Lecture 2: Firms, Jobs and PolicyEconomics 522
Esteban Rossi-Hansberg
Princeton University
Spring 2014
ERH (Princeton University ) Lecture 2: Firms, Jobs and Policy Spring 2014 1 / 34
Restuccia and Rogerson (2009)
Assess the quantitative role of resource allocation across productive uses indevelopment
Consider a version of neoclassical growth model with heterogeneous producers
Consider distortions to the prices faced by different producers (idiosyncraticdistortions)
I Credit market imperfections and non-competitive banking systemsI Public enterprisesI Trade restrictionsI Labor market regulationsI Corruption and selective government industrial policy
Resource misallocation can decrease aggregate output and TFP in the rangeof 30 to 50 percent
I A theory of measured TFP
ERH (Princeton University ) Lecture 2: Firms, Jobs and Policy Spring 2014 2 / 34
The Model
Infinitely-lived representative household:
∞
∑t=0
βtu(Ct ), 0 < β < 1
Endowments: One unit of productive time each period, K0 > 0 units of thecapital stock, and equal shares of all plants
Budget constraint:
∞
∑t=0
pt (Ct +Kt+1 − (1− δ)Kt ) =∞
∑t=0
pt (rtKt + wtNt + πt − Tt )
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Technology
Production unit —a plant:
f (s, k, n) = skαnγ, 0 < γ+ α < 1
Idiosyncratic productivity s constant over time
Exogenous probability of exit λ
Fixed cost of operation cf every period
Entry cost ce and productivity of entrants from cdf H(s)
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Policy Distortions
They focus on policies that create idiosyncratic distortions to plant-leveldecisions
Each plant faces its own output tax/subsidy denoted by τ ∈ (−1, 1)Entering plants face draws of s and τ
Given cdf H(s), policy distortions induce a joint distribution cdf G (s, τ)
ERH (Princeton University ) Lecture 2: Firms, Jobs and Policy Spring 2014 5 / 34
Incumbent and Entrant’s Problem
Incumbent:I Per-period profit function
π(s , τ) = maxn,k{(1− τ)skαnγ − wn− rk − cf }
I Let k̄(s , τ), n̄(s , τ) denote the optimal decisionsI With constant (s , τ), present value of incumbent plant:
W (s , τ) =π(s , τ)1− ρ
, ρ =1− λ
1+ R= (1− λ) β
Entrant: The expected value of a potential entrant:
We =∫(s ,τ)
max [W (s, τ), 0] dG (s, τ)− ce
I Let x̄(s , τ) be its optimal entry decision
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Invariant Distribution of Plants
Denote µ(s, τ) the distribution of producing plants this period and E themass of entrants
Next period’s distribution:
µ′(s, τ) = (1− λ)µ(s, τ) + x̄(s, τ)dG (s, τ)E
Let µ̂ be the invariant distribution associated with E = 1:
µ̂(s, τ) =x̄(s, τ)
λdG (s, τ)
ERH (Princeton University ) Lecture 2: Firms, Jobs and Policy Spring 2014 7 / 34
Labor Market Clearing
Aggregate labor demand:
N(r ,w) = E∫(s ,τ)
n̄(s, τ)d µ̂(s, τ)
Labor supply inelastic equal to one, entry E satisfies:
E =1∫
(s ,τ) n̄(s, τ)d µ̂(s, τ)
ERH (Princeton University ) Lecture 2: Firms, Jobs and Policy Spring 2014 8 / 34
Equilibrium
A steady state competitive equilibrium with entry is w , r , T , µ(s, τ), E , W (s, τ),π(s, τ), We , x̄(s, τ), k̄(s, τ), n̄(s, τ), C , and K such that:
Consumer optimization r = 1/β− (1− δ)
Plant optimization
Free-entry We = 0
Market clearing: labor, capital, output
Government budget balance
T +∫s ,τ
τf (s, k̄, n̄)dµ(s, τ) = 0
Invariant µ
µ(s, τ) = Ex̄(s, τ)
λdG (s, τ)
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Calibration
Calibrate undistorted benchmark economy to U.S. data
Model period equal to a year
Parameter Value Targetα 0.3 Capital income shareγ 0.6 Labor income shareβ 0.96 Real rate of returnδ 0.08 Investment to output ratioce 1.0 Normalizationcf 0.0 Benchmark caseλ 0.1 Annual exit rate
ERH (Princeton University ) Lecture 2: Firms, Jobs and Policy Spring 2014 10 / 34
Calibration
Key elements: range of s and H(s)
Use mapping from s to n and from H(s) to µ(s) implied by the model
n̄in̄j=
(sisj
) 11−γ−α
µ(s) =x̄(s)
λdH(s)
Number of workers per plant in U.S. Census of Manufactures impliess ∈ [1, 2.43] (given α = 0.3, γ = 0.6, and normalizing lowest s to one)
Micro evidence of TFP suggest range of 1 to 3 across plants within narrowlydefined manufacturing industries
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Distribution of Plants by Employment —U.S. Data
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Distribution of Plants by Employment —Model vs. Data
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Share of Valued Added and Employment
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Distribution Statistics of Benchmark Economy
Plant Size by Employment< 10 10 to 499 500 or more
Share of plants 0.51 0.47 0.02Share of output 0.04 0.57 0.39Share of labor 0.04 0.57 0.39Share of capital 0.04 0.57 0.39Average employment 4.2 64.8 1042.0
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Aggregate Distortions
An output tax of 0.5 implies relative steady state output(distorted/undistorted) of 0.63
In standard growth model (capital share half the labor share), same tax policyimplies relative steady state output of 0.50.5 = 0.7
Output effect 10 percent larger in model with plant heterogeneity than instandard growth model: accounted for by a fall in measured aggregate TFP
Plant heterogeneity allows another form of aggregate distortions that isempirically relevant: entry cost
I An increase in the cost of entry ce due to government regulation of 50 percentimplies a drop in aggregate measured TFP of 10 percent
I Distortion to the entry cost leave the capital to output ratio unaltered
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Idiosyncratic Distortions: Tax/Subsidy Policies
Assume a fraction of plants are taxed and the rest are subsidized
Output tax/subsidy combinations:I Tax packages of 0.1, 0.2, 0.3, 0.4, with subsidies so that the net effect onsteady state capital accumulation is zero
I Lump-sum redistribution to consumers to balance the government budget
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Ind. Idiosyncratic Distortions: Tax/Subsidy Policies
τt0.10 0.20 0.30 0.40
Relative Y 0.98 0.95 0.94 0.94Relative TFP 0.98 0.95 0.94 0.94Relative E 1.00 1.00 1.00 1.00Ys/Y 0.80 0.93 0.98 0.99S/Y 0.04 0.06 0.07 0.07τs 0.05 0.07 0.07 0.07
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Corr. Idiosyncratic Distortions: Tax/Subsidy Policies
τt0.1 0.2 0.3 0.4
Relative Y 0.87 0.78 0.73 0.72Relative TFP 0.87 0.78 0.73 0.72Relative E 1.00 1.00 1.00 1.00Ys/Y 0.57 0.83 0.95 0.99S/Y 0.20 0.32 0.38 0.40τs 0.35 0.39 0.40 0.40
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Sensitivity
Sensitivity of results:I Decreasing returns at the plant level (1− α− γ)I Fixed cost of operation (cf > 0): potential selection of entering plantsI Plant dynamics: potential selection of exiting plants
Capital and human accumulation can amplify these differences
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Hopenhayn and Rogerson (1993)
Examine the qualitative and quantitative impact of government policies thatmake it costly for firms to adjust their employment levels
Large volume of job creation and destruction at the level of the individual firm
Important to understand the effects of labor market regulation
Extend Hopenhayn (1992) to a general equilibrium setting
Finding: A tax equal to 1 year’s wages reduces utility by over 2 percentmeasured in terms of consumption
I Important reduction in average labor productivity
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The Model
Labor is the only input
Profits are given by
pt f (nt , st )− nt − ptcf − g(nt , nt−1)
where nt denotes employment, wages are normalize to one
st is a firm specific productivity shock, that evolves according to transitionprobabilities F (s, s ′)
I F (s , ·) is the distribution function for next period’s value of the shockI Shock is independent across firms
cf is a fixed operating cost
g captures the presence of adjustment costsI Policy experiments can be represented as changes in gI Firing cost of τ would imply
g (nt , nt−1) = τmax(0, nt−1 − nt )
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Entry and Preferences
Entry cost ceInitial draw s from distribution ν, iid
Continuum of agents uniformly distributed in unit interval with preferences
∞
∑t=1
βt [u (ct )− v (nt )]
where ct > 0 and nt ∈ {0, 1} denote consumption and labor supplyAs in Rogerson (1988) individuals use lotteries and diversify idiosyncratic riskso economy behaves as
∞
∑t=1
βt [u (ct )− aNt ]
where Nt is the fraction of individuals who are employed in period t
ERH (Princeton University ) Lecture 2: Firms, Jobs and Policy Spring 2014 24 / 34
Equilibrium
Look at stationary equilibrium so assume constant p
Bellman equation is
W (s, n; p) = maxn′≥0{pf (n′, s)− n′ − pcf − g(n′, n)
+βmax[EsW (s ′, n′; p),−g(0, n′)]}
Optimal decisions N(s, n; p) and X (s, n; p) with X = 1 corresponding to exitand X = 0 stay
Value of entering is
We (p) =∫W (s, 0; p)dv(s).
µ(s, n) denotes the mass of firms with s and n, and M the mass of entrants
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AggregatesTotal output is given by
Y (µ,M; p) =∫[f (N(s, n; p), s)− cf ]dµ(s, n) +M
∫f (N(s, 0; p), s)dv(s)
Individual adjustment costs
r(s, n; p) = [1−X (s, n; p)]∫g(N(s ′, n′; p), n′)dF (s, s ′)+X (s, n; p)g(0, n′)
integrate to get R(µ,M, p) the aggregate adjustment costs
Labor demand
Ld (µ,M; p) =∫N(s, n; p)dµ(s, n) +M
∫N(s, 0; p)dv(s)
Profits
Π(µ,M; p) = pY (µ,M; p)− Ld (µ,M; p)− R(µ,M; p)−Mpce
All homogenous of degree one in µ and M
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Aggregates
In SS with 1/(1+ r) = β consumer problem is
max u(c)− aNs.t. pc ≤ N +Π+ R
This implies N = LS (p,Π+ R)
A stationary equilibrium consists of an output price p∗ ≥ 0, a mass ofentrants M∗ ≥ 0, and a measure of incumbents ,µ∗, such that
I Ld (µ∗,M∗, p∗) = LS (Π(µ∗,M∗, p∗) + R(µ∗,M∗, p∗))I T (µ∗,M∗, p∗) = µ∗I W e (p∗) ≤ p∗ce with equality if M∗ > 0
Need strict concavity and INADA on f , LS2 < 0, F is continuous F1 < 0 andν has a continuous cdf
ERH (Princeton University ) Lecture 2: Firms, Jobs and Policy Spring 2014 27 / 34
Quantitative specification
Benchmark model:
f (n, s) = snθ, 0 ≤ θ ≤ 1g(nt , nt−1) = 0
log(st ) = a+ ρlog(st−1) + εt , where εt ∼ N(0, σ2ε )u (c) = ln (c) and v(n) = An
Persistence ρ < 1 important since it determines how much firms care aboutfiring costs
s exhibits mean reversion
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Optimal Rules
For this case n not a state variable so
log nt =1
1− θ(log θ + log p + log st )
X (st , nt , p) = 1 if st ≤ s∗ for some s∗
So employment of surviving firm evolves according to
log nt+1 =1− ρ
1− θ(log θ + log p +
a1− ρ
) + ρ log nt−1 +1
1− θεt (1)
ERH (Princeton University ) Lecture 2: Firms, Jobs and Policy Spring 2014 29 / 34
Calibration
Length of period is set to 5 years
Let β = .8 and θ,labor’s share of total revenue, is set to .64
Use (1) to estimate ρ and σε
Fix price p = 1I Choose cf and a to match average of log employment and exit rateI Choose ν to match size distribution in dataI Choose cε so that p = 1
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DistortionsIntroduce distortions such that
g(nt , nt−1) = τmax(0, nt−1 − nt )
If τ = .2 tax of 1 year of wages
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