Sydney Afriat arrived in Purdue in the late 60’s with a Bentley in tow. Mort Kamien described him as having walked out of the pages of an Ian Flemming novel. Why he brought the Bentley was a puzzle, as there were no qualified mechanics as far as the eye could see. In Indiana, that is a long way. Afriat would take his Bentley on long drives only to be interrupted by mechanical difficulties that necessitated the Bentley being towed to wait for parts or specialized help.

I came upon Afriat when I learnt about the problem of rationalizability. One has a model of choice and a collection of observations about what an agent selected. Can one rationalize the observed choices by the given model of choice? In Afriat’s seminal paper on the subject, the observations consisted of price-quantity pairs for a vector of goods and a budget. The goal was to determine if the observed choices were consistent with an agent maximizing a concave utility function subject to the budget constraint. Afriat’s paper has prompted many other papers asking the same question for different models of choice. There is an aspect of these papers, including Afriat’s, that I find puzzling.

To illustrate, consider rationalizing expected utility (Eran Shmaya suggested that `expected consumption’ might be more accurate). Let be the set of possible states. We are given a sequence of observations and a single budget . Here represents consumption in state and is the unit price of consumption in state in observation . We want to know if there is a probability distribution over states, , such that each maximizes expected utility. In other words, solves

subject to

The solution to the above program is obvious. Identify the variable with the largest objective coefficient to constraint ratio and make it as large as possible. It is immediate that a collection of observations can be rationalized by a suitable set of non-zero and nonnegative ‘s if the following system has a feasible solution:

This completes the task as formulated by Afriat. A system of inequalities has been identified, that if feasible means the given observations can be rationalized. How hard is this to do in other cases? As long as the model of choice involves optimization and the optimization problem is well behaved in that first order conditions, say, suffice to characterize optimality, its a homework exercise. One can do this all day, thanks to Afriat; concave, additively separable concave, etc. etc.

Interestingly, no rationalizability paper stops at the point of identifying the inequalities. Even Afriat’s paper goes a step farther and proceeds to `characterize’ when the observations can be rationalized. But, feasibility of the inequalities themselves is just such a characterization. What more is needed?

Perhaps, the characterization involving inequalities lacks `interpretation’. Or, if the given system for a set of observations was infeasible, we may be interested in the obstacle to feasibility. Afriat’s paper gave a characterization in terms of the strong axiom of revealed preference, i.e., an absence of cycles of certain kinds. But that is precisely the Farkas alternative to the system of inequalities identified in Afriat. The absence of cycles condition follows from the fact that the initial set of inequalities is associated with the problem of finding a shortest path (see the chapter on rationalizability in my mechanism design book). Let me illustrate with the example above. It is equivalent to finding a non-negative and non trivial solution to

Take logs:

This is exactly the dual to the problem of finding a shortest path in a suitable network (I believe that Afriat has a paper, that I’ve not found, which focuses on systems of the form ).The cycle characterization would involve products of terms like being less than 1 (or greater than 1 depending on convention). So, what would this add?

## 8 comments

October 10, 2014 at 3:57 pm

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October 15, 2014 at 3:51 pm

LucaI’m no expert on this, but one possible answer is that GARP ‘looks like” economics while the Afriat numbers are just… numbers. As you say, it is just a different interpretation of the same result.

October 15, 2014 at 4:49 pm

rvohraHi Luca

Actually Eran Shmaya convinced me that I overlooked something important that the GARP characterization gives. Rationalization by a complete order implies rationalization by concave utility.

October 18, 2014 at 4:06 pm

A fanRakesh,

on the topics of Farkas’ lemma and linear programming, what do you think is the next book to study after your “Advanced Mathematical Economics”?

Thanks

October 19, 2014 at 6:06 pm

rvohraGlad you found the book useful. As to what is next, depends on your objectives. The obvious candidates are Mas-Collel, Whinston & Green or Krep’s Microeconomic Foundations. Both provide an exhaustive (literally) account of the foundations of microeconomic theory. However, if you want less generality and more discussion of the modeling choices in standard microeconomic theory, I’d urge

`Baby Kreps' on you (A Course in Microeconomic Theory). Finally, less fashionable and well off the beaten track is Gale's`

Linear Economic Models’.October 19, 2014 at 11:17 pm

A fanI am a theorist who finished his PhD a few years ago but who (like many of us, unfortunately) doesn’t have a strong background in optimization. What are the “bibles” in optimal control, convex analysis or measure theory is well known. What is a good reference book for linear optimization?

Btw, thanks for suggesting Gale’s book. It looks fascinating.

October 20, 2014 at 5:38 pm

rvohraFor convex analysis, Rockafellar has been a staple, but his notation is not, shall we say, widely adopted. For a gentle introduction, Borwein and Leiws (Convex Analysis) and slightly steeper would be Hirirart-Urty and Lemarechal. Not their intimidating 2 volume set but the

`baby' one entitled`

Fundamentals of Convex Analysis’. If your interest is in Convex Optimization itself, then Boyd and Vanderberghe covers the subject.For measure theory, I like the old fashioned texts. The last chapter of Baby Rudin covers Lebesgue measure (and its not hard to find a pdf version on the web). Bartle’s

`Elements of Integration' is also good. I've always liked S. J. Taylor's`

Measure and Integration’. By coincidence, Taylor’s son and myself were undergraduates together reading mathematics!Optimal control is harder for me to give advice on. Kamien and Schwarz for lots of worked examples. For general results, one might as well take functional analysis. Semi-infinite linear programming might be a compromise between the general and specific (see Anderson and Philpott).

October 20, 2014 at 8:29 pm

A fanthanks a lot.