## Saturday, May 23, 2020

### School of sustainability

In a few recent posts, I was critical of university endowment practices. Why build up a stock of investment, rather than invest in faculty, research, or other core activities? Why wall that pile of assets from being spent, especially when budgets are cratering in a pandemic? When we see businesses with piles of cash, we infer they don't have any good investment projects, and the piles are ripe for diversion to bad ideas.

But universities are non-profits, and one major piece of being a non-profit is that the business is protected from the market for corporate control. If you see a business wasting money on bad investments, buy up the stock, fire management, and run it right. Repurchases were part of an earlier reform effort, to stop management from wasting money on aggrandizing projects.

Perhaps restrictions on endowment spending serve a somewhat parallel function for universities. Perhaps I was wrong to criticize so harshly.

These thoughts are brought to mind by Stanford's announcement of a new school "focused on climate and sustainability." A "school" is bigger than a center, an institute, a department, a division. Stanford has seven "schools," Business, Education, Engineering, Humanities & Sciences, Law, Medicine, and, yes, Earth, Energy & Environmental Sciences.

Why a new school? It will
"amplify our contributions in education, research and impact further by aligning people and resources more effectively.
Says university President Tessier-Lavigne. Vice Provost Kathryn Moller will
"lead an inclusive process designing the school’s structure....consult with key internal and external stakeholders to develop a school organization that amplifies faculty and student contributions to address the most urgent climate and sustainability challenges."
creating an
"impact-focused community, with new opportunities to enhance the impact of their work on the issues they deeply care about,”
"Impact" and "amplify" repeat quite a few times.

## Tuesday, May 19, 2020

### Reopening the economy, and aftermath, now on Youtube

My Bendheim Center talk and discussion with Markus Brunnermeier on all things Covid-19 and economics is now on YouTube, direct link here. I start at 14:08.

If you like the paintings behind me and you're getting bored, more info here. (Shameless nepotism disclaimer.)

## Monday, May 18, 2020

### Endowment humor

Steven Wood writes a wonderful letter from a university president, responding to suggestions that a university dip in to its endowment,
As president of this University, there is nothing more important to me than the health and safety of our community. Though I’m currently away from campus, summering on my private island off of Maine, my thoughts are almost always with you, and my secretary is literally always available to field your questions and hear your concerns.
...a number of you have reached out to provide us with valuable feedback regarding our recently announced budget adjustments. Specifically, many of you have asked why an institution with a $46 billion endowment is freezing salaries, rescinding job offers, refusing to adjust tenure tracks, and laying off staff instead of using an endowment the size of Iceland’s GDP to keep our community afloat. Let me say this: We hear you. You are valid. You. Matter. Secondly, and no less importantly, let me make something clear: The. Endowment. Is. Not. For. You. ...the first rule of the endowment was “Never talk about the endowment.” At the end of every quarter, they blindfold me, take me to an undisclosed location which I suspect is the Chairman of the Board’s rumpus room, show me the quarterly returns, rough me up a little, then blindfold me again, and dump me on the lawn of my University-owned home. This is as close as I’ve ever gotten to the endowment, so good luck getting anywhere near that money. Oh, I give up, just go read the whole (short) thing and have a nice chuckle. Thanks to a colleague for the pointer. ### Schmitz on monopoly Jim Schmitz has released the first salvo in what promises to be a monumental work on monopoly, titled Monopolies Inflict Great Harm on Low- and Middle-Income Americans. (I love titles with answers and no colons.) Today, monopolies inflict great harm on low- and middle-income Americans. One particularly pernicious way they harm them is by sabotaging low-cost products that are substitutes for the monopoly products. I'll argue that the U.S. housing crisis, legal crisis, and oral health crisis facing the low- and middle-income Americans are, in large part, the result of monopolies destroying low-cost alternatives in these industries that the poor would purchase. He promises more to come Legal Services, Residential Construction, Hearing Aids, Eyecare and ...Repair, Pharmaceutals, Credit Cards, Public Education... There is a huge one right there. To Jim the main characteristics of monopoly are A. Monopolies sabotage and destroy markets. They typically destroy substitutes for their products, those that would be purchased by low-income Americans. B. Monopolies also use their weapons to manipulate and sabotage public institutions for their own gains... ### Reopening the economy -- and the aftermath I'm doing a Zoom talk at the Bendheim Center, Princeton, with Markus Brunnermeier, 12:30 Eastern today (Monday May 18) on this, tune in if you're interested. It's mostly based on recent blogs and opeds. Sign up here. I'll post a link to the video when it's over. ## Thursday, May 14, 2020 ### Strategies for Monetary Policy Strategies for Monetary Policy is a new book from the Hoover Press based on the conference by that name John Taylor and I ran last May. (John Taylor gets most of the credit.) This year's conference is sadly postponed due to Covid-19. We'll have lots to talk about May 2021. At that link, you can see the table of contents and read Chapter pdfs for free. You can buy the book for$14.95 or get a free ebook.

The conference program and videos are still up.

Much of the conference was about the question, what will the Fed do during the next downturn? Here we are, and I think it is a valuable snapshot. Of course I have some self interest in that view.

As long as I'm shamelessly promoting, I'll put in another plug for my related Homer Jones Lecture at the St. Louis Fed, video here and the article Strategic Review and Beyond: Rethinking Monetary Policy and Independence here. That was written and delivered in early March, about 5 minutes before the lookouts said "Iceberg ahead." John and I don't put a lot of our own work into the conference books, but it sparked a lot of thoughts.  I am grateful to Jim Bullard and the St. Louis Fed for the chance to put those together.

Monetary policy is back to "forget about moral hazard, rules, strategies and the rest, the world is ending." This is a philosophy that happens quite regularly and now has become the rule and strategy. So strategic thinking about monetary policy is more important than ever.  This is a philosophy very much due to John Taylor.

The last part of my Homer Jones paper delves into just what risks the big thinkers of central banking were worried about on the eve of the pandemic. Pandemic was not in any stress test.  BIS, BoE, FSB and IMF  wanted everyone to start stress testing ... climate change and inequality. This is a story that needs more telling.

## Monday, May 11, 2020

### Comment apology

To my commenters: I hit the wrong button this morning while cleaning up the huge amount of spam that comes in the comments, and many good comments got deleted. I appreciate your thoughts and I apologize for inadvertently deleting them.

## Thursday, May 7, 2020

### Markets work even in crisis

A lovely result of the corona virus outbreak has been how we see stifling aspects of regulations. Right left and center are figuring out that the regulations need reform. Now, the forces for regulatory stagnation are always strong, so the insight may fade with the virus. Still, let us enjoy it while it lasts.

The trouble with regulations is that, unlike "stimulus," the action is all in minute detail not grand sweeping plan.

John Goodman writes in Forbes
The Americans for Tax Reform calculates that 397 regulations have been waived in order to fight COVID-19. That count is probably way too low. The federal Food and Drug Administration (FDA) has eliminated so many restrictions it would be hard to count them all. ...
Consider that, up until a few months ago:
·     The only tests for the coronavirus that were approved for use in the United States were produced by the Centers for Disease Control (CDC) and half of those tests turned out to be defective.
·     It was illegal to produce, sell and distribute ventilators, respirators, and other  medical equipment without complicated and burdensome government regulatory permission.

### Covid and economics publishing

The pandemic is dramatically illustrating one area in which the epidemiologists are beating the economists about 100-1: publishing. Scientific publications are reviewed and posted in days, contributing in real time to the policy debate.

Economists are writing papers in a similar flurry. They are writing really good, thoughtful, well done papers that are useful to the policy debate. See the NBER website for example, or SSRN. See my last post and previous one for several great examples.

But when will these papers be peer reviewed? Where will they be published?

## Monday, May 4, 2020

### An SIR model with behavior

Following my last post, the SIR model has been completely and totally wrong. Answers follow from assumptions. It assumes a constant reproduction rate, and the virus peters out when sick people run in to recovered and immune people. That's not what's happening -- people responded by lowering the contact rate, long before we ran in to herd immunity.

I speculated last time about a model in which people respond to the severity of the disease by reducing contacts. Let's do it. (Warning: this post uses MathJax to show equations. It may not work on all devices.)

I modify the SIR model as presented by Chad Jones and JesúsFernández-Villaverde: \begin{align*} \Delta S_{t+1} & =-\beta S_{t}I_{t}/N\\ \Delta I_{t+1} & =\beta S_{t}I_{t}/N-\gamma I_{t}\\ \Delta R_{t+1} & =\gamma I_{t}-\theta R_{t}\\ \Delta D_{t+1} & =\delta\theta R_{t}\\ \Delta C_{t+1} & =(1-\delta)\theta R_{t}% \end{align*} S = susceptible, I = infected (and infectious), R resolving, i.e. sick but not infectious, D = dead, C = recovered and immune, N = population. The lags give the model momentum. Lowering the reproduction rate does not immediately stop the disease. The model uses exponential decays rather than fixed lags to capture timing. $$\beta$$ is the number of contacts per day. A susceptible person meets $$\beta$$ people per day. $$I/N$$ of them are infected, so $$\beta S_{t}I_{t}/N$$ become infected each day. We parameterize $$\beta$$ in terms of the reproduction rate $$R_{0}$$, $R_{0}=\beta/\gamma$ The number of infections from one sick person = number of contacts per day times the number of days contacts are infectious (on average).

The standard SIR model uses a constant $$\beta$$ and hence a constant $$R_{0}$$. The disease grows exponentially, then becomes limited by the declining number of susceptible people in the population. Each infected person runs in to recovered people, not susceptible people. The whole point is, that did not happen. We lowered $$\beta$$ instead.

I model the evolution of $$\beta$$ behaviorally. First, suppose people reduce their contacts in proportion to the chance of getting the disease. As people see more infectious people around, the danger of getting infected rises. They reduce their contacts proportionally to the number of infectious people. $\log(\beta_{t})=\log\beta_{0}-\alpha_I I_{t}/N_{t}.$ This function could also model a policy response.

However, due to the lack of testing we don't really know how many people are infectious at any time. So as a second model, suppose instead people or policy reduce contacts according to the current death rate, $\log(\beta_{t})=\log\beta_{0}-\alpha_D \Delta D_{t}/N.$ $$\beta$$ is a rate, how many people do you bump in to per day. I use the log because it can't be negative. The log also captures the idea that early declines in $$\beta$$ are easy, by eliminating superspreading activities. Later declines in $$\beta$$ are more costly.

I use Chad and Jesús  numbers, $$\gamma=0.2$$ or 5 days of infectiousness on average, $$\theta=0.1$$ implying 10 more days on average with the disease before it resolves, $$\delta=0.08$$ $$(0.8\%)$$ death rate. They parameterize and estimate $$\beta$$ $$\$$via $$R_{0}=\beta/\gamma$$. I take the original $$R_{0}% =5$$, which is typical of their estimates, and implies $$\beta_{0}=\gamma R_{0}=1$$. They estimate $$R_{0}^{\ast}=0.5$$ so $$\beta^{\ast}=0.1$$, which I will use to calibrate $$\alpha$$. New York peaked at 90 deaths per million, but we will see the dynamics overshoot. So I'll pick $$\alpha$$ in that case so that $$\beta=\beta^{\ast}$$ at 50 daily deaths per milllion triggers $$R_{0}% =0.5$$, i.e. $$\alpha_D$$ solves $\log\left( 0.1\right) =\log\left( 1\right) -\alpha_D\times50/10^{6}.$ The death rate is about 1%, so I calibrate the infection model so that $$R_{0}=R_{0}^{\ast}=0.5$$ at an infection rate of 5000 per million or 0.5%. $$\alpha_I$$ solves $\log(0.1)=\log\left( 1\right) -\alpha_I \times 5000 / 10^{6}.$

Here is my assumed reproduction rate as a function of deaths per million. The red dot is the calibration point: at 50 deaths per day, people and policy will drive the reproduction rate to 0.5. The red dashed line is a much more aggressive response, which I'll investigate later.

The standard SIR model

As background, here is a simulation of the standard SIR model with these numbers, and a constant $$\beta=1$$ meaning $$R_0=5$$.

I start at day 1 with a single infected person. The virus grows exponentially. The number infected peaks at about half the population. Around day 25 however, herd immunity starts to kick in. The number infected peaks. Sick people (resolving) peaks a bit later. The pandemic goes away almost as quickly as it came and it's over after two months. With $$R_0=5$$ everyone gets it and 0.8% or 8000 people die.

This is  the nightmare scenario presented to policy makers in February and caused the economic shutdowns. It is completely wrong -- it's not what happened anywhere.

The behavioral SIR model

Here is the simulation of the behavioral SIR model, in which people (or policy) reacts by lowering the contact rate in response to the number infected.

The vertical scale is different. Only about 4000 people get infected here, not 1 million! The pandemic gets going with the same exponential speed (blue line), but now once infections get up to  1000 per million we see the sharp reduction in the reproduction rate (dashed black line).

This is a lot more like what we saw! A rapid rise, to a plateau, with a much more sensible set of numbers. That's the good news. The bad news is that it goes on and on and on. The minute infections decline people slack off just enough to get it going again. Responding to infections, even though there is a lag, produces very stable dynamics.

The reproduction rate asymptotes to $$R_0=1$$.  This is both the good news and the bad news outlined in my last post. It doesn't get worse with second waves. But it doesn't get better either.

That result not at all related to the calibration. The reproduction rate always asymptotes to one in this model, with a steady number of infections and a steady number of deaths per day, until finally after years and years we get herd immunity and all efforts to reduce contacts are turned off.

Here is the same simulation with the much stronger response, $$alpha$$ is raised by a factor of 5 to the red dashed line in my first graph. No, it's not the same graph. Notice the vertical scale. This response is much less tolerant of infections, so the overall rate of infection is much lower. But the path is exactly the same.

Being more forceful does not change the reproduction rate, which still asymptotes to one. We just trundle along with much lower infections and daily death rates.

This comparison makes nice sense of what we see, per the last post. Far different regimes give rise to essentially the same dynamics, but some at much higher and some at much lower levels.

Technology offers some hope. What happens if the costs of reducing $$\beta$$ become lower over time, so people can slowly become more careful while also letting the economy grow? Widespread individual testing and tracing, for example, are ways of distancing that are less costly. In this case, we steadily move from the second to last graph to the last graph. To model that, I let $$\alpha$$ vary over time, growing by a factor of 2 from time 0 to time 100,

This accounts for a plateau with a slow tail. The actual reproduction rate still is close to one, but it's just enough below one to gently let the virus decay.

Deaths and information

A big objection: here I keyed behavior to the infection rate -- people are more careful the more infected people are around. But we don't see how many people are infected. We do see deaths.
Here is the simulation when people respond to the death rate rather than the infection rate

Since deaths lag infections by a few weeks, responding to the death rate leads to over controlling. The pandemic quickly gets out of control before deaths crank up, causing the crash in the reproduction rate. Then people really are careful, and the infection declines quickly. As deaths lower though, people ease up, and a second wave happens and so forth.

The positive feedback does eventually control the pandemic. Each wave is smaller. And this model also trends to $$R_0 = 1$$ by the same mechanism. It just takes a lot of wiggles to get there.

Information, rational exceptions, and externalitities

The contrast between the first and second graphs gives a quick policy suggestion: good information on how many people are infected in one's local area would be really helpful to avoid waves of infections.  If we had just enough random testing to know how many people are infected in our local area, people and  officials could follow the top graphs not the bottom graph. It is not expensive. In the model, one can back out the number of people infected from the increase in the number "resolving." The rate of hospital admissions might be a widely publicized number now available that could be a very good guess.

Widespread, available (no protocol, no prescription, just go get it, free market) testing would radically reduce the economic costs of social distancing, and end this fast. (The optimal $$\alpha$$ would rise by orders of magnitude. Yes, you point to externality, why should I test myself. But you ignore economic and social demands. If such testing is available, it's really easy for customers to demand you show your test. Paul Romer is right.

Of course now we get to the delicate question of public vs. private incentives. My first model seems like a reasonable guess of how people will behave -- take actions to be careful the greater my chance of getting sick is by going out.

We want, naturally, a dynamic model in which people's actions incorporate an understanding of the dynamics. In that vein, the latter graph seems unduly pessimistic. People are pretty smart and they know that the death rate is high when the danger of going out has passed. Thus, one may well expect them to foresee the dynamics, be careful when the death rate is increasing, and slack off when it is decreasing. More generally, in this deterministic model, you can back out what the state of all the variables is if you observe one of them. Thus, the rational expectations equilibrium of this model if people want to react to the number of infections is the first one, even if they can't  see infections. They can back infections out of the death data. That may be too much to hope for, but reality is likely in between.

Being careful has an externality, of course, so people following a private optimum of costly but careful behavior vs. getting sick is not necessarily the social optimum. Most economists jump quickly from this observation to calibrated time-varying lockdown policies to try to control $$\beta$$. But let us not forget the other side of that coin: The public policy tools are sledgehammers, which do a poor job of controlling interactions $$\beta$$ at reasonable economic cost. Really, what we have are at best exhortations to be careful in the details of daily life, plus extremely expensive business shutdowns. As a concrete example, in the model one may be tempted to advocate that officials lie about the number of infected to get people to be more careful than they would be privately. But once a lie is found out, nobody believes anything anymore, and the next step is China. I still think timely and accurate information is better.

To do list

Some more thought on functional form would be useful. Do we have any data or other ways of measuring how people behave?

Obviously a real economic model would derive these behavioral responses from a maximization problem, and consider the tradeoff between more distancing $$\beta$$ and economic costs.

Optimal policy may differ most from individual behavior in the dynamics. It is not worth it to an individual to be careful early when there are few sick people around, but policy considers the effect of you getting sick on everyone who gets it from you.  Optimal control of $$\beta$$ beckons. But to be realistic we must include the fact that public control of $$\beta$$ against private wishes will be much less efficient.

On the other papers: Chad and Jesús model social distancing, whether voluntary or by policy, via a deterministic and permanent exponential decay from a state of nature $$\beta_{0}$$ to a new lower value $$\beta^{\ast}$$ over a period $\beta_{t}=\beta_{0}e^{-\lambda t}+\beta^{\ast}(1-e^{-\lambda t}).$ The point here is to realize there is feedback, and both people and policy behavior respond to facts. Their paper fits the data so far beautifully. My goal is to think about what happens next. That $$\beta$$ just sits at $$beta^\ast$$ as it does in their model, seems unrealistic because people aren't going to keep distancing voluntarily or involuntarily.

Eichenbaum, Rebelo and Trabandt,  have an economic model of $$\beta$$. People work and shop less when they are more afraid of getting sick.  But  the tradeoff is not very attractive. Here is their model (solid) vs. the basic SIR model (dash)

In their model all people can do to avoid getting sick is to avoid work or consumption, both of which offer very little protection for great economic cost. So you still see the basic -- and false -- prediction of the SIR model. I think a good direction is to modify their model, calibrating it to data as Chad and Jesús do, which would imply an economically easier reduction in reproduction rate.

Update:

1) Equilibrium social distancing by Flavio Toxvaerd is a simple economic model with endogenous social distancing. It also produces plateaus when people choose to be safer

(Thanks to a tweet from Chryssi Giannitsarou @giannitsarou)

2) Economists vs. epidemiologists has a long history. Economists point out that disease transmission is not a biological constant, but varies with human behavior. And human behavior varies predictably in response to incentives (and information).  Many beautiful facts and stories on this point are collected in Tomas Phillipson and Richard Posner's book, Private Choices and Public Health: The AIDS Epidemic in an Economic Perspective. For example, AIDS patients in clinical trials would mix their medicines together. Half of the drug for sure is better than a 50 50 chance of nothing. Thanks to a correspondent for the reminder.

3) A Multi-Risk SIR Model with Optimally Targeted Lockdown by  Daron Acemoglu, Victor Chernozhukov Michael Whinston and Ivan Werning just came out. I haven't read it yet, but it is an obvious addition to the stack for people working on epidemiology models with economic incentives. It has diverse populations and transmission mechanisms, which have long struck me as a key insight. We are not all average, and that really matters here.

4) From the comments, a many-authored paper arguing that herd immunity may be much lower. Essentially the super spreaders are more likely to get the disease, so they are more likely to be immune first. "Super spreader" includes people in nursing homes, emergency room technicians, bus drivers, etc., not just jet setting partiers.

5) I missed Macroeconomic Dynamics and Reallocation in an Epidemic by  Dirk Krueger Harald Uhlig and  Taojun Xie
...we distinguish goods by their degree to which they can be consumed at home rather than in a social (and thus possibly contagious) context. We demonstrate that, within the model the “Swedish solution” of letting the epidemic play out without government intervention and allowing agents to shift their sectoral behavior on their own can lead to a substantial mitigation of the economic and human costs of the COVID-19 crisis, avoiding more than 80 of the decline in output and of number of deaths within one year, compared to a model in which sectors are assumed to be homogeneous. For different parameter configurations that capture the additional social distancing and hygiene activities individuals might engage in voluntarily, we show that infections may decline entirely on their own, simply due to the individually rational re-allocation of economic activity: the curve not only just flattens, it gets reversed.
6) A behavioral SIR model YouTube talk from Lones Smith

7) Systematic biases in disease forecasting – The role of behavior change  by Ceyhun Eksina Keith Paarpornb Joshua S. Weitzcde
...during real-world outbreaks, individuals may modify their behavior and take preventative steps to reduce infection risk. ... we evaluate this hypothesis by comparing the dynamics arising from a simple SIR epidemic model with those from a modified SIR model in which individuals reduce contacts as a function of the current or cumulative number of cases.
Thanks to Andy Atkeson for the tip. See his A note on the economic impact of coronavirus and Talk at NBER

8)...I'm sure more updates will follow.

Code

Here is the Matlab code for my plots

close all
clear all

gam = 0.2;
thet = 0.1;
delt = 0.008;
R0 = 5;
alphaI = (0 - log(0.1))*1E4/50;

beta0 = R0*gam;
N = 1E6;

T = 100;

% plot beta

figure
drate = (0:100)'/1E6;
plot(drate*1E6, betat/gam,'linewidth',2);
hold on
plot(drate*1E6, betat1/gam,'--','linewidth',2);
hold on
plot(50, 0.5, 'o','markerfacecolor','r')
legend('Original','\alpha multiplied by 5','location','best')
xlabel('Daily deaths/million');
ylabel('Reproduction rate R0 = \beta / \gamma')
axis([0 80 0 5]);
print -dpng betafig.png

% standard SIR model

S = zeros(T,1);
I = S;
R = S;
D = S;
C = S;

S(1) = N-1;
I(1) = 1;

for t = 1:T-1;

S(t+1) = S(t) -beta0*S(t)*I(t)/N;
I(t+1) = I(t) + beta0*S(t)*I(t)/N - gam*I(t);
R(t+1) = R(t) + gam*I(t)-thet*R(t);
D(t+1) = D(t) + delt*thet*R(t);
C(t+1) = C(t) + (1-delt)*thet*R(t);

end;

figure;
plot((1:T)',[S I R 100*D C]/1E6, 'linewidth',2);
xlabel('Days')
ylabel('Millions');
axis([10 70 0 1]);
title('SIR model, constant R0 = 5');
print -dpng std_sir.png

figure;
plot((2:T)',[ I(2:T) -(S(2:T)-S(1:T-1)) 100*(D(2:T)-D(1:T-1))]/1E6, 'linewidth',2);
xlabel('Days')
ylabel('Millions');
axis([10 70 0 0.6]);
title('SIR model, constant R0 = 5');
print -dpng std_sir_diffs.png

disp('last s i r d');
disp([S(T) I(T) R(T) D(T)]);

% my model,  response to infections

S = zeros(T,1);
I = S;
R = S;
D = S;
C = S;
betat = S;

S(1) = N-1;
I(1) = 1;
I(1) = 1;

S1 = S;
I1 = I;
R1 = R;
D1 = D;
C1 = C;
betat1 = betat;

S2 = S;
I2 = I;
R2 = R;
D2 = D;
C2 = C;
betat2 = betat;

for t = 1:T-1;

betat(t) = exp(log(beta0) - alphaI*((I(t))/N));
S(t+1) = S(t) -betat(t)*S(t)*I(t)/N;
I(t+1) = I(t) + betat(t)*S(t)*I(t)/N - gam*I(t);
R(t+1) = R(t) + gam*I(t)-thet*R(t);
D(t+1) = D(t) + delt*thet*R(t);
C(t+1) = C(t) + (1-delt)*thet*R(t);

betat1(t) = exp(log(beta0) - 5*alphaI*((I1(t))/N));
S1(t+1) = S1(t) -betat1(t)*S1(t)*I1(t)/N;
I1(t+1) = I1(t) + betat1(t)*S1(t)*I1(t)/N - gam*I1(t);
R1(t+1) = R1(t) + gam*I1(t)-thet*R1(t);
D1(t+1) = D1(t) + delt*thet*R1(t);
C1(t+1) = C1(t) + (1-delt)*thet*R1(t);

betat2(t) = exp(log(beta0) - (1+1*t/T)*alphaI*((I2(t))/N));
S2(t+1) = S2(t) -betat2(t)*S2(t)*I2(t)/N;
I2(t+1) = I2(t) + betat2(t)*S2(t)*I2(t)/N - gam*I2(t);
R2(t+1) = R2(t) + gam*I2(t)-thet*R2(t);
D2(t+1) = D2(t) + delt*thet*R2(t);
C2(t+1) = C2(t) + (1-delt)*thet*R2(t);

end;

figure;
%yyaxis left
plot((2:T)',[ I(2:T) 100*(D(2:T)-D(1:T-1)) ],'linewidth',2)
xlabel('Days')
ylabel('People');
axis([0 70 0 inf]);

yyaxis right
plot((1:T)',betat/gam,'--k','linewidth',2);
ylabel('Reproduction rate R0','color','k')
axis([0 70 0 inf]);

legend('Infected','100 x Deaths/day','R0 (right scale)','location','best')

title('BSIR model, R0 varies with infection rate');
print -dpng std_sir_aI.png

figure;
%yyaxis left
plot((2:T)',[ I1(2:T) 100*(D1(2:T)-D1(1:T-1)) ],'linewidth',2)
xlabel('Days')
ylabel('People');
axis([0 70 0 inf]);

yyaxis right
plot((1:T)',betat1/gam,'--k','linewidth',2);
ylabel('Reproduction rate R0','color','k')
axis([0 70 0 inf]);

legend('Infected','100 x Deaths/day','R0 (right scale)','location','best')

title('BSIR model, R0 varies with infection rate, higher \alpha');
print -dpng std_sir_aI1.png

figure;
%yyaxis left
plot((2:T)',[ I2(2:T) 100*(D2(2:T)-D2(1:T-1)) ],'linewidth',2)
xlabel('Days')
ylabel('People');
axis([0 70 0 inf]);

yyaxis right
plot((1:T)',betat2/gam,'--k','linewidth',2);
ylabel('Reproduction rate R0','color','k')
axis([0 70 0 inf]);

legend('Infected','100 x Deaths/day','R0 (right scale)','location','best')

title('BSIR model, R0 varies with infection rate, \alpha increases over time');
print -dpng std_sir_aI2.png

% my model,  response to deaths

S = zeros(T,1);
I = S;
R = S;
D = S;
C = S;
betat = S;

S(1) = N-1;
I(1) = 1;
I(1) = 1;
betat(1) = beta0;

for t = 1:T-1;

S(t+1) = S(t) -betat(t)*S(t)*I(t)/N;
I(t+1) = I(t) + betat(t)*S(t)*I(t)/N - gam*I(t);
R(t+1) = R(t) + gam*I(t)-thet*R(t);
D(t+1) = D(t) + delt*thet*R(t);
C(t+1) = C(t) + (1-delt)*thet*R(t);

end;

figure;
%yyaxis left
plot((2:T)',[ I(2:T) 100*(D(2:T)-D(1:T-1)) ],'linewidth',2)
xlabel('Days')
ylabel('People');
axis([0 100 0 inf]);

yyaxis right
plot((1:T)',betat/gam,'--k','linewidth',2);
ylabel('Reproduction rate R0','color','k')
axis([0 100 0 inf]);

legend('Infected','100 x Deaths/day','R0 (right scale)','location','best')

title('BSIR model, R0 varies with death rate');

### Dumb reopening might just work

A smart reopening, with well worked out protocols at work, and a robust competent test and trace public health response to stamp out the embers, seems unlikely. Technology to save us -- vaccine, cure, cheap daily test, scaled up and implemented -- seems unlikely in the next few months. We seem fated to a dumb reopening. Conventional wisdom says we will then get a massive second wave in the fall, followed by a larger shutdown.

Spread in last week's tidbits and a bit of modeling over the weekend, I see hope that this dumb reopening might just work, including the steady and slowly declining new set of cases we are seeing right now.

The Theme

In February and early March, the models predicted exponential growth, massive infections, hospitalizations, and deaths, with most everyone getting the virus in a matter of months, and then the virus to quickly pass. The models were disastrously -- or, better, miraculously -- wrong.  New cases plateaued quickly and then slowly declined. In many parts of the country, hospitals have plenty of extra space.

Conventional wisdom holds this great good fortune is because of the lockdown. But, that wisdom warns,  the minute the economy reopens, it all starts again absent the above public health or a vaccine.  Conventional wisdom thus says, do not extrapolate the current trends.

The tidbits of news that give me hope, below, are that the plateau came far sooner than expected, it is lasting far longer than expected, and the shape seems quite similar across many regimes.

I hazard here a guess of why this is occurring: 1) The models do not take account that the reproduction rate R0, how many people each infected person gives it to, is immensely influenced by human behavior. And, said humans, read the news. 2) The average reproduction rate heavily influenced by super-spreading activities. The average is composed of a large majority of activities that give it to less than one other person, and a small minority of activities -- singing in choirs, beer pong at ski resorts, big loud indoor wedding parties -- that super-spread it.

Add up 1) and 2). When people hear there is a disease about, they quickly stop super-spreading activities, all on their own, because they don't want to get sick. Shutdowns only marginally affect this process. We saw, for example, massive declines in travel and restaurants long before shutdowns were announced. This action quickly and rather easily reduces the average reproduction rate to something like 5 to something below 2.

Then, as people hear news of how bad it is in their area, they adjust more. If people hear it's not so bad, they adjust less. If the virus is on the upswing, they social distance more. Do you walk or take the bus? Do you eat at a social distanced restaurant or take out? There are hundreds of little behaviors each of us take that push the reproduction rate around.

You can see a self-regulating state here, where the number of new cases sits at a steady plateau for a long time. You can easily see a self-regulation that drives the system to R0=1, or to a steady number of new cases.

That's not great news. There will still be a steady flow of new cases  per week, just enough to scare people.  But as people slowly start to adopt common sense and ignore silly shutdowns, and as people start to adopt common sense and avoid even permitted dangerous activities, the economy can recover a good deal. All we need is good information.

Now in detail.

## Friday, May 1, 2020

### Romer: if virus tests were like sodas; a modest extension

Paul Romer has a lovely post, If virus tests were like sodas. (HT Marginal Revolution.) Go enjoy the whole thing. It's short. A few excerpts and a suggested addition:

Imagine a world in which the only way to get a soda is to get your doctor to write a prescription. It costs $20 per can. Your insurance company pays. ... Because they have to keep total costs from running out of control, insurance companies, health care providers, and government regulators have cobbled together a system that limits access to soda. One part of this system is an expensive regulatory process... The only people who can get sodas are those already under the care of the health care system. They are not thirsty, but the insurance company covers the cost, so whatever. People who are thirsty start going to the hospital just to get soda. Doctors comply with their requests for a prescription. Soda producers try to increase output, but soon run into “bottlenecks.” One vendor with an approved soda delivery system that packages a straw with a can finds that its supplier of straws can not keep up with the increased demand. This soda company explains to its unhappy customers that it has FDA approval only for a product that includes a straw from its traditional supplier. The soda company says that it is applying to the FDA for an Emergency Use Authorization (EUA) that gives it permission to bundle a can with a straw from a different vendor. As it waits, it keeps repeating its excuse: “There is a straw bottleneck!”... In their experiments with drinking from the can, these same university researchers realize soda is just flavored sugar water and that they could produce millions of sodas per day at a price well under$1 per can. The researchers publicize their findings. Policy wonks urge them to get going: “Produce the sodas that a thirsty nation needs.” But these do not say anything about who will pay for all these additional sodas. The researchers are good sports, but they are not idiots. They produce some token batches of soda and go back to writing papers.
... wonks conclude that even an economic system as big, as powerful, and as innovative as the one we have established in the United States cannot rise to the challenge of producing millions of sodas per day. They settle for a stretch goal of offering one soda per month to each family.
Comment: The policy wonks as usual left out the problem: big, powerful, innovated, and regulated to death.
The facts:
Researchers affiliated with Rutgers University did discover that you do not need a swab to do an RT-PCR test for the SARS-CoV-2 virus. They even went to the trouble to get an EUA to conduct tests on saliva samples.
No one has proposed a way to pay the researchers at Rutgers, or their peers in comparable laboratories located throughout the United States, for the tests they could supply. For now, they do them because they are good sports.
The US economy produces 350 million 12 oz cans worth of soda each day.
Soda producers do not need to get regulatory approval each time they innovate around some hurdle or bottleneck.
I'm not sure soda is so lightly regulated, but we'll leave that.
Lessons
If we want to use this nation’s massive capacity – much of which, by the way, is now sitting idle – to produce tens of millions of virus tests per day, there is a way to do it:
Decide what a test should do.
As long as labs provide tests that do what a test is supposed to do, let them worry about the details.
Do not appeal to charity; be prepared to pay these labs twice as much as we spend on soda.
On the last point, the usually clear Paul ran out of steam. Who should be prepared to pay the labs? The same insurance companies and government purchasers where the whole problem started?

Let me offer a suggestion. Allow people and businesses to pay the labs whatever the labs want to charge and buy the tests themselves. Require only that they report the test result to the CDC's national database.

Lots of people and businesses will happily pay cash for a test. Spitting in a cup and sending it in -- or putting it in an Abbot Labs machine for instant results -- cannot possibly hurt anyone. There is no reason such tests should not be sold, unregulated, on the free market, like pregnancy tests.

Sure, label the test with the best estimate of its false positive and negative rate, and the same long legal boilerplate disclaimers that go on a lawnmower you buy from Home Depot.

Who gets it first? Well, those willing to pay the most. This is not a capitalist inequality outrage, this is a good idea. GDP and employment are cratering. The  people and businesses who get most economic value out of testing should get them first. And, by doing so, they fund the immense expense of test development and rapid ramp up for the rest of us.  And, of course, the higher the price, the more quickly competitors will ramp up and drop prices. We'll all get tests faster if those who "can afford it"  pay through the nose to get it first.

### Weekly Podcasts

The grumpy economist, on the university finances post and related issues. I found the embed code! The original is here.

Good fellows conversation Direct link here. On reopening, and strategic issues.