THE ANALYTICAL ECONOMIST

he war stories of corporate research laboratories are poignant at best, costly at worst. Remem-

ber when XeroxÕs Palo Alto Research Center held the pieces of the personal computer revolution on its desktops, but the corporation could not see a route to market? Or the time when IBMÕs John Cocke sketched out ideas for reduced instruction set computer (RISC) designs, but the corporation declined to take up the technology until after others had established tidy businesses?

All too often, the corporate giant that nurtures an innovative idea seems to be the last to make use of it. In many cases, companies are reluctant to undertake the arduous process of bringing a new research idea to market, particularly if it promises to cannibalize an established winner. As a result, upstart Þrms often topple industrial Goliaths by commercializing radical innovationsÑeven those invented in the labs of the giants themselves.

Such stories run counter to economic theories that paint companies as rational, proÞt-maximizing organizations. But, curiously enough, even though economists have analyzed decisions as intimate as marriage and divorce, they have spent less time delving into why companies invest inÑor eschewÑnew ideas.

A handful of economists hope to shake up the status quo. Among them, Avinash Dixit of Princeton University and Robert S. Pindyck of the Massachusetts Institute of Technology are trying to codify how a Þrm decides to invest in, say, facilities for producing a new product. Rebecca Henderson of M.I.T. has been exploring the aspects of large and small Þrms that might aÝect the likelihood they will developÑand proÞt fromÑradical innovations.

According to the ÒsimpleÓ tale of investment, a Þrm will expand production or break into a market if the revenue gained by selling the product exceeds the long-run costs of making it. But Òreality is very diÝerent,Ó Dixit notes. Because companies want their investments to earn some minimum return, they will delay initiating product development until they believe their proÞts will exceed such a Òhurdle.Ó

But what about a Þrm caught betwixt and betweenÑwhen existing products

are not especially proÞtable, but future ones fall short of corporate hurdle rates? Pindyck and Dixit suggest that decisions may then be strongly inßuenced by recent ßuctuations in product proÞtability. ÒWe call it economic hysteresis,Ó Pindyck says.

Physical hysteresis describes how a magnetic Þeld induced in a metal bar by an electric current will linger after the current is shut oÝ. Similarly, the economists suggest, once a firm begins investing, it will continue to do so even if prospective returns slide below the hurdle rate.

Decisions to enter or quit a market, Dixit and Pindyck argue, are in fact driven by the combined eÝect of three considerations, something economists have typically given short shrift. First, investments carry costs that are either partially or completely irreversible. Second, all such investments are swathed in uncertainty and may never earn the predicted return. And last, companies can control the timing of their investments.

The economists say these characteristics mean that corporate investment decisions can be modeled like decisions to exercise a stock option, a right to buy or sell a security in the future. Companies accrue such options by developing new technologies or by building up their marketing prowess. They exercise

Why are big companies typically the last

to capitalize on radical innovations?

these options by exploiting the ideas. A pharmaceutical company that holds Òsleeping,Ó or unused, patents essentially has a call option for the lifetime of the patent, Pindyck believes.

Viewed as options, corporate decisions not to invest in a novel technology look diÝerent. Unexercised options are still valuable, Pindyck points out. Delaying or ignoring an investment opportunity may simply be a rational response to market vicissitudes.

Henderson is less convinced that Þrms are making optimal choices. But neither is she willing to write oÝ managers as

ly have written was a process that forces constant revision of your assessment of its truth value, p ← 1 Ð p. If you assume that P has a particular truth-value p, then P itself tells you to replace that truth value by 1 Ð p. For example, if you started out thinking that P was 30 percent true, so that p = 0.3, the revision rule implies that p = 0.7, which in turn implies that p = 0.3 again...and you get an inÞnite sequence of truth values, oscillating periodically between the two values 0.3 and 0.7. The classical paradox, with p = 0 or 1, leads to the sequence 0, 1, 0, 1,...which faithfully reßects the logical argument if P is false, then P is true, so P is false, so P is true, so.... The logical oscillations of the paradox are captured by the dynamics of the truth value.Ó

ÒAnd p = 0.5 is the only value that doesnÕt lead to an oscillation,Ó Plato mused.

ÒPrecisely. Now, the Dualist Paradox on your business card is really a logical dynamic:

p ← q

q ← 1 Ð p.

Suppose you start out by estimating p = 0.3, q = 0.8. Then your Þrst revision is to p = 0.8, q = 0.7. A further revision leads to p = 0.7, q = 0.2, a third to p = 0.2, q = 0.3. A fourth revision gives p = 0.3, q = 0.8, and youÕre back where you started. It cycles with period fourÑunless you start at p = 0.5, q = 0.5, when everything stays unchanged.Ó

ÒOkay, IÕll buy that,Ó I said. ÒBut what about the chaos?Ó

LukasiewiczÕs face went very serious. ÒBefore I can explain that, I must be more precise,Ó he said. ÒIf you want to play around with these ideas for yourself, IÕd better tell you how to calculate fuzzy truth values for combinations of logical statements [see box at right]. Although all you really need to know at the moment is that not-P has truth value 1 Ð p if P has truth value p. Second, you must know how to assess the truth value of statements about statements.Ó

ÒIÕd like an example,Ó Socrates said. ÒOkay. Suppose I said Plato is a good golfer. How true do you think

that is?Ó

ÒOoohhhÑabout 40 percent,Ó Socrates said. Plato gave him a nasty look. ÒWell, Epimenides usually beats you, and heÕs pretty mediocre.Ó I gave him an even nastier look.

ÒFine. LetÕs call that statement S. It has a truth value s = 0.4. Suppose I make a statement about the statement S. Suppose I utter statement T: ÔS is 100 percent true.Õ How true is statement T?Ó

Fuzzy Logic

In classical logic, a statement has a truth value of either 1

for true or 0 for false. The statement “the sun is shining” has a truth value of 0 if it is cloudy. In general, statement P has a truth value p equal to 1 or 0. In fuzzy logic, a statement can have a truth value of between 1 and 0. If a cloud obscures a quarter of the sun, then statement P has a value of 0.25.

In fuzzy logic, like the classical theory, the truth value of a statement will change when applying the operators NOT, AND, OR, IMPLIES and IF AND ONLY IF.

NOT-P has a truth value of 1 – p. EXAMPLE: If the sun is shining with a truth value of 0.25, then the sun is NOT shining with a truth value of 0.75.

P AND Q has a truth value equal to the lesser of p and q where q is the truth value of statement Q. EXAMPLE: The sun is shining with a truth value of 0.25, AND Jane is getting tan with a truth value of 0.10.

The value of the example is 0.10.

P OR Q has a truth value equal to the greater of p and q. EXAMPLE: The sun is shining with a truth value of 0.25, OR Jane is getting tan with a truth value of 0.10.

The value of the example is 0.25.

P IMPLIES Q has a truth value equal to the lesser of 1 and 1 – p + q.

EXAMPLE: If the sun is shining with a truth value of 0.25, then Jane is getting tan with a truth value of 0.10.

The value of the example is 0.85.

P IF AND ONLY IF Q has a truth value equal to 1 – | p – q |, that is, one minus the absolute value of p minus q.

EXAMPLE: The sun is shining with a truth value of 0.25 IF AND ONLY IF Jane is getting tan with a truth value of 0.10.

The value of the example is 0.85.

I thought for a moment. ÒWell, itÕs certainly not 100 percent true itself. Otherwise S would be 100 percent true, and weÕve already decided it isnÕt.Ó

ÒRight. The degree of truth of my statement T, which is about S, depends on the actual truth value of S and on the truth value attributed to S by T. Here s = 0.4, but the value that T leads me to assess is 1. So T will be untrue to the extent that these two values diÝer, yes? The more inaccurate my assessment, the falser my statement becomes. Because they now diÝer by 0.6, T is false to the extent 0.6. That is, it is true to the extent 0.4.Ó

ÒWhat if youÕd said S is half true?Ó Lukasiewicz nodded happily. ÒYouÕll

see how nicely it works. That statement assesses the truth value of S as 0.5, but the actual value is 0.4. The diÝerence is 0.1, which is how false your statement is, so its truth value is 0.9. Because your assessment is only wrong by 10 percent, youÕre 90 percent correct.Ó

ÒAh. And if IÕd said S is 40 percent true, IÕd have been 100 percent right. So the truth value would have been 1. IÕve got it.Ó

ÒGood. In general, suppose I have a statement P with truth value p and a statement Q that leads you to assess the truth value of P to be p ′. Then the argument weÕve just been through says the truth value of Q is q = 1 Ð | p Ð p ′ |, where | x | means the absolute value of x (equal to x when x is positive, Ðx when x is negative). Let me call this the assessment formula.Ó

Lukasiewicz thought for a moment. ÒNow I can show you what I call the Chaotic Liar, statement C:

This statement is as true as it is assessed to be false.

If its truth value is c, then it instructs you to assess a truth value of 1 Ð c. So by the assessment formula, its truth value is 1 Ð | c Ð (1 Ð c) | = 1 Ð | 1 Ð 2c |. In short, there is a dynamic process

c ← 1 Ð | 1 Ð 2c |

of reassessment of the truth value c. Choose any starting value for c, say c = 0.12345, and calculate successive values. YouÕll Þnd they are chaotic. Actually, I should warn you that because of round-oÝ errors in your calculator or computer, the process may appear to settle down to either 0 or 1. It may help to replace the dynamic by

c ← 1 Ð | 0.999999 Ð 2c |.

You can even observe the famous butterßy eÝect of chaos theoryÑif a but-

terßy ßaps its wings, it can cause a hurricane a month later. More prosaically, small changes in initial conditions make big changes to the subsequent dynamics. If you use a start value of 0.12346 instead, you get a diÝerent image.Ó

Lukasiewicz paused. ÒNext there is the Chaotic Dualist, which involves two statements:

X:X is as true as Y is true

Y:Y is as true as X is false

ItÕs rather like your business card, Epimenides. The dynamics are

x ← 1 Ð | x Ð y |

y ← 1 Ð | y Ð (1 Ð x) |.

To see what it does, you choose an initial pair of values, say, (x,y) = (0.2, 0.9), and calculate successive pairs of values. Think of them as coordinates and plot them in the plane. You get a geometric shape, called the attractor of the dynamic system. In this case, you get a triangle, densely Þlled with points [see right illustration on page 110]. This representation can be transformed into a beautiful and intricate image known as an escape-time diagram. To create it, temporarily relax the conditions that x and y lie between 0 and 1. The idea is to watch how far (x,y) moves from the origin (0,0) and to count how many calculation steps are needed before it goes beyond some threshold value. Then the point (x,y) is plotted in a color that depends on the number of steps required. To start, you should try a threshold value just larger than 1 [see left illustration on page 110].Ó

ÒI begin to see now,Ó Socrates said. ÒYou take the train of thought involved in assessing the truth value of a set of self-referential statements and convert it into a dynamic process. Then you can apply all the techniques of chaos theory to that process. The escape-time plot is inspired by exactly the same method that creates all those wonderful multicolored images associated with the Mandelbrot set: swirling spirals, sea horses, cacti, stars and so on.Ó

ÒIndeed. HereÕs one Þnal idea for you to mull over. We can rephrase the Chaotic Liar as

The assessed falsehood of this statement is not diÝerent from its assessed truth.

In fuzzy logic, it is standard to interpret the adjective ÔveryÕ as forming the square of a truth value. So think about the rather woollier statement