Nicolas Franz-Pattillo

The Power of an Apology

Nicolas Franz-Pattillo — Sat, 26 Sep 2026 00:00:00 GMT

Photo: Joshua Clay, Unsplash via Wikimedia Commons — CC0 (public domain).

Most of us learned to apologize as children, and most of us learned it badly. An adult pointed at the person we had hurt, and we said “sorry” as fast as possible so the moment would end. The word was a toll we paid to get out of trouble.

A real apology is something else. It is one of the few tools we have for repairing harm after it has already happened. It is also hard to do well, and the difficulty is exactly what makes it work.

An apology starts with agreeing that something was wrong

An apology begins with a shared recognition that a wrong act occurred. Without it, every later step rests on two conflicting stories about what happened. The person apologizing thinks they are making amends, and the person receiving it hears a negotiation. So the first job is simply to agree, clearly and without qualification, that what happened was not okay.

The work is in the details

Say exactly what you did. A vague “I’m sorry for everything” costs nothing, and the other person knows it. Naming the specific act shows that you know what you did and aren’t hoping it gets forgotten.

Explain why, without passing the blame. This is the hardest part. The instinct is to explain our behavior by pointing at the other person: “I reacted that way because you yelled at me.” That sentence looks like an apology but works as an accusation. It moves responsibility outward. Compare “I reacted that way because I couldn’t control my anger.” The event is the same, but now the cause is yours, which it was all along.

The “why” matters to the person you hurt. It shows you’ve looked past what happened to what drove it. It is often slow and uncomfortable to find, because the honest answer is rarely flattering: fear, insecurity, pride, a habit we picked up long ago.

Sit with the suffering you caused. After you’ve explained yourself, stay present while the other person tells you what it was like for them, without defending, minimizing, or rushing to the resolution. It hurts. It is also the only way to actually understand the harm, and you can’t take responsibility for something you haven’t understood.

Take responsibility and make amends. If you’ve done the previous steps honestly, this one mostly follows. Taking responsibility means committing to do better and then showing it with concrete actions over time.

Why go through all this?

A sincere apology is a gift to the person harmed. It is also, less obviously, a way out for the person who caused the harm.

People who hurt others often carry a burden of their own: they refuse to be vulnerable, they fear looking weak, and they live with guilt they never deal with. That armor is heavy. Punishment hardens us, but it doesn’t teach us, and humiliation doesn’t bring insight. Being punished or shamed rarely changes anyone. Understanding what you did can.

Many religious traditions have a ritual built on this idea: naming your wrong out loud is what releases its weight.

Catholicism: the sacrament of reconciliation, once called confession. You name your sins to a priest, feel real sorrow for them, and take on a penance, a concrete act of repair.
Judaism: teshuvah means recognizing the wrong, regretting it, confessing it aloud, and resolving not to repeat it. Yom Kippur atones only for wrongs against God. For a wrong against another person, you must first ask that person’s forgiveness.
Islam: tawbah is made directly to God. It asks you to stop the wrong, regret it, and resolve not to return to it. If someone was harmed, you must also make it right with them.
Buddhism: monks confess their offenses to a fellow monk before the twice-monthly recitation of the monastic rules.
Other Christian churches: the Orthodox Church keeps private confession, and Anglicans and Lutherans say a general confession during the service. The Letter of James asks believers to “confess your sins to one another… that you may be healed.”
Hinduism: prāyaścitta are rites of penance meant to cleanse the consequences of a wrong act.

An apology makes both people human again. It shows the person you hurt that you can see them, and that you are more than the worst thing you did to them.

Learning to apologize

If apologizing is this hard for adults, it is worth being patient with children who are learning it. When a child struggles to apologize, the difficulty is rarely a lack of empathy or goodwill. The steps above demand skills that are still growing in them: regulating emotions, identifying their own feelings and other people’s, thinking flexibly, and tolerating vulnerability. It can be very hard to:

find words for what they felt in the moment,
tell an explanation apart from an excuse,
tolerate intense feelings of shame or guilt,
work out how the other person felt,
or stay in an emotionally demanding conversation without being overwhelmed.

That’s why learning to apologize usually needs support, time, modeling, and emotional containment more than punishment or on-the-spot demands. The patience we ask of them when they apologize is the patience we owe them while they learn.

Conclusion

Apologizing is hard because it requires exposure, self-examination, and humility. That’s also why it works. By taking responsibility, acknowledging the harm, and letting ourselves feel what the other person felt, we make it possible to rebuild trust. We also take a step, not only toward repairing a relationship, but toward becoming better people.

What is loose commitment?

Nicolas Franz-Pattillo — Wed, 01 May 2024 00:00:00 GMT

John William Waterhouse, Ulysses and the Sirens, 1891. National Gallery of Victoria, Melbourne — public domain.

Textbook monetary policy comes in two flavors. Either the central bank follows a rule, such as a Taylor rule, or it chooses policy optimally, in which case one has to say whether it can commit to its promises or acts under discretion (Kydland and Prescott (1977)). Outside of the textbook, we may find something in between.

This note explains the machinery behind a model of that middle ground, loose commitment, as used in my paper Optimal Monetary Policy under Loose Commitment and the Zero Lower Bound. Here the goal is to show how the math works: the model, the three policy problems, and why solving them requires a global (non-linear) method.

The economy

The model is a small New Keynesian economy with three ingredients: sticky prices Rotemberg (1982), monopolistic competition and a lower bound on the nominal interest rate. It is deliberately small; the point is to isolate how commitment interacts with those three frictions, not to fit the data.

Households

A representative household chooses consumption , hours and nominal bonds to maximize

where is a preference shock: when is high the household becomes more patient and wants to save. It follows . This is the shock that pushes the economy to the zero lower bound, because a sudden desire to save lowers the interest rate the economy needs.

The household’s choices give two familiar conditions, the Euler equation and the labor supply condition:

where is the gross nominal interest rate, gross inflation, the real wage and a labor subsidy.

Firms and the non-linear Phillips curve

A competitive final-goods firm combines a continuum of intermediate goods with a constant elasticity of substitution . Intermediate firm therefore faces the demand , produces with and, following Rotemberg, pays a quadratic cost to change its price.¹

Two parameters matter here. governs how sticky prices are. is the inflation rate that is free of cost; the literature usually sets (zero net inflation), and so does the benchmark calibration.

Each firm chooses its price path to maximize the discounted value of profits, using the household’s stochastic discount factor (Yun (2005)). In a symmetric equilibrium this yields the non-linear Phillips curve:

Derivation: from the firm’s problem to the Phillips curve

Firm solves

with and the nominal discount factor .

The price appears in period (revenue, labor cost and today’s adjustment cost) and in period (tomorrow’s adjustment cost, through ). Differentiating and imposing symmetry, :

revenue: ;
labor cost: ;
today’s adjustment cost: ;
tomorrow’s adjustment cost: .

Setting the sum to zero, using and multiplying by gives the equation above. With flexible prices () it reduces to : firms charge a constant markup over marginal cost.

The private-sector equilibrium

Using market clearing and the resource constraint, which subtracts the resources used up in changing prices, the economy is summarized by

Only the Euler equation and the Phillips curve involve expectations. That will turn out to be the whole story: the only reason policy is hard is that today’s outcomes depend on what people expect policy to do tomorrow.

The benchmark: flexible prices and an efficient subsidy

Before asking what a central bank can achieve, it helps to know what the best possible outcome is, and which frictions stand in the way.

The efficient allocation

A planner who could simply assign consumption and hours, with no prices involved, would maximize subject to . At the optimum, the marginal disutility of an hour of work equals the consumption it produces:

Combined with , this gives the efficient allocation:

The market economy reaches this point only if prices are flexible () and markets are competitive (). Monetary policy is then irrelevant.

Flexible prices with market power

Now keep prices flexible but let firms have market power (). With the Phillips curve says that firms set a constant markup, so the real wage is below the marginal product of labor:

Substituting into the household’s labor supply condition, , gives

which differs from the efficient condition by the factor . The flexible-price allocation is therefore

Without a subsidy (), the markup acts like a tax on work: people work too little and the economy produces too little. With the benchmark calibration (, , ), output is about 6% below its efficient level even with perfectly flexible prices.

The subsidy that undoes the markup

A labor subsidy can close this gap exactly. Setting

makes the flexible-price allocation efficient. The government pays the subsidy on wages and finances it with a lump-sum tax of the same size, so its budget balances and the tax itself distorts nothing. This is the scheme that much of the optimal-policy literature assumes, for example Siu (2004) and Yun (2005).

What this means for monetary policy

With the subsidy in place and sticky prices (), the central bank can reproduce the efficient allocation by keeping inflation at its free-of-cost level every period, . Adjustment costs are then zero, both sides of the Phillips curve collapse to , and the economy behaves as if prices were flexible. The interest rate that delivers this follows from the Euler equation:

Two things can break this. First, a large enough preference shock makes the required fall below one, which the lower bound rules out. Second, without the subsidy, keeping delivers the flexible-price allocation, not the efficient one, and the central bank is tempted to push output up with inflation.

The paper keeps on purpose. The subsidy would remove the second problem, and with it the inflation bias. Leaving the markup in place is what lets the model study how the two biases, inflationary in normal times and deflationary at the lower bound, trade off as commitment changes.

Why optimal policy is time-inconsistent

The central bank sets to maximize the household’s welfare, taking this equilibrium as a constraint. It would like to close three gaps, or wedges, but it cannot close all of them at once:

Price adjustment. Any inflation different from wastes resources. This wedge says: keep .
Monopolistic competition. Markups keep output below its efficient level. In the benchmark there is no subsidy (; see the previous section), so this wedge is open. Surprise inflation raises output, so this wedge says: inflate.
The zero lower bound. When the economy needs a negative nominal rate, the bound keeps real rates too high. This wedge says: promise higher inflation later, so that real rates fall today.

The second and third wedges create opposite temptations, and both are temptations to break a promise.

Inflation bias. Suppose the central bank promises and firms believe it. Once expectations are set, a bit of surprise inflation raises output at no cost in credibility for today. Firms anticipate this, so in equilibrium inflation ends up above with no gain in output.
Deflation bias. At the lower bound, the central bank would like to promise higher future inflation to lower real rates now. But when the future arrives, the crisis is over and it would rather deliver . If people anticipate that, the promise is worthless, and the economy spends longer at the bound with lower inflation.

This is Kydland and Prescott’s time inconsistency in its cleanest form. What the central bank can do therefore depends on what it can promise, which is what the three policy problems below differ in.

Three policy problems

For all three, write and split each forward-looking equation into the part that depends on today’s variables and the part that depends on tomorrow’s:

The Euler equation is then and the Phillips curve is .

Discretion

Under discretion no promise binds: each period the central bank re-optimizes, taking how it will behave in the future as given. Because it has no incentive to preserve a promise when the next period arrives, promises have no independent force; it is as if they did not exist. The state therefore carries no burden from previous promises: policy depends only on the current exogenous shock, not on an inherited policy plan. The right-hand sides and are then outside its control, and the problem is static:

subject to the equilibrium conditions, with and taken as given.

Discretionary models have more than one equilibrium. I focus on the Markov-perfect equilibrium in which policy depends only on the current shock and the lower bound binds occasionally; there is also a “deflation trap” equilibrium in which low expectations keep the economy at the bound, which I rule out.

The first-order condition for gives the most useful insight in the whole problem. With multipliers on the Euler equation and on the lower bound,

Away from the bound, , so : the Euler equation is not a constraint in normal times. The central bank can always set the rate the Euler equation requires. Aggregate demand only becomes a binding problem at the lower bound. Keep this in mind for the next section.

Commitment

Under full commitment the central bank chooses, once and for all, a plan for every possible history of shocks. Attach multipliers and to the Euler equation and the Phillips curve, and to the lower bound. The Lagrangian is

The difficulty is that appears in the constraints of period . Marcet and Marimon’s trick is to move those terms into period , where they belong. Because the discount weight of period is , the term is exactly . So in period the problem contains yesterday’s multiplier. Define

The problem becomes recursive, a saddle-point functional equation with the lagged multipliers as a new state:

This is where the word promise becomes literal. and are yesterday’s promises about today; they enter today’s problem as costs of deviating from what was announced. Today’s multipliers become tomorrow’s promises. The state now has two endogenous dimensions on top of the shock.

Two consequences follow from the first-order conditions.

No inflation bias in the long run. In a steady state without the bound, promises are constant, so . The terms then pull in opposite directions with equal weight, and the first-order condition for inflation reduces to

where is the multiplier on the resource constraint. Hence . Under discretion there is no term to offset , and inflation settles above : the inflation bias.

Forward guidance lives in . As under discretion, whenever the bound does not bind. So the Euler promise is positive only after the economy has been at the bound. When it is, the first-order condition for inflation acquires an extra term,

which raises inflation after the episode ends. This is the credible promise of higher future inflation that makes commitment so effective at the bound.

Loose commitment

Loose commitment, from Debortoli and Nunes (2010), sits between the two. Each period an i.i.d. coin is tossed:

with probability the plan continues (commit);
with probability the central bank reneges and a new plan is drawn up from scratch (renege).

A promise made today therefore survives until tomorrow with probability . When the central bank reneges, it starts again with no inherited promises, so the state jumps back to . The parameter moves the economy continuously from discretion () to commitment (). Promises last periods on average.

Private agents know this, so their expectations average over the two branches. The Euler equation of the committed plan becomes

and similarly for the Phillips curve, where is what a re-optimizing central bank will do. The same regrouping trick as before moves the part to the next period of the committed branch. The part stays where it is, because the central bank cannot promise anything about a plan that will have been replaced. The result is the loose-commitment saddle-point problem:

The reneging branch is not free to choose. Whoever re-optimizes tomorrow faces exactly the same problem with a clean slate, so equilibrium requires a fixed point:

A loose-commitment equilibrium is a set of functions , , and that solve the problem above and satisfy this fixed point. Setting recovers full commitment; setting removes every promise and recovers the discretionary problem.

Why there is no steady state

Under full commitment the promise state starts at zero and drifts to its long-run level. Under loose commitment it is reset to zero every time the coin comes up renege, so the economy keeps falling back to the origin and climbing away from it. Even without shocks it never settles. Its long-run distribution is a mixture of commitment spells of different lengths, with spell length geometric in , and it changes with itself.

Measuring welfare

Welfare is reported as a consumption equivalent: the percentage of consumption that would make the household indifferent between the economy with the lower bound and the efficient allocation derived above,

and its unconditional version , which takes expectations over the model’s long-run distribution on both sides.

Solving it without linearizing

The usual shortcut, a linear-quadratic approximation around a steady state, fails here for three reasons:

The lower bound is a kink. Policy functions bend sharply where the bound starts to bind.
There is no single point to approximate around. Average inflation moves from under commitment to plus the inflation bias under discretion, so any approximation point is wrong for most values of . And under loose commitment there is no steady state at all.
The results are about the shape of things. The main result is that the probability of reaching the bound is hump-shaped in . That depends on how the mean and the volatility of inflation shift along the commitment line, which is exactly what a linearization around one point throws away.

So the model is solved globally, with time iteration (Coleman (1991)): guess policy functions on a grid of states, solve the first-order conditions at each grid point given the guess for tomorrow, update, and repeat until nothing changes.

At each grid point, compute the expectations in the Euler equation and the Phillips curve. Tomorrow’s promise state is with probability and with probability ; the shock is integrated with Gauss–Hermite quadrature.
Solve the first-order conditions assuming the bound does not bind.
If the implied is below one, re-solve with .
Update the policy guess with some dampening, and stop when successive guesses agree to within a tolerance.

Discretion is the easy case: the only state is the exogenous shock, whose distribution is known, so the grid is simple. The only refinement is to find the point where the bound starts to bind by bisection and place extra nodes around it.

Loose commitment is the hard case. The state has three dimensions (the shock and two promises), and the relevant part of the promise space is unknown before solving the model, and different for every . A dense rectangular grid is both slow and unstable, because it includes promise levels the economy never reaches. The solution follows Maliar and Maliar (2015): let the model tell you where it lives.

Solve on a coarse grid.
Use that solution to approximate the long-run distribution of the promise state. Every spell starts at ; mass is pushed forward along the policy functions, and spells survive each period with probability .
Keep only the grid nodes with non-negligible probability, which gives an irregular grid concentrated where the economy actually spends its time.
Re-solve on the new grid, interpolating between the scattered nodes with a Delaunay triangulation, and repeat until the grid stops changing.

Where this leaves us

The machinery comes down to one idea: a promise is a state variable. Under discretion there are no promises, and the central bank suffers from an inflation bias in normal times and a deflation bias at the lower bound. Under commitment, promises are permanent and both biases disappear. Under loose commitment, promises decay at rate , and with them the value of making them.

Because the model is solved globally, one can move smoothly from zero to one, holding everything else fixed, and examine how inflation, welfare and the lower bound change along the way.

The recursive formulation of the policy problem follows the loose-commitment and recursive-contracts literature (Debortoli and Nunes (2010); Marcet and Marimon (2019)), while the treatment of the lower bound and the welfare comparison is related to Siu (2004).

This is my personal website. The views and opinions expressed here are my own and do not necessarily represent those of the Central Bank of Chile or its Board. Any errors, unfinished thoughts, questionable code, and overly complicated solutions found here are, of course, my own.

References

Calvo, G. A. (1983): “Staggered prices in a utility-maximizing framework,” Journal of Monetary Economics, 12, 383–98.

Coleman, W. J. (1991): “Equilibrium in a production economy with an income tax,” Econometrica, 59, 1091–1104.

Debortoli, D., and R. Nunes. (2010): “Fiscal policy under loose commitment,” Journal of Economic Theory, 145, 1005–32.

Dotsey, M., R. G. King, and A. L. Wolman. (1999): “State-dependent pricing and the general equilibrium dynamics of money and output,” The Quarterly Journal of Economics, 114, 655–90.

Kydland, F. E., and E. C. Prescott. (1977): “Rules rather than discretion: The inconsistency of optimal plans,” Journal of Political Economy, 85, 473–91.

Maliar, L., and S. Maliar. (2015): “Merging simulation and projection approaches to solve high-dimensional problems with an application to a new keynesian model,” Quantitative Economics, 6, 1–47.

Marcet, A., and R. Marimon. (2019): “Recursive contracts,” Econometrica, 87, 1589–1631.

Rotemberg, J. J. (1982): “Sticky prices in the united states,” Journal of Political Economy, 90, 1187–1211.

Siu, H. E. (2004): “Optimal fiscal and monetary policy with sticky prices,” Journal of Monetary Economics, 51, 575–607.

Yun, T. (2005): “Optimal monetary policy with relative price distortions,” American Economic Review, 95, 89–109.

Footnotes

I use Rotemberg pricing because it gives a smooth, fully nonlinear price-adjustment problem that is convenient for global solution and for studying an occasionally binding lower bound (Rotemberg (1982)). It also avoids introducing an additional endogenous distribution of firms’ prices as a state variable in this formulation. The familiar Calvo model instead gives each firm a fixed probability of being able to reoptimize its price in each period (Calvo (1983)), which generally requires tracking price dispersion. The two formulations generate the same New Keynesian Phillips curve only locally, after a first-order linearization around a steady state and with an appropriate mapping between parameters; they are not the same model away from that approximation. In particular, their nonlinear welfare costs, price dispersion, and responses near the ZLB can differ. A third possibility is state-dependent pricing, in which firms adjust when their individual state makes it worthwhile rather than according to a fixed probability (Dotsey et al. (1999)), adding a distribution of pricing states to the aggregate state.↩︎

COVID-19 Pandexit and the effects on economic activity

Nicolas Franz-Pattillo — Sun, 01 Aug 2021 00:00:00 GMT

Photo: Lisa Ferdinando, U.S. Department of Defense — public domain.

Introduction

Chile’s vaccination campaign is progressing rapidly, raising a natural question: when can we expect mobility—and economic activity—to return to normal? The answer depends not only on how quickly the population is vaccinated, but also on how effective the vaccines are at preventing contagion.

Using the methodology proposed by Rungcharoenkitkul (2021), we project the evolution of the pandemic and the associated optimal mobility response under a vaccination process similar to Chile’s. We consider three scenarios in which vaccines prevent contagion 95%, 60%, and 5% of the time. The exercise allows us to ask how vaccine efficacy affects the expected timing of the return to normality—and what a slower pandexit could cost in terms of economic activity.

shows a sizable difference across scenarios. With 95% efficacy, mobility returns to normal roughly a month and a half earlier than with 60% efficacy. Over the following six months, the average mobility gap between the two scenarios is approximately 4.7%.

That delay has an economic cost. We estimate that a 10% reduction in mobility is associated with GDP growth that is roughly 1.7 to 3 percentage points lower. Combining this estimate with the projected mobility paths suggests that a slower return to normality could subtract around 0.4 to 0.6 percentage points from annual GDP growth. For an economy expected to grow 8.5% under the 60%-efficacy scenario, this implies growth could instead reach roughly 8.9% to 9.1% if vaccines prevented contagion 95% of the time.

The effectiveness of vaccination therefore matters for more than the evolution of the pandemic. It also determines how quickly we can return to normal—and how much economic activity we lose while we wait.

Epidemiological model, mobility, and activity

The mechanism is a feedback loop: mobility supports economic activity, but also creates opportunities for infection. Rising health risks make restrictions more attractive. Effective vaccination reduces the susceptible population, allowing mobility to recover with a smaller health cost.

The framework follows the 12-compartment model of Rungcharoenkitkul (2021). People move from susceptible to exposed and then infectious. They subsequently isolate at home, enter quarantine, or go to hospital, before recovering or dying. The six isolated groups distinguish these three settings and the two eventual outcomes; only the infectious group transmits in the model. Vaccination provides a route from susceptibility to immunity without passing through infection.

The economic block chooses mobility by balancing a quadratic cost of restrictions against a mortality-loss measure. A separate empirical relationship translates mobility into activity. The vaccine percentages below are scenario assumptions about protection against infection, rather than estimates of protection against hospitalisation or death.

Explore the model: states, equations, and mobility choice

Vaccine effectiveness and alternative scenarios

In this model, effective vaccination removes people from the susceptible group without them having to get sick. Protection does not arrive immediately. For this 2021 exercise, the vaccination chart assumes two doses 30 days apart, with full effectiveness reached 14 days after the second dose. shows first and second doses alongside the population effectively protected under 100%, 75%, and 50% effectiveness assumptions. The first dose contributes half of eventual effectiveness by day 30, building linearly. These illustrative percentages differ from the 95%, 60%, and 5% assumptions used in the mobility scenarios.

The effects of mobility restrictions on economic activity have not been homogeneous over time.

The observed mobility figures show that there are factors that have changed drastically relative to the base scenario.¹ These factors cannot be identified without assuming a vaccine efficacy and, therefore, are not estimated in our analysis. The contagion rate estimated when assuming 95% efficacy is 6 times higher than the maximum recorded before the vaccination process began. We interpret this as the model rejecting the 95%-efficacy hypothesis.

The vaccination chart illustrates a timing issue that the mobility counterfactuals do not separately model: the delay between inoculation and effective protection. Under the two-dose schedule assumed here, full protection takes about a month and a half from the first injection. The vaccination illustration should therefore not be read as a separate dose-by-dose model underlying the mobility projections.

Conclusion

Chile’s rapid vaccination campaign means that a return to normality may finally be getting close. But how close depends critically on how effective vaccines are at preventing contagion. If vaccination substantially weakens the link between mobility and infections, restrictions can be relaxed sooner and economic activity can recover faster. If it does not, the pandexit will take longer.

Our estimates suggest that the difference could be economically meaningful. Under the 95%-efficacy scenario, mobility returns to normal roughly a month and a half earlier than under the 60% scenario. Over the following six months, this translates into an average mobility gap of about 4.7 percentage points and, using our estimated relationship between mobility and activity, around 0.4–0.6 percentage points of annual GDP growth.

That last calculation should be taken with a grain of salt. The regressions relating mobility to economic activity are deliberately simple and leave out many factors that changed simultaneously during the pandemic—fiscal support, uncertainty, firms’ and households’ adaptation, external conditions, and the composition of restrictions, among others. These omitted variables may bias the estimated relationship between mobility and activity. The GDP figures should therefore be read as an illustration of the possible economic magnitude of a delayed pandexit, rather than as a causal estimate.

The data arriving as the vaccination campaign progresses also give us reasons to be cautious about the most optimistic epidemiological scenario. Matching observed mobility under the assumption of 95% effectiveness against contagion requires a contagion rate around six times higher than anything estimated before vaccination began. Within the model, this makes the 95% scenario increasingly difficult to reconcile with what we are seeing.

The next few months will therefore be particularly informative. As vaccination advances and more data become available, we should learn much more about how strongly vaccines break the connection between mobility and contagion. That connection will help determine not only when Chile can return to normal, but also how much economic activity is lost on the way there.

Our World in Data

References

Rungcharoenkitkul, P. (2021): Macroeconomic Consequences of Pandexit, BIS Working Papers, Bank for International Settlements.

Footnotes

The base projection scenario uses estimates with data through December 20, 2020. After that, all shock innovations are assumed to disappear.↩︎

The Pandexit model: states, equations, and mobility choice

Nicolas Franz-Pattillo — Sun, 01 Aug 2021 00:00:00 GMT

This is a companion piece to COVID-19 Pandexit and the effects on economic activity, covering the underlying epidemiological model in full.

This is a restatement of the core framework in Rungcharoenkitkul (2021), sections 2.1–2.2, checked against the author’s public implementation. The notation below groups the six isolation equations to make the population accounting easier to follow.

States and notation

Each state is a number of people; one period is one day. Write .

State	Interpretation
	Susceptible
	Exposed, not yet infectious
	Infectious and able to transmit
	Isolated and eventually recovering: undetected, quarantined, hospitalised
	Isolated and eventually dying, in the same three settings
	Recovered and immune
	Cumulative deaths
	Effectively vaccinated and immune

The twelve compartments sum to the initial population , including deaths. The public code also tracks auxiliary series, such as cumulative detected cases; these are not additional mutually exclusive population compartments.

Infection and vaccination

Let be new exposures and the daily flow of susceptible people gaining vaccine protection. With the incubation-transition rate, the isolation-transition rate, and the immunity-loss rate:

Here measures effective protection, not injections: an efficacy assumption scales the vaccination input. A delay can be represented in that input without adding compartments. The displayed core equations allocate vaccination to susceptible people; the upstream code also offers allocation across susceptible and recovered people. Setting rules out waning immunity. Flows must be bounded so no compartment becomes negative.

Isolation, recovery, and deaths

Let be detection probability, hospitalisation probability conditional on detection, and eventual death probability after infection. Define the allocation weights

For each , the two equations

represent six state transitions. The recovery rates satisfy , while hospital recovery uses ; is the death-transition rate. The remaining two equations are

Summing all twelve changes gives zero. The outcome superscripts are accounting categories, not assumptions that an individual’s outcome is observed in advance.

Mobility and the policy rule

Mobility is a fractional deviation from normal: means 10% below normal. Its contribution to transmission is exponential:

The residual captures transmission changes not explained by mobility. The paper’s baseline projection uses a mean-reverting residual, , with future innovations set to zero. This still allows the inherited residual to decay gradually.

Writing and , the reduced policy block uses the mortality-loss measure

This is the policy rule’s mortality measure, not observed daily deaths . The mobility decision balances it against the cost of restrictions:

An interior choice satisfies

Boundary choices must also be considered. The upstream numerical routine searches a mobility grid for the smallest first-order-condition residual. This is the source’s reduced policy rule, rather than a newly solved full dynamic-planning problem.

A useful summary of epidemic pressure is

Reducing mobility lowers ; effective vaccination lowers . Both can push this quantity below one, when the combined exposed-and-infectious pool starts shrinking.

From mobility to activity

The activity calculation is a separate empirical step. The rolling regressions below estimate an intercept and a mobility coefficient using eight monthly observations. For a scenario comparison, the approximation is , with mobility and activity changes expressed in consistent percentage-point units. It is an estimated association, not an additional epidemiological equation or a causal identification result.

The 95%, 60%, and 5% mobility paths keep the vaccination scenario comparable while varying assumed protection against infection. This section documents their model framework; it does not supply a new calibration or re-estimate the historical charts.

References

Rungcharoenkitkul, P. (2021): Macroeconomic Consequences of Pandexit, BIS Working Papers, Bank for International Settlements.

Git for economists

Nicolas Franz-Pattillo — Wed, 27 Jan 2021 00:00:00 GMT

Railway Clearing House, junction diagram of Clapham Junction, Stewarts Lane, Lavender Hill & Longhedge, 1912 (detail) — public domain, via Wikimedia Commons.

Here is a folder I used to keep on my computer. I suspect you have one like it.

A research project before version control: one folder per date, and inside the latest one, “Project.tex”, “Project final.tex” and “Project final V2.tex”, next to “solution_chevicehv(not working).m”.

Every folder is a snapshot of the project on some date. Every file with “final” in its name was final until it wasn’t. One script is labelled “not working”, and I no longer remember why I kept it. Somewhere in there is the version of the code that produced the figures in the draft I sent to my advisor, and I couldn’t tell you which one.

This post is a short version of a talk I gave to economists about version control, and about Git in particular. The original slides are in Spanish.

Our projects are software projects now

Not long ago a research project was a paper and maybe a do-file. Today it is rarely one file. It has text, data, tables and figures, and programs in Stata, MATLAB, Python, R or EViews. It usually has more than one author. More and more, the output is not a static document but a web app or a dashboard.

That is the situation software developers have been in for decades: many files, many people, and a constant need to move forward without breaking what already works. They have spent those decades building tools for it. Economists mostly haven’t adopted them, and we pay for it in three ways:

Innovating is risky. Trying a new specification means editing code that works. If the experiment fails, getting back to where you were depends on memory and on how disciplined you were about copying folders.
Collaboration and replication suffer. Emailing files, sharing a Dropbox folder, or using Word’s track changes all work up to a point. None of them tells you which version of the code produced which result.
History gets lost. Usually in a folder called old. Or in the code itself, as hundreds of commented-out lines that “might be useful later”. I call this computational Diogenes syndrome: hoarding dead code because deleting it feels like losing it.

Version control solves all three. One caveat before going further: version control is designed for code and text, not for data. Large or binary files belong somewhere else.

A little vocabulary

Version control systems share a common vocabulary, the same way most programming languages share if, for and while. Three words go a long way:

A commit is a snapshot of the project at a moment in time, with a message saying what changed and why.
A repository (“repo”) is the whole project, including its full history of commits.
The working copy is the version of the project you are currently looking at and editing.

The important word is commit. It is not an autosave. You decide when a change is ready and commit to it, the way you would sign off on a version. Your editor can keep saving temporary copies as it likes; version control works on top of that and doesn’t interfere.

Systems differ in where the repository lives:

Local systems keep the repository on your computer only. They track history but don’t help you collaborate.
Centralized systems keep the repository on a server and give each person a working copy. Everyone commits to the server and updates from it. Collaboration is easy, but everything depends on that one server.
Distributed systems give every person a complete copy of the repository, history included. You commit locally, then push your commits to a shared server; your coauthors pull them from there. There are a couple more steps, but nothing is lost if the server dies, and you can browse the whole history offline.

The distributed model won. Git, the system almost everyone uses today, is distributed.

Git is not GitHub

Git is software that you install on your computer. It was written in 2005 by Linus Torvalds to manage the development of Linux, and it is now by far the most widely used version control system. You don’t need a server or an internet connection to use it. You may already have it: it ships with macOS’s command line tools, with Anaconda, and with recent versions of MATLAB. Type this in a terminal to check:

$ git --version
git version 2.39.5 (Apple Git-154)

GitHub is a website built on top of Git. It hosts repositories so people can share them, and adds what a team needs around the code: access control, issue tracking, task boards, automated builds and wikis. Think of it as a social network for Git repositories. GitLab and Bitbucket do the same job. You can use Git without any of them, but a shared server is what makes collaboration painless.

A worked example

You can use Git through the command line or through a graphical interface. The basic workflow needs only five to ten commands. Most documentation is written for the command line, so it is worth seeing what they are even if you end up clicking buttons.

Say we have a small MATLAB project in ~/Desktop/project: a script main.m that runs a model, a function solution.m that solves it, and a report Report.tex.

Start a repository(steps 1–2). Go to the folder and initialize it:

~ $ cd Desktop/project
~/Desktop/project $ git init
Initialized empty Git repository in /Users/you/Desktop/project/.git/

That creates a hidden .git folder, where Git keeps the entire history. Nothing else in your folder changes. If you ever want to stop using Git, delete that folder. If the project already exists somewhere else, such as on GitHub, you clone it instead, which downloads the files along with their full history: git clone .

Make the first commit(steps 3–4). Committing is a two-step process. First you choose which changes go into the snapshot by adding them to the staging area. Then you commit them with a message:

~/Desktop/project $ git add main.m solution.m Report.tex
~/Desktop/project $ git commit -m "First version of the model and report"
[main (root-commit) f026d7e] First version of the model and report
 3 files changed, 48 insertions(+)

Experiment on a branch(steps 5–8). Now we want to see how alternative parameter values change the results. In the old world, this is the moment we would copy the folder or create main_alternative calibration.m. Instead, we create a branch, a parallel line of work that leaves main untouched:

~/Desktop/project $ git switch -c alt-parameters
Switched to a new branch 'alt-parameters'
# ...edit main.m...
~/Desktop/project $ git add main.m
~/Desktop/project $ git commit -m "Try alternative parameter values"

Fix a bug in the right place(steps 9–12). While experimenting we find a bug in solution.m. The bug isn’t specific to the experiment: it is in the code everyone uses. So the fix belongs on main, especially if other people are working on the project. We fix the file, switch to main (the uncommitted change comes with us), and commit it there:

~/Desktop/project $ git switch main
M       solution.m
Switched to branch 'main'
~/Desktop/project $ git add solution.m
~/Desktop/project $ git commit -m "Fix sign error in solution.m"

Bring the fix into the experiment(steps 13–14). Finally we go back to our branch and merge main into it, so the experiment also runs on the corrected code:

~/Desktop/project $ git switch alt-parameters
~/Desktop/project $ git merge main -m "Merge main: fix in solution.m"

If the experiment works out, we merge alt-parameters into main. If it doesn’t, we delete the branch and main is exactly as we left it. Either way, there’s no old folder, no “final V2”, and every result can be traced back to the commit that produced it.

The animation in the figure below runs the whole example, from an empty folder to a merge. The right panel shows what Git sees at each step: the history, the files you have changed, and what is staged for the next commit. The step links in the paragraphs above jump it to the matching point.

Habits that help

Version control doesn’t replace good organization. The bad habits we have in our digital lives tend to come along into Git, and they can get worse if we don’t notice them. There is no single right way to work: two coauthors need less structure than a team of fifty. A few practices help almost everywhere:

Commit often, and write messages that mean something. “Final version” says nothing. “Add robustness check with quarterly data” tells your future self, and your coauthors, what changed and why.
Use the staging area to keep commits tidy. You will often make progress on several things at once. Rather than committing everything together, stage and commit related changes separately, so each commit tells one story.
Use branches for experiments. A new specification, a new estimator, a referee’s suggestion: each can live on its own branch until you know whether it stays.
Fix shared problems on the shared branch. As in the example, fix bugs where everyone will get the fix, then merge it into your branches.
Tell Git what to ignore. A .gitignore file lists what shouldn’t be versioned: raw data, compiled PDFs, LaTeX auxiliary files, MATLAB .mat outputs.

How to learn it

The best way to learn Git is to use it. Start with your next project, even if you work alone.

You don’t have to live in the terminal. Graphical interfaces such as VS Code, GitKraken and GitHub Desktop show the history as a graph, highlight the differences between versions, and turn most commands into a button. Still, it is worth understanding the handful of commands above. Each interface implements Git in its own way, and learning one without understanding what it does underneath is like learning Excel without knowing arithmetic. The commands are what every interface has in common.

Conclusion

I started with three costs of working without version control: experimenting is risky, collaborating and replicating are hard, and history gets lost. Git deals with each of them. A branch lets you try something new without putting working code at risk. A shared repository means everyone works on the same history instead of emailing copies around. Every commit records what changed, when, and why, so you can trace any result back to the code that produced it.

None of this happens automatically. Git is a habit more than a tool, and it only pays off once committing becomes as routine as saving a file. The cost is a few commands and a few weeks of slight discomfort. The return grows with every project, coauthor and referee report, and with every time a journal asks for a replication package.

So start small: run git init in your next project, and commit when something works. After a few weeks the vocabulary stops feeling foreign, and the 20200925Project folders start to look like a relic. If you’d rather go through it in Spanish, the original slides are still online.