Firm Dynamics in the Neoclassical Growth Model

Aims The Economy Stationary Equilibrium Technical Progress Summary

Firm Dynamics in the Neoclassical Growth Model

Omar LicandroUniversity of Nottingham

2015 RIDGE December Forum

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Ramsey-Hopenhayn Model

• A continuous time optimal growth model with heterogeneousfirms based on Hopenhayn (1992)

An extension of the Neoclassical growth model

• Firms’ productivity is heterogeneous; the dynamics of firms isgoverned by firms’ entry and exit

Selection tends to eliminate low productive firms andreallocate resources towards high productive firms

• As a consequence, more selective economies are moreproductive, produce more and have larger welfare

• The analysis is restricted to steady state

• We study an economy with zero growth (easy to generalize)

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The model

Competitive dynamic general equilibrium model with entry and exit

• There is only one good

• A continuum of heterogeneous, competitive firms

• Firms technology employs capital (fixed factor) and labor

• Free entry: the value of entry is equal to the investment cost

• The initial productivity of entrants is random

• When productivity is too small, firms optimally exit

• Capital is partially irreversible: its scrap value is θ ∈ (0, 1)

More efficient economies have larger θ

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The model










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Preferences

• A mass of measure one of identical households

• A representative household offers inelastically one unit of labor

• The Euler equation is

ctct

= σt(rt − ρ)

ρ > 0 is the discount rate

σt > 0 is the intertemporal elasticity of substitution

• At steady state rt = ρ

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Preferences




ctct

= σt(rt − ρ)




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Preferences




ctct

= σt(rt − ρ)




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Preferences




ctct

= σt(rt − ρ)




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Preferences




ctct

= σt(rt − ρ)




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Technology

• A firm is characterized by a firm-specific productivity z

• It requires one unit of capital to produce

• The technology of a firm with productivity z is

x = A zα`1−α

A > 0 and α ∈ (0, 1)

• At equilibrium, output y = y(z) and employment ` = `(z)

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Technology




x = A zα`1−α

A > 0 and α ∈ (0, 1)


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Technology




x = A zα`1−α

A > 0 and α ∈ (0, 1)


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Technology




x = A zα`1−α

A > 0 and α ∈ (0, 1)


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Technology




x = A zα`1−α

A > 0 and α ∈ (0, 1)


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Entry and Exit

• Endogenous entry and exit

• Entering firms buy (one unit of) capital before entering

• Then, draw productivity z from density ϕ(z)

Expected z at entry is one (without any lost of generality)

• If z is too small, the firm immediately exit and recovers θ < 1

• Notation: the marginal firm has productivity z∗

z∗ is endogenously determined at equilibrium

• Exogenous exit

• Incumbent firms exogenously exit at the rate δ > 0In this case, capital fully depreciates

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Entry and Exit








• Exogenous exit


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Entry and Exit








• Exogenous exit


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Entry and Exit








• Exogenous exit


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Entry and Exit








• Exogenous exit


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Entry and Exit








• Exogenous exit


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Entry and Exit








• Exogenous exit


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Equilibrium Distribution

Since firms with productivity z < z∗ immediately exit

the equilibrium distribution is the truncated entry distribution

φ(z) =ϕ(z)

1− Φ(z∗)

Φ(z) is the cumulative entry distribution, s.t., ϕ(z) = Φ′(z)

Average productivity:

z =

∫ ∞z∗

z φ(z) d(z) > 1

Selection makes average productivity larger than expected productivity at

entry

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φ(z) =ϕ(z)

1− Φ(z∗)



z =

∫ ∞z∗

z φ(z) d(z) > 1


entry

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φ(z) =ϕ(z)

1− Φ(z∗)



z =

∫ ∞z∗

z φ(z) d(z) > 1


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Firms’ Problem

• Both goods and labor markets are competitive

• A firm with productivity z ≥ z∗ solves

max`

A zα`1−α − w`

given the wage rate w

• The optimal labor demand is

`(z) =

((1− α)A

w

) 1α

z

decreasing in wages and increasing in firm’s productivity

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Firms’ Problem



max`

A zα`1−α − w`



`(z) =

((1− α)A

w

) 1α

z


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Firms’ Problem



max`

A zα`1−α − w`



`(z) =

((1− α)A

w

) 1α

z


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Firms’ Problem



max`

A zα`1−α − w`



`(z) =

((1− α)A

w

) 1α

z


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Labor market equilibrium

• Equilibrium in the labor market

k

∫ ∞z∗

`(z) φ(z) dz = 1

k is the total number of firms, equal to aggregate capital per capita

• Labor market clearing implies

w =(1− α

)A(zk)α︸︷︷︸

marg. prod. of labor

Selection raises productivity and wages

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k

∫ ∞z∗

`(z) φ(z) dz = 1



w =(1− α

)A(zk)α︸︷︷︸



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k

∫ ∞z∗

`(z) φ(z) dz = 1



w =(1− α

)A(zk)α︸︷︷︸



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k

∫ ∞z∗

`(z) φ(z) dz = 1



w =(1− α

)A(zk)α︸︷︷︸



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Aggregate Output

• Aggregate output per capita:

y = k

∫ ∞z∗

A zα`(z)1−α︸︷︷︸y(z)

φ(z) dz (1)

• Aggregate technology is Neoclassical at equilibrium

y = A (zk)α

Selection makes the economy more efficient in transforming inputs

into output

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Aggregate Output


y = k

∫ ∞z∗

A zα`(z)1−α︸︷︷︸y(z)

φ(z) dz (1)


y = A (zk)α


into output

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Aggregate Output


y = k

∫ ∞z∗

A zα`(z)1−α︸︷︷︸y(z)

φ(z) dz (1)


y = A (zk)α


into output

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Aggregate Output


y = k

∫ ∞z∗

A zα`(z)1−α︸︷︷︸y(z)

φ(z) dz (1)


y = A (zk)α


into output

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Aggregate Output: General Case

• Assumex = F (z , `)

where F (.) is a Neoclassic production function

• Aggregate output per capita is

y = F (zk, 1) = f (zk)

Aggregate technology is Neoclassic

Capital is measured in efficiency units

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y = F (zk, 1) = f (zk)



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y = F (zk, 1) = f (zk)



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y = F (zk, 1) = f (zk)



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Firm’s Value

• Equilibrium profits are linear on z/z

π(z) =(αA zαkα−1

)︸︷︷︸

marg. prod. of capital

z

z

Profits are the return to capital

• Firms’ value function at steady state is

v(z) =

π(z)/(ρ+ δ) if z ≥ z∗

θ otherwise

v(z) is monotonically increasing on z

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Firm’s Value



)︸︷︷︸


z

z



v(z) =

π(z)/(ρ+ δ) if z ≥ z∗

θ otherwise


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Firm’s Value



)︸︷︷︸


z

z



v(z) =

π(z)/(ρ+ δ) if z ≥ z∗

θ otherwise


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Equilibrium Exit

• The exit condition determines the threshold z∗

π(z∗) = αA (zk)α−1z∗ = θ(ρ+ δ)

• A firm exits when z < z∗

When the value of staying in the market is smaller than the

scrap value of capital θ

• Notice thatz∗

θz=

ρ+ δ

αA zαkα−1(EC)

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Equilibrium Exit


π(z∗) = αA (zk)α−1z∗ = θ(ρ+ δ)




• Notice thatz∗

θz=

ρ+ δ

αA zαkα−1(EC)

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Equilibrium Exit


π(z∗) = αA (zk)α−1z∗ = θ(ρ+ δ)




• Notice thatz∗

θz=

ρ+ δ

αA zαkα−1(EC)

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Equilibrium Exit


π(z∗) = αA (zk)α−1z∗ = θ(ρ+ δ)




• Notice thatz∗

θz=

ρ+ δ

αA zαkα−1(EC)

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Equilibrium Entry

• The expected value at entry is equal to the investment cost

Φ(z∗) θ +(

1− Φ(z∗))π(z)/(ρ+ δ) = 1

• Free entry condition becomes

1− Φ(z∗)

1− θΦ(z∗)=

ρ+ δ

αA zαkα−1(FE)

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Equilibrium Entry


Φ(z∗) θ +(

1− Φ(z∗))π(z)/(ρ+ δ) = 1


1− Φ(z∗)

1− θΦ(z∗)=

ρ+ δ

αA zαkα−1(FE)

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Equilibrium Entry


Φ(z∗) θ +(

1− Φ(z∗))π(z)/(ρ+ δ) = 1


1− Φ(z∗)

1− θΦ(z∗)=

ρ+ δ

αA zαkα−1(FE)

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Equilibrium Cutoff

• Combining the exit (EC) and free entry (FE) conditions

z∗

θ=

1− Φ(z∗)

1− θΦ(z∗)z ≡ A(z∗)

Remind that z is increasing in z∗

z =1

1− Φ(z∗)

∫ ∞z∗

zϕ(z)dz

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Equilibrium Cutoff

• Combining the exit (EC) and free entry (FE) conditions

z∗

θ=

1− Φ(z∗)

1− θΦ(z∗)z ≡ A(z∗)

Remind that z is increasing in z∗

z =1

1− Φ(z∗)

∫ ∞z∗

zϕ(z)dz

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Equilibrium Cutoff

• Proposition 1: z∗ exists and is unique

• Corollary: z∗ ≥ θ

It is always better to close down when productivity is smaller than

the scrap value

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Equilibrium Cutoff




the scrap value

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Equilibrium Cutoff




the scrap value

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Existence and Unicity

0.2 0.4 0.6 0.8 1.0 1.2 1.4z*

0.5

1.0

1.5

2.0

AHxL

x�Θ

ë

z*

ë

Figure: Existence and unicity of z∗

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Selection and Capital Irreversibility

• Proposition 2:dz∗

dθ> 1

• More reversible capital is, more the economy is selective

• Efficiency

• A reduction in the degree of irreversibility (θ)

• Generates productivity gains through selection (z)

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dθ> 1


• Efficiency



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dθ> 1


• Efficiency



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dθ> 1


• Efficiency



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Selection and Productivity Dispertion


dσ> 0 (σ = variance of the entry

distribution)

Larger the variance is, more likely is to get a high productivity

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Selection and Productivity Dispertion


dσ> 0 (σ = variance of the entry

distribution)

Larger the variance is, more likely is to get a high productivity

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Neoclassical Model A

• The Neoclassical Model corresponds to the extreme case of adegenerate entry distribution with zero variance

There is a representative firm with productivity z = z = 1

• Per capita output is y = Akα

• Wages w = (1− α)Akα

• Profits are π = αAkα−1

The free entry condition makes ρ+ δ = αAkα−1

• Irreversibility plays no role: (FE) ⇒ v(1) > θ

The representative firm never exits

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Neoclassical Model B

• Let us assume the support of the productivity distribution isbounded above by zmax

• The Neoclassical model is the limit case of θ = 1

• Selection will make z∗ = z = zmax

• Per capita output y = A(zmaxk

)α• Wages w = (1− α)A

(zmaxk

)α• The exit condition makes ρ+ δ = αA

(zmaxk

)α−1

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(zmaxk


(zmaxk

)α−1

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(zmaxk


(zmaxk

)α−1

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(zmaxk


(zmaxk

)α−1

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)α

• Wages w = (1− α)A(zmaxk


(zmaxk

)α−1

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(zmaxk

)α

• The exit condition makes ρ+ δ = αA(zmaxk

)α−1

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(zmaxk


(zmaxk

)α−1

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Equilibrium Capital per capita

From the entry condition (EC)

zk =

(αA

ρ+ δ

) 11−α

︸︷︷︸Neoclassical Model A

(z∗

θ

) 11−α

︸︷︷︸selection

<

(αA zmax

ρ+ δ

) 11−α

︸︷︷︸Neoclassical Model B

• Since dz∗/dθ > 1

More selective economies cumulate more capital

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zk =

(αA

ρ+ δ

) 11−α


(z∗

θ

) 11−α


<

(αA zmax

ρ+ δ

) 11−α




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zk =

(αA

ρ+ δ

) 11−α


(z∗

θ

) 11−α


<

(αA zmax

ρ+ δ

) 11−α




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Output

• Output per capita is

y = (zk)α =

(αA

ρ+ δ

) α1−α

︸︷︷︸Neoclassical model A

(z∗

θ

) α1−α


More selective economies produce more

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Output

• Output per capita is

y = (zk)α =

(αA

ρ+ δ

) α1−α

︸︷︷︸Neoclassical model A

(z∗

θ

) α1−α


More selective economies produce more

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Consumption and Welfare

• Selection raises per capital consumption and welfare

Consumption per capita is

c = y − i =

((αA

ρ+ δ

) α1−α

− δ(αA

ρ+ δ

) 11−α

)︸︷︷︸

Neoclassical model A

(z∗

θ

) α1−α

Selection is welfare improving

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Consumption and Welfare

• Selection raises per capital consumption and welfare

Consumption per capita is

c = y − i =

((αA

ρ+ δ

) α1−α

− δ(αA

ρ+ δ

) 11−α

)︸︷︷︸

Neoclassical model A

(z∗

θ

) α1−α

Selection is welfare improving

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Technical Progress

The model can be extended to the case of exogenous growth

Assume the productivity of incumbent firms grow at the exogenous rate

γ > 0 and the entry distribution Φt(z) has expected productivity at entry

equal to eγt

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Summary

• The Ramsey-Hopenhayn model has the Neoclassical model asa limit case

• Idiosyncratic uncertainty increases the probability of highproductivity events to occur, raising selection

• Capital reversibility is good, raising selection too

• More selective economies are more productivity, produce moreoutput and welfare

• More selective economies reallocate labor from less to moreproductive firms through a higher wage rate

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Documents

Firm Dynamics in the Neoclassical Growth Model