Feb 18

Information economics I: Complete, incomplete information

Complete information and incomplete information are terms widely used in economics, especially game theory and behavioural economics. We say that there is complete information when each agent knows the other agent’s utility function and the rules of the game. As Luce and Raiffa put it in their “Games and Decisions: Introduction and Critical Survey“, 1957, complete information, understood as the situation where “each player is fully aware of the rules of the game and the utility functions of each of the players”, is a central assumption of game theory. However, this situation and its definition does not consider the awareness of each player, which is covered by the term “common knowledge“, which means that each player is aware that the other players know the rules and every utility function.

Incomplete information, also known as asymmetric information, refers to the contrary, where not all players know each other’s utility functions. John Harsanyi developed the theory of incomplete information in his “Games with Incomplete Information Played by ‘Bayesian’ Players”, 1967.

Feb 18

Lope Gallego

Information economics I: Information economics

Economics of information, or information economics, belongs to the field of microeconomics and it studies the importance of information in Economics. The neoclassical theory was developed around the assumptions of perfect information and the absence of uncertainty, and although this simplification allows modelling the reality, it is unrealistic. Economics of information is the result of an alteration in the usual neoclassical analysis, and it integrates the posibility of imperfect information that will result in asymmetric information.

Asymmetric information occurs in those situations in which an agent has greater information, either before or after, the interaction between agents happens. The study of these situations is of special relevance mainly to avoid the conflicts that will arise from them it. If an agent has inside information previous to the economic relations, ex-ante assymetry, adverse selection may occur. If the inside information is after the economic relations, post-ante, it will beconsidered moral hazard.

The importance and value of information within economics is huge. It eliminates risk and uncertainty, and it makes it possible to take better choices that will report higher yields. The less risk and uncertainty there is, the higher the utility will be valued.

The study of economics of information is key for many economic fields including game theory, agency theory or contract theory.

Feb 18

Lope Gallego

Economics of information

The study of economics of information is key for many economic fields including game theory, agency theory or contract theory.

Feb 18

Lope Gallego

Moral hazard

Moral hazard is a case of asymmetric information. It occurs when both parties (usually an agent and a principal) assign or are subject to a different probability of a same (normally adverse) event occurring. The behaviour of the agent changes ex-post, after a contract is signed and as a consequence of their new, advantageous position. As economist Paul Krugman puts it, moral hazard refers to “any situation in which one person makes the decision about how much risk to take, while someone else bears the cost if things go badly”.

In order to clarify this concept, let’s take a quick look at an example: insurance premiums. The person taking out an insurance policy is at an advantage versus the insurer, who therefore charges a premium for the risk derived from their imperfect information, aside from any mark-up designed purely in order to generate benefits.

Lucy has never smoked in her life, because she has always been concerned about expensive hospital stays taking away disposable income that could be better spent on her butterfly-collecting hobby. However, her cautious friend who works for a large insurance company talks her into subscribing to a healthcare policy. After this, Lucy decides that as her health is now a fixed cost (she pays the same premium whether she becomes ill or not), she might as well start smoking, drinking, and driving like a lunatic. In fact, she swaps her butterfly collecting hobby for cliff jumping. Lucy’s advantage is ex-post.

Moral hazard also appears in other markets, such as in the credit market. In their article about credit rationing, economists Joseph E. Stiglitz and Andrew Weiss explain how raising interest rates will make indebted people take higher risks.

Feb 14

Lope Gallego

Adverse selection

Adverse selection is a case of asymmetric information. It occurs when both parties assign or are subject to a different probability of a same (normally adverse) event occurring. In this case, the agent that has the best information is clearly at an advantage. We say that this advantage is ex-ante because, contrary to moral hazard, the advantage occurs before the ‘contract’ (real or otherwise) is signed.

Lucy takes out a health insurance plan but ‘forgets’ to tell the insurer that she has been a heavy smoker, drinker and lunatic driver for 50 years. Lucy knows that her probability of contracting heart or lung disease is significantly greater than the national average. The insurance company does not know that she is a hell raising grandmother and base their calculation for her premium based on the probability of an average person being hospitalised for a long stay, an average person who is probably much younger and much healthier than Lucy. Lucy’s advantage is ex-ante. At the same time, healthy Hanna who is 30 will be put off by the price of the premiums because of people like Lucy and will probably never subscribe to a policy.

Of course, insurance companies have ways of getting around this to a certain extent. But an insurance company is never going to be able to segment their policies fully: when was the last time you were asked about your driving style when subscribing to a dental plan? And most sane people won’t risk their life just because their healthcare is covered (in fact, most countries with a free, national health service have longer life expectancies and shorter hospital stays than countries that don’t), but how many of us park our friend’s car more carefully because we are not insured in it?

There are ugly consequences deriving from this example, which can be generalised to markets other than the insurance market, as seen in The Market for Lemons, an article by George Akerlof, or in the article about credit rationing by Joseph E. Stiglitz and Andrew Weiss. Since people like Hanna are asked for premiums higher than the ones they would get with perfect information, this people will decide not to take insurance. The lack of healthy and careful people will make the ‘average person’ considered by insurers be less healthy and careful, which will force insurers to raise premiums. This will encourage people healthier than this ‘average person’ to stop paying insurance, which would start the process all over again. Theoretically, the healthcare insurance market would disappear, since no one wants to pay premiums that high.

Feb 14

Lope Gallego

Joseph Bertrand

Joseph Bertrand Joseph Louis François Bertrand, 1822-1900, was a French mathematician and economist who was Professor at the École Polytechnique and a member of the Collège de France.

With regards to economics, Bertrand is mostly known for reviewing the articles “Théorie mathématique de la richesse sociale” and “Recherches sur les principes mathématiques de la théorie des richesses” by Leon Walras and Antoine Cournot respectively, which were published at the Journal des Savants in 1883. He criticized Walras’ process of tâttonnement, as he argued that in real life there are transactions that take place under non-balanced situations and so we can consider a non-determination in pricing. The critique to Cournot had greater value for economics, as it further developed both game theory and oligopoly theory, creating another model for duopolies, known as the Bertrand duopoly model. The main difference between Cournot’s view on duopolies and Bertrand’s is that the latter assured that duopolies competition would be linked to prices and not to their production quantity, as was proposed by Cournot.

Feb 13

Lope Gallego

Credit Rationing in Markets with Imperfect Information

In their 1981 paper, “Credit Rationing in Markets with Imperfect Information”, Joseph E. Stiglitz and Andrew Weiss define a situation similar to the case of The Market for Lemons, an article by George Akerlof, except in the financial markets. In this case, it is the ‘seller’ of credit who pulls out of the market because of adverse selection.

Credit rationing implies that credit is not available for all those who want it, at any interest rate. If markets worked perfectly, supply and demand for credit would fix a level of interest rates at which lenders would be happy to assume the risk, and borrowers would be happy to accept it in order to fund their projects. However, accepting high interests can also be a signal: if we have a good credit rating and are not desperately in need of funds, we probably won’t accept excessively high interest rates, we will simply put off our projects or purchases. Really needing money is intrinsically a bad sign for lenders, which can mean that when interest rates rise above a certain level, financial agents can assume that you are a ‘lemon’, and that the risk of you defaulting on your credit is too high. The chances are you are either acquiring the loan in good will but in dire financial straits (which means that despite your best efforts you’ll probably default anyway) or that you simply don’t care about the interest rate you are charged because you have no intention of paying the loan back in the first place.

Supposing there are two types of borrowers, As and Bs, who both want to invest in similar projects with a different level of risk, and assuming that As invest in ‘safer’ projects than Bs although the mean return is the same, say 110. As have a range of possible ROIs, ranging between 90 and 120 for their investment, meaning that if they invest 100, then for interest rates over 20% their profit will be 0 even if their project is successful. However, Bs also have a mean return of 110 for their 100 invested, but with a range of between 70 and 180, with a distribution that skews their average ROI to 10%, just like the As. This means that, if they are really optimistic (as riskier investors tend to be), they can potentially accept interest rates of up to 80% and still break even.

Assuming both As and Bs default on their credit if their ROI is negative (below 100), which is an assumption that banks are fond of making, then they will have a natural cut off point, assuming that all those willing to accept rates of over 20% are therefore risky investors, Bs. When interest rates approach 20%, credit will simply dry up as all those willing to accept higher rates and still borrow will be singled out as probable defaulters.

Feb 12

Lope Gallego

The Market for Lemons

“The Market for ‘Lemons’” is a key article written by George Akerlof in 1970, which aims to explain some of the market failures derived from imperfect information, in this case asymmetry. The paper itself is available on the bibliography and is characterised by its approachability and humour: as Akerlof himself stated, he lacked the mathematical dexterity to fully model the problem (although this has later been done by a host of economists, of which Hal Varian’s analysis is probably the most well known).

We are presented with the problem of someone who wants to buy a car, and decides to scout the used car market for a bargain. The market itself is composed of two types of cars: those that are being sold in good faith and those that are being sold off because they are known to be unreliable: these are the ‘lemons’ (in US slang). The seller, of course, knows how good the car is: they’ve had time to decide. The buyer, however, comes to the market blind: all they have to go on is the average quality of the used car market (which Akerlof defines as μ) and the price of the car, p. Obviously, all similar models of cars need to be sold for an identical price, p. If we sell below market price, be it a lemon or not, we are sending out signals that the car is worth less than the market price, leading buyers to assume it is a lemon. However, it is also logical to assume that, if we know the car we are selling is reliable, we will want more than the average market price, because that average includes a proportion of lemons.

Therefore, two prices are naturally set up: one for the lemons (say 1000$), and one for the good cars (sometimes called ‘cherries’ or ‘plums’), say 2500$. Below these prices, the seller will not be willing to let go of their vehicle and would derive greater utility from keeping it. Above this price, they would rather sell. The problem is, therefore, that competition will naturally drive down the price to one below which the honest seller are willing to go. Buyers will not be willing to pay 2500$ knowing that there is a probability that what they are buying is really a lemon: they need to reflect the premium related to the uncertainty derived from the asymmetrical information in a lower price. Lemon seller will be able to assume this drop in price, all the way down to 1000$, but honest sellers will not. Eventually, all that will be left in the market will be an orchard of lemons, but no cherries nor plums.

Analytically, if the quality of lemons is 0 and the quality of cherries is 1, and there are equal amounts of each, then

Buyers are willing to pay up to 3/2 q for a car with quality equal to q, and sellers are willing to sell for q (we are assuming p=q for simplicity’s sake, but the ratio could be any we decide to set). This means that, as

then

which means that no cars will be sold.

Feb 11

Lope Gallego

Game theory III: Repeated games

In game theory, repeated games, also known as supergames, are those that play out over and over for a period of time, and therefore are usually represented using the extensive form. As opposed to one-shot games, repeated games introduce a new series of incentives: the possibility of cooperating means that we may decide to compromise in order to carry on receiving a payoff over time, knowing that if we do not uphold our end of the deal, our opponent may decide not to either. Our offer of cooperation or our threat to cease cooperation has to be credible in order for our opponent to uphold their end of the bargain. Working out whether credibility is merited simply involves working out what weighs more: the payoff we stand to gain if we break our pact at any given moment and gain an exceptional, one off payoff, or continued cooperation with lower payoffs which may or may not add up to more over a given time. Therefore, each player must consider their opponent’s possible punishment strategies.

This means that the strategy space is greater than in any regular simultaneous or sequential game. Each player will determine their strategies or moves taking into account all previous moves up until that moment. Also, since each player will take into account this information, they will play the game based on the behaviour of the opponent, and therefore must consider also possible changes in the behaviour of the latter when making choices.

Repeated games provide different payoffs at each repetition, depending on each player’s moves. Since these payoffs are given at different points in time, in order to analyse repeated games, we must compare each player’s discounted sum of payoffs, which for infinite repetitions and finite repetitions are calculated using the following formulae:

Where: -P: the discounted sum of payoffs;

-t: the number of the repetition being considered;

-n: the total number of repetitions for finite repeated games;

-p_t: the payoff at the repetition being considered;

-r: the discount rate.

Repeated Prisoner’s dilemma:

In the game known as the Prisoner’s dilemma, the Nash equilibrium is Confess-Confess (defect-defect). In order to see what equilibrium will be reached in a repeated game of the prisoner’s dilemma, we must analyse two cases: the game is repeated a finite number of times, and the game is repeated an infinite number of times.

When the prisoners know the number of repetitions, it’s interesting to operate a backwards induction to solve the game. Consider the strategies of each player when they realise the next round is going to be the last. They behave as if it was a one-shot game, thus the Nash equilibrium applies, and the equilibrium would be confess-confess, just like in the one-time game. Now consider the game before the last. Since each player knows in the next, final round they are going to confess, there’s no advantage to lie (cooperate with each other) on this round either. The same logic applies for prior moves. Therefore, confess-confess is the Nash equilibrium for all rounds.

The situation with an infinite number of repetitions is different, since there will be no last round, a backwards induction reasoning does not work here. At each round, both prisoners reckon there will be another round and therefore there are always benefits arising form the cooperate (lie) strategy. However, prisoners must take into account punishment strategies, in case the other player confesses in any round.

Collusion agreement games:

If we assume the game can be played ad infinitum, we can apply it in a collusion agreement game, where two firms collude, forming a cartel. Consider two firms (a duopoly) that may either behave as Cournot duopolists earning profits π_Cournot each, or collude and act as a cartel, earning π_Cartel each, which correspond to the profits of a monopoly divided into the number of firms colluding (two in our example).

In this case, we simply need to apply the formula for calculating an infinite sequence and a discount factor to compensate for the fact that the gains to be derived are over time (accounting for impatience, inflation, loss of interest, etc.):

The left hand side represents the payoff derived from collusion, which can be held infinitely over time, with δ being the discount factor to bring future benefits forward to the present. For our threats or offers to be credible, this left hand side must be greater than the right hand side, which represents the one off payoff to be gained from deviating and breaking our cartel. The higher δ is, the higher the value assigned to future benefits and therefore the greater the chances of collusion. It is worth reminding here that fair competition is regulated in almost all countries, with cartels being banned, so most markets that lend themselves to reduced competition and price fixing are closely monitored.

Although this example is widely used in game theory and for the analysis of market structures, it can be easily seen that it does not represent a real situation. Let’s consider the same example: any of the colluding firms might deviate, in order to dump more output in the market at lower prices, in order to gain market share. This move will allow that firm to sell more products than the other firms, which directly contradicts Cournot’s premise that each duopolist will produce the same quantity. Therefore, considering a Stackelberg duopoly might seem more realistic. This would obviously change the analysis and outcome of the game.

Feb 11

Lope Gallego

Game theory III: Collusion

Collusion makes allusion to the cooperation between different firms. This cooperation leads to a restrain of market competition, in any of its forms, which translates into higher profits for the firms in detriment of consumer’s welfare. A cartel is an example of firms belonging to the same industry structure which collude to some degree in setting prices and/or output levels. Agreements which have as their object or effect the prevention, restriction or distortion of perfect competition are prohibited. Such agreements include, but are not restricted to, activities such as:

-fixing purchase or selling prices or any other trading condition, directly or indirectly;

-controlling or limiting production levels, markets, technological advances or investments;

-sharing markets or resources supplies.

Legislation in different countries may consider different scenarios and penalties for such agreements, but the main idea is clear: firms behaviour shall not affect the correct functioning of market forces.

A clear example is to consider an industry where there are only two firms (duopoly). Both firms will set their levels of output and prices with the objective of maximizing their joint profits. Many strategies can be used in order to maximize profits which would lead to a multiple Nash equilibria solution. As seen in the figurebelow, collusion maximises aggregate profits for both firms, since the isoprofit curves are tangent. It’s a better equilibrium than the one in Cournot duopolies or Stackelberg duopolies.

However, cartels are not stable. There will always be incentives for each firm to trick the other, and change their output and/or price level in such a way, that they’ll increase their own profit in detriment of the other’s. To avoid this practice, any deviation by any party should be instantly punished; this is known as a trigger strategy. James W. Friedman demonstrated in his paper “A Non-cooperative Equilibrium for Supergames”, 1971, that in this context of infinite interactions, it is possible that collusion occurs due to this punishment strategy. That is, the cartel may endure as long as punishment strategies are so devastating that the benefits derived form deviation would end up being smaller than the benefits of keep colluding. J. W. Fridman put this idea in what is known in game theory as Folk theorem:

The sustainability of the equation will depend mainly in two factors: the credibility of the threat of punishment, and the discount factor. The former is easily understood as a credible threat will ensure no deviations are made, and the latter is related with how much does each party value the profits obtained from the results of following a collusive strategy, compared to the possible profits of changing their strategy.

Factors that guarantee collusion stability:

There are number of factors that affect this collusive equilibrium, such as:

-Number of firms in the market: the higher the degree of concentration in a market the higher the incentives to collude. Firms in highly concentrated markets will tend to collude since all the profits will be distributed amongst fewer firms.

-Multimarket contact: if firms compete in more than one market, the collusive agreement will be more stable. Firms that compete with other firms over many markets can establish trigger strategies that can be applied in all these markets, which will create a more devastating punishment strategy.

-Market transparency: the more transparent a market is, the easier it is to ensure that every firm is following the same strategy and is not deviating from the deal. Collusion will be more difficult in industries where it is harder to detect changes in firm’s prices or output.

-Asymmetry between firms: the bigger the asymmetry between firms, the harder it is for collusion to take place. If firms have different cost structures, the one with the lowest costs will be incentivised to lower its prices, and thus cause the other firm to have to exit the market.

Collusion agreement games:

In game theory, collusion agreements can be described using the extensive form, as depicted in the adjacent game tree. In this case, two firms share the market, already colluding and maintaining high prices. Each firm can decide to stop colluding and start a price war, in order to increase their market share, even force the other to quit the market. Firm 1 can either keep colluding with firm 2, or start a price war. If firm 1 decides to keep colluding, firm 2 will need to make a decision. If they both agree to collude, they will get 5,5. However, if one of them decides to start a price war, the set of payoffs will be either 4,3 or 3,4, depending on which one starts the war (and therefore acquires a greater market share). It’s easy to see that collude-collude is both the Nash equilibrium and a Pareto optimum situation. This result may change when considering repeated games, as seen before.