Friday, October 19, 2012

The Diffeq From Hell

As I sit here composing the morning schtick, ungraded midterm exams quietly beckon to reveal their hidden truth.  Today I prefer ignorance.  The truth may be too much to bear.  Instead I cling to the belief that with clever use of Excel to exposit the math models and let the students get their fingers dirty with them, students can learn some interesting economics, stuff that 20 years ago we'd only teach to graduate students and then only in a formal way.

The current topic is bargaining, not unlike what you do at a flea market or when buying or selling a house.  The buyer knows her value.  The seller knows his cost.  Each is uncertain about the player on the other side of the table.  The operative questions are: (1) will they end up walking away or reaching a deal, and (2) if they do reach a deal, at what price?

Last time when I taught the course I was perfectly content to cover the model about bargaining as it is done in textbook.  In that setup there are two types of potential buyers, high value and low value, and there are two types of potential sellers, low cost and high cost.  In other words, on each side of the market there is a good type and a bad type.  The interesting economics happen when the parameters are set so good types should always reach a deal but bad types should only reach a deal when their counterpart is a good type.  Further, the setup should be symmetric so as not to confer bargaining power on one side or the other.

Considered in isolation, the 2x2 model is fine to illustrate the ideas.  But this time around in looking at it I find it clunky.  We've already done a model of insurance (that I did not cover last semester) where each potential insured is one of two types, high risk or low risk.  In that model it is reasonable to assume that the insurance providers know the probability distribution across types, as that comes from population averages and insurance companies are in the business of knowing what the population averages look like.  In the bargaining context, however, it is far less clear from where beliefs over the counterpart's type emerges.  Yet those beliefs are a critical determinant of the outcome, which starts to make the model look like a house of cards.  There is a further issue that in the insurance model the insureds are risk averse, but in the bargaining model the players are assumed risk neutral.  One might want to respond that risk neutrality is assumed to keep the model tractable but, really, the math is not much harder under risk aversion.  The real reason is that one does not want differences in risk attitudes to confer bargaining power, yet risk aversion typically varies with circumstance, so that one side or the other has an idiosyncratic bargaining advantage seems unavoidable.  To eliminate the bargaining power issue, each player should have the same risk aversion, independent of circumstance.  Risk neutrality is then a reasonable approach in the modeling to achieve that end.

So I start to ask myself whether I can do a more elegant version of the model than the textbook authors provide.  This might seem like hubris on my part.  The authors, Paul Milgrom and John Roberts, are very good and well known economists.  In that dimension, I couldn't carry their jockstraps.  But their textbook doesn't have the insurance model I mentioned above, so building a bridge between the two models is not an issue they confront.  Further, they are explicating in the traditional way while I  have Excel at my disposal.  The development of an alternative model puts me in the mindset I had as an assistant professor when I was doing econ theory research --- if for whatever reason you don't like how a model is designed, then build an alternative model more toward your liking.  (You can take the boy out of economics....)   But I fear this going back to a research perspective is deleterious to the teaching, because this urge toward a better model is driven by an aesthetic that is transparent to me but is likely opaque to the students.  The students want some intuition that helps them penetrate the ideas.  Beyond that the details of the model are extra baggage they could do without.

Let me get to the economics punchline (though not yet to the punchline of the story).  At the 60,000 foot level, the message is that to get the incentives right (meaning that the choice the individual makes reveals the type of the individual in a way anticipated by the mechanism), it is necessary to produce some inefficiency relative to the case where the individual's type is known by others ahead of time.  Further, the inefficiency manifests in a particular form - quality deterioration for low end customers.  In the insurance example the result is that low risk individuals face very high deductibles or find no insurance available whatsoever.  This is so high risk types don't pretend to be low risk.  The insurance example is an instance of Akerlof's Market for Lemons or what economist's have termed adverse selection. The same underlying economics is afoot when talking about why when flying on an airplane coach is so often such a crappy experience or why in purchasing a new cable tv package the basic coverage offers such a limited variety of channels.  Having modeled this once in the insurance case to show how the mechanism separates the types, one does not need to model it again and again as the context changes.  The other contexts can be discussed without the aid of a model.  Doing  so helps to emphasize the underlying principle in operation.

This brings me back to bargaining and two sided asymmetric information.  The operative pedagogic question is whether this represents a significant enough change to justify doing a new model from scratch and what lessons are to be learned from it that aren't in the single sided asymmetric information model.    I believe that, yes, doing the bargaining model from scratch is justified and that the main lesson from it is that the inefficiency - parties walk away from a deal when there are actually gains from trade to be had - disproportionately impacts middling types.

In the bargaining model it is necessary to focus on couplets, each comprised  of a buyer and a seller, and make comparisons across them.  For example, if when trade occurs the price was such as to split the difference between the buyer's value and the seller's cost, then holding the sum of value and cost fixed so that if trade occurs the price would remain unaltered, how does the likelihood of trade change as the difference between value and cost increases?  Since that difference measures the gains from trade, one is apt to intuit that as the difference gets larger, trade gets more likely.  That turns out to be true.  If this were the main insight to be had from the bargaining model, it would hardly be worth doing.

What of this other comparison; holding the gains from trade fixed how does increasing the sum of cost plus value impact the likelihood of trade?  One might intuit that it shouldn't matter, that only the gains from trade matter.  It sounds good, but it is wrong.  Something else happens.  The parties are not symmetric in their contribution to the likelihood of trade.  But to even begin to consider why, it is necessary to go beyond the simple two types model.  There needs to be a type  in the middle, preferably several types in the middle.  Having made that concession, however, it really is better to go all the way to a continuum of types approach, since then one can use geometry in the representation to facilitate understanding of the issues and to rely on calculus techniques to perform the analysis when that is needed.

The model I had in mind has the cost realization, c, a draw from the uniform distribution on the interval [0,1] and the value realization, v, also a draw from the same distribution, with the distributions of the two random variables independent.  Thus the pairs, (c,v), can be found in the unit square and under complete information the pairs where the players walk away from the deal are below and to the right of the main diagonal of the square, since for such pairs cost exceeds value.

In this model it is straightforward to consider the benchmark case where the seller can set a take it or leave it offer for the buyer.  This translates to monopoly pricing under constant marginal cost and linear demand.  In that model the monopoly price is midway between the marginal cost, c, and the demand intercept, 1.  Hence buyers with v > (1 + c)/2 purchase at this price.  Otherwise the buyers walk away.  One can trace out the set where trade occurs as c varies.  It too is a a triangle, one with half the area of the triangle where trade occurs under complete information.

This benchmark is useful, as is the analogous benchmark that arises when it is the buyer who can make a take it or leave it offer for the seller.  Again the triangle where trade occurs has half the area of the triangle where trade occurs under complete information.  What is unclear from these benchmark cases, however, is whether the underlying cause for the excessive amount of walking away from the deal is the asymmetric information itself or the price setting power.  Getting at the answer to that issue motivates the remainder of the analysis, where neither party has a price setting advantage.

Mumbling about something called the Revelation  Principle, which means that for whatever mechanism you might come up with to resolve the bargaining model there is an equivalent "direct revelation mechanism" where the buyer announces her value and the seller announces his cost and then based on that the mechanism specifies whether the players walk away or trade and if they do trade at what price, an economist teaching this would wave his hands and say we're going to model the bargaining as a direct revelation game.  Non-economists who read this might prefer to personalize the concept called a mechanism in the form of an arbitrator whose word on the matter is final.  The arbitrator makes this decision based on what is communicated by the buyer and seller about their circumstance.  In turn, so the buyer and seller each know what message to send, the arbitrator lets both of them know how their joint message translates into the arbitrator's decision.

It is instructive to start out with a naive but hopeful arbitrator whose dual goals are to achieve the same efficiency in trade that occurs under complete information and do so with fair pricing.  Letting cm denote the cost message the seller sends and vm the value message the buyer sends, the obvious rule for the arbitrator to come up with is trade only if vm > cm and when that is the case set the price, p, so that p = (cm + vm)/2.  Under this rule would it be optimal for the buyer and the seller to tell the truth, which means cm = c and vm =v?  Or would it be better for each of them individually, taking account of their own benefit only and ignoring the consequences for their counterpart, to "strategically misrepresent" their circumstance?  When truth telling is optimal, the mechanism is said to be incentive compatible.

It turns out that the naive arbitrator's mechanism is not incentive compatible, but before explaining why a little aside.  In a nation where every school child is taught that the Father of our Country and first President, George Washington, never told a lie shouldn't it be that telling the truth is its own reward and therefore that every possible mechanism is incentive compatible?  Further, isn't economics as a discipline morally reprehensible for suggesting otherwise?    The responses to these questions are as follows.  In the model the message sent has no direct impact on the player's payoff.  All that matters is the expected surplus generated for the player.  If a message that is other than the truth yield's a higher expected payoff, then truth telling is not optimal.  Shouldn't the model then be changed to reflect more preferred ethical behavior?  The models purpose is positive, to predict what will be observed.  The model is not normative.  It should not be taken as a guide of behavior.  The acid test of whether it is a good model, then, is whether the predictions of the model are in accord with experience.  Anyone who observes when driving on the interstate that traffic slows down a cop car is in view implicitly knows that incentives matter, at least in certain settings.  Without much effort, the reader likely can come up with several other scenarios where extrinsic rewards (or punishments) influence behavior in a predictable way.  That is sufficient to justify the modeling approach.

Let's get back to the naive arbitrator's mechanism.  We'll consider the choice problem of the seller under the assumption that buyers are truthful about the value they report in their message.  Then sending a message of cm means the probability of trade is 1 - cm and the expected price at which trade occurs is (1 + 3cm)/4.  In other words, the seller will treat the mechanism like a demand curve.  For truth telling to be optimal, marginal revenue for this demand curve at cm = c must equal marginal cost, which is c.  That doesn't happen here.  Instead, marginal revenue is less than marginal cost at cm = c, which means the seller would prefer to sell less but at a higher price.  Indeed, it is not hard to show that the optimal message is cm = (1 + 2c)/3.

What does an incentive  compatible mechanism with split the difference pricing look like?  From the discussion in the previous paragraph it must be that the implied marginal revenue when setting cm = c must equal the marginal cost, c, and that must be identically true for all c.  Treating the mechanism from the seller's perspective, it specifies the minimum value at which trading occurs given the cost message of the seller.  It is then not hard to derive that this function must satisfy:

dv/dc = (1 - v)/(v - c).

This disarmingly simple looking equation is the diffeq from hell mentioned in the title of my post.  So let me switch gears and leave the explanation of the model to give the little history from which the title emerged.  Then I will close.

There is a practical aspect to teaching in more or less staying on track subject-wise and doing so at a reasonable pace for the students.  On Sunday or Monday when I started to work on the alternative model, I implicitly told myself that I've got one day to solve this thing.  If I do it, great.  But if not I should fall back on the 2 types example and leave it at that.  I'm more than a little rusty with differential equations, but I'm optimistic that once I get rid of the rust I'll get to a solution quickly.  After one day nada, but I can't let go of solving this thing, so I don't honor my previous commitment but instead keep at trying to find a solution.  Did I mention that this equation is not seperable?  I know how to solve them when they are seperable.  Sometime Wednesday morning it occurs to me that even if I can't find a closed form solution, I can do a numerical approximation in Excel.  After working on that for a while it occurs to me that I'm confused about the boundary condition, v(0) = 0, which means for the lowest cost seller there is no walk away from buyers and for this one type (and the other extreme where v(1) =1) trade is at the efficient level.  I somehow got it into my head that if the seller sets cm = 0 then the expected price would also be zero, but I was ignoring the contribution of the buyer's value in determining the expected price, which actually turns out to be .25, half the expected value.  And here it is Friday and I'm still not done with the numerical approximation.  It's close but the thing is off in some ways I'd like to fix.  In particular, the area of the region where trade occurs should be greater than 1/4.  I wasn't getting that initially.  There are some other changes I'd like to make as well. For what it's worth, the graph of what I have at present is below.  Now I think it's time to contemplate grading those exams.

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