case study perfect competition in credit card industry

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How Competitive Is the Credit Card Market?

T he credit card market is highly competitive, particularly among card issuers but also among payment networks based on several different metrics. Industry competition keeps prices low, promotes innovation, and gives consumers the power to choose the card that works best for them.

In a landmark U.S. Supreme Court case confirming the “two-sided market” as a fundamental framework for analyzing the credit card market, the Court found that the credit card market lacked any hallmarks of a noncompetitive sector. In fact, there are at least four parties involved in most credit card transactions. There are the cardholder, the cardholder’s bank (the “issuing bank”) that extends credit on the card, the merchant, and the merchant’s bank. A credit card network uses its brand to communicate to the customer where their card will be accepted, and various processors can play a role in ensuring data reaches the right place.

The card brands (or “networks”) act as intermediaries, balancing and connecting the needs of everyone in the transaction through established rules, so that transactions happen in a fraction of a second. There are actually two dynamic markets that banks serve (issuers market cards to consumers and commercial cardholders and acquiring banks provide card acceptance services to processors and card-accepting businesses). In the middle are the card brands, working to attract both issuing banks and merchants to their network.

These distinct segments of the value chain provide significant space to innovate and compete. Over the past 10 years, 13 issuers have been in the top 10, 32 issuers have been in the top 20, 80 issuers have been in the top 50, and 163 issuers have been in the top 100. This movement over the last decade, including among the largest issuers, is indicative of a competitive market. New card acceptance solutions like Checkout.com, Toast and Square have created options for retailers, competing aggressively with the traditional merchant processor ecosystem. All of this change has taken place without government intervention, demonstrating that the foundations of the card marketplace foster rather than inhibit innovation and that the marketplace is flexible, rather than protective of incumbents.

A common, generally accepted measure of market concentration is the Herfindahl-Hirschman Market Concentration Index. The U.S. Department of Justice uses the HHI to analyze market concentration when, for example, a merger might affect industry competition. According to DOJ, markets in which the HHI is between 1,500 and 2,500 points are considered “moderately concentrated,” while markets where the HHI is higher than 2,500 points are considered “highly concentrated.”

As shown in Figure 1, neither the credit card issuing industry nor the financial transactions processing, reserve and clearinghouse activities industry—which includes credit cards, financial transaction processing and electronic financial payment and funds transfer services—meet DOJ’s threshold of a concentrated market. Indeed, several other industries that rely heavily on credit cards (for example, department stores, bookstores, wireless carriers and passenger car rental) are significantly more concentrated.

As shown in Figure 2, the top 50 credit card issuers account for nearly all the credit card market in both 2012 and 2017. For a national marketplace where regulations provide guardrails that standardize some (but far from all) product features, this is not surprising. In comparison to domestic commercial airlines, for instance, the card issuance market has many more major players. There are also thousands of card-issuing financial institutions in the US, many more than in similarly-sized foreign countries, and barriers to issuing cards are virtually nil for regulated lenders. Again, card issuance options for consumers compare well to trying to find thousands of airlines selling tickets. Of course these are different products, which is why it makes sense to delve into the particularities of each market (like the card market being a two-sided market) before jumping to any conclusions. The reality that the card market is not actually concentrated is demonstrated by the HHI scores in Figure 1.

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Reinventing credit cards: Responses to new lending models in the US

Credit cards have long been one of the most popular methods of making payments and accessing unsecured borrowing in the United States, accounting for 37 percent of consumer purchases by dollar value in 2021. 1 Data from the McKinsey US Payments Map, calculated as share of consumer purchases by dollar value (excluding bill pay). Credit cards include private-label as well as general-purpose cards. But their market position is gradually being undermined by the growth of point-of-sale (POS) financing offerings that combine installment lending with the convenience of card payments. US issuers could by 2025 lose up to 15 percent of incremental profits to newer forms of borrowing, based on our simulation of the potential impact of buy now, pay later (BNPL).

Alarming as that may sound to credit-card issuers, it is far from the whole story. Issuers have decades of experience and well-established relationships with consumers and merchants to help them compete. What’s more, credit cards have several deep-rooted advantages over other credit products. Many consumers engage with credit cards daily when purchasing goods and services. The spending data generated in this way gives issuers valuable insights into consumers’ preferences and needs. And credit cards are often part of a complementary suite of offerings alongside deposits, consumer loans, and other products, helping to reinforce customer loyalty.

Issuers can tackle the challenges they face by building on these strengths. They can reimagine their products to meet consumer needs, introduce tailored solutions to reach younger consumers, drive engagement, and rethink card economics.

A strong track record—but can it be sustained?

In the United States, credit cards are one of the best-performing businesses in financial services, with a return on assets of 3.6 percent in 2020. Credit cards are also a primary method of unsecured borrowing for US consumers, accounting for 78 percent of balances. 2 McKinsey Panorama, Global Banking Pools. Over the past few years, transaction volumes have grown by 10 percent per year, reaching $49 trillion in 2021.

Modeling revolver and transactor economics

Our analysis uses data from the Consumer Financial Protection Bureau, the McKinsey US Payments Map, the McKinsey Consumer Financial Life Survey, and McKinsey estimates. To model revolver and transactor economics, we assumed that revolvers are responsible for all of issuers’ interest income, losses, late fees, and debt suspension fees and that transactors are responsible for the majority of annual fees. We also assumed that revolver spending earns fewer rewards and generates lower interchange per dollar than transactor spending and that per-account operating expense is higher for revolvers than for transactors.

However, today’s issuers face circumstances that make profitable growth harder to sustain. Their profits rely mainly on revolvers, or customers who carry a balance on their credit-card account from month to month (see sidebar “Modeling revolver and transactor economics”). Revolvers make up around 60 percent of credit-card accounts, but they generate 85 to 90 percent of issuers’ revenues, net of rewards. Profit per account stands at around $240 for revolvers but at just $25 for transactors, or customers who pay off their balance every month (Exhibit 1). The difficulty for issuers is that the share of revolvers has started to decline over the past few years. 3 For general-purpose credit cards, the share of revolvers has fallen by as much as six to eight percentage points from its 2015 level. See The consumer credit card market , Consumer Financial Protection Bureau, September 29, 2021. At the same time, reward spend is growing, low loss rates are heading back toward normal levels, and funding costs are rising. The net effect of these trends is a squeeze on issuers’ margins.

Enter the BNPL players

Risks associated with pos financing.

Point-of-sale financing is associated with a set of risks for consumers and financial systems. In late 2021, the US Consumer Financial Protection Bureau opened market-monitoring orders for select BNPL providers. The orders focus on three themes: debt accumulation arising from the regular nature of purchases using POS financing; regulatory arbitrage , leading to limited transparency and consumer protection; and practices for data collection and usage . 1 “Consumer Financial Protection Bureau opens inquiry into ‘buy now, pay later’ credit,” Consumer Financial Protection Bureau news release, December 16, 2021. Regulators in other countries, including Australia, Singapore, and the United Kingdom, have also expressed concerns about possible risks associated with BNPL.

Meanwhile, credit bureaus and other market participants are moving to create more transparency around POS financing. Organizations such as Equifax, Experian, and TransUnion are expanding their existing products to include BNPL financing or working with BNPL providers to create new credit-reporting products geared to BNPL transactions. 2 Alicia Adamczyk, “Equifax is adding buy now, pay later payments to credit reports. Here’s how it could affect your score,” CNBC, December 31, 2021; “TransUnion to maximize financial inclusion opportunities for the nearly 100 million consumers using BNPL loans,” TransUnion news release, February 24, 2022; “Buy now pay later,” Experian, accessed June 2022.

Traditional sales finance, commonly called layaway, has been available in the US for decades, but for credit-card issuers, the risk to profitable growth comes from the rapid growth of a relative newcomer to the payments arena: technology-enabled BNPL . Consumers are choosing BNPL for a variety of reasons, including lower APR (starting at 0 percent for some purchases), predictable repayments, and the convenience of using a payment method that is integrated into online customer journeys and shopping apps. The sustainability of POS financing is subject to debate: pay-in-4 providers have historically made a loss despite positive unit economics, BNPL players now face a more challenging macroeconomic environment with rising interest rates and defaults, and questions have been raised about the risk associated with BNPL (see sidebar “Risks associated with POS financing”). Nevertheless, it seems clear that BNPL has changed consumers’ expectations of the borrowing experience and expanded the role lenders can play in shopping journeys.

Providers such as Affirm and Afterpay offer consumers seamless borrowing at the point of sale for small- and mid-ticket purchases. In doing so, they could erode a fraction of issuers’ volumes. 4 In this article, BNPL and POS financing are used interchangeably to denote all types of financing at point of sale, from small-ticket “pay-in-4” offerings from providers such as Afterpay and Sezzle to mid-ticket financing from providers such as Affirm and Uplift. The exact size of that fraction is hard to establish. McKinsey’s US Digital Payments Survey indicates that 39 percent of BNPL users making a purchase would otherwise have paid with a credit card. In another survey, 62 percent of users expressed the belief that BNPL could replace their credit card—although only about a quarter said they would want it to do so. 5 Maurie Backman, “Study: Buy now, pay later services continue explosive growth,” Ascent , updated March 22, 2021.

What is certain is that credit-card holders are adopting BNPL. Among the users of mid-ticket POS financing—typically consumers with loans of $300 to $3,000 on purchases of furniture, appliances, electronics, and other durable goods—almost 95 percent have credit cards (Exhibit 2). So do 85 to 90 percent of pay-in-4 users, who typically have six-week merchant-funded loans of less than $300 on purchases of apparel, beauty products, and accessories.

As well as capturing transaction volumes, BNPL providers are doing something else that could undermine issuers’ business: acting as an entry product for younger consumers who are new to credit. Although use among older customers is growing, BNPL attracts a predominantly young audience: 37 percent of Gen Z and 30 percent of millennials are reportedly users, compared with 17 percent of Gen X and 6 percent of baby boomers. 6 “Almost 75% of BNPL users in the US are Gen Z or millennials,” Emarketer , June 25, 2021. Issuers have traditionally relied on younger consumers as a source of growth. Since 2017, credit-card spending has increased by 11 percent a year among those under 40 while remaining flat among those over 40, who account for 62 percent of this spending. If BNPL providers continue to attract large numbers of young consumers and are able to retain them as they grow older, credit-card volume growth is likely to suffer.

BNPL providers such as Affirm, Afterpay, Klarna, and Sezzle are also starting to shape the wider retail ecosystem by developing shopping apps that drive consumer traffic and stickiness. Users of Afterpay and Klarna are engaged and loyal, making transactions via these apps almost every month. Klarna reports that customers who use its shopping app make purchases via Klarna three times more often than nonusers.

In parallel, established payments providers are expanding into BNPL and developing comprehensive financing and payments offerings for merchants and consumers. Examples include Block’s acquisition of Afterpay 7 “Block, Inc. completes acquisition of Afterpay,” Block news release, January 31, 2022. and PayPal’s introduction of credit and pay-in-4 options. 8 “PayPal introduces new interest-free buy now pay later installment solution,” PayPal news release, August 31, 2020.

How BNPL could change the payments landscape

We see four trends in BNPL that are likely to affect—or are already affecting—the strategies of issuers, as well as banks, fintechs, and other payments providers. BNPL apps are playing a greater role early in shopping journeys and offering a broader range of services. At the same time, payment networks are making POS financing widely available, and financial institutions are getting into the game.

Trend 1: BNPL apps are becoming a starting point for consumers’ shopping journeys

BNPL providers are starting to position themselves primarily as integrated apps that combine shopping with consumer financing. This strategy enables them to build customer loyalty and generate affiliate fees from nonintegrated merchants. This trend is likely to intensify as rising interest rates push up the cost of funds and merchant discount rates continue to decline, squeezing BNPL providers’ margins and prompting them to turn to affiliate fees as an additional source of revenue. 9 According to a McKinsey BNPL merchant survey conducted in 2022, the median merchant discount rate (MDR) in the US stands at 2.5 percent. Affiliate fees vary widely, from 0.5 to 1.0 percent on appliances in Home Depot and Best Buy to 20 percent on Amazon Games, according to these companies’ websites.

Trend 2: BNPL providers are venturing beyond installment lending

As BNPL players continue to expand their customer base, they are introducing new financial and loyalty products to meet their young customers’ evolving needs and to maximize customers’ lifetime value. Early examples include Klarna’s credit card and Affirm’s Debit+ card, which allow consumers to make staged payments in offline channels and at nonintegrated merchants. Other examples include Klarna’s checking accounts in Germany and Afterpay’s Money app in Australia, which offers savings accounts and a debit card. Over time, moves like these could extend to other products: high-yield savings accounts, loyalty programs, and other financial or shopping-related services.

Trend 3: Payment networks are providing access to consumer POS financing at scale

Capitalizing on their access to merchants and ownership of credit-card transaction processing, payment networks are rolling out solutions that enable greater use of BNPL. For example, Mastercard Installments allows customers to access a BNPL product via a virtual card issued by a bank or fintech, 10 “Introducing Mastercard Installments,” Mastercard, accessed June 2, 2022. and with Visa Installments, customers can split purchases on eligible Visa cards into equal installments at the point of purchase. 11 “What is Visa Installments?,” Visa, accessed June 2, 2022. Mastercard Installments technology will be used by Apple Pay for their recently announced BNPL product, Apple Pay Later. 12 Kif Leswing, “A wholly owned subsidiary of Apple will extend loans for its Pay Later service,” CNBC, June 8, 2022. Network BNPL solutions could make BNPL more accessible for consumers, small merchants, and merchants from categories with lower BNPL penetration. Payment networks wanting to raise the standard for customer experience could also allow customers to choose the best payment method—say, credit card, on-card BNPL, or virtual-card-enabled BNPL—for any transaction, depending on ticket size, credit-card limit, pricing, and other factors.

Trend 4: Financial institutions are expanding their reach by entering POS lending

Credit-card issuers and other financial institutions are exploring participation in POS lending. Some lenders are setting up their own offerings, such as Citizens Pay; others are entering the market via acquisitions, such as Goldman with GreenSky, Regions Bank with EnerBank, and Truist with Service Finance. Lenders’ robust balance sheets, strong brands, ability to underwrite big-ticket installment loans, and a large and loyal consumer base give them a competitive advantage in this new arena. In time, POS financing could become a customer-acquisition channel for lenders, as well as a means to increase their share of wallet by cross-selling traditional banking products to POS financing users.

The extent to which these trends will reshape POS financing, and consumer lending more broadly, will depend on multiple factors, including consumers’ willingness to start their shopping journey on BNPL shopping apps, the ability of networks and issuers to provide a compelling user experience and drive adoption, and lenders’ ability to integrate and grow the POS financing businesses they acquire.

How BNPL could affect issuers’ volumes and profits

Three key risks associated with BNPL could significantly affect issuers’ volumes and profits. First, issuers could lose younger consumers who prefer financing to be embedded in the shopping experience. Second, BNPL providers could take away some of the revolvers, who are issuers’ most profitable consumer segment. Third, as BNPL providers start to own customer relationships, issuers may find they must spend more on customer acquisition to compete.

Simulating BNPL’s potential impact on credit-card volumes and profits

We modeled three scenarios of declining growth in credit-card spending (exhibit), based on different assumptions for how credit-card spending would change between 2016–19 and 2020–25. A moderate scenario assumed that the 9 percent CAGR in credit-card spending in 2016–19 would decline by a half percentage point, and the base case assumed a one-percentage-point decline. In the third, or aggressive, scenario, spending would fall two percentage points. In calculating the impact of the potential incremental reduction in spending, we controlled for changes in GDP growth between 2016–19 and 2020–25, and we excluded 2020 and 2021 from the analysis because of pandemic-related distortions.

To calculate how the decline in the share of revolvers might affect pretax profit, we multiplied the incremental reduction in spend volume by the operating-profit margin, which was itself calculated on the assumption that the share of revolvers would decline by three to seven percentage points between 2020 and 2025.

This what-if analysis suggests that the growth of BNPL has the potential to reduce US credit-card spending by 2 to 12 percent by 2025. What’s more, US issuers are likely to see their profits squeezed even more than their volumes, since the customers who switch to BNPL tend to be the revolvers—the most profitable customers. If growth in credit-card spend were to decline by a half to two percentage points from its 9 percent CAGR in 2016–19, and if the share of revolvers were to fall by three to seven percentage points, issuers could see their profits decline by 2 to 15 percent.

To understand the potential impact of BNPL on US issuer volumes and profits, we ran a simulation based on three different scenarios for credit-card spending over the next few years. The simulation revealed that US issuers could lose between 2 and 15 percent of incremental profits to newer forms of borrowing by 2025 (see sidebar “Simulating BNPL’s potential impact on credit-card volumes and profits”).

In markets with more mature POS financing offerings, significant volumes have already shifted from credit cards to BNPL. In Australia, for instance, credit-card accounts have declined by about 6 percent a year, and BNPL accounts have grown by more than 40 percent a year since 2017 (Exhibit 3). Because of its higher interchange fees and different market fundamentals, the US may see a more muted shift than in Australia, but it is evident that replacement is under way.

Findings from the 2021 McKinsey Digital Payments Survey suggest that the credit-card business is more likely to be cannibalized by mid-ticket POS financing than by pay-in-4 providers. That’s because users of mid-ticket POS financing are more likely to have a credit card and to use it if BNPL is not available, as shown in Exhibit 2.

Private-label credit cards are popular among merchants because of their favorable economics, but they are likely to see more impact on their volumes than general-purpose cards. For one-off purchases at a particular merchant, BNPL tends to offer consumers experiences that are more seamless, more transparent, and in some cases more affordable than using a private-label credit card.

Finally, BNPL’s impact on credit cards is likely to vary by industry and product category. In travel, where cobranded cards offer generous rewards for customer loyalty, BNPL represents only about 2 percent of consumer transactions. In contrast, furniture, mattresses, electronics, and appliances could see considerable inroads from BNPL providers as purchases continue to shift to online channels and private-label card penetration stagnates.

How issuers could respond

As issuers face a changing consumer-lending landscape and the possibility of losing credit-card business to BNPL providers, they should prepare a thoughtful response. Options they might consider to sustain and grow their unsecured consumer lending could include reimagining their products to meet customer needs, reaching younger consumers with tailored solutions, driving consumer engagement, and rethinking the economics of their card product.

Reimagining products

Issuers could consider rolling out POS financing products and on-card installment solutions that meet consumers’ need for predictability and demand for financing offered as part of the shopping journey. Fintechs have entered this arena with products such as the Upgrade Card, a hybrid between installment lending and a traditional revolving credit card. When designing their own offerings, issuers will need to carefully consider how a product can deliver sustainable profits while remaining competitive with fintech solutions. That will involve assessing the lifetime value of potential customers, which depends on the issuer’s ability to move customers to offerings with a higher return on assets (ROA) and/or to develop multiproduct relationships with customers.

Reaching younger consumers

Issuers could offer innovative types of credit cards geared to consumers who are new to credit. In Australia, for instance, CommBank and NAB have launched cards that allow consumers to subscribe to a line of credit without being charged interest, although they may in some cases end up paying more in monthly card fees. The appeal of products like these lies in their transparency and simplicity.

Driving consumer engagement

Some issuers and payment providers have acquired e-commerce players that allow them to reduce their customer-acquisition costs or offer new forms of value to boost consumer engagement. Examples include Capital One’s acquisition of Wikibuy, a price-comparison solution, and PayPal’s acquisition of Honey, a coupon-finder service. By becoming a starting point in a shopping journey and offering consumers distinctive value, issuers can increase their chances of staying top of wallet while creating a new revenue stream from affiliate marketing.

Rethinking card economics

Issuers could consider moving toward partly or fully merchant-funded on-card financing offers, rewards, or both to help them sustain their profitability in the face of mounting margin pressures. The key will be to deliver value not only to transactors but also to revolvers, who benefit from BNPL products that are partly or fully funded by merchants.

For US credit-card issuers, the prospect of losing a substantial share of volume and profits to BNPL over the next few years should act as a spur to action. With the right strategic moves, planned and implemented without delay, issuers can give themselves the best chance of stemming likely losses and positioning their business for success in an increasingly competitive arena.

Amit Garg is a senior partner in McKinsey’s New York office, where Diana Goldshtein is an expert, Udai Kaura is a partner, and Roshan Varadarajan is an associate partner.

The authors wish to thank Phil Bruno, Aaron Caraher, Soma Cserhati, Amit Gandhi, Joe Nutter, Oliver Søe, and Jon Steitz for their contributions to this article.

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• Published: 14 November 2014
• Volume 47 , pages 153–175, ( 2015 )

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I explain the credit card market’s observed systematic pricing patterns by examining time-inconsistent consumers. I find that time inconsistency steers the competition from long-term borrowing contingent prices to short-term noncontingent ones. This pattern occurs because the consumer in the contracting period underestimates the future charges, and therefore pays attention only to short-term price elements, such as annual fees. The consumer’s risk of default also plays a role in determining who gets which contract.

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Dynamic Pricing of Credit Cards and the Effects of Regulation

Insurer commitment and dynamic pricing pattern

Consumer Demand for Credit Card Services

Mester ( 1994 ) analyzes the credit card market by using a screening model and shows that interest-rate stickiness may be because of asymmetric information between consumers and banks. Parlour and Rajan ( 2001 ) show that default possibility and multi-contracting may result in positive interest rates and non-competitive profits. Brito and Hartley ( 1995 ) show that transaction costs for other loans can explain high credit card prices. Lastly, Evans and Schmalensee ( 2005 ) point out the default risk for seemingly high interest rates.

Strotz ( 1956 ) defines the time-inconsistent consumer as one who does not obey his optimal plan of the present moment when he reconsiders his plan in future periods.

Ausubel ( 1991 ) and Laibson ( 1997 ) were the first to point out the importance of incorporating consumer time inconsistency in models of this market.

For convenience, I consider the consumer making decisions at different periods as different “selves” of the same consumer.

Ausubel ( 1991 ) finds evidence for noncompetitive profits, but (Evans and Schmalensee 2005 ) criticize this evidence on the basis of not adjusting for the differences in the risk factors. Moreover, credit card companies argue that the interest rates are high because of the high default risks, not because of noncompetitive prices (Rougeau 1996 ).

In Section  5.2 , I also analyze an extension where the consumer has a private type, which determines if he is hyperbolic or exponential.

O’Donoghue and Rabin ( 2001 ) introduce a model to represent a partially naive hyperbolic consumer who is aware of his time inconsistency but underestimates its severity. The partially naive hyperbolic consumer knows that future discounting today is {1, β δ , β δ 2 , β δ 3 ,..}, and incorrectly believes that it will be \( \{1,\beta ^{\prime }\delta ,\beta ^{\prime }\delta ^{2},\beta ^{\prime }\delta ^{3},..\}\) from tomorrow onward with \(\beta <\beta ^{\prime }.\)

I can get the same qualitative results for a subset of partially naive consumers, as demonstrated in Section  5.4 . I do not get the same results with sophisticated hyperbolic consumers, since they correctly calculate their future debt just like time-consistent consumers.

This cost is the cost of bankruptcy proceedings and of receiving unfavorable terms in any contract in the future after declaring bankruptcy. The higher credit quality implies a higher cost of default because the higher credit quality consumers have more to lose in terms of forgone favorable terms in future contracts.

It has been documented that over 70 percent of the credit card issuers’ revenue is from revolvers (Chakravorti 2003 ).

My results do not change as long as the upper bound for the interest rate is finite.

Contrary to the standard subgame perfect equilibrium, a naive hyperbolic consumer has incorrect beliefs about his future decisions. This does not create a problem for the definition of the equilibrium in my case, as there is no strategic game after the contracting period, just a decision-making problem.

See Section A.1 in Appendix.

See Section A.2 in Appendix.

See Section A.3 in Appendix .

Secured credit cards typically require a cash deposit and give the owner a small credit limit. They are intended for users with bad credit or no credit as implied in my model.

When the consumer has both contracts in hand and if he accumulates interest-bearing debt then the consumer pays the company with the higher interest rate within the grace period to minimize the interest payment. Foreseeing this, the companies compete on interest rates resulting in zero-profit equilibrium.

When a consumer starts using a credit card for the first time, there is a grace period of about 21 days, during which time no interest accrues. Therefore, at minimum, during these first 21 days, everyone is a convenience user.

For a time-inconsistent consumer, “no default contraint” is different according to each period’s self, and therefore the analysis is more involved, as shown in Section A.3 in Appendix. Adding incomplete information at the top of it would complicate the analysis prohibitively in the current setting. Therefore, I look at the simplest possible case, allowing to see the mechanism at work.

From Section A.1 in Appendix recall that n 1 ( r ) solves the following maximization problem

and \(\frac {\partial n_{1}}{\partial r}<0\) .

I have a numerical example for each case. These are available on request.

I would like to thank an anonymous referee for raising these points.

It is possible to show that \(\beta _{r=1}^{\ast \ast }(\delta )<\beta _{r}^{\ast \ast }(\delta )<\beta _{r=0}^{\ast \ast }(\delta )=\beta ^{\ast }(\delta )\) for δ > δ ∗ .

Note that G c ( l i , r )= C is closer to the origin for higher values of r .

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Tepper School of Business, Carnegine Mellon University, 5000 Forbes Ave, Pittsburgh, PA, 15213, USA

Elif Incekara-Hafalir

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Correspondence to Elif Incekara-Hafalir .

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I would like to thank the co-editor, an anonymous referee, as well as Kalyan Chatterjee, Edward Green, Susanna Esteban, Isa Hafalir, Jeremy Tobacman, and Neil Wallace for their valuable comments and suggestions.

1.1 A.1 Naive hyperbolic consumer underestimates future consumption

If the consumer is not planning to default, the consumer pays all of his debt back by the last period. Therefore, only the first period borrowing will create interest revenue, but not the second period borrowing. Therefore, I analyze only first-period new borrowing. First, I analyze the exponential consumer as the benchmark (0≤ δ ≤1 and β =1). The exponential consumer is time consistent, and therefore the first-period borrowing according to the contracting-period self ( \({n_{1}^{0}},\) believed amount of borrowing) and according to the period-one self ( \({n_{1}^{1}},\) actual amount of borrowing) is the same ( \({n_{1}^{0}}={n_{1}^{1}}\) ). Exponential consumers can either be borrowers or convenience users, but cannot switch between these roles. There is a δ ∗ such that the contracting-period self correctly knows that he will not pay interest for δ ≥ δ ∗ (convenience user). Therefore, this self is unresponsive to interest rates. Nevertheless, being unresponsive to interest rates does not hurt him because he will not pay interest in the future anyway. The companies earn zero profit even without competition on interest rates. If δ < δ ∗ , then the contracting-period self correctly knows that he will pay interest (borrower). Therefore, this self looks for the lowest interest rate. As a result, a Bertrand competition drives the interest rates down to zero. A naive hyperbolic consumer, on the other hand, has a self-control problem and is not aware of it ( β <1). Consequently, this consumer underestimates his future borrowings \(\left ({n_{1}^{0}}<{n_{1}^{1}}\right )\) . The following proposition shows that there is a naive hyperbolic consumer (specified by δ and β ) who has a contracting-period self that plans to use the card for transactions only, but who has a period-one self that ends up using it for borrowing.

Proposition 5

For a naive hyperbolic consumer, there is a δ ∗ , such that \({n_{1}^{0}}\leq m\) for all δ≥δ ∗ . There is also a \(\beta _{r=1}^{\ast }(\delta )>0,\) such that \({n_{1}^{0}}\leq m<{n_{1}^{1}}\) for all \(\left (\delta ,\beta \right ) \) where δ≥δ ∗ and \(\beta <\beta _{r=1}^{\ast }(\delta )\) .

We start the analysis with an exponential consumer. It is a dominant strategy for the consumer to pay off as much of his borrowing in the grace period as possible. Therefore, if \(~{n_{1}^{t}}\leq m,\) then \({p_{2}^{t}}={n_{1}^{t}}\) and \({p_{3}^{t}}={n_{2}^{t}}\) ; if \(\ {n_{1}^{t}}>m,\) then \({p_{2}^{t}}=m\) and \({p_{3}^{t}}=\left ({n_{1}^{t}}-m\right ) \left (1+r\right ) +{n_{2}^{t}}\) . As a result, the consumer’s utility function is as follows:

If \({n_{1}^{t}}\leq m,\) then \(U_{{n_{1}^{t}}\leq m}=\delta \left [u\left (m+{n_{1}^{t}}\right )+\delta u\left (m-{n_{1}^{t}}+{n_{2}^{t}}\right )+\delta ^{2}u\left (m-{n_{2}^{t}}\right )\right ]\) .

If \({n_{1}^{t}}>m,\) then \(U_{{n_{1}^{t}}>m}=\delta \left [u\left (m+{n_{1}^{t}}\right )+\delta u\left ({n_{2}^{t}}\right )+\delta ^{2}u\left (m-\left ({n_{1}^{t}}-m\right ) \left (1+r\right ) -{n_{2}^{t}}\right )\right ]\) .

The solutions to these two utility functions show that

\(\max U_{{n_{1}^{0}}\leq m}\geq \max U_{{n_{1}^{0}}>m}\) for δ ≥ δ ∗ , irrespective of the interest rate, and

\(\max U_{{n_{1}^{0}}\leq m}\leq \max U_{{n_{1}^{0}}>m}\) for \(\delta <\delta _{r=1}^{\ast }\) .

Completing the proof demonstrates that \(\delta _{r=1}^{\ast }<\delta ^{\ast }\) .

For \(U_{{n_{1}^{0}}\leq m}\) , the consumer’s maximization problem at time zero is given by:

The Lagrangian is given by

The set of FOCs is as follows:

If \({n_{1}^{0}}=m,\) then λ 1 ≥0 and λ 2 =0. If \( {n_{2}^{0}}=0,\) then Eq. 5 creates a contradiction; therefore \({n_{2}^{0}}>0\) and λ 3 =0. Then, by Eqs. 4 and 5 ,

There is a δ ∗ such that Eqs. 6 and 7 both hold for δ ≤ δ ∗ where δ ∗ is determined by \(u^{\prime }(2m)=\delta u^{\prime }\left ({n_{2}^{0}}\right )\) and \(u^{\prime }\left ({n_{2}^{0}}\right )=\delta u^{\prime }\left (m-{n_{2}^{0}}\right )\) . However, Eqs. 6 and 7 cannot simultaneously hold for δ > δ ∗ . Hence, \({n_{1}^{0}}\leq m\) for δ ≥ δ ∗ .

For \(U_{{n_{1}^{0}}>m}\) , the consumer’s maximization problem at time zero is given by:

If \({n_{1}^{0}}=m,\) then λ 1 ≥0 and λ 2 =0. If \( {n_{2}^{0}}=0,\) then Eq. 10 cannot hold; therefore \({n_{2}^{0}}>0\) and λ 3 =0. Then, by Eqs. 9 and 10 :

For a given r , there is a \(\delta _{r}^{\ast }\) such that Eqs. 11 and 12 both hold for \(\delta \geq \delta _{r}^{\ast }\) where \(\delta _{r}^{\ast }\) is determined by \(u^{\prime }(2m)=\delta (1+r)u^{\prime }\left (m-{n_{2}^{0}}\right )\) and \(u^{\prime }\left ({n_{2}^{0}}\right )=\delta u^{\prime }\left (m-{n_{2}^{0}}\right )\) . However, Eqs. 11 and 12 cannot simultaneously hold for \(\delta <\delta _{r}^{\ast }\) . Thus, \({n_{1}^{0}}>m\) for \(\delta <\delta _{r}^{\ast }\) .

Therefore, \(\delta _{r}^{\ast }\) decreases with r and \(\delta _{r=1}^{\ast }<\delta _{r}^{\ast }<\delta _{r=0}^{\ast }=\delta ^{\ast }\) . In summary, the consumer borrows less than or equal to m if δ ≥ δ ∗ and more than m if \(\delta <\delta _{r}^{\ast }\) .

An exponential consumer with δ ≥ δ ∗ correctly believes that he will borrow less than his income. A naive hyperbolic consumer with an exponential discount factor δ ≥ δ ∗ possesses the exact same belief \(\left ({n_{1}^{0}}\leq m\right )\) , but his belief might not be correct, depending on his hyperbolic discount factor. I now analyze the first-period-self of the naive hyperbolic consumer with δ ≥ δ ∗ . As before, the consumer’s utility function is as follows:

If \({n_{1}^{t}}\leq m,\) then \(U_{{n_{1}^{t}}\leq m}=u\left (m+{n_{1}^{t}}\right )+\beta \delta u\left (m-{n_{1}^{t}}+{n_{2}^{t}}\right )+\beta \delta ^{2}u\left (m-{n_{2}^{t}}\right )\) .

If \({n_{1}^{t}}>m,\) then \(U_{{n_{1}^{t}}>m}=u\left (m+{n_{1}^{t}}\right )+\beta \delta u\left ({n_{2}^{t}}\right )+\beta \delta ^{2}u(m-\left ({n_{1}^{t}}-m\right ) \left (1+r\right ) -{n_{2}^{t}})]\) .

I follow similar steps as before and solve these two utility functions separately and show that \(\max U_{{n_{1}^{1}}>m}\geq \max U_{{n_{1}^{1}}\leq m}\) for all \(\left (\delta ,\beta \right ) \) where δ ≥ δ ∗ and \(~\beta <\beta _{r=1}^{\ast \ast }(\delta )\) .

For \(U_{{n_{1}^{1}}\leq m}\) , the consumer’s maximization problem at time one is as follows:

For δ > δ ∗ and β =1, the constraints are not binding, and hence λ 1 , λ 2 , λ 3 =0. Moreover, one can show that \(\frac {\partial {n_{1}^{1}}}{\partial \beta }<0\) and that there is a β ∗ ( δ ) such that the constraint ( \({n_{1}^{1}}\leq m \) ) is binding for β < β ∗ ( δ ).

For \(U_{{n_{1}^{1}}>m}\) , the consumer’s maximization problem at time one is as follows:

For δ > δ ∗ , r =1, and β =1, the constraint \(\left (-{n_{1}^{1}}\leq -m\right )\) is binding, and it can also be shown that \(\frac { \partial {n_{1}^{1}}}{\partial \beta }<0\) . Moreover, there is a \(\beta _{r=1}^{\ast }(\delta )\) such that the constraint \(\left (-{n_{1}^{1}}\leq -m\right )\) is not binding for \(\beta <\beta _{r=1}^{\ast }(\delta )\) . Footnote 23

Thus, I conclude that the contracting-period self believes that he will borrow less than or equal to m in the future consumption periods, but the period-one self ends up borrowing more than m for all ( δ , β ) where δ ≥ δ ∗ and \(\beta <\beta _{r=1}^{\ast }(\delta )\) .

In Fig.  2 , I demonstrate how a naive hyperbolic consumer’s period-one self might end up borrowing more than his income even though his contracting-period self plans not to. The x-axis shows the δ discount factor in [0,1]. The y-axis shows the β hyperbolic discount factor in [0,1]. The contracting-period self believes that the β discount factor does not affect his future consumption plans. If the parameter values that define the consumer are in region A 1 or A 2 , then the consumer believes that he will not accumulate interest-bearing debt. If the parameter values are in region C , the consumer believes that he will accumulate interest-bearing debt and pay interest even if the interest is at the highest possible rate. On the other hand, the period-one self takes β into account when deciding how much to borrow. For the period-one self, the interest rate ( r ) and the exponential discount factor ( δ ) are no longer the only determinants of borrowing; the hyperbolic discount factor ( β ) plays a role as well. Therefore, the vertical line at \(\delta _{r=1}^{\ast }\) separating the interest payers from convenience users at the highest interest rate transforms into the downward sloped line in the diagram. If the consumer is in region A 2 , B 1 , or C , his period-one self accumulates interest-bearing debt even at the highest interest rate. However, there is a conflict between what the contracting-period self believes and what the period-one self ends up doing if the parameter values are in region A 2 , irrespective of the interest rate. Proposition 1 shows the existence of consumers in this region. I analyze only the consumers in region A 2 throughout the paper. Although the contracting-period self is unresponsive to interest rates, the period-one self ends up paying interest.

β and δ cutoffs for a naive hyperbolic consumer’s borrowing

1.2 A.2 Contract choice depending on credit limits

In summary, if the offered credit limits are l 1 and l 2 :

if \(\max \left \{ l_{1},l_{2}\right \} \geq n^{0}>\min \{l_{1},l_{2}\}\) , then the consumer accepts the contract with the higher credit limit only.

If \(\min \left \{ l_{1},l_{2}\right \} \geq n^{0}\) , then the consumer accepts one contract randomly.

If \(\max \left \{ l_{1},l_{2}\right \} <n^{0}\) , then the consumer accepts both contracts.

1.3 A.3 Default consideration according to different period selves

Suppose that l 1 and l 2 are in the appropriate ranges, such that the consumer chooses only one contract. Then, the unselected contract’s interest rate does not affect the gain from default. The gain from default is G 0 (l 1 ,l 2 ) with \(\frac {\partial G_{0}}{\partial l_{1}}=\frac {\partial G_{0}}{\partial l_{2}}>0,\) according to the contracting-period self, and is G c (l 1 ,r 1 ) with \(\frac { \partial G_{c}}{\partial l_{1}}>0\) and \(\frac {\partial G_{c}}{\partial r_{1}} >0,\) according to the consumption-period selves if the selected contract is (l 1 ,r 1 ). Moreover, G 0 (l 1 ,l 2 )<G c (l 1 ,r 1 ) for low enough values of l 2 and for all r 1 ∈[0,1].

In the contracting period, the consumer’s total utility is

if he plans to default, and

if he does not plan to default.

Therefore, the gain from defaulting according to the contracting-period self is given by \(G_{0}=\frac {1}{\beta \delta ^{3}}\left (U_{t=0,~d=-1}-U_{t=0,~d=0}\right ) \) and \(\frac {\partial G_{0}}{\partial l_{1} }=\frac {\partial G_{0}}{\partial l_{2}}>0\) (adjusted according to the last period self).

When the consumer reaches the first period with the chosen contract (which is ( l 1 , r 1 )), he realizes that his actual debt is more than his income. Therefore, the consumer’s total utility is as follows:

Therefore, the gain from defaulting planned in the first period is

\(G_{1}=\frac {1}{\beta \delta ^{2}}\left (U_{t=1,~d=-1}-U_{t=1,~d=0}\right ) \) .

When the consumer reaches the second period, the total utility is

Therefore, the gain from defaulting planned in the second period is \(G_{2}= \frac {1}{\beta \delta }\left (U_{t=2,~d=-1}-U_{t=2,~d=0}\right ) \) . As a result, the gain from defaulting according to the consumption period selves is \(G_{c}=\max \{G_{1}(l_{1},r),G_{2}(l_{1},r)\}\) with \(\frac {\partial G_{c} }{\partial l_{1}}>0\) and \(\frac {\partial G_{c}}{\partial r_{1}}>0\) .

If l 2 =0 and β =1; then U t =0, d =−1 and U t =1, d =−1 are equal, and consequently any difference between G 0 and G 1 is due to the difference between U t =0, d =0 and U t =1, d =0 . From Proposition 1, the profit maximizing n 1 is less than m for δ > δ ∗ . Therefore, U t =0, d =0 is greater than U t =1, d =0 for δ > δ ∗ . Consequently, G 0 < G 1 for β =1. Additionally, \(\frac {\partial G_{1}}{\partial \beta }=-\frac { \left [ u(m+n_{1}^{1\ast })-u(m+n_{1}^{1\ast \ast })\right ] }{\left (\beta \delta \right )^{2}}<0\) such that \(n_{1}^{1\ast }\) and \(n_{1}^{1\ast \ast }\) represent the profit maximizing n 1 in the case of planning and not planning default, respectively. As a result, G 0 ( l 1 , l 2 =0)< G 1 ( l 1 , r 1 )≤ G c ( l 1 , r 1 ) for all r 1 ∈[0,1].

The reasoning behind this lemma is as follows. The contracting-period self believes that he will not pay interest; therefore, the gain from defaulting is not affected by interest rates. Moreover, a marginal gain from each credit limit increase is the same because the credit limits always appear as a sum in the gain function. Consumption-period selves, on the other hand, realize that they pay interest; therefore, the gain from defaulting increases with the interest rate of the chosen contract. The credit limit offered by the contract that is not chosen does not affect the gain because the contracting period has already passed, and the consumer has only the chosen contract on hand. As a result, the set of credit limits that do not induce the consumer to default are shown by the shaded area in Fig.  3 . Footnote 24

The no-default-region if only one contract is chosen

If I further investigate the gain functions from different selves’ point of view, I find that two different effects determine the gain. One is “the credit effect” and the other is “the spending effect”. The credit effect is the effect of the potential credit limit to default at; it may be larger in the contracting period than in the consumption period depending on other companies’ offers. The spending effect is the effect of the expenditure level, and it is larger in the consumption periods since the consumer spends more than what he had planned earlier. Depending on the contract terms, the credit effect might offset the spending effect or vice versa. For example, G c ( l i , r ) is the restrictive constraint for the lower values of l j (the credit effect is smaller). However, if the consumer chooses both contracts, then the credit effect is the same in all periods (the potential credit limit to default at is constant at l 1 + l 2 ), although the spending effect is greater in the consumption periods. Therefore, the gain from defaulting is determined according to the consumption-period selves, namely, G c ( l 1 , l 2 , r ).

Suppose that l 1 and l 2 are in the appropriate ranges such that the consumer chooses both contracts, and suppose that r 1 ≤r 2 . Then, the gain from defaulting is G c (l 1 ,l 2 ,r 1 ), with \(\frac {\partial G_{c}}{\partial l_{1}}, \frac {\partial G_{c}}{\partial l_{2}},\frac {\partial G_{c}}{\partial r_{1}} >0 \) and \(\frac {\partial G_{c}(l_{1},l_{2},r_{1}=0)}{\partial l_{1}}=\frac { \partial G_{e}(l_{1},l_{2},r_{1}=0)}{\partial l_{2}}>0\) .

The gain from defaulting according to the contracting-period self is the same as before, namely, G 0 ( l 1 , l 2 ). But, the gain from defaulting according to the consumption-period selves depends on the credit limit offers of both companies instead of only one, namely, G c ( l 1 , l 2 , r 1 ) if the lower interest rate contract is ( l 1 , r 1 ). The consumer pays the contract with the higher interest rate fully to minimize the cost of borrowing, which eliminates the higher interest rate from the gain function.

By using the same arguments as in the previous proof: \(G_{0}(l_{1},l_{2})<G_{c}(l_{1},l_{2},r_{1}),\frac {\partial G_{c}(l_{1},l_{2},r_{1})}{\partial r_{1}}>0,\) and \(\frac {\partial G_{c}(l_{1},l_{2},r_{1}=0)}{\partial l_{1}}=\frac {\partial G_{c}(l_{1},l_{2},r_{1}=0)}{\partial l_{2}}>0\) . □

When the consumer has two contracts on hand, the no-default-region is determined by G c ( l 1 , l 2 , r 1 =0)≤ C . This is shown in Fig.  4 .

The shaded region is the no-default-region if two contracts are chosen

1.4 A.4 Equilibrium results

Proof (proposition 1).

By referring to Fig.  1 , I will first illustrate the number of contracts that the consumer will choose under different contract offers and determine whether the consumer will pay interest on them. Afterward, I will determine the equilibria. If the offered contracts lie in region one, the consumer chooses only one contract and pays the agreed interest on that contract. If the offered contracts are in region two, the consumer chooses only one contract but does not pay interest, because the chosen contract’s credit limit prevents him from spending more. If the contracts are in region three, the consumer chooses both contracts but does not pay interest (even the total credit limit offered by these contracts does not allow the consumer to accumulate interest-bearing debt). If the contracts are in region four, the consumer chooses both contracts and pays interest on the contract with the lower interest rate (the total credit limit allows the consumer to accumulate interest-bearing debt and the first-period self pays the contract with higher interest rate within the grace period).

Each company offers a monopoly contract ( l ∗ , r ∗ ) if there is no risk of default. But, when there is risk of default, the companies might not be able to offer the credit limits they want without triggering default. For convenience, from this point on, I analyze the problem from the first company’s point of view. The second company’s problem is similar. The default constraints are G c ( l 1 , r 1 )≤ C and G 0 ( l 1 , l 2 )≤ C assuming that only one contract is chosen in equilibrium. Additionally, G 0 ( l 1 , l 2 )≤ G c ( l 1 , r 1 )≤ C for lower values of l 2 . Suppose that \((l_{1}^{\ast \ast },r_{1}^{\ast \ast })\) is the first company’s profit-maximizing contract offer when the other company offers a zero credit limit. Let \(l_{2}^{\prime } \) be the solution to \(G_{0}(l_{1}^{\ast },l_{2})=C\) and \(r_{2}^{\prime }= \underset {G_{c}(l_{2},r_{2})\leq C}{\arg \max }(l_{2}^{\prime }-m)r_{2}\) . Each company’s contract offer is affected by the other company’s offer only if G 0 ( l 1 , l 2 )= C .

Let C ∗ = G c ( l 1 = m , r 1 =0) and C ∗∗ = G c ( l 1 = n 0 , r 1 =0).

Consider the case C > C ∗ ; I now demonstrate that a company offers a large enough credit limit with a positive interest rate when the other company offers a zero credit limit. If C > C ∗ , then G c ( l ∗ , r ∗ )≤ C might or might not be satisfied. If \( G_{c}(l_{1}^{\ast }=l^{\ast },r_{1}^{\ast }=r^{\ast })\leq C,\) then the monopoly contract is feasible. Otherwise, the company offers \(l_{1}^{\ast }>m \) and \(r_{1}^{\ast }>0\) as the profit-maximizing contract with \( G_{c}(l_{1}^{\ast },r_{1}^{\ast })=C\) . The argument is as follows. Suppose that \(l_{1}^{\ast }=m\) and \(r_{1}^{\ast }=0\) . The company can obtain a higher profit by slightly increasing l 1 and r 1 because the constraints C > G c ( l 1 = m , r 1 =0)> G 0 ( l 1 = m , l 2 =0) are not binding for \(l_{1}^{\ast }=m\) and \(r_{1}^{\ast }=0\) . Suppose that \( l_{1}^{\ast }=m\) and \(r_{1}^{\ast }>0\) . The company can increase its profit by slightly decreasing the interest rate and slightly increasing the credit limit. Last, suppose that \(l_{1}^{\ast }>m\) and \(r_{1}^{\ast }=0\) . The company can increase its profit by slightly decreasing the credit limit and slightly increasing the interest rate. I now show the existence of different equilibria depending on where \(G_{0}(l_{1}^{\ast },l_{2})\leq C\) starts to bind.

If \(l_{2}^{\prime }\geq l_{1}^{\ast },\) then \((l_{1}^{\ast \ast },r_{1}^{\ast \ast })\) and \((l_{2}^{\ast }=l_{1}^{\ast \ast },r_{2}^{\ast }=r_{1}^{\ast \ast })\) is the unique symmetric equilibrium in region 1. This is because \(G_{0}(l_{1}^{\ast \ast },l_{2}^{\ast })<C,\) and each company offers the profit-maximizing contract without triggering default.

If \(m<l_{2}^{\prime }<l_{1}^{\ast },\) then \((l_{1}^{\ast \ast },r_{1}^{\ast \ast })\) and \((l_{2}^{\ast },r_{2}^{\ast })\) is an equilibrium such that \(m<l_{2}^{\ast }\leq l_{2}^{\prime }\) and \(r_{2}^{\ast }>0\) . This is because the second company cannot offer more than \(l_{2}^{\prime }\) and does not have an incentive to offer less than m . Moreover, the second company can offer a positive interest rate without triggering default, as \( G_{0}(l_{1}^{\ast \ast },l_{2}^{\ast })\leq C\) and \(G_{c}(l_{2}^{\ast },r_{1}^{\ast })<G_{c}(l_{1}^{\ast \ast },r_{1}^{\ast })\leq C\) .

If \(l_{2}^{\prime }\leq m,\) then \((l_{1}^{\ast \ast },r_{1}^{\ast \ast })\) and \((l_{2}^{\ast },r_{2}^{\ast })\) is an equilibrium such that \( l_{2}^{\ast }\leq l_{2}^{\prime }\) and \(r_{2}^{\ast }\geq 0\) . This is because the second company cannot offer more than m , and consequently makes zero profit.

If C ∗∗ ≤ C ≤ C ∗ . \((l_{1}^{\ast },r_{1}^{\ast })\) and \((l_{2}^{\ast },r_{2}^{\ast })\) are then an equilibrium where \(0\leq l_{1}^{\ast }\leq l_{1}^{\ast \ast },r_{1}^{\ast }\geq 0,\) and \(0\leq l_{2}^{\ast }\leq l_{2}^{\prime },r_{2}^{\ast }\geq 0\) . The companies cannot offer more than m without triggering default, and consequently there is no profitable deviation (region 2 or region 4). Companies make zero profits with or without competition on interest rates. In region 4, the equilibrium contracts are zero-interest contracts, because the consumer accepts two contracts in this region and pays the higher interest rate within the grace period.

If \(C\leq C^{\ast \ast },(l_{1}^{\ast },r_{1}^{\ast })\) and \( (l_{2}^{\ast },r_{2}^{\ast })\) are then an equilibrium where \(l_{1}^{\ast }+l_{2}^{\ast }\leq n^{0},r_{1}^{\ast }\geq 0,\) and \(r_{2}^{\ast }\geq 0\) . There is no profitable deviation, as the total credit limit to be offered to the consumer without triggering default is not more than the consumer’s income (region 3).

Proof (Proposition 2)

The naive hyperbolic consumer’s contracting-period self does not plan to borrow more than his income at any consumption period, but his period-one self ends up borrowing. The exponential consumer, on the other hand, correctly believes that he will not borrow on the credit card. But, in equilibrium, noncompetitive interest rates can be observed depending on the default risk. This follows from the fact that neither the exponential consumer nor the naive consumer’s contracting-period self are responsive to interest rates. Being unresponsive to interest rates neither hurts the exponential consumer nor benefits the companies since the consumer does not borrow anyway and consequently there are no profits from lending. But, when it comes to naive hyperbolic consumers, being unresponsive to interest rates hurts the consumer and benefits the companies as showed in 1. With probability p , the consumer is naive hyperbolic, and the company earns profit with the of probability \(\frac {p}{2}\) by offering a positive interest rate and a high enough credit limit. With probability 1− p , the consumer is exponential, and the company does not earn a profit from him independent of the interest rate. As a result, the companies do not have an incentive to compete on the interest rates. □

Proof (Proposition 3)

If companies offer positive interest rates for one-period-loans, then the company with the higher interest for the one-period loan can profitably deviate by offering a smaller interest for the one-period-loan. Therefore, no equilibrium contract will have a positive one-period-loan interest. The consumer is not responsive to interest rates for two-period loans, the companies can safely offer a monopoly interest rate for two-period loans if the default risk is low enough. □

Proof (Proposition 4)

According to both the contracting-period self and the period-one self, the consumer’s utility when he reaches the period-two is given by

\(u(m-n_{1}+n_{2})+\widehat {\beta }u(m-n_{2})\) if n 1 ≤ m , and

\(u(n_{2})+\widehat {\beta }u(m-(n_{1}-m)r-n_{2})\) if n 1 > m .

I denote the optimal period-two borrowing as \(n_{2}^{\ast }(n_{1}, \widehat {\beta })\) .

According to the period-one self, the consumer’s utility when he reaches this period is:

\(u(m+{n_{1}^{1}})+\beta \left [ u(m-{n_{1}^{1}}+n_{2}^{\ast }({n_{1}^{1}}, \widehat {\beta }))+u(m-n_{2}^{\ast }({n_{1}^{1}},\widehat {\beta }))\right ] \) if \({n_{1}^{1}}\leq m,\) and

\(u(m+{n_{1}^{1}})+\beta \left [ u(n_{2}^{\ast }({n_{1}^{1}},\widehat {\beta } ))+u(m-({n_{1}^{1}}-m)(1+r)-n_{2}^{\ast }({n_{1}^{1}},\widehat {\beta }))\right ] \) if \({n_{1}^{1}}>m\) .

I denote the optimal period-one borrowing according to the period-one self as \(n_{1}^{1\ast }(\beta ,\widehat {\beta })\) .

But, according to the contracting-period self, the consumer’s utility when he reaches the first period is:

\(u(m+{n_{1}^{0}})+\widehat {\beta }\left [ u(m-{n_{1}^{0}}+n_{2}^{\ast }({n_{1}^{0}},\widehat {\beta }))+u(m-n_{2}^{\ast }({n_{1}^{0}},\widehat {\beta })) \right ] \) if \({n_{1}^{0}}>m,\) and

\(u(m+{n_{1}^{0}})+\widehat {\beta }\left [ u(n_{2}^{\ast }({n_{1}^{0}}, \widehat {\beta }))+u(m-({n_{1}^{0}}-m)(1+r)-n_{2}^{\ast }({n_{1}^{0}},\widehat { \beta }))\right ] \) if \({n_{1}^{0}}\leq m\) .

I denote the optimal period-one borrowing according to the contracting-period self as \(n_{1}^{0\ast }(\widehat {\beta })\) .

Let us consider the case when \(\beta =\widehat {\beta }\) (sophisticated hyperbolic consumer). If \(\widehat {\beta }=1,\) then \({n_{1}^{0}}={n_{1}^{1}}=0\) . As \(\widehat {\beta }\) decreases, \({n_{1}^{0}}={n_{1}^{1}}\) increase and there is a cutoff β ∗ such that the \({n_{1}^{0}}={n_{1}^{1}}\leq m\) for \( \widehat {\beta }\geq \beta ^{\ast }\) and \({n_{1}^{0}}={n_{1}^{1}}>m\) for \( \widehat {\beta }<\beta ^{\ast }\) . Now, \(\widehat {\beta }\geq \beta ^{\ast }\) stays the same but β decreases starting from \(\widehat {\beta }\) . Decreasing β does not affect the optimal period-one borrowing according to the contracting-period self, but does increase the optimal period-one borrowing according to the period-one self. As \(\beta \rightarrow 0, {n_{1}^{1}}\rightarrow \infty \) . Therefore, there is a \((\beta _{r=1}^{\ast \ast }(\widehat {\beta }),\beta ^{\ast })\) such that \( {n_{1}^{0}}\leq m\) and \({n_{1}^{1}}>m\) for \(\beta <\beta _{r=1}^{\ast \ast }(\widehat {\beta })\) and \(\widehat {\beta }>\beta ^{\ast }\) . This means that there are partially hyperbolic consumers with a contracting-period self who believes he will not borrow more than his income and with a period-one self who ends up borrowing more than his income even at the highest interest rate of r =1. □

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Incekara-Hafalir, E. Credit Card Competition and Naive Hyperbolic Consumers. J Financ Serv Res 47 , 153–175 (2015). https://doi.org/10.1007/s10693-014-0208-4

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Received : 01 May 2012

Revised : 01 October 2014

Accepted : 08 October 2014

Published : 14 November 2014

Issue Date : April 2015

DOI : https://doi.org/10.1007/s10693-014-0208-4

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Case study: a perfect competition.

• a large number of small firms,
• identical products sold by all firms,
• perfect resource mobility or the freedom of entry into and exit out of the industry, and
• perfect knowledge of prices and technology.
• Market Structure
• Perfect Competition
• Equilibrium of the Firm
• Short Run Equilibrium of the Price Taker Firm
• Short Run Supply Curve of a Price Taker Firm
• Short Run Supply Curve of the Industry
• Long Run Equilibrium of the Price Taker Firm
• Long Run Supply Curve For the Industry
• Price Determination Under Perfect Competition
• Market Price
• Determination of Short Run Normal Price
• Long Run Normal Price and the Adjustment of Market Price to the Long Run Normal Price
• Distinction between Market Price and Normal Price
• Interdependent Prices
• Joint Supply
• Fixation of Railway Rates
• Composite or Rival Demand

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Monday, august 31, 2015, case study: perfect competition in credit card industry .

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14 comments:

#15BAL120# Credit cards facilitates the buyer(consumer) in purchasing things, even though he/she has no balance in the account up to certain limit. However,this facility may draw him/her to spend in purchasing unnecessary things and makes him/her extravagant. He/she has to pay the debt and recoup his/her account. Isn't it against the saving habits of Indian society and the accepted norms that one should spend according to his needs only?

Credit cards do more harm than good to the consumer as a whole. The credit card industry has caused more people to go into debt than any other industry. They have been the source of more heartache, grief, and deceit than any other industry save perhaps the collection companies. However, credit card companies have their own collections departments so they still fall into that category. the one gets in a habit of purchasing unnecessary

Yes credit card company has power to influence people by the offer that without having money you can buy. It leads to people stop thinking rationally and spend money without caring about their budget.

I agree that Credit card industry largely resembles the perfectly competitive market. India has also entered this almost-perfect market. We have launched our own card scheme called RuPay with the help of National Payments Corporation of India (NPCI).

As we do today discussed in the class that what do we mean my large when it comes to large number of buyers and sellers in a perfect competition. So in my opinion ,large number implies that each firm is very small in comparison to the total market and if one firm has to become significantly large it would dominate the market and competition would be eliminated or at least diminished.

Credit cards market is flourishing one reason is there flexibility in dealing with the worldwide transactions whether you have money in your account or not you can fulfill your desires to a great extent ,but this laxity is also because it is actually helping the credit card firms to prosper with huge usage. Hence we can say both the market and buyers of the desired product compliment each other to let the. Flow of economy to grow.

Credit card market resembles a perfectly competitive market.But question arises whether the banks who are providing these cards also thinks that they are in a perfectly competitive market? Because in these market the advertising of products is just a waste because it will not influence the market instead of it we can see a lot of advertisement and promotions they do to influence the market by influencing the mindset of the buyers,in case of India these promotions obviously works because here more you highlight your product,more the chances of your product will increase .So its true that credit card is a perfectly competitive market but up to what level if not in the world but what in India??

Credit card companies are the example of perfectly competitive market as no single buyer or seller influences the market. There are many sellers and buyers are even greater in number. Because of Advertising, certain company might get an edge over the other co. but it doesn't completely throw out the other co. in the credit industry.Advertising doesnt create a monopolistic situation so credit industries will still be called the perfectly competitive market even if certain co. Are advertising their cards. 15bal105

Credit card companies can easily allure the consumers to purchase unnecessary items. The facility to buy a good without worrying about the cash tends to hamper the prudency of a person. This ultimately results in making a purchase beyond the budget. The credit card scheme has an extremely high tendency to bring down the savings of a person. 15BAL099

as you have mentioned that this is a perfect competition,why cant this be a oligopoly market? 1) as we know that the major market share of the credit card market is shared by few sellers and major thing is 2) the product can be differentiated by the payment gateway they were offered like master card or visa card. 3)considering the natural barrier the company needs to have high capital to enter in to the credit card market.

If you are also one of the customers who are in search for americanexpress.comconfirmcard or American express verify card by an online process, then here is the post of americanexpress .com /confirmcard Americanexpress.com/confirmcard is the official website from American Express for their customers to confirm their credit cards. American Express Confirm Card americanexpress.comconfirmcard

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