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 04 245 05 15 5 415 1 3
BENCHMARK
.60 .10 .17 ( ) = 13 23 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
