RiskSharing and Labor Supply Elasticity by Rachel Soloveichik

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Risk-Sharing and Labor Supply Elasticityby
Rachel Soloveichik
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Cargo System in Mexico

• Individuals provide public goods for community
• Administration and maintenance done without pay
• Rich people throw huge parties for community
• Cancian finds the largest party in the village he
  studied costs 10x annual earnings for unskilled
  laborer
• Men expected to exhaust their savings and go into
  debt
• Significant redistribution from rich to poor
• Different from Government programs
• Does not distort labor supply
• Leisure is taxed as well as earned income
• Rich people get status in return for payments

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Why is Cargo System Important?

• Can increase social welfare
• Expected utility higher if cargo system transfers
  resources from rich to poor without distorting
  behavior
• Social institutions substitute for insurance
  contracts
• Risk-sharing changes behavior
• Income must be adjusted for risk-sharing
• Lottery winners not necessarily richer than
  lottery losers
• Different price elasticity with and without
  risk-sharing
• Two elasticities have different economic
  interpretations
• Elasticity may change even when technology and
  preferences are fixed, if risk-sharing changes

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Outline of Paper

• Discuss how risk-sharing changes labor supply
  elasticity
• Develop a model of labor supply with partial
  risk-sharing
• Use my model to predict labor supply elasticity
  estimated with IV and OLS
• Test my predictions with data from 2000 Mexican
  census

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Two Price Elasticities

• Frisch elasticity, EF
• Changes prices, holds marginal utility of
  consumption constant.
• Marshallian elasticity, EM
• Changes prices, holds non-labor income constant.
• EF and EM similar for goods with small income
  share, like cellphones

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What Does Each Price Elasticity Mean?

• Increasing price has two effects
• Substitution effect people buy less of the more
  expensive good.
• Income effect total consumption falls
• EF is only the substitution effect, EM is both.
• EF used for short-term elasticity
• Lifetime income effect from price for one day is
  tiny
• EF also applies for insured price changes.
• Effect of prices changes for only one person in a
  large group is small.
• Intuitively, smoothing across states of the world
  and time is similar

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Labor Supply Elasticity With Respect to Wages

• Substitution Effect
• Cost of leisure is foregone income
• Leisure is more expensive for people with high
  wages
• Income Effect
• Leisure is a normal good
• High wages increase total income
• Economic theory clear EF gt0
• Economic theory ambiguous for EM
• Substitution effectgt0
• Income effectlt0
• Empirically estimated at close to 0

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What Determines Lifetime Wages?

• Ln(w) a0aS
• a0 is ability
• Assumed to be fixed at birth
• Unobservable
• S is schooling
• Endogenous
• Exogenous shocks, Z
• Mandatory schooling laws at 16
• College tuition at 18
• Z uncorrelated with ability
• Economists use Z as an instrument for S

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Model of Education with Risk-Sharing

• Education Sso Z
• so is initial schooling endowment for community
• Z random individual schooling shock
• Cov(Zj,Zk)0
• Cov(Z,s0)0 Cov(Z,a0)0 E(Z)0
• Ln(w) a0a(s0Z)
• Schooling shocks are equivalent to wage shocks
• w?ea0as0 (1aZ)w0(1aZ)
• Main Assumption
• Full risk-sharing within communities, so marginal
  utility of consumption is independent of Z
• No risk-sharing across communities

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Consumption and Labor Supply Equations 1

• U(c,h)dln(c)(1-d)ln(1-h)
• Budget constraint cw(1-h)F wt
• Total time available is 1, h is fraction spent
  working
• F is full income, including the value of leisure
• t is transfer income, can be positive or negative
• Solve to get cFd
• Income effect ?c/?Fdgt0
• Solve to get h1-F(1-d)/w
• Income effect ?h/?F-(1-d)/wlt0
• Substitution effect ?h/?w F(1-d)/w2gt0

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Consumption and Labor Supply Equations 2

• Full risk-sharing within a community
• ?F?w, so F?w0
• F ?w0(1aZ)t?w0, so t?-aw0Z
• h 1-w0 (1-d)/w0(1aZ)? da(1-d)Z
• Only substitution effect, no income effect for Z
• No risk-sharing across communities
• Cov(t,s0)0
• Cov(h,s0)0
• Income and substitution effects exactly cancel
  out

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Transfer Income

• Direct test of risk-sharing model
• Transfer income is observable
• A few datasets ask about transfers directly
• Other datasets have spending and income, so
  transfers can be computed indirectly

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Variables of Interest

• Log wages, ln(w)
• ln(w) a0a(s0Z)
• Hours worked, h
• hda(1-d)Z
• h h/E(h) ? 1a(1-d)/dZ
• Relative transfer income, t
• t t/c-a/dZ

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OLS Regressions

• Wages
• aOLSCov(ln(w),S)/Var(S)aCov(a0,s0)/Var(s0)
• Biased upwards if Cov(a0,s0) gt 0
• Biased downwards if Cov(a0,s0)lt0
• Labor supply
• hOLSCov(ln(h),S)/Var(S) a(1-d)/dVar(Z)/Var(S)
• Var(Z) and Var(S) are known population constants
• Relative transfer income
• tOLSCov(t,S)/Var(S) -a/dVar(Z)/Var(S)

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IV Regressions

• Wages
• aIV Cov(ln(w),Z)/Cov(S,Z)a
• Unbiased estimate of a
• Labor Supply
• hIVCov(ln(h),Z)/Cov(S,Z) a(1-d)/d
• Relative transfer income
• tIVCov(t,Z)/Cov(S,Z)-a/d

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Predictions under Imperfect Risk-Sharing

• Within community risk-sharing of r
• Fw0(1-r)aZ
• Income effect is (1-r) of uncompensated income
  effect
• r1 full risk-sharing
• aOLS and aIV unchanged
• hOLSra(1-d)/dVar(Z)/Var(S)
• hIVra(1-d)/d
• tOLS-ra/dVar(Z)/Var(S)
• tIV-ra/d

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Predictions from Risk-Sharing Model

• In Mexico, Var(Z)/Var(S).2
• 20 of variation in OLS is smoothed
• Test if hIVgt hOLSgt0 and tIVlt tOLSlt0

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Outline of Empirical Work

• Discuss Data
• Describe Instruments used
• Regression of Hourly Wages on Education
• Regression of Labor Supply on Education
• Regression of Transfer Income on Education

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Data from 2000 Mexican Census

• Data set is non-Indian men 20-74
• Large age range to get elastic labor supply
• Exclude men in school for labor supply and
  transfer income
• Dataset contains
• Earned income last month
• Cash transfer income received last month
• Average 4.6 of total household income
• Reduce inequality within a family by at most 17
• Non-cash transfers not observed
• Hours worked last week
• Calculate wages for workers with positive
  earnings and hours
• Demographic information

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OLS Estimates of Education

• Four levels of school
• Primary grades 1-6. Will be designated S1
• Lower Secondary grades 7-9. Designated S2
• Upper Secondary grades 10-12. Designated S3
• College grades 12. Designated S4
• Different effects for each schooling level in OLS
• Not clear which schooling level IV should be
  compared to
• Report separate coefficients for each level

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Description of Instruments

• First instrument uses school calendar age 6-17
• Second instrument uses weather age 6-17
• Both instruments use within community variation
• Why two instruments?
• More robust test of model
• Similar results for both instruments

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Calendar Changes An Instrument for Education

• Two school calendars before 1966
• Type A schools started in February and ended in
  November
• Type B schools started in September and ended in
  June
• Calendars synchronized 1966 -1970
• Type A states lost one month of school each year,
  and Type B states unchanged

• Gray is Type A.
• Type A states are temperate
• Type B states are tropical

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Type A Primary School Statistics

• From Department of Education official reports
• Drop-out rates calculated from enrollment and
  graduation, negative number indicate migration
  into cities.
• Cutting school year by 11 in 1966 increases
  failure by 5 and drop-out in rural areas by 5

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First Stage for Calendar IV
S g1CaleU1 g2CaleU2 g3CaleU3 Age fixed
effects state fixed effects

• Cale is of shortened school years ages 6-17
• Assume people go to school in their birth state
• Dummies for 3 separate urbanization categories
• U1 is lt50 rural
• U2 is 50-65 rural
• U3 is gt 65 rural
• Errors are clustered by state-age cohort

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Temperature Changes as an Instrument for Education

• Plants grow faster in warmer temperatures
• Each crop stage requires a fixed amount of labor,
  so warm months require more labor
• Total amount of labor per year is fixed.
• Children attend more school if peak labor demand
  during school vacations.
• Deschennes and Greenstone find weather changes
  have little effect on profits
• So, temperature affects education only through
  substitution effect, no income effect.

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First Stage for Temperature IV
Sg1VaceU1 g2SchooleU1g3VaceU2g4SchooleU2
g5VaceU3g6Schoole U3Age fixed effects state
fixed effects

• Vace is mean C during school vacations ages 6-17
• Schoole is mean C during school year ages 6-17
• U1, U2 and U3 same as before
• Errors are clustered by state-age cohort.

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Predictions from Model

• Hourly wages Ln(w)aS
• Ambiguous
• aIV gt aOLS if Cov(s0,a0)lt0
• aIV lt aOLS if Cov(s0,a0)gt0
• Labor Supply ln(h) hS
• hIVgthOLSgt0
• hIV is larger for men with better risk-sharing
• Transfer Income t tS
• tIVlttOLSlt0

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Regression of Hourly Wage on Education
OLS Regression Ln(w) a1S1 a2S2a3S3a4S4 IV
Regression ln(w) aS

• Regression includes age fixed effects and state
  fixed effects
• IV and OLS estimate similar returns to education.
• Consistent with Cov(a0,s0) small

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Labor Supply Predictions

• Risk-sharing of r
• hIV ar(1-d)/d
• hOLS .2ar(1-d)/d
• a is the return to schooling, estimated at 5-9
• Why is hIVgt hOLS?
• IV estimates only rEF (1-r)EM
• OLS estimates .2rEF (1-.2r)EM
• EF, the Frisch elasticitygt EM, the Marshallian
  elasticity

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Regression of Relative Labor Supply on Education
OLS Regression h h1S1 h2S2 h3S3 h4 S4 IV
Regression h hS

• Regression includes age fixed effects and state
  fixed effects
• h is hours worked/average hours for age
• Reject hIVhOLS for both instruments
• Cannot reject that .2hIVhOLS for primary and
  lower secondary

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Incomplete Risk-sharing for Movers

• Cargo system provides local public goods
• No transfers between men who move and old
  neighborhood
• Cargo system enforced through social pressure
• No punishment for men outside community who
  default
• Insurance impossible for already resolved risks
• Community joined as an adult cant insure against
  education shocks in childhood
• Predictions
• hIV large for men currently living in birth
  state
• hIV small for men who moved since birth

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Labor Supply Elasticity By Location
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How Large is the Frisch Labor Supply Elasticity
with Respect to Wages?

• Model of risk-sharing for wage shocks
• Ln(w)aS, so schooling shocks are equivalent to
  wage shocks
• aIV 5-9, use 7 as true a
• hIV 7-14 for men in birth state
• EF is hIV/xaIV
• 1-2 if x1
• Higher if xlt1

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Explanation for Puzzle in Returns to Education
Literature

• Previous returns to education literature
• Ln(Y) bS ae Y is income, Y hw
• Economists want b
• Survey of literature by Card finds bIV gt bOLS
• Economists expect bIV lt bOLS
• Assume Cov(a,S)gt0
• Not explainable by measurement error
• My model
• True return to education is a
• bOLSaOLShOLS bIVaIVhIV
• bOLS 30 larger than a bIV 100-200 larger than
  a
• Other authors find hIV gt hOLS
• Duflo finds hIV 10 and hOLS 3 for labor force
  participation
• Angrist and Kruger find hIV 3 and hOLS 1.5 for
  weeks worked

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Transfer Income Predictions

• Model predicts t-a/dse
• tIV -a/d
• tOLS -.2a/d
• a and d estimated from wage and labor supply
  equations, can calculate total transfers required
• Transfers only partially observed
• Only observe cash transfers received
• Cannot observe cash transfers sent
• Cannot observe indirect transfers from cargo
  system
• Will test if tIVlt tOLSlt0, not tIV -a/d

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Regression of Income from Transfers on Education
OLS Regression t t1S1t2S2t3S3t4S4 IV
Regression t tS

• Regression includes age fixed effects and state
  fixed effects
• t transfers received/total income
• Reject tIVtOLS for 3 out of 4 specifications

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How Much Risk-Sharing Do Cash Transfers Provide?

• tIV for transfers received is -.9 to -.3
• Assume tIV for net transfers is doubled
• Assume aIV 7
• Risk-sharing from cash transfers depends on labor
  supply elasticity
• Assume functional form from model
• Utility of leisure and utility of consumption
  independent
• Holding labor supply fixed, 10-30 risk-sharing
• EF is 1, 5-15 risk-sharing
• Compare with 17 risk-sharing estimated from size
  of household transfers

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Alternative Explanations

• Measurement Error in Schooling
• Effects of schooling are different in urban and
  rural communities
• Habit Formation

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Measurement Error in Schooling

• Biases OLS towards 0, so IV ?OLS
• Labor supply effects require 80 error in
  measured schooling
• aIV ? aOLS, indicating low measurement error
• Cannot explain labor supply differences between
  movers and stayers

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Urban and Rural Differences

• Instruments used have largest impact on rural
  areas
• IV estimates effects of schooling in rural areas
• OLS estimates weighted average of effects in
  urban and rural areas
• Might explain non-linear OLS
• Education correlated with urbanization
• Urban children get much more schooling
• Test by comparing OLS for urban and rural birth
  states
• Coefficients for wages, labor supply and transfer
  income similar across areas
• Non-linearities persist in OLS
• OLS coefficients for heavily rural areas still
  different from IV

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Habit Formation

• Education teaches good work habits, so drop-outs
  are lazy adults.
• Rural children who drop out of school are
  working, not watching TV
• Does not explain differences in transfer income

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Conclusions

• Predictions from Risk-Sharing Model
• hIVgt hOLSgt0
• People with more schooling than average for their
  community work more hours
• hIV small when r is small
• Schooling shocks have little effect on labor
  supply for men outside the cargo system
• tIVlt tOLSlt0
• Redistribution is observable people with more
  schooling than average for their community get
  less cash transfers
• Empirical estimates suggest hIV is large
• bIV is 2-3 times aIV
• A regression that confuses a and b will overstate
  returns to education