Optimal Public Expenditure

with Inefficient Unemployment

Pascal Michaillat (Brown)

Emmanuel Saez (Berkeley)

May 2017

1 / 22

should macroeconomic concerns affect policy?

• in practice, they already do:

– monetary policy

– unemployment insurance

• but other policies could depend on macroeconomic conditions

• in this paper: public expenditure

2 / 22

existing theories of optimal public expenditure

• Samuelson (1954):

– public goods financed by lump-sum taxation

– efficient level of production

– rule: spend until marginal utilities are equalized

– what if production is inefficient?

• Keynes:

– in recession, multiplier of public expenditure > 1

– rule: spend to fill output gap

– what happens with multiplier < 1?

– role of social value of public good? of taxation?3 / 22

our theory of optimal public expenditure

• blends theories of Samuelson + Keynes

• by embedding Samuelson’s framework into matching model from

Michaillat & Saez (2015)

• outcome: a formula linking optimal stimulus spending to

– unemployment gap

– unemployment multiplier

– social value of government consumption

4 / 22

informal description of the model

5 / 22

this is a service economy, without firms

6 / 22

this is a service economy, without firms

6 / 22

this is a service economy, without firms

6 / 22

there is an asset for saving

7 / 22

there are private services (c) and public services (g)

8 / 22

there are private services (c) and public services (g)

8 / 22

matching: it is costly to purchase services5/14/2016 San Francisco Nannies, Nanny, Housekeepers, Butlers, and Personal Assistants - SF, CA

http://www.inhousestaffing.com/ 1/2

           Contact   Site Map    Home 

Home Services Employers Job Seekers About Contact

Services

Childcare Staffing

Baby NursesFamily AssistantsGovernessesNanniesParents HelpersOn Call Nannies & BabySitters

HomeHotelCorporate

Household Staffing

HousekeepersExecutive HousekeepersHouseman/HandymanLaundressesCooksCooks/HousekeepersCompanions

Estate Staffing

Estate ManagersHousehold ManagersButlersChefsChauffeursGardnersHousehold CouplesValets

Private Office Staffing

Executive AssistantsPersonal Assistants

Eldercare Staffing

San Francisco & Marin & Sonoma Counties: 415 7493650Peninsula Counties: 650 5731000 East Bay Counties: 925 9332273

National Placements: 866 7292667

Welcome to Aunt Ann's In House Staffing Nanny, Housekeeper and EstateAgency of San Francisco, CA  the nationwide leader in Household Staffing.

EMPLOYERSBrowse Candidates >>

  JOB SEEKERSBrowse Jobs >>

Aunt Ann's In House Staffing has beenspecializing in exclusive domestic staffingservices for 50 years. We have worked withsome of the world's finest families andestates, representing highly trained andqualified domestic household personnel.

Aunt Ann's In House Staffing is a full-service private service placement firm thatspecializes in the staffing of key privateservice positions including nannies,housekeepers, baby nurses, doulas,executive house keepers, private chefs,chauffeurs, butlers, household managers,domestic couples, estate managers,companions, personal assistants, executiveassistants and more.

Aunt Ann's In House Staffing of San Francisco is staffed withseasoned, accomplished and professional placement consultants.

Our professional consultants are individualswho are trained to listen to your uniqueneeds and are experienced in private service.We understand high standards and the art ofservice, absolute discretion and the specialcircumstances of our client's lives.

Whether you are experienced in domestic service, or are a client searching to filla position, we welcome the opportunity to offer our world class services to assistyou.

Aunt Ann's In House Staffing Staffing News & Events

As Seen on CBS TV! [ Read More... ]

 Aunt Ann's Denise Collins Quoted in the Chicago Tribune [ Read More... ]

 [ More News and Events... ]

Our San Francisco office serves San Francisco and the greater Bay Areaincluding the following communities:

9 / 22

matching: there is unemployment (high or low)

10 / 22

matching: there is unemployment (high or low)

10 / 22

the efficient rate of unemployment is positive

• too much unemployment is bad

– too many services are idle

• too little unemployment is bad

– too many services are devoted to recruiting

• there is a positive efficient rate of unemployment (u∗)

– the number of services enjoyed (y = g+ c) is maximized

• when the unemployment rate is efficient, Samuelson rule holds

11 / 22

optimal stimulus spending

12 / 22

formula for optimal stimulus spending

stimulus = A · ε ·m1+B · ε ·m2

· (u−u∗)

• stimulus: public spending - Samuelson spending

• u−u∗: initial unemployment gap

• ε : elasticity of substitution between g and c

– marginal social value of public spending

• m: unemployment multiplier

– decrease in u when g increases by 1% of y

– same as output multiplier13 / 22

sign of stimulus spending, for different

unemployment gaps and unemployment multipliers

m < 0 m = 0 m > 0

u > u∗ − 0 +u = u∗ 0 0 0u < u∗ + 0 −

14 / 22

sign of stimulus spending, for different

unemployment gaps and unemployment multipliers

m < 0 m = 0 m > 0

u > u∗ − 0 +u = u∗ 0 0 0u < u∗ + 0 −

14 / 22

sign of stimulus spending, for different

unemployment gaps and unemployment multipliers

m < 0 m = 0 m > 0

u > u∗ − 0 +u = u∗ 0 0 0u < u∗ + 0 −

14 / 22

sign of stimulus spending, for different

unemployment gaps and unemployment multipliers

m < 0 m = 0 m > 0

u > u∗ − 0 +u = u∗ 0 0 0u < u∗ + 0 −

14 / 22

numerical illustration:

Great Recession in the US

15 / 22

starting point: winter 2008–2009

• unemployment = 6% and public spending = 16.5% of GDP

– for illustration: we take these values as efficient

• unemployment is forecast to increase to 9%

– initial unemployment gap = 9%−6% = 3%

• we compute optimal stimulus for various elasticities of

substitution and unemployment multipliers

16 / 22

optimal stimulus spending (% of GDP)

0 0.5 1 1.5 2unemployment multiplier

0%

2%

4%

6%

✏ = 1

17 / 22

optimal stimulus spending (% of GDP)

0 0.5 1 1.5 2unemployment multiplier

0%

2%

4%

6%

0.1

2%

17 / 22

optimal stimulus spending (% of GDP)

0 0.5 1 1.5 2unemployment multiplier

0%

2%

4%

6%

0.4

3.7%

17 / 22

optimal stimulus spending (% of GDP)

0 0.5 1 1.5 2unemployment multiplier

0%

2%

4%

6%

$520 billion

17 / 22

optimal stimulus spending (% of GDP)

0 0.5 1 1.5 2unemployment multiplier

0%

2%

4%

6%

1.4

2%

17 / 22

optimal stimulus spending (% of GDP)

0 0.5 1 1.5 2unemployment multiplier

0%

2%

4%

6%✏ = 2

✏ = 0.5

17 / 22

more on the elasticity of substitution between

public and private services

• ε = 0: public services = digging holes

– stimulus = 0

• ε =+∞: public services = private services

– entirely fill unemployment gap

• 0 < ε 0

– but only partially fill unemployment gap

18 / 22

unemployment with optimal stimulus

0 0.5 1 1.5 2unemployment multiplier

6%

7%

8%

9%

✏ = 2

✏ = 1

✏ = 0.5

19 / 22

summary and discussion

20 / 22

1. multiplier > 1 is not necessary for stimulus

– stimulus requires unemployment multiplier > 0 (as in data)

2. bang-for-the-buck logic does not hold

– same stimulus for m = 0.1 and m = 1.4

3. completely filling the unemployment gap is not optimal

– optimal to partially fill unemployment gap

– except if public services = private services

4. low marginal social value of g does not imply no stimulus

– optimal to reduce unemployment gap

– except if public services = digging holes

21 / 22

distortionary taxes do not imply smaller stimulus

• with distortionary taxation, formula remains valid

– but Samuelson spending is lower

• however, the output multiplier is not useful anymore

– dy/dg = m + labor-supply response to taxes

– labor-supply distortion reduces dy/dg but not m

– so m > dy/dg, and possibly dy/dg < 0 while m > 0

5. distortionary taxation does not imply smaller stimulus

– only average public spending is lower

22 / 22