-indices indicate partial derivatives. Therefore, from equation (5), any contingent claim

F (before it has been exchanged for the terminal payment E(t, S)) must satisfy the following partial

differential equation (PDE)

C + Ft + αFS +1

2σ2FSS = ρF. (7)

If there are no limits on the termination time of this security and if neither cash flows nor ter-

mination value explicitly depend on time, then neither does F and PDE (7) becomes an ordinary

differential equation (ODE):

C + αF ′ +1

2σ2F ′′ = ρF. (8)

From the theory of ODEs (e.g., Tenenbaum and Pollard 1985) the solution of (8) is

F (S) = P (S) + k1e−γS + k2e

−δS , (9)

where P is the particular solution of non-homogeneous equation (8) and k1e−γS + k2e

−δS is the

general solution of homogeneous equation (8) (if C = 0 then P = 0) with k1 and k2 some constants,

γdef=α+

√α2 + 2ρσ2

σ2> 0, (10)

δdef=α−

√α2 + 2ρσ2

σ2< 0. (11)

Coefficients k1 and k2 are determined using boundary conditions and the terminal (exercise) value

of the claim. For instance condition that F is bounded at ∞ implies k2 = 0 and condition that F

is bounded at −∞ implies k1 = 0.

With the contingent claim valuation tool at our disposal, in the next section we analyse a

simpler model—Franchise system without financing.

3 Analysis of Model II—Franchise System without Financing

We analyze the simplest model (Model I — integrated system) in the Appendix. Here we study

Model II and assume that both the entrepreneur and the franchisor have access to unlimited capital,

10

and that financial markets are perfect (Modigliani and Miller 1958). In other words, franchise

contracting matters, but financial considerations are irrelevant. In the sequel, we shall denote

functions and quantities related to the franchisor and the entrepreneur using subscripts “F” and

“E”, respectively. We shall use superscript “fra” if it is necessary to emphasize that the quantities

refer to the model of a franchise system without financing.

The following is our plan for this section. We shall conduct analysis in the reverse chronological

order (see Figure 2), starting with the entrepreneur’s problem. First, we derive the values of the

operating store VE(S) to the entrepreneur, given any store opening threshold So and any contract

(x, y). Our problem is time-homogeneous, therefore, we omit time variable from the value functions.

Value VE(S) is a contingent claim on store profit S. Second, once VE is computed, we turn to valuing

the option of the entrepreneur to open the store, denoted by WE(S). This value is a contingent

claim on VE and it depends on the projected store profit S (see §2 for the relationship between S

and S). Third, armed with the expression for WE , we determine the optimal threshold Sfrao for

the entrepreneur to open the store, given any contract (x, y). Then, forth, we calculate the value

of the operating store to the franchisor VF (S). Fifth, knowing the entrepreneur’s optimal response

Sfrao to a contract (x, y), we compute the value to the franchisor WF (S|(x, y)) comprising the value

of cash flows before the store is opened and the claim on the operating store cash flows. Sixth, we

find the contract (xfra, yfra) that maximizes franchisor’s total value WF (S|(x, y)). The franchisor’s

value under this optimal contract is denoted by W fraF and the corresponding entrepreneur’s value is

W fraE . Finally, we discuss the optimal contract properties and highlight various managerial insights.

3.1 Entrepreneur’s value from the operating store

Recall from Table 3 that for the operating store (t ∈ (τo,∞)), the cash flows to the entrepreneur

are C(t, S(t)) = (1 − y)S(t) − cE . Using the contingent claim analysis (§2.1), the value of the

entrepreneur’s claim on these cash flows VE(S) (the boundary conditions are VE is bounded at

infinities) is:

VE(S) = (1− y)(Sρ + α

ρ2

)− cE

ρ . (12)

The first term corresponds to the fraction of perpetual store profit (1− y)S(t) and the second term

corresponds to the perpetual cost cE . Thus, this value has an easy intuitive interpretation as the

difference between perpetual profit and perpetual costs. (An alternative and equivalent derivation of

expression (12) follows from the Feyman-Kac formula: VE(S) = E[∫∞

0 e−ρt [(1− y)S(t)− cE ] dt].)

However, the approach using ODEs (as in §2.1) is convenient for more complex analysis.

11

For simplicity, we shall allow both positive and negative VE(S) (and VF (S) in the sequel). One

could relax this assumption and assume that the store is liquidated if S(t) < SL. Liquidation

boundary SL can be either exogenous or endogenous. This does not affect the qualitative insights

from the model, but it increases the complexity of the analysis, requiring solution of systems of

non-linear equations. With positive drift α, it is unlikely that the abandonment barriers will be

reached. Therefore, for simplicity we shall ignore abandonment options in this paper. Additional

discussion on this issue is provided in §5.

3.2 Entrepreneur’s option to open a store and the optimal threshold

As discussed in §1, an important feature of franchise contracting captured in our model is that

the entrepreneur has the option to decide when to open the store. Similar to McDonald and

Siegel (1986), the entrepreneur will open the store when the projected store profit S(t) exceeds the

threshold So, which we shall characterize in this subsection.

If the store is closed, S < So, from Table 3, there are no cash flows to the entrepreneur,

C(t, S) = 0. When the entrepreneur exercises his option to open the store, he pays CAPX + x

and receives a claim to its share from the open store E[VE(So + η)], given by (12) and where the

expectation is taken over η (recall that at the time the store opens, S = S + η). The value to the

entrepreneur WE(S) satisfies ODE (8) with boundary conditions that this value is bounded at the

negative infinity and WE(So) = E[VE(So + η)] − CAPX − x. The solution of the corresponding

ODE is

WE(S) = [VE(So)− CAPX − x]πo(S, So), (13)

where πo(S, So)def= eδ(So−S) ∈ (0, 1) for S < So (recall that δ =

α−√α2+2ρσ2

σ2 < 0). Expression

πo(S, So) is the price of the Arrow-Debreu security that pays out $1 when the store is opened. Less

rigorously, one can also interpret πo(S) as the probability of eventually opening a store, adjusting

for discounting. With this interpretation, equation (13) affords a nice explanation — the value to

the entrepreneur is the discounted expectation of payoff V (So)− CAPX − x.

The optimal threshold for opening the store (So) is determined from the smooth-pasting con-

dition (Dixit and Pindyck 1994):

W ′E(So) = V ′E(So). (14)

Using W ′E(S) = −δ [VE(So)− CAPX − x]πo(S) and V ′E(S) = 1−yρ , condition (14) is equivalent to

(1− y)(Soρ + α

ρ2

)− cE

ρ − CAPX − x = −1−yδρ , (15)

12

which yields:

Sfrao = cE+(CAPX+x)ρ1−y − α

ρ −1δ . (16)

From equation (16), as franchise fee x or royalty y increase, the threshold for opening the store

Sfrao increases, and, thus the opening of the store is delayed. Comparing the decentralized system

threshold Sfrao (in model II) given in (16) with the integrated system threshold Sinto (in model I)

given in (45), we observe that either one can be larger for an arbitrary contract (x, y). But we shall

prove that under the optimal contract, the threshold in the franchise system Sfrao is always higher

than that in the integrated system Sinto .

3.3 Franchisor’s value from the operating store

From Table 3, for the operating store (t ∈ (τo,∞)), the cash flows to the franchisor are C(t, S(t)) =

yS(t)− cF − fup. The value of the franchisor’s claim VF (S) is bounded at infinities, and its value

from §2.1 is

VF (S) = y(Sρ + α

ρ2

)− cF+fup

ρ . (17)

The first term corresponds to the perpetual store profit of yS(t) and the last term corresponds to

the perpetual costs cF + fup.

3.4 The value of the franchisor’s rights

During time interval [0, τo] the franchisor holds a portfolio (whose value is denoted by W (S))

comprising costs of non-operating store and a claim on the value from the operating store plus

franchise fee. It is important to reiterate that the entrepreneur affects the franchisor’s value through

So, but the franchisor can control the choice of So through contract (x, y). Given any contract (x, y)

and the corresponding optimal threshold So(x, y) (computed by the entrepreneur using (16)), the

franchisor’s value is defined differently, depending on whether S < So(x, y) or S ≥ So(x, y).

For Case 1: When S < So(x, y), the franchisor incurs cash flows C(t, S) = −fdown until time t =

τo (see Table 3). At time t = τo, the franchisor receives x and access to cash flows from the operating

store, valued as VF and given in (17). Thus, the franchisor’s value function WF satisfies boundary

conditions of being bounded at negative infinity and WF (So(x, y)) = x+E[VF (So(x, y)+η)], where

the expectation is taken over η. Applying analysis in §2.1, this value is

WF (S|(x, y)) = −fdownρ

[1− πo(S, So(x, y))

]+ [x+ VF (So(x, y))]πo(S, So(x, y)), (18)

where as before, πo(S, So) = eδ(So−S) and δ =α−√α2+2ρσ2

σ2 < 0 and So(x, y) is given in (16).

13

For Case 2: When S ≥ So(x, y), the franchisor immediately receives x and the claim to her

share of the cash flows from the operating store VF . Therefore, the franchisor’s value function is

WF (S|(x, y)) = x+E[VF (S+η)]. We now use WF to determine the optimal contract (x, y). Going

forward, we need the following assumption.

Assumption 1. The expected perpetual profit from the store is sufficiently high: Sρ + α

ρ2 > − 1δρ .

Assumption 1 is not very restrictive. One way to think about this assumption is to consider

the expression for the threshold So in (16). Assumption 1 asserts that if the entrepreneur kept

all profit to himself and did not have any costs of either investing in or running the store (i.e.,

x = cE = CAPX = 0), then the entrepreneur would open the store immediately. Thus Assump-

tion 1 eliminates the need to consider non-interesting cases, where the entrepreneur would not find

investing in the store attractive, even under the best possible conditions.

3.5 The optimal franchise contract

Having thus computed function WF , we now can find the optimal contract (x, y). This contract

must satisfy x ≥ 0 and y ∈ [0, 1] and maximize value WF (S|(x, y) for the franchisor, as follows:

maxx≥0,0≤y≤1

WF (S|(x, y)) (19)

The entrepreneur’s individual rationality constraint is automatically satisfied because the en-

trepreneur has an option to delay accepting the contract and employ his resources in alternative

ventures, earning reservation value. Thus, in choosing the contract, the franchisor faces a trade-off

between earning higher franchise fees and royalties and having the contract accepted sooner (but

franchise fees and royalties are lower). The following Proposition describes the optimal contract.

Proposition 1. The optimal contract is xfra = 0 and yfra = 1 −cEρ

+CAPX

Sfraoρ

+ αρ2

+ 1δρ

, where Sfrao =

max(S, So), and So is the solution of the following problem:

maxSo

{(fdownρ +

cEρ

+CAPX(Soρ

+ αρ2

+ 1δρ

)δρ

+ Soρ + α

ρ2 − c+fupρ − CAPX

)eδSo

}(20a)

Soρ + α

ρ2 + 1δρ ≥

cEρ + CAPX (20b)

The entrepreneur’s value under the optimal contract satisfies

W fraE (S) = −

cEρ

+CAPX

δρ

(Sfraoρ

+ αρ2

+ 1δρ

)eδ(Sfrao −S) > 0. (21)

14

3.5.1 Insights from the optimal contract

Proposition 1 leads to several observations about the optimal franchise contract (xfra, yfra). First,

computing the optimal contract is a matter of solving a simple optimization problem (20). Second,

zero franchise fee (xfra = 0) is consistent with the empirical evidence. According to Blair and

Lafontaine (2011), franchise fees are tiny fraction of the franchise value. Third, the royalty rate

yfra is positive.

As we discussed in §1, static economics models rely on various explanations, including fran-

chisor’s moral hazard and entrepreneur’s risk-aversion to explain why positive royalty rates are

observed in practice. Our work offers an alternative explanation, driven by the entrepreneur’s op-

tion to delay the signing of the contract. To better appreciate results in Proposition 1, it helps

to compare them with those from a static model (using our notation). In a static model, the

entrepreneur would have to decide immediately and irreversibly whether to accept the contract.

Thus, using (17) and (12), the contract design problem for the franchisor would be

maxx≥0,0≤y≤1

{E[VF (S + η|(x, y))] + x = y

(Sρ + α

ρ2

)− cF+fup

ρ + x}, (22a)

s.t., E[VE(S + η|(x, y))]− x− CAPX = (1− y)(Sρ + α

ρ2

)− cE

ρ − x− CAPX ≥ 0. (22b)

For the static problem (22), any values (x, y) that satisfy

(1− y)(Sρ + α

ρ2

)− cE

ρ − x− CAPX = 0 (23)

constitute an optimal contract. In particular, x = Sρ + α

ρ2 − cEρ − CAPX > 0 and y = 0 is a

solution. Furthermore, unlike the result in Proposition 1, the entrepreneur has zero value in the

static problem (22).

Why is there a difference between results of the static and our models? In a static model,

the franchisor can raise up-front fee x to expropriate the entire surplus of the entrepreneur. In

our dynamic model, if the franchisor raises x, the entrepreneur responds by raising his optimal

threshold So (see expression (16)). The benefit for the franchisor of higher x is offset by the longer

wait for the entrepreneur to sign the contract.

One may think that the franchisor could raise x while lowering y to keep the opening threshold

So constant. The following is a mathematical argument for why this would be suboptimal. Let’s

focus on the interesting case of S < So. Using (18), (17), and (16) we derive expressions (48) in

the Appendix for the contracting problem. Keeping So constant means that the only non-constant

expression in the franchisor’s objective function is

x+ y(Soρ + α

ρ2

). (24)

15

Using (16), expression (24) simplifies to

const1 − yδρ × const2, (25)

where const1 and const2 are constants and const2 > 0. Because δ < 0, for the optimal contract, it

is best to increase y (and thus decrease x to keep So constant).

Intuitively, when solving contracting problem (19), the franchisor would like to increase both

fee x and royalty rate y. The downside of increasing either one is that the entrepreneur will raise

the threshold So in response (thus delaying store opening). But the upsides from raising y in (24)

are multiplied. The royalty rate y itself increases and the profit to which royalty rate is applied(Soρ + α

ρ2

)increases as well. This compounding of benefits is missing from static models (like model

(22)) because in static models the store is opened or not immediately and the benefit of higher So

is not incurred.

3.5.2 Contract properties

From Proposition 1, the optimal threshold Sfrao can be found by solving a relatively easy opti-

mization problem (20), which does not depend on S. This optimization model affords a number of

useful insights as presented in the following corollary.

Corollary 1. The optimal solution to the franchisor’s problem (without financing) has the following

properties:

1. If c + fup − fdown + CAPXρ > 0, then the optimal threshold Sfrao > Sinto , where Sinto is the

threshold for Model I—integrated system—given by (45).

2. As fup, cF , cE, or CAPX increase or as fdown decreases, the optimal threshold for opening

the store Sfrao increases.

3. The value of the entrepreneur W fraE (S) > 0 given in (21) peaks at S = Sfrao .

4. The royalty rate yfra is increasing in S (non-strictly). Furthermore, yfra is increasing in

fdown and decreasing in fup and cF .

5. The noise term η does not affect the optimal solution.

As part 1 of Corollary 1 states, the optimal threshold in the current model is greater than that

of the integrated system, provided some conditions on the parameters values are satisfied. These

conditions are easy to satisfy, especially when one considers that the cost to the franchisor from

16

the operating store fup+ cF is likely to be higher than the cost from the non-operating store fdown.

Part 3 of Corollary 1 highlights the fact that the entrepreneur’s value under the optimal contract

is positive. This is due to the franchisor using (as is done in practice) a static contract to manage

dynamic behavior of the entrepreneur. The following numerical example illustrates the analytical

results above and provides additional insights.

Example 1. We use the following parameter values: discount rate ρ = 0.1, growth rate of profit

α = 0.05, volatility of profit σ = 0.5, cost of the store paid by the entrepreneur cE = 0.5, cost of the

store paid by the franchisor cF = 0.5, initial capital of the entrepreneur m0 = 0, capital investment

to open a store CAPX = 10, cost to the franchisor of running a franchise when store is operational

fup = 0.1, cost to the franchisor when store is not operational is fdown = 0.05. Because noise η

does not appear in the solution, we assume that η = 0.

Given these values, the optimal equilibrium threshold to open the store is Sfrao ≈ 3.95. The

optimal franchise contract is (xfra, yfra) = (0, 0.5082).

As Corollary 1 proves, Sfrao is greater than the threshold for the integrated system (whose

solution is in the Appendix), Sinto ≈ 2.95, which, in turn (as we proved in the Appendix), greater

than the static NPV threshold SNPV = 1.55. Thus, the lack of coordination in the franchise system

manifests itself in the timing of store opening: Sfrao > Sinto > SNPV .

The left panel of Figure 3 shows the value of the integrated company W int, the value of the

franchisor W fraF , the value of the entrepreneur W fra

E (under the optimal contract), and the value

of the franchise system, W fra def= W fra

F +W fraE with respect to the current store profit S.

One interesting observation from Figure 3 is that the value of the entrepreneur is positive and

it peaks at S = Sfrao , as discussed in Corollary 1. The economic intuition is that the franchisor

uses a static contract to control the dynamic behavior of the entrepreneur. At S ≈ Sfrao , the

entrepreneur deviates the most from the integrated system’s decision to open a store. This is the

only lever the entrepreneur has in the franchise model and it involves the greatest transfer of value

to the entrepreneur. Specifically, consider that for S < Sinto , neither the integrated system nor the

franchise system will have store open. For S > Sfrao both the integrated system and the franchise

system will have an open store. For S ∈ [Sinto , Sfrao ) the integrated system has an open store but

the franchise system does not. At S ≈ Sfrao the franchisor wishes the most that the entrepreneur

would open the store they can begin operating as the integrated system.

Another and related observation from the left panel of of Figure 3 is that for S > Sfrao > Sinto

integrated and franchise systems earn the same value, W fra(S) = W int(S). The intuition is that for

S > Sfrao > Sinto the two systems do not differ in decisions (to open the store). As S increases after

17

S > Sfrao , the optimal royalty rate yfra increases as well (Corollary 1), lowering the entrepreneur’s

value, W fraE . This is intuitive—the franchisor will offer more aggressive terms (higher royalty fees)

when stores are very profitable and entrepreneurs are eager to open them.

WEHSL

WFHSL

S

W intHSLW fraHSL

1 2 3 4 5

5

10

15

20

25

30

W fraHSL

WEHSL

S

W comHSL

WFHSL

1 2 3 4 5

5

10

15

20

25

30

Figure 3: Value of the integrated system, W int (left panel), the company store W com (right panel), the

franchise system W fra = W fraF +W fra

E , the franchisor, W fraF , and the entrepreneur, W fra

E (both panels)vs the franchise store profit S.

Legend: Parameter values for the numerical example: ρ = 0.1, α = 0.05, σ = 0.5, cE = 0.5, cF = 0.5,fup = 0.1,fdown = 0.05, CAPX = 10, ψ = 0.85

Not surprisingly, the decentralized franchise system earns less (non-strictly) than the integrated

company, i.e., W fra(S) ≤W int(S). Furthermore, everything else being equal, the franchisor earns

less than the integrated company would, W ∗F (S) ≤ W int(S). A question might arise: why would

the franchisor not take over the entrepreneur and open a company store. The answer is that

the entrepreneur has unique talents, higher motivation, or special market knowledge that adds to

the value of the partnership more than the company store would. One way to model this is to

assume that the company store profit is a fraction of the franchise store profit, that is, for all t, the

company store profit is C(t) = ψS(t). This assumption not only translates into a different initial

value C(0) = ψS(0), but also into different parameters for the stochastic process for company store

profit C(t). Specifically, from (1)

dC(t) = ψαdt+ ψσdW (t) (26)

Repeating the analysis with C(t) replacing S(t), as the underlying, we compute the value of the

integrated system that runs company store, W com. The right panel of Figure 3 shows the value of

the system that operates the company store, W com, for ψ = 0.85.

The value of parameter ψ was chosen intentionally to demonstrate that the preference for

operating a franchise vs. a company store might change in S. Opening a company store is preferable

(W com(S) > W fraF (S)) for lower S, because the lack of coordination between the franchisor and

the entrepreneur results in a loss in value (a company store would be operational whereas a store

18

in the decentralized franchise system would still be closed). Opening a franchise store is preferable

(W com(S) < W fraF (S)) for higher S, because the lack of coordination is not a significant factor

and the loss of value from the entrepreneur’s talents is a greater concern (both a company store

and a franchise store would be running, but the company store would be earning a fraction ψ of

the profit). For ψ > 0.85 operating company store might be uniformly preferred. For ψ < 0.85,

operating a franchise might be uniformly preferred.

Left panel of Figure 4 shows the relationship between the value of the integrated system,

W int(S), the decentralized franchise system, W fra(S), the franchisor, W fraF (S), and the entrepreneur,

W fraE (S), and the volatility of store assets, σ. All values are increasing in the volatility of store

profit σ. In the franchise system the franchisor’s value W fraF (S) is increasing at a greater rate than

the entrepreneur’s value W fraE (S). This may seem counterintuitive because the entrepreneur is the

one who owns the option to open the store. However, it is the franchisor who possesses the contract

power.

W int HSL

W fra HSL

WF HSLWE HSL

Σ

0.2 0.4 0.6 0.8 1.0

2

4

6

8

10

Sofra So

int

Πoint

Πofra Σ

0.0 0.2 0.4 0.6 0.8 1.0

1

2

3

4

5

Figure 4: Left panel: Value of the integrated system, W int, the franchise system W fra = W fraF +W fra

E ,

the franchisor, W fraF , and the entrepreneur, W fra

E vs the volatility of store profit σ, when S = 2.Right panel: Thresholds for opening the store for franchise and integrated systems, Sfra

o and Sinto , the

corresponding probabilities of eventually opening the store, πfrao and πint

o vs the volatility of store profit σ,when current value of profit S = 2.

Legend: Parameter values for the numerical example: ρ = 0.1, α = 0.05, cE = 0.5, cF = 0.5,fup = 0.1,fdown = 0.05, CAPX = 10

Right Panel of Figure 4 shows the relationship between the optimal thresholds to open the store

in the integrated and franchise systems, the corresponding probabilities of eventually opening the

store, and the volatility of store profit, σ. Both the integrated and the franchise systems’ thresholds

are increasing in σ. However, the franchise threshold is higher and is increasing at a higher rate than

the integrated threshold. Thus, the corresponding probability of opening the store πo is lower in

the franchise system. Interestingly, as σ increases, the probabilities converge, despite the thresholds

diverging. The explanation for this is that the probabilities are affected by σ directly, in addition

to being functions of thresholds Sfrao and Sinto .

19

Next we study comparative statics with respect to the franchisor’s cost cF relative to the

entrepreneur’s cost cE . We assume that the total cost remains c = 1 so that cE + cF = 1.

Left panel of Figure 5 shows the relationship between the value of the integrated system, W int,

the decentralized franchise system, W fra, the franchisor, W fraF , and the entrepreneur, W fra

E , and

the cost, cF . The franchise system value W fra and the franchisor’s value W fraF are increasing

in the franchisor’s cost cF , while the entrepreneur’s value W fraE is decreasing. This may seem

counterintuitive, but by taking over more of the operational costs, the franchisor can expedite the

opening of the store and make herself and the entire system better off. The optimal thresholds for

opening the store are shown in the right panel of Figure 4. Thus, if possible, the franchisor should

implement profit sharing rather than revenue sharing. This would increase system’s efficiency and

benefit the franchisor. �

W int HSLW fra HSL

WF HSL

WE HSL

c f

0.2 0.4 0.6 0.8 1.0

1

2

3

4

5

6 Sofra

Soint

c f

0.2 0.4 0.6 0.8 1.0

1

2

3

4

Figure 5: Left panel: Value of the integrated system, W int, the franchise system W fra = W fraF +W fra

E ,

the franchisor, W fraF , and the entrepreneur, W fra

E vs the franchisor cost cF , such that cE = 1− cF .Right panel: Thresholds for opening the store for franchise and integrated systems, Sfra

o and Sinto , the

corresponding probabilities of eventually opening the store, πfrao and πint

o vs the franchisor cost cF , suchthat cE = 1− cF .

Legend: Parameter values for the numerical example: S = 2, ρ = 0.1, α = 0.05, σ = 0.5, fup = 0.1,fdown = 0.05,CAPX = 10.

4 Analysis of Model III—Franchise System with Financing

In this section we study model III and compute the optimal franchise contract under financing

considerations — the entrepreneur’s personal savings, bank loans, competitive lending markets,

bankruptcy and bankruptcy costs. We use subscripts “F”, “E”, and “B” to denote functions

and quantities related to the franchisor, the entrepreneur, and the bank, respectively. We use

superscript “fin” if it is necessary to emphasize that the quantities refer to the model of a franchise

system with financing.

The following is our plan for this section. As in §3, we solve the problem in the reverse

chronological order (see Figure 2). However, the analysis is more complex, because in addition to

20

the franchisor and the entrepreneur, we now also have to account for valuation and actions of banks.

First, for any contract (x, y), any threshold choice So, and any interest rate i on the bank loan, we

value cash flows to the bank, VB. Second, we discuss interactions between noise η and bankruptcy.

This is a new consideration that becomes important because of non-linearities that financing brings.

Third, assuming that banking system is perfectly competitive, we calculate the equilibrium loan rate

i given any contract (x, y) and threshold So. Fourth, we value the entrepreneur’s claim on the cash

flows from the operating store VE(S) for any i and the equilibrium value VE(S) for the equilibrium

rate i. Fifth, we compute the entrepreneur’s value to open the store WE (under the equilibrium

rate i). Sixth, for any contract (x, y) we determine the optimal threshold for the entrepreneur to

open the store, Sfino . Seventh, we find the franchisor’s value from the operating store VF (S), the

value VF (S) under the equilibrium rate i, and the franchisor’s value at time zero with the option

to open the store, WF (S) (accounting for the entrepreneur’s optimal response Sfino ). Eighth, we

compute the contract (xfin, yfin) that maximizes the franchisor’s value WF . Under the optimal

contract, the franchisor’s value is W finF and the entrepreneur’s value is W fin

E . Finally, ninth, we

study properties of the solution.

4.1 Value of the bank’s cash flows from the loan to the entrepreneur

Assume that interest rate on the loan i is given. Then the value of the bank’s claim, VF (S), on the

stream of interest payments C(t, S) = iL (see Table 2), while the entrepreneur is not in default,

S > SB, satisfies boundary conditions that the value is bounded at infinity and VB(SB) = −bB.

The latter condition says that at default, the bank incurs bankruptcy costs. Applying analysis

from §2.1,

VB(S) = iLρ

[1− e−γ(S−SB)

]− bBe−γ(S−SB). (27)

where from §2.1 γ =α+√α2+2ρσ2

σ2 > 0. Define πB(S)def= e−γ(S−SB) ∈ (0, 1) for S ≥ SB. Then

VB(S) = iLρ [1− πB(S)]− bBπB(S). (28)

Similar to the interpretation of πo, quantity πB(S) is the price of the Arrow-Debreu security that

pays $1 when bankruptcy occurs. Less rigorously, one can think of πB(S) as a discounted probability

of the eventual bankruptcy. If no bankruptcy happens, then the first term represents the banks’s

claim on a perpetuity of interest payments from the entrepreneur. If bankruptcy happens, then the

second term represents the banks’s bankruptcy cost bB.

21

4.2 Noise η and nonlinear effects of bankruptcy

Expression for VB (and later VE and VF ) include a non-linear term with πB(S) = e−γ(S−SB). In the

subsequent analysis it will be necessary to manipulate expression E[πB(S + η)] = E[e−γ(S+η−SB)].

Define νdef= E[e−γη]. Then E[πB(S + η)] = νE[e−γ(S−SB)]. The fact that ν can be separated is

convenient because a single variable contains all effects of noise η. These effects are summarized in

the following Lemma (proofs are standard application of definitions of stochastic order relations,

see Shaked and Shanthikumar 1994).

Lemma 1. Properties of variable ν = E[e−γη]:

(i) ν2 = E[e−γη2 ] ≤ ν1 = E[e−γη1 ] for all η2 ≥st η1 (i.e., η2 stochastically larger than η1).

(ii) ν2 = E[e−γη2 ] ≥ ν1 = E[e−γη1 ] for all η2 = η1 + ε where E[ε|η1] = 0.

In plain words, the first point in Lemma 1 states that as noise η increases (in the stochastic

sense) quantity ν decreases. The second point states that as noise η becomes more variable (in the

mean-preserving way), quantity ν increases.

4.3 Equilibrium loan rate

Having valued the cash flows post loan offer, we shall now determine the equilibrium rate i that the

bank will set for the loan. Assume that the contract (x, y) is known. Knowing the value (28) of the

cash flows from the loan extended to the entrepreneur, the bank offers a “perfectly competitive”

loan rate to the entrepreneur, that is, i(S) = min{i ≥ ρ : E[VB(S + η)] ≥ L}.

The equilibrium rate i has to be computed numerically. However, several properties of the

equilibrium rate facilitate these calculations. First, observe that there is one-to-one correspondence

between the loan rate i and the bankruptcy threshold SB through SBdef= cE+iL

1−y as defined in (3).

This enables the following result.

Lemma 2. Suppose cE+ρL1−y < S. The equilibrium bank rate is i = (1−y)SB−cE

L , where SB is the

solution of the following optimization problem (assuming there is a feasible solution):

minSB{SB} (29a)

(1−y)SB−cEρ

[1− νe−γ(S−SB)

]− bBνe−γ(S−SB) ≥ L (29b)

cE+ρL1−y ≤ SB < S (29c)

If there is no feasible solution, then i = SB = +∞ (i.e., no loan will be offered).

We can simplify calculations of the equilibrium bankruptcy threshold SB in (29), as follows.

22

Corollary 2. Suppose cE+ρL1−y < S. Let

f∗def= max

cE+ρL

1−y ≤SB<S

{f(SB)

def= (1−y)SB−cE

ρ

[1− νe−γ(S−SB)

]− bBνe−γ(S−SB)

}(30)

Then,

1. If f∗ < L, then there is no feasible solution for problem (29) and i = SB = +∞ (i.e., no loan

will be offered).

2. If f∗ ≥ L, then there is a feasible solution for problem (29) with i < +∞, SB < +∞ and the

equilibrium SB satisfies

(1−y)SB−cEρ

[1− νe−γ(S−SB)

]− bBνe−γ(S−SB) = L, (31)

and i = (1−y)SB−cEL .

Note that equation (31) may have more than one solution, in general. If that is the case, the

smallest solution is used.

Proposition 2. Holding everything else constant, the equilibrium bankruptcy threshold SB (equiv-

alently, loan rate i) increases in franchise fee x, royalty rate y, capital expenditures CAPX, the

entrepreneur’s store operating cost cE, bank’s bankruptcy cost bB, or as noise η increases (stochas-

tically). The equilibrium bankruptcy threshold SB (equivalently, loan rate i) decreases as projected

profit S increase or as noise η becomes more variable (in the mean-preserving way).

Results Proposition 2 are intuitive. For example, because franchise fee x enters the calculations

of SB via L = x + CAPX only, as x or CAPX increase, so does the smallest solution of (31).

It is interesting to observe the implications of Lemma 1. Because of non-linearities introduced

by financing, banks are sensitive to noise η in observations of the store profit S. As observations

become noisier (i.e., η is stochastically increases) the interest rate on loans rises. But as noise

becomes more variable (in the mean-preserving way), interest rate declines.

Having found the equilibrium rate i(S) (equivalently, bankruptcy threshold SB(S)) as a function

of projected store profit S, it is convenient to define equilibrium bankruptcy probability as

πB(S) = E[πB(S + η)|SB = SB(S)]def= νe−γ[S−SB(S)] (32)

4.4 Value of the entrepreneur’s cash flows from the operating store

Assume that the contract (x, y) and the interest rate i are given and the store is operating. When

the entrepreneur is not in default, that is for S > SB, the value VE(S) of the entrepreneur’s claim on

23

cash flows (C(t, S) = (1− y)S− cE − iL) from the operating store (see Table 2), satisfies boundary

conditions that this value is bounded at infinity and VE(SB) = −bE . Applying analysis from §2.1

VE(S) =[(1− y)

(Sρ + α

ρ2

)− cE

ρ

]− iL

ρ −[(1− y)

(Sρ + α

ρ2

)− cE

ρ −iLρ + bE

]e−γ(S−SB). (33)

The first term in [ ] in (33) is the same as the expression for the entrepreneur’s value without

financing (12). The next term is the value of interest payments to the bank, and the last term

represent effects of bankruptcy. Using definition of πB above it is instructive to rewrite (33) as

VE(S) =[(1− y)

(Sρ + α

ρ2

)− cE+iL

ρ

][1− πB(S)] +

[(1− y)S−SBρ − bE

]πB(S). (34)

In (34), if no bankruptcy happens, the entrepreneur derives value from a perpetual stream of profit

minus royalty payments to the franchisor, minus interest payments and costs. If bankruptcy hap-

pens, the entrepreneur incurs addition cost bE , but benefits from profit “on the way to bankruptcy”.

Expression (34) can be simplified, using the bankruptcy threshold definition SB = cE+iL1−y (defined

in (3)) as follows.

VE(S) = (1− y)(S−SBρ + α

ρ2

)−[

(1−y)αρ2 + bE

]πB(S). (35)

This representation will be convenient to use when describing the value of the store to the en-

trepreneur in equilibrium. Specifically, the expression for the entrepreneur’s value VE(S) from (35)

under equilibrium bank rate i becomes

VE(S) = E[VE(S + η)|SB = SB(S)] = (1− y)[S−SB(S)

ρ + αρ2

]−[(1− y) α

ρ2 + bE

]πB(S). (36)

where πB(S) is defined in (32).

4.5 Entrepreneur’s value of the option to open a store

As we have seen in §3, the entrepreneur may not accept the contract to take over the store as soon

as the NPV value of the investment is positive, that is, when VE(S) > m0. Similar to the insights

from the seminal work by McDonald and Siegel (1986), there is a value for the entrepreneur to

delay the entry because it involves an irreversible investment. In this subsection we shall determine

the optimal threshold on profit, So, for the entrepreneur to open the store. The contract (x, y) is

assumed to be given.

For S < So, the entrepreneur does not recieve any cash flows (C(t, S) = 0) and value to open

a store WE(S) satisfies boundary conditions that this value is bounded at negative infinity and

WE(So) = VE(So)−m0, where VE(S) is the entrepreneur’s equilibrium value of operational store,

given by (36). Applying analysis from §2.1,

WE(S) = [VE(So)−m0]eδ(So−S), (37)

24

where, as before, δ =α−√α2+2ρσ2

σ2 < 0. The optimal threshold So satisfies the smooth-pasting

condition:

W ′E(So) = V ′E(So). (38)

Equivalently, instead of solving (38) directly, we can think that the entrepreneur chooses So

to maximize its value with the option WE(S), as given in (37). Numerically, this optimization

approach proves to be easier to handle. Furthermore, it allows us to derive properties of the

optimal threshold So, as presented in Lemma 3.

Lemma 3. Expression WE(S) given by (37) is supermodular in variables So and SB.

Supermodularity property (Lemma 3) implies that the optimal threshold So increases in SB

(Topkis, 1998). From Proposition 2, SB increases in franchise fee x, and various parameters.

Therefore, it follows that

Corollary 3. Everything else held constant, the optimal threshold to open the store So increases in

the franchise fee x, capital expenditures CAPX, the entrepreneur’s store operating cost cE, bank’s

bankruptcy cost bB, and the entrepreneur’s bankruptcy cost bE.

The following example demonstrates calculations of the optimal threshold So and the option

value WE(S), given any contract (x, y).

Example 2. In this numerical example we use the following parameter values: discount rate

ρ = 0.1, growth rate of profit α = 0.05, volatility of profit σ = 0.5, cost of the store paid by the

entrepreneur cE = 0.5, cost of the store paid by the franchisor cF = 0.5, initial capital of the

entrepreneur m0 = 0, capital investment to open a store CAPX = 10, fixed cost the franchisor

pays in case of bankruptcy bF = 5, fixed cost the entrepreneur pays in case of bankruptcy bE = 5,

fixed cost of bankruptcy to the bank is bB = 5, cost to the franchisor of running a franchise when

store is operational fup = 0.1. We assume that noise η = 0.

Note that we are not yet computing the optimal contract terms (x∗, y∗) and values (x, y) =

(0, 0.2) in this example are arbitrary.

Figure 6 shows the entrepreneur’s value with the option WE(S) (thicker line), the value of

the open store to the entrepreneur in equilibrium minus the entrepreneur’s personal investment

VE(S)−m0 (thinner line), and the optimal threshold to open a store for the entrepreneur So. �

4.6 Value of the Franchisor’s cash flows from the operating store

Assume that contract (x, y) is fixed and the store is operating. When the entrepreneur is not

in default (i.e., when for S > SB), the value of the franchisor’s claim on cash flows (C(t, S) =

25

SSo

WEHSL

VE` HSL -m0

0 1 2 3 4 5 6

10

20

30

40

Figure 6: The entrepreneur’s value with the option value to the open the store WE(S) (thicker line), thevalue of the open store to the entrepreneur in equilibrium minus the entrepreneur’s personal investment

VE(S)−m0 (thinner line), and the optimal threshold to open a store for the entrepreneur So.Legend: Parameter values for the numerical example: ρ = 0.1, α = 0.05, σ = 0.5, cE = 0.5, cF = 0.5, bF = 5,

bE = 5, bB = 5, fup = 0.1, fdown = 0.05, CAPX = 10, m0 = 0, η = 0. Contract (x, y) = (0, 0.2).

yS−cF −fup) from the operating store, VF (S) (see Table 2), satisfies the boundary conditions that

the value is bounded at infinity and VF (SB) = −bF . Applying analysis from §2.1,

VF (S) =[y(Sρ + α

ρ2

)− cF+fup

ρ

]−[y(SBρ + α

ρ2

)− cF+fup

ρ + bF

]e−γ(S−SB) (39)

The first term in [ ] in equation (39) is the same as the value of the operating store for the franchisor

without financing (17). The second term in (39) represents effects of bankruptcy. Going forward

we need the expected value to the franchisor from the operating store VF (S) under the equilibrium

bank rate i

VF (S) = E[VF (S + η)|SB = SB(S)] =[y(Sρ + α

ρ2

)− cF+fup

ρ

]−[y(SB(S)ρ + α

ρ2

)− cF+fup

ρ + bF

]πB(S). (40)

where πB(S) is defined in (32).

4.7 The total franchisor’s value and the optimal franchise contract

The last step in the analysis (but the first step chronologically), is to determine the optimal contract

and the corresponding value of the franchisor’s rights.

As discussed in §3, during time interval [0, τo] the franchisor holds a portfolio (whose value is

denoted W (S)) comprising costs of non-operating store (C(t, S) = −fdown) and a claim on the

value from the operating store plus franchise fee. Given any contract (x, y), and the corresponding

optimal threshold, determined by the entrepreneur, So(x, y), for S < So(x, y), the franchisor’s

value function WF satisfies boundary conditions that this value is bounded at negative infinity and

WF (So(x, y)) = x+ VF (So(x, y)). Recall that VF is the equilibrium value of the operating store to

26

the franchisor (40). Analysis from §2.1 yields

WF (S|(x, y)) = −fdownρ

[1− πo(S, So(x, y))

]+[x+ VF (So(x, y))

]πo(S, So(x, y)), (41)

where as before, πo(S, So) = eδ(So−S) and δ =α−√α2+2ρσ2

σ2 < 0.

In deciding on the optimal contract the franchisor solves:

max(x,y)≥0

WF (S|(x, y)). (42)

Solving (42) is non-trivial, because connection between (x, y) and So(x, y) is given only numerically

(through (38)) and equilibrium rates i (or, equivalently, equilibrium bankruptcy threshold SB) in

(40) are computed numerically as well. The following numerical analysis presents our results.

Example 3. Continuing from Example 2. Numerically we find that the optimal franchise fee

x∗ = 0. This is realistic. In practice franchise fees are quite small (see Blair and Lafontaine 2011,

Chapter 3). The franchisor does not want to burden the entrepreneur with the additional debt,

on top of already high capital investment expenditures CAPX. Instead, the franchisor takes an

‘equity’ stake in the entrepreneur, claiming a fraction of profit through royalty rate y. This is also

consistent with the optimal contract for the franchise system, without financing considerations (§3).

The left panel of Figure 7 demonstrates how the optimal royalty rate, yfin, in the current model

(Model III) changes with profit volatility σ, and how yfin compares with yfra, rate in the franchise

system where financing considerations were ignored (Model II). As one might expect, with financing

considerations and bankruptcy costs, the franchisor should be less aggressive in setting terms of

the franchise contract. In this example, when σ = 0.5, yfin = 0.45. Recall that without financial

concerns (§3), the optimal franchise contract is (x∗, y∗) ≈ (0, 0.51). Furthermore, the higher the

profit volatility σ, the greater the gap between yfra and yfin. Thus, when profit is very volatile,

the miscalculation of the optimal contract terms is the worst.

Σ

y finy fra

0.2 0.4 0.6 0.8

0.1

0.2

0.3

0.4

0.5

Σ

SointSo

fraSofin

0.2 0.4 0.6 0.8

1

2

3

4

5

6

Σ

WF

WE

W fin

0.2 0.4 0.6 0.8

1

2

3

4

Figure 7: Effects of profit volatility, σ, on the optimal royalty rates, y (left panel); optimal thresholds for

opening the store So (middle panel); and the values of the franchisor, W finF , the entrepreneur, W fin

E , and

the system, W fin = W finF +W fin

E (right panel).Legend: Parameter values for the numerical example: S = 2, ρ = 0.1, α = 0.05, cE = 0.5, cF = 0.5, bF = 5, bE = 5,

bB = 5, fup = 0.1, fdown = 0.05, CAPX = 10, m0 = 0.

27

The middle panel of Figure 7 compares the optimal thresholds at which the store will be opened

(So) for our three models. As we have already seen in §3, decentralization of the franchise system

(Model II compared with Model I) leads to a higher threshold for opening a store. Financing

considerations raise this threshold even higher (despite the fact that the entrepreneur gets to keep

greater fraction of of the profit, as yfin < yfra). For example, when σ = 0.5, under the optimal

contract Sfino = 5.12 > Sfrao = 3.95 > Sinto = 2.95 > SNPV = 1.55. The gaps among thresholds

increase in the profit volatility σ.

The right panel of Figure 7 shows values to the franchisor, W finF , the entrepreneur, W fin

E ,

and the system, W fin = W finF + W fin

E , under the optimal contract in the model with financing.

Interestingly, these values are non-monotone in profit volatility σ. This is in contrast with the

corresponding values in the integrated (Model I) and franchise without financing (Model II) models

(see Figure 4), where these values were monotonically increasing in σ. The non-monotonicity is

caused by bankruptcy costs. On one hand, due to the option to open the store, the overall values to

the franchisor and the entrepreneur should be increasing in σ. On the other hand, when volatility

of the profit is very small, the probability of bankruptcy is small and so is the risk of incurring

bankruptcy costs. Consequently, we observe that values W finE , W fin

F , and W fin increase as σ

decreases below σ < 0.2.

What are the repercussions if the franchisor ignores financing considerations? Suppose σ = 0.5

and the franchisor used (a suboptimal) contract (x, y) ≈ (0, 0.51) (from Model II), but in the

model where financing matters (Model III). Then, the opening of the store would be delayed until

S(t) = So(y = 0.51) = 5.53, compared with the optimal threshold in Model III So(y = 0.45) = 5.12.

Furthermore, under (a suboptimal) contract (x, y) ≈ (0, 0.51), the equilibrium bankruptcy threshold

would be set at SB = 3.36 (equilibrium bankruptcy threshold is determined by banks at the time

the store opens), which is higher than the equilibrium bankruptcy threshold SB = 3.01 under the

optimal contract (x, y) ≈ (0, 0.45). Having a higher bankruptcy threshold is a problem because,

for a given value of profit S (once the store is operating), the higher the bankruptcy threshold SB,

the higher the bankruptcy probability. Therefore, it is detrimental both for the franchisor and the

entrepreneur to ignore the financing concerns when selecting contract terms.

Next we study comparative statics of this problem with respect to the franchisor’s share of the

costs cF , such that the entrepreneur’s share is cE = 1− cF . The left panel of Figure 8 shows that

the optimal royalty rate yfin increases in cost cF . If the franchisor pays a larger part of the cost,

she can optimally charge a larger royalty rate. This is to be expected.

The middle panel of Figure 8 demonstrates that the optimal thresholds of opening the store

28

y fra

y fin

c f

0.2 0.4 0.6 0.8 1.0

0.1

0.2

0.3

0.4

0.5

0.6 Sofin

Sofra

Soint

c f

0.2 0.4 0.6 0.8 1.0

1

2

3

4

5

W fin

WE

WF

c f

0.2 0.4 0.6 0.8 1.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Figure 8: Effects of the franchisor’s cost cF , such that cE = 1− cF , on the optimal royalty rate, y (left

panel); optimal threshold for opening the store So (middle panel); and the values of the franchisor, W finF ,

the entrepreneur, W finE , and the system, W fin = W fin

F +W finE (right panel).

Legend: Parameter values for the numerical example: S = 2, ρ = 0.1, α = 0.05, σ = 0.5, m0 = 0, bF = 5, bE = 5,bB = 5, fup = 0.1, fdown = 0.05, CAPX = 10, σ = 0.5.

in the franchise system with and without financing decrease with cF . Intuitively, because the

entrepreneur pays less to operate the store, he will open the store earlier.

The right panel of Figure 8 presents the values of the franchisor W finF , the entrepreneur W fin

E ,

and the system W fin with respect to cF . Interestingly, the entrepreneur’s value does not change

much, but the franchisor’s and the system’s values increase. Recall that the optimal royalty rate

yfin increases in cF . Thus, whatever assistance the franchisor shows to the entrepreneur by paying

higher cF , the franchisor extracts back through yfin. Interestingly, in the model with financing

(Figure 8), the entrepreneur’s value W finE is slightly increasing in cF , while in the model without

financing (Figure 5), the entrepreneur’s value W fraE decreases with cF . A possible explanation for

this difference is that in Model II, sharing costs addresses inefficiencies due to decentralization only.

But in Model III, the franchisor must consider not only decentralization but costs of financing and

bankruptcy. Due to the latter, the entrepreneur value must be kept at a stable level.

Next, let us consider effects of the entrepreneur’s personal wealth m0. The left panel of Figure 9

m0

y fin

2 4 6 8

0.1

0.2

0.3

0.4

0.5

Sofin

m0

2 4 6 8

1

2

3

4

5

W fin

WE

WF

m0

2 4 6 8

1

2

3

4

Figure 9: Effects of the entrepreneur’s wealth m0 on the optimal royalty rate, y (left panel); optimal

threshold for opening the store So (middle panel); and the values of the franchisor, W finF , the entrepreneur,

W finE , and the system, W fin = W fin

F +W finE (right panel).

Legend: Parameter values for the numerical example: S = 2, ρ = 0.1, α = 0.05, σ = 0.5, cE = 0.5, cF = 0.5,bF = 5, bE = 5, bB = 5, fup = 0.1, fdown = 0.05, CAPX = 10.

shows that the optimal royalty rate yfin increases in the entrepreneur’s personal wealth. This is

to be expected. Intuitively, the greater the entrepreneur’s personal wealth, the less he needs

29

to borrow, the smaller is the bankruptcy risk, and the more the franchisor can charge without

risking the entrepreneur’s bankruptcy. The middle panel of Figure 9 demonstrates that the optimal

threshold of opening the store in the franchise system decreases with m0. Initially, one may think

this is surprising because the more of his money the entrepreneur puts on the table when opening

the store, the longer the entrepreneur would wait to do so. But greater personal wealth of the

entrepreneur m0 reduces loan size and reduces risk of incurring bankruptcy costs. With less risk,

the entrepreneur is willing to open the store earlier.

The right panel of Figure 9 presents the values of the franchisor W finF , the entrepreneur W fin

E ,

and the system W fin with respect to m0. Interestingly, the entrepreneur’s value does not change

much, but the franchisor’s value increases in the entrepreneur’s personal wealth m0. Recalling the

increase in the optimal royalty rate yfin, we can conclude that the franchisor expropriates all benefits

of increasing m0. This resonates with the policies that some of the franchises employ, requiring a

certain level of personal wealth from the franchisees. Requiring the entrepreneurs to put more “skin

in the game” both serves as incentive for the entrepreneur to exert effort (a consideration outside

of our model) and benefits the franchisor because financial risks are reduced (a consideration that

we incorporated). In practice, however, the franchisors need to weigh the benefits of working with

wealthier entrepreneurs against the costs of searching and attracting such entrepreneurs.

The benefits of greater initial wealth m0 stem from the reduction of entrepreneurs’ reliance on

external financing. The franchisor’s can achieve the same outcome by providing financing to the

entrepreneurs. The benefits of doing so exceed the discount rate ρ for the franchisors. In fact,

empirical evidence (Shane et al. 2006) suggests that offering financing to entrepreneurs stimulates

growth of franchise systems. �

5 Conclusions

Our analysis captures two salient features that have not been studied rigorously in the franchise

contracting context before: (1) dynamic decision of the entrepreneur on when to accept the contract,

and (2) the entrepreneur’s financing (and bankruptcy) considerations.

We prove that the optimal franchise fee must be zero and the optimal royalty rate must be

positive. We present a simple optimization model for computing the optimal royalty rate. Thus,

one contribution of this paper to the franchising literature is to offer a new explanation for the

dominance of royalties over fees in practice. Under the optimal contract, the entrepreneur’s value

is positive, which also matches empirical observations. We show that an intuitive attempt to

extract more value from the entrepreneur by raising franchise fee (as static contracting models

30

might suggest), has a side-effect of delaying the entrepreneur’s signing of the contract.

In the effort to highlight new features, we have made several assumptions that, if relaxed, could

lead to both fee and royalty rate being positive. For example, we did not model the moral hazard

problem of the entrepreneur. In the presence of the moral hazard problem, it might be preferable

to increase the size of the on-going payments kept by the entrepreneur to provide him with a

strong incentive to exert effort. Nor (like most economics papers) did we consider the screening

of entrepreneurs, where a small application fee could serve to select “serious” applicants. Another

feature that could produce a solution with both fee and royalty rate being positive is risk-aversion

of the franchisor (or the risk-loving behavior of the entrepreneur). Depending on the franchisor’s

overall portfolio, it might be preferable to limit her exposure to the variable payments from the

store and use the fee instead. On the other hand, the existing explanations for the use of royalties

over fee, such as franchisor’s moral hazard, or the entrepreneur’s risk-aversion could tilt the optimal

contract back to featuring zero fees.

We show that the opening of the stores will be delayed, if the entrepreneur has has an option

time the opening. In the integrated system with a single decision maker this mirrors findings in

McDonald and Siegel (1986). The decentralization through franchise contracting results in further

delays. If the entrepreneur needs to raise external financing, this causes even greater delays. These

are not necessarily obvious results because, the franchisor can control the threshold for store opening

through the contract, and could choose to expedite the opening. For example, the franchisor

optimally lowers royalty rates, if the entrepreneur needs external financing. This makes opening of

the store more attractive, but financial frictions still cause overall delays.

There are ways for the franchisors to reduce waiting times and extract greater value from the

franchise system. Comparative statics show that franchisors can benefit by accepting a greater

share of the operating cost and by partnering with the entrepreneurs with greater initial wealth.

The latter observation is consistent with the minimum net worth requirement observed in franchise

agreements in practice. At the same time, franchisors need to be cautious about moral hazard

consequences if the costs of operating a store are not observable and in the presence of asymmetric

information about entrepreneurs’ abilities.

We study the effect of noise (prior to store opening) in store profits process. We find that

forecast-based uncertainty reduces franchisor’s value and delays store opening, but the effect is the

greatest if the entrepreneur must use external financing, due to the intrinsic non-linearities that

financing brings.

This paper studies franchise contracts specifically. However, some of the ideas that our analysis

31

generated could be extended to other business contracts, where the agent has option to optimally

time the contract signing and where external financing is important.

The focus of this paper is on choosing values of franchise contract terms, accepting as given

from practice the structure of the contract, the contracting process, and the franchise practice

existence in the first place. Franchising is a cornucopia of research problems and features (see Blair

and Lafontaine 2011), including questions of territory restrictions, number of franchise outlets,

vertical integration, dual distribution channels, quality control, termination provisions, product

tying, advertising and promotion, cost sharing, risk sharing, and others. We intentionally overlooked

these problems in this paper. Practical settings would dictate whether our findings can be extended

to address other or more general questions. For example, consider a model with multiple outlets. A

common franchise practice is to give stores rights to exclusive territory (so as to limit competition

among stores). Also, entrepreneurs negotiate financing terms individually. Therefore, from the

entrepreneurs perspective one can ignore interactions with other stores. From the franchisors

perspective, if conditions at different locations are homogeneous and the franchisor is risk-neutral,

than simplifying the problem and focusing on one store is not restrictive. But, if stores operate in

very heterogeneous conditions, choosing terms of the single contract that is applied to all outlets

requires weighing cash flows from all stores. In that case, one can think of our problem as a

subroutine for a larger problem where the franchisor also decides on the number of stores.

Lastly, a few observations about possible extensions and robustness of our assumptions.

We assumed that cash flows from the store are immediately consumed by the franchisee. An

alternative approach would be to solve a consumption/savings model, but this dooms the paper

to rely exclusively on the numerical analysis (and highly challenging analysis). The current paper

already has a three-player game with the stopping time problem, which is subject to the market

equilibrium constraint and (after this revision) noisy observations about the underlying process.

To solve the consumption/savings problem, one would need to add: the utility of the entrepreneur

for consumption, a new decision variable (consumption vs saving rate), and a new state variable

(amount of cash). Just the consumption/savings model with bankruptcy is non-trivial (e.g., Sethi

1997). If in addition, one considers equilibrium loan rates, the optimal timing of the store opening,

the franchise contract, the problem becomes extremely challenging and we leave it for future re-

search. Qualitatively, we do not believe that the insights from the current model would change with

the addition of consumption/savings tradeoffs. The entrepreneur who maintains a cushion of cash

would be less likely to experience a liquidity crisis. Thus, loan rates would be lower. This should

encourage earlier opening of the store. But, consumption will be reduced, thus the entrepreneur

32

would not be as willing to open a store. The intrinsic non-linearities of financing will still be present.

Thus, the noise in observations would still play a role in the model with financing considerations,

as we discuss in the paper. The tradeoffs between charging higher franchise fee or higher royalty

rates would still be the same.

We allow positive and negative values for process (1) and do not allow abandonment options.

Qualitatively, the existing results would not changed if we did, but the analysis would become

more complex. For example, for the simplest model of the integrated system, one would need to

introduce a threshold Sopenl at which the store is liquidated, after it is operating. This barrier

could be exogenous or, similar to the analysis of the threshold So, we can solve for the optimal

value of the threshold. There is another barrier for abandoning the store before the store is opened

Sclosedl . Before the store is open, the owner of the integrated system holds a portfolio of claims

on the cash flows from non-operating store, an option to open, and an option to liquidate. We

can solve for the optimal barriers to open and abandon, but the solution is a system of non-linear

equations. Extension to include liquidation options is even more challenging with multiple decision

makers because each player may have an option to walk away, before or after the store is opened.

Thus, the number of thresholds which need to be determined quickly becomes very large. There

are potential new insights to be gleaned from abandonment options, and we postpone exploring

them for future research.

We do not believe that a more complex process for (1), such as, mean-reverting, would add

new insights. We do not have any evidence that mean-reverting process is more appropriate for

Subway, Quiznos, Krispy Kream Doughnuts, Supercuts, Anytime Fitness, Hampton Hotels or other

franchise store sales. In fact, instead of reverting to the mean, higher S(t), could bring higher future

profits through word of mouth advertisement, on-line reviews, cost reduction (due to learning), and

other factors. We do not have evidence that mean-reverting is inappropriate either. One can also

consider adding other features to the model: jumps, stochastic volatility, multi-factor modeling,

stochastic interest rate process, etc. The downside of including any and all of these features is the

loss of analytical tractability.

Studies of asymmetric information, whether with respect to store profits, or abilities of either

party, present interesting possibilities, but due to their complexities in the dynamic settings are

left for future research. It is not clear who would have informational advantage in the franchise

relationship. The law requires that the franchisor shares information with the potential franchisee.

Moreover, entrepreneurs are encouraged to look not only for the franchise information in aggregate,

but contact local store owners for more location specific information. Perhaps the entrepreneur is

33

better informed of the local demand conditions, due to his geographic proximity, etc. One can

make arguments for either the franchisor having superior information, or the franchisor and the

entrepreneur being equal, or for the entrepreneur having superior information, or for double-sided

asymmetric information. Depending on the circumstances all of these would be equally valid.

Similarly, while one can foresee how our findings will interact with models of hidden effort, actual

rigorous studies of moral hazard, either one-sided or double-sided, in our dynamic framework are

left for the future.

References

Bhattacharyya, S., F. Lafontaine 1995. Double-sided moral hazard and the nature of share contracts.

RAND Journal of Economics 26 761–781.

Black, F., J. Cox 1976. Valuing corporate securities: some effect of bond indenture conditions.

Journal of Finance 31 351 – 367.

Blair, R. D., D. L. Kaserman 1982. Optimal franchising. Southern Economic Journal pp. 494 – 505.

Blair, R. D., F. Lafontaine 2011. The Economics of Franchising. Cambridge University Press. 32

Ave. of the Americas, New York, NY 10013, USA.

Brickley, J. A. 2002. Royalty rates and upfront fees in share contracts: evidence from franchising.

Journal of Law, Economics, and Organization 18(2) 511–535.

Caves, R. E., F. Murphy, William 1976. Franchising: Firms, markets, and intangible assets. South-

ern Economic Journal 42(4) 572 – 586.

Clifford, C. 2010. For franchisees, a new lender steps up to bat. CNNMoney.com. http://money.

cnn.com/2010/04/07/smallbusiness/franchise_lending/.

Dixit, A. K., R. S. Pindyck 1994. Investment Under Uncertainty. Princeton University Press. Prince-

ton, New Jersey.

Geske, R. 1977. The valuation of corporate liabilities as compound options. Journal of Financial

and Quantitative Analysis 12 541 – 552.

Ho, T. S., R. F. Singer 1982. Bond indenture provisions and the risk of corporate debt. Journal of

Financial Economics 10(4) 375 – 406.

Kaufmann, P. J., F. Lafontaine 1994. Costs of control: The source of economic rents for McDonald’s

franchisees. JL & Econ. 37 417.

Kim, I. J., K. Ramaswamy, S. Sundaresan 1993. Does default risk in coupons affect the valuation

of corporate bonds?: A contingent claims model. Financial Management pp. 117–131.

Klein, B. 1995. The economics of franchise contracts. Journal of Corporate Finance 2(1) 9–37.

34

Lafontaine, F. 1992. Agency theory and franchising: some empirical results. Rand Journal of Eco-

nomics 23(2) 263–283.

Leland, H. E. 1994. Corporate debt value, bond covenants, and optimal capital structure. Journal

of Finance 49(4) 1213 – 1252.

Longstaff, F. A., E. S. Schwartz 1995. A simple approach to valuing risky fixed and floating rate

debt and determining swap rates. Journal of Finance 50 798 – 819.

Mathewson, G. F., R. A. Winter 1985. The economics of franchise contracts. Journal of Law and

Economics 28(3) 503–526.

McDonald, R., D. Siegel 1986. The value of waiting to invest. Quaterly Journal of Economics

101(4) 707 – 727.

Merton, R. 1974. On the pricing of corporate debt: The risk structure of interest rates. Journal of

Finance 29 449 – 470.

Modigliani, F., M. Miller 1958. The cost of capital, corporation finance, and the theory of invest-

ment. American Economic Review 48 261 – 297.

Opler, T. C., S. Titman 1994. Financial distress and corporate performance. The Journal of Finance

49(3) 1015–1040.

Radner, R., L. Shepp 1996. Risk vs. profit potential: A model for corporate strategy. Journal of

Economic Dynamics and Control 20 1373 – 1393.

Sen, K. C. 1993. The use of initial fees and royalties in business-format franchising. Managerial and

decision Economics 14(2) 175–190.

Sethi, S. P. 1997. Optimal Consumption and Investment with Bankruptcy. Kluwer Academic Pub-

lishers. Boston, MA.

Shaked, M., J. G. Shanthikumar 1994. Stochastic Orders and Their Applications. Academic Press.

525 B Street, Suite 1900, San Diego, CA 92101-4403.

Shane, S., V. Shankar, A. Aravindakshan 2006. The effects of new franchisor partnering strategies

on franchise system size. Man. Sci. 52(5) 773–787.

Tenenbaum, M., H. Pollard 1985. Ordinary Differential Equations. Dover Publications.

Topkis, D. M. 1998. Supermodularity and Complementarity. Princeton University Press. Princeton,

New Jersey.

Trigeorgis, L. 1996. Real Options: Managerial Flexibility and Strategy in Resource Allocation. MIT

Press. Cambridge, MA.

35

Appendix

A Analysis of Model I — Integrated System

In this section, we present the valuation of cash flows and the optimal policy in the benchmark model

where franchisor and the entrepreneur are integrated and there are not financing considerations.

Similar to the other models, we shall conduct analysis in the reverse chronological order (see

Figure 2). First, we shall derive the value of the operating store, V (S) (recall that the process for

S(t) is given in (1)). Second, once we computed V (S) in §A.1, we shall turn to the question of

the optimal choice of the time to open the store τo (equivalently, the optimal threshold So) and

the total value of all cash flows, before and after the store is open, assuming that the decisions are

made optimally (§A.2). This total value (denoted by W (S)) depends on the value function V (S).

One can think of W as a contingent claim on V , which itself is a contingent claim on S.

A.1 Valuation of cash flows from the operating store

Recall from Table 3 that, when the store is operating, the integrated system collects C(t, S) =

(S − c − fup). The value V (S) of a claim on these cash flows satisfies the boundary conditions

stipulating that V does not blow up at infinities. Applying valuation process from §2.1,

V (S) = Sρ + α

ρ2 − c+fupρ . (43)

The first two terms Sρ + α

ρ2 correspond to the expected value of a perpetuity that pays out S(t) at

time t. The last termc+fupρ is the value of a perpetuity that costs a constant amount c + fup per

unit of time.

A.2 The optimal time to open the store and the optimal total value of theintegrated system

The integrated company might not open the store right away, because this requires an irreversible

investment CAPX and profit S(t) is uncertain. The company holds an option to open a store.

The company will open the store only if projected store profit S(t) exceeds a threshold So, which

we shall characterize.

From Table 3, the integrated system cash flows are C(t, S) = −fdown when the store is not

operating, S < So,. When the integrated system exercises its option to open the store, it receives the

value of the open store V (computed in (43)) in return for the investment costs CAPX (see Table 3).

Thus, the total value of the integrated system, W , is a combination of the value of cash flows while

36

the store is closed and a claim on V—the value of cash flows from an open store. The boundary

conditions on W are that this value does not blow up at −∞ and W (So) = E[V (So+η)]−CAPX,

where V (S) is given in (43) and expectation is taken over η (recall from §2 that S(τo) = S(τo) +η).

From equation (43), E[V (So + η)] = V (So). Therefore, from §2.1

W (S) = −fdownρ

[1− eδ(So−S)

]+ [V (So)− CAPX]eδ(So−S) (44)

where δ =α−√α2+2ρσ2

σ2 < 0. As before, expression πo(S, So) = eδ(So−S) ∈ (0, 1) for S < So is the

price of the Arrow-Debreu security that pays out $1 when the store is opened. Less rigorously, one

can also interpret πo(S) as the probability of eventually opening a store, adjusting for discounting.

With this interpretation, equation (44) affords a nice explanation: the value to the integrated

system is the discounted expectation of payoff (V (So) − CAPX) when the store is opened and

payoff (−fdown/ρ) if the store is never opened.

The optimal threshold for opening the store (So) is determined from the smooth-pasting con-

dition (Dixit and Pindyck 1994): W ′(So) = V ′(So). The solution to this equation (using (44) and

(43)) is

So = c+ fup − fdown − αρ + CAPXρ− 1

δ (45)

We shall denote the threshold in the integrated system as Sinto . It is instructive to compare Sinto

with the static “positive NPV” investment rule. By investing in the store, the integrated company

exchanges a stream of perpetual costs fdown for the perpetual cash flows from store profit S(t)

minus costs c + fup, by making investment CAPX. Thus, the NPV investment threshold SNPV

satisfies:

E[V (S + η)]− CAPX = Sρ + α

ρ2 − c+fupρ − CAPX = −fdown

ρ . (46)

Comparing (45) with (46) and using the fact that δ < 0, we conclude that

Sinto = SNPV − 1δ > SNPV . (47)

This is a classical result from the real options literature (e.g., McDonald and Siegel 1986): the

optimal investments should happen later than the NPV rule suggests, even in the integrated system.

The following proposition presents properties of the optimal solution for the integrated system.

Proposition 3. Properties of the optimal solution for the integrated system:

(i) The threshold Sinto (given in (45)) is increasing in σ, CAPX, c, and fup. Also, Sinto is

decreasing in fdown and α.

(ii) Sinto > SNPV

37

(iii) If V (Sinto )−CAPX+ fdownρ ≥ 0, then the integrated system value W int(S) (given in (44))

increases in profit volatility σ.

Statements in Proposition 3 are intuitive. As costs of running the store increase or costs of not

running it decrease, the optimal threshold for opening the store Sinto increases. The greater the

volatility of the store profit, the more valuable the timing option (of opening the store) and hence

the larger the threshold Sinto . The greater the rate of growth for store profit, the more attractive

opening the store is and the lower the optimal threshold Sinto .

The final note on the integrated system is that because the value function V (S) given in (43)

is linear, the noise η does not appear in the solution.

38

Online Appendix

Proofs and Technical Lemmas

Proof of Proposition 1:

The franchisor’s value function WF is defined differently depending on whether S < So or S ≥ So.

Case 1: Consider S < So(x, y). Using (18), (17), and (16), the optimization problem for the

optimal contract is

maxx,y

{−fdown

ρ (1− πo) +[x+ y

(Soρ + α

ρ2

)− cF+fup

ρ

]πo

}(48a)

cE+(CAPX+x)ρ1−y − α

ρ −1δ ≥ S (48b)

x ≥ 0, y ≥ 0, y ≤ 1 (48c)

To facilitate analysis, we replace decisions (x, y) with decisions (So, x), using the expression for the

optimal threshold (16). Specifically, 1− y =cEρ

+CAPX+x

Soρ +

αρ2 +

1δρ

. Recall that c = cF + cE .

maxx,So

{−fdown

ρ (1− πo) +

[cEρ

+CAPX+x(Soρ +

αρ2 + 1

δρ

)δρ

+ Soρ + α

ρ2 − c+fupρ − CAPX

]πo

}(49a)

So ≥ S (49b)

0 ≤cEρ

+CAPX+xSoρ

+ αρ2

+ 1δρ

≤ 1 (49c)

x ≥ 0 (49d)

Recall that δ < 0. From Assumption 1 and S0 ≥ S, it follows that(Soρ + α

ρ2 + 1δρ

)> 0. Therefore,

x∗ = 0 and S∗o solves

maxSo

{−fdown

ρ

[1− eδ(S0−S)

]+

(cEρ

+CAPX(Soρ

+ αρ2

+ 1δρ

)δρ

+ Soρ + α

ρ2 − c+fupρ − CAPX

)eδ(S0−S)

}(50a)

Soρ + α

ρ2 + 1δρ ≥

cEρ + CAPX (50b)

So ≥ S (50c)

Case 2: Consider S ≥ So(x, y). Using (17) and (16), the optimization problem for the optimal

contract is

maxx,y

{x+ y

(Sρ + α

ρ2

)− cF+fup

ρ

}(51a)

cE+(CAPX+x)ρ1−y − α

ρ −1δ ≤ S (51b)

x ≥ 0, y ≥ 0, y ≤ 1 (51c)

A-1

If Sρ + α

ρ2 + 1δρ <

cEρ + CAPX, then the problem above does not have a feasible solution and the

solution of the problem (50) is the optimal solution overall.

Therefore, assume that Sρ + α

ρ2 + 1δρ ≥

cEρ +CAPX. Observe that the objective function (51a) is

increasing in both x and y. Left hand side of constraint (51b) goes to infinity as y ↑ 1. Therefore,

constraint (51b) will be binding. Thus, the optimization problem becomes

maxx,y

{1−yρδ + S

ρ + αρ2 − CAPX − c+fup

ρ

}(52a)

cEρ

+CAPX+x

1−y = Sρ + α

ρ2 + 1ρδ (52b)

x ≥ 0, y ≥ 0, y ≤ 1 (52c)

Solution of this problem is x = 0 and y = 1−cEρ

+CAPX

Sρ

+ αρ2

+ 1δρ

. The objective value is

cEρ

+CAPX(Sρ

+ αρ2

+ 1δρ

)δρ

+ Sρ + α

ρ2 − c+fupρ − CAPX (53)

which is the same as (50a) at So = S. Therefore, the solution of the problem (50) is the optimal

solution overall.

Next, we shall simplify problem (50). Ignoring additive constant terms and removing positive

constants multiplying the entire objective function, re-write optimization problem (50) as follows.

maxSo

{(fdownρ +

cEρ

+CAPX(Soρ

+ αρ2

+ 1δρ

)δρ

+ Soρ + α

ρ2 − c+fupρ − CAPX

)eδSo

}(54a)

Soρ + α

ρ2 + 1δρ ≥

cEρ + CAPX (54b)

So ≥ S (54c)

From Lemma 4, function f(So) =

(fdownρ +

cEρ

+CAPX(Soρ

+ αρ2

+ 1δρ

)δρ

+ Soρ + α

ρ2 − c+fupρ − CAPX

)eδSo is uni-

modal. Therefore, S∗o = max(S, So), where So solves (20).

Finally, to complete the proof of this Proposition, substitute the optimal contract terms into

(13), to obtain the expression for the entrepreneur’s value (21). �

Lemma 4. Function f(So)def=

(fdownρ +

cEρ

+CAPX(Soρ

+ αρ2

+ 1δρ

)δρ

+ Soρ + α

ρ2 − c+fupρ − CAPX

)eδSo is uni-

modal.

Proof of Lemma 4:

Rewrite function f(x) = g(x)eδx, where g(x) = fdownρ +

cEρ

+CAPX(xρ

+ αρ2

+ 1δρ

)δρ

+ xρ + α

ρ2 − c+fupρ −CAPX. The

first and the second derivatives of f are f ′(x) = [g′(x)+δg(x)]eδx and f ′′(x) = [g′′(x)+δg′(x)]eδx+

A-2

δ[g′(x)+ δg(x)]eδx. To prove unimodality (quasiconcavity), it is sufficient to show that for x, which

satisfies the first order condition f ′(x) = 0, the second derivative is negative f ′′(x) < 0.

Specifically, it is sufficient to show that g′′(x) + δg′(x) < 0. By Assumption 1, xρ + α

ρ2 + 1δρ > 0.

In this set of x, function g is increasing and concave. Therefore, g′(x) > 0 and g′′(x) < 0. Because

δ < 0, it follows that g′′(x) + δg′(x) < 0. �

Proof of Corollary 1:

We shall discuss parts 1, 2, 3, and 4 only. Part 1 proof is based on the observation that for unimodal

f(So) =

(fdownρ +

cEρ

+CAPX(Soρ

+ αρ2

+ 1δρ

)δρ

+ Soρ + α

ρ2 − c+fupρ − CAPX

)eδSo , f ′(Sinto ) > 0 if c+fup−fdown+

CAPXρ > 0.

For part 2, results follow by invoking supermodularity of function f (see Topkis 1998).

For part 3, observe that for S < So, S∗o = So is constant in S and eδ(S

∗o−S) is increasing in S.

For S ≥ So, S∗o = S and −cEρ

+CAPX

δρ(Sρ

+ αρ2

+ 1δρ

) is decreasing in S because δ < 0.

For part 4, for S < So, S∗o = So is constant in S and so is y∗. For S ≥ So, S

∗o = S and

y∗ = 1−cEρ

+CAPXSρ

+ αρ2

+ 1δρ

is increasing in S. �

Proof of Lemma 2:

From SBdef= cE+iL

1−y and L > 0, it follows that i = (1−y)SB−cEL . There is one-to-one correspondence

between i and SB and between i and SB. Furthermore, minimizing i to find i(S) = min{i ≥

ρ : E[D(S + η)] ≥ L} is equivalent to minimizing SB. Using SBdef= cE+iL

1−y in (28), we rewrite

E[D(S + η)] in terms of the decision variable SB

E[D(S + η)] = (1−y)SB−cEρ

[1− νe−γ(S−SB)

]− bBνe−γ(S−SB) (55)

Finally, again, from SBdef= cE+iL

1−y , inequality i ≥ ρ is equivalent to cE+ρL1−y ≤ SB. Thus, min{i ≥ ρ :

E[D(S + η)] ≥ L} is equivalent to problem (29). �

Proof of Corollary 2:

Consider part 2. At SB = cE+ρL1−y , E[D(S + η)] = L

[1− νe−γ(S−SB)

]− bBνe−γ(S−SB) < L. By

assumption f∗ ≥ L. From continuity of function f , there is a number SB ∈ [ cE+ρL1−y , S) such that

SB satisfies (31) and for all SB ∈ [ cE+ρL1−y , SB), f(SB) < L. �

A-3