Topic 2-3 :Policy Design: Unemployment Insuranceand Moral...

Introduction Trade-off Optimal UI Empirical

Topic 2-3: Policy Design: Unemployment Insurance and Moral

Hazard

Johannes Spinnewijn

London School of Economics

Lecture Notes for Ec426

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This Lecture: UI & Moral Hazard

1 Moral Hazard: Insurance vs. Incentives

2 Optimal level of UI benefits [Baily-Chetty model]

1 Model of Moral Hazard - generalizes for other applications

2 Suffi cient Statistics Approach - use of envelope conditions

3 Empirical estimation to test for optimality of program

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Unemployment Insurance: Basic Trade-off

Insurance against unemployment

loss of current (and potentially future) earningsuninsured unemployed experience drop in consumption

If fully insured, unemployed has no (monetary) incentive tokeep/get a job

moral hazard on the job and during unemployment

Central trade-off: insurance vs. incentives ⇒ optimalgenerosity

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Generosity of the Program: Replacement RateCommon measure of program’s size is its “replacement rate”

r =(net) benefit(net) wage

UI reduces agents’effective wage rate to (1− τ)w × (1− r)Benefit level depends (non-linearly) on pre-unemploymentearnings and employment, age, children, etc. Often featuresmaximum and minimum.Example: Michigan’s benefit schedule

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UI in the USFederally mandated, but implemented by the states

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Generosity of the Program: Duration of Eligibility

Source: Gruber’s book

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Introduction Trade-off Optimal UI Empirical Baily-Chetty First Best Second Best

Baily-Chetty model

Canonical analysis of optimal level of UI benefits: Baily(1978)

Shows that the optimal benefit level can be expressed as a fnof a small set of parameters in a static model

Once viewed as being of limited practical relevance because ofstrong assumptions

Chetty (2006) shows formula actually applies with arbitrarychoice variables and constraints

Parameters identified by Baily are “suffi cient statistics” forwelfare analysis ⇒ robust yet simple guide for optimal policy

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Baily-Chetty model: Setup

Static model with two states: an agent is either

employed and earns wage wor unemployed and has no income

Agent is initially unemployed. Controls probability ofremaining unemployed by exerting search effort

If the agent searches at cost e, the probability of finding a jobequals π (e) with π′ > 0, π′′ < 0

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Baily-Chetty model: Setup

UI system that pays constant benefit b to unemployed agents

Benefits financed by lump sum tax τ paid by the employedagents

Govt’s balanced budget constraint:

π (e) · τ − (1− π (e)) · b = 0

Agent’s expected utility, with u(c) utility over consumption, is

π (e) u(w − τ) + (1− π (e))u(b)− e

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Baily-Chetty model: First Best Solution

In first best, there is no moral hazard problem

Government chooses b and e (determining τ) to maximizeagent’s welfare:

maxb,e

π (e) u(w − 1− π (e)

π (e)b)+ (1− π (e))u(b)− e

Solution to this problem is

FOCb : u′(ce ) = u′(cu)⇒ full insurance

FOCe : π′ (e) [u (ce )− u (cu)]− 1+ π′(e)π(e) bu

′ (ce ) = 0

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Baily-Chetty model: Second Best Solution

In second best, effort is unobserved by govt. ⇒ moral hazard

Problem: agents only consider private marginal benefits andcost when choosing e

agent does not internalize the effect on the govt’s budgetconstraint

e I (b, τ) : π′ (e) [u (ce )− u (cu)]− 1 = 0eS (b, τ) : π′ (e) [u (ce )− u (cu)]− 1+ π′(e)

π(e) bu′ (ce ) = 0

hence, agent searches too little from a social perspective ⇒source of ineffi ciency

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Government’s problem is to maximize agent’s expected utility,taking into account agent’s behavioral responses:

maxb,τ,e

π (e) u(w − τ) + (1− π (e))u(b)− e

such that

BC : π (e) τ − (1− π (e))b = 0IC : π′ (e) [u (w − τ)− u (b)]− 1 = 0

Denote by e (b) and τ (b), the functions satisfying BC and IC

The (unconstrained) problem of the government is

maxbV (b) = π (e (b)) u(w − τ (b))+ (1−π (e (b)))u(b)− e (b)

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Two Approaches to Optimal Policy ProblemsFocus in public finance is on deriving an empirically implementablesolution to this problem:

1 Structural: specify complete models of economic behaviorand estimate the primitives

identify b∗ as a fn. of deep parameters: returns and cost of jobsearch, discount rates, nature of borrowing constraints,informal ins. arrangements.challenge: diffi cult to identify all primitive parameters in anempirically compelling manner

2 Suffi cient Statistics: derive formulas for b∗ as a fn. ofhigh-level elasticities

these elasticities can be estimated using reduced-formmethodsestimate statistical relationships using research designs thatexploit quasi-experimental exogenous variation.Baily-Chetty model is an example of this approach

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Baily-Chetty model: Second Best SolutionAt an interior optimum, dV/db(b∗) = 0

⇔ (1− π (e))u′(b)− π (e) u′(w − τ)dτ

db

+{

π′ (e) [u (w − τ)− u (b)]− 1} dedb= 0

Since the expected utility has been optimized over e,the Envelope Thm implies:

(1− π (e))u′(cu)− π (e) u′(ce )dτ

db= 0

Key here is that we can neglect the dedb term

given the agent’s optimization, the impact on expected utilitythrough effort is of second orderthis holds for any optimal behavior by the agent, e.g.endogenous consumption (Chetty 2006)

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The change in effort does have a first order effect on thegovernment’s UI budget

With τ (b) = (1−π(e(b)))π(e(b)) b, we find

dτ

db=

1− π (e)π (e)

− π′ (e)

π (e)2dedbb

=1− π (e)

π (e)(1+

ε1−π(e),b

π (e))

⇒

dV (b)db

= (1− π (e)){u′(cu)− (1+ε1−π(e),b

π (e))u′(ce )}

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Baily-Chetty model: Second Best SolutionThis yields the optimality condition

u′(cu)− u′(ce )u′(ce )︸︷︷︸MB

=ε1−π(e),b

π (e)︸︷︷︸MC

LHS is marginal social benefit of UIbenefit of transferring $1 from high to low state due toincreased insuranceMB is decreasing in insurance coverage

RHS is marginal social cost of UIcost of transferring $1 due to decreased search effortMC is constant (or decreasing less) with insurance coverage

Comparative statics; ceteris paribus,if MC is higher, optimal UI benefits should be lowerif MB is higher, optimal UI benefits should be higher

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Implementation: Consumption-Based FormulaCan we identify suffi cient statistics to test for the optimalityof the current system?

Write marginal utility gap using a Taylor expansion:

u′(cu)− u′(ce ) ≈ u′′(ce )(cu − ce )

Defining coeffi cient of relative risk aversion γ = −u ′′(c )cu ′(c ) , we

can writeu′(cu)− u′(ce )

u′(ce )≈ −u

′′

u′ce

∆cc

= γ∆cc

Gap in marginal utilities is a function of curvature of utility(risk aversion) and consumption drop from high to low states

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Implementation: Consumption-Based Formula

TheoremThe optimal unemployment benefit level b∗ satisfies

γ∆cc(b∗) ≈

ε1−π(e),b

π (e)

where

∆cc

=ce − cuce

= consumption drop during unemployment

γ = −u′′(ce )u′(ce )

ce = coeffi cient of relative risk aversion

ε =d log 1− π (e)

d log b= unemployment elasticity

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Introduction Trade-off Optimal UI Empirical MH Elasticity Consumption Smoothing

Estimating the Moral Hazard Cost

Lots of empirical work on labor supply effect of socialinsurance. Overview by Krueger and Meyer (2002)

Early literature used cross-sectional variation in replacementrates. Problem: this implies a comparison of high and lowwage earners, whose employment prospects may be verydifferent!

This gave way in late 80s/early 90s to modern methods usingmore exogenous variation/quasi-experiments

difference in UI generosity across states, across time, acrossgroup...state experiments with UI bonuses (Meyer 1995)

Evidence suggests elasticity of around 0.5.

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Difference-in-Differences EstimatesCompare a group affected by a change in the unemploymentpolicy (T ) to a group for which the unemployment policy isunchanged (C ). Let B and A denote before and after thereform.

The effect on the exit probability can be estimated by thedifference-in-differences

∆πT − ∆πC =[πTA − πTB

]−[πCA − πCB

].

before-after estimator[πTA − πTB

]is biased by time effects

a group comparison[πTA − πCA

]is biased by group effects

The dif-in-dif removes (group-invariant) time effects and(time-invariant) group effects. The identification assumptionis that groups follow parallel trends over time, absent thepolicy change.

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“Spike” in hazard rate

Most striking evidence for moral hazard effect ofunemployment insurance: “spike” in hazard rate at benefitexhaustion.

Source: Schmieder et al. QJE 201122 / 27


Estimating the Consumption Smoothing Benefits

The smoothing benefits can be estimated as well, but weshould take into account that UI crowds out self-insurance

some people use their savings when unemployedsome people borrow from banks of family

cu = b+ savings

ce = w − τ − savings

however, many unemployed have no savings and faceborrowing constraints

Gruber analyzes drop in food consumption ce−cuce

andestimates how this is affected by a change in the benefit ratiobw .

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Simulated InstrumentsSame problem: the difference in consumption drop forindividuals with high and low replacement rates is not onlydue to the replacement rate differential.

Alternative solution: Simulated Instrumentstake a representative subsample of individuals S sub

for each individual i in state j at year t in the original samplecalculate the subsample’s average replacement rate if allindividuals of the subsample had lived in state j at year t(

bw

)simulatedj ,t

= ΣsεS subbsimulateds ,j ,t

ws

use(bw

)simulatedj ,t

as an instrument for(bw

)i ,j ,t

The approach exploits only variation in the generosity of thestate UI system over time (∼ difference-in-difference).Underlying the identification is a similar parallel-trendsassumption.

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Estimating the Insurance ValueGruber runs IV regression(

ce − cuce

)i ,j ,t

= β1 + β2

(bw

)i ,j ,t+ β3δj + β4τt + εi

and finds:

β1 = 0.24, β2 = −0.28without UI, cons drop would be about 24%a 10 pp increase in UI replacement rate causes 2.8 ppreduction in cons. drop.with current replacement rate (b/w = 0.5), cons drop isabout 10%

Is current level optimal?

γ× 10% ?= 0.5

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Calibrating the Model

We can find the optimal level using our estimates

γ∆cc≈ ε/π

γ(β1 + β2bw

∗) ≈ ε

Results: bw∗varies considerably with γ

γ 1 2 3 4 5 10bw∗

0 0 0.20 0.41 0.50 0.68

Consumption smoothing benefits seem small relative to themoral hazard cost of unemployment insurance?

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SummaryPolicy maker faces trade-off between the provision ofinsurance and the provision of incentives.

Simple model with search efforts can capture this trade-off.Model generalizes for other behavioral responses like saving,moral hazard on the job, quality of job matches,...

if behavior is optimal, change in behavior has second-ordereffect on welfareonly the effect on the government’s budget constraint isimportant and this is captured by the unemploymentprobability

Empirical evidence suggests that job seekers are quiteresponsive to monetary incentives, implying that consumptionbenefits need to be large to justify generous unemploymentbenefits

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