A Growth Model with Developing Capital Markets: Can Firm Productivity Heterogeneity Lower the Speed of Convergence?

A Growth Model with Developing Capital Markets: Can Firm Productivity

Heterogeneity Lower the Speed of Convergence? ^∗

Quoc Hung Nguyen ^† This version: February, 2017

Abstract

This paper addresses the effects of firm productivity heterogeneity on the aggregate dynamics of economies with developing capital mar- kets. It first constructs a tractable growth model with steady state equilibrium and endogenous aggregate productivity and then analyt- ically shows that firm productivity heterogeneity can lower the speed of convergence. The paper also discusses the conditions under which the model can have a balanced-growth equilibrium.

JEL Classification: E22, E32, F34, F41

Keywords: Heterogeneous Agent; Externalities; Financial Frictions

∗

This is a preliminary and incomplete draft.

†

School of Economics, University of Hyogo. Address: 8-2-1 Gakuen-Nshimachi, Nishi- ku, Kobe, Hyogo 651-2197 Japan. Tel.: +81-78-794-5442; Fax: +81-78-794-6166. Email:

[email protected]

1 Introduction

This paper investigates the impact of firm productivity heterogeneity on the aggregate dynamics, particularly the speed of convergence in an econ- omy with developing capital markets. It is first motivated by the observa- tion that developing economies usually have underdeveloped capital markets with a significant degree of financial frictions but the quality of these markets tends to improve along with the overall economic development. Second, the neoclassical growth models with exogenous aggregate productivity are un- able to offer an satisfactory explanation to the transition dynamics of many developing economies. The convergence speed implied by calibrated neoclas- sical growth models is about three times faster than those of economies with miracle growth like Japan, Korea, etc.

To that end, this paper focuses on the interaction between entrepreneur- ship and financial frictions. In particular, the quality of financial markets by impacting on each individual firm’s capital accumulation and operation activities affects the aggregates such as GDP and economy wide capital stock and their transition dynamics. Moreover, since each entrepreneur is endowed with heterogeneous productivity and faces a borrowing constraint, the chan- nel via that the financial markets allocate capital resources will influence the aggregate productivity. Consequently, this will determine how the economy grows and transitions.

This paper is organized as follows. Section 2 setups the model economy and then describes the equilibrium. Section 3 derives the aggregate equilib- rium dynamics and demonstrates how to construct a tractable growth model with steady state equilibrium and endogenous TFP and then analytically shows that productivity heterogeneity can lower the speed of convergence.

Section 4 concludes.

Related Literature Review.

2 The Model

I extend the model of Moll (2014) by introducing Arrow-Romer type knowl- edge externalities and endogenous quality of capital markets. As in Moll (2014), time is continuous and there is a continuum of entrepreneurs and a mass L of hand-to-mouth workers. Each entrepreneur owns a private firm and entrepreneurs are indexed by their productivity, z, and their personal wealth, a. Consequently, at each time, t, the state of the economy state is characterized by the joint cumulative distribution, Φ(a, z).

2.1 The Economic Setup

Entrepreneurs have preferences:

E

0

Z

∞ 0

e

^−ρt

logc(t)dt (2.1)

Each entrepreneur possesses a firm that produces homogenous goods used for both consumption and capital accumulation. At each time, t, this firm employs l units of labor from workers and rents k units of capital subject to a financial constraint discussed later to produce y unit of final goods with the following production technology:

y = f (z, K, k, l) = (zk)

^α

(K

^β

l)

^1−α

(2.2) where α ∈ (0, 1], β ∈ [0, 1] and z denotes idiosyncratic productivity and can be interpreted as individual entrepreneurial ability or efficiency level of capital. Following Romer (1986), the labor efficiency units are defined as the product of labor and the economy-wide level of capital stock, K, which each individual producer takes as given but in equilibrium determined endogenously from the aggregation of all private capital stock.

A mass L of hand-to-mouth workers also have the same preferences as

(2.1). Each of worker is endowed with one unit of labor and he supplies

inelastically at a competitive labor market at a wage w(t). At each point of time, t, entrepreneurs hire l units of labor at the wage rate, w(t) and rent capital from a competitive capital market at the rental rate, r(t) + δ, where r(t) is the interest rate and δ is the depreciation rate. Therefore, an entrepreneurs wealth a(t) evolves as follows:

¹

˙

a = f(z, K, k, l) − wl − (r + δ)k + ra − c (2.3) Entrepreneurs face the following borrowing constraint:

k

a ≤ λ(K), λ(K) ≥ 1; dλ(K)

dK ≥ 0 (2.4)

where λ(K) ≥ 1 is the endogenous maximum borrowing leverage ratio, which captures the degree of financial market development.

²

When λ(K ) = 1, we have financial autarky where entrepreneurs are completely capital self- financed while λ(K ) = ∞ implies a perfect financial markets where en- trepreneurs can borrow freely regardless of their wealth level.

The borrowing constraint, (2.4), which is equivalently to k ≤ λ(K)a states that at each time the amount of capital available to an entrepreneur is limited both his personal assets but also the degree of financial market development characterized by the maximum borrowing leverage ratio λ(K). The signifi- cant difference between this paper and others in the literature such as Buera and Shin (2013) and Moll (2014) is that the maximum borrowing leverage ratio, instead of being assume as exogenous, is endogenized as an increasing function of economy wide level of capital stock , a proxy for the degree of aggregate economic development. Alternatively, the endogenous maximum leverage ratio could be motivated as resulting from positive externalities and

1

I follow this rental capital market setting as in Moll (2014) for simplicity. Moll (2014) show in the Appendix C that it is equivalent to a setup where entrepreneurs own and accumulate capital and are allowed to trade a risk-free bond.

2

Buera and Shin (2013) and Moll (2014) for further discussions of this borrowing con-

straint.

spillover of the economic development to the quality and efficiency of the capital markets and institutions.

Each entrepreneur maximizes the profit of his private firm subject to the technology (2.2) and his borrowing constraint (2.4), hence the profit of an entrepreneur can be expressed as follows:

Π(a, z) = max

k,l

{f (z, K, k, l) − wl − (r + δ)k}

s.t. k ≤ λ(K)a

(2.5) The first order condition with respect to labor reads:

(1 − α)(zk)

^α

K

^β(1−α)

l

^−α

= w (2.6) The implied labor demand then is:

l

^∗

=

1 − α w

¹_α

K

^β(1−α)α

(zk) (2.7)

Substitute the optimal labor demand into (2.2) we then obtain the pro- duction function that is linear in capital as follows:

F (z, K, k) =

1 − α w

^1−α_α

K

^β(1−α)α

(zk) (2.8)

Consequently, an entrepreneur’s profits can be re-written as follows:

Π(a, z) = max

k

{F (z, K, k) − wl

^∗

− (r + δ)k}

s.t. k ≤ λ(K)a

(2.9) Equivalently:

Π(a, z) = max

k

n

πK

^β(1−α)α

(zk) − (r + δ)k o s.t. k ≤ λ(K)a

(2.10) where

π ≡ α

1 − α w

^1−α_α

(2.11)

As a result, for a given level of aggregate capital stock, K, individual entrepreneurs’ factor demands and profits are linear in personal wealth, and there is a productivity cutoff for active entrepreneurs z as follows:

k(a, z, K) =

( λ(K)a for z ≥ z

0 for z<z (2.12)

l

^∗

(a, z, K) =

1 − α w

¹_α

K

^β(1−α)α

(zk) (2.13)

Π(a, z, K) = max n

zπK

^β(1−α)α

− (r + δ), 0 o

λ(K )a (2.14)

zπK

^β(1−α)α

= (r + δ) (2.15)

Entrepreneurs also maximize the expected sum of discounted utilities from consumption (2.1) subject to the budget constraint (2.3), which given the above results, can now be rewritten as:

˙ a = h

λ(K) max n

zπK

^β(1−α)α

− (r + δ), 0 o + r i

a − c (2.16) which states that the law of an entrepreneur’ personal wealth is also linear in wealth.

Let V (a, z, t) be the value function of this optimality problem, then the Hamilton-Jacobi-Bellman equation is set as follows

³

ρV (a, z, t) = max

c(t)

log(c(t)) + 1

dt E

^a,z

[dV (a, z, t)] s.t. (2.16)

(2.17) This linearity property, together with the log utility (2.1) for the entrepreneur implies the optimal consumption rule such that c(t) = ρa(t).

⁴

Consequently,

3

See the Appendix for the derivation of this Hamilton-Jacobi-Bellman equation.

4

This optimal consumption rule can be confirmed by the guess and verify strategy.

First guess that the value function takes the form V (a, z, t) = B [v(z, t) + loga], where B

is an undermined constant. Substitute this form back to the Hamilton-Jacobi-Bellman

equation (2.17) and take first order condition to obtain c = a/B. Substitute back in and

then apply the envelop theorem to obtain B = 1/ρ. See the Appendix A2 in Moll (2014)

for detailed derivation.

we obtain the following optimal saving policy function that is also linear in wealth:

˙

a =s(z, K)a, where : (2.18)

s(z, K) =λ(K) max n

zπK

^β(1−α)α

− (r + δ), 0 o

+ r − ρ (2.19) is the optimal saving rate of productivity type z given the economy-wide level of capital stock K .

2.2 The Market Equilibrium

An equilibrium in this model economy, for a given initial aggregate capital stock K(0) > 0, is consequences of factor prices r(t), w(t) and corresponding quantities, such that (i) entrepreneurs maximize their expected sum of dis- counted utilities (2.1) subject to the budget constraint (2.3) taking as given the economy wide level of capital stock K (t) and equilibrium prices and (ii) the capital and labor markets clear at each point in time

Z

k

t

(a, z, K)dΦ

t

(a, z) = Z

adΦ

t

(a, z) ≡ K(t) (2.20) Z

l

^∗

(a, z, K)dΦ

_t

(a, z) =L (2.21)

Following Moll (2014), I introduce a new state variable representing the wealth distribution as follows:

ω(z, t) ≡ the share of wealth held by productivity type z

≡ 1 K(t)

Z

∞ 0

aφ

_t

(a, z)da (2.22)

Equivalently, the total amount of wealth held by productivity type z is

K (t)ω(z, t). In equilibrium, the aggregate amount of capital demanded by

all active entrepreneurs with productivity z greater than the productivity

cutoff z is, therefore:

K

^d

(t) ≡ Z

∞

z

λ[K(t)]K(t)ω(x, t)dx

=λ (K(t)) K(t) Z

∞

z

ω(x, t)dx = K

^s

(t)

(2.23)

At equilibrium, K

^d

(t) = K

^s

(t) = K(t), therefore:

1 = λ (K(t)) Z

∞

z

ω(x, t)dx = λ (K (t)) (1 − Ω(z, t)) (2.24) where Ω(z, t) is the corresponding distribution function with respect to ω(z, t) and is then defined as:

Ω(z, t) ≡ Z

z

0

ω(x, t)dx (2.25)

3 The Aggregate Equilibrium Dynamics

In this economy, the aggregate output denoted by Y can be obtained by adding all final goods produced by active entrepreneurs:

Y = Z

f (z, K, k, l)dΦ(a, z)

= Z Z

1 − α w

^1−α_α

K

^β(1−α)α

zλ(K)aφ(a, z)dadz

= π

α λ(K)K

^β(1−α)α

K Z 1

K Z

aφ(a, z)dazdz

= π

α λ(K)K

^β(1−α)α

K Z

∞

z

zω(z, t)dz

= π

α λ(K)XK

^β(1−α)α

K, where X ≡ Z

∞

z

zω(z, t)

(3.26)

Substituting the optimal labor demand (2.7) into the labor market clear- ing condition 2.21, we obtain

L = Z

l

^∗

(a, z, K )dΦ(a, z) = π α

_1−α¹

λ(K)XK

^β(1−α)α

K

which then implies that:

π = α (λ(K )X)

^α−1

K

^{α−1−β(1−α)2α}

L

^1−α

(3.27) Plugging this equation back to (3.26) we obtain the aggregate output as follows:

Y = (λ(K)X)

^α

K

1+^β(1−α)_α +α−1−^β(1−α)2_α

L

^1−α

=AK

^{[α+β(1−α)]}

L

^1−α

(3.28) where A is the endogenized TFP

A(t) ≡ (λ(K)X)

^α

= R

∞

z

zω(z, t) (1 − Ω(z, t))

!

α

= E

^ω,t

[z|z ≥ z]

^α

(3.29) Note that unlike Moll (2014), A(t) is endogenous and depends not only on the distribution of wealth but also on the aggregate capital stock K(t) since the productivity cutoff z is defined as: 1 = λ (K(t)) (1 − Ω(z, t))

The wage rate, w, can be obtained by substituting (3.27) into the defini- tion of π (2.11):

w = (1 − α)AK

^{[α+β(1−α)]}

L

^−α

(3.30)

Similarly, the capital return, r, can be expressed as:

r = α z

E

ω,t

[z|z ≥ z] AK

[α+β(1−α)−1]

L

^1−α

− δ (3.31) We then derive the dynamic equation of the aggregate capital stock, K, by first aggregating wealth of all entrepreneurs

K ˙ K = 1

K Z

˙

adΦ(a, z) = 1 K

Z Z

˙

aφ(a, z)dadz

= Z Z

λ(K) max n

zπK

^β(1−α)α

− (r + δ), 0 o

+ r − ρ 1

K aφ(a, z)dadz

= Z

∞

0

λ(K) max n

zπK

^β(1−α)α

− (r + δ), 0 o

+ r − ρ

ω(z, t)dz

and then dividing entrepreneurs into the inactive group (z < z) and the active group (z ≥ z), therefore

K ˙

K = (r − ρ) + Z

∞

z

λ(K) n

zπK

^β(1−α)α

− (r + δ) o

ω(z, t)dz

=(r − ρ) + πλ(K)K

^β(1−α)α

Z

∞

z

zω(z, t)dz − (r + δ)λ(K) Z

∞

z

ω(z, t)dz

=πλ(K)XK

^β(1−α)α

+ r − ρ − (r + δ)

=α (λ(K)X)

^α−1

K

^{α−1−β(1−α)2α}

L

^1−α

λ(K )XK

^β(1−α)α

− ρ − δ

=αAK

[α+β(1−α)−1]

L

^1−α

− ρ − δ

where the third and forth equal signs are implied by the definition of X in (3.26), z in (2.24), and π in (2.11). We sum up our results so far in a Lemma.

Lemma 1. For a given evolution of wealth shares ω(z, t), the aggregate out- put and capital stock, measured TFP, and the productivity cutoff can be ex- pressed as:

Y = AK

^{[α+β(1−α)]}

L

^1−α

(3.32)

K ˙

K = αAK

[α+β(1−α)−1]

L

^1−α

− ρ − δ (3.33)

A(t) ≡ R

∞

z

zω(z, t) (1 − Ω(z, t))

!

α

= E

ω,t

[z|z ≥ z]

^α

(3.34)

1 = λ (K(t)) (1 − Ω(z, t)) (3.35)

No Knowledge Externatilities, Exogenous Maximum Leverage Ra-

tio When β = 0, i.e., the individual technology production has no knowl-

edge externatilities, and the maximum leverage ratio λ does not depend on

the aggregate level of capital stock, the model economy is reduced exactly to

the model considered by Moll (2014) (Proposition 1) as follows:

Y = AK

^α

L

^1−α

(3.36)

K ˙ (t)

K (t) = αAK

^α−1

L

^1−α

− ρ − δ (3.37)

A(t) ≡ R

∞

z

zω(z, t) (1 − Ω(z, t))

!

α

= E

ω,t

[z|z ≥ z]

^α

1 = λ (1 − Ω(z, t)) (3.38)

The critical difference between the aggregate dynamics in this paper (re- sults in Lemma 1) and Moll (2014) is the determination of the productivity cutoff. As in Moll (2014), given wealth shares, the equation (3.38) pins down the productivity cutoff z as a function of the quality of the capital markets, λ, which is exogenously given. Therefore, the measured TFP and the speed of convergence in Moll (2014) are exogenous when the wealth share is station- ary. By contrast, when the quality of the capital markets are endogenous, the productivity cutoff and the aggregate capital stock are simultaneously determined as in the equation (3.35). As a result, the measured TFP and the speed of convergence, if existed, are endogenous with K even under a stationary wealth shares as we will explore in details below.

In order to obtain further analytical results and to address the effects of productivity heterogeneity on the dynamic system, we assume that idiosyn- cratic productivity z is i.i.d over time as well as across entrepreneurs.

⁵

In each moment entrepreneurs draw idiosyncratic productivity z from a Pareto distribution whose distribution function is given by

G(z) =

( 1 −

^z_z^˜

ϕ

z ≥ z ˜

1 z < z ˜ (3.39)

5

As a result, owing to the law of large numbers, the population share of type z en-

trepreneurs z is stationary and deterministic.

where the slope/shape parameter, ϕ > 1, represents the degree of heterogene- ity in productivity; a lower value of ϕ implies a higher level of productivity heterogeneity among entrepreneurs. The parameter ˜ z represents the mini- mum level of productivity that each entrepreneur can draw. A higher value of ˜ z implies on average a higher productivity at the firm level.

The productivity density function g(z) for z ≥ z ˜ is hence ϕ˜ z

^ϕ

z

^−ϕ−1

. Since z is i.i.d over time as well as across entrepreneurs, from the definition of wealth shares (2.22) we obtain ω(z, t) = g (z) so

Z

∞ z

zω(z, t)dz = Z

∞

z

zϕz

^−ϕ−1

dz = ϕ

ϕ − 1 z ˜

^ϕ

z

^1−ϕ

(3.40) Consequently, weighted average of productivity of active entrepreneurs and the endogenous TFP are as follows:

E

ω,t

[z|z ≥ z] = R

∞

z

zω(z, t)

(1 − Ω(z, t)) = ϕ ϕ − 1

z

^1−ϕ

z

^−ϕ

= ϕ

ϕ − 1 z (3.41) A(t) ≡ (λ(K)X)

^α

= E

ω,t

[z|z ≥ z]

^α

=

ϕ ϕ − 1 z

α

(3.42) For simplicity and tractability, we also assume that the endogenous max- imum leverage ratio, λ(K), takes the following form:

λ(K) = BK

^γ

, (3.43)

where 0 ≤ γ ≤ 1, B = K(0)

^−γ

(3.44) where the coefficient B is normalized so that the initial maximum leverage ratio, λ(0), is equal to one, i.e., the economy is initially at financial autarky.

As a result, the aggregate dynamics of the economy model are expressed

as follows:

Y = AK

^{[α+β(1−α)]}

L

^1−α

(3.45)

K ˙

K = αAK

[α+β(1−α)−1]

L

^1−α

− ρ − δ (3.46)

A ≡ ϕ

ϕ − 1 z

α

(3.47)

1 = BK

^γ

z ˜

^ϕ

z

^−ϕ

(3.48)

The implied TFP, A, is now a function of the aggregate level of capital and also the parameter ϕ that captures productivity heterogeneity.

A = ϕ

ϕ − 1 zB ˜

^ϕ1

K

^γϕ

α

(3.49) The law of motion for the aggregate capital is then:

K ˙ K =α

ϕ

ϕ − 1 zB ˜

^ϕ1

K

^γϕ

α

K

[α+β(1−α)−1]

L

^1−α

− ρ − δ

=α ϕ

ϕ − 1 zB ˜

^ϕ1

α

L

^1−α

K [

^αγ_ϕ+α+β(1−α)−1

] − ρ − δ

=ΓK

^Θ−1

− ρ − δ

(3.50)

where:

Γ ≡ α

ϕ

ϕ−1

zB ˜

^ϕ1

α

L

^1−α

(3.51)

Θ ≡

^αγ_ϕ

+ α + β(1 − α) (3.52)

Assumption 1. Θ ≡

^αγ_ϕ

+ α + β(1 − α) ≤ 1

Equivalent, we only consider either the economy model with a balanced-

growth equilibrium, Θ = 1, or the economy with a convergent steady state

equilibrium, Θ < 1.

3.1 Steady State Equilibrium: Θ < 1

Next, we focus on the case where Θ < 1 so that the model economy converges to a steady state with the speed of convergence, (1 − Θ) (ρ + δ).

⁶

Proposition 1. When the steady-state equilibrium condition is satisfied, i.e., Θ < 1, heterogeneity in productivity lowers the speed of convergence of the economy to the steady state.

Proof. The proof is straightforward from the definition of Θ in (3.52) and the fact that the slope/shape parameter, ϕ > 1, represents the degree of heterogeneity in productivity; a lower value of ϕ implies a higher level of productivity heterogeneity among entrepreneurs.

Note that from the definition of Θ in (3.52), productivity heterogeneity characterized by the parameter ϕ affects the speed of convergence, 1−Θ, only when γ 6= 0, i.e., the maximum leverage ratio is endogenous with K. When λ is exogenous and idiosyncratic productivity z is i.i.d as in Moll (2014), the degree of productivity heterogeneity has no impact on the dynamic system characterized by folowing equations:

Y = AK

^α

L

^1−α

K ˙

K = αAL

^1−α

− ρ − δ A ≡

ϕ ϕ − 1 z

α

z = (λ)

^ϕ1

(3.53)

which is a standard Solow model with constant exogenous TFP.

By contrast, this paper demonstrates that when the quality of capital markets is endogenous then even if idiosyncratic productivity z is i.i.d, we can construct a steady state equilibrium growth model with endogenous TFP in which heterogeneity in productivity can lower the speed of convergence.

6

See for example Acemoglu (2009) for the derivation of this speed of convergence.

3.2 Balanced-growth Equilibrium: Θ = 1

When Θ ≡

^αγ_ϕ

+ α + β(1 − α) = 1 then we obtain the AK economy model with the balanced-growth rate, Γ. From the definition of Γ in (3.52), the balanced-growth rate is higher the more heterogeneous are producer pro- ductivity. There are 3 notably familiar cases where this balanced-growth equilibrium condition is satisfied.

The first case is when γ = 0 and β = 1, i.e. when the maximum leverage ratio is exogenous and the production technology of each producer/entrepreneur has Arrow-Romer knowledge externalities as follows:

⁷

y = f(z, K, k, l) = (zk)

^α

(Kl)

^1−α

(3.54) The second case is when γ = 0 and α = 1, i.e. when the leverage ratio is fixed and the production technology of each producer/entrepreneur an Ak property as follows:

⁸

y = f (z, K, k, l) = zk (3.55)

Finally, the balanced-growth equilibrium condition can also be (asymptot- ically) satisfied if the production technology of each producer/entrepreneur has Arrow-Romer knowledge externalities as in (3.54), β = 1, and there is no productivity heterogeneity, ϕ → ∞, hence, Θ → 1. Consequently, the balanced-growth rate Γ → α z ˜

^α

B

^αϕ

L

^1−α

.

4 Concluding remarks

This paper constructs a tractable growth model where heterogeneous pro- ducers face endogenous quality of capital markets and analytically addresses

7

Mino (2016) uses a similar technology production to address the effects of fiscal policy in a balanced growth equilibrium.

8

Mino (2015) specifies the same AK technology in his paper to construct an endogenous

growth model with a transition dynamics.

the impact of firm productivity heterogeneity on the aggregate dynamics.

It shows that there exists a steady state equilibrium growth with endoge-

nous TFP in which firm heterogeneity can slower the speed of convergence

and identifies conditions where the economy can obtain the balanced-growth

equilibrium. In order to obtain analytical solutions, idiosyncratic productiv-

ity is assumed to be i.i.d. across both producers and over time in this paper

but this assumption can be relaxed in further research with an emphasis on

numerical analysis.

Appendix: Derivation of Bellman Equation

Let U (c) be the instantaneous utility function with C

²

property and ρ be the discount rate. The general optimality problem can be expressed as

V (a, z, t) = max

c(t)

E

a,z

Z

∞ 0

e

^−ρt

U (c(t))dt s.t. da = [Ψ(z, t) − c] dt

(4.56) The principle of optimality states that:

V (a, z, t) = max

c(t) 0≤t≤∆t

E

^a,z

Z

∆t 0

e

^−ρt

U (c(t))dt

+ max

c(t)

∆t≤t≤∞

E

a(∆t),z(∆t)

Z

∞

∆t

e

^−ρt

U(c(t))dt







(4.57)

By the intermediate value theorem, the first integral of (4.57) the can be written as, w.p.1,

Z

∆t 0

e

^−ρt

U (c(t))dt = e

^ρζ∆t

U (c

_ζ∆t

)∆t (4.58) where ζ ∈ [0, 1] such that c

ζ∆t

→ c as ∆t → 0. Using the change of variables s = t − ∆t, the second integral (4.57) becomes

max

c(t)

∆t≤t≤∞

E

a(∆t),z(∆t)

Z

∞

∆t

e

^−ρt

U (c(t))dt

= max

c(s+∆t) 0≤s≤∞

E

a(∆t),z(∆t)

Z

∞ 0

e

^{−ρ(s+∆t)}

U (c(s + ∆t))ds

=e

^−ρ∆t

max

c(s) 0≤s≤∞

E

a(∆t),z(∆t)

Z

∞ 0

e

^−ρs

U (c(s))ds

=e

^−ρ∆t

V (a(∆t), z(∆t), ∆t)

(4.59)

As a result, the equation (4.57) can be re-written as:

0 = max

c(t) 0≤t≤∆t

E

^a,z

e

^ρζ∆t

U (c

_ζ∆t

)∆t + e

^−ρ∆t

V (a(∆t), z(∆t), ∆t) − V (a, z, t)

(4.60)

For sufficiently small ∆t, we have e

^−ρ∆t

= 1 − ρ∆t + o(∆t). Hence, e

^−ρ∆t

V (a(∆t), z(∆t), ∆t) − V (a, z, t)

= [V (a(∆t), z(∆t), ∆t) − V (a, z, t)] − ρ∆tV (a(∆t), z(∆t), ∆t) + o(∆t) (4.61) Substituting back in (4.60) and divide both sides by ∆t we obtain

0 = max

c(t) 0≤t≤∆t

E

a,z

e

^ρζ∆t

U (c

_ζ∆t

) − ρV (a(∆t), z(∆t), ∆t) + 1

∆t [V (a(∆t), z(∆t), ∆t) − V (a, z, t)] + 1

∆t o(∆t)

(4.62) Let ∆t → 0. As ∆t → 0, we have a(∆t) → a, z (∆t) → z, e

^ρζ∆t

→ 1, c

_ζ∆t

→ c, w.p.1, the Bellman equation can be expressed as follows:

0 = max

c(t)

E

a,z

U(c(t)) − ρV (a, z, t) + 1

dt [dV (a, z, t)]

(4.63) Therefore, the Bellman equation of the original optimality problem is:

ρV (a, z, t) = max

c(t)

U (c(t)) + 1

dt E

a,z

[dV (a, z, t)]

s.t. da = [Ψ(z, t) − c] dt

(4.64)

References

[1] Acemoglu, D., 2009. Introduction to Modern Economic Growth. Prince- ton University Press.

[2] Buera, F., Shin, Y, 2013, “Financial Frictions and the Persistence of His- tory: A Quantitative Exploration,” Journal of Political Economy 121(2), 221-272.

[3] Mino, K., (2015), “A Simple Model of Endogenous Growth with Financial Frictions and Firm Heterogeneity,” Economics Letters, 127, 20-23.

[4] Mino, K., (2016), “Fiscal Policy in a Growing Economy with Finan- cial Frictions and Firm Heterogeneity,” The Japanese Economic Review, 67(1), 3-30.

[5] Moll, B., (2014), “Productivity Losses from Financial Frictions: Can Self- Financing Undo Capital Capital Misallocation?” American Economic Review 104 (10), 3186-3221.

[6] Romer, P.M. 1986, “Increasing Returns and Long-Run Growth. Journal

of Political Economy,” 94, 1002-37.