Directed Search Lecture 2: Matching Patterns and Inequality October 2012 ° Shouyong Shi c
Main sources of this lecture: • Becker, G., 1973. “A Theory of Marriage: Part I,” JPE 81, 813—846. • Shi, S., 2001, “Frictional Assignment I: E ffi ciency,” JET 98, 232-260. • Shi, S., 2005, “Frictional Assignment, Part II: In fi nite Horizon and Inequality,” RED 8, 106-137. • Shimer, R. and L. Smith, 2000, “Assortative Matching and Search,” ECMA 68, 343—370 • Shi, S., 2002, “A Directed Search Model of Inequality with Heterogeneous Skills and Skill-Biased Technology,” RES 69, 467-491. 2
1. Motivation and Issues • many markets have heterogeneity on both sides: — labor market: workers di ff er in skills, fi rms in capital and size — loan market: borrowers di ff er in project quality, lenders in funds — marriage market: men and women di ff er in income, beauty, etc. 3
• positive assortative matching (PAM): individuals are matched according to their ranking: — workers with higher skills match with better fi rms; — projects with higher quality match with better loans; — rich people marry rich people; handsome men marry beautiful women, etc. • two questions about the matching pattern: — positive: is PAM an equilibrium? — normative: is PAM socially e ffi cient? 4
• answer by Gary Becker (73, JPE) and Tinbergen (51): — PAM is an equilibrium and it is socially e ffi cient when markets are frictionless — necessary and su ffi cient condition for this result: joint surplus of a match is complementary (supermodular) in the two sides’ attributes • think again: — most matching markets are frictional — not all observed matching patterns are PAM 5
Main questions: when there are search frictions, • does the e ffi cient allocation have PAM? • how to decentralize the e ffi cient allocation? • how does matching a ff ect inequality? With undirected search, Shimer and Smith (00) fi nd that complementarity is not enough for PAM to arise in eqm • but their equilibrium is ine ffi cient, generically; is this ine ffi ciency responsible for non-PAM? • still need to answer the normative questions above 6
Directed search: • makes sense with homogeneous individuals • makes even more sense with heterogeneity: observable heterogeneity helps directing search — job ads typically specify worker quali fi cations; workers can observe fi rms’ attributes — di ff erentiated loan terms target di ff erent borrowers — people may date selectively 7
Roadmap: • analyze a market with TU and matching between workers who di ff er + machines that di ff er in skill levels in qualities • eqm and e ffi cient allocation with no friction • with search friction and directed search, characterize: e ffi cient allocation decentralization, inequality • extend to in fi nite horizon; dynamics • calibrate to examine e ff ects of skill-biased technology 8
2. Frictionless Economy and Assignment One-period environment • risk-neutral workers: exogenous supply; observable skill ∈ ⊂ R + : number = ( ); • machine quality ∈ K ⊂ R + : costs ( ); endogenous supply determined by free entry • one worker operates one machine; • output of the pair ( ): ( ) 9
Assumptions on : • complementarity (supermodularity): 0 • both inputs are necessary: (0 ) = ( 0) = 0 • every skill is employable with some machine quality: ( ) − ( ) 0 for some ∈ K • regularity conditions: 0; ≥ 0, ( − ) ( − ) 2 10
Frictionless assignment • no frictions: all pairs are matched instantaneously • e ffi cient assignment : → K max [ ( ) − ( )] , for each ∈ , ( ( ) ) = ( ( )) i.e., • ( ) exists and is unique for each • PAM: 0 ( ) = 0 i ff 0 − 11
Decentralization: • wage function: ½ ( ) − ( ) if ( ) − ( ) ≥ 0 ( ) = 0 otherwise • a fi rm chooses for each to attract workers: ⇒ solution: = ( ) max ∈ K ( ) = ( ) = ( ( ) ) − ( ( )) • equilibrium wage: • Does PAM increase wages for higher ? (relevant for identifying sorting through wage or value added) 12
Answer: Not marginally. • assignment pattern has NO fi rst-order e ff ect on wage: ( ) = ( ( ) ) − ( ( )) 0 ( ) = ( ( ) ) e ff ect of by itself + 0 ( )[ ( ( ) ) − ( )] e ff ect of a better match (but this is = 0) 13
3. E ffi cient Assignment with Frictions Frictional economy qualities in skill unit subset ( ) workers # : ( 1 ) → # : ( 1 ) ( 1 ) 1 ( 1 ) : workers/machines . . . . . . . . . ( ) # : ( ) → # : ( ) ( ) Matching probability in a unit ( ): for a worker: 1 − − ( ) for a machine: 1 − − ( ) ; ( ) 14
E ffi cient allocation : The planner chooses • ( ) ⊆ K : machine qualities assigned to ∈ • ( ): # of machines created for the unit ( ) • ( ): worker/machine ratio in the unit ( ) h³ 1 − − ( ) ´ i X X max ( ) ( ) − ( ) | {z } ( ) ∈ ∈ ( ) expected surplus of a match ( ) X s.t. ( ) ( ) = ( ) ∈ ( ) | {z } # of skill workers assigned to 15
Component problem of the e ffi cient allocation: For each ∈ , the e ffi cient allocation ( ( ) ( )) solves: ( ) ( ) − ( ) ( ) max social value of a worker h 1 − (1 + ( )) − ( ) i s.t. ( ) = ( ) | {z } social value of a machine in unit ( ) • FOC of ( ) leads to the constraint in ( ) • FOC of coincides with that of ( ) • if 1 ∈ ( ) does not solve ( ), welfare can be increased 16
Why can the planner’s problem be decomposed so? • The planner chooses machines for each separately; there is no direct interaction between di ff erent • For each , the planner should — maximize the worker’s social marginal value, which is the objective function in ( ) — create as many machines in each unit ( ) as to equate: social marginal value of a machine = the cost; (this is the constraint in ( )) 17
B : w o rk er/m ach in e ratio zero n et p ro fit Z N P (k ) d irectio n o f h ig h er so cial m arg in al n et v alu e o f a m ach in e IN D (k ) in d ifferen ce cu rv e b o E d irectio n o f h ig h er so cial m arg in al v alu e o f a w o rk er Ф ο m ach in e q u ality k E ffi cient allocation 18
E ffi cient allocation: solution • Assignment is distinct: ( 1 ) ∩ ( 2 ) = ∅ if 1 6 = 2 — suppose 1 and 2 are both assigned to , with 2 1 . Let = ( ) and = ( ). Then, − 1 1 = − 2 2 = ⇒ 2 1 | {z } social value of 1 and 2 — contradiction: net value of using skill 2 is higher: h i h i 1 − (1 + 2 ) − 2 1 − (1 + 1 ) − 1 2 − ( ) 1 − ( ) • assignment is one-to-one: ( ) is unique for each if ( − ) ( − ) 2 19
E ffi cient allocation: solution (continued) • e ffi cient choice of for (where ( ) = ( ( ) )): h 1 − − ( ) i ( ( ) ) ( ( )) = | {z } | {z } expected marginal product of marginal cost recall: frictionless assignment ( ( ) ) = ( ( )) = ⇒ ( ) ( ) • e ffi cient choice of for : h 1 − (1 + ( )) − ( ) i ( ( ) ) = ( ( )) | {z } social value of a machine ( ) 20
E ffi cient allocation: solution (continued) Write these conditions more explicitly: ∙ ¸ 1 − ( ( )) ( ) = − ln ( ( ) ) ( ( )) ∙ ¸ ( ( ) ) ( ( ) ) − ( ( )) 1 − ( ( )) ln = ( ( ) ) ( ( ) ) − ( ( )) 21
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