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2014-05-01

Essays on Capital Structure and Product Market

Competition

Song, Yang

Song, Y. (2014). Essays on Capital Structure and Product Market Competition (Unpublished

doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/25942

http://hdl.handle.net/11023/1468

doctoral thesis

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UNIVERSITY OF CALGARY

Essays on Capital Structure and Product Market Competition

by

Yang Song

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

GRADUATE PROGRAM IN ECONOMICS

AND

GRADUATE PROGRAM IN MANAGEMENT

CALGARY, ALBERTA

April, 2014

© Yang Song 2014

Abstract

This volume examines the interaction of capital structure and product market competition.

The first chapter investigates how upstream firms use trade credit to affect downstream

firms behavior in imperfect competition. The second chapter explores the impact of price

matching on firms’ advertising investments in a price duopoly game and the last chapter

focuses on how an incumbent firm adopting a price matching strategy changes its investments

to accommodate a new entrant.

Chapter 1 explains trade credit financing as a strategic tool for a supplier to influence her

retailer behavior in a product market, provides a new rationale for the existence as well as the

contract structure of trade credit financing, and shows why financially unconstrained firms

occasionally finance their inventory with expensive trade credit. In our model competing

supply chains deliver a homogeneous good to a market with imperfect competition where

retailers have to make inventory decisions before demand is realized. When demand is weak

trade credit financing makes the retailer more aggressive as he avoids having to finance

unsold inventory at the high trade credit interest rate. The ex-ante expected cost of having

to finance excess inventory at the high trade credit rate when demand is weak reduces

retailers’ optimal ex-ante inventory levels. When demand is high sales are constrained by

inventory and competition is less intense. The modified product market behavior induced

by trade credit financing increases the producer surplus at the expense of consumer surplus

in oligopoly markets, while we find no benefit for producers in either monopoly or perfect

competition.

Chapter 2 examines how a price matching strategy affects a firm’s advertising decision

under price duopoly competition. Price matching serves as a double-edged sword for firms’

investments in advertising, profits and social welfare. Specifically, if a firm’s advertising

benefits to both firms, a price matching strategy increases advertising investments, profits,

i

and consumer surplus relative to the Bertrand equilibrium. This arises as price matching

effectively reduces market competition and mitigates the free-riding problem of advertising,

giving both firms strong incentives to invest. Conversely, if advertising is predatory, price

matching harms both firms and consumers due to over-investment. Price matching moves

competition between firms solely into the realm of advertising, thereby exacerbating the

externality effects.

Chapter 3 studies how price matching affects an incumbent firm’s investment to accom-

modate a new entrant prior to competition. A simple theoretical model is developed to

investigate the interaction among price matching, the incumbent firm’s investments (R&D

and advertising) as well as product market competition. The price matching policy has

a significant effect on such investments but the impact works totally oppositely on invest-

ments for demand enhancement and cost reduction. Compared to Bertrand competition,

price matching facilitates advertising investment but impedes R&D investment.

iii

Acknowledgements

I owe a great debt to many people who have helped me make the completion of my thesis

possible.

I would like to express my deepest gratitude to my supervisors, Drs. Alfred Lehar, Robert Oxoby

and Lasheng Yuan, for their guidance, encouragement and patience over the past years. I

benefited greatly from their advice, insights, valuable time and especially the opportunity to

work with them. I would also like to thank Dr. Alex David for his help on my job market and his

crystal clear finance lectures.

Many thanks also go to Drs. Christina Atanasova, John Boyce, Eugene Beaulieu, Eugene Choo,

Jess Chua, Aidan Hollis, Joanne Roberts, J-F Wen and Scott Taylor, for their insightful

comments and suggestions which served to improve my papers. These graduate students, Rui

Wan, Libo Xu, Liang Chen, Jevan Cherniwchan, Kent Fellows, Ian Herffernan, Razieh Zahedi

and Matt Krzepkowski also deserve thanks for helping me to improve my English writing and

practice job talks. In addition, the Department of Economics and Finance area at Haykayne

School of Business, their faculty members and staff also deserve my gratitude for accepting me

into the program and providing me with an exceptional education---I am extremely proud have

been part of them.

I would like to express my heart-felt gratitude to my family. My parents, my uncles and my

Canadian family, the Turleys, have provided me with continued encouragement, kindness and

support for which I am extremely grateful.

Finally, without my Ailin and the sacrifices she has made, none of this would have been possible.

iv

Dedication

To Ailin and our forthcoming baby

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiDelication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Industry Structure and the Strategic Provision of Trade Credit by Upstream

Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Financing choice and product market behavior . . . . . . . . . . . . . . . . . 131.4 Financing choice and industry structure . . . . . . . . . . . . . . . . . . . . 181.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Price Matching & Strategic Investment in Advertising . . . . . . . . . . . . . 252.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.2 Game Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.3 Game Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Price Matching—A Double-Edged Sword . . . . . . . . . . . . . . . . . . . . 332.3.1 Cooperative Advertising . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.2 Predatory Advertising . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.3 Comparison of Cooperative and Predatory Advertising . . . . . . . . 42

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 Puppier Puppy Dog and Fatter Fat Cat: Strategic Investment for Incumbent

Firm under Price Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.2 Game Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60A Chapter One Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65B Chapter Two Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73C Chapter Three Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

v

List of Tables

1.1 State Dependent Cashflows . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

vi

List of Figures and Illustrations

1.1 Product Market Equilibria, Marginal Revenues and Costs under AlternativeFinancing Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 Price Matching and Cooperative Advertising . . . . . . . . . . . . . . . . . . 352.2 Price Matching and Predatory Advertising . . . . . . . . . . . . . . . . . . . 39

3.1 Price Matching and Demand Enhancement . . . . . . . . . . . . . . . . . . . 553.2 Price Matching and Cost Reduction . . . . . . . . . . . . . . . . . . . . . . . 57

vii

Chapter 1

Industry Structure and the Strategic Provision of

Trade Credit by Upstream Firms

1.1 Introduction

Trade credit financing is one of the largest and most important short-term financing options

in the United States and other countries. Cunat (2007) shows that trade credit accounts for

25% of total assets and 47% of total short term debt for a US representative firm.1 Trade

credit financing is also structured differently from straight debt as retailers borrow goods

from their suppliers free of charge for a certain period of time. When retailers can sell the

goods within that period they get free inventory financing. Once the free financing period

has expired, however, trade credit becomes a very expensive source of financing as suppliers

charge very high effective interest rates.2 In this paper, we offer a novel explanation why trade

credit contracts are optimally structured with a very high interest rate following a period of

free financing, why financially unconstrained firms occasionally finance their inventory with

expensive trade credit, and why only suppliers but not banks or debt markets can effectively

offer trade credit financing.

We argue that trade credit financing distorts product market competition and acts as a

collusion mechanism amongst competing supply chains. To illustrate our intuition imagine

two competing car dealerships in a city facing uncertainty about demand for cars by local

1Trade credit accounts for 17% of total assets and 50% of short debt for a representative UK firm.2For example, the commonly found scheme of 2/10 net 30 means that the retailer has to pay 2% more

if he pays within 30 days rather the first 10 days, which is equivalent to an annual interest rate of around46%—a huge penalty for the delayed payment. See also Smith (1987) and Petersen and Rajan (1997). Ng,Smith, and Smith (1999) report that most firms in their survey claim to demand payment within 30 days.Examining actual trade credit contracts Klapper, Laeven, and Rajan (2012) document that payment termsare often much longer and that for 30% of the contracts in their sample the discount period ends exactlyone day before the payment is due indicating that the discount is an incentive to pay on time.

1

residents. Suppose that the car manufacturer provides trade credit to the dealership under

which the latter gets free financing when they sell all cars this period but face a high interest

rate for all cars that the dealer rolls over for sale in the future. When consumer demand is

low the dealer realizes that he has to roll over some inventory for sale in the next period. Yet

for every additional car that he sells this period he can save the high financing costs. Thus

he will be more aggressive in the market, sell more cars at a lower price than a standard

Cournot model would suggest, and make a lower profit.

Ex-ante, before consumer demand is realized, the dealer anticipates that he could end up

in the unprofitable low demand state and he therefore orders a smaller inventory from the

manufacturer. When consumer demand turns out to be strong he can only supply a limited

number of cars to the market but his competitor, who follows the same inventory policy, also

has only limited supply. Prices for cars are, due to the limited inventory of both dealers,

higher than in a standard Cournot game and dealers can earn fat profits.3 We show that the

distortions that trade credit financing creates in the product market competition result in

higher combined expected profits for the manufacturer and the dealer compared to straight

debt financing.

The discrete jump in the interest rate that the manufacturer charges after the free fi-

nancing period, which is unique to trade credit financing, is essential for the mechanism of

our model. New inventory has a different financing cost to the retailer than inventory that

was rolled over from the previous period. The dealer can sell all his inventory when demand

is high and restocks with new inventory that comes without financing costs. When demand

is low the part of the inventory that gets rolled over into next period has to be financed at

a very high rate. The trade credit interest rate is only applied in the low demand state and

can thus be seen as a state contingent financing cost that allows the supply chain to fine

tune the retailers optimal product market strategy. Under straight debt financing the cost

3This prediction of our model is consistent with the findings of Zettelmeyer, Morton, and Solva-Risso(2007) who find that car dealers earn scarcity rents when demand for cars is high.

2

of newly stocked inventory is the same as for rolled over inventory and financing costs are

not conditional on consumer demand.

The distortions trade credit financing creates in the product market competition allow

suppliers and retailers to extract rents from consumers relative to an equilibrium where all

inventory is financed with straight debt. Trade credit financing can be seen as a collusion

mechanism that increases total producer surplus and reduces welfare. The increase in total

producer surplus from trade credit financing relative to bank financing depends on the de-

gree of competition and is highest in oligopoly markets; there is no benefit of trade credit

financing for producers under monopoly or perfect competition. Our model thus implies an

inverse U-shaped relationship between the benefit of extending trade credit and the degree

of competition.

This novel prediction of our model is consistent with the findings of previous empirical

studies. Analyzing trade credit policy of Indonesian companies, Hyndman and Serio (2010)

find an inverse U-shaped pattern exactly as predicted by our model with a very sharp increase

in trade credit when moving from monopoly to duopoly. Our predicted positive relationship

of trade credit use and competition in highly concentrated markets is consistent with the

findings of Fisman and Raturi (2004), who examine supply chain relationships in five African

countries and find that monopoly power is negatively associated with credit provision. Our

predicted negative relationship of trade credit use and competition in more competitive

markets is consistent with McMillan and Woodruff (1999), who find trade credit to decrease

as competition intensifies for a sample of Vietnamese firms, and Giannetti, Burkart, and

Ellingsen (2011), who find that sellers of differentiated goods, which are subject to less

competition, carry higher receivables than producers of homogeneous goods.

While it is hard to get data on specific trade credit or floor plan financing contracts we

can find anecdotal evidence mostly from court cases that is consistent with our main idea.

A home appliance retailer4 obtained a floor plan financing contract with a free financing

4see Romine vs. Philco Finance Corporation, United States Court of Appeals, Eighth Circuit, No 76-1535,

3

period of three to six months after which the rate would jump to 18%. The court notes

that ‘... the free floor plan program created and interest-free span of time which served as an

incentive for a dealer to rapidly sell his inventory and pay off his obligation to the company.

If the dealer failed to dispose the merchandise within the designated time, he was penalized

for not moving it quickly enough...’. A similar incentive program by Fiat motors offered

a 120 day free financing period.5 A recent industry publication notes that car dealerships

can take ‘...advantage of programs in which factories repay them for interest [of inventory

financing]. By selling a vehicle faster than a factory-set target number of days, which varies

by manufacturer, a dealer can actually make money on floorplanning.’ 6

We contribute to the existing literature in three ways: First, we offer a novel explanation

for the existence of trade credit under symmetric information, and in the absence of financial

constraints, inability to access bank financing, default, or agency problems.7 The only friction

we need is that retailers have to make their inventory decision before knowing consumer

demand. In the collusive trade credit equilibrium financially unconstrained firms optimally

pay the higher trade credit interest rate even though they would have access to cheaper

sources of financing because producer surplus is higher under trade credit financing, allowing

both firms to be better off.

Second, we document a new channel through which alternative types of debt financing

influence product market competition in different ways. Because of its unique structure with

1977.5see Fiat Motors of North America vs. Mellon Bank, United States Court of Appeals, Third Circuit, Nos

86-3588 and 86-3606, 1987.6Jamie LeReau, Interest spike would trim inventories, Automotive News, July 15, 2013.7Our paper adds to a large body of literature that explains the existence of trade credit in the presence

of competitive banking system (see Petersen and Rajan (1997) for a survey). Previous studies point outthat suppliers have a comparative advantage to control their retailers (suppliers can stop supplying goodsto retailers, see Cunat (2007); it is easier for suppliers to re-possess collateral than banks, see Frank andMaksimovic (2005) ; it is costly for a retailer to find a new supplier, see Boyer and Gobert (2009), suppliersalso have an informational advantage relative to outside financiers since it is less costly for suppliers monitorretailers’ financial status (Jain (2001)). In addition, trade credit can mitigate a moral hazard problem onthe side of retailers (Cunat (2007) and Burkart and Ellingsen (2004)), trade credit might also serve as aquality-guarantee mechanism for intermediate goods (Lee and Stowe (1993)), relaxes budget constraint dueto the possibility of a postponed debt payment (Ferris (1981)), and help retailers overcome credit rationingproblems if asymmetric information makes banks unwilling to lend to retailers (Biais and Gollier (1997)).

4

free financing followed by a very high interest rate, trade credit effectively allows suppliers to

charge demand state specific financing costs. The influence of trade credit debt on product

market competition therefore differs from that of straight debt. Our model therefore also

provides an explanation on why the trade credit is structured differently than straight debt.

Finally we contribute to the debate of separation of commerce and banking. In many

industries, like the automotive industry, producer sponsored financial institutions offer in-

ventory or floor plan financing to their retailers. Our model offers an explanation for the

wide spread use of vendor financing and in our context trade credit can be seen as a collusion

mechanism by producers to distort competition and extract rents from consumers. Allowing

manufacturers to conduct extensive financing activities can thus be welfare reducing.

Our paper builds on the literature that examines trade credit as a strategic tool for

price discrimination following Brennan, Maksimovic, and Zechner (1988). In their model, a

producer price-discriminates between consumer types. Low type consumers finance goods

with expensive trade credit but default with a high probability on their debt, effectively

making a low expected payment to the vendor. High type customers never default and

prefer to pay cash to avoid the high interest rate, and thus pay an ex-ante higher price

for the good. In our model, suppliers price discriminate under symmetric information over

demand states. The price discrimination mechanism of trade credit in our model also requires

no default.

Our paper is also tied into the literature on the interaction of financial structure and

product market competition that builds on Brander and Lewis (1986). They show that

debt financing makes firms with limited liability more aggressive in Cournot competition.

While most of the work in this field examines how levels of debt change firms’ behavior in

imperfect competition, our paper analyzes how different types of debt affect firms’ behavior

in a strategic setting. Our approach also differs because we do not utilize default or conflicts

between shareholders and bondholders in our model.8

8Our paper is also related to the huge literature on contracting and competition in vertical relationships

5

The rest of the paper is organized as follows: Section 1.2 sets up the model; Section

1.3 discusses how bank and trade credit financing affect a retailer’s behavior in imperfect

competition; Section 1.4 analyzes how the incentive of a supplier to offer trade credit varies

with industry structure, and Section 1.5 concludes the paper. All proofs are in the Appendix.

1.2 The Model

We consider an infinitely repeated three-stage game in which n supply chains, each consisting

of one supplier selling to one retailer (or dealer) that produce and sell a homogeneous,

non depreciable good to consumers. Consumer demand is either high (good state) with

probability q or low (bad state) with probability 1 − q. The price in the product market is

given by As −Q, where the intercept is state dependent and Q denotes aggregate quantity.

In stage 1, at the beginning of each period, each upstream supplier, given the financing

scheme (bank financing or trade credit financing), sets a wholesale price P for the good

as well as the trade credit interest rate rs, if applicable. Suppliers can produce unlimited

quantities of the good at zero marginal cost. In stage 2, each retailer orders goods from his

own upstream supplier to fill his inventory, taking the price (and trade credit interest rate

if applicable) as given.9 In stage 3, at the end of each period, consumer demand is realized

and retailers sell their goods to the product market, competing in quantity.

The only friction we assume is that a retailer cannot acquire inventory from his supplier

instantly (e.g. goods take time to build or require transportation); each retailer’s end of

period sales are therefore bound by his inventory. However, retailers can store any unsold

inventory for the next period at no cost, except financing costs. We assume retailers to have

zero fixed costs. To simplify the exposition of the paper and to create a need for financing we

assume that profits are distributed to shareholders at the end of each period so that retailers

based on Hart and Tirole (1990) and to papers identifying other mechanisms for price discrimination suchas resale price maintenance (e.g. Chen (1999)), or slotting allowances (e.g. Shaffer (1991) ).

9The contract between a supplier and a retailer is exclusive: each retailer can only purchase inventoryfrom their own supplier not the other one and verse visa.

6

need to find external financing for their inventory.10 To rule out trivial solutions we assume

that the demand state is verifiable but not contactable.

There are two available external financing choices for the retailers, straight debt provided

by a competitive banking sector (or a bond market) and trade credit financing offered by

their own suppliers. Under bank financing, the retailer pays the supplier in cash at the time

of the order and finances the inventory with a bank. Since retailers have no fixed costs and

are on average profitable, they will never default. Under bank financing retailers can thus

borrow at the risk free rate. Under trade credit financing, the retailer gets free financing

from the vendor for the goods that are sold at the end of the period, while he has to pay

the trade credit interest rate rs, which is optimally chosen by the supplier, to finance any

unsold inventory that is rolled over to the next period. In this infinite horizon game, the

end of current period equals the beginning of the next period. We assume that all agents

are risk-neutral.

1.2.1 Solution

Our model is a dynamic game with demand uncertainty and the solution could be path

dependent as current orders depend on last period sales and inventory levels. Because of

the infinite horizon we are able to rearrange and reinterpret the cash flows and inventory

valuation in such a way that each period is identical. With identical periods one possible

strategy that maximizes the overall expected profit is to maximize the profit in each period.

We therefore solve the game as a time independent static game, which is much more tractable.

We will explain our approach in more detail in the following subsections. We solve for the

subgame perfect Nash equilibrium by backward induction starting with the retailers’ decision

10This assumption can be relaxed without changing the findings of the model. All we need for the modelis that the marginal good sold in the bad state is financed externally. Allowing the firm to finance a partof the inventory with equity does not change our main result but complicates the exposition of the papersubstantially as we would have to keep track of current leverage and the realizations of profits in past periods.To simplify the exposition we abstracted from incentives to take on leverage such as tax benefits of debt andassume that inventory is externally financed and that all profits are paid out to shareholders at the end ofeach period.

7

problem.

Stage 3: The retailer’s end of period problem —ex-post competition At the end of the

period, each retailer i maximizes its profit ωi by competing in quantity Qfs given the demand

state s, the chosen form of financing f , and the amount of inventory If obtained before the

state of the demand is realized.11 The retailer’s problem is

maxQf

s

ωfs = (As −Qfs −Q−i,fs )Qf

s − Cfs , f ∈ {B, T}; s ∈ {b, g} (1.1)

s.t. Qfs ≤ If (1.2)

where Q−i,fs is the aggregate quantity offered by the other retailers except i and Cfs is the

retailer’s total cost measured in the end of period values. The total cost for the retailer

include the purchase cost of the good as well as the financing cost of keeping the good in

inventory until it is sold. Holding inventory is costly under both forms of financing and

therefore the retailer will never optimally hold more inventory than what he can sell in the

good state.12 In the good state, the constraint (1.2) is therefore binding and Qfg = If . In the

bad state, the sales in equilibrium should be below the sales in a good state or the inventory.

Thus, the constraint should not be binding ( i.e., If > Qfb ). We solve the third stage game

assuming the inventory constraint to be binding in good states and not binding in bad states

and later verify that this assumption is indeed true.

We start by analyzing the bank financing case first. The upper part of Panel A in Table

1.2.1 provides an overview of the end of period cash flows and inventory levels under bank

financing. At the end of each period the retailer starts out with an inventory of QBg that is

fully financed with bank debt having a face value equal to the cost of the inventory PBQBg

where PB is the price charged by the supplier under bank financing. The retailers total cost

11Superscript f ∈ {B, T} indicates a retailer’s financing choices, either bank financing (B) or trade creditfinancing (T ); subscript s ∈ {b, g} denotes the demand states, either a good state (g) or a bad state (b). In abad state, the choke price is Ab; in a good state, the choke price is Ag, where Ag > Ab.To avoid a degeneratesolution with zero sales in the bad state we assume that the choke price in the bad state is sufficiently high,Ab > (Ag −Ab)q.

12Under bank financing excess inventory would have to be financed at the bank rate r, and under tradecredit financing excess unsold inventory is financed at the trade credit interest rate rs.

8

Table 1.1: State Dependent Cashflows

State dependent cashflows for the retailer at the end of each period (panel A) and at thebeginning of each period (panel B) under bank and trade credit financing, respectively. Byallocating cashflows and valuing inventory as presented in the tables we make the total wealthat the beginning of the period independent of the state in the previous period as shown inPanel A. The total wealth at the end of the period only depends on the demand state that isrealized in that period. We can therefore see each period as identical. Qg and Qb denote thequantity sold in the good state and the bad state, P denotes the wholesale price charged bythe supplier, Pm is the price achieved in the retail market, and ωg and ωb are the retailer’sprofit in the good and bad state, respectively. All superscripts, which are used to indicatebank financing or trade credit financing throughout the text, are omitted for simplicity.

Panel A: End of each periodBank Financing

Good state Bad stateGoods Cash Flow Loan balance Goods Cash Flow Loan Balance

Starting value Qg −QgP Qg −QgPSale −Qg QgPm −Qb QbPm

Interest −rQgP −rQgPRepay Loan −QgP +QgP −QbP +QbPTotal wealth 0 ωg 0 Qg −Qb ωb −(Qg −Qb)P

Trade Credit FinancingGood state Bad state

Goods Cash Flow payables Goods Cash Flow payablesStarting value Qg −QgP Qg −QgPSale −Qg QgPm −Qb QbPm

Interest 0 0 − rs1+r (Qg −Qb)P

Pay supplier −QgP +QgP −QbP +QbPTotal wealth 0 ωg 0 Qg −Qb ωb −(Qg −Qb)P

Panel B: Beginning of each periodBank financing

Previously good state Previously bad stateGoods Cash Flow Loan balance Goods Cash Flow Loan Balance

Starting value 0 0 Qg −Qb −(Qg −Qb)POrder Qg −QgP Qb −QbPDraw loan +QgP −QgP +QbP −QbPTotal wealth Qg 0 −QgP Qg 0 −QgP

Trade Credit FinancingPreviously good state Previously bad state

Goods Cash Flow payables Goods Cash Flow payablesStarting value 0 0 0 Qg −Qb −(Qg −Qb)POrder Qg 0 −QgP Qb −QbPTotal wealth Qg 0 −QgP Qg 0 −QgP

9

CBs consists of the interest paid to the bank for financing the inventory and the repayment of

the principal of the loan for the sold units. Since the inventory is always fully financed with

bank debt, the financing costs of the inventory are the product of the interest rate and the

cost of the inventory, rPBQBg , where r is the bank interest rate. If a good state occurs, the

retailer sells his whole inventory QBg and re-pays the full face value of the outstanding loan,

PBQBg , to the bank. His total cost in the good state is therefore CB

g = rPBQBg +PBQB

g . If a

bad state occurs the retailer sells only QBb goods to consumers and still keeps QB

g −QBb unsold

goods in hand for sale in the next period. He then repays the loan for the sold QBb goods

and thus reduces the face value by PBQBb . Thus, his total cost is CB

b = rPBQBg + PBQB

b .13

Now we turn to analyze the trade credit financing case as shown in the lower part of

Panel A in Table 1.2.1. If a good state occurs, the retailer clears out his inventory, pays

the supplier for all goods, and total cost is CTg = P TQT

g , where P T is the price charged by

each supplier under trade credit financing. There is no financing cost because the retailer

obtains free-financing when selling all the goods within the period. However, if a bad state

occurs, the retailer only sells QTb goods, cannot repay the supplier in full, and has to finance

the unsold inventory at the trade credit interest rate rs. Specifically the retailer will pay an

amount of P TQTb to the supplier for the sold goods and will finance the unsold inventory with

face value P T (QTg −QT

b ) at the trade credit interest rate rs until the end of the next period

when product markets open again. Since interest payments are made in arrears we discount

the trade credit interest payment for one period at the risk free rate r. The retailer’s total

cost is then given by CTb = P TQT

b + P T rs(QTg −QT

b )/(1 + r).

As noted above in a good state the constraint (1.2) is binding and the sales are constrained

13Our model would also work if the retailer used a part of his profit to reduce his debt by more thanPBQB

b to save on future financing costs as long as the marginal unit sold in the bad state is still financedwith debt. Our debt repayment policy is consistent with industry practice. For example the U.S. SmallBusiness Administration defines floor plan financing as ”Floor plan financing is a revolving line of creditthat allows the borrower to obtain financing for retail goods. These loans are made against a specific pieceof collateral (i.e. an auto, RV, manufactured home, etc.). When each piece of collateral is sold by the dealer,the loan advance against that piece of collateral is repaid.” (see U.S. Small Business Administration, SpecialPurpose Loans Program, http://www.sba.gov/content/what-floor-plan-financing)

10

by the inventory If which is determined in stage 2. In a bad state, however, the constraint

is not binding and the first order condition to solve Qfb is derived from Equation (1.1). In

the appendix we derive the first order conditions and the optimal quantity of sales in the

bad state.

Stage 2: The retailer’s start of period problem —ex-ante inventory decision Given the

price of the good, P f , (and trade credit interest rate, rs, if applicable) charged by each

supplier, each retailer makes an ex-ante inventory decision to maximize their expected total

profits. As illustrated in Panel B in Table 1.2.1, no cash flows occur for the retailer at the

beginning of the period because inventory is externally financed either through straight debt

or through trade credit financing. At the beginning of the period each retailer stocks up

his inventory to the optimal level Qfg and thus ends up with the same inventory level and

payment obligations, either to the bank or to the supplier. Independent of the demand state

in the previous period after ordering their inventory retailers have the same inventory level,

the same payment obligations, and no cash flows. We can therefore transform the potentially

complex dynamic optimization problem to a state-independent static optimization problem

in which all periods are ex-ante identical. One strategy to maximize the retailer’s overall

profits in this infinitely repeated game and the one we will focus on in this paper is to

maximize the profit in each period ωf .

To determine the inventory and thus the sales quantity in the good state the retailer

maximizes the expected payoff, which is the probability weighted average of the retailers

profit in the good state, ωfg , and in the bad state, ωfb , respectively. Given the sales quantity

in the bad state, Qfb , as the solution of the optimization problem (1.1) and thus ωfb , the

retailer solves the following maximization problem to determine Qfg .

maxQf

g

[(1− q)ωfb + qωfg

]= 0 f ∈ {B, T}, (1.3)

Stage 1: The supplier’s problem Given the optimal inventory policy of the retailer the

supplier sets the price (and the trade credit interest rate, if applicable) to maximize profits.

11

We again structure cash flows such that each period is identical for the supplier and examine

the strategy to maximize each period’s profit, πf , as one way to maximize overall profits.

Under bank financing, each supplier only has the wholesale price as a choice variable to

maximize their expected profits

maxPB

πB = (1 + r)PBQBg − (1− q)PB(QB

g −QBb ) (1.4)

When selling QBg goods to their retailer, the supplier immediately obtains the cash payment,

which will be worth (1 + r)PBQBg at the end of the period. However, with probability 1− q

a bad state will occur and the retailer will buy fewer goods in the next period as he keeps

Qg − Qb goods in hand. Thus, the supplier will indirectly lose the expected profit of the

amount (1− q)PB(QBg −QB

b ).14

Under trade credit financing, each supplier maximizes the expected profit through simul-

taneous choice of price P T and trade credit interest rate rs.

maxPT ,rs

πT = qP TQTg + (1− q)

(P TQT

b +rsP

T (QTg −QT

b )

1 + r

)(1.5)

With probability q, a good state occurs and each supplier obtains a cash payment of

P TQTg at the end of the period for all the goods she has lent to the retailer. With probability

1− q, a bad state occurs and each supplier gets paid for the sold goods P TQTb in the current

period and collects the penalty payment in the next period, which has a present value of

rsPT (QT

g −QTb )/(1 + r).

Given the solutions for the optimal sales quantities for the good and the bad states

from the optimization problems (1.1) and (1.3), respectively, we can determine the suppliers

optimal wholesale price under bank financing by solving problem (1.4). Similarly, under

Trade credit financing we can find the optimal wholesale price and trade credit interest rate

by solving the optimization problem (1.5). All calculations can be found in the appendix.

14Recognizing the loss in next period’s revenue of PB(QBg −QB

b ) when the bad state occurs in this periodmakes each period again ex-ante identical. We can therefore again avoid any path dependencies in thesolutions.

12

To simplify the exposition of the paper, we limit ourselves to the case when the risk free rate

goes to zero.15 The following propositions summarize the game solutions for bank financing

and trade credit financing, respectively.

Proposition 1 There exists a subgame perfect Nash equilibrium under bank financing such

that each supplier charges PB = 2(qAg+(1−q)Ab)

3+n, each retailer orders an inventory QB

g =

(3+n−2q)Ag−2(1−q)Ab

(n+3)(n+1), and sells exactly QB

g in a good state and QBb = 2qAg+(1+n+2q)Ab

(n+3)(n+1)in a bad

state, respectively.

Proposition 2 There exists a subgame perfect Nash equilibrium under trade credit financing

such that each supplier charges P T = 2(qAg+(1−q)Ab)

3+nand sets the trade credit interest rate at

rs = q(Ag−Ab)

qAg+(1−q)Ab, each retailer orders an inventory QT

g = Ag

n+3, and sells exactly QT

g in a good

state and QTb = Ab

n+3in a bad state, respectively.

We can immediately see that the form of financing has an effect on the quantities firms

offer and will influence the overall profitability of the firms in the supply chain.16 It is also

noteworthy that the trade credit interest rate is always positive as Ag > Ab by assumption.

We will explore the intuition for the firms’ optimal product market strategy and its empirical

implications in the following sections.

1.3 Financing choice and product market behavior

In this section, we first illustrate how financing choices affect a retailer’s behavior in imperfect

competition, how this affects profits, and why both suppliers and retailers prefer trade credit

to bank financing.

15Our results also hold for a positive risk free rate but the expressions are significantly more complexwithout providing any major insights. A Mathematica workbook with the solutions for the general case isavailable from the authors upon request.

16The price the supplier charges is identical under both forms of financing, but this holds only when therisk free rate goes to zero.

13

The retailer’s decision in the bad state

We start with a retailer’s ex-post decision problem in the bad state. At the optimal

quantity determined by the first order condition of Equation (1.1), the marginal revenue

(MRb) from selling one more unit must equal the total marginal cost (MCb), which we can

write as the sum of the marginal purchasing (MPC) and marginal financing costs (MFCb),

MRb = MCb = MPC +MFCfb . (1.6)

Under both forms of financing marginal revenue and marginal purchase costs are given by

MRs = As−2Qfs −Q−i,fs , and MPC = P f , respectively. Under bank financing the marginal

financing cost is zero in a bad state, because the financing cost depends only on the inventory

level and is independent the retailers’ bad-state sales. Under trade credit financing, however,

the value of any unsold inventory has to be financed at the trade credit interest rate rs. Selling

an additional good saves the trade credit interest payment which is due at the end of next

period. Since rs > 0, the marginal financing cost is negative, MFCTb = −P T rs/(1 + r).

Trade credit financing therefore effectively lowers the total marginal cost for the retailer in

bad states and makes him more aggressive in sales. The higher the trade credit penalty

rate is, the lower the retailer’s total marginal cost in bad states is and the more severe the

competition is. By changing the trade credit interest rate, the supplier can strategically

influence the retailer’s aggressiveness in a bad state.

The retailer’s inventory decision and behavior in a good state

The trade credit penalty also affects the retailer’s behavior in a good state through the

ex-ante inventory decision. We will show that sales are always bound by inventory in good

states and that constraint (1.2) is binding. The retailer’s quantity choice in good states is

actually made when he chooses inventories before the state of demand is realized.

We can rewrite the first order condition of Equation (1.3) for the retailer’s optimal level

14

of inventory as

qMRg = qMCfg = q(MPC +MFCf

g ) + (1− q)MFCfb . (1.7)

The retailer obtains a marginal revenue from an increased unit of inventory only in the

good state because not all of the inventory is sold in a bad state. Increasing inventory incurs

two ex-ante marginal costs: the marginal cost of purchasing and financing the additional

good when it gets sold in the good state and the marginal cost of financing the additional

good in the bad state when it stays in inventory. Similar to the analysis of the bad state,

the marginal revenue and purchase costs are MRs = As − 2Qfs − Q−i,fs , and MPC = P f .

Under bank financing, the ex-ante marginal financing cost equals to the interest paid for the

value of the additional good in both demand states, MFCBg = MFCB

b = rPB. Substituting

into Equation (1.7) we get

qMCBg = q(PB + rPB) + (1− q)rPB.

The total marginal cost is then MCBg = PB(1 + 1

qr).

Under trade credit financing, the retailer gets free financing for the good state in which

all goods are sold but has to finance the unsold goods at the trade credit interest rate in the

bad state: MFCTg = 0, MFCT

b = P T rs/(1 + r) and MCTg = P T (1 + rs

1+r1−qq

).

In good states, if we were to ignore the inventory constraints, the retailer’s optimal level

of sales should be given by MRg = MPC + MFCfg . Comparing to the inventory decision

problem, the retailer would choose to sell more than the inventory under both, bank financing

and trade credit financing. The shadow price of the inventory is r 1qPB and rs

(1+r)1−qqP T ,

respectively, for bank financing and trade credit financing. The retail competition in good

states therefore is softened under both financing schemes. However, there are two differences

between bank financing and trade credit financing. First, rs is a choice variable optimally set

by the supplier while r is exogenously given. As a result, the trade credit financing scheme

enables the supplier to strategically influence her retailer’s aggressiveness in both demand

15

states, specifically how much to intensify the competition in a bad state and how much to

soften the competition in a good state. Second, under trade credit financing, rs explicitly

reduces the marginal cost in bad states and increases the marginal cost in good states, while

r does not have an explicit effect on the marginal cost in the bad state under bank financing.

We summarize the above results in the following proposition.

Proposition 3 Under trade credit financing, the retailers’ marginal cost is lower (higher)

in the bad (good) state, competition is intensified (softened), and aggregate supply increases

(decreases) relative to bank financing.

The trade credit interest rate

From Proposition 2, we know that the trade credit interest rate is rs = q(Ag−Ab)

qAg+(1−q)Abas

r goes to zero, which has several noteworthy properties: First, rs is always positive, i.e.,

the solution is consistent with the empirical facts that the retailers have to pay a penalty

rate if they cannot repay their suppliers on time. Second, rs increases when a good state

becomes more significant (q or Ag −Ab is high) or a bad state is less significant (Ab or 1− q

is low). As the relative importance of the good state increases, protecting the good state

profit (softening the competition) becomes more important, which requires a rise in rs.

Corollary 1 Under trade credit financing, the penalty rate rs increases with the probability

of a good state and the gap of choke prices in both states.

Since demand states are assumed to be observable but not contactable, the retailer can

theoretically avoid paying the high trade credit interest rate by financing the unsold inventory

in the bad state with bank debt at the lower rate r and repaying the supplier in full. In our

model the retailer is financially unconstrained, never defaults, and has always access to debt

markets. However, while such a strategy would be rewarding in the short term, in the long

term the game would revert back to the bank financing equilibrium. We will show in Section

1.4 that with at least two competing supply chains, aggregate profits of each supply chain

16

as a whole are higher under trade credit financing than under bank financing. The supplier

can make the retailer therefore always indifferent with respect to the financing scheme with

transfer payments. Car manufacturers, for example, make ex-post transfer payments called

”holdbacks” to their retailers on a monthly or quarterly basis which are based on past

sales volume.17 Suppliers can then refuse to pay holdbacks when retailers deviate to bank

financing. Another potential transfer mechanisms is advertising which is often paid by the

supplier and benefits the retailers’ sales.

Trade credit financing as a supplier’s price discrimination scheme against her retailer

There are potentially two stages of price discrimination. First, retailers price discriminate

against consumers in different states of demand. Second, and the main focus of this paper, the

suppliers price discriminate against their retailers. Trade credit financing implicitly allows

the supplier to charge the retailer state contingent marginal costs and thus price discriminate

between demand states. On appearance, the supplier seems to charge the retailer a low price

(free financing) in good states and a high price (due to a higher trade credit interest rate)

in bad states, which seems to contradict the typical pricing pattern in price discrimination

theory (a high (low) price is charged when a demand is high (low)). The misconception

arises from the incorrect use of the average price instead of unit price or marginal price. To

find the correct state contingent prices rewrite the supplier’s profit function

17The IRS describes holdbacks in the following way: ’When dealers acquire their new car inventory frommanufacturers, usually the invoice includes a separately coded charge for ”holdbacks.” Dealer holdbacksgenerally average 2-3 percent of the Manufacturer’s Suggested Retail Price (MSRP) excluding destinationand delivery charges. These amounts are returned to the dealer at a later date. The purpose of the”holdbacks” is to assure the dealer of a marginal profit.’, see New Vehicle Dealership Audit Technique Guide2004 - Chapter 14 - Other Auto Dealership Issues (12-2004), Internal Revenue Service.

17

πT = qP TQTg + (1− q)

(P TQT

b +rsP

T (QTg −QT

b )

1 + r

)

=

(qP TQT

g + (1− q)rsP

TQTg

1 + r

)+

((1− q)

(P TQT

b −rsP

TQTb

1 + r

))

= q

(P T +

1− qq

rsPT

1 + r

)QTg + (1− q)

(P T − rsP

T

1 + r

)QTb

= qP Tg Q

Tg + (1− q)P T

b QTb

where P Tg = P T (1 + 1−q

qrs1+r

) and P Tb = P T (1 − rs

1+r) are the supplier’s effective prices

in good and bad states, respectively. The price charged by the supplier in the good state is

clearly higher than that in the bad state. Notice that P Tg and P T

b are exactly the retailer’s

total marginal costs for the equilibrium sales in good and bad states, respectively.

The incentive for a supplier to provide trade credit is hence to price-discriminate to

her retailer between strong and weak demand states. In our model the rationale for a

supplier to set state contingent prices is to change the marginal cost for the retailer to

influence his behavior in the final product market. The supplier’s price discrimination in our

analysis is a double price discrimination, or a price discrimination to influence the retailer’s

price discrimination against the consumers. We summarize our findings in the following

proposition:

Proposition 4 Under trade credit financing, the supplier optimally price discriminates the

retailer between the states of demand: charging a high effective price P Tg = P T (1+ 1−q

qrs1+r

) in

good states and a low effective price P Tb = P T (1− rs

1+r) in bad states. As a result, compared

to bank financing, the profits of the supplier are higher under trade credit financing.

1.4 Financing choice and industry structure

We now look into the incentive of a supplier to offer trade credit financing, the effect of

industry structure, and the producer surplus.

18

Figure 1.1: Product Market Equilibria, Marginal Revenues and Costs under AlternativeFinancing Arrangements

The graph shows the inverse demand curve (bold line), the marginal revenue of the inte-grated monopolist, and the equilibrium marginal revenue line for an oligopoly, for whicheach point corresponds to an equilibrium in an oligopoly game and represents the marginalrevenue and aggregate supply in that equilibrium. The point M , O , B, and T denotethe equilibrium points where marginal revenue equals marginal cost of the integrated mo-nopolist, the integrated oligopolist, the n supply chains under bank financing, and the nsupply chains under trade credit financing, respectively. The parameters for the graph are:Ag = 10, Ab = 7, q = 1/2, r = 0, n = 3.

P

∑Qg

∑Qb

Good stateBad state

PM = PT

M

T

Ag

PO

O

P = MCB

MCTg

B

PB

MR

gintegrated

monop

olist

equilibriumMRgoligopoly

PM = PT

Ab

M

TPO

O

P = MCB

PB

B

MCTb

19

Figure 1.4 illustrates the basic mechanics of our model by means of an example. The

graph shows the marginal cost curves under trade credit and bank financing (horizontal lines),

the inverse demand curve (bold line), and the marginal revenue for the integrated monopolist.

Each point in the line labeled ”equilibrium marginal revenue oligopoly” corresponds to an

equilibrium in a Cournot oligopoly game and shows the marginal revenue (along the vertical

axis) and aggregate output (along the horizontal axis) in that equilibrium. To derive this

line we solve a simple Cournot game for marginal costs ranging from zero to the choke price.

We then plot for each equilibrium a point defined by marginal revenue and the aggregate

output.

We start as a reference case with a single, vertically integrated monopolist. Since pro-

duction cost of the good is assumed to be zero, the optimal quantity that the vertically

integrated monopolist offers can be found where the marginal revenue line hits the x-axis

(point M) and the corresponding price in the consumer product market is given by PM .

When more firms enter the industry, competition flattens the equilibrium marginal revenue

curve under oligopoly and firms offer more in aggregate (point O) which decreases their

equilibrium revenue as products in the consumer market are sold for PO.

The retailers of an oligopoly supply chain face the same marginal revenue function but

their marginal costs increase because they have to purchase the intermediate goods at the

wholesale price P from the supplier. When r goes to zero, as in the example of the graph, the

retailers using bank financing pay no financing costs and thus their marginal cost equals the

price set by the supplier, P , and the overall equilibrium in the product market is at point B.

The aggregate output of the supply chain comes closer to the quantity that is offered by the

vertically integrated monopolist, however under bank financing we see that relative to the

integrated monopolist the output is too high in the good state and too low in the bad state,

respectively. This is is exactly the problem that trade credit can overcome. By optimally

choosing the trade credit interest rate the supplier can increase her retailer’s marginal cost

20

in the good state to MCTg and lower the marginal cost in the bad state to MCT

b . In some

cases it is possible – as in this specific example – to achieve exactly the output of a vertically

integrated monopolist. In general trade credit financing, with its ability to charge state

dependent marginal costs, can make the retailer choose an output closer to the output of

the integrated monopolist than bank financing.

Trade credit financing allows producers to price discriminate across demand states, moves

industry output closer to the integrated monopolist’s optimal choice and thereby increases

producer surplus at the expense of consumers. Price discrimination results in a less effi-

cient outcome and under trade credit financing consumer surplus as well as welfare decrease

relative to bank financing. This intuition is summarized in the following proposition.

Proposition 5 In imperfect competition expected consumer surplus and total welfare are

lower under trade credit financing than under bank financing. With at least two supply

chains the expected producer surplus per supply chain is higher under trade credit financing.

For producers the relative advantage of trade credit financing over bank financing depends

on industry concentration. In the case of a monopoly supply chain, under bank financing, we

have a typical double marginalization problem. The retailer will then set prices higher than

the integrated monopoly prices and the aggregate profits fall. Under trade credit financing,

the supplier and the retailer still face the double marginalization problems. In addition,

to strategically price discriminate against the retailer, the supplier charges state contingent

marginal costs to the retailer and thus introduce a further distortion to the market. As a

result aggregate profits are further reduced.

With two or more competing supply chains competition lowers prices. When the number

of supply chains increases initially, the competition mitigates the effects of double marginal-

ization, brings the equilibrium prices closer to the monopoly price, and the aggregate profits

grow. As the number of the supply chains increases further, competition dominates and

the prices and aggregate profits fall. In perfect competition, we can see from from Propo-

21

sitions 1 and 2 that the suppliers’ wholesale price P is zero, producer surplus is zero, and

thus the form of financing becomes irrelevant. We summarize our intuition in the following

proposition:

Proposition 6 The difference in expected producer surplus per supply chain between trade

credit financing and bank financing is an inverse U-shaped function in the number of supply

chains.

Trade credit is a collusion mechanism that allows producers to extract rents from con-

sumers. Like for most other collusion mechanisms there exists an incentive for firms to

deviate from the collusive equilibrium for short term gain. One way to deviate in our

equilibrium is that a whole supply chain might move from trade credit financing to bank fi-

nancing. However, we believe that trade credit is a very robust collusion mechanism because

firms can observe at the beginning of each period, before the product market opens, whether

other firms offer vendor financing or not. In practice companies also create an institutional

framework for vendor financing which can be seen as a commitment device to the trade

credit equilibrium. For example, almost all car companies have financing arms, separate

corporations, that offer favorable financing for car dealers’ inventory (floor plan financing).

Set up costs of these finance companies and long term financing contracts with dealers make

it very costly for car manufacturers to deviate from the trade credit equilibrium for short

term gain. We also find that for a wide range of parameter values the benefit of short term

deviation is far smaller than the present value of future gains obtained under the trade credit

equilibrium.

Another issue to consider is why banks cannot offer a contract that replicates the trade

credit contract. By offering a trade credit contract with free initial financing and a payment

of rs in the bad state banks would earn a positive profit. Since we assume free entry in

the banking sector banks would compete that profit away and such a contract would not be

22

sustainable.18

1.5 Conclusion

We investigate how different kinds of debt affect output market equilibrium by comparing

bank and trade credit financing, and offer a novel explanation that why trade credit exists

even though it is often viewed as an expensive financing option. We argue that trade credit

financing modifies a retailer’s ex ante inventory policy and ex post product market strategy,

respectively, in an uncertain demand environment. When demand is low the retailer sells

more to avoid financing the unsold inventory at the high trade credit rate ex ante the

possibility of having to pay the high trade credit interest rate induces the retailer to reduce

his optimal inventory level, which in turn limits competition in the product market when

demand is high.

The distortions that trade credit financing introduces to product markets allows pro-

ducers to increase their profits at the expense of consumer surplus. We can therefore see

trade credit as a collusion mechanism between supply chains that mitigates competition and

reduces welfare. We offer a novel explanation why financially unconstrained firms finance

their inventories with expensive trade credit and why suppliers are able to offer a financing

contract that cannot be replicated by banks.

Our findings also have important policy implications for industries that rely heavily on

vendor financing, often through institutionalized finance companies, such as the automobile

industry. All major car producers own finance companies that provide financing of their

retailers’ inventories, often referred to as floor plan financing. Our analysis shows that

allowing commercial firms to engage in financing activities can mitigate competition and

18Banks could in theory offer a step-function contract with a negative interest rate for one period and apositive rate for any unsold goods such that the expected profit of the contract is zero. However, we believethat such a contract would be hard to sustain in equilibrium. Unless demand states are contractible theretailer could claim that a high demand state has occurred and refinance the inventory with another bank.Since banks and retailers have no long term relationship there are no negative consequences for the retailer.

23

reduce welfare. Our paper also contributes to the long ongoing discussion in the U.S. on

the separation of banking and commerce. The recent Dodd-Frank enacted a three year

moratorium on the creation of industrial loan companies (ILCs) in the U.S., which are often

owned by large industrial producers and provide financing to the firms clients. Our analysis

provides and argument that separation of banking and commerce is welfare increasing by

shutting down a potential collusion mechanism amongst producers.

24

Chapter 2

Price Matching & Strategic Investment in Advertising

2.1 Introduction

Price matching, commonly adopted in many retail markets, is a promise by firms that the

lowest price will be guaranteed if a consumers finds a cheaper price for a good in another

retailer’s store. It seems it incurs strong price competition among retailers and consumers

are made better off under this policy. Economics research however shows a totally opposite

opinion that retailers use such a promise as a strategic tool to extract consumer surplus in

imperfect competition.

There are three main streams of arguments in this literature. The first argument focuses

on collusion theory. Price matching serves as a collusion facilitating device, reducing retailers’

undercutting incentives and then facilitating cooperation. A collusive equilibrium in price

competition is not sustainable as each rival always would lower its price to gain a larger

market share. Under the price matching policy however undercutting is eliminated as other

firms will match the same price and then no one can increase its own profits. As a result, a

collusive equilibrium is achieved under the price matching policy (see Salop (1986), Belton

(1987), Chen (1995) and Dugar (2007)). The second one focuses on price discrimination

theory, arguing that given asymmetric information among consumers, retailers earn more

profits by charging different prices between informed and uninformed ones. (see Png and

Hirshleifer (1987), Corts (1997) and Lim and Ho (2008)). Signalling argument is initiated by

Moorthy and Winter (2006), showing that retailers use price matching to signal consumers

that they have low costs and more competitive in the market. Low cost firms can always

match or even beat any prices listed by high cost firms but not vice versa. The larger the

cost difference between high and low cost firms, the more notable the cost advantage for the

25

low cost firms. Therefore by promising the lowest price, low cost firms advertise customers

that they are offering a cheaper price than others.

We are motivated in two ways from the literature on price matching. First, most papers

only focus on the competition stage (i.e. the impact of price matching on product compe-

tition); however, prior to competition, retailers in oligopolistic markets always incur huge

advertising expenditures on their products to enhance demands—firms advertise almost ev-

erywhere in our daily life—on TV, magazines, newspapers, internet and so on. Of course

the associated advertising expenditures are always very large—for example, in 2006 the total

amount of advertising expenditures in the US was around 285.1 billion, accounting for 2.2%

of US GDP (Bellefflamme and Peitz (2010)); General motors in 2003 spent 3.43 billion to

advertise its cars and trucks while Proctor and Gamble devoted 3.32 billion to advertise its

detergents and cosmetics (Bagwell (2007)).1

Based on these statistics, there is no doubt that advertising is an extremely important

strategic tool for retailers prior to product competition. As mentioned above, price matching

is also a commonly adopted strategy for most retailers at the competition stage. Thereby the

question of interest is as both important strategic tools, whether they have some unexplored

but important interaction with each other: does a price matching strategy have a significant

impact on firms’ advertising investments and vice versa? To our best knowledge few papers

investigate the effect of price matching on firms’ advertising investments and this is the first

paper to formally consider this issue.

By modeling a two stage game, we are interested in investigating the interaction among

price matching, firms’ advertising investments as well as product market competition. More

1There is a large literature on advertising and basically there are two broad kinds of advertising—informative advertising and persuasive advertising. The former one conveys product information for con-sumers including its existence, price, properties and location of a product and also saves consumers’ searchingcosts (e.g. Grossman and Shapiro (1984); Bester and Petrakis (1995); and Dukes (2004) ). The latter onechanges a consumer’s taste and then alters the willingness to pay and reservation price (e.g. Dixit andNorman (1978); Slade (1995); von der Fehr and Stevik (1998); Bloch and Manceau (1999) and Kim and Shin(2007)). In this paper, we mainly focus on the persuasive perspective and two subcategories are discussedin the following section—cooperative and predatory advertising.

26

specifically, we try to answer the three questions as follows:

(1) How does a firm in oligopolistic competition make or change its advertising investment

decision if adopting a price matching strategy?

(2) Given the types of advertising, does a price matching strategy facilitate or impede

firms’ advertising investments and why?

(3) How do the advertising investment and price matching jointly affect product market

competition, firms’ profitability and social welfare?

Given the above motivations, we complement the existing literature in two ways: first

we provide a new explanation for the strategic effect of price matching on a firm’s advertis-

ing investment decision in oligopolistic competition. Two kinds of advertising are broadly

discussed in the literature: one is cooperative advertising, meaning that one firm’s advertis-

ing not only increases its own demand also enhances its rival firm’s demand. The other is

predatory, showing one firm’s advertising increases its own demand but attracts consumers

away from its rival and thereby reduces the rival firm’s demand. The two kinds of adver-

tising however have their own problems. The problem arising from cooperative one is that

each firm’s advertising imposes a positive externality on the other. The advertising firm

only cares about its own profits, resulting in the amount (intensity) of advertising under-

supplied relative to the amount maximizing total industry profits. In other words, each firm

would always wait for its rival firm’s investment and then free-ride on such a contribution.

Predatory advertising however shows us a totally different scenario. As each firm’s advertis-

ing investment harms the rival’s demand, imposing a negative externality, each firm has to

advertise excessively to mitigate this negative externality and thereby the amount of adver-

tising is wasteful from the industry perspective as a whole (i.e. the socially optimal amount

of advertising would be zero).

We argue that price matching has a significant effect on firms’ advertising investments

prior to competition stage, which in turn affect product competition. We find that price

27

matching serves as a double-edged sword for firms’ advertising investments, profits and con-

sumer surplus. More specifically, under cooperative advertising, price matching facilitates

both firms’ advertising investments and increase their profits and consumer surplus relative

to the Bertrand equilibrium. Price matching effectively weakens firms’ free-riding incentive,

encourages both of them to contribute more efficiently in this ”public good” and facilitates

advertising investment. This occurs because price matching reduces price competition in the

second stage, giving both firms a stronger incentive to invest more in advertising moving

closer to the optimal amount from an industry perspective. The effective advertising con-

tribution from both firms increase firms’ profits and consumer surplus in terms of Bertrand

equilibrium. Both firms and consumers are made better off under the price matching policy.

Conversely, price matching under predatory advertising makes both firms’ overinvest in

advertising but decreases firms profits and consumer surplus compared to Bertrand equi-

librium. We find price matching strengthens firms’ incentives to overinvest and makes the

wasteful advertising competition more wasteful. The main reason arises from the nature

of predatory advertising. Each firm facing an increased demand after advertising invest-

ment always implicitly undercuts its rival to take up a larger market share by matching its

rival’s price at the product competition stage. This gives both firms a stronger incentive

to overinvest further and the resulting intensity of advertising is much more excessive from

the industry perspective. Eventually each firm is caught in a prisoner’s situation where

both firms equally overinvest in advertising but their market shares keep unchanged and net

profits are reduced by advertising costs. In addition, as there is no change on consumer will-

ingness to pay but consumers are charged at a high collusive equilibrium price, consumers

are definitely worse off. Price matching harms both firms and consumers and reduces social

welfare under predatory regime.

We also challenge the traditional argument that price matching always makes firms better

off but consumers worse off. The type of advertising plays an important role to determine

28

whether both consumers and firms can be better off and worse off from such advertising

investments. Under both types of advertising, price matching always facilitates firms’ ad-

vertising investments; however, the results from such investments are totally different. The

nature of each type of advertising or the consumer preference determines whether such in-

vestments are more efficient or more wasteful. An interesting empirical question associated

with this prediction is whether retailers adopting a price matching policy advertise more

heavily than those who do not adopt such a policy.

The rest of the paper is organized as follows: Section 2 sets up the model; Section 3

discusses and compares how price matching affects firms’ investment behavior in advertising

under cooperative and predatory regimes; and Section 4 concludes the paper. All proofs are

in the Appendix.

2.2 The Model

2.2.1 Assumptions

Consider an industry in which two rival firms compete in price on differentiated products

(i.e. Bertrand competition) after making an investment on advertising.2 Each firm has

no fixed cost, and a constant and symmetric marginal cost, c, which is assumed to be 0.

There is no asymmetric information between the two firms. Consumers are fully informed

and have no hassle cost. Assuming a consumer’s utility function is a consumption function

of the two differentiated goods, qi and qj, and a numeraire good n (i.e. U(qi, qj, q0) =

a(qi + qj)− 12(q2i + 2dqiqj + q2j ) + q0), we have the following demand function, which includes

an advertising effect for the two goods:

qi = a+ θAi ± βAj − pi + dpj (2.1)

where a denotes the size of the market; Ai is the advertising provision by firm i; d

2Following the literature on price matching (e.g. Logan and Lutter 1989), the goods under price matchingare identical but with retailing firms differentiated.

29

shows the degree of differentiation between the two goods. If d = 1, the two goods are

homogenous; if d = 0; the two goods are independent. If d ∈ (0, 1), the two goods are

imperfect substitutes and strategic complements to each other. θ, between 0 and 1, is the

advertising factor measuring the effect on demand; β is the product of d and θ, which means

when the two goods are more substitutable, the effect of firm i′s investment of advertising

on firm j′s demand becomes stronger. However, there are two types of advertisement. One

is cooperative, meaning firm i′s advertising increases both firms’ demands, then demand is

given by qi = a+ θAi +βAj − pi + dpj; however, the other one is predatory, meaning firm i′s

advertising only increases its own demand but reduces firm j′s demand, and verse visa, then

the demand is given by qi = a+ θAi − βAj − pi + dpj. Finally, each firm incurs a quadratic

advertising cost, which is assumed to be 12mA2

i , where Ai is the advertising intensity and m

is assumed to be greater or equal to 1.

2.2.2 Game Sequence

The formal game set-up progresses over two stages:

1. Advertising investment

Firm i and j make their advertising investments prior to product market com-

petition.

2. Product market competition

Firm i and j make a decision on either competing in price or price matching to other’s

price to maximize its own profits. If they compete in price, it gives rise to Bertrand equilib-

rium otherwise, price matching equilibrium is generated.

2.2.3 Game Solution

We analyze and compare how price matching affects a firm’s investment decision on adver-

tising and then product market competition in terms of Bertrand competition for two cases.

30

More specifically, we discuss how price matching affects each firm’s decision under cooper-

ative and predatory advertising, respectively. Backward induction is employed to solve this

game.

Stage 2: Competing or price matching

Consider that both firms make a decision on either competing in price or price matching

after making their investment decision on advertising. The advertising cost to each firm is

12mA2

i thus the profit function for each firm is given by:

maxpi

Πi = (pi − ci)qi −1

2mA2

i (2.2)

st : pi = pj if price matching (2.3)

The best response functions on price for both firms at the equilibrium in terms of adver-

tising levels under Bertrand competition can be derived as3:

p∗i =1

2(a+ dpj + θAi ± βAj) (2.4)

Similarly the best response functions on price for both firms at the equilibrium in terms

of advertising levels under price matching is:

p∗i =a+ θAi ± βAj

2− 2d(2.5)

Stage 1: Advertising Investment

In the first stage, both firms simultaneously choose the amount of advertising prior to product

competition and the profit function becomes:

maxAi

Πi(Ai,Aj) = p∗i (a+ θAi ± βAj − p∗i + dp∗j) (2.6)

At the equilibrium, under a cooperative regime the advertising intensity by each firm in

Bertrand competition and price matching are:

3If the sign before β is positive, it means the advertising is cooperative; otherwise it is predatory. Thisalso applies for the case under price matching.

31

A∗iB =2aθ (2 + d2)

m (2− d)2 (2 + d)− 2θ2 (1 + d) (2 + d2)(2.7)

and

A∗iPM =aθ

(1 + d)θ2 − 2(1− d)m(2.8)

Similarly, under a predatory regime the advertising intensity by each firm in Bertrand

competition and price matching are:

A∗iB =2aθ (2− d2)

m (2− d)2 (2 + d)− 2θ2 (1− d) (2− d2)(2.9)

and

A∗iPM =aθ

(1− d)(2m− θ2)(2.10)

The following propositions summarize the game solutions for Bertrand competition and

price-matching under cooperative and predatory advertising, respectively.

Proposition 7 There exists a subgame perfect Nash equilibrium under Bertrand competition

such that each firm makes an investment in cooperative advertising A∗i =2aθ(2+d2)

m(2−d)2(2+d)−2θ2(1+d)(2+d2)

in the first stage, charges p∗i =am(4−d2)

m(2−d)2(2+d)−2θ2(1+d)(2+d2) and earns profits π∗i =a2m

((4−d2)

2m−2(2+d2)

2θ2

)(m(2−d)2(2+d)−2(1+d)(2+d2)θ2)

2

in the second stage.4

Proof. See appendix.5

Proposition 8 There exists a subgame perfect Nash equilibrium under price matching such

that each firm makes an investment on cooperative advertising A∗i = aθ2m(1−d)−(1+d)θ2 in the

4First, d cannot be equal to 1 as products will become homogenous and Bertrand paradox will occur; dalso cannot be 0 as the two products will become totally unrelated.

Second, to guarantee price and ads intensity are positive, we also need θ2 < m(1−d)(2−d2)2(1+d)(2−d)2 .

5All proofs can be found in the appendix. Detailed model derivations are done by ”mathematica” andavailable upon request.

32

first stage, charges p∗i = am2m(1−d)−(1+d)θ2 and earns profits π∗i =

a2m(2(1−d)m−θ2)2(2m(1−d)−(1+d)θ2)2 in the second

stage.

Proof. See appendix.

Proposition 9 There exists a subgame perfect Nash equilibrium under Bertrand competition

such that each firm makes an investment on predatory advertising A∗i =2aθ(2−d2)

m(2+d)(2−d)2−2θ2(1−d)(2−d2)

in the first stage, charges p∗i =am(4−d2)

m(2+d)(2−d)2−2θ2(1−d)(2−d2) and earns profits π∗i =a2m

(m(4−d2)

2−2θ2(2−d2)

2)

(m(2+d)(2−d)2−2θ2(1−d)(2−d2))2

in the second stage.

Proof. See appendix.

Proposition 10 There exists a subgame perfect Nash equilibrium under price matching such

that each firm makes an investment on predatory advertising A∗i = aθ(1−d)(2m−θ2) in the first

stage, charges p∗i = am(1−d)(2m−θ2) and earns profits π∗i =

a2m(2m(1−d)−θ2)2(1−d)2(2m−θ2)2 in the second stage.

Proof. See appendix.

From the above propositions, we find that price matching affects firms’ investments and

pricing decisions as well as the profitability for both types of advertisement. We investigate

the intuition on how price matching changes firms behavior in the following sections.

2.3 Price Matching—A Double-Edged Sword

In this section, we formally analyze how price matching affects a firm’s advertising invest-

ment and product market equilibrium in oligopolistic competition. Compared to the existing

literature, we find price matching serves as a double-edged sword on firms’ profits, consumer

surplus and social welfare contingent upon the types of advertising. Under a cooperative

regime, price matching facilitates firms’ advertising investments, increases profits and con-

sumer surplus, and improve social welfare. On the other hand, firms under a predatory

regime are induced to overinvest more excessively in advertising by a price matching policy

33

and thereby firms’ profits are reduced due to high investment costs in this wasteful adver-

tising competition. Consumers are also worse off due to the high collusive equilibrium price

caused by a price matching policy. We will discuss the two cases respectively in the following

sections.

2.3.1 Cooperative Advertising

We first consider how price matching affects firms’ advertising investments under a cooper-

ative regime.

Cooperative advertising means advertising investment made by one firm not only in-

creases its demand but also its rival’s demand. As the two products are strategic comple-

ments given price competition, such investment shifts both firms’ best response functions

outward. Since this kind of advertising serves as a public good, it imposes a positive exter-

nality on the other firm’s demand, giving the rival firm a strong inventive to free-ride on such

efforts. As a result no one would like to invest more in advertising but each of them would

wait for the other’s contribution. In addition, as both firms know that they will compete in

price in the second stage after advertising investment, price matching strengthens each firm’s

free-riding incentive. Each firm would underinvest in advertising and the total amount of

advertising contributed by both firms would be much less than the intensity that maximizes

total industry profits (i.e. the optimal or efficient level from the industry perspective).

Now we start to illustrate how price matching affects firms’ investment decisions on ad-

vertising and product competition. We find that a price matching policy serves as an effective

device to facilitate each firm’s advertising investment and mitigates the free-riding problem,

making both firms invest more efficiently in advertising relative to Bertrand competition.

Figure 2.1 helps describe the detailed intuition. The initial Bertrand equilibrium occurs

at point B0 where the advertising intensity is zero. Under Bertrand competition, their best

response functions shifts outward symmetrically after advertising investments and the new

34

Figure 2.1: Price Matching and Cooperative Advertising

Bertrand Nash Equilibrium moves from point B0 to point B1. However due to the free-riding

problem and following competition after this investment, both firms have a weak incentive

to invest in advertising so that the level of advertisement at point B1 is still much less than

the social efficient level.

To illustrate the mechanism of price matching, we separate the price matching effect into

two effects. The first one is an indirect effect—”advertising effect” and the second one is a

direct effect—”colluding effect”. The two effects are analyzed respectively in the following

discussion.

The advertising effect is shown from point B1 to point B2. As mentioned before, the

free-riding problem always occurs to cooperative advertising under a cooperative regime

as each firm knows the advertising contribution from the other side will increase its own

demand and they will compete in price in the second stage so that each of them would invest

insufficiently and wait for the other contribution. However, we find that price matching

can effectively mitigate this problem and induce each firm to invest more efficiently than

Bertrand competition. The intuition is that under a price matching policy, price matching

is an effective practice for collusion, largely softening the competition between firms. Each

35

firm knows that there is no competition between them in the second stage (i.e. the two

firms seem to be operated by one monopolist) and if they could cooperate in the first stage

and make sufficient investment in advertising, both of their demands will increase, ending

at a high equilibrium price. Thereby each firm takes this positive externality into account

and makes more investment in advertising, increasing mutual demands and earning higher

profits together. Price matching gives both of them a stronger incentive to invest more in

advertising. As shown in Figure 2.1, the best response functions shift outward further in

terms of Bertrand competition (i.e. from point B1 to point B2).

The distance from B2 to PM measures the colluding effect in Figure 1. The increased

advertising investments under price matching effectively increase both firms’ profits. Besides

the advertising effect, price matching also serves as a colluding device so that each firm earns

further profits, commonly charging a colluding price under price matching policy.

We use a hypothetical monopoly price as the reference price to explain the colluding

mechanism of price matching . Given that the two firms’ best response functions shift

outward after advertising investments, if the two firms could cooperate with each other, the

collusive equilibrium occurs at point M where the products’ prices match those charged by

a hypothetical monopolist. The problem arising from collusion is: point M is not a stable

equilibrium since each firm has an incentive to gain a larger market share by reducing its

own price. As a result, each firm has an incentive to deviate and then the equilibrium comes

back to Bertrand competition. However, price matching serves as an effective practice of

coordinating price, mitigating the competition between rivalry firms. Being matched by its

rival’s price, neither firm has an incentive to lower price since there are no benefits for them

to do so. Under a price matching policy, the best response functions for both firms always

overlap and converge to the 45 degree line. The same amount of investment by both firms

increases their demands symmetrically and thereby the equilibrium under price matching

policy is exactly the same as the one when both firms collude. Price matching effectively

36

induces firms to commonly charge a monopoly price.

Hence, we can see as price matching softens the competition between firms, it generates

two benefits to firms: motivating both firms to invest more in advertising to enhance de-

mand and stably colluding in price. We summarize the above discussion in the following

proposition.

Proposition 11 Under a cooperative regime both firms adopting price matching invest more

efficiently in advertising, charge a higher price and then earn more profits compared to

Bertrand competition.

Proof. See appendix.

Consumer Surplus and Welfare

When both firms more efficiently contribute on advertising, such an investment increases

consumers’ willingness to pay and market size. This enhances consumer surplus and social

welfare, and thereby price matching may make both firms and consumers better off.

However, as discussed before, there are two forces generated from price matching. On one

hand the indirect advertising effect of price matching increases consumers’ willingness to pay

and market size; on the other hand the direct colluding effect of price matching facilitates

firms collusion and a monopoly price is charged on consumers.

Therefore, when the two forces interact together, whether cooperative advertising can

make consumers better off or worse off under a price matching policy depends on the adver-

tising factor, θ. If θ is large enough, the indirect effect dominates the direct effect and then

consumer surplus increases; otherwise consumers are still worse off. The following statement

summarizes this finding.

Proposition 12 There exists a critical value θ∗coop such that, if the cooperative advertising

factor θ is above θ∗coop, consumer surplus and total welfare are higher under price matching

than Bertrand competition.

37

Proof. See appendix.

Corollary 2 If the degree of differentiation increases, the critical value θ∗coop decreases.

Proof. See appendix.

The degree of differentiation plays an important role to affect firm’s pricing choices, affect-

ing the critical value of advertising factor. Given a certain level of advertising, if the degree

of differentiation increases, the two products become more similar or more substitutable. It

follows that the benefits from price matching for both firms become more obvious, and that

both firms would invest more in advertising. Therefore, the requirement for θ∗coop decreases.

From the above discussion, we conclude that price matching encourages firms’ advertising

investment and facilitates firms collusion. As a result, both firms and consumers are better

off under such a policy.

2.3.2 Predatory Advertising

We now consider how price matching affects firms’ predatory advertising investment. The

scenario of predatory advertising is totally opposite to the cooperative one.

Predatory advertising means an advertising investment made by one firm only increases

its own demand at the expense of its rival’s demand, attracting customers from its rival.

Given the nature of strategic competition, such an investment shifts the investing firm’s

best response curve outward but its rival’s best response inward. As predatory advertising

imposes a negative externality on rival’s demand, it entails a business stealing effect and

motivates each firm to excessively overinvest to increase its own demand but at the same

time decrease its rival’s. As a result, both firms face a prisoners’ dilemma such that they

have the same market share and gross profits but have to pay high advertising costs. In

this wasteful advertising competition, the total amount of advertising from both sides is

excessive from the industry perspective. Actually, if the two firms could cooperate and take

the negative issue into account, the optimal level of advertising would be zero. In addition,

38

Figure 2.2: Price Matching and Predatory Advertising

predatory advertising from either side is eventually mitigated by the other side and then

cannot enhance consumers’ willingness to pay so that consumers also cannot benefit from

such competition.

Now we start to analyze the price matching effect on predatory advertising investment.

We find price matching makes this existing prisoners’ dilemma even much worse: each firm

adopting price matching overinvests more excessively and competition becomes more severe

and wasteful compared to Bertrand competition.

Figure 2.2 can help us understand the detailed intuition. Without advertising investment,

the initial Bertrand equilibrium occurs at point B0. After predatory advertising investment,

both firms’ best response functions symmetrically shift outward and the temporary new

Bertrand Nash Equilibrium moves from B0 to B1. As predatory advertising only enhances

its own demand at the expense of the rival firm, both firms’ advertising has no effect on their

demands eventually and then their best response functions symmetrically shift back to the

initial Bertrand equilibrium. Since neither firm wants to be ”hurt” by rival firm’s advertising,

they have a strong incentive to over invest and thereby the intensity of advertising contributed

39

by both firms is much higher than the social efficient level (i.e. no advertising).

Similar to the discussion for cooperative advertising, we separate the price matching

effect into two effects—advertising effect (indirect effect) and colluding effect (direct effect)

to analyze price matching effect. The advertising effect is shown from point B1 to point B2.

As discussed before, excessive investments in predatory advertising always occur to both

firms as such advertising from one side attracts some consumers and decreases demand for

the rival. However, we find price matching even worsens the problem by inducing both firms

to overinvest more excessively. The basic intuition is that each firm knows its investment in

advertising only enhances its own demand but lowers its rival’s demand. If the firm invests

more than its rival in the first stage, its own demand is higher than the rival’s. If the firm

facing the higher demand in the competition stage automatically matches its rival’s price,

this firm can implicitly undercut its rival’s price to gain a larger market share and higher

profits. As a result, each firm adopting price matching has the same strong incentive to

overinvest further in advertising in terms of Bertrand competition and the total investment

in advertising is much more excessive than the social efficient level. However, their advertising

efforts will eventually be neutralized by each other and no one’s demand will increase. Each

firm keeps the same market share and gross profits but has to pay higher advertising costs.

The already wasteful competition becomes more wasteful. As shown in Figure 2.2, the best

response functions shift outward further in terms of Bertrand competition (i.e. from point

B1 to point B2).

The distance from B2 to PM measures the colluding effect in Figure 2.2 The colluding

effect here is the same as we discussed in cooperative case and the intuition is the same. Being

matched by rival’s price, no firm has an incentive to deviate, and the best response functions

for both firms always overlap, converging to the 45 degree line. In addition, symmetric

investments makes the equilibrium under price matching policy the same as the one when

both firms collude. We summarize the above discussion in the following proposition.

40

Proposition 13 Under a predatory advertising regime both firms adopting price matching

invest more excessively in advertising and charge a higher price than Bertrand competition.

Proof. See appendix.

From the above analysis, we can see that the two effects generated from price matching

go in opposite directions. Due to the nature of predatory advertising, each firm’s increased

demand will be reduced by the other’s advertising efforts and vise versa. Thereby both of

their best response functions will shift back by rival firm’s advertising contribution while the

advertising costs are already sunk. Thus the advertising effect is negative for both firms. On

the other hand, the colluding effect is always positive—it softens competition and increases

firms’ profits. When the two forces interact together, whether each firm can earn higher

profits depends on which force dominates the other. The following proposition summarizes

the finding.

Proposition 14 There exists a critical value θ∗non−coop such that if predatory advertising

factor θ is above θ∗non−coop, both firms’ profits are lower under price-matching than Bertrand

competition.

Proof. See appendix.

If the advertising factor θ is big enough, then each firm under a price matching policy

has a stronger incentive to overinvest and undercut its rival. Both of them harm each

other’s demand but have to incur a huge investment cost. As a result, the colluding effect is

dominated by the advertising effect and each firm is worse off under a price matching policy.

Corollary 3 If the degree of differentiation increases, the critical value θ∗non−coop decreases.

Proof. See appendix.

The degree of differentiation plays an important role to affect the critical value of adver-

tising factor. If the degree of differentiation increases, the two products become more similar

41

or more substitutable, the benefits from price matching for both firms become more obvious.

Under this situation, the colluding effect dominates the advertising effect and therefore, the

critical value of θ∗non−coop decreases.

Under predatory advertising the increased demands for both firms are reduced by the

other’s advertising investment so that the market size and consumers’ willingness to pay do

not increase. However, consumers are charged a monopoly price under price matching and

therefore consumers are definitely worse off.

Proposition 15 Consumer surplus and total welfare are lower under price matching than

Bertrand competition.

Proof. See appendix.

2.3.3 Comparison of Cooperative and Predatory Advertising

From the discussion above we find that price matching serves as a double-edged sword on

firms’ profits, consumer surplus and social welfare. The common finding under two kinds of

advertising regimes shows that price matching facilitates firms advertising investments prior

to product market competition—firms invest more in advertising compared to Bertrand

competition; however, the underlying incentive is totally different. Firms are encouraged to

advertise more efficiently under a cooperative regime and the level of advertising is close to

the intensity maximizing total industry profits and thereby both firms and consumers are

made better off if price matching is adopted. On the other hand, firms using price matching

invest much more excessively under a predatory regime in which the optimal advertising

level would be zero. Price matching thus makes the prisoners dilemma even worse and the

advertising competition between firms is more wasteful. Consumers do not benefit from such

competition and are definitely worse off as advertising did not increase consumer willingness

to pay but the equilibrium price is driven too high under collusive equilibrium. Thereby

42

price matching can either be beneficial or harmful for firms and consumers contingent upon

the types of advertising.

2.4 Conclusion

We consider a two-stage game such that two firms each make an investment in advertising

first and then compete in a product market. We find that a price matching strategy affects

firm’s advertising and pricing behavior in imperfect competition, serving as a double-edged

sword. The types of advertising play an important role on firms profits and consumer surplus.

Both firms and consumers are made better off under a cooperative regime as price matching

induces firms to advertise more efficiently; however, price matching can also make firms

invest more wastefully, resulting in both firms and consumers worse off under a predatory

regime.

In this paper, we assume consumers’ preference determines the types of advertisement.

However, the types of advertising are also sometimes determined by firms’ strategies. Firms

in oligopolistic competition may choose to compete by offering predatory advertising or

choose to collude by offering cooperative advertising. Therefore, for future research we may

consider when, how and why a firm chooses a certain type of advertising and related policy

implications.

43

Chapter 3

Puppier Puppy Dog and Fatter Fat Cat: Strategic

Investment for Incumbent Firm under Price Matching

3.1 Introduction

Price matching guarantees are commonly adopted in many retail markets. These guarantees

are a contract that if a consumer can find a lower price for a particular good in another

retailer’s store, she can be promised to enjoy the same price from her retailer. Consumers

might be better off from such announcements since they seem to induce severe competition

among retailers. Economists, however, hold an opposite opinion of price matching guarantees

that they are used by retailers as a strategic tool to extract consumer surplus in imperfect

competition.

There are three main streams of argument to explain price matching guarantees serve as

a strategic tool for retailers in the literature. The first argument treats such a promise as a

collusion facilitating device since firms offering a price matching guarantee have no incentive

and no benefit to undercut the price. If someone lowers the price, her rival will automatically

match it, therefore no one can increase their own profits by deviating. As a result, price

matching guarantees facilitate firms collusion (see Salop (1986), Belton (1987), Chen (1995)

and Dugar (2007)). By considering the asymmetric information, the second argument is

that retailers price discriminate between informed and uninformed consumers by using such

guarantees because consumers having the perfect information know the difference between

list and transaction prices while the uninformed ones do not (see Png and Hirshleifer (1987),

Corts (1997) and Lim and Ho (2008)). The last argument by Moorthy and Winter (2006)

shows that price matching guarantees convey a low cost signal to consumers and then make

44

the retailers more competitive in the market. Low cost firms can always offer a guarantee

to match any price set by high cost firms while high cost firms cannot mimic as low cost

ones. The larger the cost difference between high and low cost firms, the more notable the

cost advantage for the low cost firms. Therefore, by offering such guarantees low cost firms

signal that they are offering a cheaper price than others.

We are motivated in two ways by the literature on price matching. First, most papers only

focus on the competition stage (i.e. how a price matching strategy softens competition among

firms); however, we try to expand the analysis by looking at the investment stage. Prior to

competition, to maintain their competitive positions in oligopolistic competition, firms may

strategically invest either to reduce cost (e.g. R&D) or enhance demand (e.g. advertising).

Second, Logan and Lutter (1989) and Belton (1986) extend the literature from homogenous

products with an identical demand and cost to differentiated products involving asymmetric

costs or demands. We expand their work by exploring the main driver for demand or cost

asymmetry (i.e. firms’ investments prior to competition stage). Therefore, by modeling a

two stage game, we are interested in investigating the interaction among a price matching

strategy, firms’ investments as well as product market competition. More specifically, we

try to answer the three questions as follows. First how does a price matching policy affect

an incumbent firm’s investment to strategically accommodate an entrant? Second, why and

under which conditions would the incumbent firm adopt a price matching strategy? Third,

given the investment made by the incumbent firm, how does the price matching policy affect

product market competition after the entrant comes in the market. To our best knowledge,

this is the first paper to consider the interaction between firms’ strategic investments in

oligopolistic competition and price matching policies.

We contribute to the existing literature in three ways: First, we provide a novel expla-

nation for the interaction between a price matching policy and the investment made by an

incumbent prior to competition. We find that a price matching policy has a significant ef-

45

fect on such an investment. More interestingly, we find it has totally opposite impacts on

investments for demand enhancement and cost reduction.

The pioneering work by Tirole and Fudenberg (1984) argues that given price competition

in the second stage, strategic accommodation by the incumbent firm works in two ways. On

one hand, the incumbent firm would overinvest if devoting to demand enhancement since

such an investment shifts the incumbent’s best response curve outward, ending at a higher

equilibrium price associated with larger profits. In addition, given the nature of strategic

complement relationship between two firms, such an investment also makes the entrant better

off at the higher equilibrium price. As the investment made by the incumbent benefits both

firms in the market, the incumbent is referred to as a Fat-Cat. On the other hand, to

maintain its competitive stance in the market, the incumbent incurs a R&D expenditure

to reduce its marginal cost.1 However, it would underinvest. Although investment lowers

the marginal cost, it also shifts the incumbent’s response curve inward—the wrong way to

improve its profits, resulting in a lower equilibrium price at which both firms are made worse

off. The incumbent under this scenario is referred to as a Puppy Dog.

By investigating the impact of a price matching strategy on the incumbent’s investment

behavior, we find that it works in two ways. On one hand the price matching strategy

facilitates the incumbent advertising investment—the incumbent is encouraged to advertise

more under the price matching policy than Bertrand competition; in other words, the Fat

Cat becomes fatter by price matching. On the other hand, price matching induces the

incumbent firm to invest less in R&D compared to price competition and then the Puppy

dog looks puppier. The following subsection documents the detailed intuition.

We find for both cases price matching creates two strategic effects—colluding and un-

dercutting to affect the incumbent firm’s investment behavior. As to demand enhancement,

1There are two kinds of R&D expenditures. The first one is called process R&D focusing on productioncost-reducing. (e.g. Arrow (1962); Brander and Spencer (1983); D’Asremont and Jacquemin (1988); Ziss(1994); Lin and Saggi (2000)). The second one is product R&D aiming on product quality promotion (e.g.Symeonidis (2003) and Bonanno and Haworth (1996)). We only focus on process R&D in this paper.

46

the two effects work in the same direction: both of them induce the incumbent firm to

invest more in advertising. A colluding effect means price matching facilitates collusion.

As neither firm can benefit by reducing price under the price matching policy, it induces

both firms to cooperate instead of competing, effectively mitigating the competition, which

creates a stronger incentive for the incumbent firm to invest more to enhance demand pre-

entry. Second, the undercutting effect means the incumbent firm implicitly lowers its price

to gain a larger market share by matching its rival’s price. The two firms’ demands become

asymmetric caused by the advertising investment—the best response of the incumbent shifts

outwards and the incumbent firm’s price is higher than the entrant at the collusive equilib-

rium. We show that a price matching strategy provides an opportunity for the incumbent

firm to undercut its rival’s price from the collusive equilibrium: by matching its rival’s price,

the incumbent firm deviates from the collusive equilibrium and implicitly lowers its price,

earning further profits. Therefore, the incumbent firm would invest more to enlarge this

demand asymmetry.

The two effects for cost reduction still work in the same direction but both of them

induce the incumbent to invest less. As price matching softens the competition between

firms, resulting in a weaker incentive for the incumbent firm to reduce cost, and the colluding

effect makes the incumbent firm invest less. In addition, the incumbent firm’s best response

shifts inward after cost reduction and thereby its price is lower than its rival’s at the collusive

level. However, by matching its rival’s price, the incumbent firm has to charge a price higher

than the collusive level, lowering its profits and making it worse off. In other words, the

incumbent is undercut by its rival under the price matching policy. As a result, to avoid

the cost incurred by matching its rival’s price, the incumbent would underinvest further to

reduce the undercutting effect. Therefore, the investment for cost reduction under price

matching is much less than at the Bertrand competition level. Our finding supports Arrow’s

argument (1962) that firms in Bertrand competition have stronger incentives on R&D than

47

in a monopoly setting. The competition under price matching is much weaker than pure

price competition so that the incumbent firm would invest less on R&D. The arguments by

Symenidis (2003) and Bonanno and Haworth (1998) that R&D in Bertrand competition is

less than that in Cournot competition is also confirmed by our result. Under price matching,

the incumbent firm even invests less further due to the two effects mentioned above and

becomes puppier.

We also challenge the traditional argument that price matching prevents firms undercut-

ting price. Consistent with the literature, we find price matching is an effective practice of

collusion between firms. No matter the type of investment made by the incumbent, price

matching always mitigates the competition after the entrant comes in and drives the equi-

librium price above the Bertrand competition level. However, we find undercutting does

exist under a price matching policy. On one hand a price matching strategy can help the

incumbent firm undercut its rival due to demand asymmetry. On the other hand, given cost

asymmetry, the equilibrium price under price matching could be higher than collusive price

under which the incumbent firm is undercut by its rival.

Finally, we find the degree of product differentiation plays as the key driver to determine

whether the incumbent firm would adopt a price matching policy. More highly related

products create a stronger incentive for the incumbent firm to adopt a price matching policy.

The rest of the paper is organized as follows: Section 2 sets up the model; Section 3

discusses how a price matching policy affects an incumbent firm’s behavior in imperfect

competition for both advertising and R&D investment; and Section 4 concludes the paper.

All proofs are in the Appendix.

48

3.2 The Model

3.2.1 Assumptions

Consider an industry in which two rival firms—one incumbent and one entrant producing

differentiated products—compete in price after the entrant enters the market. Assume that a

fully-informed consumer’s utility function is a consumption function of the two differentiated

goods, q1 and q2, and a numeraire good n, then we have the following demand function for

the two goods:

qi = a− pi + dpj (3.1)

where a denotes the size of the market; d shows the degree of differentiation between the

two goods. If d = 1, the two goods are homogenous; if d = 0; the two goods are independent,

meaning the two firms are monopolists; if d ∈ (0, 1), the two goods are imperfect substitutes

and strategic complements to each other.

The incumbent firm who cannot or does not want to deter entry would maintain its

profitable positioning by strategically accommodating the entry—thus prior to competition it

makes an investment either to reduce cost (e.g. R&D) or enhance demand (e.g. advertising).

By advertising in the investment stage, the incumbent firm can increase its market size from

a to a + A1, where A1 is the advertising intensity associated with a quadratic advertising

cost, 12A2

1. The incumbent firm makes R&D investment in the amount of 12K2

1 to reduce the

marginal cost from c to c−K1.2

In the competition stage, the incumbent firm has two choices: it may announce that it

will match the entrant price or compete in price post entry. Finally, there is no asymmetric

information between two firms and no demand or cost uncertainties.

2First, we assume initially both firms have the common market size, a, and the same marginal cost, c.Second, the motivation for this research is to investigate how price matching affects an incumbent firm’s

investments, therefore, we assume there is no spillover effect either from advertising or R&D on the entrant’sdemand or cost.

49

3.2.2 Game Sequence

The modeled game proceeds in the following stages:

1. The incumbent firm (firm 1) makes an investment either to reduce cost (e.g.

R&D) or enhance demand (e.g. advertising) before the entrant (firm 2) comes

into the market.

2. Firm 1 makes a decision on whether it uses a price matching strategy by

announcing its price always matches firm 2 prices.

3. After the decision made by the incumbent firm, firm 2 enters the market and

then the two firms either compete in price (Bertrand competition) or match

prices.

3.2.3 Solution

We investigate how price matching affects the incumbent firm’s investment decision in two

ways. We first focus on how price matching affect the incumbent firm’s investment decision

on demand enhancement (e.g. advertising). We then investigate the effect of price matching

on cost reduction investment (e.g. R&D) for the incumbent firm.

Preliminary

Proposition 16 If both firms have a symmetric cost (demand), price matching gives rise

to the same outcome of collusive equilibrium.

Price matching is an effective practice of collusion, which fully internalizes cross price

effect. If firm 1 always matches firm 2’s price, there is no interest conflict between the two

firms and competition is totally minimized. As a result, both firms gain a half of monopoly

profits but consumers are worse off due to the higher price charged by both of them.

50

Proposition 17 Both firms earn higher profits under price matching when the two goods

become less differentiated.

When the degree of differentiation decreases, the two products become more similar or

more substitutable and the cross-price elasticity of demand becomes larger. Price matching

is an effective practice of colluding, so if firm 1 price always matches her rival’s price, both

of them will make higher profits when the two goods become highly substitutable.3

End of Preliminary

Price Matching and Strategic Investment in Advertising

We now analyze and compare how price matching affects a firm’s investment decision on

advertising and then product market competition by solving this game through backward

induction starting with the competition stage.

Stage 2: Product Market Competition

Under a price matching policy, firm 2 knows firm 1 will match its price after firm 1 makes

its investment decision. Firm 2 takes this into account and optimally chooses the price to

maximize her profits.4 Therefore, firm 2’s objective function is shown as:

maxp2

Π2 = (p2 − c)(a− p2 + dp1) (3.2)

st : p1(p2) = p2 if PMG (3.3)

Firm 1’s objective function is shown as:

maxp1

Π1 = (p1 − c)(a+ A1 − p1 + dp2)−1

2A2

1 (3.4)

3If the two goods are independent (i.e. d = 0), there is no benefit for firm 1 to match firm 2’s price. Ifthe two goods are independent, each firm charges the monopoly price according to their demand and cost.Hence, if firm 1 still matches his rival price, he cannot gain any benefits from doing so.

If the two goods are homogenous (i.e. d = 1), there is also no benefit to match price. Bertrand paradoxmay occur if the costs are the same. Both of them will charge the same imaginal cost price. Otherwise, thelow cost firm will monopolize the whole market.

4The result of Bertrand competition is shown in appendix.

51

After firm 2 makes its decision, firm 1 automatically matches firm 2’s price and both

firms charge the same price p∗1 = p∗2 = a2(1−d) and quantities for firm 2 and firm 1 are q∗2 = a

2

and q∗1 = a2

+ A1, respectively.

Stage 1: Advertising Investment

Given the price charged by both firms, firm 1 makes an investment on advertising prior to

price competition.

maxA1

Π1(A1) = (p∗1 − c)(a+ A1 − p∗1 + dp∗2)−1

2A2

1 (3.5)

Equation 3.5 shows firm 1’s profit function, indicating firm 1 makes an advertising in-

vestment before they compete. The following propositions summarize the game solutions for

Bertrand competition and price matching cooperation, respectively.

Proposition 18 There exists a subgame perfect Nash equilibrium under Bertrand compe-

tition such that an incumbent firm makes an investment on advertising A∗1 = 2a(2+d)12−8d2+d4

in the first stage; the incumbent firm and new entrant charge p∗1 = a(2−d)(2+d)212−8d2+d4 and p∗2 =

a(6+d(4−d(2+d)))12−8d2+d4 in the second stage.

Proof. See appendix.

Proposition 19 There exists a subgame perfect Nash equilibrium under the price matching

policy such that an incumbent firm makes an investment on advertising A∗1 = a4(1−d) in the

first stage; the incumbent firm and new entrant charge p∗1 = p∗2 = a2(1−d) in the second stage.

Proof. See appendix.

Price Matching and Strategic Investment in R&D

Stage 2: Product Market Competition

Now we turn to analyze how a price matching policy affects the incumbent firm’s investment

on R&D. Under Bertrand competition, each firm competes in price after the incumbent

52

firm’s investment. The solution is shown in appendix. Under the price matching policy, firm

2 knows firm 1 will match its price after firm 1 makes its investment decision. Firm 2 takes

this into account and optimally chooses the price to maximize her profits. Therefore, firm

2’s objective function is shown as:

maxp2

Π2 = (p2 − c)(a− p2 + dp1) (3.6)

st : p1(p2) = p2 if price matching (3.7)

Firm 1’s objective function is:

maxp1

Π1 = [p1 − (c−K1)](a− p2 + dp1)−1

2K2

1 (3.8)

After firm 2 has made its decision, firm 1 automatically matches firm 2 price and therefore

both firms charge p∗1 = p∗2 = a−c(1−d)2−2d and quantity for firms 1 and 2 are q1 = q2 = 1

2(a −

c(1− d)), respectively.

Stage 1: R&D Investment

Given the price charged by both firms, firm 1 makes an investment decision on R&D prior

to market competition.

maxK1

Π1(K1) = [p∗1 − (c−K1)](a− p∗2 + dp∗1)−1

2K2

1 (3.9)

Equation 3.9 shows firm 1’s profit function, indicating firm 1 makes a R&D investment

prior to competition. The following propositions summarize the game solutions for Bertrand

competition and price matching, respectively.

Proposition 20 There exists a subgame perfect Nash equilibrium under Bertrand competi-

tion such that an incumbent firm makes an investment on R&D K∗1 = (a−c(1−d))(2+d)(2−d2)4(3−d2) in

the first stage; the incumbent firm and new entrant charge p∗1 = a(2+d)+c(4+d−d2)6−2d2 and p∗2 =

a(6+(2−d)d)+c(6+d(4−d−d2))4(3−d2) in the second stage.

Proof. See appendix.

53

Proposition 21 There exists a subgame perfect Nash equilibrium under the price matching

policy such that an incumbent firm makes an investment on advertising K∗1 = 14(a− c(1−d))

in the first stage; the incumbent firm and new entrant charge p∗1 = p∗2 = a2(1−d) in the second

stage.

Proof. See appendix.

Price Matching, Investment Decision and Product Market Competition

In this section, we first illustrate how price matching affects the incumbent firm’s investment

in advertising and R&D, how this policy affects its profits, and then explain why and under

what conditions the incumbent would adopt a price matching strategy.

Price Matching and Fatter Fat-Cat

If firm 1 makes an investment in advertising to enhance its demand, its best response function

shifts outward, ending at a higher equilibrium price. As the two products are strategic

complements in price competition, it means if firm 1 unilaterally increases its price, firm 2’s

demand will also increase, making firm 2 better off. The firm making such an investment is

called a ”Fat Cat” by Tirole and Fudenberg (1984), meaning that firm 1 would overinvest

to increase its demand and such an investment not only increases its own profits but also

improves firm 2’s profits.

Now we start to illustrate how price matching affects firm 1’s investment decision as to

demand enhancement and then product competition. We find that firm 1 would overinvest

even more in advertising if adopting price matching in order to earn higher profits. Further-

more, firm 1 uses a price matching strategy to reduce its price, undercutting its rival, firm

2, to gain further profits. Figure 3.1 describes the detailed story.

As shown in Figure 3.1, after firm 1 makes an investment in advertising, the best response

function shifts outward and the new Bertrand Nash Equilibrium moves from B to B1 where

54

Figure 3.1: Price Matching and Demand Enhancement

both firms are better off at point B1. Under a price matching policy, the best response

functions for both firms always overlap and converge to the 45 degree line, the equilibrium

under price matching occurs at point PMG. To explore how price matching works, we

separate the price matching effect into two effects—colluding effect and undercutting effect.

In Figure 3.1, the distance from B1 to M measures the colluding effect. We use a hypo-

thetical monopoly price as the reference price to explain the mechanism of price matching.

Given firm 1’s best response function shifts outward after its investment, the collusive equi-

librium occurs at point M if two firms could cooperate with each other, where the products’

prices were charged by a hypothetical monopolist and both firms are better off but con-

sumers are worse off. However, point M is not a stable equilibrium since both firms have

an incentive to take up a larger market share to gain more profits by reducing their own

price. As a result, each firm will deviate from point M and the equilibrium comes back to

Bertrand competition. However, price matching effectively solves this conflict, mitigating

the price competition between rivalry firms. By matching its rival’s price, neither firm has

an incentive to reduce its price since there are no benefits for them to do so. This gives firm

1 a stronger incentive to invest more in advertising to enhance demand and therefore the

55

”Fat Cat” becomes fatter.

The undercutting effect is shown from point M to point PMG. Such an advertising

investment made by firm 1 also makes firm 2 better off as the equilibrium price is driven

higher by the increased demand, meaning that firm 2 can free-ride on the effort offered by

firm 1. How does firm 1 mitigate this free-riding problem? We find that price matching

provides an opportunity for firm 1 to undercut or cheat by matching its rival’s price. As

the advertising investment gives rise to a higher demand for firm 1, the coordinated price of

firm 1 is higher than that of firm 2 at point M . By sticking to firm 2’s price firm 1 reduces

its price from its own hypothetical monopoly price to its rival’s hypothetical monopoly price

and it thereby indirectly undercuts its price to take up a larger market share, meaning that

this ”Fat-Cat” would not allow firm 2 to fully enjoy the benefits it created so that it would

deviate from the collusive equilibrium to obtain higher profits by price matching.

However, the advertising intensity that firm 1 would invest and its profitability totally

depends on the degree of differentiation. Since the nature of price matching is to soften com-

petition between rival firms, the degree of differentiation plays is the key factor to determine

whether firm 1 would adopt price matching. If the two products are very different or the

demand for firm 1 is much higher than firm 2, then there is no benefit for firm 1 to choose

to match price. Therefore, price matching is only adopted when the two products are highly

substitutable. The following propositions summarize the above discussion.5

Proposition 22 There exists an upper bound d̄ such that for all d > d̄ , the incumbent firm

adopts a price matching strategy, overinvests more in advertising, charges a higher price and

gains higher profits compared to Bertrand competition.

Proof. See appendix

5This mechanism also works if the rival strategically rise firm 1’s cost. By doing this, it would reducethe profits earned by firm1 but firm 1 can use such strategy to protect himself. If the rival firm strategicallyrises the firm marginal cost, for example, the firm facing such situation can adopt price-matching policy tomitigate such threat.

56

Figure 3.2: Price Matching and Cost Reduction

Price Matching and Puppier Puppy Dog

Now we turn to analyze how price matching policy affects firm 1’s R&D investment. The

investment on cost reduction makes firm 1’s best response function shift inwards, resulting

in a lower equilibrium price at which both firms are worse off given the nature of Bertrand

competition. As firm 1 chooses the wrong way to increase its profits, firm 1 would underinvest

to reduce cost, and thereby it is referred to as a ”Puppy Dog” in Tirole and Fudenberg (1984).

We start to illustrate how price matching affect firm 1’s investment decision as to cost

reduction, and then product market competition. We find firm 1 would underinvest further

in R&D compared to Bertrand competition and we argue firm 1 when using a price matching

strategy becomes even ”puppier”. Figure 2 describes the detailed story.

As shown in Figure 3.2, the best response function of firm 1 after the R&D investment

shifts inwards and the new Bertrand Nash Equilibrium moves from B to B1 at which both

firms are worse off in terms of point B. Under the price matching policy, the best response

functions for both firms always overlap and converge to the 45 degree line. The Nash

Equilibrium occurs at point PMG. Thus, the total effect of price matching is shown from

57

point B1 to point PMG. We also separate the price matching effect into a colluding effect and

an undercutting effect to explain how price matching affects the cost reduction investment.

The colluding effect is shown from B1 to M . Given firm 1’s best response function

shifts inward after its investment, the collusive equilibrium occurs at point M at which

the price is the same as one charged by a hypothetical monopolist. This is also originated

from the nature of price matching—it serves as an effective practice of coordinating price,

mitigating the competition between rivalry firms. Since the competition between firms is

much weaker, the incentive for firm 1 to invest to reduce cost is also weaker under price

matching than Bertrand competition. Hence, firm 1 would underinvest further in terms of

Bertrand equilibrium.

The distance fromM to PMGmeasures the undercutting effect that is opposite to the one

for demand enhancement. Price matching generates the same equilibrium as the one when

two firms collude conditionally on the symmetric cost or demand. However, the collusive

equilibrium price for firm 1 after the R&D investment is lower than the price matching price.

By matching its rival price, firm 1 has to increase its price to a much higher one than the

price charged by a monopolist at which firm 1 can earn the highest profits. As a result, firm

1 gains less profits and a lower market share. Therefore, by matching a higher price, firm 1 is

undercut by its rival—the ”negative” undercutting effect makes firm 1 worse off. Therefore,

to avoid this loss as much as possible, we find firm 1 would underinvest further to reduce

this undercutting effect.

On one hand, the colluding effect makes firm 1 better off, on the other hand, the un-

dercutting effect makes it worse off. Hence, as long as the colluding effect dominates the

undercutting effect, firm 1 would adopt this policy. The key factor to determine whether firm

1 would adopt price matching still depends on the degree of differentiation. Since the nature

of price matching is to soften competition between rivalry firms, the degree of differentia-

tion determines how much the competition can be mitigated. If the two products are very

58

different or the cost for firm 1 is much lower than firm 2, then there is no benefit for firm 1

to choose to match price. Therefore, price matching is only adopted when the two products

are highly substitutable. The following propositions summarize the above discussion.6

Proposition 23 There exists an upper bound d̄ such that for all d > d̄ , the incumbent firm

adopts a price matching strategy, under-invests more in R&D, charges a higher price and

gains higher profits compared to Bertrand competition.

Proof. See appendix

3.3 Conclusion

We consider a two-stage price matching game in which product market competition occurs

after the incumbent firm makes an investment either to enhance demand or to reduce cost.

By investigating the impact of price matching on the incumbent’s investment behavior, we

find that price matching works in two ways but totally opposite directions on such invest-

ments. On one hand price matching facilitates the incumbent’s advertising investment—the

incumbent is encouraged to advertise more under a price matching policy than Bertrand

competition; in other words, the Fat Cat becomes even fatter by adopting price matching.

On the other hand, price matching induces the incumbent firm to invest less in R&D com-

pared to price competition and then the Puppy dog looks puppier. However, the degree

of differentiation plays an important role affecting the incentive for the incumbent firm to

adopt such a strategy. Highly differentiated products discourage the firm to use such pol-

icy since when the two products become more irrelevant, the coordination effect becomes

weaker. Therefore, price matching is only adopted if the products are highly substitutable.

6This mechanism also works if the rival strategically rise firm 1’s cost. By doing this, it would reducethe profits earned by firm1 but firm 1 can use such strategy to protect himself. If the rival firm strategicallyrises the firm marginal cost, for example, the firm facing such situation can adopt price-matching policy tomitigate such threat.

59

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64

Appendix A

Chapter One Proofs

Proof of Proposition 1:

There exists a subgame perfect Nash equilibrium under bank financing such that each sup-

plier charges PB = 2(qAg+(1−q)Ab)

3+n, each retailer orders an inventory QB

g = (3+n−2q)Ag−2(1−q)Ab

(n+3)(n+1),

and sells exactly QBg in a good state and QB

b = 2qAg+(1+n+2q)Ab

(n+3)(n+1)in a bad state, respectively.

Proof. According to Equation (1.1), the first order condition to solve QBb is

∂

∂QBb

[(Ab −QB

b −Q−i,Bb )QB

b − CBb

]= 0 (A.1)

Solving the partial derivative and substituting the cost CBg , Equation (A.1) becomes

Ab − 2QBb −Q

−i,Bb + PB = 0 (A.2)

which yields the best response function

QBb =

Ab −Q−i,Bb + PB

2(A.3)

Assuming that all firms are symmetric we set Q−i,Bb = (n− 1)QBb and solve for the optimal

quantity that the retailer offers in the bad state which is

QBb =

Ab − PB

n+ 1(A.4)

According to Equation (1.3), the first order condition to solve QBb is

∂

∂QBg

[(1− q)ωBb + qωBg

]= 0 (A.5)

By substituting the cost CBg and the optimal quantity that the retailer offers in the bad

state, QBb , and solving the partial derivative, Equation (A.5) becomes

65

Ag − 2QBg −Q−i,Bg + PB = 0 (A.6)

which yields the best response function

QBg =

Ag −Q−i,Bb + PB

2(A.7)

Assuming that all firms are symmetric, we set Q−i,Bg = (n− 1)QBg and solve for the optimal

quantity that the retailer offers in the good state which is

QBg =

Ag − PB

n+ 1(A.8)

According to Equation (1.4), the first order condition to solve PB is:

∂

∂PB

[(1 + r)PBQB

g − (1− q)PB(QBg −QB

b ))]

= 0 (A.9)

By substituting the optimal quantities, QBb , QB

g , that the retailer offers in a bad state

and a good state, respectively, and solving the partial derivative, Equation (A.9) becomes

Agq + Ab(1− q)− 2PB

n+ 1= 0 (A.10)

which yields the price charged by a supplier under bank financing,

PB =2

n+ 3[Agq + Ab(1− q)] (A.11)

By substituting the equilibrium price into the Equation (A.4) and Equation (A.8), the

quantities sold in the good state and the bad state are (3+n−2q)Ag−2(1−q)Ab

(n+3)(n+1)and 2qAg+(1+n+2q)Ab

(n+3)(n+1),

respectively.

This completes the proof.

Proof of Proposition 2

There exists a subgame perfect Nash equilibrium under bank financing such that each

supplier charges P T = 2(qAg+(1−q)Ab)

3+nand sets the trade credit interest rate at rs = q(Ag−Ab)

qAg+(1−q)Ab,

66

each retailer orders an inventory QTg = Ag

n+3, and sells exactly QB

g in a good state and

QTb = Ab

n+3in a bad state, respectively

Proof. According to Equation (1.1), the first order condition to solve QTb is:

∂

∂QTb

[(Ab −QT

b −Q−i,Tb )QT

b − CTb

]= 0 (A.12)

Solving the partial derivative and substituting the cost CTb , Equation (A.12) becomes:

Ab − 2QTb −Q

−i,Tb + P T (1− rs) = 0 (A.13)

which yields the best response function

QTb =

Ab −Q−i,Tb + P T (1− rs)2

(A.14)

Assuming that all firms are symmetric we set Q−i,Tb = (n− 1)QTb and solve for the optimal

quantity that the retailer offers in the bad state which is

QTb =

Ab − P T (1− rs)n+ 1

(A.15)

According to Equation (1.3), the first order condition to solve QTb is

∂

∂QTg

[(1− q)ωTb + qωTg

]= 0 (A.16)

By substituting the cost CTg and the optimal quantity that the retailer offers in the bad

state, QTb , and solving the partial derivative, Equation (A.16) becomes

q(Ag − 2QTg −Q−i,Tg − P T )− (1− q)P T rs = 0 (A.17)

which yields the best response function

QTg =

q(Ag + P T )− (1− q)P T rs −Q−i,Tb

2(A.18)

67

Assuming again that all firms are symmetric, we set Q−i,Tg = (n − 1)QTg and solve for the

optimal quantity that the retailer offers in the good state which is

QTg =

(Ag − P T )q − (1− q)P T rsn+ 1

(A.19)

According to Equation (1.5), the first order condition to solve P T and rs is:

∂

∂P T

[qP TQT

g + (1− q)

(P TQT

b +rsP

T (QTg −QT

b )

1 + r

)]= 0 (A.20)

∂

∂rs

[qP TQT

g + (1− q)

(P TQT

b +rsP

T (QTg −QT

b )

1 + r

)]= 0 (A.21)

By substituting the optimal quantities, QTb , QT

g , that the retailer offers in a bad state and

a good state, respectively and solving the partial derivative, Equations (A.20) and (A.21)

become

Ab(1− q)q(1− rs) + Agq(q + rs − qrs)− 2P T (q + (1− q)r2s)n+ 1

= 0 (A.22)

P T (1− q)[(Ag − Ab)(1− q) + 2P T rs]

(n+ 1)q= 0 (A.23)

Simultaneously solving Equations (A.22) and (A.23) yields the price and trade credit

interest rate charged by a supplier under trade credit financing,

P T =2(qAg + (1− q)Ab)

3 + n(A.24)

rs =q(Ag − Ab)

qAg + (1− q)Ab(A.25)

By substituting the equilibrium price and trade credit interest rate into the Equation

(A.15) and Equation (A.19), the quantities sold in the good state and the bad state are Ag

n+3

and Ab

n+3, respectively.

This completes the proof.

Proof of Proposition 3

68

Under trade credit financing, the retailers’ marginal cost is lower (higher) in the bad

(good) state, competition is intensified (softened), and aggregate supply increases (decreases)

relative to bank financing.

Proof. The marginal cost in the bad state

Under bank financing, Equation (1.6) becomes

MCBb = PB +MFCB

b (A.26)

The marginal financing cost is zero in the bad state, thus, the marginal cost is equal to

PB.

Under trade credit financing, Equation (1.6) becomes

MCTb = P T +MFCT

b (A.27)

The marginal financing cost is − PT rs(1+r)

in the bad state, thus, the marginal cost is equal to

P T − PT rs(1+r)

,since PB = P T at the equilibrium, the marginal cost under trade credit is lower

in the bad state.

The marginal cost in the good state

Rearrange Equation (1.7), it becomes

MCg = MPC +MFCfg +

(1− q)q

MFCfb . (A.28)

under bank financing, the marginal cost is equal to PB since the marginal financing costs in

both states are zero by assuming r = 0.

Under trade credit financing, the marginal product cost is P T , the marginal financing

cost in the good state is 0; and the marginal financing cost in the bad state is (1−q)q

PT rs(1+r)

,

thus the marginal cost in a good state is P T (1 + (1−q)q

rs(1+r)

). Again, as PB = P T at the

equilibrium, the marginal cost under trade credit is higher in a good state.

This completes the proof.

Proof of Corollary 1

69

Under trade credit financing, the penalty rate rs increases with the probability of a good

state and the gap of choke prices in both states.

Proof. As rs = q(Ag−Ab)

qAg+(1−q)Ab, thus the partial derivative with respect to q is given by

∂rs∂q

=Ab(Ag − Ab)

(qAg + (1− q)Ab)2> 0 (A.29)

and the partial derivative with respect to (Ag − Ab) is given by

∂rs∂(Ag − Ab)

=q

qAg + (1− q)Ab> 0 (A.30)

This completes the proof.

Proof of Proposition 4 Under trade credit financing, the supplier optimally price

discriminates the retailer between the states of demand: charging a high effective price

P Tg = P T (1 + 1−q

qrs1+r

) in good states and a low effective price P Tb = P T (1 − rs

1+r) in bad

states. As a result, compared to bank financing, the profits of the supplier are higher under

trade credit financing.

Proof. see the main text.

This completes the proof.

Proof of Proposition 5

In imperfect competition of at least two supply chains, the expected total producer

surplus is higher, and the expected consumer surplus and total welfare are lower under trade

credit financing than under bank financing.

Proof. By using the solutions obtained from Proposition 1 and Proposition 2, we have

the explicit solutions of supplier and retailer’s profits as well as consumer surplus for both

bank financing and trade credit financing cases. Under bank financing, by substituting the

equilibrium price, quantities sold by the retailer in both good and bad states into Equation

4 and 6, the profits obtained by the retailer and the supplier as well as the total producer

70

surplus are shown as:

ωB =1

(1 + n)2(3 + n)2[A2

g(9 + n2 + n(6− 4q)− 8q)q −

8AgAb(2 + n)(1− q)q + A2b(1− q)(1 + n2 + 8q + n(2 + 4q)] (A.31)

πB =2(Ab(1− q) + Agq)

2

(3 + n)2(A.32)

ωB + πB =1

(1 + n)2(3 + n)2[A2

g(9 + 6n− 6q + n2(1 + 2q)) +

4AgAb(n2 − 3)(1− q) + A2

b(1− q)(3 + 6n+ 6q + n2(3− 2q)] (A.33)

Similarly, under trade credit financing, by substituting the equilibrium price, trade credit

interest rate and quantities sold by the retailer in both good and bad states into Equation

4 and 7, the profits obtained by the retailer and the supplier as well as the total producer

surplus are shown as:

ωT =A2b(1− q) + A2

gq

(3 + n)2(A.34)

πT =2(A2

b(1− q) + A2gq)

(3 + n)2(A.35)

ωT + πT =3(A2

b(1− q) + A2gq)

(3 + n)2(A.36)

Thus, the difference of producer surplus between trade credit financing and bank financing

is

(ωT + πT )− (ωB + πB) =2(Ag − Ab)2(n2 − 3)(1− q)q

(3 + 4n+ n2)2(A.37)

From the above equation, we can find that when n = 1 (one supply chain or monopoly

industry), the total producer surplus is smaller under trade credit financing than bank financ-

ing; however, if n ≥ 2 (at least two supply chains or duopoly industry), the total producer

surplus is greater under trade credit financing than bank financing.

71

This completes the proof.

Proof of Proposition 6

The difference in aggregate profits between trade credit financing and bank financing is

an inverse U-shaped function in the number of supply chains.

Proof. For Equation (A.37), we take the partial derivative with respect to n,and then the

first order condition is shown as:

∂[(ωT + πT )− (ωB + πB)]

∂n=

4(Ag − Ab)2(12 + 9n− n3)(1− q)q(3 + 4n+ n2)3

(A.38)

then, when n = 1,we have ∂[(ωT+πT )−(ωB+πB)]∂n

= 88(Ag−Ab)2(1−q)q

3375> 0 and when n = ∞,

then ∂[(ωT+πT )−(ωB+πB)]∂n

= 0. From this finding, we can see that the partial derivative is rising

initially when n is very small but declines when n becomes very large, therefore, there exists

an inverse U-shaped relationship between the advantage of trade credit financing and the

number of supply chains.

This completes the proof.

72

Appendix B

Chapter Two Proofs

Proof of Proposition 7: There exists a subgame perfect Nash equilibrium under Bertrand

competition such that each firm makes an investment in cooperative advertising A∗i =

2aθ(2+d2)m(2−d)2(2+d)−2θ2(1+d)(2+d2) in the first stage, charges p∗i =

am(4−d2)m(2−d)2(2+d)−2θ2(1+d)(2+d2) and earns

profits π∗i =a2m

((4−d2)

2m−2(2+d2)

2θ2

)(m(2−d)2(2+d)−2(1+d)(2+d2)θ2)

2 in the second stage.1

Proof. According to Equation (2.2), the first order condition to solve pi is

∂

∂pi[(pi − ci)qi −

1

2mA2

i ] = 0 (B.1)

Solving the partial derivative, Equation (B.1) becomes

a− 2pi + dpj + θ(Ai + dAj) = 0 (B.2)

which yields the best response functions for both firms:

pi =1

2(a+ dpj + θ(Ai + dAj)) (B.3)

By substituting pj function into Equation (B.3), solve the price for each firm in terms of

the level of advertising:

pi =a(2 + d) + θ((2 + d2)Ai + 3dAj)

4− d2(B.4)

Substituting Equation (B.4) into Equation (2.6) and then solving the partial derivative,

the best response function in terms of advertising for each firm is shown as:

Ai =2θ (2 + d2) (a (2 + d) + 3dθAj)

(4− d2)2m− 2 (2 + d2)2 θ2(B.5)

1First, d cannot be equal to 1 as if products will become homogenous, Bertrand paradox will occur; dalso cannot be 0 as the two products will become totally unrelated.

Second, to guarantee price and ads intensity are positive, we also need θ2 < m(1−d)(2−d2)2(1+d)(2−d)2 .

73

By substituting Aj function into Equation (B.5), solve the advertising intensity for each

firm:

A∗i = A∗j =2aθ (2 + d2)

m (2− d)2 (2 + d)− 2θ2 (1 + d) (2 + d2)(B.6)

By substituting the equilibrium advertising intensity into the Equation (B.3), the price

charged by each firm is pi = pj =am(4−d2)

m(2−d)2(2+d)−2θ2(1+d)(2+d2) ; and substituting the equilibrium

price and advertising intensity into Equation (2.2), the profit earned by each firm is πi =

πj =a2m

((4−d2)

2m−2(2+d2)

2θ2

)((2−d)2(2+d)m−2(1+d)(2+d2)θ2)

2 .

This completes the proof.

Proof of Proposition 8:

There exists a subgame perfect Nash equilibrium under price matching such that each

firm makes an investment on cooperative advertising A∗i = aθ2m(1−d)−(1+d)θ2 in the first stage,

charges p∗i = am2m(1−d)−(1+d)θ2 and earns profits π∗i =

a2m(2(1−d)m−θ2)2(2m(1−d)−(1+d)θ2)2 in the second stage.

Proof.

According to Equation (2.2), the first order condition to solve pi subject to pi = pj is

∂

∂pi[(pi − ci)qi −

1

2mA2

i ] = 0 (B.7)

Solving the partial derivative, Equation (B.7) yields the best response functions for both

firms in terms of the level of advertising:

pi = pj =a+ θ(Aj + dAi)

2(1− d)(B.8)

Substituting Equation (B.8) into Equation (2.6), and then solving the partial derivative,

the best response function of advertising for each firm is shown as:

Ai =θa+ dθ2Aj

2m (1− d)− θ2(B.9)

By substituting Aj function into Equation (B.9), solve the advertising intensity for each

firm:

A∗i = A∗j =aθ

2m (1− d)− (1 + d) θ2(B.10)

74

By substituting the equilibrium advertising intensity into the Equation (B.8), the price

charged by each firm is pi = pj = am2m(1−d)−(1+d)θ2 ; and substituting the equilibrium price

and advertising intensity into Equation (2.2), the profit earned by each firm is πi = πj =

a2m(2(1−d)m−θ2)2((1+d)θ2−2m(1−d))2 .

This completes the proof.

Proof of Proposition 9: There exists a subgame perfect Nash equilibrium under Bertrand

competition such that each firm makes an investment on predatory advertising A∗i =2aθ(2−d2)

m(2+d)(2−d)2−2θ2(1−d)(2−d2)

in the first stage, charges p∗i =am(4−d2)

m(2+d)(2−d)2−2θ2(1−d)(2−d2) and earns profits π∗i =a2m

(m(4−d2)

2−2θ2(2−d2)

2)

(m(2+d)(2−d)2−2θ2(1−d)(2−d2))2

in the second stage.

Proof. According to Equation (2.2), the first order condition to solve pi is

∂

∂pi[(pi − ci)qi −

1

2mA2

i ] = 0 (B.11)

Solving the partial derivative, Equation (B.11) becomes

a− 2pi + dpj + θ(Ai − dAj) = 0 (B.12)

which yields the best response functions for both firms

pi =1

2(a+ dpj + θ(Ai − dAj)) (B.13)

By substituting pj function into Equation (B.13), solve the price for each firm in terms

of the level of advertising:

pi =a (2 + d) + (2− d2)θAi − dθAj

4− d2(B.14)

Substituting Equation (B.14) into Equation (2.6), and then solving the partial derivative,

the best response function in terms of advertising for each firm is shown as:

Ai =2θ (2− d2) (a (2 + d)− dθAj)m (4− d2)2 − 2θ2 (2− d2)2

(B.15)

By substituting Aj function into Equation, solve the advertising intensity for each firm:

A∗i = A∗j =2aθ (2− d2)

m (2− d)2 (2 + d)− 2θ2 (1− d) (2− d2)(B.16)

75

By substituting the equilibrium advertising intensity into the Equation (B.13), the price

charged by each firm is pi = pj =am(4−d2)

2θ2(1−d)(2−d2)−m(2+d)(2−d)2 ; and substituting the equilibrium

price and advertising intensity into Equation (2.2), the profit earned by each firm is πi =

πj =a2m

(m(4−d2)

2−2θ2(2−d2)

2)

(m(2−d)2(2+d)−2θ2(1−d)(2−d2))2 .

This completes the proof.

Proof of Proposition 10: There exists a subgame perfect Nash equilibrium under price

matching such that each firm makes an investment on predatory advertising A∗i = aθ(1−d)(2m−θ2)

in the first stage, charges p∗i = am(1−d)(2m−θ2) and earns profits π∗i =

a2m(2m(1−d)−θ2)2(1−d)2(2m−θ2)2 in the

second stage.

Proof. According to Equation (2.2), the first order condition to solve pi subject to pi = pj

is

∂

∂pi[(pi − ci)qi −

1

2mA2

i ] = 0 (B.17)

Solving the partial derivative, Equation (B.17) yields the best response functions for both

firms in terms of the level of advertising:

pi = pj =a+ θ(Aj − dAi)

2(1− d)(B.18)

Substituting Equation (B.18) into Equation (2.6), and then solving the partial derivative,

the best response function of advertising for each firm is shown as:

Aj =θa− dθ2Ai

2m (1− d)− θ2(B.19)

By substituting Aj function into Equation (B.15), solve the advertising intensity for each

firm:

A∗i = A∗j =aθ

(1− d) (2m− θ2)(B.20)

By substituting the equilibrium advertising intensity into the Equation (B.18), the price

charged by each firm is pi = pj = am(1−d)(2m−θ2) ; and substituting the equilibrium price

and advertising intensity into Equation (2.2) the profit earned by each firm is πi = πj =

a2m(2m(1−d)−θ2)2(1−d)2(2m−θ2)2 .

76

This completes the proof.

Proof of Proposition 11: Under a cooperative regime both firms adopting price match-

ing invest more efficiently in advertising, charge a higher price and then earn more profits

compared to Bertrand competition.2

Proof. According to the results in propositions 7 and 8, the differences in advertising

intensities, prices, and profits between price matching equilibrium and Bertrand equilibrium

are shown respectively:

A∗ipmg − A∗iBer =adθ(4− 6d+ 5d2)

(4− θ2 − d(4 + θ2))((2− d)2(2 + d)− (1 + d)(2 + d2)θ2)(B.21)

To guarantee advertising intensities are always positive, (4 − θ2 − d(4 + θ2)) and ((2 −

d)2(2 + d)− (1 + d)(2 + d2)θ2) are required to be positive. In addition, as (4− 6d+ 5d2) is

always positive, the difference in advertising intensities is positive.

p∗ipmg − p∗iBer =2a

4− θ2 − d(4 + θ2)− a(4− d2)

(2− d)2(2 + d)− (1 + d)(2 + d2)θ2(B.22)

π∗ipmg − π∗iBer = a2[4− 4d− θ2

(4− θ2 − d(4 + θ2))2− (4− d2)2 − (2 + d2)2θ2

((2− d)2(2 + d)− (1 + d)(2 + d2)θ2)2] (B.23)

By re-arranging Equation (B.22), the key term determining the sign of the differences is

ad(8− d(2d+ 3(1 + d)θ2)). As we assume d and θ are between 0 and 1, the price difference

is always positive.

As both prices increase under price matching policy under cooperative advertising, their

price converge to a monopoly price so that their profits are larger than under Bertrand

competition.

This completes the proof.

Proof of Proposition 12: There exists a critical value θ∗coop such that, if the cooperative

advertising factor θ is above θ∗coop, consumer surplus and total welfare are higher under price

matching than Bertrand competition.

2For simplicity, m is normalized to 1 without losing generality.

77

Proof. The consumer surplus is calculated based on the utility function, U(qi, qj, q0) =

a(qi + qj)− 12(q2i + 2dqiqj + q2j ) + q0. The difference in consumer surplus under price matching

and Bertrand competition is shown as:

CS∗ipmg − CS∗iBer =4a2(1− d)

(4− θ2 − d(4 + θ2))2− a2(4− d2)2

(1− d)((2− d)2(2 + d)− (1 + d)(2 + d2)θ2)2

(B.24)

By setting the difference equal to zero, the critical value θ∗coop is shown as:

θ∗coop =√

2(4− d2

4 + d− 2d2)1/2 (B.25)

This completes the proof.

Proof of Corollary 2: If the degree of differentiation increases, the critical value θ∗coop

decreases.

Proof. As θ∗coop =√

2[ (2+d)(2−d)(1−d)4+d−d2+2d3

]1/2 and d is between 0 and 1 by assumption, the partial

derivative with respect to d is given by:

∂θ∗coop∂d

= −√

2(20− d2(17− d(18 + d))

2(4 + d− d2 + 2d3)2< 0 (B.26)

This completes the proof.

Proof of Proposition 13: Under a predatory advertising regime both firms adopting price

matching invest more excessively in advertising and charge a higher price than Bertrand

competition.

Proof. According to the results in propositions 9 and 10, the differences in advertising

intensities, prices, and profits between price matching equilibrium and Bertrand equilibrium

are shown respectively:

A∗ipmg − A∗iBer = aθ[1

(1− d)(4− θ2)− 2− d2

(2− d)2(2 + d)− (1− d)(2− d2)θ2] (B.27)

p∗ipmg − p∗iBer = 2a[1

(1− d)(4− θ2)− 2− d2

(2− d)2(2 + d)− (1− d)(2− d2)θ2] (B.28)

78

By re-arranging Equation (B.27) and Equation (B.28), a common term from these equa-

tions determining the sign of the differences is (4 + 2d− 3d2). As we assume d is between 0

and 1, the differences for advertising and price are always positive.

This completes the proof.

Proof of Proposition 14: There exists a critical value θ∗non−coop such that if predatory

advertising factor θ is above θ∗non−coop, both firms’ profits are lower under price-matching

than Bertrand competition.

Proof. The profit difference is shown as:

π∗ipmg − π∗iBer = a2[4− 4d− θ2

(1− d)2(4− θ2)2− (4− d2)2 − (2− d2)2θ2

[(2− d)2(2 + d)− (1− d)(2− d2)θ2]2] (B.29)

By setting the difference equal to zero, the critical value θ∗non−coop is shown as:

θ∗non−coop =√

2(4− d2

4 + d− 2d2)1/2 (B.30)

This completes the proof.

Proof of Corollary 3: If the degree of differentiation increases, the critical value θ∗non−coop

decreases.

Proof. As θ∗non−coop =√

2( 4−d24+d−2d2 )1/2, the partial derivative with respect to d is given by:

∂θ∗non−coop∂d

= −√

2(4− d2)2(4 + d− 2d2)2

< 0 (B.31)

This completes the proof.

Proof of Proposition 15: Consumer surplus and total welfare are lower under price match-

ing than Bertrand competition.

Proof. The consumer surplus is calculated based on the utility function, U(qi, qj, q0) =

a(qi + qj) − 12(q2i + 2dqiqj + q2j ) + q0. The difference of consumer surplus between price

matching and Bertrand competition is shown as:

CS∗ipmg − CS∗iBer =4a2

(1− d)[

(4− d2)2 − (2− d2)2θ2

[(2− d)2(2 + d)− (1− d)(2− d2)θ2]2− 1

(4− θ2)2] (B.32)

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A key term determining the sign of the differences is −d(8− 2d2− 4θ2− dθ2 + 2d2θ2). As

we assume d and θ are between 0 and 1, the difference in consumer surplus is negative.

This completes the proof.

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Appendix C

Chapter Three Proofs

Proof of Proposition 18: There exists a subgame perfect Nash equilibrium under Bertrand

competition such that an incumbent firm makes an investment on advertising A∗1 = 2a(2+d)12−8d2+d4

in the first stage; the incumbent firm and new entrant charge p∗1 = a(2−d)(2+d)212−8d2+d4 and p∗2 =

a(6+d(4−d(2+d)))12−8d2+d4 in the second stage.

Proof. According to Equation (3.2) and Equation (3.4), the first order condition to solve

p1, p2 are shown as:

∂

∂p1[(p1 − c)(a+ A1 − p1 + dp2)−

1

2A2

1] = 0 (C.1)

∂

∂p2[(p1 − c)(a− p2 + dp1)] = 0 (C.2)

Solving the partial derivatives, Equation (C.1) and Equation (C.2) become

a+ A1 − 2p1 + dp2 = 0 (C.3)

a+ dp1 − 2p2 = 0 (C.4)

which yields the best response functions for both firms:

p1 =a+ A1 + dp2

2(C.5)

p2 =a+ dp1

2(C.6)

By substituting Equation (C.5) into Equation (C.6) , solve the price for each firm in

terms of the level of advertising:

p1 =2A1 + a(2 + d)

4− d2(C.7)

p2 =A1d+ a(2 + d)

4− d2(C.8)

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Substituting Equation (C.7) and Equation (C.8) in to Equation (3.5), and then solving

the partial derivative, the advertising intensity for firm 1 is:

A1 =2a(2 + d)

12− 8d2 + d4(C.9)

By substituting the advertising intensity into the Equations (C.7) and (C.8), the price

charged by each firm is p∗1 = a(2−d)(2+d)212−8d2+d4 and p∗2 = a(6+d(4−d(2+d)))

12−8d2+d4 in the second stage and the

profit earned by each firm is π1 = a2(2+d)2

12−8d2+d4 and π2 = a(6+d(4−d(2+d)))12−8d2+d4 .

This completes the proof.

Proof of Proposition 19: There exists a subgame perfect Nash equilibrium under a

price matching policy such that an incumbent firm makes an investment on advertising

A∗1 = a4(1−d) in the first stage; the incumbent firm and new entrant charge p∗1 = p∗2 = a

2(1−d)

in the second stage.

Proof. According to Equation (3.2), the first order condition to solve p2 is shown as:

∂

∂p2[(p1 − c)(a− p2 + dp1)] = 0

st : p1 = p2

Solving the partial derivatives, the price for each firm under price matching policy is

shown as:

p1 = p2 =a

2(1− d)(C.10)

By substituting Equation (C.10) into Equation (3.5), the first order condition to solve

A1 is shown as:

a

2(1− d)− 2A1 = 0 (C.11)

which yields the advertising intensity for firm 1:

A1 =a

4(1− d)(C.12)

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By substituting the equilibrium price into Equation (3.2) and Equation (3.4), the price

charged by each firm is p∗1 = a2(1−d) and p∗2 = a

2(1−d) in the second stage and the profit earned

by each firm is π1 = a2(5−4d)16(1−d)2 and π2 = a2

4(1−d) .

This completes the proof.

Proof of Proposition 20: There exists a subgame perfect Nash equilibrium under Bertrand

competition such that an incumbent firm makes an investment on R&D K∗1 = (a−c(1−d))(2+d)(2−d2)4(3−d2)

in the first stage; the incumbent firm and new entrant charge p∗1 = a(2+d)+c(4+d−d2)6−2d2 and p∗2 =

a(6+(2−d)d)+c(6+d(4−d−d2))4(3−d2) in the second stage.

Proof. According to Equation (3.8) and Equation (3.6), the first order condition to solve

p1, p2 are shown as:

∂

∂p1[[p1 − (c−K1)](a− p2 + dp1)−

1

2K2

1 ] = 0 (C.13)

∂

∂p2[(p1 − c)(a− p2 + dp1)] = 0 (C.14)

Solving the partial derivatives, Equation (C.13) and Equation (C.14) become

a+ c−K1 − 2p1 + dp2 = 0 (C.15)

a+ c+ dp1 − 2p2 = 0 (C.16)

which yields the best response functions for both firms

p1 =a+ c−K1 + dp2

2(C.17)

p2 =a+ c+ dp1

2(C.18)

By substituting Equation (C.17) into Equation (C.18), solve the price for each firm in

terms of the level of advertising:

p1 =[a− c(1− d)](2 + d) + (2− d2)K1

4− d2(C.19)

p2 =[a− c(1− d)](2 + d)− dK1

4− d2(C.20)

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Substituting Equation (C.19) and Equation (C.20) in to Equation (3.9), and then solving

the partial derivative, the advertising intensity for firm 1 is:

K1 =(a− c(1− d))(2 + d)(2− d2)

4(3− d2)(C.21)

By substituting the intensity of R&D into the Equations (C.19) and (C.20), the price

charged by each firm is p∗1 = a(2+d)+c(4+d−d2)6−2d2 and p∗2 = a(6+(2−d)d)+c(6+d(4−d−d2))

4(3−d2) in the second

stage and the profit earned by each firm is π1 = (a−c(1−d))2(2+d)24(3−d2) and π2 = (a−c(1−d))2(6+(2−d)d)2

16(3−d2)2 .

This completes the proof.

Proof of Proposition 21: There exists a subgame perfect Nash equilibrium under a price

matching policy such that an incumbent firm makes an investment on R&D K∗1 = 14(a−

c(1 − d)) in the first stage; the incumbent firm and new entrant charge p∗1 = p∗2 = a2(1−d) in

the second stage.

Proof. According to Equation (3.6), the first order condition to solve p2 is shown as:

∂

∂p2[(p1 − c)(a− p2 + dp1)] = 0 (C.22)

st : p1 = p2 (C.23)

Solving the partial derivatives, the price for each firm under price matching policy is

shown as:

p1 = p2 =a+ c(1− d)

2(1− d)(C.24)

By substituting Equation (C.24) into Equation (3.9), the first order condition to solve

K1 is shown as:

1

2(a− c(1− d)− 4K1) = 0 (C.25)

which yields the R&D intensity for firm 1:

K1 =1

4(a− c(1− d)) (C.26)

By substituting the equilibrium price and R&D intensity into the Equation (3.8) and

Equation (3.6), the profit earned by each firm is π1 = (a−c(1−d))2(5−d)16(1−d) and π2 = (a−c(1−d))2

4(1−d) .

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This completes the proof.

Proof of Proposition 22: There exists an upper bound d̄ such that for all d > d̄ , the

incumbent firm adopts a price matching strategy, overinvests more in advertising, charges a

higher price and gains higher profits compared to Bertrand competition.

Proof. The profit difference is shown as:

π∗ipmg − π∗iBer =a2(5− 4d)

16(1− d)2− a2(2 + d)2

12− 8d2 + d4(C.27)

By setting the difference equal to zero, the critical value function is shown as:

5d3 + 7d2 − 3d− 1 = 0 (C.28)

And the critical value d̄ is 0.51.

This completes the proof.

Proof of Proposition 23: There exists an upper bound d̄ such that for all d > d̄ , the

incumbent firm adopts a price matching strategy, under-invests more in R&D, charges a

higher price and gains higher profits compared to Bertrand competition.

Proof. The profit difference is shown as:

π∗ipmg − π∗iBer =(a− c(1− d))2(5− d)

16(1− d)− (a− c(1− d))2(2 + d)2

4(3− d2)(C.29)

By setting the difference equal to zero, the critical value function is shown as:

d4 + 8d− 4 = 0 (C.30)

And the critical value d̄ is 0.49.

This completes the proof.

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