# Expected values vs uncertainty

Most models I have seen use expected values. Why is this a better economic model than uncertainty and economic agents thus having to make 'best guesses', with the result of 'animal spirits' playing a large role. For example, it seems like no one knows where bitcoin would head for sure, hence the rife animal spirits-eqsue speculation involved and variable price.

One can distinguish between probabilistic randomness (which can be quantified) and uncertainty. This article by a Fed Researcher discusses this: “The Stock Market: Beyond Risk Lies Uncertainty” (Frank Schmid). As the tag to the question suggests, this is associated with Keynes, and post-Keynesian economics at present. Therefore, if you look at the post-Keynesian literature, you will see more discussion of uncertainty.

“Mainstream” economics and finance focus on quantifiable models. As a result, they rely on probability theory, and expected values are very common. However, option pricing theory does take into account the whole probability distribution, and so “uncertainty” might be thought of as the properties of the tail of the distribution. Rather than describing the price of bitcoin as uncertain, the tendency would be to find a model (such as a “rational bubble”) that might fit observed data. That is, one can argue that it is possible to analyse bitcoin using a standard probabilistic framework.

Whether the probabilistic approach or the more qualitative post-Keynesian approach is “better” is going to be opinion-based, and so I do not want to pursue that. (As a disclaimer, I am in the post-Keynesian camp.)

Why is this a better economic model than uncertainty and economic agents thus having to make 'best guesses', with the result of 'animal spirits' playing a large role?

Your question is ill-posed, but I will try and answer what I think you are struggling with. Let us go very simple to coin tosses. We are going to simplify it even further in that our coin will be "fair." You have two choices. If you bet \$1,000, then you will either win \$1,000 or lose the $1,000 you bet. We will ignore why someone would offer this bet. We will assume the offer has positive marginal utility to the offeror. Your alternative is not to bet. We will also assume linear utility of wealth. We will denote$\delta_a$as the decision to bet and$\delta_b$the decision to not bet. You must solve $$\max_{\{\delta_a,\delta_b\}}\tilde{w}$$ subject to $$\tilde{w}=1,000\text{ dollars, if }\delta_b,$$ otherwise $$\tilde{w}=2,000\text{ if }\delta_a\text{ and coin is "heads"}$$ and $$\tilde{w}=0\text{ if }\delta_a\text{ and coin is "tails."}$$ The problem has no solution. It isn't the nature of the bet. It is also not due to the nature of the utility function. It isn't even because the actor is indifferent between betting and not betting. It is due to the fact that to maximize utility you have to know now how the coin toss will come out. If it is heads, then you must bet. If it is tails, then you must not bet. Since you cannot know the outcome before you bet, you cannot solve the problem at all as posed here. Now let us consider a more "real" utility function such as$U(\tilde{w})=\log(\tilde{w})$. It still cannot be solved, but something else can be solved. That something is minimizing the cost of being wrong. For the coin tossing example, the lowest risk method of calculating the solution happens to be to minimize the squared loss. Other solutions have a greater amount of risk. If you want to know why then you should look at https://www.stat.washington.edu/~pdhoff/courses/581/LectureNotes/admiss.pdf or https://en.wikipedia.org/wiki/Admissible_decision_rule. Expected utility, in this case, is the risk-minimizing rule. You can solve $$\max_{\{\delta_a,\delta_b\}}\mathbb{E}(\log(\tilde{w})),$$ subject to the above constraints. Your utility of wealth is concave and so the utility of uncertainty should be concave. Now let's assume you want to include uncertainty in an explicity manner. To do this you would not assume the coin is "fair," but rather the coin is biased in an unknown manner. You would still solve this problem with expectations, but it would now be based on a Bayesian rather than Frequentist model. Frequentist models lack uncertainty because they condition on the null hypothesis. All effects are due to chance alone. Bayesian methods condition on the data and so there is no such thing as chance, there is only uncertainty. A discussion of Bayesian analysis is too complex to post here. It is a very different style of question. However, there is a decent comparison of Frequentist confidence intervals and Bayesian credible intervals at Keith Winstein's cookie intervals. There would be a lot of additional math you would have to learn beyond what you likely know. I am basing that assumption on the way you are asking the question as it doesn't appear you have approached this issue and this issue is at the base of any discussion of probability. If you are interested in Bayesian analysis of economic questions, then I would suggest reading Stata's primer on Bayesian methods. For example, it seems like no one knows where bitcoin would head for sure, hence the rife animal spirits-eqsue speculation involved and variable price. In this question you are confusing the short-run behavior and the limiting behavior. An expectation is a calculus concept. It is a discussion of how people would behave at the limit, but not in any specific instance. I haven't derived the density for Bitcoin, but for stocks that are not merged out of existence or going bankrupt, the density is $$\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{\mu}{\sigma}\right)\right]^{-1}\left(\frac{\sigma}{\sigma^2+(x-\mu)^2}\right).$$ While the distribution has no expectation, the log of the distribution does have an expectation. Further, 99.99\% of the mass is within$\pm{636}\sigma.\$ That is an incredibly wide range of behavior. At the limit, the behavior should revolve around expected utility, in the immediate case there is no such boundary condition. Wide swings of behavior are not inconsistent with an expected utility hypothesis.

You should note that expectations are not always the risk-minimizing choice. Expectations do tend to get overused or inappropriately used.

Adding uncertainty to models can be a material improvement in many cases, but it comes at a cost. Not only do you have the uncertainty of movements in prices and volumes, but an additional uncertainty to the nature of the parameter.

Instead of a fair coin as above, consider the case where there are two types of casinos. The first kind always engages in fair bets. The other one says it is, but you are suspicious because the croupiers go by names such as "Mandrake the Magician," and "Slick Eddy." If it is not a fair coin, it is probably close to fair or people would easily notice. Now you need to solve both the distribution given a density with parameters, and the distribution of the parameters.

There is a ready math to solve this type of problem, but you will likely land back in the land of expectations.

Because it is easier to model. If the main point were the role of risk preferences, or if risk preferences would affect the conclusions in some important way, further extensions would presumably be provided.