University of Amsterdam
Master’s Thesis
Econometrics
On the possibility of Government Intervention to protect
Consumers with Dynamically Inconsistent Preferences
Author: Andrés Méndez Ruiz
Supervisor: Jan Tuinstra
Second Marker: David Kopanyi
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1. Introduction
Experience and empirical evidence (for example DellaVigna and Malmendier 2006) show that contrary to the standard assumption made throughout the classic models of consumer choice, individual’s preferences display time-inconsistencies, which can lead to distorted predictions about future consumption decisions. Firms, on the other hand, have more resources and incentives to predict the actual future behavior of consumers. This means that they are able to exploit their costumer’s dynamic inconsistencies in their favor, affecting consumer’s welfare. This paper considers possible government interventions to protect consumers from this type of exploitation by the firms.
One of the favorite examples of behavioral economists to illustrate temptation, or dynamic inconsistency of preferences, is the one of Odysseus and the sirens (see for example Thaler and Sunstein 2008). The classic tale goes as follows. After the Trojan War, Odysseus has to travel trough the domains of the sirens in order to get back home to Ithaca. These creatures have an irresistible voice, and every sailor who hears them sing suddenly feels an urge to get closer to them, crashing his ship into the rocks and drowning. In order to overcome temptation while still being able to hear the music of the sirens, Odysseus instructs his crew to fill their ears with wax, so that they cannot be seduced by their music, and to tie him to the mast, so that he will not steer the ship towards the rocks.
We can split Odysseus in the last example into two: present Odysseus (before hearing the sirens) and future Odysseus (once he is enjoying the beautiful melodies of these creatures). From this example it is clear that the preferences of both are different: present Odysseus prefers to stay away from the rocks while future Odysseus wants to get close. The aforementioned partition of a decision maker into different “selves” with different preferences is the most widely used
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modeling principle when analyzing decisions under changing tastes (Spiegler 2011). Moreover, we can view Odysseus’ petition of being tied to the mast as a commitment device. He has the incentive to do this request because he suspects his future self will have different preferences. In this sense Odysseus is a sophisticated decision maker. If he was a naïve one, he would not be aware of his changing preferences and would probably end up drowning, as many sailors before him.
In the classic microeconomic models of consumer choice and in the standard models of industrial organization the assumption is that agents have time unvarying preferences between the moment they make the decision and consumption. This is implied by the assumption that consumers have rational preference relations. If a decision maker reveals sometimes that he strictly prefers to and sometimes that he strictly prefers to , his behavior is not rational because he lacks stable preferences (Spiegler 2011). Therefore, if Odysseus were rational he would not need to be tied to the mast in order to be able to stay away from the rocks because future Odysseus would have the same preferences as present Odysseus.
However, there are many market situations in which the consumer resembles the less rational version of Odysseus, and the aforementioned assumption of time consistent preferences fails to hold. For example, a consumer attending a restaurant might wish to eat a healthy dish. Before going to the restaurant he decides that the option that maximizes his utility is eating a salad instead of a pizza. Still, once the consumer is in the restaurant he is tempted by the delicious smell of the freshly baked pizza and decides to order a pizza instead of a salad. If the consumer were able to foresee himself succumbing to temptation he could “tie himself to a mast” and go to a restaurant in which only salads are served (Spiegler 2011). The latter type of
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consumer is referred to in the literature as sophisticated consumer, whereas those that are unaware that they might have self control problems are called naïve consumers.
Let us denote by “self 1” the consumer before he is at the restaurant and by “self 2” the consumer at the restaurant. From the view point of self 2’s utility, the decision of eating pizza is the optimal one. Nevertheless, in most of the literature that applies the multi-selves approach, when it comes to analyzing the effects of time inconsistent preferences on welfare, welfare is equated to self 1’s utility (Heidhues and Kőszegi 2010). According to Spiegler (2011), and in the context of our example, the preferences of self 1 are more important because they reflect long-run planning, while self 2’s considerations reflect visceral urges. From this perspective the consumer’s time inconsistencies can result in welfare losses.
So far I have briefly described some aspects that arise when departing from the conventional assumption of microeconomic theory that preferences are dynamically consistent. I have mentioned only issues arising from the consumer’s side. Yet, assuming away time consistency of consumer’s preferences has also an implication for the behavior of the firms.
According to Smallwood and Conlisk (1979), unlike the consumers, firms have more at stake when it comes to solving optimization problems. The penalty for not finding the optimal solution to the firm’s profit maximization problem is usually higher than the consequences for an individual that is unable to correctly solve the consumer’s utility maximization problem. Therefore, according to Ellison (2006), assuming a market in which a rational firm faces an irrational consumer has become the norm in the literature of behavioral industrial organization. The latter implies that the main focus when analyzing the firm’s behavior lies in what they do to exploit the irrationality of the consumers. Furthermore, it means that there is room for policy interventions that aim to protect consumers by limiting the extent of the exploitation by the firm.
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This thesis considers the effects on welfare of a restriction by the government regarding the spread of the payments corresponding to the different actions available to the consumer at the time of actual consumption. To do so, I extend a simple two-period model by Spiegler (2011) in which the consumer has dynamically inconsistent preferences. A naïve consumer believes that the utility of choosing action in period 2 will given by . However, his actual second period utility from action will be given by . Naivety is modeled using the notion of frequency naivety. According to this definition of naivety, a partially naïve consumer is one who assigns probability to having his preferences represented by utility function in period 1 and probability to having them instead represented by utility function .
According to this model, the firm offers two tariffs: and . The former payment is a continuous function of , the action that maximizes the utility of the consumer from period 1 self’s perspective, and the latter payment is a function of , the action that maximizes consumer’s utility according to his second period self’s point of view. The consumer chooses the actions and from a continuous menu in the first period. Then, in the second period, the individual decides whether to consume action or , and makes the payment corresponding to the action chosen. To capture the government policy, I include the additional restriction to the firm’s profit maximization problem that the difference, resulting from the subtraction of payment minus , must be smaller or equal to some quantity, , chosen exogenously by the government: . Therefore, parameter sets an upper limit to the spread between the two tariffs.
The solution to this model suggests that from the view point of period 1 self, which better reflects long term preferences, consumer’s welfare is decreasing with respect to . In other words, the optimal from period 1 self’s perspective is . In practice, it is difficult to
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implement a policy which restricts the possibility of the firm to set different tariffs. However, the results also suggest that reducing , in the relevant interval, might be good enough as a welfare-improving policy that is possible to implement.
The following sections of this paper are organized as follows. Section 2 reviews the economic literature that deals with the behavior of firms when facing consumers with dynamically inconsistent preferences. In Section 3, I present the original model by Spiegler which does not include the extra constraint imposed by the government. Section 4 presents my extended model and its implications. Section 5 concludes.
2. Background
Assuming away time consistency of preferences has implications for the behavior of the consumer and of the firms interacting with them in the market. In the present section and in light of the existing literature my aim is to address briefly mainly the following issues:
What are the main methods and concepts to model dynamic inconsistencies in preferences?
What types of price schemes are offered by firms to consumers with changing tastes?
How is consumer surplus and in general welfare affected by the contracts offered by firms who know they face dynamically inconsistent and naïve consumers?
What are the possibilities for governmental intervention?
What happens when firms face consumers with different degrees of naïveté?
I will proceed as follows. In Section 2.1, I will briefly describe some of the most relevant concepts and methods for modeling time-inconsistent preferences. In Section 2.2, I will review DellaVinga and Malmendier’s (2004) article which was the first that analyzed firm behavior when facing time-inconsistent consumers. Then, in Section 2.3, I will discuss an article by
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Botond Kőszegi (2005) which explores the opportunities for beneficial governmental intervention in markets with time-inconsistent consumers. In Section 2.4, I will briefly comment Heidhues and Kőszegi’s (2010) paper which analyses the welfare effects of consumer’s taste for immediate gratification in the credit market and possible policy interventions to improve welfare. Thereafter, in Section 2.5, I will review Gottlieb’s (2008) article that focuses on the role of exclusivity in contracts with time-inconsistent consumers and how this affects welfare. Finally, in Section 2.6, I will comment on Eliaz and Spiegler’s (2006) article which models contracting with diversely naïve agents.
2.1 Important methods and concepts for modeling dynamic inconsistencies
First, I want to discuss briefly some of the most important tools and concepts employed in modeling time-inconsistency of preferences. Therefore, I will introduce the multi-selves approach, two different notions of naivety and the quasi-hyperbolic discounting preferences model.
2.1.1 The multi-selves approach
The multi-selves approach was first introduced by Strotz (1956) and further developed by Peleg and Yari (1973). Therefore, I will shortly review these two papers.
The main goal of Strotz’s paper is to present a theory to explain three phenomena: overconsumption, commitment devices to regiment future behavior and “preventive” savings behavior. These phenomena are analyzed in the context of a consumer with changing tastes. According to Strotz, if a consumer is not able to anticipate the aforementioned change in his preferences he may end up overconsuming. However, if he correctly foresees that he will have different tastes in the future, he can either use commitment devices that exclude the undesirable options from the view point of the present, or he can adapt his plan, for example to include
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savings to be able to afford his future overconsumption. The most important aspect of Strotz’s paper in the context of the present review is the way he models the individual facing a dynamic decision-making problem as a collection of infinitely many individuals (one for each time period) with different utility functions.
In Peleg and Yaris’s article the behavior of an agent with time varying preferences is also analyzed. Moreover, they pose the question of whether such an agent can decide in a manner that his past actions are also optimal from the perspective of his present preferences. To examine the above issues these authors adopt a similar approach as the one introduced by Strotz: they model the individual as a sequence of decision-makers, one for each period, each of these having idiosyncratic preferences. The novelty introduced by these authors is that they model the decision maker’s choices as the outcome of the game theoretic solution concept of Nash equilibrium.
2.1.2 Naivety
A time-inconsistent agent is naïve when he is unaware about his changing preferences. On the other hand, he is sophisticated if he is fully conscious about them. Finally, an agent is partially naïve if the degree of awareness is intermediate. According to Spiegler (2011), the set of the consumer’s beliefs over his second period preferences that can be viewed as partially naïve is intractable. Therefore, following his presentation, I will discuss only two of the possible restrictions on the set of partially naïve beliefs.
The notion of magnitude naivety was conceived by Loewenstein, O’Donoghue and Rabin (2003). With their notion the authors attempt to capture the fact that people tend to understand qualitatively the direction in which their preferences will change but underestimate the magnitude of the change. Therefore, naivety is modeled as a projection bias by the consumers, in which they exaggerate the degree to which their present preferences will resemble the future ones.
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Consider an individual in state attempting to predict his future utility from consumption, , in state . Denote by his actual future utility and by the prediction he makes while in state . The individual’s predicted utility exhibits simple projection bias if there exists such that for all , , and , it holds that that the predicted utility is
a convex combination of the true future utility in state and the utility in the current state : . The parameter can be viewed as the consumer’s type. If , the individual presents no projection bias. On the other hand if , he is partially naïve. Moreover, notice that the degree of the agent’s naivety increases with . Finally, if the individual is fully naïve and believes his future tastes will be identical to his current ones.
An alternative definition of naivety is given by the notion of frequency naivety introduced by Eliaz and Spiegler (2006). Under this definition, in contrast to the previous, a (partially) naïve consumer does not underestimate the extent to which his future preferences will change but instead underestimates the likelihood that this will happen. Under this notion of naivety, a partially naïve consumer is one who assigns probability to the probability of having his preferences represented by utility function in period 2 and probability that they will be instead represented by utility function . If the agent’s true future utility function is , his degree of naivety is increasing in the parameter .
2.1.3 The (β,δ) quasi-hyperbolic discounting preferences model
One functional form regularly used by economists to capture the phenomenon of present bias is the model. This model was introduced originally by Phelps and Pollack (1968), and has gained popularity especially in long-horizon models of consumption and saving (see for instance Laibson 1997). Nevertheless, in this paper I will follow closely DellaVigna and Malmendier’s (2004) presentation of this discounting model.
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The discount function for time , when evaluated at period , equals 1 for and equals for with . This means that the present value of a flow of
future utilities as of time is given by . Parameter can be viewed
as the short-run discounting and as the long-run discounting parameter. For this model is identical to the time-consistent exponential model. However, for the discount factor between the present period and the next period is , while the discount factor for any subsequent adjacent periods is only . This difference between the short-run and long-run discount factors represents the time-inconsistency in the preferences of the individual.
2.2 A first model of contracting with time inconsistent consumers
In DellaVigna and Malmendier (2004) the pricing behavior of a firm that faces consumers with dynamically inconsistent preferences is analyzed for the first time. These authors consider markets for goods with delayed benefits (investment goods) or delayed costs (leisure goods). The effect of consumers’ time-inconsistent preferences on welfare is analyzed. In addition, the authors discuss the possible actions that a benevolent government could take in light of the conclusions derived from their model.
The main model in the paper for investment goods is as follows. In period 0, the monopolistic firm proposes the consumer a two-part tariff . The quantity represents a lump-sum fee that the consumer has to pay if he chooses to accept the contract offered by the firm, and represents the price of consumption implied by the contract. In period 1, the consumer has to decide whether to reject or accept the two-part tariff offered by the firm. If he rejects, the firm makes zero profit and the consumer attains his reservation utility . On the other hand, if the consumer accepts he pays to the firm, learns his cost type , and chooses if he wants to consume or not. If he chooses to consume, he has to pay to the firm and incurs a cost
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. If he decides not to consume, he attains a payoff of zero in period 1. In period two the consumer receives a delayed payoff if he chose to consume, and of if he chose not to consume.
To model the time-inconsistent preferences of the consumers DellaVigna and Malmendier rely on the quasi-hyperbolic discounting model discussed before. This means that consumers are modeled having a higher discount rate between the present and the next period than between any of the subsequent periods. The first discount rate corresponds to and the subsequent are . The latter implies time inconsistency because the discount rate between two periods depends on the time of evaluation.
To capture the different degrees of awareness that consumers can have regarding their time inconsistent preferences the authors introduce a third parameter: . This parameter captures the degree of sophistication or naivety of the consumer in the case he presents time-inconsistent preferences, i.e. if . Accordingly, a consumer can be described by the triplet . This results in the following types of consumers. First, there is the exponential agent which has time-consistent preferences and is aware of it . Then, there is the sophisticated agent which has time-inconsistent preferences and is aware of it . Additionally, there is the (fully) naïve agent which has time-inconsistent preferences and is completely unaware of it . Finally, an agent can also be partially naïve, meaning that he is not fully aware of his time inconsistency . In the case an agent is of the latter type, the difference between the perceived and actual future short-run discount factor represents his overconfidence about future self-control.
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The implications of the model for consumer behavior are the following. In the case of investment goods (with delayed benefits) an important effect on consumer behavior of having time-inconsistent preferences is that the agent consumes less in period 1 than he originally wanted to consume in period 0. In the case of leisure good (with delayed costs) the effect is the opposite: agents end up consuming more in period 1 than they originally intended to in period 0.
First, the authors consider the behavior of monopolist firms when they face a market with homogeneous consumers that are either sophisticated, naïve or partially naïve. The contracts offered by the firms to these consumers are contrasted against the contracts that they would offer consumers with time-consistent preferences. In the case of investment goods, DellaVigna and Malmendier’s model predicts that if consumers are fully rational, i.e. exponential, the firms will try to extract the entire consumer surplus via the lump-sum fee and set the consumption price equal to the marginal cost of the firm.
I will now explain the predictions of the model for markets with consumers with changing preferences. If the firms face a market of time-inconsistent consumers they set the consumption price lower than the marginal cost. The reason for this is different in the case of a market with sophisticated agents and in the case of a market with naïve agents. For sophisticated agents the following commitment rationale applies. An individual who is aware of his time inconsistency looks for ways to increase his future investment. A way to do this is by choosing a contract with a low consumption price . Moreover, he is willing to accept a contract with an increased lump-sum fee because he values the commitment device.
In the case of naïve agents, firms offer a price lower than the marginal cost because they know naïve consumers overestimate future consumption. Therefore, by offering a discount on and an increase in , relative to the contract for the time-consistent agents, they can make the
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contract more attractive for naïve consumers. Ultimately, this results in higher profits for the firms since the agent takes advantage of the discount less often than he anticipated. It is easy to understand that when dealing with markets with partially naïve agents the prediction of the model that firms offer contracts with a price below marginal cost and an increased holds because the two effects on the contracts of facing naïve and sophisticated agents go in the same direction.
The prediction of the model when the market in question is one for leisure goods is that firms offer a contract with a consumption price above marginal cost and a lump-sum fee lower than the one offered to time-consistent agents. Prices are set above marginal cost for the same commitment and overconfidence reasons as in markets for investment goods.
DellaVigna and Malmendier consider additionally the case of a competitive market. Results remain the same for this type of market as the results they find for the monopolistic case. The authors also try to answer the question of how the time-inconsistencies in the agents’ preferences affect welfare. According to their model, if consumers are sophisticated, welfare is unaffected by the degree of time inconsistency. The market offers these consumers a commitment device that allows them the time 0 self to achieve his first-best. However, when agents are partially naïve they have a lower surplus than sophisticated agents with the same parameter values and .
In regard to possible government intervention, the authors argue that it is unnecessary in the case in which consumers are sophisticated because the market itself leads to a socially optimal outcome. On the other hand, if consumers are naïve government intervention could in principle make consumers better off, but the implementation of the adequate policy for this purpose could be difficult. In general, the authors consider that a better policy is to educate the partially naïve consumers to make them aware of their time-inconsistencies.
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2.3. Can the market alone provide solutions for self-control?
The extent in which market forces can correct self-control problems is analyzed by Botond Kőszegi (2005) in the context of competitive markets for harmful and beneficial goods. His article focuses on the choice behavior of dynamically time-inconsistent consumers and its effects on welfare. Its main goal is to establish if the market itself can provide solutions that lead to the optimal level of consumption or if this is only achievable with the aid of government intervention.
I will start by briefly describing the basic three-period model of individual choice of the article. Each consumer lives for three periods: Consumption occurs in period 1 only. In this period the individual can either choose to consume a standard good , i.e. a good that has no implications for future utility, or an intertemporal good . The consumption of the latter type of good is pleasant or unpleasant in period 1, but also has consequences, , for utility in period 2. This can be positive for harmful goods or negative for beneficial goods. To capture the time inconsistent and present biased preferences of consumers Kőszegi uses the multi-selves approach and the quasi-hyperbolic discounting preferences model. To capture different degrees of awareness about consumer’s time-inconsistency the discount factor of consumer’s period 2 self is δ as in O’Donoghue and Rabin (2001). This means that if the consumer is perfectly sophisticated, while if the consumer is perfectly naïve. According to this model, consumers end up systematically overconsuming harmful products and underconsuming beneficial ones from the point of view of period 0 self.
Under the above framework the article discusses possible welfare enhancing interventions, from the view point of period 0 self, which the government could undertake. The context considered is one of competitive markets. A simple possibility for the government to
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correct self-control problems would be to introduce a price increase and then to lump-sum-redistribute the revenues to consumers. Since firms make no profits and the government would return all revenue, it follows that the social impact of this type of intervention would equal its impact on consumer surplus.
The aforementioned price increase could in principle be also provided via the market by a variety of ex-ante contracting or other commitment arrangements. Kőszegi first considers an environment similar to DellaVigna and Malmendier (2004) in which he assumes that in period 0 competitive firms offer two-part tariffs that specify a lump-sum payment and a per-unit price for consumption in period 1. Moreover, it is also assumed that the contract is exclusive, meaning that if a consumer accepts the firm’s contract he will not be able to purchase the good from competitor firms.
I will present the conclusions for beneficial goods. Bear in mind that the conclusions for harmful goods go in the opposite direction. In the case of beneficial goods, for example exercise, sophisticated consumers are aware that they will underconsume this good when period 1 arrives. Therefore, they are willing to pay high upfront tariffs for reduced per-use prices that will provide the right incentives for them to consume more often the good. On the other hand, naïve consumers believe that they will consume optimally in period 1. Hence, because naïve consumers are overconfident they do not demand any commitment devices. Nevertheless, this misperception of the naïve consumers results in the firm offering them a contract similar to the one they offer sophisticated consumers. The firm, knowing that the naïve consumer believes he will take advantage more often than in reality of the lower per-use price, can lure him into a contract with a upfront payment. It follows that the contract intended by the firm to exploit naïve consumers serves sophisticated consumers as a commitment device. The latter implies that in this type of
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market environment government intervention aimed at attacking self-control problems by making consumers more patient would be less effective than helping them overcome their naivety.
The last conclusion does not hold for all market environments. In the case that exclusive contracts are not possible or switching costs between supplier firms are low, consumers cannot be prevented from recurring to the spot market in period 1. Because we are considering competitive markets, the latter means that the per-usage price of the good in period 1 cannot exceed its marginal cost. This means that optimal contracts for beneficial goods, which imply below marginal cost pricing, survive in this alternative setting while optimal contracts for harmful goods, which imply above marginal cost pricing, do not. No consumer will be willing to pay above marginal per-use prices for the harmful good in period 1 if they can buy from another firm charging a price equal to the marginal cost. This implies that the market is better in providing commitment devices in the case of beneficial goods than in the case of harmful ones. According to Kőszegi this means that the government is in a privileged position to solve the problem, because it can increase the price in such way that it applies to every transaction in a determined market.
Finally, in the context of Kőszegi’s model the need for government intervention to help consumers overcome self-control problems becomes again important, even in the context of contracts that are exclusive, if firms can offer more complicated contracts in period 0 which include non-linear per-unit prices. In this case, if consumers are perfectly sophisticated the equilibrium contract results in an optimum level of consumption according to the view point of period 0 self. Nevertheless, for those consumers who are even slightly naïve about their ability to self-control the market does not offer any self-control devices.
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The latter arises because if agents are slightly naïve, the firm can offer the optimal period 0 consumption level at a slightly lower total price and introduce an option to pay a penalty and switch to an alternative barely preferred choice in period 1. The slightly naïve consumer does not anticipate his period 1 desire to switch and accepts this contract over the one without a possibility to switch. In period 1, the firm makes switching tempting in such way that the alternative option gives consumer’s self 1 the same level of consumption he would choose in a spot market. Firms have the incentive to proceed in this manner because it allows them to profit the most. They are able to capture both the profits from the demand for commitment devices and from the consumers’ short-sightedness in period 1 using non-linear prices to make consumer believe they are purchasing self-control while in reality they are not. Here again, the room for welfare enhancing governmental intervention is open.
2.4 Is there room for possible welfare-improving policy interventions?
The welfare effects of consumer’s time-inconsistent taste for immediate gratification and possible welfare-improving interventions are analyzed by Paul Heidhues and Botond Kőszegi in the context of competitive credit markets.
One of the advantages of the model proposed by these authors is that it is consistent with real-life credit-card and subprime mortgage contracts. In their three-period model, the consumer borrows an amount in period 0 and repays amounts and in periods 1 and 2, respectively. Consumer’s self 0’s utility is given by . Notice that represents the cost of repayment. Self 1 maximizes , where , which satisfies 0 , is the parameter that measures the degree of time-inconsistency, i.e. the intensity of the taste for immediate gratification. If , the consumer puts lower relative weight on the period-2 cost of repayment than in period 0. The latter can be interpreted as the consumer being tempted or
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having less self-control than in period 1. Note that self 0, unlike self 1, does not discount the future cost of repayment relative to the utility from consumption . The reason is that the borrowing motivating the analysis is for future consumption and, from the perspective of self 0, , , and are all in the future.
To model different degrees of awareness about consumer’s time-inconsistent preferences, the authors follow Ted O´Donoghue and Mathew Rabin’s (2001) formulation of partial naivety. Accordingly, self 0 believes with certainty that self 1 will maximize . This implies that , which satisfies represents the consumer’s beliefs about or, in other words, the consumer’s degree of naivety. Note that corresponds to perfect sophistication, and corresponds to complete naivety about having dynamic inconsistent preferences.
The aforementioned consumers interact with profit-maximizing suppliers of credit in a completive market. They can sign exclusive nonlinear contacts in period 0 with the firms, in which they agree on the consumption level and the menu of installment plans that will be available to consumer’s self 1. Moreover, a possible policy intervention is analyzed. It consists in forbidding disproportionately large penalties. The latter is achieved model-theoretically by restricting the contracts to be linear. Hence, in the restricted model, a borrower can shift repayment between periods 1 and 2 according to a single interest rate, as established by the contract.
In the basic version of the model, the authors assume that and are known to firms. Moreover, it is assumed that the partially naive borrowers are over optimistic , i.e. they over estimate their degree of self-control. The results of the basic model are the following. On the one hand, a sophisticated consumer , which correctly predicts his future behavior,
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chooses a contract that maximizes his ex ante utility. On the other hand, a nonsophisticated borrower mispredicts his future behavior. This type of consumer believes he will choose a cheap front-loaded repayment schedule, which makes the contract attractive, but in reality ends up choosing an expensive back-loaded repayment schedule. The latter enables firms to break even. The consequence of a partially naïve consumer failing to understand that he will pay a large penalty and back-load repayment is that he borrows too much. Therefore, a consumer that has a degree of naivety greater than zero, no matter how small, has discontinuously lower welfare than a sophisticated one. Notice that welfare analysis is conducted equating consumer’s welfare with self 0’s utility, following most of the behavioral industrial organization literature. Hence, when contracts are nonlinear, even small mispredictions of future preferences by the consumers can have large welfare effects.
In line with the proposed welfare-improving intervention, in a restricted market all contracts have to be linear. This implies that the borrowers have the option of paying a small fee for deferring a small amount of repayment. Hence, consumers with a small degree of naivety do not drastically mispredict their future behavior. Therefore their utility is greater than in the unrestricted market. Moreover, sophisticated borrowers achieve the highest possible utility regardless of the type of market. All together, this means that a restricted market Pareto dominates the unrestricted one.
2.5 What happens when contracts with time inconsistent consumers are non exclusive?
In the present section I will review an article that resembles Kőszegi’s (2005) article because it also considers competitive markets with time-inconsistent consumers and non exclusive contracts. However, according to Gottlieb (2008) his article distinguishes itself from Kőszegi’s one because it is more explicit in the modeling of competition and because it studies
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the welfare properties of the resulting equilibrium. Nevertheless, he does recognize certain similarities in their predictions, especially regarding the asymmetry between markets for leisure and investment goods.
DellaVigna and Malmendier’s (2004) model describes accurately observed behavior in markets for investment goods and in some markets for leisure goods. However, their model is not consistent with observations from some leisure good markets such as the tobacco or alcohol markets. According to Gottlieb, the reason for the lack of explicative power of the model in the aforementioned markets originates in a key component missing in it. He argues that the missing element in the model is that firms are typically not able to offer exclusive contracts. For example, one reason for this inability could be that the costs which the consumer has to incur in order to switch between different firms are very low.
In leisure good markets DellaVigna and Malmendier’s model predicts that the optimal contract is such that consumers receive from the firm a lump-sum transfer and then pay per-unit consumption prices above marginal cost. However, if the contract offered to the consumer is non-exclusive there is no way of preventing the consumer, after the contract is signed, from buying from another firm that charges a lower per-unit price. This means that leisure good markets in which consumers have changing preferences are not just merely the inverse reflection of investment good markets.
Let me now briefly present Gottlieb’s model. This model is a competitive version of DellaVigna and Malmendier’s model. It consists of three periods of interaction between firm and consumer. In the context of this model, consumption takes place in period 2 and it provides an immediate payoff of – in period 2 and a delayed payoff of in period 3. According to the payoffs, the goods can be classified in investment goods – or leisure goods
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– . In period 1, consumers make a “take it or leave it” offer of a contract, consisting in the two-part tariff , to the firms. is the lump-sum prices and the usage price. Both and are paid in period 2. In period 2, the consumer observes and makes the firms another “take it or leave it” offer of a contract. A second-period contract is a price contingent on consuming the good. Afterwards, the consumer chooses whether he wishes or not to consume the good. If consumption takes place in period 2, the consumer gets a payoff with expected value in period 3.
Consumers in this model also have quasi-hyperbolic discounting preferences. If β=1 consumers are time-consistent (exponential) and if β<1 they are time inconsistent (hyperbolic). A third parameter, , represent the degree of awareness that a hyperbolic agent has regarding his time-inconsistency. When the agent is naïve, when he is partially naïve and when he is sophisticated.
The incorporation of non exclusivity into the model of contracting with time-inconsistent consumers in a competitive market results in the same predictions for investment goods’ markets as the ones resulting from DellaVigna and Malmendier’s model. However, the implications for leisure goods markets do differ. In this case, Gottlieb´s model predicts per-unit consumption prices equal to the marginal cost. Moreover, it predicts welfare losses for both types of time inconsistent consumers, sophisticated and naïve, when compared to the time-consistent ones. This arises because time-inconsistent consumers have the possibility to circumvent ex-post any acquired commitment devices and it is profitable for firms to offer contracts that incentivize consumers to do so. This only arises in the case of leisure goods because the commitment device is a per-unit price above marginal cost. The latter means that another firm could offer the consumer a slightly lower price to win him over. On the other hand, in the case of investment
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goods the commitment device remains useful because it implies per-unit consumption prices below marginal cost. Hence, no other firm in the competitive market will have ex-post an incentive to offer better prices to the consumer.
An interesting implication of Gottlieb’s model is that in markets for leisure goods the effect of competition could be to decrease total surplus. The effect of competition in this case is to offer consumers an alternative in period 2 (when they consume the good) to circumvent any commitment devices acquired via the period one contract. This would not happen under monopoly because there would be no competitors offering alternative usage prices in period 2. Hence, a monopoly could effectively provide above marginal cost usage prices as a commitment device for time-inconsistent consumers in the leisure goods market.
According to Gottlieb’s model and in the case of competitive markets for leisure goods, government intervention could help to obtain an efficient allocation as a market outcome. This could be achieved by the means of a sales tax that provides consumers the incentives to act according to their long run preferences. According to the author, this policy would be relatively easy to implement since it does not require much information from a regulator and it does not depend on the degree of naivety. On the other hand, in the case of investment goods to find the appropriate tax much more information is required because it depends on the degree of consumer’s naivety. In the case of sophisticated consumers, for example, no tax is required at all. From the aforementioned, the model implies that much more opportunities for successful governmental intervention are available in the case of leisure goods’ markets than in the case of investment goods’ markets.
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According to Ellison (2006), one notable limitation of DellaVigna and Malmendier’s (2004) article is that consumers in a market are assumed to be of only one type, i.e. only markets of homogeneous consumers are considered. The following paper I will review, by Eliaz and Spiegler (2006), considers a model in which diversely naïve agents participate in the same market.
The main concern of Eliaz and Spiegler’s paper is to model contracts offered by monopolies to agents with time inconsistent preferences which differ in their degree of sophistication, i.e. the ability to forecast the change in their future tastes. According to the authors, their paper introduces the following differences with respect to DellaVigna and Malmendier’s article. In the first place, Eliaz and Spiegler focus on the problem of discriminating between diversely naïve types, whereas in DellaVigna and Malmendier’s model the firm knows the degree of naivety of the homogeneous agents in the market it faces. Secondly, DellaVigna and Malmendier restrict the firm’s contract space to two-part tariffs while in Eliaz and Spiegler’s paper the domain of feasible contracts is not restricted.
The model presented in Eliaz and Spiegler’s paper relaxes two standard assumptions: time consistency and common priors of the agents and the firm. Moreover, it allows isolating the effects of each of these two assumptions. In the model the firm or principal is the sole provider of a certain set of actions . The cost of providing this action for the firm is , which is increasing in its argument. In period 1, the agent signs a contract which specifies, for every second-period action, a (possibly negative) monetary transfer from the agent to the principal. The agent can also refuse to sign the contract and to choose some other outside option. In the latter case he is restricted to the default action . In period 2, the agent chooses an action from the set , and the principal is perfectly capable to monitor the agent’s decision.
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The agent has quasi-linear preferences over the action-transfer pairs. However, his utility changes from period 1 to period 2. In period 1 his utility from second- period actions is given by a continuous function with . On the other hand, at period 2 the agent’s utility function changes and is given by . The degree of naivety of an agent is modeled by a parameter as follows. An agent of type believes that with probability his second-period utility function will remain , and that with probability it will change to . This means that an agent of type is fully sophisticated while an agent of type is fully naïve. Partially naïve agents are those with values of strictly between 0 and 1.
As I have mentioned the model relaxes the assumption of common priors. This is a resource to model that the principal and the agent are likely to disagree on the degree of time-inconsistency of the agent’s preferences. The principal is assumed to know correctly that in period 2 the agent’s utility will be given by . The latter means that the forecast of the principal is unbiased. Nevertheless, the principal does not know the agent’s type . He only knows that is distributed according to a continuous cumulative distribution function with support . The principal’s problem is then to find the optimal menu of contracts that
maximizes his expected profits.
In the context of this model a partially naïve agent does not know whether he will maximize utility function or in the second period. Therefore, he associates with each contract two actions and two corresponding transfers: one which maximizes the agent’s net utility according to while the other one according to . The indirect utility of an agent from a contract is given by the expected u-value of each of these two actions minus the expected sum of the corresponding transfers. That the agent has time inconsistent preferences is captured by the fact that the agent evaluates his second period actions according to his first period utility function.
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The principal’s problem is to find a menu that offers action-transfer pairs to each agent type such that his profit is maximized. From the view point of the principal, the action chosen by the agent to maximize his net utility according to is “imaginary” while the one chosen according to is the “real” action that the agent will actually choose in period two. This means that according to the principal only the real action-transfer pair affects his revenue. Therefore, he uses the imaginary action-transfer pair to seduce agents into signing an “exploitative” contract which extracts more than the agent’s willingness to pay from his first period perspective. An agent accepts an exploitative contract because he believes he is getting a “gift” with probability that compensates him for the excessive payment for the real action . The value of the “gift” that an agent associates with the imaginary action increases with his degree of naivety. Therefore, the principal’s profit from the real action is also an increasing function of the agent’s naivety.
According to the model, the optimal menu of contracts offered by the principal is such that the set of agent’s types is partitioned in two intervals. Those agents that are relatively sophisticated (low ) choose a contract that commits them to choosing an action that maximizes the difference between his first period willingness to pay, and the principal’s costs . On the other hand, relatively naïve agents (high ) choose exploitative contracts. An interesting feature of the model is that it implies that the relatively sophisticated agents, who choose the non-exploitative contract, do not exert any informational externalities on the partially naïve agents. The latter implies that as long as there is a positive surplus in the interaction with the more sophisticated types of agents, the optimal menu does not exclude any type of agents.
As mentioned above, in Eliaz and Spiegler’s model the assumptions of time-consistency and common priors are relaxed. The authors also try to disentangle the effects of each of these two assumptions. According to their analysis, time consistency by itself does not preclude the use
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of exploitative contracts. However, if agents are time-consistent the principal’s contracts with the sophisticated types might exert informational externalities on the naïve types. Therefore, the optimal menu might be unable to include all types as it is possible when agents have changing tastes. On the other hand, if dynamic inconsistency is retained while restoring common priors the principal’s menu consists only of a single contract. This implies that there is no discrimination between types and also no exploitation.
The authors also discuss the possible effects to welfare implied by their model. However, they are very cautious about this type of analysis when agents have dynamically inconsistent preferences. They claim that in this context welfare judgments become conceptually problematic because it is unclear if the point of view of the first or second period self should be applied. Additionally, the non-common-prior assumption poses a further ambiguity by making unclear whose prior should be applied in the evaluation of ex ante welfare. The authors’ choice to adopt the principal’s prior belief, i.e. they assume that the agent ends up choosing according to . When taking the view point of the first period self the model unequivocally predicts that consumer surplus decreases with the degree of naivety as more naïve agents are more heavily exploited by the principal.
The present paper introduces a model in which agents in the market have different degrees of naivety. However, the assumption is made that all agents experience a change in their utility. According to the authors themselves, an interesting extension to their model would be to allow for agents to differ in two dimensions: their subjective probability that their tastes will change and the objective probability of this event happening.
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In this section I will introduce a model similar to the one presented in Section 2.5. The main difference is that in the model of this section the firm knows the degree of the consumer’s frequency naivety whereas in the model of Section 2.5 this was the consumer’s private information. This is the model that I will extend, in Section 4, to explore the effect of a policy which restricts the payment differential for the actions available to the individual at the time of consumption.
3.1 Optimal price schemes for sophisticated consumers
First, it will be convenient to examine the optimal price scheme that result from the firm’s problem when it faces a fully sophisticated consumer. This will serve as a benchmark to contrast the results that arise from the same problem but with partially naïve consumers. I will follow Spiegler (2011) in the derivation of the optimal action-payment pair, , for a fully sophisticated consumer. Note that is the action that the consumer chooses and is the payment he makes in period 2. Additionally, the cost of production incurred by the firm in period 2 is given by .
Since the firm is fully rational and the consumer is sophisticated, they both agree that the consumer’s second-period preferences will be given by the utility function . Moreover, since is the action that the consumer will choose in period 2, the firm can set to be arbitrarily large for any , without loss of generality. By doing so, the firm provides a commitment device that forces the consumer to choose provided that he accepts the price scheme in period 1.
The latter implies that the firm’s problem is reduced to the following maximization problem:
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subject to
(1)
Notice that according to constraint (1) the agent’s net utility, from the perspective of period 1, has to be greater than or equal to zero. This constraint also implies that the payment, , that the firm chooses has to meet the condition Since the profit of the firm is monotonically increasing in , it follows that (1) has to hold with strict equality.
Therefore, the solution to this problem is given by
3.2 Optimal price schemes for time-inconsistent and partially naïve agents
Now I will introduce a model on first-best monopoly pricing with time-inconsistent agents (Spiegler 2011). In the context of this model the firm knows it faces a partially naïve consumer of type . At its core, the model incorporates the notion of frequency naivety introduced by Eliaz & Spiegler (2006).
Under this notion, a partially naïve consumer is one who assigns probability to having his preferences represented by utility function in period and probability that they will be instead represented by utility function . Moreover, because the agent’s second period utility function is , his degree of naivety is increasing in the parameter .
The monopolist’s problem is to derive the optimal price scheme against a partially naïve consumer. The price scheme can be reduced to a 4-tuple , where is the action-payment pair in the “imaginary” event that the consumer’s second-period utility function
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is and is the action-payment pair in the “real” event that the consumer’s second-period utility function is .
As before, the firm incurs the cost of production , which is a function of the action chosen by the individual in period 2. Notice that the firm is assumed to be fully rational in contrast to the consumer. This means that the firm knows that the individual is going to have the utility function in the second period. According to Smallwood and Conlisk (1979) this assumption is reasonable because firms have more resources to better make predictions and they have more incentives to do so.
The setting of the two-period interaction between the firm and the consumer is as follows. In period 1, the firm offers the consumer a continuous menu of actions and to choose from. To each of the aforementioned actions corresponds a payment and respectively, which the individual incurs at the moment of consumption in period 2. Once the consumer chooses actions and , an exclusive contract is signed between the agent and the firm. Note that fulfillment of the contract is enforced. Then, in period 2, the consumer decides between actions and and makes the payment corresponding to the action chosen.
Having said the latter, the monopolist’s maximization problem is given by:
subject to the following incentive compatibility constraints
(2)
(3)
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(4) The first constraint, (2), is an incentive compatibility constraint for the case that the consumer has utility function in the second period. It basically implies that, given , the utility of choosing the “real” action minus its payment, , should be greater than the utility of choosing the “imaginary” action minus its payment, . Moreover, it ensures that the consumer indeed chooses when his preferences are given by . The second constraint, (3), is the corresponding incentive compatibility constraint for the case that the consumer has utility function in the second period. It ensures that the consumer chooses when his preferences are given by (Spiegler 2011).
Finally, the third constraint, (4), is just an individual rationality constraint. It implies that the agent’s net expected utility, according to his first period preferences given by , has to be greater or equal to zero in order for him to enter into a transaction.
I will now solve this model as it is done by Spiegler (2011). First, notice that the participation constraint, (4), must be binding in the optimum. This follows from the fact that if this were not the case, the firm could raise both and by an arbitrarily small while all the constraints would still continue to hold. This would result in an increase of the firm’s profit.
Moreover, the first incentive compatibility constraint, (2), must also be binding in the optimum because the firm could raise by an arbitrarily small and this would raise its profit while satisfying the remaining constraints. The interpretation of (2) being binding is that the firm is able to fully extract the consumer’s second-period willingness to pay for the “real” action relative to the “imaginary” action . This means that the consumer’s second-period surplus, whenever the agent’s utility function is , is fully extracted by the firm relative to action .
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From the fact that the participation constraint, (4), is binding, it follows that
(5) Moreover, because the first incentive compatibility constraint, (6), is also binding it follows that
(6)
Substitution of (5) into (6) results in
(7) Then, the combination of equations (7) and (5) yields
(8) Using (7), the firm’s maximization problem can be simplified to
subject to the following incentive compatibility constraint
(9)
Ignoring, for the moment, constraint (9) the solution to the firm’s maximization problem is given by
(10)
(11)
The next proposition shows that this indeed is the solution to the problem.
Proposition 1. Given the solution to the monopolist’s problem, as given by equations (7), (8), (10) and (11), the incentive compatibility constraint, given by (9), will always hold.
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Proof. Substitution of the expressions for and , given by (8) and (7) respectively, into the incentive compatibility constraint, given by (9), results in the following expression:
(12)
Notice that , as given by (11). Therefore, the inequality given by (12) will always hold. ■
The actual action that the consumer will choose in period 2 is given by (10). According to this equation the consumer will choose an action that maximizes a weighted surplus. Since and are continuous functions, as approaches zero the action-payment pair converges to the action-payment pair that results from the interaction of a firm with a fully sophisticated consumer1. We can observe this by comparing equations (10) and (7) with the corresponding solution to the monopolist’s problem when facing a sophisticated consumer.
Spiegler (2011) leaves it as an exercise to show that the monopolist’s maximal profit is increasing with . I show this here.
Proposition 2. The monopolist’s maximal profit is increasing with .
Proof. The profit of the monopolist is given by
. According to the envelope theorem only the direct effects of change in an exogenous parameter need to be considered, even if the parameter may enter the benefit function indirectly through the optimal values of the endogenous variables. Therefore, using this theorem, the effect total derivative of with respect to is given by:
1 Similarly, as approaches one, the resulting price scheme from the firm’s maximization problem
converges to the price scheme that would result from the interaction of a firm with a fully naïve consumer. This is intuitive, since the firm’s problem when facing a fully naïve consumer is the same up to the participation constraint which reduces to . For further discussion on optimal price schemes for fully naïve consumers see Spiegler (2011).
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Notice that this will always be positive if constraint (9) is to hold. This follows from the fact that substitution of the expressions for , given by (7), and for , given by (8), into (9) yields:
Hence, we observe that the derivative with respect to of the monopolist’s maximal profit is greater or equal than zero. This means that the monopolist’s maximal profit is increasing with respect to , i.e. the degree of naivety of the consumer. ■
4. Can the government protect the consumers?
Now, I will consider the effects of a proposed policy intervention. Namely, I will analyze the effects of restricting the spread, or difference, between the payments and . Formally the latter implies adding the following additional constraint to the firm’s maximization problem:
Notice that is an exogenous parameter set by the government.
I conduct my analysis using two examples which specify concrete functional forms for , and . I proceed in this manner because solving the model with the additional constraint imposed by the government for the general case, in which the utility and cost functions are left unspecified, is a task that is better suited as a second step, once some understanding of the effects of adding the extra constraint is gained.
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In the present section I will assume the following simple functional forms for , and : , and . This implies that a naïve consumer underestimates the utility that the consumption of the good will provide him in period 2 in the case that and underestimates the utility that the good will provide him in period 2 in the case that . A fully sophisticated consumer on the other hand knows that his utility function will be given by in the second period. Moreover, I will assume the following values for the parameters and : and
4.1.1 Optimal price schemes for time-inconsistent consumers with a restriction by the government
I will start analyzing what happens if the government decides to set the restriction that the difference between the two payments and should not be greater than . Given the functional forms we have assumed for , and , the monopolist’s maximization problem when the government imposes a restriction is given by:
subject to the following incentive compatibility constraints the following participation constraint
(13)
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(14)
Notice that the participation constraint, given by (13), is binding in the optimum. If it were not, the firm could raise both and by a small , increasing the profits of the firm. Moreover, all the constraints would still hold, including the restriction imposed by the government and given by (14).This yields the following expression for :
(15)
Using the fact that we have an expression for that will always hold, given by (15), we can reexpress the monopolist’s problem as follows:
subject to the following incentive compatibility constraints
(16)
and the additional constraint imposed by the government
I will now solve the monopolist’s problem by making some of the restrictions binding while leaving the remaining ones “turned off”. The latter means that there are in principle eight different types of solutions. However, the solution must be such that at least one restriction is binding, because otherwise the “real” action payment pair that maximizes the firm utility is given by . The latter cannot be a solution, because it never satisfies the three constraints. Therefore, I will find only the remaining seven possible solutions.
35 4.1.2 First incentive compatibility constraint binding
If the first incentive compatibility constraint, given by (16), is binding, this means that the monopolist’s problem is given by
(17)
subject to the following incentive compatibility constraint
(18)
and the additional constraint imposed by the government given by
(19)
Proposition 3. The optimal tuple that results from solving the firm’s problem, as given by expression (17) is the following:
(20) (21) (22) (23)
Proof. The monopolist chooses such that the profit function, given by
, is maximized. This implies that the two arguments of the benefit function must satisfy and
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respectively. The solution to these two maximization problems is given by expressions (20) and (21).
Since the first incentive compatibility constraint, given by (16), is binding it follows that Substitution of (20) and (21) into the latter expression for results in and new expression for in terms of the parameters only; this is given by (23). Substitution of the expressions that I have obtained for , and , given by (20), (21) and respectively, into the expression for , given by (15), that I derived in Section 4.1.1 results in a new expression for , given by (22), in terms of the parameters. ■
Note that this solution is the same as the one presented in Section 3, with the difference that it is now possible, given that I have specified concrete functional forms for the utility functions, to further simplify the expressions for the optimal tuple. Additionally, note that this solution can be considered an exploitative one, because the fact that the first constraint is binding implies that the consumer’s second-period surplus in state is fully extracted, relative to the “imaginary” action .
Proposition 4.2 The constraint , given by (18), will be satisfied by the optimal tuple
, given by (20)- (23), for all values of the parameters and .
Proposition 5. The constraint , given by (19), will be satisfied by the optimal tuple , given by (20)- (23), for values of that satisfy
(24)
2
The proof of this proposition is presented in the appendix. In general, if the proof of a proposition does not follow right away, it can be found in the appendix.
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Proposition 5 implies that the solution to the firm’s maximization problem presented in the present section, i.e. the solution when only the first incentive compatibility constraint is binding, is only available to the monopolist for values of , imposed by the government, that satisfy (24). In Figure 1, I present the threshold value of for which the solution is valid as a function of for different values of .
Figure 1: Threshold value of for which the solution holds as a function of for different values of
The benefit of the firm in period 2 is given by: Substitution of the expressions for and , from the solution to the monopolist’s problem, into this expression results in:
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(25)
Substitution of the parameter values and in equations (20) - (23) results in
the following optimal tuple: Furthermore, with the aforementioned parameter values, has to satisfy the following inequality in order for this solution to the firm’s maximization problem to hold: . Moreover, it is straightforward to see that the optimal tuple fulfills the latter condition , because computation of the solution’s implied delta results in: . Finally, substitution of and into (25)
results in the following profit for the firm: .
4.1.3 Second incentive compatibility constraint binding
If the second incentive compatibility constraint, , is binding, this means that the firm’s problem reduces to the following
(26)
subject to the following incentive compatibility constraint
(27)
and the additional constraint imposed by the government given by
(28)
Proposition 6. The optimal tuple that results from solving the firm’s problem, as given by expression (26), and using constraint (27), is the following:
39 (29) (30) (31)
Proposition 7. The constraint , given by (28), implies that the solution to the firm’s maximization problem will hold for values of that satisfy
(32)
When the solution is available to the monopolist, the resulting profit is given by
