Book Viii Supportive Notions

37. On the nature and significance of a free lunch

It has been a cause of wonder for the present author why other economists are not more outspoken on the Tax Void, and why above theorem on the possibility of returning to full employment meets such disbelief as it apparently does. In the course of time, I found that the following issue forms part of the explanation.

Many economists think that there are no free lunches. It may even be a dogma or mantra to them. With this general attitude, they close their eyes to the free lunch that presently exists in the inefficient labour market. They adhere to their ‘no free lunch’ philosophy regardless of what arguments other people forward. My diagnosis is that this is one of the reasons why the debate on unemployment is rather stuck.

It actually can be shown that the economy is full of free lunches. We will discuss two examples below, namely the examples of the consumers surplus and economic growth. By regarding these examples we will better appreciate the nature and significance (as Robbins might say) of a free lunch. When the possibility of a free lunch is accepted, then we can discuss unemployment in more realistic terms.

Some quotes

The American science fiction writer Robert Heinlein once created a rough Moon Colony where the rules of the free market are exploited to their limits. In this colony the phrase “Your money or your life” is not a criminal threat but a sound business proposal - and a bargain for many as well. In the same vein all incidents in the novel are subject to bets - and after some consideration, the reader of this novel may well accept this as a useful system of rational contingent forward markets. Then, properly, the slogan & law of this Moon Colony is TANSTAAFL: “There Aint No Such Thing As A Free Lunch”.

TANSTAAFL is rather “accepted wisdom” in the economics profession, and not something that is subject to critical discussion. There are only few explicit statements on the supposed absence of a free lunch. A recent statement is by Cnossen & Van Ewijk (1995):

“No society limited in resources can for a moment proceed from the premise that there is such a thing as a free lunch. Dispassionate analysis of the problem and hard-headed calculation of the costs of alternative courses of action are called for. This applies especially to the economics discipline, which gives center stage to the concept of opportunity costs.”

So, evidently, in the views of these authors, people disagreeing to their views on this issue are emotional or soft-headed !

Coase (1994:200) has a fine anecdote:

“Charles Walgreen in 1936 withdrew his niece from the University of Chicago because he had been informed that the university taught free love and communism. I know nothing about the university’s teaching on communism but presumably Mr. Walgreen would not have been mollified to learn that the true Chicago view is that there is no such thing as a free love. Eventually, however, Mr. Walgreen was convinced that he had been misinformed (...)”

The British newspaper The Economist (1994b) and the Dutch economist Van Bergeijk (1994) state, in reaction to proposals by Snower, that there would be no free lunch on the labour market. Even with current unemployment, it would not be possible to change taxes, contributions and benefits in such manner that this would raise employment opportunities for the unemployed without other agents having to pay some bill.

These latter authors use arguments for their views. So their judgement does not seem dogmatic. However, their arguments have been refuted. Authors like Snower and myself, and many others, have also pointed to the possibilities for improvement in the labour market, and these arguments have not met with convincing rejections. So it may well be that TANSTAAFL works its ways in the back of the minds and hinders proper balancing of arguments.

We somehow might welcome the Cnossen & Van Ewijk statement, since it makes explicit what often is only implicit. In the following I shall deal with the problem in general. I hope to banish TANSTAAFL to the domain of science fiction, so that thereafter we can discuss the labour market in more useful terms.

Consumers surplus

The more innocent examples of free lunches happen around us every day. For example, in a free country, a transaction occurs only when both parties get something out of it. TANSTAAFL adepts will hold that when there is a transaction, and people pay for their lunch, then there clearly is no free lunch. However, the theory of the consumers surplus reminds us that you may pay for your lunch, but likely not as much as you might be willing to pay. If you would not get more out of it, there would be little point is actually doing the transaction. In everyday life, we see few people exchanging dollars for dollars, just for the fun of it. So if p is what you pay for your lunch, and if wtp is your willingness to pay, then wtp - p is your free lunch.

One might argue that the TANSTAAFL conjecture properly reads that p 0. Thus TANSTAAFL-ists accept that wtp > p, but the point would be that you have to invest a nonzero amount before you can reap greater benefits. It would seem to me that the following is the proper reaction to this:

1. We might accept a definition that ‘no free lunch’ means p 0.

2. However, that definition does not warrant universal truth. Some goods have p = 0, notably endowments, ideas and, in a sense, public goods.

3. So, please then, do not use this mal-definition to kill arguments on the labour market that concern new ideas.

4. And, please see the point that it may be advisable to define ‘p 0’ ‘there are some costs’, and ‘wtp > p’ ‘there is a free lunch’.

In a sense, the discussion might only be about words. But there are also emotional connotations involved, that should cause us to be rather careful in that choice.

Economic growth

Economic growth is another instance of manna from heaven, and also a phenomenon that has been with us since the dawn of mankind.

An invention in one industry will generally have consequences for the entire economy. The industry of origin can seldom claim all proceeds. When the optimal ratio of production factors changes, then prices change. E.g. just by mentioning the possibility of other prices, one signals to the other parties that there is room for discussion. The other parties will use that room, and their knowledge and possessions, to claim part of the economic value of any innovation. Other parties have had no effort in bringing about the innovation, but they consider themselves partners in the industry, they know their leverage, and, thus, exploit it. Their advantage not only concerns the consequences of a better product, but also an improvement of their income position.

Model

In a general equilibrium framework we consider an economy with 400 units of labour and 600 units of capital. The economy produces food and clothing, and a social welfare function (SWF) determines the optimal combination. Here, our SWF will be a Cobb-Douglas function that neglects the distribution of income:

(SWF)

Labour a en capital k are allocated to the food (v) and clothing (k) industries via av + ak = 400 and kv + kk = 600. Industrial output is determined by the production functions. Here we take CES-functions, that have a constant elasticity of substitution between capital and labour:

Equilibrium and the optimum are found at 278 units of food and 253 units of clothing, with a distribution of the factors of production of av/ak = 299/101 and kv/kk = 210/390.

The allocation can be shown using two figures. Figure 36 confronts the social welfare function with the Production Possibility Curve (PPC).

Figure 36: Social Welfare and the Production Possibility Curve

The PPC gives those combinations of food and clothing that can be produced with the scarce resources. The choice of the highest possible value of the SWF generates a tangent of a contour of the SWF with the PPC. The tangent gives the optimal price ratio (thus trading ratio) of food and clothing.

Figure 37 confronts the production functions of the separate industries in an Edgeworth-Bowley diagram. The food industry has its origin in the lower left-hand corner, and the clothing industry has its origin in the top right-hand corner. The amounts of capital and labour that are not allocated to the food industry are allocated to the clothing industry. The drawn contour for the food industry gives those combinations of capital and labour that produce the same amount of food. That contour is touched in a tangent by a contour of the clothing industry. The collection of all tangency points is called the contract curve. The tangent drawn here passes through the optimum selected by the SWF. This tangent thus also determines the price ratio of wages and capital rent.

Now we assume that there is an innovation in the clothing industry. This innovation can be of technical or organisational origin, and it causes that the same garment can be produced with a little less labour but a little more capital. To be concrete: the production possibility is discovered that can be stated in the production function clothing = CES[0.2, 0.5]. Is this innovation useful ? The answer appears to be that labour is the factor that is relatively scarce and that this innovation allows its better use, so that welfare can rise to 282 units of food and 269 units of clothing. The allocation of factors of production becomes av/ak = 309/91 and kv/kk = 202/398.

Figure 37: Edgeworth-Bowley diagram for the factors of production

Figure 38 and Figure 39 present the same plots as before so that one may see how the economy changes. The figures speak for themselves. It will be clear that our analysis is comparative statics. How quickly the prices change, and how quickly the agents react, will be a question of dynamics.

Figure 38: SWF and PPC of two situations

Figure 39: Edgeworth-Bowley of two situations

The free lunch

Above model was not perfect but helps us to understand how a free lunch percolates through the economy. It helps us to understand what a free lunch actually is.

In above model, the innovation falls from heaven like manna. The innovation is the free lunch. One may see the tautology: If you accept the model, then there is a free lunch; and you accept the model if you see innovation as a free lunch.

One may hold that above model is incomplete. One would want to introduce a separate R&D sector, and then there will be a balancing of R&D costs and the expected increase in national income. As an economist, I’m very much in favour of developing such models. However, actually doing this only moves the question one station further, and does not answer the proper question. For, it is possible that an economy spends 99% of its resources to R&D, and still does not come up with innovations. Good ideas remain like manna from heaven.

You may hold the view that agents already expect economic growth, so that they will not regard it as a free lunch. This reminds of the attitude of some children of rich parents who expect a rich inheritance and who don’t show gratitude for their daily bread. The point to note, though, is that the concept of a free lunch is not an expectational variable, but one of circumstance. There is a free lunch or not, whatever one expects. Indeed, as another example, our wealth is a cumulation of free lunches in the past. That we don’t experience this as a free lunch anymore, is more a sign that we are spoiled, rather than a sign of our dynastic rationality.

And even if we would design a revised expectational concept of a free lunch: then perfect foresight or rational expectations are only assumptions. There is always the possibility of a surprise idea. The future is uncertain (though predictable) - even though our scientific predisposition is deterministic.

Let me rephrase the point that I want to make here. There are data (exogenes or endowments such as soil, sun, technical relations and the like), the economy depends on the use of these, and the development of the economy can be described in terms of the developments in these data. The data are for free. Ideas are part of these data, and the (major) source of uncertainty. In this terminology, there are free lunches by definition. That is the crux. When economists better deal with their definitions, we get better economics.

Conclusion

Our discussion on the consumers surplus showed that much may be a matter of words. However, using an abstract argument and a concrete small general equilibrium model, we showed that innovation and economic growth are an example of a free lunch for the whole economy. Our intention was to refute the attitude of “there aint no such thing as a free lunch”. Hopefully, this refutation creates more room for discussion of proposals concerning the present immense inefficiency on the labour market. The latter discussion is especially important, since the major proposals for solving the inefficiency concern ideas by impartial economists.

Note 1999: I was afraid that I would clash with Paul Krugman on this issue, since he has a Fortune column ‘No Free Lunch’. To my great relief, Krugman (1999:167) however writes: “And this brings us to the deepest sense in which depression economics has returned. The quitessential economic sentence is supposed to be “There is no free lunch.”; it says that there are limited resources, that to have more of one thing you must accept less of another, that there is no gain without pain. Depression economics, however, is the study of situations where there is a free lunch, if we can only figure out how to get our hands on it, because there are unemployed resources that could be put to work. In 1930 John Maynard Keynes wrote that “we have involved ourselves in a colossal muddle, having blundered in the control of a delicate machine, the working of which we do not understand.” The true scarcity in his world - and ours - was therefor not of resources, or even of virtue, but of understanding.” Hurray!

38. Proper definitions for uncertainty and risk

This discussion will present proper definitions for uncertainty and risk. Such definitions are required since the current definitions in common use are rather erroneous and generate conceptual problems.

Uncertainty

The new definitions are - see also Figure 40:

(1) First there is the distinction between certainty and uncertainty.

(2) Uncertainty forks into known categories and unknown categories.

(3) Known categories forks into known and unknown probabilities.

(4) Unknown probabilities forks into assuming a uniform distribution (Laplace) or use non-probabilistic techniques like minimax or neglect.

Note that these definitions only use certainty, knowledge and the distinction about categories (category-uncertainty), and that they do not use the term ‘risk’. Thus an independent definition of ‘risk’ is possible.

A.S. Hornby (1985) “Oxford Advanced Learner’s Dictionary of Current English” defines ‘uncertain’ as: “1 changeable; not reliable: ~ weather; a man with an ~ temper. 2 not certainly knowing or known: be/feel ~ (about) what to do next; a woman of ~ age, one whose age cannot be guessed”. The above fits this.

Figure 40: A diagram of the new definitions

Risk

Hornby (1985) defines ‘risk’ as: “(instance of) possibility or chance of meeting danger, suffering loss, injury, etc.” Also: “at the ~ risk of / at ~ of, with the possibility of (loss etc.)”.

Thus, if there are possible outcomes O = {o1, o2, ..., on}, then the situation is risky if at least one of the o’s represents a loss. The risks are the oi that are losses, thus Risks[O] = {oi O | oi is a loss}. The risk factors are the positions or index numbers of the risky outcomes, the i’s, or the dimensions (the causes that make such positions to be filled).

We will use the term ‘valued risk’ when a risk is valued with money or utility. When all risks have been made comparable by valuing them, then we can add them, and we will use the term expected risk value for the expected value of the ‘valued risks’. Then, crucially, once these definitions are well understood, then we may also use ‘the risk’ for the expected risk value.

With such understanding, risk will be r = -Ex<0[x] or for short r = -E[x < 0].

Valued risk deals with the cases when probabilities are known or when unknowns are assumed to be uniformly distributed over known categories. It is not customary to use the term ‘risk’ for unknown categories. For example, it is uncommon to say, or write economics papers about this, that “all our lives are at risk of a suddenly imploding universe, or black hole hitting Earth, or waking up as a cockroaches”. Such real ‘Acts of God’ are commonly neglected. Note though that it still remains possible to say that a situation is risky even though one cannot put a number to it. Above expectation may be indeterminate since one may lack knowledge about the probability distribution or even the categories.

Relative risk is defined as r(t) = t - E[x < t] for some target level t. Risk (or absolute risk) takes t = 0, and relative risk would allow for a different target level.

An interesting application is when x is a stochastic rate of return and r the certain rate, so that there is relative risk r(r) = r - E[x < r]. This relative risk answers the question: What is the probable loss with respect to a target return of r ? Here, r - r(r) = E[x < r] gives the weight of underperformance in the total target return (which weight has to be compensated by probable profits to achieve the target).

Conditional (relative) risk is defined as k(t) = t - E[x | x < t] for some target level t. With respect to rates of return, conditional risk k(r) answers the question: What would one expect to lose with respect to r, if earnings actually underperform and fall below r. Indeed, r - k(r) would give your expected return when actually underperforming.

Conditional risk is related to relative risk by the property that E[x | x < t] = E[x < t] / Pr[x < t]. The probable loss thus is corrected for the probability of the loss. Or, the probability measure in the expectation is corrected so that a density is taken that sums to 1.

Example

In everyday parlance, profit and loss are nonnegative concepts. For example, if the difference between revenue and costs is $-10, then your loss is $10. It is only in mathematical economics that profits are defined as a general profit function such that ‘negative profits’ are possible. To understand risk, we however return to the everyday parlance convention.

Let us have a prospect that can give profit with probability p, and loss with probability 1 - p. We denote this as Prospect[profit, -loss, p]. We call profit * p ‘probable profit’ and loss * (1 - p) ‘probable loss’. Then the following definitions apply:

· Expected Value = = p profit + (1 - p) (-loss) = probable profit - probable loss

· Risk = risk value = expected value of the risks = probable loss = (1 - p) loss

· Risk Ratio = Risk / (ExpectedValue + Risk) = (1 - p) loss / (p profit)

· Thus: Expected Value = p profit (1 - Risk Ratio)

· Risk Probability = cumulative probability of all losses (in this case 1-p)

Risk is the (absolute value of the) down side of a bet. A venture is judged to be risky if the probable loss is large. Note that this notion still is somewhat vague. A probable loss can be large because of the probability or because of the sum of money involved. This vagueness is unfortunate, in some respects, but here is little to be done about it, since this vagueness is inherent in working with probabilities. In fact, this vagueness is an essentially positive aspect of working with probabilities. For, when we have different prospects, then we can order and evaluate them on risk, neglecting differences in losses and probabilities.

Colignatus (1999, 1999a) further develops these notions for simple binary prospects, multidimensional prospects, joint prospects, and continuous probability densities. An interesting application is the ‘Markowitz efficiency frontier’, but now with risk rather than the spread.

Wrong use in economics 1921-2005

The above definitions are proper in the sense that they conform to every day parlance and the definitions provided by Hornby’s dictionary op. cit.. The definitions provided here however differ from the use within the economics literature. First there are the definitions of Knight (1921) that have been adopted widely in economics, as for example in The New Palgrave (1998:III:358). Or it has become custom in finance to associate risk with the standard deviation. And some mathematical statisticians use another concept of risk. Let us discuss these in turn.

Uncertainty and risk

The New Palgrave, Eatwell c.s. (1998:III:358), gives the current common view:

“The most fundamental distinction in this branch of economic theory, due tot Knight (1921), is that of risk versus uncertainty. A situation is said to involve risk if the randomness facing an economic agent can be expressed in terms of specific numerical probabilities (these probabilities may either be objectively specified as with lottery tickets, or else reflect the individual’s own subjective beliefs). On the other hand, situations where the agent cannot (or does not) assign actual probabilities to the alternative possible occurences are said to involve uncertainty.”

Indeed, most economic texts use this distinction in this manner (at least, up to now). However, I cannot disagree more. The objections to Knight’s concept are:

(a) Certainty and uncertainty are binary. So, if a situation is not uncertain, then we have certainty, and there is no assigning of probabilities.

(b) If I am uncertain about a situation and assign equal probabilities to all cases - the Laplace suggestion - then according to Knight this no longer is uncertainty!

(c) In Hornby’s definition, the distinction is not between known and unknown probabilities, but the distinction is between events and human thought.

Figure 41 contains a diagram of the objectionable use of terms 1921-2005.

Figure 41: A diagram of the current but objectionable use of terms

The diagram clarifies the inconsistency with the binary character of certainty/uncertainty, the curious treatment of “Laplace”, and the over-use of terms by introducing the term ‘risk’ where there already is the qualification that the probabilities are known.

Risk is not the variance

The finance literature often uses the term ‘risk’ for the variance or spread (standard deviation) of the distribution of the rates of return of investments. This would be an improper use of the term. Suppose that one has a very profitable venture without the possibility of a loss. Suppose that the rate of return of this venture has a large variance, from mildly profitable to highly profitable. Is this a risky venture ? No, not in the usual understanding of the term.

Risk is not the negative of expected revenue

In mathematical statistics, some authors, like Ferguson (1967), define ‘risk’ as ‘expected loss’. However, it appears that they actually regard ‘loss’ as the negative of total returns (i.e. - revenue), so the definition used is -(p profit + (1-p) (-loss)), which is the negated expected value. This use of the term ‘risk’ is inappropriate. My proposal is to use the word “due” to stand for the negative of expected value, so that the standard statistical decision theory (with the game against nature) can be described as minimising due.

Note on Bernstein’s “Against the gods”

I came across Bernstein (1996) “Against the gods”, and found it equally entertaining as his “Capital Ideas”. One comment is that Bernstein indeed emphasises Knight’s and Keynes’s statements on “uncertainty”. My answer to that is, again, that unknown probabilities or even unknown categories indeed are serious cases of uncertainty, so that earlier writers on the subject were right in emphasising that seriousness. However, we should not be tempted to reserve the word “uncertainty” to only those cases. So with all due respect to Knight and Keynes, the definitions provided here are the proper ones.

Note on Wilson & Crouch (2001)

Wilson & Crouch (2001), “Risk-benefit analysis”, adopt the same definition of “risk” as discussed here. I saw this only after the first edition of this book. Since professor Wilson has been teaching on the subject for decades and his book only collects his teaching material I apparently only rediscovered what was already clear to him. Perhaps my presentation is a bit clearer since I use the formal E[.] notation. This chapter remains useful since it clarifies the confusions from the other definitions. Where risk is the product of probability and severity, this book also benefits from the emphasis on this definition, since, where I started to develop this argument after the Fall of the Berlin Wall in 1989, we have to deal with a future where there are huge dangers: though with only a small probability but on balance a relevant risk.