relational contracts in competitive labor markets · relational contracts in competitive labor...
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Relational Contracts in Competitive LaborMarkets
Simon Board, Moritz Meyer-ter-Vehn
UCLA
November 7, 2012
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Motivation
Firms face incentive problems
I Employment contracts are typically incomplete.
I Firms motivate workers via long-term relationships.
Micro and macro interactions
I Longevity of firm’s relationship depends on other firms’ offers.
I We solve for equilibrium in optimal self-enforcing contracts.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Summary of Results
Firm-optimal self-enforcing contractsI Stationary wage and effort.
I No back-loading.
Industry equilibrium
I Identical firms offer different contracts.
I Entry can lead to full employment.
I On-the-job search erodes productivity.
ApplicationsI Heterogeneous firms and firm location decision.
I Heterogeneous workers and over-qualification.
I Policy experiments.
Pictures
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Literature
Shapiro and Stiglitz (1984), MacLeod and Malcomson (1998).
I All firms offer same job.
I Unemployment necessary in equilibrium.
Burdett and Mortensen (1998)
I Wage posting with on-the-job search
I Higher wage attracts more employees.
I Non-degenerate wage distribution.
Empirics
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Wage Distribution
Krueger, A. B., and L. H. Summers (1988): “EfficiencyWages and the Inter–Industry Wage Structure,” Econometrica,56(2), p. 261.
“If all firms were identical, one would not expect to seedifferent firms paying different wages even if efficiencywages were important.”
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Outline
1 Introduction
2 Firm’s Problem
3 Matching
4 Industry Equilibrium
5 Free Entry
6 Internship Matching
7 Heterogeneous Firms and Workers
8 Conclusion
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Firm’s Problem
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Model Overview
EconomyI Mass 1 identical workers and n ≤ 1 identical firms.
I Firm has one job each period.
I Time {1, 2, . . .}; discount rate δ ∈ (0, 1).
Job (stage game)1 Worker receives outside offers; firm fills vacancy immediately.
2 Firm pays wage w ∈ <+.
3 Worker exerts effort at cost η ∈ <+ and produces output φ(η).
4 Separation with prob. 1− α, and if either party terminates.
-Time t Time t+ 1
Match Wage w Effort η Separation
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Model Overview
EconomyI Mass 1 identical workers and n ≤ 1 identical firms.
I Firm has one job each period.
I Time {1, 2, . . .}; discount rate δ ∈ (0, 1).
Job (stage game)1 Worker receives outside offers; firm fills vacancy immediately.
2 Firm pays wage w ∈ <+.
3 Worker exerts effort at cost η ∈ <+ and produces output φ(η).
4 Separation with prob. 1− α, and if either party terminates.
Stage payoffs
I Utility u := w − η; Profit π := φ(η)− w.
I Assume φ(0) = 0, φ′(0) =∞, φ′(∞) = 0, φ′′(η) < 0.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Perspective of Single Firm
Matching Stage
I W ∼ F e is cont. value of best offer; may have atom at 0.
I Firm fills vacancy instantly.
Restrictions:
I F e stationary and anonymous.
I Contract 〈wt, ηt〉 only depends on history within relationship.
Self-enforcing contracts
I SPNE in pure strategies.
I No voluntary terminations.
I Harshest penal code off equilibrium.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Firm’s Problem
Firm’s problem is to choose 〈wt, ηt〉 to maximise Π1 s.t.
wt − ηt + δαVt+1 + δ(1− α)V ∅ ≥ wt + δV ∅ (IC)
w1 − η1 + δαV2 + δ(1− α)V ∅ ≥ δV ∅ (IR)
Πt ≥ Π1 (FIC)
I Worker’s pre- and post-matching value functions
Vt =
∫max{W,Wt}dF e(W )
Wt = ut + δαVt+1 + δ(1− α)V ∅
I Firm’s pre-matching profit function
Πt = F e(Wt)[φ(ηt)− wt + δ(αΠt+1 + (1− α)Π1)] + [1− F e(Wt)]Π1
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Firm’s Problem
Firm’s problem is to choose 〈wt, ηt〉 to maximise Π1 s.t.
−ηt + δαVt+1 ≥ δαV ∅ (IC)
w1 − η1 + δαV2 ≥ δαV ∅ (IR)
Πt ≥ Π1 (FIC)
I Worker’s pre- and post-matching value functions
Vt =
∫max{W,Wt}dF e(W )
Wt = ut + δαVt+1 + δ(1− α)V ∅
I Firm’s pre-matching profit function
Πt = F e(Wt)[φ(ηt)− wt + δ(αΠt+1 + (1− α)Π1)] + [1− F e(Wt)]Π1
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Stationary Contracts
I Contract is stationary if independent of tenure, 〈w, η〉.
Theorem 1.For any self-enforcing contract there is a stationary self-enforcingcontract with weakly higher profits.
Idea
I Firm would like to backload to extract worker’s rent.
I But firm would fire old workers, so not self-enforcing.
Notation
I Utility of job u = w − η sufficient statistic for job.
I Value of job V (u); outside offers F e(u).
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Proof Sketch
Consider original contract 〈wt, ηt〉I Let φ(η∗)− η∗ = maxt{φ(ηt)− ηt}.I Let V ∗ = Vτ∗ = maxt{Vt} and let w∗ be corresponding wage.
New contract 〈w∗, η∗〉I (IC): Follows from ηt ≤ αδ[Vt+1 − V ∅] for all t.
I (IR): Follows from V ∗ ≥ V1.
I (FIC): Follows from stationarity.
Profits are higherI Π∗1 ≥ Πτ∗ : Higher surplus, same worker rents.
I Πτ∗ ≥ Π1: Firm IC.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
First-Order Conditions
Firm’s problem is to choose 〈u, η〉 to maximize
π = φ(η)− η − us.t. − η + δαV (u) ≥ δαV ∅ (IC)
u+ δαV (u) ≥ δαV ∅ (IR)
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
First-Order Conditions
Firm’s problem is to choose η to maximize
π = φ(η)− η − u∗(η)
s.t. η = δα[V (u∗(η))− V ∅] (IC)
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
First-Order Conditions
Firm’s problem is to choose η to maximize
π = φ(η)− η − u∗(η)
s.t. η = δα[V (u∗(η))− V ∅] (IC)
Value of job
V (u) =
∫ u
umax{u+ δαV (u);x+ δαV (x)}dF e(x) + δ(1− α)V ∅
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
First-Order Conditions
Firm’s problem is to choose η to maximize
π = φ(η)− η − u∗(η)
s.t. η = δα[V (u∗(η))− V ∅] (IC)
Value of job
V ′(u) = (1 + δαV ′(u))F e(u) =F e(u)
1− δαF e(u)
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
First-Order Conditions
Firm’s problem is to choose η to maximize
π = φ(η)− η − u∗(η)
s.t. η = δα[V (u∗(η))− V ∅] (IC)
Value of job
V ′(u) = (1 + δαV ′(u))F e(u) =F e(u)
1− δαF e(u)
First-order condition
d
dη(η + u∗(η)) = 1 +
1
δαV ′(u∗(η))=
1
δαF e(u∗(η))
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
First-Order Conditions
Firm’s problem is to choose η to maximize
π = φ(η)− η − u∗(η)
s.t. η = δα[V (u∗(η))− V ∅] (IC)
Value of job
V ′(u) = (1 + δαV ′(u))F e(u) =F e(u)
1− δαF e(u)
First-order condition
φ′(η) =1
δαF e(u∗(η))
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Job Market Matching
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Job Market Matching
Initially: αn filled jobs, (1− α)n vacancies with cdf F (u).Axioms: Individual rationality, Anonymity, Market clearing.
I Offers to employed: F e(u)
I Offers to unemployed: F∅(u)
I Market clearing:
(1− αn)(1− F∅(u))︸ ︷︷ ︸unemployed
+αnF (u)(1− F e(u))︸ ︷︷ ︸employed below u
= (1− α)n(1− F (u))︸ ︷︷ ︸vacancies above u
Matching function ψ : [0, 1]→ [0, 1], weakly increasing.
I Comparative advantage of employed: F∅(u) = ψ(F e(u)).
I Retention rate F e(u) = β(F (u)) on range of F (·).
I Let q = F (u) and expand β(q) to all q ∈ [0, 1].
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Job Market Matching
Initially: αn filled jobs, (1− α)n vacancies with cdf F (u).Axioms: Individual rationality, Anonymity, Market clearing.
I Offers to employed: F e(u)
I Offers to unemployed: F∅(u)
I Market clearing:
(1− αn)(1− ψ(F e(u)))︸ ︷︷ ︸unemployed
+αnF (u)(1− F e(u))︸ ︷︷ ︸employed below u
= (1− α)n(1− F (u))︸ ︷︷ ︸vacancies above u
Matching function ψ : [0, 1]→ [0, 1], weakly increasing.
I Comparative advantage of employed: F∅(u) = ψ(F e(u)).
I Retention rate F e(u) = β(F (u)) on range of F (·).
I Let q = F (u) and expand β(q) to all q ∈ [0, 1].
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Job Market Matching
Initially: αn filled jobs, (1− α)n vacancies with cdf F (u).Axioms: Individual rationality, Anonymity, Market clearing.
I Offers to employed: F e(u)
I Offers to unemployed: F∅(u)
I Market clearing:
(1− αn)(1− ψ(β(F (u))))︸ ︷︷ ︸unemployed
+αnF (u)(1− β(F (u)))︸ ︷︷ ︸employed below u
= (1− α)n(1− F (u))︸ ︷︷ ︸vacancies above u
Matching function ψ : [0, 1]→ [0, 1], weakly increasing.
I Comparative advantage of employed: F∅(u) = ψ(F e(u)).
I Retention rate F e(u) = β(F (u)) on range of F (·).
I Let q = F (u) and expand β(q) to all q ∈ [0, 1].
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Job Market Matching
Initially: αn filled jobs, (1− α)n vacancies with cdf F (u).Axioms: Individual rationality, Anonymity, Market clearing.
I Offers to employed: F e(u)
I Offers to unemployed: F∅(u)
I Market clearing:
(1− αn)(1− ψ(β(q)))︸ ︷︷ ︸unemployed
+αnq(1− β(q))︸ ︷︷ ︸employed below u
= (1− α)n(1− q)︸ ︷︷ ︸vacancies above u
Matching function ψ : [0, 1]→ [0, 1], weakly increasing.
I Comparative advantage of employed: F∅(u) = ψ(F e(u)).
I Retention rate F e(u) = β(F (u)) on range of F (·).
I Let q = F (u) and expand β(q) to all q ∈ [0, 1].
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
More Matching
Examples
I Shapiro-Stiglitz: ψ(z) = 0.
I Fully anonymous: ψ(z) = z.
I Intern matching: ψ(z) = 1.
Properties
I We assume unemployed are better searchers: ψ(z) ≤ z.
I ψ(·) obeys OJS if it is continuous (i.e. β(q) strictly inc. in q).
I More OJS under ψ̃(·) than ψ(·) if ψ̃(z) ≥ ψ(z).
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium
An industry equilibrium is mass n of contracts 〈u, η〉 s.t.
(a) Every contract 〈u, η〉 is firm-optimal w.r.t. F e and V ∅.
(b) F e and V ∅ derived from matching function ψ and rents F .
The value of unemployment is
V ∅ =
∫(u+ δαV (u)) dF∅(u) + δ (1− αθ)V ∅
where θ = (1− α)n/(1− αn)
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Construction
Equilibrium 〈u(x), η(x)〉x∈[0,1] is defined by three conditions:
1 First-order condition (or marginal IC constraint)
φ′(η(x)) =1
δαβ(x)
Uniquely determines η(x).
2 Constant profits: There is π such that
u(x) = φ(η(x))− η(x)− π
Uniquely determines u(x) up to constant π.
3 IC constraint for highest firm
η(1) = δα(V (u(1))− V ∅).
Uniquely determines π.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Construction
Equilibrium 〈u(x), η(x)〉x∈[0,1] is defined by three conditions:
1 First-order condition (or marginal IC constraint)
φ′(η(x)) =1
δαβ(x)
Uniquely determines η(x).
2 Constant profits: There is π such that
u(x) = φ(η(x))− η(x)− π
Uniquely determines u(x) up to constant π.
3 IC constraint for highest firm
η(1) = δα(V (u(1))− V ∅).
Uniquely determines π.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Construction
Equilibrium 〈u(x), η(x)〉x∈[0,1] is defined by three conditions:
1 First-order condition (or marginal IC constraint)
φ′(η(x)) =1
δαβ(x)
Uniquely determines η(x).
2 Constant profits: There is π such that
u(x) = φ(η(x))− η(x)− π
Uniquely determines u(x) up to constant π.
3 IC constraint for highest firm
η(1) = δα(V (u(1))− V ∅).
Uniquely determines π.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Characterization
Theorem 2.(a) Industry equilibrium exists and is unique(b) Equilibrium effort is determined by
φ′(η(x)) =1
δαβ(x)
with support
φ′(η(0)) =1
δαβ(0)and φ′(η(1)) =
1
δα
(c) With OJS, F (u) is strictly increasing and continuous.
If F (·) has an atom and β(·) increasing
I Retention rate β(F (u)) jumps up, so MC jumps down.
I Profits kink upwards, contradicting local optimality.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Example: Shapiro-Stiglitz Matching
Shapiro-Stiglitz matching
I Only unemployed receive offers: ψ ≡ 0 and β ≡ 1.
I Theorem 2: All firms offer same job, with φ′(η) := 1/δα.
Profits in Shapiro-Stiglitz
I Profit as function of effort:
π∗(η) = φ(η)− 1
δαη − (1− δ)V ∅.
I Overall profit
πSS = φ(η)− 1
δα (1− θ)η
with market tightness θ := (1− α)n/ (1− αn).
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Example: Shapiro-Stiglitz Matching
Shapiro-Stiglitz matching
I Only unemployed receive offers: ψ ≡ 0 and β ≡ 1.
I Theorem 2: All firms offer same job, with φ′(η) := 1/δα.
Profits in Shapiro-Stiglitz
I Profit as function of effort:
π∗(η) = φ(η)− 1
δαη − (1− δ)V ∅.
I Overall profit
πSS = φ(η)− 1
δα (1− θ)η
with market tightness θ := (1− α)n/ (1− αn).
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Example: Fully Anonymous Matching
Fully anonymous matchingI Employed and unemployed receive same offers: ψ(z) = z.
I Retention rate: β(q) = (1− n(1− q))/(1− αn(1− q)).
Lowest JobI Retention rate β (0) = 1− θ.
I Theorem 2: Effort is φ′(η) = 1/δα(1− θ).
I Profit as function of effort:
π∗(η) = φ(η)− 1
δα (1− θ)η
I Overall profit
πFA = φ(η)− 1
δα (1− θ)η.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Example: Fully Anonymous Matching
Fully anonymous matchingI Employed and unemployed receive same offers: ψ(z) = z.
I Retention rate: β(q) = (1− n(1− q))/(1− αn(1− q)).
Lowest JobI Retention rate β (0) = 1− θ.
I Theorem 2: Effort is φ′(η) = 1/δα(1− θ).
I Profit as function of effort:
π∗(η) = φ(η)− 1
δα (1− θ)η
I Overall profit
πFA = φ(η)− 1
δα (1− θ)η.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Shapiro-Stiglitz - No OJS
φ(η)−π
u∗(η)+η
Wa
ge
w ICφ’(η)
Marginal Cost and Benefit Contract Space
Effort η
−π
η(1)
Profit > π
Effort ηη(1)
MC1(η)
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
OJS - Downward Deviation
φ(η)−π
u∗(η)+η
Wa
ge
w
φ’(η)
MC0(η)
Marginal Cost and Benefit Contract Space
Effort η
−π
η(1)η(0)
−π’
Effort ηη(1)
MC1(η)
η(0)
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
OJS - Equilibrium Contract Distribution
φ(η)−π’
u∗(η)+η
Wa
ge
w
φ’(η)
MC0(η)
Marginal Cost and Benefit Contract Space
φ(η)−π’
Effort ηη(1)η(0)
−π’
Eqm
Effort ηη(1)
MC1(η)
MC0(η)
η(0)
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Comparative Statics of OJS
Theorem 3.When on-the-job search increases:
(a) Retention rates β(x) decrease for all jobs.
(b) Effort η(x), output and surplus decrease for all jobs.
(c) Rents u(x) decrease for all employed/unemployed workers.
(d) Profits π increase for all firms.
Idea
I Increase in OJS increases turnover and MC of effort.
I Firms substitute good jobs for bad.
I Lowers V ∅ and introduces slack into IC.
I Firms lower wages until IC binds, raising profits.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Free Entry
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium
An industry equilibrium is a distribution of contract 〈u, η〉 s.t.
(a) Every contract 〈u, η〉 is firm-optimal w.r.t. F e and V ∅.
(b) Each contract yields zero profits, π = 0.
(c) F e and V ∅ derived from matching function ψ and rents F (u).
I Retention rate βn(q) depends on n via market clearing
(1− αn)(1− ψ(βn(q))) + αnq(1− βn(q)) = (1− α)n(1− q)
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Construction
Equilibrium {〈u(x), η(x)〉}x∈[0,1] is defined by three conditions:
1 First-order condition (or marginal IC constraint)
φ′(η(x)) =1
δαβn(x)
Uniquely determines η(x) up to constant n.
2 Zero Profits:u(x) = φ(η(x))− η(x)
Uniquely determines u(x) up to constant n.
3 IC constraint for highest firm
η(1) = δα(V (u(1))− V ∅).
Uniquely determines n.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Construction
Equilibrium {〈u(x), η(x)〉}x∈[0,1] is defined by three conditions:
1 First-order condition (or marginal IC constraint)
φ′(η(x)) =1
δαβn(x)
Uniquely determines η(x) up to constant n.
2 Zero Profits:u(x) = φ(η(x))− η(x)
Uniquely determines u(x) up to constant n.
3 IC constraint for highest firm
η(1) = δα(V (u(1))− V ∅).
Uniquely determines n.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Construction
Equilibrium {〈u(x), η(x)〉}x∈[0,1] is defined by three conditions:
1 First-order condition (or marginal IC constraint)
φ′(η(x)) =1
δαβn(x)
Uniquely determines η(x) up to constant n.
2 Zero Profits:u(x) = φ(η(x))− η(x)
Uniquely determines u(x) up to constant n.
3 IC constraint for highest firm
η(1) = δα(V (u(1))− V ∅).
Uniquely determines n.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Characterization
Theorem 4.
(a) Equilibrium with free-entry exists and is unique.
(b) With FA matching, there is full employment n = 1.
(c) With less OJS, there is some unemployment, n < 1.
Idea
I Workers must be compensated for opp. cost of searching.
I Creates fixed cost to employ a worker.
I With fully anonymous matching, the fixed cost is zero.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Contracts - Fully Anonymous
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Policy Experiment - Unemployment Benefits
I Firms exit until IC is met.
I Equilibrium value of unemployment V ∅ unaffected.
η(1) = αδ(V (u(1))− V ∅)
V (u(1)) = u(1) + δ(αV (u(1)) + (1− α)V ∅)
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Policy Experiment - Minimum Wage
I Atom of jobs paying minimum wage.
I Increases variance at the bottom.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Comparative Statics of OJS
Theorem 5.If η ∈ [η(0), η(1)] then an increase in OJS:
(a) Increases the number of jobs n.
(b) Decreases the number of good jobs with rent above u(η).
Increasing OJS . . .
I Leads firms to replace good jobs with bad jobs.
I This lowers V ∅ and leaves IC slack.
I Firms enter at bottom until IC tight.
I Welfare: loss of good jobs balanced by lower unemployment.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Intern Matching
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Intern Matching (with n < 1)
Intern matchingI Employed prioritized: ψ(z) ≡ 1, so β(q) = 0 on [0, 1− α].
I Internship with w(0) = 0, η(0) > 0 and u(0) < 0.
I Internships have mass F (u(0)) > 1− α.
I Entry jobs are gatekeepers for better jobs.
Characterizing job distribution
I For η > η(0), F (·) characterized by FOC.
I Since IR binds in unemployed worker’s first job, V ∅ = 0.
I Firms make monopoly profits: πIM = φ(η)− η/αδ
Increase in nI Scales up distribution of contracts.
I Free entry leads to n = 1 and same effort distribution.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Intern Matching - Fixed n
Wa
ge
w
IC &IR
Real jobs
Effort ηInternships
Real jobs
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Intern Matching - Free Entry
IC &IR
Real jobs
Wa
ge
w
Effort η
Internships
Real jobs
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Heterogeneous Firms
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Construction
Firm productivity p ∼ G[p, p] and φ(η, p) is supermodular.
1 First-order condition (or marginal IC constraint)
∂
∂ηφ(η, p) =
1
δαβ(G(p))
Uniquely determines η(p).
2 Profits determined by envelope condition:
π(p) = π(p)−∫ p
p
∂
∂pφ(η(p̂), p̂)dp̂
Utilities given by
u(p) = φ(η(p), p)− η(p)− π(p)
Uniquely determines u(p), with free parameter π(p).
3 IC constraint for highest firm uniquely determines π(p).
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Construction
Firm productivity p ∼ G[p, p] and φ(η, p) is supermodular.
1 First-order condition (or marginal IC constraint)
∂
∂ηφ(η, p) =
1
δαβ(G(p))
Uniquely determines η(p).
2 Profits determined by envelope condition:
π(p) = π(p)−∫ p
p
∂
∂pφ(η(p̂), p̂)dp̂
Utilities given by
u(p) = φ(η(p), p)− η(p)− π(p)
Uniquely determines u(p), with free parameter π(p).
3 IC constraint for highest firm uniquely determines π(p).
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Construction
Firm productivity p ∼ G[p, p] and φ(η, p) is supermodular.
1 First-order condition (or marginal IC constraint)
∂
∂ηφ(η, p) =
1
δαβ(G(p))
Uniquely determines η(p).
2 Profits determined by envelope condition:
π(p) = π(p)−∫ p
p
∂
∂pφ(η(p̂), p̂)dp̂
Utilities given by
u(p) = φ(η(p), p)− η(p)− π(p)
Uniquely determines u(p), with free parameter π(p).
3 IC constraint for highest firm uniquely determines π(p).
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Wage as a Function of Productivity
0.2 0.4 0.6 0.8 10
1/32
2/32
3/32
4/32
5/32
(a) Wage functions
Productivity
p~U[0.25,1]p~U[0.5,1]
p~U[0.9,1]
p=1
0 1/32 2/32 3/32 4/32 5/32
(b) Wage distributions
Wage
p~U[0.25,1]
p~U[0.5,1]
p~U[0.9,1]
p=1
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Externalities
Shapiro-Stiglitz matchingI Increase in competitors prod. raises effort and rents.
I This tightens IC, reducing profits.
I Low productivity American firm should move to India.
Fully anonymous matchingI Increase in lower firms’ prod. lowers π(p).
I Increase in higher firms’ prod. does not affect π(p).
Intern matchingI Increase in lower firms’ prod. does not affect π(p).
I Increase in higher firms’ prod. raises π(p).
I Low productivity studio should move to LA.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Heterogeneous Workers
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Construction
Workers have effort cost η/κ for κ ∈ {κL, κH}. Contracts〈uκ(x), ηκ(x)〉 offered by nκ firms, where nL + nH = n.
1 First-order condition (or marginal IC constraint)
φ′(ηκ(x)) =1
δακβ(x)
Uniquely determines ηκ(q).
2 Constant profits: There are {π, nL, nH} such that
uκ(x) = φ(ηκ(x))− ηκ(x)− π
Uniquely determines uκ(x) up to {π, nL, nH}.
3 IC constraint for highest firm for each type κ,
ηκ(1) = δα(V (uκ(1))− V ∅).
Uniquely determines {π, nL, nH}.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Construction
Workers have effort cost η/κ for κ ∈ {κL, κH}. Contracts〈uκ(x), ηκ(x)〉 offered by nκ firms, where nL + nH = n.
1 First-order condition (or marginal IC constraint)
φ′(ηκ(x)) =1
δακβ(x)
Uniquely determines ηκ(q).
2 Constant profits: There are {π, nL, nH} such that
uκ(x) = φ(ηκ(x))− ηκ(x)− π
Uniquely determines uκ(x) up to {π, nL, nH}.
3 IC constraint for highest firm for each type κ,
ηκ(1) = δα(V (uκ(1))− V ∅).
Uniquely determines {π, nL, nH}.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Equilibrium Construction
Workers have effort cost η/κ for κ ∈ {κL, κH}. Contracts〈uκ(x), ηκ(x)〉 offered by nκ firms, where nL + nH = n.
1 First-order condition (or marginal IC constraint)
φ′(ηκ(x)) =1
δακβ(x)
Uniquely determines ηκ(q).
2 Constant profits: There are {π, nL, nH} such that
uκ(x) = φ(ηκ(x))− ηκ(x)− π
Uniquely determines uκ(x) up to {π, nL, nH}.
3 IC constraint for highest firm for each type κ,
ηκ(1) = δα(V (uκ(1))− V ∅).
Uniquely determines {π, nL, nH}.
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Three Types of ContractsW
ag
e
ηL(0) η
L(1)η
H(0) η
H(1) Effort, η ®
Isoprofit
IC High
IC Low
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Introduction Firm’s Problem Matching Equilibrium Entry Interns Heterogeneity The End
Summary
Relational Contracts in Competitive Labor Markets
I Endogenous wage and productivity dispersion.
I Free entry can lead to full employment.
I On-the-job search erodes productivity.
Flexible Framework
I General class of matching technologies.
I Intern matching.
I Heterogeneous firms and workers.
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Empirical Support for Model
Higher wages encourage effort
I High wage plants have fewer disciplinary actions.
I Wages are positively correlated with self-reported effort.
I Firms refuse to cut pay in order to sustain morale.
Relational contracts matter
I Effort declines at the end of a relationship
I Employment protection reduce effort.
I Unemployment increases effort
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Empirics: Predictions
Predictions
I Large wage differentials across firms
I High wage firms have lower turnover and more applications.
I Large amount of wage growth occurs at job transitions.
I Wage jumps more frequent and larger at start of career.
Example: Professional industries
I Effort is more subjective and frequent job-to-job transitions.
I Explains higher levels of residual wage inequality.
I Job ladders common
Back