An Equilibrium Theory of Learning, Search and Wages Francisco Gonzalez Shouyong Shi U. Calgary U. Toronto RSW NBER Group Macro Perspectives on Labor Market (2009) Federal Reserve Bank of Minneapolis
1. Tasks and Motivation • Formulate the problem of learning by unemployed workers about themselves • characterize equilibrium with such learning • examine how reemployment wages and rates depend on search history 2
What’s the story? • workers do not know their ability/productivity — some are lucky to fi nd jobs = ⇒ revise beliefs upward — some are not so lucky = ⇒ discouragement • divergence in histories = ⇒ endogenous heterogeneity in: — workers’ beliefs about their job- fi nding process — search decisions — job- fi nding rates and wages 3
Speci fi c facts: • average job- fi nding prob decreases with duration • wage losses increase with unemployment duration: — US Displaced Worker Survey (Addison and Portugal 89): increasing duration by 100% reduces wages by 10% — UK Labour Force Survey (Gregg and Wadsworth 00): duration of 7-12 months = ⇒ wage loss of 27 log points 4
Other complementary theories: • unobserved worker heterogeneity, and long duration is a signal of low productivity • skill depreciation during unemployment • declining wealth/bene fi t during unemployment 5
Why use an equilibrium? • need to explain the above facts as market outcomes • fi rms can adjust o ff ers and vacancies to respond to learning: — with exogenous wages, low-wage jobs would be fi lled more quickly as reservation wages fall 6
2. Model Environment Workers and jobs: • fi rms or jobs: free entry • workers (risk neutral): — unemployed workers search — employed workers produce y > 0, shock of separation into unemployment: δ — shock of exit from market: σ 7
Worker’s unknown ability: • new worker draws ability i ∈ { H, L } : — unknown, permanent, prob ( H ) = p • worker’s productivity is a random variable: ½ y > 0, prob a i 0, prob 1 − a i — H is more “productive”: 0 < a L < a H < 1 — realized immediately after contact 8
Directed search: • continuum of submarkets x ∈ X = [0 , 1 /a H ] x : prob of getting productive match (per search unit); W ( x ): wage level; λ ( x ): tightness • search choice: a submarket x to enter (tradeo ff between x and W ) 9
Matching in submarket x : • total number of (productive) matches: F ( u e ( x ) , v ( x )) • total productive units of search in submarket x : u e ( x ) = a H × u H ( x ) + a L × u L ( x ) u i ( x ): # of type- i workers in submarket x 10
Matching in submarket x (continued): • matching probability per productive (search) unit: = F (1 , v ( x ) x = F ( u e ( x ) , v ( x )) ) u e ( x ) u e ( x ) | {z } tightness λ ( x ) • matching probabilities for participants: type- H type- L vacancy F x v = a H x a L x λ ( x ) 11
Wage function W ( x ): • free-entry of vacancies: J v ( x ) ≤ 0 and v ( x ) ≥ 0 for all x ∈ X with complementary slackness 12
• fi rm’s expected pro fi t of vacancy in market x : x J v ( x ) = − c + λ ( x ) × (1 − σ ) × J f ( W ( x )) • fi rm’s value of employing worker at w , discounted to the end of previous period: £ ¤ 1 J f ( w ) = y − w + (1 − σ ) × (1 − δ ) × J f ( w ) 1 + r 13
Wage function W ( x ): • free entry implies wage function: W ( x ) = y − cA λ ( x ) A ≡ δ + r + σ x , 1 − σ • W 0 ( x ) < 0 (tradeo ff between W and matching prob x ) 14
3. Learning in directed search equilibrium Information and learning: • match success and failure contain info about a i — info content depends on x : a L x < a H x • fi rms do not face signal extraction: matching prob x/λ ( x ) and wage W ( x ) are known • all participants know all statistics in all submarkets 15
Worker’s beliefs: expected value of his a • common initial belief: μ 0 = p a H + (1 − p ) a L • belief before search in a period: μ = P H a H + P L a L • posterior prob after search outcome: P ( a i | x, success) = a i x μ x P i = a i μ P i P ( a i | x, failure) = 1 − xa i 1 − xμ P i 16
Updating beliefs: • beliefs before and after search: ⎧ ⎪ E ( a | x, success) = a H + a L − a H a L /μ ≡ φ ( μ ) ⎨ μ → ⎪ E ( a | x, failure) = a H − 1 − xa L ⎩ 1 − xμ ( a H − μ ) ≡ H ( x, μ ) • properties of updating: — beliefs obey a Markov process — μ is su ffi cient statistic for search history — search in market with higher x is more informative 17
Rule out experimentation (su ffi cient condition): y − b > [ A + a H x ∗ ] λ 0 ( x ∗ ) − a H λ ( x ∗ ) c x ∗ is de fi ned by: λ 0 ( x ∗ ) = a H λ ( 1 ) a H 18
Search decision of a worker with belief μ : • value of being employed at wage w , discounted to the end of previous period: 1 J e ( μ, w ) = 1 + r { w + (1 − σ ) [(1 − δ ) J e ( μ, w ) + δ V ( μ )] } • return to search in market x : R ( x, μ ) ≡ xμ J e ( φ ( μ ) , W ( x )) + (1 − xμ ) V ( H ( x, μ )) 19
Search decision of a worker with belief μ (continued): • search decision: (1 + r ) V ( μ ) = b + (1 − σ ) × max x ∈ X R ( x, μ ) • policy functions: — search choice (of submarket): x = g ( μ ) ∈ G ( μ ) — desired wage: w ( μ ) = W ( g ( μ )) 20
Stationary symmetric equilibrium: • Block 1: individual decisions and market tightness (i) given W ( . ), workers with belief μ choose x = g ( μ ) ∈ G ( μ ) (ii) workers update beliefs according to φ ( μ ) and H ( g ( μ ) , μ ) (iii) W ( . ) satis fi es free-entry condition (iv) consistency: λ ( x ) = v ( x ) u e ( x ) for all x with v ( x ) > 0 • Block 2: (v) distribution of workers consistent with law of motion Equilibrium is block recursive (as in Shi 09) 21
4. Monotonicity of desired wages Want to show: • policy function, w ( μ ) = W ( g ( μ )), is strictly increasing — i.e., wages fall as beliefs deteriorate — i.e., search decision x = g ( μ ) strictly decreases in μ Problems: • value V ( μ ) is convex; • V 0 ( μ ) may not exist; FOC may not be applicable 22
A map of our approach: • use lattice-theoretic methods to prove: policy function is monotone • monotone policy function + convex value function = ⇒ validate fi rst-order condition • the above results + fi rst principles of calculus = ⇒ envelope condition + di ff erentiability of V 23
Topkis’ Theorems (98): z ∈− X f ( z, μ ) , max z = − x ; μ ∈ M • If f is supermodular in ( z, μ ), (and if ( − X ) × M is a lattice), then max Z ( μ ) and min Z ( μ ) are increasing in μ • If f is strictly supermodular in ( z, μ ), then every selection z ( μ ) ∈ Z ( μ ) is increasing in μ . 24
Use lattice-theoretic techniques: • transform payo ff function: R ( z, μ ) ≡ R ( x, μ ) ˆ , z ≡ − x μ • optimal search decision z ( μ ) ∈ Z ( μ ): ˆ (1 + r ) V ( μ ) = b + (1 − σ ) × μ × max R ( z, μ ) z ∈− X 25
Theorem 4.1: monotonicity of desired wages Assume separation rate satis fi es 0 ≤ δ ≤ ¯ δ . Then • ˆ R ( z, μ ) is strictly supermodular in ( z, μ ) • every selection z ( μ ) ∈ Z ( μ ) is an increasing function; every selection x = g ( μ ) is a decreasing function • w ( μ ) = W ( g ( μ )) is an increasing function 26
Why is ˆ R ( z, μ ) strictly supermodular? ⎧ ⎪ expected value: ⎪ ⎪ ⎨ prob. xμ φ ( μ ); δ × xμφ ( μ ) + μxW ( x ) − − − − − − − − − → μ ⎪ ⎪ ⎪ ⎩ prob. (1 − xμ ) → H ( x, μ ); m ≡ (1 − xμ ) H ( x, μ ) − − − − − − − − − − High x submarkets have low wages: ⇒ deeper discouragement: ∂m • failure in higher x = ∂x < 0 h i ∂ ∂m • marginal “damage” of x increases in μ : < 0 ∂μ ∂x 27
Why is ˆ R ( z, μ ) strictly supermodular? (cont’d) • convexity of V is important: properties above carry over to payo ff only for convex V : R = − z W ( − z ) A (1 − σ ) − δ + (1 ˆ A z V ( φ ( μ )) μ + z ) V ( H ( z, μ )) | {z } | {z } expected payo ff to success to failure • assumption δ ≤ ¯ δ is needed 28
Theorem 4.1 (continued): strict monotonicity Assume 0 < δ ≤ ¯ δ . Statements below are equivalent: • (i) V ( μ ) is strictly convex for all μ • (ii) every selection z ( μ ) ∈ Z ( μ ) is strictly increasing • (iii) corner z = − 1 /a H is not optimal for any μ > a L • (iv) corner z = − 1 /a H is not optimal for μ = a H • (v) y − b < ( A + 1) λ 0 ( 1 a H ) − a H λ ( 1 a H ) c 29
Why linear V over some beliefs = ⇒ even most optimistic workers search for lowest wage? • V ( μ ) being linear in [ μ a , μ b ] = ⇒ decision problem is strictly concave for such μ = ⇒ optimal choice of z is unique for such μ • strict supermodularity of ˆ R = ⇒ monotonicity of optimal decisions = ⇒ unique maximizer is corner, { − 1 /a H } , for such μ 30
Why linear V over some beliefs = ⇒ even most optimistic workers search for lowest wage? • V ( μ ) linear in [ μ a , μ b ] = ⇒ unique maximizer is { − 1 /a H } £ ¤ φ i ( μ a ) , φ i ( μ b ) • Same argument applies to μ ∈ , i ≥ 1: unique maximizer is { − 1 /a H } for all such μ • lim i →∞ φ i ( μ ) → a H , and Z ( μ ) is upper hemicontinuous = ⇒ { − 1 /a H } ∈ Z ( a H ). 31
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