A Model of TFP

Ricardo Lagos Minneapolis Fed and NYU

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1 Motivation • Differences in TFP levels account for a large fraction of the variation in output per worker across countries Hall and Jones (1999), Parente and Prescott (2000)

• QUESTION: What determines the level of TFP? • USUAL ANSWER: “Institutions” • – Hall and Jones (1999): “social infrastructure” (law and order, bureaucratic quality, corruption, risk of expropriation, government repudiation of contracts, openness to trade) – Parente and Prescott (1994, 1999, 2000): “barriers to riches” (any institution or government policy that increases the cost of technology adoption; e.g. monopoly rights as modelled in Parente and Prescott, 1999)

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• This paper: add the LABOR-MARKET to the list of “Institutions” that determine the level of TFP Focus on the mix and magnitude of: – unemployment benefits – employment subsidies – hiring subsidies – firing taxes

• Basic idea: Work on the theory underlying the aggregate production function and show how policy affects this relationship

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• Specifically: – Build on Mortensen and Pissarides (1994) and Houthakker (1955-1956) to derive a relationship between aggregate inputs and output by aggregating across production units – Through aggregation: productivity distribution of active firms ⇒ TFP – And through the equilibrium: policy ⇒ productivity distribution of active firms

• Additional result (extension of Houthakker, 1955-1956): From the perspective of aggregate output, inputs and productivity, the “standard” search model (with its meeting frictions and simple fixed-proportions micro-level production technologies) can look just like the “standard” neoclassical model

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2 Environment As in Mortensen and Pissarides (1994)

• Infinite horizon, continuous time (will focus on steady states) • Large labor force (size 1)

• Large population of firms (endogenous number) • All agents risk-neutral and infinitely-lived • One-to-one matching

• States – Firms: vacant (and searching) or filled – Workers: unemployed (and searching) or employed

• No on-the-job search

• Meeting frictions summarized by a “nice”  ( ) • Notation:  = 

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3 Firm-level technology  (  ) =  min ( ) • Notation:  : match-specific productivity shock  : hours supplied by the matched worker to her employer  : capital needed for firms to search and produce • All projects have the same “scale” 

• Rental rate of capital, , given • Hours chosen by the match

• Innovations to the idiosyncratic productivity  follow a Poisson process with rate   ∞

• When match of productivity  suffers a change, the new value 0 is a draw from a fixed density  (0|) •  (0|) is (weakly) stochastically increasing in 

• Exogenous terminations: Poisson process with rate   ∞

• Poisson processes and draws are independent across matches

Observation: The stochastic productivity process induces a cross-sectional distribution of productivities over active matches Result: there is a productivity level  such that existing matches dissolve (and new matches don’t form) if   

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4 Flows  () : fraction of matches with productivity  or lower (1 − )  () : number of matches with productivity  or lower Z ∞  [(1 − )  ()] [ (|) −  (|)]  () =  (1 − )  Z + () 

− (1 − ) − (1 − )



[ (|) −  (|)]  ()

−∞ Z 

Z−∞  −∞

[1 −  (|)]  ()  (|)  ()

− (1 − )  () Z

˙ =  (1 − ) +  (1 − )  (|)  () Z − ()  [1 −  (|)]  ()

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5 Steady States R

 +   (|)  () R =  +   (|)  () +  () [1 −  (|)]  () R

and for  ≥  :

 () =

R

[ (|) −  (|)]  () R [1 −  (|)]  ()

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6 Value Functions (I) (case with uncorrelated shocks:  (|) =  () for all )

 =  +  ()

Z

max [ () −  0]  () Z  = − +  () max [ () −  0]  ()  () =  () Z −  [ () − ] + max [ () −  ()   −  ()]  ()  () =  () Z −  [ () −  ] + max [ () −  ()   −  ()]  ()    ()  ()     ()  ()

: : : : : : : : :

value of unemployment value of a vacancy ( = 0 in equilibrium) value of employment at  value of a filled job with productivity  discount rate flow utility while unemployed rental rate of capital flow wage flow profit

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7 Flow Profit (I)  () =  min ( ) −  −  −  ( )  −  ()  : variable cost  ( ) : fixed cost

Bargaining ⇒  () =

½

 if    0 if  ≤ 

⇒  () = [max ( −  0) −  −  ( )]  −  () Special case:



 ( ) = max ( −  0)  () = ( −  − )  −  ()

why introduce  ? why introduce  ?

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8 Bargaining max [ () − ] [ () −  ]1− 

hours are chosen efficiently: ½  if     () = 0 if  ≤  wage solves:

 [ () −  ] = (1 − ) [ () − ] ⇔  () −  =  ()  () −  = (1 − )  () where  () ≡  () +  () −  − 



 () =  ( −  − )  + (1 − )   () = (1 − ) [( −  − )  − ]  =  +

  1−

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9 Separations (I) Nash solution and value functions imply

( +  + )  () = ( −  − ) −+  0 () =

Z

max [ ()  0]  ()

  0 ⇒ ∃! s.th.  () ≥ 0 iff  ≥  ++

⇒ ( +  + )  () = ( −  − )  −  + 

Z

 ()  ()



evaluating at  =  ⇒



Z



 ()  () =  − ( −  − ) 

⇒ −  () =  ++

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10 Equilibrium (I) An equilibrium is a list [      ] satisfying

−−−

  +  ++

Z

Z



( − )  () = 0

(JD)

( +  + )  ( − )  () = (1 − )  ()   () =

 () −  () 1 −  ()

 =  +

(JC) (SSH)

  1−

 +  () =  +  () +  () [1 −  ()]  () =  ( −  − )  + (1 − ) 

(U) (SSU) (wage)

 = [1 − (1 − ) ]  Note: in equilibrium we could have    or   

(MC)

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11 Value Functions (II) (case with correlated shocks:  (|)   (|0) if   0)

 =  +  ()

Z Z

max [ () −  0]  (|)  () Z Z  = − +  () max [ () −  0]  (|)  ()  () =  () Z −  [ () − ] + max [ () −  ()   () − ]  (|)  () =  () Z −  [ () −  ] + max [ () −  ()   () −  ]  (|)

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12 Hours, Profit and Wages (II) max [ () − ] [ () −  ]1− 

 () =

½

 if    0 if  ≤ 

 () =  [max ( −  0)  − ] + (1 − )   () = (1 − ) [max ( −  0)  −  − ]    =  + 1−

13 Separations (II)  () =

R [max(−0)−]−+ max[()0](|) ++

 0 ()  0 all  ⇒ ∃! s.th.  () ≥ 0 iff  ≥ 

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14 Aggregation    

≡ : : =

max ( ) aggregate output aggregate hours worked (1 − )  : capital at all matched firms

 = (1 − )  = (1 − )

Z

Z

 [  ()  ]  ()  ()  ()



Recall:  () =

½

 if    0 if  ≤ 

 z }| {Z  ( ) = (1 − )   () 

= [1 −  ()] Z  (| ≥ )

 = (1 − )  | {z } 



 () = [1 −  ()] 

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 ( ) = [1 −  ()]  (| ≥ )  = [1 −  ()] 

(1) (2)

(II) ⇒  ( ) in (I) ⇒  [  ( )] ≡  ( ) Follow Houthakker: assume primitive “heterogeneity” has a Pareto distribution

 () =

½

⇒  () =

0¡ ¢ if     1 −  if  ≤ 

½

0¡ ¢ if     1 −  if  ≤ 

µ ¶   =  ⇒  ( )    ( ) = 1− −1  ( ) =   1− ≡

 and  ≡ 1 1−

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15 Aggregation with Correlated Shocks  (|) =

(

i if    () if  () ≤  1 − ()  h0

The functional equation for  () has a closed-form solution:

 () = Remarks:

½

0¡ ¢ if     1 −  if  ≤ 

•  (·) is a continuously differentiable function, 0  0 Note: 0 = 0 ⇒  shocks

• Assumptions: – there is an   0 such that  () =  and  () = 0 if    – lim  () = 1 +  →∞

• Note: only equilibria with   1 +  have endogenous destruction • An example:  () = 1 +  − −, for any   0

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16 Effects of LM Policies on TFP    

: : : :

 =  +  ()

employment subsidy hiring subsidy firing tax unemployment benefits

Z

 = − +  ()

max [ () −  0]  ()

Z

max [ () −  +   0]  ()

 () =  () Z +   −  [ () −  ] + max [ () −  ()   −  () −   ]  ()  () =  () Z −  [ () − ] + max [ () −  ()   −  ()]  ()

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with  = 

µ  =  +



   1−

Job-Destruction

µ −−+ +  −   +

¶ Z  −  +  () = 0 1− ++

Job-Creation

Z

 −  () + [1 −  ()] (  −   ) = (1 − )  () ++

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17 Two Additional Aggregation Results 17.1 Model with state-dependent destruction shocks Suppose 0   ()  ∞ and  0  0

 [(1 − )  ()] =  (1 − ) [1 −  ()] [ () −  ()]  + ()  [ () −  ()] − (1 − )  ()  () )  () [1 −  ()] − (1 − Z − (1 − )

Value functions:

 () =  () + 

Z

 ()  ()



max [ () −  0]  ()

− [ () +Z] [ () − ]

 () =  () + 

max [ () −  0]  ()

− [ () + ] [ () −  ]

Surplus:

 () =

R

[max ( −  0) − ]  −  +  max [ ()  0]  ()  +  () + 

Note:  0 ()  0 for all  (even   )

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Proposition 1 Suppose  () = − , and the primitive density of shocks is ( 0 if    ¡ ¢  () = (+)  − 1 + −−1 if  ≤  − (+)+ 

where     0, and   1. Then in equilibrium, the aggregates  ,  and  satisfy

 =   1− with

  = 1−  = 1

23 17.2 CES Aggregate Suppose the primitive distribution of shocks is ⎧ ⎨0 if    h ¡ ¢ i−1  () = ⎩ 1 − 1  1− − 1− if  ≤   with   0 and   ∈ (0 1)

The corresponding steady-state productivity distribution of active matches is ⎧ ⎨0 if    h i−1  ¡  ¢ 1−  () = 1 1− ⎩1−   −  if  ≤  with  ≡ [1 −  ()]−1

Proposition 2 If the primitive distribution of shocks is given by , then in equilibrium, the aggregates  ,  and  satisfy ¤ £   1 ¯  =    + (1 − )  with

" µ ¶ # 1− 1 1−  ¯  = −      = 1−

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18 Measurement  : demand for capital from all matched firms (not all are active)  =  1 − (1 − )  {z } | 

 : hours worked (no missmeasurement)   ;  ≡ 1 = ()1 1 − (1 − )   : number of employed workers = 1−  ( ) =

  1− ;  ∙

 = 1−

¸ 1 −  ˆ   1− ; ˆ = ˆ ( ) =   1 − (1 − )  " #1− 1 ˜   1− ; ˜ = ()  ˆ ˜ ( ) =  1 − (1 − ) 

25 Note: This is NOT a productivity-differences-are-due-to-mismeasurement paper

 : capital at all matched firms  = [1 −  ()]  : capital being used in production 1 1− 



}| { z }| {z  ( ) = [1 −  ()]  (| ≥ )  = [1 −  ()]   ≡ max ( ) • Equilibrium with no hoarding ( ≥ ) ½  =   = ⇒  =    =  with  = 1− • Equilibrium with hoarding (  ) – Some mismeasurement (measure , not )

 =   1− with  = – No mismeasurement (measure )

 =  ⇒  =   =  with  =

 1−

 1−

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19 Concluding Remarks • Presented a model of TFP differences based on the interaction between “institutions” (LM policies) and the microeconomics underlying the aggregate production function • Model suggests that: – Employment subsidies and Firing restrictions ⇒ lower TFP

– Unemployment benefits and Hiring subsidies ⇒ higher TFP

• Future work: use the model as a guide and look again at TFP differences Are LM policies quantitatively important to understand TFP differences among a homogeneous set of countries? How large are the TFP differences implied by reasonable differences in the mix and magnitude of labor market policies?

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Some Preliminary Work  () =

½

0¡ ¢ for     1 −  for  ≥ 

 ( ) =   1− 

         ˆ  5 5 04 04 245 05 15 5 415 1 3

BENCHMARK

    .60 .10 .17  ( ) = 13 23 3  =  2

28 Unemployment Benefits

  % of avg. yearly wages       0 0% .60 .10 .17 1 0 36 26% .42 .15 .11 1.09 .012 22 18% .50 .13 .13 1.06 .006 Firing Taxes

  % of avg. yearly wages        0 0% .60 .10 .17 1 0 83 100% .69 .07 .16 .97 0 Hiring Subsidies

  % of avg. yearly wages       0 0% .60 .10 .17 1 0 .54 50% .59 .12 .17 1.03 .01 Employment Subsidies

  % of avg. yearly wages       0 0% .60 .10 .17 1 0 .05 5% .66 .10 .19 .99 .01

A Model of TFP

of technology adoption; e.g. monopoly rights as modelled ... This paper: add the LABOR-MARKET to the list of .... J (x) : value of a filled job with productivity x.

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