Strategic Implications of Competing For Consumers with Time - - PDF document

Strategic Implications of Competing For Consumers with Time Inconsistent Preferences∗

Alexei Alexandrov† University of Rochester‡ March 13, 2009

Abstract I examine oligopolistic competition for time inconsistent consumers. The two cases of in- vestment (health clubs) and leisure goods (credit cards) have different implications for strategy. For leisure goods the firms offer introductory rates at the fully rational consumer level, but consumers end up paying higher fixed rates later. In the limit, the markups go to zero. For investment goods there is a non-trivial cutoff of consumer naivete above which the market equi- librium is as if the consumers are rational. Below the cutoff the firms offer schedules such that consumers pay the membership fee, but do not attend.

1 Introduction

Behavioral literature has grown considerably during the past ten years1. Yet, one of more interesting questions about not fully rational behavior – how does it affect firm competition and market structure – has not received enough attention. I take one of the least controversial assumptions about not fully rational consumer behavior, quasilinear hyperbolic discounting, and examine firms’ incentives and market outcomes in oligopolistic competition for time-inconsistent consumers. I examine two settings - goods with immediate benefits and later payments (credit cards) and goods with immediate payments and postponed benefits (gyms). The two applications that I am interested in the most in the first setting are the payday lending and the credit card industries2. In the second setting my work applies to personal-investment industries such as health clubs and education.

∗Keywords: time inconsistent consumers, imperfect competition, product differentiation, credit cards. JEL Codes:

D03, D14, G21, L13.

†Thanks to be added. ‡Assistant Professor of Economics and Management, email: Alexei.Alexandrov@Simon.Rochester.edu 1See Camerer (2008) for a summary of the advances. See Ellison (2006) for the summary of the advances in

Industrial Organization literature in particular. This paper fills Ellison’s ”Hyperbolic Discounting”-”Static Oligopoly” box with not trivial results. See Hendricks (2006) for a discussion.

2According to a review of the payday lending industry by Stegman (2007), ”Most reforms of payday lending

revolve around efforts to reduce serial borrowing”.

1

In the first setting (credit cards) I find that with nave consumers (the ones that do not realize they are time-inconsistent), firms offer an introductory rate which they would have offered to the fully rational discounting consumers, and a much higher fixed rate. The consumers sign up believing that they will pay off during the introductory period, but end up paying off the fixed rate in the last period. Increased competition makes the difference between the rates smaller, and in the limit it goes to zero - consumers pay marginal cost, despite being effectively locked up for the last period. If some of the consumers are sophisticated (realize that they are time inconsistent), than firms lower both rates, with the introductory rate approaching the fully rational non-discounting con- sumer rate, and the fixed rate approaching the introductory rate with nave consumers. In the limit, as all of the consumers are sophisticated, they pay the fixed rate which is the same as the fully rational discounting consumers would have paid. The second setting is the one where consumers have to pay before they get the benefit (for example gyms). Firms announce contracts, consumers first sign up for gym membership, then given the per-visit rate decide whether to visit the gym, and if they do visit, they get the benefit (health) the period after. The results are different in this case. If the consumers are sufficiently rational, the equilibrium prices are as in the fully rational equilibrium, with consumers signing up and attending in the next period – imperfect competition completely mitigates the effects of time-inconsistent behavior, even with just nave consumers in the market. However, if once the consumer gets to the attendance decision, and his discounted benefits of attendance are less than firms’ marginal cost, then the firms play a deceiving equilibrium - consumers pay the membership fee, but do not attend. Adding some sophisticated consumers to the mix does not alter the threshold above. However, below the threshold there is a region where a mixed equilibrium is played - some firms catering to sophisticated consumers and others catering to nave. The qualitative differences between the two regimes are due to the situation in the second period when time-inconsistency kicks in and the consumer chooses between either paying now or later (credit cards) or attending or not (gyms). In the case of credit cards, it is always best for the firms to push it to later – consumers are willing to pay a lot for that option (at that point in time), and the firms’ marginal costs are much smaller than the consumers’ willingness to pay for this postponement. In case of the gyms, if the firms entice the consumers not to attend, the firms do not have to pay the marginal cost, however they lose the value of the discounted benefits which they could have extracted from consumers, and sometimes the firms are better off making sure that the consumer actually attends. I assume that firms are symmetric, and that consumers have random i.i.d. preferences for each

• f the firms (Perloff and Salop (1985)). I assume that these random preferences are realized in the

first period, and thus the consumers are homogenous by the time time-inconsistent behavior kicks

• in. This results in an unattractive feature of having either all consumers attend the gym or none
• f them. This is a trade-off for the generalized random preferences structure. In the appendix, I

use the standard Salop’s circle (1979) to show that even when the consumers are not homogenous 2

at that point, the results still go through. An added complexity is an additional Salop-style kink in the demand function. Many consumers end up paying more when they sign up for low introductory offers as opposed to signing up for fixed rates. Also, the credit card industry is differentiated – consumers do not necessarily pick the credit card with the lowest interest rate, rather there is a whole list of options which might or might not be there: particular airline miles, particular discount, cash back on a particular set of purchases, a whole menu of fees for cash advances, international transactions, and so on3. Ausubel (1991) empirically examines profits of the credit card companies, and suggests that consumer irrationality might be one of the reasons for abnormally high profits. Payday lenders are product differentiated at least in location, adding frequent advertisements on top of that. Skiba and Tobacman (2008) structurally estimate the demand side of the payday industry, and conclude that the consumers are (at least partially) naive about their time preferences. The health club industry is well-covered in the literature, see DellaVigna and Malmendier (2006). In relation to the recent housing crisis, the same question arises for the negative amortization mortgages, with the infamous catch phrases like 999 dollars per month for a 300,000 dollar house. While this is not exactly the same as the introductory rate in credit cards, many consumers perceive it to be the same, and similar reasoning applies there4. The paper that is the closest to mine is Shui and Ausubel (2005), exaiming switching costs in the credit card industry, with presence of hyperbolic behavior, both theoretically and empirically, and finding that naive consumers like introductory offers because they underestimate what they will own, and sophisticated consumers like introductory offers because it provides them with a commitment device. There is a stream of literature in the optimal contract design with time- inconsistent consumers with perfectly competitive firms (some papers also look at monopolists and/or switching costs). DellaVigna and Malmendier (2004) examine optimal contract structure (for perfect competition) for investment goods (i.e. gyms) and for leisure goods (i.e. credit cards) and find that firms price the first kind below the marginal cost, and the latter above the marginal

• cost. I find that while there is an incentive to do this with imperfect competition, depending on

the degree of competition and differentiation both of these results might not hold in oligopolistic

• competition. Heidhues and K¨
• szegi (2008b) are more interested in the contracts in the credit card

industry in particular, and in the effects of consumer heterogeneity. Gottlieb (2008) introduces non-exclusive contracts, and explains consumer behavior in the industries like alcohol and tobacco, again under assumption of perfect competition. Incekara (2006) allows consumers to hold more than one (two) cards at the same time, with companies potentially competing on the interest rates, and finds that, despite perfect competition between the two firms, they might derive positive profits under some parameter values. Time inconsistent, in particular hyperbolic, behavior was heavily covered by economists lately, starting with the seminal Laibson (1997) article. One of the more recent empirical papers is

3See Chapter 7 of Evans and Schmalensee (2000). 4In this case, the interest rate is the same, but the minimum payment is only 999. If one thinks of it as consumers

being too optimistic, then Manove and Padilla (1999) seems to apply more.

3

DellaVigna and Malmendier (2006), who had examined signing up for gym membership, and found some hyperbolic behavior on the consumer part (signing a contract and not going afterwards). The question that I examine is what do the firms do when consumers are hyperbolic, and what effect does this have on the standard comparative statics. There is already some work bringing behavioral assumptions and industrial organization theory together – see, for example, Heidhues and K¨

• szegi (2008a) looking at the strategic implications
• f competition for loss-averse consumers. Manove and Padilla (1999) examine banks competing

for (too) optimistic entrepreneurs. Their question relates more to the informational issues, and figuring out how a bank can distinguish between a too optimistic and just a good entrepreneur, with the result being that banks are overly conservative. This does not look like the case in either payday lending or credit card markets where (at least until very recently) the firms did not require credit checks for a payday loan, and barely anything to get an introductory credit card. Gabaix and Laibson (2006) examine whether firms have more incentives to reveal all the add-on prices with competition (no). Grubb (2009) looks at firms responding optimally to consumers being

• verconfident in their forecasts, resulting in three-part tariffs even with perfect competition. Al-

Najjar et. al. (2008) take it a step further, and examine firms where managers are subject to sunk-cost fallacies, and show that the market does not necessarily correct this behavior. Oster and Scott-Morton (2005) find (empirically) that firms in the magazine industry might be taking advantage of consumers’ time-inconsistency.

2 Credit Cards

2.1 Model

There are three periods. In period one, consumers draw their loan valuations and decide on which credit card to sign up for. In period two, consumers can pay back the loan. If a consumer does not pay back in period 1, he must pay back in period 2. There is mass 1 of consumers. Consumers are hyperbolic discounters and are naive about it. At any given period they discount the future at the rate of βδτ , where τ ≥ 1 is the period when the money is paid and/or the good is consumed5. For the ease of exposition, I set δ = 1 for the rest of the paper. The naive part is that the consumer does not realize that next period he will also discount the future by β < 1 on top of the standard discounting. For an example of naive discounting, suppose β = .5, and δ = 1. In period zero, the consumers think that a dollar in period 1 is equivalent to .5 × 1 = .50 dollars today, and dollar in period two is equivalent to .5 × 12 = .50 dollars today - meaning a dollar the day after tomorrow is worth as much as the dollar tomorrow. But when the consumer gets to period 2, his hyperbolic behavior kicks in again, and he value a period two dollar at .5 × 1 = .50 period one dollars, β multiplied by the same ratio in period zero. The lower the β,

5While not the only form of hyperbolic discounting, this one has been prevalent in the literature lately. It was

introduced by Phelps and Pollak (1968). See Frederick et.al. (2002) for a review of time discounting in the literature.

4

the farther from rational the consumers are, with β = 1 being fully rational, and β = 0 means that consumers do not see beyond this period. There are N symmetrical firms. The firms’ marginal cost is normalized to zero6. A firm can lend the consumer a dollar at period zero, and can set the rate at rintro for repayment in the next period, and rate rfixed in the period after the next. Consumer can borrow the dollar, and pay back either rintro if they pay back in the first period, or rfixed if they pay back in the second period. From now on, rfixed is the overall interest rate that consumer pays in period 2 (as opposed to the interest on top of the introductory rate). There is no outside option. Each consumer has to sign up for a credit card and borrow. The

• nly choice that the consumers have is when to pay back - in the introductory rate period or in the

final period. A given consumer gets the following utility from firm i: u0 =

• B + tθi − βrintro

if paid back in introductory period B + tθi − βrfixed if paid back in the final period, (1) where B is the constant benefit of getting a loan, θi is how much the consumer likes this particular firm, and t is the intensity of consumers’ preferences. Each consumer draws N θs from a probability distribution function g(•). The draws are independent, and the p.d.f.s do not vary accross brands

• r consumers, so the draws are identically distributed as well.

2.2 Parameter Interpretation

The interpretation in this section is the same as in the rest of the paper. The model has to have three time periods, because this is the lowest number of periods so that time inconsistent consumers can be distinguished from rational consumers. The constant benefits (B) are the common value of getting a loan. The intensity of prefences parameter t corresponds to how much a consumer cares about the firm-specific part of the choice versus the common value part. This t is in many ways a counterpart to the standard Hotelling/Salop travel cost, and if the reader interprets it as such, no intuition will be lost7. Higher β means that the consumers are more rational8. Also, higher β means that consumers discount more, so a fair comparison with a fully rational individual is with a consumer who in the first period discount the future by β. Thus, the fully rational solutions in the propositions (and graphs) below depend on β. The choice in (1) seems clear - the consumer should just choose whichever rate is lower. That’s what the naive consumers believe they will do. However, once they get to the introductory period,

6It is a normalization if the consumers (accounting for hyperbolic behavior and standard discount rate) discount

more than the firms do. See conclusion on more about this issue.

7The preference distribution is from Perloff and Salop (1985). The only difference is that the t in this paper is

denoted by β in their paper.

8Another way to proceed would be to have partially sophisticated consumers - the ones who believe that they

are hyperbolic with the rate β, such that β ∈ (β, 1). I believe that the qualitative results still go through, but this addition would lead to one more variable floating around. In a later section I deal with the case where some of the consumers are fully sophisticated.

5

their time inconsistency kicks in, and thus the choice becomes to pay now and lose rintro or pay later and lose βrfixed. I assume that in case of a tie consumer goes for the later option.

2.3 Solution

Proposition 1 In a symmetric equilibrium with naive consumers the firms offer the following rates: rfixed = t β2M(N) (2a) rintro = βrfixed, (2b) where M(N) = N(N − 1)

• GN−2(θ)g2(θ)dθ.
• Proof. A consumer is going to believe that he is paying off at the inroductory rate, but actually

pays off the fixed rate if βrfixed ≤ rintro ≤ rfixed. Therefore, if the firms play an equilibrium where this type of behavior happens, they compete on the introductory rates, but derive the profits from the fixed rates. Thus, the fixed rates are going to be as high as possible given the introductory rates, or rfixed = rintro β . (3) By deviating from this scheme, a firm gains no new consumers (since they do not believe they’ll pay the fixed rates) and loses markup. It also would not make more money by deviating to an equilibrium where consumers pay back in the first period (by making rfixed a little higher). Thus, knowing that the consumers care about the introductory rate, but actually pay back the fixed rate in the last period, a consumer signs up for credit card i iff, ui > uj for all j = i. Or, tθi + B − βri−intro > tθj + B − βrj−intro (4a) θj < θi + Fj − Fi t + β(rj−intro − ri−intro) t . (4b) The probability of this event for a given j is G(θi + β(rj−intro−ri−intro)

t

). Then firm i’s demand is Qi =

j=i

G(θi + β(rj−intro − ri−intro) t )g(θi)dθi, (5) and the profit is rfixedQi since the consumers pay back in the last period. Assuming symmetric strategies, the profit equation becomes Πi = ri−fixed

• GN−1(θi + β(rintro − ri−intro)

t )g(θi)dθi. (6) 6

Rate β Last period rate Intro rate 1

t M(N)

Figure 1: Equilibrium interest rates for time inconsistent consumers. From (3), plug in rfixed = rintro

β

, and the FOC is: Qi − ri−intro β

• GN−2(θi + β(rintro − ri−intro)

t )g2(θi)dθi = 0, (7) solving for ri−intro, and since in a symmetric equilibrium Qi = 1

N for all i, we get the result in the

• proposition. For the second order conditions and the conditions under which M(N) is increasing in

N, the interested reader is encouraged to look at Perloff and Salop(1985) - the article’s appendix for M(N) derivations, and endnote 6 for second order conditions. Value M(N) is exactly the same as in Perloff and Salop (1985) and it represents a combination

• f the particular functional form of the probability distribution function of consumer valuations

and the number of firms in the market. If G(•) is log-concave, then M(N) increases in N (see Weyl (2009) and Gabaix et. al. (2009) for more detailed results about the Perloff and Salop (1985) framework). Corollary 1 Consumers believe that they will pay back in the introductory period, but actually pay back in the last period. Corollary 2 Each firm’s profit is Π∗ = t β2NM(N). (8) Both markup and profit are increasing in preference intensity, and decreasing in the number of firms (assuming log-concavity of the complement of the c.d.f.) - standard oligopoly results. As firms 7

become barely differentiated, or very close to perfect competition (t → 0), the markups and profits

• disappear. The more interesting result, is that the closer to rational the consumers are (bigger β),

the less profits the firms derive. The figure shows the equilibrium rates given consumers’ rationality. Consumers believe they pay the bottom rate, but actually pay the top one. As consumers become more rational, both rates converge to the rate in the fully rational equilibrium. With positive marginal costs, it is possible that the firms offer an introductory rate that is lower than the fully rational rate.

2.4 Sophisticated Consumers

Assume that some (σ < 1) of the consumers realize that they are hyperbolic - they know that in period one, they will discount period two at βδ, and therefore they avoid the trap of initially low interest rates. For the discussion above, this means that these consumers know they’ll end up agreeing to an interest rate of rfixed = rintro

β

, which takes advantage of their less than perfect rationality. Since the sophisticated consumers know that they will choose not to repay once they get to the next period, they only care about the last period interest rate, the opposite of the naive consumers. This means that firms can either ignore the sophisticated consumers, or compete on the first period interest rates for naive consumers, and on the second period interest rates for the sophisticared

• consumers. For the rest of this subsection I assume that there are enough sophisticated consumers

so that all the firms would want to compete for both types. Since the sophisticated consumers care only about the second period rate, their utility of bor- rowing money from a lender is B + tθi − βrfixed. Using a similar probability argument as in the proof of Proposition (1), firm i faces the following two demands Qi−naive = (1 − σ)

j=i

G(θi + β(rj−intro − ri−intro) t )g(θi)dθi (9a) Qi−sophisticated = σ

j=i

G(θi + β(rj−fixed − ri−fixed) t )g(θi)dθi. (9b) The lender therefore faces two different demand functions on two different rates, making the optimal profit calculations more complicated. One observation greatly simplifies the task. Only the naive consumers care about the introductory interest rate, and since everyone ends up repaying in the last period, the lenders only care about the last period interest rate, rfixed. Then, for a given rfixed it is optimal for any lender to offer the lowest introductory rate - the naive demand goes up, the sophisticated demand and the markup do not change. There is a limit on the period 1 rate however. If the lender offers a really low introductory rate relative to rfixed, then the consumers pay back everything before the last period begins. Therefore, for a given rfixed, the optimal introductory rate is βrfixed, or rfixed = r1

β . This transforms the demand function of sophisticated consumers

8

(equation 9a) into Qi−sophisticated = σ

j=i

G(θi + rj−intro − ri−intro t )g(θi)dθi, (10) which differs from the naive demand (equation 9b) only by not having a β in the fraction defining the slope - sophisticated consumers are more rate sensitive than the naives. Proposition 2 In a symmetric equilibrium with a covered market, σ sophisticated consumers, and 1 − σ naive consumers firms offer rintro = βrfixed, (11a) rfixed = t M(N)β((1 − σ)β + σ). (11b)

• Proof. Let’s begin by transforming the profit function into a function of rintro, without any rfixed’s

floating around. We can use the demand function from (10) for the sophisticated consumers, and the demand function from (9b) for the naive consumers, labeling them Qs and Qn respectively. Also, since rfixed = rintro

β

, we plug that into the markup term as well. Π = 2(σxs + (1 − σ)xn) × (rintro β ), (12) then the FOC is Qi−s + Qn−s − ri−intro β(1 − σ) + σ

• GN−2(θi + β(rintro − ri−intro)

t )g2(θi)dθi = 0, (13) solving for ri−intro, and since in a symmetric equilibrium Qi = 1

N for all i, we get the result in the

• proposition. For the second order conditions and the conditions under which M(N) is increasing

in N, the interested reader is again encouraged to look at Perloff and Salop(1985) - the article’s appendix for M(N) derivations, and endnote 6 for second order conditions. Corollary 3 Consumers believe that they will pay back in the introductory period, but actually pay back in the last period. Corollary 4 Each firm’s profit is Π∗ = t β((1 − σ)β + σ)NM(N). (14) The profits are just the fixed rate times the market share ( 1

N ). The interesting comparison of

this markup with the one in equation (8) from the previous section is that instead of the β2 we have β((1 − σ)β + σ). In other words, one of the betas have turned into the weighted sum of β and 1 (the rational beta), with the weights being the population proportions. Since β < 1, the consumers pay lower interest rates. 9

Rate β Last period rate Intro rate 1

t M(N)

Figure 2: Solid lines - equilibrium interest rates for naive consumers. Dotted lines - equilibrium interest rates with some of the consumers being sophisticated. As almost all consumers become sophisticated (σ → 1), the introductory rate approaches the fully rational rate. However, consumers still end up paying a big premium since firms take advantage

• f consumers in the last period. The sophisticated consumers know that this is coming, however

they cannot do much about it.

3 Investment Goods (Gyms)

3.1 Setup

Again, I take the Perloff-Salop (1985) model, and see what happens when consumers are time inconsistent. There is a mass 1 of consumers. Each consumer is a naive hyperbolic discounter with δ = 1 and β < 1. There are three time periods. In the first time period, the consumer chooses which gym to sign up for, pays the fixed membership cost (if any), and derives the benefits for being associated with the particular gym. The benefits for each brand are drawn from some independently and identically distributed p.d.f. g(•), and are weighted by t > 0, where t indicates the strength of preferences9. In the second period, the consumer chooses whether to attend the gym or not. He has to pay pi for attendance. If the consumers attends, he gets the benefit of B in the third period. The utility function of consumer for a given gym i is described below.

9The parameter t is similar to the standard Hotelling transportation costs.

Interestingly, it’s called β in the Perloff-Salop article.

10

ui =

• tθi − Fi − βpi + βB

if the consumer attends tθi − Fi if the consumer doesn′t attend (15) I assume all the consumers sign up for a gym, and that the product differentiation happens before the attendance decision10. There are N symmetric firms. Each firm simultaneously chooses the fixed membership fee (F) and the attendance price (p). If a consumer attends, the firm incurs a cost of c. There are two possible symmetric equilibria in this game. The first one is an equilibrium where the firms take advantage of consumers - consumers pay a membership fee, but end up not coming. I call this the deceiving equilibrium. For this equilibrium to happen, consumers must decide in the second period that getting B in third period is not worth the price of attendance in the second, or βB ≤ p. However, when the consumers are making their decisions in the first period, they think that they will attend, and therefore will take the price into account. Thus, all the firms charge p = βB in this equilibrium - just enough to scare off the consumers once they get into the second

• period. Since the consumers are not actually going to attend, the firms do not receive this price,

and the whole profit will come from the fixed membership fee (F). The other equlibrium is the one where the consumers attend the gym. I call this the rational

• equilibrium. In this case, the firms do not care whether they get their payment from the fixed fee
• r from the per attendance price. Consumers do care - their β works like a standard discount rate

in this case. Therefore, the firms have the incentive to shift all of the markup from the fixed fee to the attendance fee - this makes the overall plan more attractive to the consumers, yet does not hurt the firms. The competition becomes a familiar product differentiated competition on prices, and since there are no other tools to fool consumers, the prices end up at the fully-rational consumer level. I derive the equilibria first, and then I go on to find the line that separates the two of them. I start with the deceiving equilibrium. Lemma 1 If the consumers are too far from being rational and derive relatively small benefits of attendance, then firms charge such fees and prices that consumers sign up and pay the membership fee, but do not attend in equilibrium. p∗ = βB (16a) F ∗ = t M(N), (16b) where M(N) = N(N − 1)

• GN−2(θ)g2(θ)dθ.
• Proof. A consumer signs up for gym i iff (given the attendance decision), ui > uj for all j = i.

10See appendix for the full derivation (monopoly, oligopoly, and duopoly with some sophisticated consumers) of the

case where product differentiation happens is in the travel costs of attendance, and thus happens in the same period as the attendance decision. The results do not change drastically. Within the proofs in the text, I derive bounds on when a consumer would choose not to sign up at all.

11

Or, tθi − Fi − βpi + βB > tθj − Fj − βpj + βB (17a) θj < θi + Fj − Fi t + β(pj − pi) t . (17b) The probability of this event for a given j is G(θi + Fj−Fi

t

+ β(pj−pi)

t

), and since the firms all set the same price of βB by the argument in the text, the probability becomes G(θi + Fj−Fi

t

). Then firm i’s demand is Qi =

j=i

G(θi + Fj − Fi t )g(θi)dθi, (18) and the profit is QiFi since the consumers are not going to attend in the second period. Assuming symmetric strategies, the profit equation becomes Πi = Fi

• GN−1(θi + F − Fi

t )g(θi)dθi, (19) and the FOC is: Qi − Fi N − 1 t

• GN−2(θi + F − Fi

t )g2(θi)dθi = 0, (20) solving for Fi and since in a symmetric equilibrium Qi =

1 N for all i, we get the result in the

• lemma. For the second order conditions and the conditions under which M(N) is increasing in N,

the interested reader is again encouraged to look at Perloff and Salop(1985) - the article’s appendix for M(N) derivations, and endnote 6 for second order conditions. Note that the amount that the consumers pay does not depend on how naive they are. The consumers do not attend, so they just pay the fixed membership fee

t M(N). Once the firms are

in the bad equilibrium, they (effectively) do not have a choice over which price to set. Thus, the competition is on the fixed fee, subject to brand preferences, which are all known in the first period, and do not suffer from discounting. Lemma 2 If the consumers are not too hyperbolic and derive sufficiently large benefits, then the firms play the same symmetric equilibrium as they would play with fully rational consumers. p∗ = c + t βM(N) (21a) F ∗ = 0, (21b) where M(N) = N(N − 1)

• GN−2(θ)g2(θ)dθ, and the equilibrium is played if

B > t β2M(N) + c β (22)

• Proof. Begin by noting that the equilibrium price must be p∗ ≤ βB, otherwise we are back to the

12

deceiving equilibrium11. In this case the firms do not have an obvious strategy for p, as they had in the case above. However, since the firms do not care whether consumers pay in p or in F, but consumers do care (since they discount by β), it is optimal for each firm to shift the payment to the second period, i.e. to make F = 0. Then with F = 0 the utility inequality that has to hold for each j = i becomes θj < θi + β(pj − pi) t (23) Since the consumers attend, the firms incur the marginal cost of attendance, c, and thus the profit function is Πi = (pi − c)

j=i

G(θi + β pj − pi t )g(θi)dθi. (24) Assuming all firms play symmetric strategies, the first order condition is Qi − Fi β(N − 1) t

• GN−2(θi + β p − pi

t )g2(θi)dθi = 0, (25) Solving for the equilibrium p, we set p = p, and since in a symmetric equilibrium Qi = 1

N , we get

the value in the equilibrium. For now, assume that p∗ < βB, so that the equilibrium condition holds. The mark-up of each firm in this equilibrium is

t βM(N), which is strictly bigger than the mark-up

under the deceiving equilibrium from previous proposition (

t M(N), since β < 1. Also, note that if

a firm were to deviate to the deceiving equilibrium, it would lose market share, since the price is the same, but the fixed costs are positive in the deceiving case. Higher F than the deceiving equilibrium would result in an even lower market share, and lower F would result in an even lower

• markup. Thus, as long as p∗ = c +

t βM(N) < βB, the rational equilibrium occurs.

If the equilibrium is the same as the one with fully rational consumers, one would expect that β should not be in the price equation. In this case this is due to consumers still doing the standard discounting of the next period by β, regardless of the naivete, and if we were to have a standard discounting model with just δ discounting factor, it would have been in the price equation instead

• f β.

Lemma 3 Firms are in a constrained rational equilibrium with the following fixed membership fee and attendance price: p∗

rational constrained = βB

(26a) F ∗

rational constrained = max

• c +

t M(N) − βB; 0

• ,

(26b)

11I treat p = βB differently in this equilibrium (consumers attend) versus the other equilibrium (consumers do not

attend). This is entirely for convenience. Requiring the inequality to be strict in either of the cases would create problems with open sets, so instead I could assume discrete prices with an increment of ǫ which goes to zero. In this case p∗

deceiving = βB + ǫ, and p∗ rational = βB − ǫ, assuming both inequalities are strict.

13

Discounted Benefit

• Cost

Deceiving Eq Rational Eq Figure 3: Equilibria of the investment good competition with naive time inconsistent consumers. if the discounted benefits of attendance (βB) are higher than the marginal cost to the firm (c). Proof. What happens to the rational equilibrium if p∗ > βB? The firms cannot charge p∗ anymore, since then the consumers do not attend in the second period, so the price must be p∗

rational constrained = βB. From the FOC for Fi, plugging in the new p, we get

F = t M(N) − βB + c (27) If the RHS is less than zero, than it makes sense for the firms to charge F = 0 and the constrained p = βB. The intuition about losing marketshare by switching to the deceiving equilibrium still holds (since the perceived price is β2B vs ), thus the constrained equilibrium is going to be played. As soon as the RHS of 27 is bigger than zero the new fixed fee is, as the FOC on Fi tells us, F ∗

rational constrained =

t M(N) − βB + c. (28) Proposition 3 The firms play the deceiving equilibrium if and only if the discounted benefits of attendance (βB) are lower than the marginal cost to the firm (c). Proof. If the discounted benefits of attendance are lower than the marginal cost to the firm, then any firm would not lose market share from the constrained equilibrium by slightly raising the attendance price, to make sure that consumers do not attend. However, it would save c − βB per

• consumer. For the opposite reason, the constrained equilibrium is going to happen if the discounted

benefits are higher.

3.2 Sophisticated Consumers

Assume that some (σ < 1) of the consumers realize that they are hyperbolic - they know that in period one, they will discount period two at βδ, and therefore they avoid the trap of signing up thinking they will go, but actually not going. For the discussion above, this means that these consumers know they’ll end up not attending if p ≥ Bβ. 14

Discounted Benefit

• Cost

Deceiving

Mixed

Rational Figure 4: Equilibria of the investment good competition with some sophisticated consumers. For the fully rational equilibrium nothing changes. The firms have even more incentives not to deviate, and the sophisticated consumers know that they will actually attend. The big difference is in the region where firms used to play the deceiving in equilibrium. Consider discounted benefits just below the firms’ marginal cost (βB = c − ǫ). All the firms play the deceiving equilibrium (p∗ = βB, F ∗ =

t M(N)). The sophisticated consumers realize that they are not going to attend.

Thus if a firm slightly lowers its attendance price, all the sophisticated consumers have a big incentive to switch (they get β(B −βB) in initial period money). The problem is that a symmetric equilibrium is not going to happen for the same reasons as in the section above. Therefore we have a mixed equilibrium as long as the marginal costs are not so much higher than the discounted benefits that it is not profitable for the firms to entice the sophisticated consumers to switch. The more sophisticated consumers there are, the more firms want to deviate from the deceiving equilibrium

• the payoff is higher. More firms in the market has a similar effect.

4 Conclusion

I have examined how oligopolistic firms react to time-inconsistent consumers. I found that cases of leisure goods (credit cards) and investment goods (gyms) have different implications. In the case of leisure goods, competition does not change the fact that the consumers end up paying larger fixed interest rate as opposed to the lower introductory rate (set at the fully rational level for discounting consumers). Competition does change the difference between the two rates, which disappears in the limit, as the market becomes almost perfectly competitive. Bringing in some sophisticated consumers lowers both the introductory and the fixed rate; however neither ever goes below the rate that fully rational non-discounting consumers would pay (let alone the marginal cost), and all the consumers end up paying the higher fixed rate. In the case of investment goods, if the consumers are sufficiently (not necessarily fully) rational, then the equilibrium is played as if the consumers are fully rational - competition completely mitigates time-inconsistency, and the threshold does not depend on the number of firms in the

• market. Below the threshold firms charge access fees lower than marginal cost, yet consumers do

not attend anyway. Addition of sophisticated consumers to the mix does not change the threshold, but below the threshold a new region appears where some firms cater to the sophisticated consumers, 15

and others cater to the nave. The message of the paper is what matters is what the consumers’ choices are in the period when time-inconsistency kicks in, and whether it is worth for the firms to entice consumers to choose the unanticipated action. In case of the credit cards, the firms have all the incentives to try to make the consumer postpone the payment – the marginal cost of waiting until tomorrow for the firm is much lower than that of the consumer. In case of the gym, even the discounted benefit of attendance to a consumer is likely higher than the marginal cost to the gym, and thus the gyms should make the right choice for the consumers. 16

Appendix A - Monopoly and Salop (1979) Oligopoly with Invest- ment Goods

Monopoly

Again, we work with three periods, and the same quasilinear hyperbolic discounting function. In period zero each consumer chooses the gym to maximize their current self’s utility. Each gym has a membership fee of F which consumer has to pay in period 0. In period 1 consumer may or may not attend the gym. Attendance hassle cost is td > 0 - consumer d units away from the gym must drive there to get the training. Also, the consumer needs to pay p to the gym, if the consumer chooses to visit. If the consumer visits the gym in period 1, then in period 2 she derives the benefit

• f B > 0. If the consumer attends, the gym bears additional cost of c in period 1. The firm is

located at 0, and the consumers are spread uniformly on a Hotelling-like ray which starts at 0. The consumer is doing backwards induction - if she comes to the gym in period 1, how much is she willing to pay to sign up in period 0? Since the consumer is naive, she thinks that her discount rate at period 1 is going to be δ. Therefore, she thinks that she will attend iff δB > p + td

• the discounted benefits from attendance are bigger than the costs of attendance (the price and

the hassle). However, the consumer’s discount rate is actually βδ in period 1, and therefore the consumer actually attends only if βδB > p + td. To sign up, the consumer’s discounted benefits must be bigger than the price of membership. The discounted benefits only happen if the consumer thinks she is going to come in period 1 (δB > p + td). Then, for the consumer to sign up in period zero, the following must be true: βδ2B > βδ(p + td) + F, where d is the distance to the gym. Say the timing is as follows: the gym picks the membership fee (F) and the per-visit price (p), then consumers decide whether to sign up in period 0. Proposition 4 The monopolist gym charges the following memebrship fee and per-visit price: F = βδ 4 − βδ ((2δ − β)B − c) , and (29a) p = (2 − δ)δβB + 2c 4 − βδ . (29b) Both increase in the consumers’ inconsistency and the benefit that the consumers derive.

• Proof. From the paragraph above, we know that the consumers sign up for the membership iff

dmember ≤ δB − p t − F βδt, (30) where d is the distance from the monopolist. Similarly, we know that consumers actually attend the gym iff dattend ≤ βδB t − p t . (31) Define by xmember and xattend respectively the consumers who are indifferent between getting the 17

membership or not, and actually coming or not. Assume that xattend < xmember - out of people who sign up some of them do not come to the gym. For this to happen, F < δ2β(1 − β)B needs to

• hold. Then the firm’s profit is

Π = Fxmember + (p − c)xattend. (32) Plugging in (30) and (31), and differentiating with respect to both p and F we get ∂Π ∂p = 1 t (−F + βδB − 2p + c) , (33a) ∂Π ∂F = 1 t

• δB − p − 2F

βδ

• .

(33b) Second order conditions are satisfied, therefore from the system of two equalities above we get the values in the proposition. Differentiating again we get the Hessian H = 1 t

• −2

−1 −1 − 2

βδ

• .

(34) The determinant of the Hessian is positive iff βδ < 4, which is always satisfied since both of the variables are between 0 and 1. F increases in β only when 2δ > β, which we have assumed before. For xattend < xmember to hold, plug in the optimal F into the condition to get β(1 − β)δ2 + 4βδ < 2δ + β + c

• B. To put it in other words, the marginal cost to the gym of customer attendance must

be sufficiently big relative to the benefit that the customer will get out of the visit.

Oligopoly

Keep the basic setup from the above, but now consumers are located on the Salop circle of circum- ference 1, and there are N firms, spaced equidistantly from each other. I assume that the market is covered to make the case interesting, if it is not covered we just get N monopolists from the previous subsection, and if it’s just touching then we get a Salop-like kink. Consumers sign up for the firm which they think is going to maximize their net utility, so the firm which has a bigger βδ2B − βδ(pi + tdi) − Fi, where p and F are the usage and subscription rates each firm is charging and d is the distance to the respective firm. There are going to be three regimes - under the first regime, when consumers are sufficiently hyperbolic, the firms are going to charge a positive subscription fee (F), and in equilibrium some of the consumers sign up and pay the fee, but do not go to the gym afterwards. In the intermediate range of β we have the Salop’s kink - the equilibrium price is constrained by what the consumer in the middle can pay given his valuation. Under the last regime, the equilibrium is the same as with standard consumers (β = 1) - the firms do not have a subscription fee, instead everything is in the usage fee (p), and everyone comes to the gym. While this regime setup looks a lot like Salop (1979), there are subtle differences. I am actually 18

assuming away the standard Salop regime switch both here and in the credit card part. So if I would not have done that, all these three regimes would have happened in Salop’s competitive regime, so there would be two kinks. In Salop’s paper the regimes happen because the firms go from monopoly to competition. The firms here are never monopolists. They only have the consumers captured in the second period, however that is enough to create this regime switch with time inconsistent

• consumers. The regime switch would not have happened if the consumers were time consistent,

because in this case everyone who signs up ends up actually going to the gym. The problem with time inconsistent consumers is that some of them do not come after they pay the membership fee, effectively creating monopoly-like regions for the firms which do not touch the other firms’ demand regions. Proposition 5 If the consumers are sufficiently naive, such that β < β∗ = 2 t + cN 2δBN + δt, (35) then the two firms charge the following membership and per-usage fees: F = βδ t N , and (36a) p = δβB 2 + c 2 − δβt 4N . (36b) Both decrease in the consumers’ inconsistency, per usage price increases in the benefit that the consumers derive and the number of firms, and decreases in the differentiation parameter. The

• verall fee increases in the differentiation parameter and decreases in the number of firms in the

market.

• Proof. By the standard Salop-like argument, the consumer indifferent between the two firms is

going to be at xmember = 1 2N + p − p 2t + F − F 2βδt , (37) with the normal p and F being the ones charged by the active firm, and the overlined ones are the competitor’s fees. From the consumers who sign up for the firm’s service not everyone attends. Since the other firm is not an option anymore, the condition here is exactly the same as in the monopoly case: xattend = βδB t − p t . (38) Therefore, the active firm’s profit is Π = 2 (Fxmember + (p − c)xattend) . (39) 19

Plugging in the equations from above, we get ∂Π ∂p = 2 t

• −F

2 + βδB − 2p + c

• ,

(40a) ∂Π ∂F = 1 t t N + p − p + F − 2F βδ

• .

(40b) Second order conditions are satisfied, therefore from the system of two equalities above we get the values in the proposition12. Comparative statics are straightforward to show. The equilibrium above unfold only if xattend ≤ xmember - out of the people who sign up, some will not show up in

• equilibrium. We know that xmember =

1 2N in equilibrium, and using the values of equilibrium p

and F, this regime happens iff βδB t − βδB 2t + c 2t − βδt 4Nt

• <

1 2N (43a) β < 2 t + cN 2δBN + δt. (43b) Corollary 5 Firms charge per usage price less than their marginal cost iff β < 2c δ(2B − t

N ) (<< β∗) ,

(44) the inequality holds if the consumer benefits are sufficiently small, consumers are sufficiently hyper- bolic, there are not too many firms, or if the firms are sufficiently differentiated or have sufficiently high marginal costs. The cutoff value is smaller than β∗ - regardless of the parameters there is a region where firms charge a usage price higher than marginal cost. This is in contradiction with DellaVigna and Malmendier (2004) results. Here the usage price is so high to discourage some of the consumers from attending the gym after paying the membership fee. The next regime is when consumers are closer to being time-consistent, with β > β∗∗. First, if the consumers were standard time-consistent consumers, then the firms would not charge a

12Differentiating again we get

∂2Π ∂F∂p = −1 t , (41a) ∂2Π ∂p2 = −4 t , (41b) ∂2Π ∂F 2 = − 2 βδt, (41c) and then the Hessian is H = 1 t −4 −1 −1 − 2

βδ

• .

(42) The determinant of the Hessian is positive iff βδ < 8, which is always satisfied since both of the variables are between 0 and 1.

20

subscription fee. Consumers value postponing the payment, and for firms the interest rate is 0, so they do not sacrifice any markup by transferring some of the F to p. Therefore for the competitive purposes, the subscription fee immediately goes to 0. Moreover, if everyone shows up to a gym in the second period, their reservation utility (δB) does not matter, as in the standard Hotelling

• model. We then get the familiar p = t + c result. This is exactly what happens in the second

regime, despite the fact that the consumers are still somewhat naive. Proposition 6 When consumers are closer to being time-consistent (β > β∗∗), firms do not charge a subscription fee, and charge p =

t N + c for usage.

• Proof. If the market is covered in the second period, then by the argument above the subscription

fee should be zero13. Then with the complete coverage the model becomes Hotelling with reservation value of βδB, which does not matter in equilibrium anyway, and by the standard argument (see the proof of Proposition 1 for example) we get the result in the proposition. In between the first and the second regime, we have a Salop-like kink. Note that we have assumed the actual Salop kink away by assuming high enough benefits, B. This kink is due to some of the consumers signing up for the membership, but potentially not showing up. Notice that this is entirely due to the assumption that the travel costs (product differentiation) is incurred in the second period. If we were to assume that the differentiation happens simultaneously with the membership decision, the consumers would have been hetergoeneous by the following period, and therefore either all of them or none of them would attend, getting rid of the kink. Proposition 7 For intermediate values of β firms are in a kinked part of the demand curve - the equilibrium price is constrained by how much the marginal consumer is able to pay for attendance. β∗ < β < β∗∗, where β∗∗ is defined by (45a) β∗∗ =

3t 2N + c

δB . (45b)

• Proof. The left hand side of the inequality was discussed already. The right side happens if the

price in the last regime (p t

N + c) is too high for the marginal person, or βδB − p < 0. Plugging in

the value of p. Corollary 6 Both cutoffs, β∗ and β∗∗, increase in marginal cost of the firm and product differen- tiation; decrease in benefits for the consumers, the discount rate, and the number of firms.

Sophisticated Consumers and Gyms - Symmetric Duopoly

Let’s take the setup of the previous subsection, and assume that σ of the consumers are sophisticated

• realize how hyperbolic they are. The sophisticated consumers sign up only if they know that they

13A firm’s profit becomes Π = (p + F − c)xmember, where xmember is defined in the proof above. The derivatives

with respect to p and F cannot be zero at the same time, which leads to the result that F must be zero in equilibrium.

21

will attend, so the firms cannot extract a subscription fee from them, and have the consumers not show up after. Proposition 8 When consumers are sufficiently hyperbolic, or there are sufficiently many naive consumers, the firms charge the following subscription (F) and usage (p) fees: p = δβB 2 + c 2 − F 1 + σ 4 , (46a) F = βδ (1 − σ)t − 2σc 1 − σ − βδσ(1 + σ). (46b)

• Proof. All the sophisticated consumers who sign up for the membership will attend. Then for

them xmember = xattend, which was not the case for the naive consumers. Therefore, the profit function becomes Π = πσ=0 − σF(xmember − xattend), (47) where xmember and xattend are defined in equations (37) and (38), and πσ=0 is the profit defined in equation (39) - the profit when everyone is naive. The relevant derivatives are: ∂Π ∂p = 1 t

• −F

2 + βδB − 2p + c

• − σF

2t , (48a) ∂Π ∂F = 1 2t

• t + p − p + F − 2F

βδ

• − σ

1 2 + p + p 2t + F − 2F 2βδt − βδB t

• .

(48b) From the symmetry assumption and the first order conditions we get the result of the proposition14.

14Differentiating again we get

∂2Π ∂F∂p = −1 + σ 2t , (49a) ∂2Π ∂p2 = −2 t , (49b) ∂2Π ∂F 2 = −1 − σ βδt , (49c) and then the Hessian is H = 1 t

• −2

− 1+σ

2

− 1+σ

2

− 1−σ

βδ

• .

(50) The determinant of the Hessian is positive iff βδ < 8

1−σ (1+σ)2 . The right hand side is bigger than 1 if σ < 4

√ 2−5 ≈ .66, so the inequality is satisfied in this case regardless of what happens on the left hand side. Given that the left hand side is actually strictly less than 1, and for sufficiently high σ this regime does not happen, I assume that the SOCs are satisfied.

22

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