Notes related to economic fluctuations

by Hans O. Melberg

Introduction
Last week I ended by quoting Charles I. Plosser who wrote that:

The purpose of this short papers, is to examine Plosser's argument. I want to understand what it means and if it is correct. (Plosser is not alone in making claims about the usefulness of first modeling a perfectly working economy and using this as a benchmark. Consider, for instance, Backhouse 1995: 122 and Blanchard & Fisher quoted in Sayer 1997: 328).

An example
Before I enter the conceptual issues (what is an explanation and so on), I want to give one concrete example. The example is meant to illustrate one possible interpretation of logically impossible explanations.

Imagine that an expert witness testifies that a child probably has been abused because he/she "shows all the multiple characteristics of abuse" (the example and the quotation is from Dawes 1998). When asked how she knows the multiple characteristics of abuse, the expert responds that she has conducted a large survey among children who were abused to find the most common characteristics.

This might sound convincing, but there is one member of the jury who argues that it is logically impossible that the expert's conclusion is valid. He argues as follows: Statistically speaking we want to know the probability that something (say t - a theory or a hypothesis) is true after being presented with evidence from the expert (use e as a symbol for this evidence). Formally, we want to know: P(t|e).

How do we find P(t|e)? Mathematically, P(t|e) is defined by the following formula:

(1) P(t|e) = P(t e) / P(e)

But, we do not normally have the information need to use this formula. To solve this problem, we note that P(t e) is the same as P(e t) and that P (e t) is:

(2) P(t e) = P(e t) = P(e|t) P(t)

This means that we can write P(t|e) as:

(3) P(t|e) = P(e|t) P(t) / P(e)

Now, p(e) is often difficult to find, but we can find an equivalent expression by noting that:

(4) P(-t|e) = P(e|-t) P(-t) / P(e)

(- means the complement)

(5) P(t|e) / P(-t|e) = [P(e|t) / P(e|-t)] [P(t) / P(-t)]

At this point it is worth recalling the original question: How probable is it that the child has been abused given the fact that the expert witness based on the mentioned research says the child has the typical signs? Or, formally, what is P(t|e)? Equation (5) tells us what kind of information we need to answer the question. Knowing that the child exhibits the typical signs, means that P(e|t) is larger than 0.5. However, as we can see this is only one of several pieces of information needed to find P(t|e). We also need P(e|-t) and P(t). If the expert has done all her research on abused children, she has no basis for estimating the probability that children who are not abused also exhibit the signs she claims are typical for children being abused [i.e. she needs to know P(e|-t)]. And without this probability, we cannot use (5) to find an estimate for how likely it is that the child has been abused given the expert evidence.

The example should illustrate one possible meaning of logically impossible inferences. We want to know the probability estimate for something. We then show what kind of information we need to find the estimate we want. Finally, we conclude that if we do not have this information there is no logical basis for estimation. In our case, we need to do research on both abused and non-abused children to know what distinguishes the two. Putting it this way makes it sound self-evident, but empirical evidence shows that people are often fooled by the representativeness bias [i.e. simply going from P(e|t) to P(t|e) ignoring that you also need P(e|-t) and P(t)].

[For a short, intuitive and concrete example of the theory above, see Bayes rule in Hargreaves Heap et al. (1992: 295-296)].

Relevance to Plosser and economic fluctuations
Plosser's "logical impossibility" is similar to the one described above. Clearly, to say that something is caused by a market imperfection requires that you have an idea about what a perfect market looks like. Things are, however, slightly more complicated.

First, the argument does not imply that if you can cancel the search for alternative explanations (based on imperfections) if you are able to construct a successful model based on a perfectly working economy. (Successful in the sense that is can account for the empirical variances and covariances and compatible with other empirical facts). The problem is that different models may explain the same phenomena. Hence, even if you happen to find one model that fits many of the facts, there is no guarantee that it is the "correct" explanation since alternative explanations may fit the fact equally well. [The problem is often called observational equivalence.]

Second, the phrase "a perfectly working economy" is easily misunderstood. It does not refer to something that actually exists. It simply describes the workings of an imagined economy of no imperfections (no externalities, no economies of scale, perfect competition, no missing markets, perfect information, instrumental rationality). Once this is acknowledged, we may question to what extent a model based on these assumptions can "explain" a phenomenon in the real world. If we can prove that the assumptions are wrong, it does not make sense to say that they explain fluctuations. Only assumptions that are true can explain!

Of course, the assumptions are not easily divided into true and false. They are more or less true; the world fits the assumptions in degrees. Some may argue that the problems involved in testing assumptions (they are, for instance, always joint tests of the hypothesis and all the assumptions) means that we should not worry about the realism of our assumptions. Friedman is well known for his insistence that only predictions matter when we judge theories (regardless of the realism of the assumptions?). Similarly, Sargent has written that the assumption of rationality is not amenable to empirical testing because of "the logical structure of rational expectations as a modeling strategy, the questions that it invites researchers to face, and the standards that it imposes for acceptable answers to those questions" (quoted in Rosser, 199?: 199. Originally in Sargent 1982: 382). This is, in my opinion, to overstate a valid point. It is true that we should be reluctant of resorting too easily to explanations based on irrationality and imperfections. Yet, the reluctance should not be translated into "never." Sometimes the evidence for imperfections and/or irrationality is so overwhelming that the plausibility of models ignoring these imperfections is very small.

Third, Plosser's argument is not exactly the same as the one with abused children. To say how abused and non-abused children differed we had to do research on both groups. We could see this in detail because the relevant probability required us to know P(e|-t). Do we have to do research on a "perfect" model in the same way before we can claim to have confidence in a model of fluctuations based on imperfections? To be concrete, assume that you start with a model with the following imperfections: it is costly to adjust prices (but these menu-costs are small), and there are real rigidities in the wage (for instance, because people react by shirking and working less hard when you reduce their wage). D. Romer then demonstrates that these two imperfections combined have consequences that fit some of the stylized facts of economic fluctuations. We do not need knowledge of the perfect model to work out these results. One might argue that we should work out the implications of an economy without these imperfections before we claim that it is these two imperfections that explain fluctuations. This may be sensible to increase our understanding of the model, but is it logically necessary in order to claim that the imperfections are important?

Assume the following:

- We can use empirical evidence that the imperfections exists

- We can show the details of the causal mechanism that demonstrate how the imperfections together can create large fluctuations

- The micro-evidence demonstrates that the links and implications involved in this causal mechanism are of sufficient strength to create the observed fluctuations.

I have so far not assumed that you have knowledge of how a perfect economy works. It seems to me that if I have the information above it is perfectly valid to have some confidence in the explanation based on imperfections. But, suppose somebody came to me and argued that I also should work out a "perfect" model to compare the two. Now, to be of any use this comparative model would have to be quite similar to the first. Only the two assumptions of nominal price and real wage rigidities should be changed. If you change the whole approach, using different assumptions and different approaches, it is impossible to isolate the effects of our two assumptions. Also, even if the alternative model also accounts for the facts, we cannot claim that it is explanatory as long as the assumptions do not fit the facts. In sum, the examples are not quite similar.

A final world, I do agree that pedagogically it is often useful to start from a simple model and then make extensions (label this the simple-to-the-general method). Theoretically there are, however, some problems with his methodology as D. Hendry has argued (see, for instance, Gilbert 1986). He argues, instead, that we should focus on general-to-simple models (as opposed to simple models that are extended). That is, we should first list all the possible variables and mechanisms and then eliminate those that have weak explanatory power. The argument is interesting and convincing. We are more likely to converge on the correct model and we have a better knowledge of the probabilities involved when we go from the general to the simple. For instance, if we both start from a simple model we may well extend the model in two different directions and ultimately arrive at two different models that both explain the facts relatively well. The-general-to-simple methodology tries to avoid this.

Conclusion
This short observation has brought out the following points. First, we need to define what it means to "explain" something. Second, we have to find out the information needed to satisfy this demand. Third, we have to ask how we best can satisfy that demand given inherent limitations (resources, data, technical and so on).