Behavior change in economic and epidemic models

This post is for epidemiologists to understand what economists mean when they say that epidemic models should be “forward-looking”. And it is for economists to try and persuade them that incorporating behavior change in an “ad-hoc” fashion is just fine. I argue that all differences boil down to the type of mathematics that the two disciplines typically use – economists are used to “fixed-point mathematics”, epidemiologists to “recursive mathematics”. All in all, behavior change is incorporated by default in economic models, although in a highly unrealistic way; on the contrary, epidemiologists need to remind themselves to explicitly introduce behavior change, but when they do so they have the flexibility to make it much more realistic.

(When I talk about economists, I mean the vast majority of economic theorists, that is, economists that mostly reach their conclusions by writing mathematical models, as opposed to analyzing the data without a strong theoretical prior. When I talk about epidemiologists, I mean the ones that I know – mostly coming from the complex systems and network science communities. In this post I express strong opinions, but I try to be factual and fair; if you think I mischaracterized something, please let me know and I’ll be happy to revise.)

I would argue that modeling how humans respond to changes in their environment is as important in epidemiology as it is in economics. Recessions induce people to be cautious with spending out of fear that they could lose their job, in the same way that pandemics induce people to limit their social contacts out of fear of infection. Humans care about health as they care about their economic well-being, devoting at least as much attention to nonpharmaceutical interventions by the government as to monetary and fiscal policies. Yet, economists have been obsessed to model how people react to government policy, while epidemiologists have spent comparatively less attention. Finding out why scientific conventions in the two fields became so different is a super interesting epistemological question.

Let’s start with how economists deal with behavior change. Suppose that households’ income is 100 and taxes are normally 30% of that, letting disposable income be 70. Suddenly, the government cuts taxes to 15%, leaving households with an income of 85. To repay the deficit it created, two years later the government raises taxes to 45%, reducing households’ disposable income to 55. It then repeats this policy every four years: in years 4, 8, 12, …, it reduces taxes to 15% of income, in years 6, 10, 14, … it raises them to 45% of income.  Households want to smooth their consumption over time. If their behavior does not change, they fail, as their consumption is 85 in years 0-2, 4-6, 8-10, … and 55 in years 2-4, 6-8, 10-12, …

A simple way to model behavior change in this setting is to assume that households adaptively form expectations on government policy. After a few years, they learn to anticipate the pattern of taxes, and keep their consumption at 70. Indeed, they save when taxes are low, and repay what they saved in periods when taxes are high. If government policy changes, say taxes increase or decrease every four years instead of every two years, households take time to adapt to the new policy. If government policy changes frequently, households get wrong tax estimates most of the time and systematically overconsume or underconsume.

Economists have never been happy about agents being systematically wrong. Since the 70s, models with adaptive expectations have been replaced by models with so-called “rational expectations”. Rational agents [1], the argument goes, would discover even hard-to-predict patterns in government policy, and replace naïve agents that are unable to do so. Rational agents are “forward-looking” in the sense that they know the equations that drive policy. Therefore, they are able to make consumption decisions in year t based on government policy in year t+1. What if these consumption decisions impact government policy, too?

A rational expectations equilibrium is an infinite sequence of consumption decisions and government policies that are consistent with one another. Finding the equilibrium amounts to finding a fixed point in the (infinite-dimensional) space of consumption and policy sequences. Discovering such a fixed point turns out to be easier if the modeler assumes that households maximize a utility function. Using the mathematics of intertemporal optimization and Bellman equations, the modeler can find these sequences. I call this approach “fixed-point mathematics”.

In contrast, the learning process based on adaptive expectations is simply a difference equation in which households update their beliefs based on past tax values. Variables in year t are only determined based on variables in years t-1, t-2, … I call this approach “recursive mathematics”, and argue that it makes it much easier to include realistic assumptions.

Let’s come to behavior change in epidemiological models. These review articles show that there are quite a few papers trying to incorporate behavior change in basic SIR models. This article from 1976 and some subsequent articles consider non-linear variations to the basic SIR model, capturing for example the idea that a high number of infected makes susceptible individuals more cautious, lowering the transmission rate. The same idea is applied to this paper modeling the COVID-19 pandemic. The authors assume that individuals reduce social contacts when the number of deaths rises; because deaths occur with a delay with respect to infections, this leads to oscillatory dynamics in the number of infections as individuals ease or tighten social distancing. This nice paper assumes that awareness about diseases is transmitted in a social network, but fades with time. Again, this has clear implications for disease dynamics.

All these ways to deal with behavior change remind of the adaptive expectations framework of learning about government fiscal policy. Indeed, these approaches are rooted in recursive mathematics, which epidemiologists coming from biology or physics are well versed in.

Of course, economists aren’t happy of these ways to deal with behavior change in epidemic models, as they aren’t happy about adaptive expectations in economic models. Especially in the last few years, quite a few papers came out that tried to apply the rational expectations framework to epidemiological models. This paper, for example, assumes that individuals receive utility from social contacts, but utility goes down if they become infected. Thus, individuals trade-off utility from contacts with infection risk [2]. “Rational” individuals know the underlying SIR model and so are able to perfectly forecast epidemic paths conditional on their level of social contacts (see these notes for a very accessible explanation on this point).

In the figure below, the solid line is the rational expectations equilibrium, in which the epidemic path optimally satisfies the contact-infection tradeoff. In other words, at all times individuals choose the number of social contacts that they have, taking the optimal level of risk. Now look at the dotted line (ignore the dashed one). This is what happens when individuals don’t respond at all, as in the baseline SIR model. Does this figure look familiar? It should, it really looks like the “flatten the curve” picture that contributed convincing several governments to impose lockdown measures in March 2020. Under these assumptions, though, lockdown was useless, as individuals would have flattened the curve by themselves. In some sense, this is the Swedish approach. I leave it to the reader to judge whether it was a good idea to provide policy recommendations based on this model.







In the last months, the number of epidemiology papers written by economists has exploded [3]. The nice thing about models with rational expectations is that you cannot forget about behavior change. In a sense, you get it for free with the build-up of the model. The bad thing is that, in my opinion, this type of behavior change is clearly unrealistic. Even if real people had been able to act optimally at the onset of the COVID-19 pandemic, the scarcity of data would have prevented them to properly forecast the epidemic trajectory. And I have strong doubts about individuals acting optimally in any case. Thus, let me end this blog post with the following plea.

Epidemiologists, please remember to introduce behavior change in your models. To be fair, the models that had most policy impact were clearly unrealistic in not including any behavioral response. (From looking at the report, I assume that the Imperial study by Ferguson et al. did not have it, but I am not sure as I could not find a full description of the model.) But, please do not include behavior change in the way that economists mean it. In this recent paper on the HIV epidemic published on a top economics journal, individuals decide whether to have protected sex optimally trading off reduced pleasure from using condoms and infection risk. Policy recommendations are drawn from it. Aside from too easy ironies about agents maximizing a utility function before having sex [4], this completely ignores realistic elements such as social norms, decentralized information traveling in social networks of infected people, altruism, etc. These are also key elements characterizing behavior change in the COVID-19 pandemic. These elements could certainly be included in “rational” models, but it is very hard when you have to respect intertemporal fixed point conditions. Indeed, none of the at least 15 papers of epidemiology by economists that I’ve seen so far departs from the baseline assumption of homogenous households maximizing their own utility independently of social pressure. These papers will come, including one deviation from the baseline framework at a time, but most papers will provide policy recommendations based on the baseline. Instead, I hope epidemiologists will keep following the literature on behavior change that they already developed – see below.

Economists, if you must build epidemic models, please accept that you can introduce behavior change in a “reduced-form” way [5]. Some of you are already doing that. This nice paper builds essentially an agent-based model with spatial features, leading to realistic outcomes such as local herd immunity. The authors model behavior change simply by assuming that the transmission rate decreases linearly with the rate of infections. I don’t think they could find a rational expectations equilibrium that is fully consistent with the spatial structure, at least without oversimplifying other aspects of the model. This other paper, modeling behavior change essentially in the same way [5], considers infection spillovers across US counties, with a very accurate calibration based on county-level daily infection data. Instead, papers that go full steam towards rational, forward-looking agents, unavoidably ignore realistic aspects such as space. I understand that models with rational expectations are elegant and comparable and that there is a wilderness of reduced-form behavior-change epidemic models that is difficult to navigate. But, at least for epidemic models, please explore various boundedly-rational, adaptive, “ad-hoc” ways to respond to infection risk: you have a universe of realistic assumptions at your fingertips.

And, if you enjoy being able to play with reduced-form assumptions without the fear to be shot down by a referee, please consider such assumptions for economic models, too. It is so interesting to explore the world of “backward-looking” reactions to the economic environment. In our COVID economics paper, for example, we have sophisticated consumption decisions that depend on “ad-hoc” estimates of permanent income. Having “smart” agents that react to their environment should not almost always mean having optimizing and forward-looking agents in a rational expectations equilibrium.


Endnotes, or the corner of this blog post where I grumble about the state of economics, except in endnote 4, where I defend practice in economics from a misplaced criticism.

[1] I hate this use of the word “rational”. Here it means two things: that agents are able to maximize an objective function, and to correctly guess what every other agent and the entire economy do. While I agree that maximizing an objective function is consistent with the notion of rationality, I think that guessing what other agents do is a matter of prediction. Rationality and prediction can be in contrast. Rational expectations are effectively “correct expectations”. But using the word “rational” is a great selling point, because it makes “boundedly rational” decision rules look suboptimal to many eyes.

[2] Many people argue that taking decisions under incentives and constraints is what defines “economics”. So epidemiological models in which agents maximize a utility function subject to infection risk are “economic-epidemiological models”. I really really dislike this use of the word “economics” and what it implies. Economics should be the study of the economy. Reaction to incentives under constraints should be a branch of psychology. Economics should be neutral to which psychological theory it uses to model human behavior. Using the word “economics” to mean reaction to incentives under constraints makes it sound like that is the only way to model human behavior to study the economy. It is not.

[3] Interestingly, I haven’t seen any epidemiologist write an economics paper. This is known as economic imperialism: with the hammer of rational choice, every other social science looks like a nail for an economist. After all, economics is the queen of the social sciences, no?

[4] Saying that it is unrealistic that individuals maximize utility somehow misses the point of rational choice theory. Maximizing utility is only a tool to make a point prediction about what individuals do given incentives and constraints. It is a very general way to say, for example, that out of risk of infection individuals will be more cautious. A boundedly rational rule could still be expressed as the optimization of a modified utility function. I personally find utility a convenient analytical device; my real problems with economic theory have to do with equilibrium.

[5] In the 70s, at the same time that economists started to care about rational expectations, they also started caring about “microfoundations”. Every decision rule needed to be rooted in first principles, namely so-called preferences, technology, and resource constraints. By contrast, a “reduced form” assumption is a decision rule that is just postulated. For example, deriving decreases in the contact rate of a SIR model from maximizing a logarithmic utility function is consistent with microfoundations; simply postulating that contacts decrease linearly with the number of infectious individuals is not. While microfoundations are laudable in principle, they are often a straightjacket in practice. Many economists start with reduced-form expressions, and then reverse-engineer microfoundations. This is an art; it too often does not matter if microfoundations are just made up without being based on empirical evidence, as long as they are consistent with axioms of decision theory.

This paper is exemplary in the class of epidemic models by economists. To capture behavioral response, the authors assume a non-linear form on the infection rate, as in the 1976 paper mentioned above. But they justify it from first principles of economic theory. “We assume that all agents receive stochastic shocks z that we interpret as economic needs. The shocks are drawn from a time-invariant distribution F(z) with support z ∈ [0, ∞). […] Facing risk of infection during an excursion, Susceptibles optimally choose to satisfy a given need z only if the benefit exceeds the expected cost of taking an excursion.” In practice, agents go shopping only if z is a larger than an exogenously postulated level depending on the number of infectious individuals. By further assuming that the CDF of the stochastic shocks is z/(1+z), the authors obtain the functional form of the SIR model that they wanted. They will have less problems with referees as they apparently comply to academic social norms, but I find it hard to see the value added of such a build-up, at least in this case. (Note that I think that other than that it is a pretty good paper, especially in the way it is calibrated to data.)