THE HAMMER AND THE DANCE - Simon MongeyÞ shadow price of infection risks 2. Break analysis into...

THE HAMMER AND THE DANCE:EQUILIBRIUM AND OPTIMAL POLICY DURING A PANDEMIC CRISIS

Toulouse School of Economics Macro Group(T. Assenza, F. Collard, M. Dupaigne, P. Fève, C. Hellwig, S. Kankanamge, N. Werquin)

Covid, Search and Matching, Chicago, 05/28/2020

ECONOMIC POLICY DURING A PANDEMIC CRISIS

• 2 Specific features/challenges related to COVID-19:

1. Fast propagation: short horizon between infection and recovery/death

2. Asymptomatic transmissions: large fraction of transmissions/infections go undetected

• Dynamic model linking economic interactions and infection risksÞ study trade-offs between

1. mortality costs induced by pandemic2. adverse economic costs of policy interventions

• Keep it at a fairly high level of generality

2

• 2 Specific features/challenges related to COVID-19:

1. Fast propagation: short horizon between infection and recovery/death

2. Asymptomatic transmissions: large fraction of transmissions/infections go undetected

• Dynamic model linking economic interactions and infection risksÞ study trade-offs between

1. mortality costs induced by pandemic2. adverse economic costs of policy interventions

• Keep it at a fairly high level of generality

2

• Equilibrium and Central Planner’s allocations Þ “The Hammer and The Dance”

• The Hammer: Strong initial lockdown to bring infections under control

• The Dance: Slow gradual deconfinement towards herd immunity

• Results = interplay between static and dynamic externalities1. Static: instantaneous economic and infection risk spill-overs

2. Dynamic: linked to dynamic immunization and infection externalities

• Long-run: Strong infection externalities Þ Excess mortality in equilibrium

• Short-run: Strong immunization externalities Þ too strong/weak equil. lockdown dependingon welfare cost of initial infection peak

3

BUILDING ON THE SHOULDERS OF GIANTS

• Alvarez, Argente and Lippi;• Atkeson• Eichenbaum, Rebelo and Trabandt;• Farboodi, Jarosch, and Shimer;

• Kaplan, Moll and Violante; Bethune and Korinek; Atkeson; Chang and Velasco;Garibaldi, Moen and Pissarides; Gonzalez-Eiras and Niepelt; Jones, Philippon andVenkateswaran; Krueger, Uhlig and Xie; Toxvaerd; Ritschl; Abaluck, Chevalier,Christakis, Forman, Kaplan, Ko and Vermund; Glover, Heathcote, Krueger andRíos-Rull; Greenstone and Nigam; Cleevely, Susskind, Vines, Vines and Wills; Beck andWagner; Rowthorn; von Thadden; Beenstock and Dai; Avery, Bossert, Clark, Ellison andEllison; Berger, Herkenhoff and Mongey; Maloney and Taskin; Zhixian and Meissner;Victoria, Menzio and Wiczer, Forslid and Herzing, Moser and Yared, Hornstein,Bodenstein, Corsetti and Guerrieri, Deb, Furceri, Ostry and Tawk, and many others…

• Too numerous to mention

4

BUILDING ON THE SHOULDERS OF GIANTS

• Alvarez, Argente and Lippi;• Atkeson• Eichenbaum, Rebelo and Trabandt;• Farboodi, Jarosch, and Shimer;• Kaplan, Moll and Violante; Bethune and Korinek; Atkeson; Chang and Velasco;Garibaldi, Moen and Pissarides; Gonzalez-Eiras and Niepelt; Jones, Philippon andVenkateswaran; Krueger, Uhlig and Xie; Toxvaerd; Ritschl; Abaluck, Chevalier,Christakis, Forman, Kaplan, Ko and Vermund; Glover, Heathcote, Krueger andRíos-Rull; Greenstone and Nigam; Cleevely, Susskind, Vines, Vines and Wills; Beck andWagner; Rowthorn; von Thadden; Beenstock and Dai; Avery, Bossert, Clark, Ellison andEllison; Berger, Herkenhoff and Mongey; Maloney and Taskin; Zhixian and Meissner;Victoria, Menzio and Wiczer, Forslid and Herzing, Moser and Yared, Hornstein,Bodenstein, Corsetti and Guerrieri, Deb, Furceri, Ostry and Tawk, and many others…

• Too numerous to mention4

THE MODEL

Time is discrete,Perfect foresightMeasure Λt of Agents

Actions: x ∈ X ⊆ RkX Compact, Convex

Economic Stage Game

Instantaneous Payoffs:U(x, X) ∈ [V, V], V > V ⩾ 0

Assumption:∃X⋆ ∈ X , U(X⋆, X⋆) = V

Þ Equilibrium Efficiency(2nd Welfare Thm)

Confinement Stage Game

Instant. Infection Risk: Rπ(i)with R = R(x, X) ⩾ R ⩾ 0

Assumption:∃X̂ ∈ X , R(̂X, X̂) = R

Þ Extreme Confinement Equil.

SIR DYNAMICS

R · π(i)

γ

δ

Susceptibleπt(s)

Infectedπt(i)

Recovered1 − πt(s) − πt(i)

DeadΛt+1 = (1 − δπt(i))ΛtAssumption (Asymptomatic Transmission)

Only Death is observable

Dynamic Game Payoffs

(1 − β)∑∞

t=0 βtΛ(xt−1, Xt−1)U(xt−1, Xt−1)

5

THE MODEL

Economic Stage Game

SIR DYNAMICS

R · π(i)

γ

δ

Susceptibleπt(s)

Infectedπt(i)

(1 − β)∑∞

5

THE MODEL

Economic Stage Game

SIR DYNAMICS

R · π(i)

γ

δ

Susceptibleπt(s)

Infectedπt(i)

(1 − β)∑∞

5

THE MODEL

STATIC GAME

Economic Stage Game

SIR DYNAMICS

R · π(i)

γ

δ

Susceptibleπt(s)

Infectedπt(i)

(1 − β)∑∞

5

THE MODEL

Economic Stage Game

SIR DYNAMICS

R · π(i)

γ

δ

Susceptibleπt(s)

Infectedπt(i)

DeadΛt+1 = (1 − δπt(i))Λt

Assumption (Asymptomatic Transmission)Only Death is observable

(1 − β)∑∞

5

THE MODEL

Economic Stage Game

SIR DYNAMICS

R · π(i)

γ

δ

Susceptibleπt(s)

Infectedπt(i)

(1 − β)∑∞

5

THE MODEL

Economic Stage Game

SIR DYNAMICS

R · π(i)

γ

δ

Susceptibleπt(s)

Infectedπt(i)

R

(1 − β)∑∞

5

THE MODEL

Economic Stage Game

SIR DYNAMICS

R · π(i)

γ

δ

Susceptibleπt(s)

Infectedπt(i)

(1 − β)∑∞

5

KEY STEPS

1. Summarize trade-off btwn economic activity and infection risks (future mortality)Þ shadow price of infection risks

2. Break analysis into static and dynamic part:Þ Dynamic problem as a sequence of static reduced form interaction game: infection risk vs utility.

(i) Static part: which sectors to open or close?• Align private and social MRS btw economic activity and infection risks to the shadow price.

(ii) Dynamic part: timing of interventions?• The dynamics of the shadow price of infection risks dictate the timing

6

KEY STEPS

6

KEY STEPS

6

KEY STEPS

6

STATIC INTERACTION GAME

Economic Stage GameU(x, X), V(X) ≡ U(X, X)

Confinement Stage GameR(x, X), R(X) ≡ R(X, X)

Period tHybrid Interaction Game

U(x, X)− ΦeqR(x, X)V(X)− Φ⋆R(X)

Φeq

Φ⋆

Shadow Priceof Infection Risks

Equate MRS to Shadow Cost of Infection Risk

Uxi (X, X) + UXi (X, X) = Φ⋆(Rxi (X, X) + RXi (X, X))

For any activity i

Static Implementation RuleXeq(Φeq), X⋆(Φ⋆)

Reduced form GameEQ: U(r, R)− ΦeqrCP: V(R)− Φ⋆R

7

Φeq

Φ⋆

For any activity i

7

Φeq

Φ⋆

For any activity i

7

Φeq

Φ⋆

Uxi (x, X) = Φeq Rxi (x, X)For any activity i

7

Φeq

Φ⋆

For any activity i

7

Φeq

Φ⋆

For any activity i

7

Φeq

Φ⋆

For any activity i

7

EQUILIBRIUM AND PARETO FRONTIERS

V(R)

R

V

R R

Pareto Frontier(Planner, V∗(R))

Equilibrium Frontier(Veq(R))

8

STATIC EFFICIENCY CONDITIONS Þ OPTIMAL POLICY

• Focus policy on sectors with largeabsolute spill-overs

• Protect/subsidize sectors if

Social MRS>Private MRS

• Restrict sectors if

Social MRS

STATIC EFFICIENCY CONDITIONS Þ OPTIMAL POLICY

• Focus policy on sectors with largeabsolute spill-overs

• Protect/subsidize sectors if

Social MRS>Private MRS

• Restrict sectors if

Social MRS

DYNAMIC INTERACTION GAME

Economic Stage GameU(xt, Xt), V(Xt)

Static Implementation RuleXeq(Rt), X⋆(Rt)

Confinement Stage GameR(xt, Xt), R(Xt)

Period tReduced-Form Game

U(rt, Rt)− Φeqt rtV(Rt)− Φ⋆t Rt

Reduced Form Policy RuleReq(πt) = Req(Φeq(πt))R⋆(πt) = R⋆(Φ⋆(πt))

SIR Dynamicsπt = (πt(s), πt(i))

Shadow Price of Infection RiskΦeqt (πt) Φ

⋆t (πt)

10

Reduced Form Policy RuleUr(rt, Rt) = Φeqt Þ Req(Φ

eqt )

V ′(Rt) = Φ⋆t Þ R⋆(Φ⋆t )

⋆t (πt)

10

Reduced Form Policy RuleUr(rt, Rt) = Φeqt Þ Req(Φ

eqt )

V ′(Rt) = Φ⋆t Þ R⋆(Φ⋆t )

⋆t (πt)

10

Reduced Form Policy RuleReq(πt) = Req(Φeq(πt))R⋆(πt) = R⋆(Φ⋆(πt))

⋆t (πt)

10

DYNAMIC EFFICIENCY CONDITIONS

• Focus on reduced form trade-off between R and V :

EQ : Ur(Rt,Rt) = ΦeqtCP : V⋆′(Rt) = Φ⋆t

• Generically

Φt ≡ β1−β πt(s)πt(i)(

∂v(πt+1)∂π(s) −

∂v(πt+1)∂π(i)

)

Proportional topropagation

rate ofInteraction

Speed ofPropagation

Marginal Cost ofAdditional Infection

Marg. lossno Immunization

Marg. gainrecovery

11

• Generically

∂v(πt+1)∂π(s) −

∂v(πt+1)∂π(i)

)

rate ofInteraction

Speed ofPropagation

Marg. gainrecovery

11

• Generically

∂v(πt+1)∂π(s) −

∂v(πt+1)∂π(i)

)Proportional topropagation

rate ofInteraction

Speed ofPropagation

Marg. gainrecovery

11

• Generically

∂v(πt+1)∂π(s) −

∂v(πt+1)∂π(i)

rate ofInteraction

Speed ofPropagation

Marg. gainrecovery

11

• Generically

∂v(πt+1)∂π(s) −

∂v(πt+1)∂π(i)

rate ofInteraction

Speed ofPropagation

Marg. gainrecovery

11

• Generically

∂v(πt+1)∂π(s) −

∂v(πt+1)∂π(i)

)

rate ofInteraction

Speed ofPropagation

Marg. gainrecovery

11

• Generically

∂v(πt+1)∂π(s) −

∂v(πt+1)∂π(i)

)

rate ofInteraction

Speed ofPropagation

Marg. gainrecovery

11

• Generically

∂v(πt+1)∂π(s) −

∂v(πt+1)∂π(i)

)

rate ofInteraction

Speed ofPropagation

Marg. gainrecovery

11

MARGINAL VALUES

• Social Marginal Value of Recovery (iÞr)

−∂v (π)∂π (i) = vr (π)− vi (π)−π (s)

∂vs (π)∂π (i) − π (i)

∂vi (π)∂π (i) − (1− π (s)− π (i))

∂vr (π)∂π (i)

direct benefit(Internalized)

Dynamic Infection Externalities:Indirect effects of a marginal decrease in infection rate on s,i,r

• Social Marginal Value of Immunization (sÞr)

−∂v (π)∂π (s) = vr (π)− vs (π)−π (s)

∂vs (π)∂π (s) − π (i)

∂vi (π)∂π (s) − (1− π (s)− π (i))

∂vr (π)∂π (s)

Dynamic Immunization Externalities:Higher immunization lowers the need for economic restrictions

12

MARGINAL VALUES

−∂v (π)∂π (i) = vr (π)− vi (π)−π (s)

∂vs (π)∂π (i) − π (i)

∂vi (π)∂π (i) − (1− π (s)− π (i))

∂vr (π)∂π (i)

−∂v (π)∂π (s) = vr (π)− vs (π)−π (s)

∂vs (π)∂π (s) − π (i)

∂vi (π)∂π (s) − (1− π (s)− π (i))

∂vr (π)∂π (s)

12

MARGINAL VALUES

−∂v (π)∂π (i) = vr (π)− vi (π)−π (s)

∂vs (π)∂π (i) − π (i)

∂vi (π)∂π (i) − (1− π (s)− π (i))

∂vr (π)∂π (i)

−∂v (π)∂π (s) = vr (π)− vs (π)−π (s)

∂vs (π)∂π (s) − π (i)

∂vi (π)∂π (s) − (1− π (s)− π (i))

∂vr (π)∂π (s)

12

• Generically

EQ : Ur(Rt,Rt) = Φeqt =β

1− β πt(s)πt(i)

vs(πt+1)− vi(πt+1)︸︷︷︸̸=0

CP : V⋆′(Rt) = Φ⋆t =

β

vs(πt+1)− vi(πt+1)︸︷︷︸̸=0

+ External Costs

• Useful to focus on β → 1 (Instantaneous propagation).

13

• Generically

EQ : Ur(Rt,Rt) = Φeqt =β

vs(πt+1)− vi(πt+1)︸︷︷︸̸=0

CP : V⋆′(Rt) = Φ⋆t =

β

vs(πt+1)− vi(πt+1)︸︷︷︸̸=0

+ External Costs

• Useful to focus on β → 1 (Instantaneous propagation).

13

A USEFUL LIMIT CASE: INSTANTANEOUS PROPAGATION (β → 1)

• Equilibrium:

• Hammer: Extreme confinement: Req0 = R+ η, η > 0until πt(i) ∈ (0, η), then

• Dance: keep πt(i) within (0,η) until herd immunity isreached Þ very slow (πt(s))

• Central Planner:

• Initially (in t = 0) let the infection propagate• strict Hammer Þ bring infection and recovery to LongRun optimal level

• Never ending Dance: keep infection under control toLong Run optimal level (R = R̃ < R)

• SR: Immunization externality: hold-out motive thatinefficiently slows the propagation

• LR: infection externality: agents exit confinement too fast,relative to the CP Þ increases LR mortality.

V(R)

R

V

R RR0 = 1

Pareto Frontier

Equilibrium Frontier

R̃Req0

14

• Equilibrium:• Hammer: Extreme confinement: Req0 = R+ η, η > 0until πt(i) ∈ (0, η), then

V(R)

R

V

R RR0 = 1

R̃

Req0

14

V(R)

R

V

R RR0 = 1

R̃

Req0

Slow

14

• Central Planner:• Initially (in t = 0) let the infection propagate

• strict Hammer Þ bring infection and recovery to LongRun optimal level

V(R)

R

V

R RR0 = 1

R̃Req0

14

• Central Planner:• Initially (in t = 0) let the infection propagate• strict Hammer Þ bring infection and recovery to LongRun optimal level

V(R)

R

V

R RR0 = 1R̃

Req0

14

• Central Planner:• Initially (in t = 0) let the infection propagate• strict Hammer Þ bring infection and recovery to LongRun optimal level

V(R)

R

V

R RR0 = 1R̃

Req0

14

THE HAMMER AND THE DANCE

PropositionStart from an (arbitrarily) small, positive fraction π0 (i) > 0 of infected agents in the population, thesequential planner’s solution and equilibrium

{R∗t , π∗t

}and

{Reqt , π

eqt}both satisfy:

1. Flatten the Curve (Short Run): Starting from R∗0 and Req0 arbitrarily close to R, both policy sequences are

initially decreasing to ”flatten the curve” and delay infections Þ The Hammer phase

2. Herd Immunity (Long-run): In the long run, R∗t and Reqt converge to R, and the economy returns to the

pre-pandemic equilibrium in a state of herd immunity Þ The Dance phase

15

{R∗t , π∗t

}and

{Reqt , π

eqt}both satisfy:

15

{R∗t , π∗t

}and

{Reqt , π

eqt}both satisfy:

15

DYNAMICS

20 40 60 80 100Weeks

0.00

0.25

0.50

0.75

1.00Susceptible

20 40 60 80 100Weeks

0.0

0.1

0.2

0.3

0.4Infected

20 40 60 80 100Weeks

0.00

0.25

0.50

0.75

1.00Recovered

20 40 60 80 100Weeks

0.000

0.002

0.004

Share of Deads

20 40 60 80 100Weeks

0

1

2

3

0

20 40 60 80 100Weeks

0.00

0.05

0.10

0.15Shadow Cost ( )

Equilibrium Optimal Allocation Epidemiological

Pareto Frontier

16

EXTERNALITIES

20 40 60 80 100Weeks

0.005

0.000

0.005

0.010

0.015

0.020

0.025Social Marginal Cost of Infection

20 40 60 80 100Weeks

0.005

0.000

0.005

0.010

0.015

0.020

0.025Direct Cost (vs( ) vi( ))

20 40 60 80 100Weeks

0.005

0.000

0.005

0.010

0.015

0.020

0.025External Cost

Equilibrium Optimal Allocation

17

POLICY DYNAMICS CONCLUDING REMARKS

UX(x,X)Ux(x,X)

RX(x,X)Rx(x,X)

Econ

omic

Externalities

Infection Risk Externalities

Protect/Subsidize

Essential, Non Critical:Banking and financePharmaceuticalsBasic researchOnline services

RestrictCritical & Inessential:EntertainmentLarge Social EventsRestaurantsNon-essential travel

Leave AloneNeither criticalNor essential:Home activities,Online video …

Manage CarefullyCritical but Essential:GroceriesHealthcareEducationEssential travel

ϕ∗ > ϕeq

ϕ∗ < ϕeq

ϕ∗ = ϕeq

18

UX(x,X)Ux(x,X)

RX(x,X)Rx(x,X)

Econ

omic

Externalities

Protect/Subsidize

18

UX(x,X)Ux(x,X)

RX(x,X)Rx(x,X)

Econ

omic

Externalities

Protect/Subsidize

18

UX(x,X)Ux(x,X)

RX(x,X)Rx(x,X)

Econ

omic

Externalities

Protect/Subsidize

18

UX(x,X)Ux(x,X)

RX(x,X)Rx(x,X)

Econ

omic

Externalities

Protect/Subsidize

18

UX(x,X)Ux(x,X)

RX(x,X)Rx(x,X)

Econ

omic

Externalities

Protect/Subsidize

18

EXTENSIONS

• Additional Externalities

• Static Externalities Show

• Congestion Externalities Show

• Hope for vaccine and cure Show

• Transitory immunization Show

• Shifts in Pareto and Equilibrium Frontiers

• Face Masks Show

• Testing and contact tracing Show

19

CONCLUDING REMARKS

• Dynamic model integrating economic interactions and infection risk

• Fast Propagation Þ Hammer and Dance Dynamics• Highlight the role of static and dynamic externalities

• Clear policy implications:• The hammer:

• Early, decisive action is warranted if saves lives in LR.• If only delays infections in SR Þ just lengthens the recovery and inflicts higher economic costs

• The dance: Optimal deconfinement• Keeps the pandemic’s transmission rate close 1.

Þ Policy must control the epidemic, not the other way around.

20

CONCLUDING REMARKS

• Dynamic model integrating economic interactions and infection risk

• Fast Propagation Þ Hammer and Dance Dynamics• Highlight the role of static and dynamic externalities

• Clear policy implications:• The hammer:

• Early, decisive action is warranted if saves lives in LR.• If only delays infections in SR Þ just lengthens the recovery and inflicts higher economic costs

• The dance: Optimal deconfinement• Keeps the pandemic’s transmission rate close 1.

Þ Policy must control the epidemic, not the other way around.

20

THANK YOU!

20

SIR DYNAMICS

• Λt is the mass of agents• Susceptible: St = πt(s)Λt, Infected: It = πt(i)Λt, Recovered: Rt = πt(r)Λt

St+1 = St − RtStIt+1 = RtSt + (1− γ − δ)ItRt+1 = Rt + γIt

where Rt = R(xt, Xt)πt(i).• Dynamics of population: Λt+1 = (1− δπt(i))Λt. Dividing the previous system by Λt

πkt+1(s) =πkt (s)− rtπkt (s)πt(i)

1− δπkt (i)

πkt+1(i) =πkt (i) + rtπkt (s)πt(i)− γπkt (i)− δπkt (i)

1− δπkt (i)

πkt+1(r) =πkt (r) + γπkt (i)1− δπkt (i)

Back

SIR DYNAMICS

where Rt = R(xt, Xt)πt(i).

• Dynamics of population: Λt+1 = (1− δπt(i))Λt. Dividing the previous system by Λt

1− δπkt (i)

Back

SIR DYNAMICS

where Rt = R(xt, Xt)πt(i).• Dynamics of population: Λt+1 = (1− δπt(i))Λt. Dividing the previous system by Λt

1− δπkt (i)

Back

EXTENSIONS: STATIC EXTERNALITIES)

20 40 60 80 100Weeks

0.2

0.4

0.6

0.8

1.0Susceptible

20 40 60 80 100Weeks

0.00

0.05

0.10

0.15

0.20Infected

20 40 60 80 100Weeks

0.0

0.2

0.4

0.6

0.8

Recovered

20 40 60 80 100Weeks

0.000

0.001

0.002

0.003

0.004

Share of Deads

20 40 60 80 100Weeks

1

2

3

0

20 40 60 80 100Weeks

0.00

0.05

0.10

0.15Shadow Cost ( )

Eq. (Static Infection Ext.) Eq. (Static Economic Ext.) Eq. (benchmark) Opt. (benchmark)

Back

EXTENSIONS: CONGESTION EXTERNALITIES

• Congestion of the medical sector: death probability is an increasing and convexfunction of share of infected

• Equilibrium:• Stronger hammer to avoid infection and face a higher risk of death• But do not understand the congestion Þ not strong enough, larger death toll

• Planner:• Much higher shadow cost of infection due to infection externality

Þ Stronger Hammer to delay infection and save lives by avoiding congestion

Back

EXTENSIONS: CONGESTION EXTERNALITIES

20 40 60 80 100Weeks

0.2

0.4

0.6

0.8

1.0Susceptible

20 40 60 80 100Weeks

0.00

0.05

0.10

0.15

Infected

20 40 60 80 100Weeks

0.0

0.2

0.4

0.6

0.8Recovered

20 40 60 80 100Weeks

0.0000

0.0025

0.0050

0.0075

0.0100Share of Deads

20 40 60 80 100Weeks

1

2

3

0

20 40 60 80 100Weeks

0.00

0.05

0.10

0.15

Shadow Cost ( )

Equilibrium Optimal Allocation Eq. (benchmark) Opt. (benchmark)

Back

EXTENSIONS: HOPES FOR VACCINES AND CURES

• Probability that all susceptible are moved to recovery state in each period• Comes too late for those who are infected

• Equilibrium:• The private benefits of a vaccine are remote

Þ No significant change in behavior.

• Planner:• Very high initial shadow cost of infection

Þ Strong Hammer to delay infection and save lives while waiting for vaccine

• Depends very much on expected time to a vaccine/cure.

Back

EXTENSIONS: HOPES FOR VACCINES AND CURES (1 YEAR)

20 40 60 80 100Weeks

0.2

0.4

0.6

0.8

1.0Susceptible

20 40 60 80 100Weeks

0.00

0.05

0.10

0.15

Infected

20 40 60 80 100Weeks

0.0

0.2

0.4

0.6

0.8Recovered

20 40 60 80 100Weeks

0.000

0.001

0.002

0.003

0.004Share of Deads

20 40 60 80 100Weeks

1

2

3

0

20 40 60 80 100Weeks

0.00

0.05

0.10

0.15

Shadow Cost ( )

Back

EXTENSIONS: TRANSITORY IMMUNITY (1 YEAR)

0 100 200 300 400 500Weeks

0.25

0.50

0.75

1.00Susceptible

0 100 200 300 400 500Weeks

0.0

0.1

0.2

0.3

Infected

0 100 200 300 400 500Weeks

0.0

0.2

0.4

0.6

0.8

Recovered

0 100 200 300 400 500Weeks

0.000

0.005

0.010

0.015

0.020Share of Deads

0 100 200 300 400 500Weeks

1

2

3

0

0 100 200 300 400 500Weeks

0.00

0.05

0.10

0.15Shadow Cost ( )

Back

EXTENSIONS: FACE MASKS

• Wearing face mask does not affect utility butlowers infection risk.

• Costly to produce masks

Þ flattens and expands the set of payoffs/actions

Þ Gives the option to push infection risk below R ÞFaster control of epidemics

Þ Relaxes the Pareto frontier to achieve higherutility during the dance phase.

• Lowers consumption losses.

V(R)

R

V

R R

Pareto Frontier(Planner)

EquilibriumFrontier

Pareto Frontier(Planner)with Masks

EquilibriumFrontier

with Masks

R0 = 1

Veq(R)

V⋆(R)

∆V(R)

Back

• Costly to produce masksÞ flattens and expands the set of payoffs/actions

V(R)

R

V

R R

EquilibriumFrontier

with Masks

R0 = 1

Veq(R)

V⋆(R)

∆V(R)

Back

V(R)

R

V

R R

EquilibriumFrontier

with Masks

R0 = 1

Veq(R)

V⋆(R)

∆V(R)

Back

V(R)

R

V

R R

EquilibriumFrontier

with Masks

R0 = 1

Veq(R)

V⋆(R)

∆V(R)

Back

20 40 60 80 100Weeks

0.2

0.4

0.6

0.8

1.0Susceptible

20 40 60 80 100Weeks

0.00

0.05

0.10

0.15

Infected

20 40 60 80 100Weeks

0.0

0.2

0.4

0.6

0.8Recovered

20 40 60 80 100Weeks

0.000

0.001

0.002

0.003

0.004Share of Deads

20 40 60 80 100Weeks

1

2

3

0

20 40 60 80 100Weeks

0.00

0.05

0.10

0.15Shadow Cost ( )

Back

EXTENSIONS: TESTING AND CONTACT-TRACING

• test Þ quarantine (temp. exit from game)• Reduce undetected infections to π(i)(1−P(test|i)

Þ Improves static efficiency frontier (better controlof infections)

Þ But ̸= face masks as it lowers threshold level ofrecoveries for

• herd immunity,• virus eradication,• elimination of economic restrictions.

• Shift V to the left.

V(R)

R

V

R R

EquilibriumFrontier

R(1 − P(test|i))

with Testing

EquilibriumFrontier

with Testing

R0 = 1

Veq(R)

V∗(R)

Back

EXTENSIONS: TESTING AND CONTACT-TRACING

• test Þ quarantine (temp. exit from game)• Reduce undetected infections to π(i)(1−P(test|i)

Þ Improves static efficiency frontier (better controlof infections)

Þ But ̸= face masks as it lowers threshold level ofrecoveries for

• herd immunity,• virus eradication,• elimination of economic restrictions.

• Shift V to the left.

V(R)

R

V

R RR(1 − P(test|i))

with Testing

EquilibriumFrontier

with Testing

R0 = 1

Veq(R)

V∗(R)

Back

DYNAMICS ALONG THE PARETO FRONTIER

0 20 40 60 80 100weeks

0.00

0.05

0.10

0.15Shadow Cost ( )

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1Rt

0.96

0.97

0.98

0.99

1.00

(Rt)

Pareto Frontier

back

0 20 40 60 80 100weeks

0.00

0.05

0.10

0.15Shadow Cost ( )

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1Rt

0.96

0.97

0.98

0.99

1.00

(Rt)

Pareto Frontier

Hamm

er Phase

back

0 20 40 60 80 100weeks

0.00

0.05

0.10

0.15Shadow Cost ( )

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1Rt

0.96

0.97

0.98

0.99

1.00

(Rt)

Pareto Frontier

Hamm

er Phase

Dance

Phase

back

UTILITY

• Individual utility function(Ueq (r,R)− V

V̄− V

)2+ α

(r− R̄R̄− R

)2+ (1− α)

(R− R̄R̄− R

)2= 1.

Þ Leads to (V (R)− VV̄− V

)2+

(R− R̄R̄− R

)2= 1.

back

Appendix

Date post:	25-Jan-2021
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THE HAMMER AND THE DANCE - Simon MongeyÞ shadow price of infection risks 2. Break analysis into...

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