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FEDERAL RESERVE BANK o f ATLANTA
WORKING PAPER SERIES
Entry Costs, Financial Frictions, and Cross-Country
Differences in Income and TFP
El-hadj Bah and Lei Fang
Working Paper 2010-16a
Revised July 2014
Abstract: This paper develops a model to assess the quantitative effects of entry costs and financial
frictions on cross-country income and total factor productivity (TFP) differences, with a primary focus on
the interaction between entry costs and financial frictions. The model is calibrated to match the
establishment level statistics for the U.S. economy, assuming a perfect financial market. The simulations
based on the calibrated model show that entry costs and financial frictions together account for 55 percent
and 46 percent of the cross-country variation in output and TFP in the data. Moreover, a substantial
portion of the variation is accounted for by the interaction between entry costs and financial frictions. The
main mechanism is that financial frictions amplify the effect of entry costs.
JEL classification: O11, O43
Key words: entry costs, financial frictions, GDP per capita, TFP
The authors thank seminar participants in the 2010 Midwest Macroeconomics Meetings, the 2010 Tsinghua Macroeconomics
Workshop for Young Economists, the Society for Economic Dynamics Annual Meetings 2010, the Southern Economic Association
Annual Meetings 2010, the Australasian Macroeconomics Workshop 2012, the Southern Workshop in Macroeconomics 2012, the
CASEE Macroeconomics Reunion Conference 2012, and the Reserve Bank of New Zealand 2011. They are also grateful to the
associate editor and two anonymous referees for their valuable comments. The views expressed here are the authors’ and not
necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors’
responsibility.
Please address questions regarding content to El-hadj Bah, The University of Auckland, 12 Grafton Road, Auckland, New
Zealand, e.bah@auckland.ac.nz, or Lei Fang, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street NE,
Atlanta, GA 30309-4470, 404-498-8057, lei.fang@atl.frb.org.
Federal Reserve Bank of Atlanta working papers, including revised versions, are available on the Atlanta Fed’s website at
frbatlanta.org/pubs/WP/. Use the WebScriber Service at frbatlanta.org to receive e-mail notifications about new papers.
1
Introduction
Income per capita differs by more than a factor of thirty between rich and poor countries.
Research on growth accounting finds that the majority of the differences are the result of
cross-country differences in total factor productivity (TFP).1 It is worth noting also that
many poor countries have poorly developed financial markets as well as substantial costs
for starting new businesses. Both of these factors have been found to be negatively correlated with income per capita across countries. For example, Djankov et al. (2002) find a
negative correlation between GDP per capita and the ratio of entry cost to GDP per capita,
Nicoletti and Scarpetta (2003, 2006) find that entry costs are negatively related to TFP in
OECD countries, and Beck et al. (2000) establish a negative relationship between financial
development and economic growth. Accordingly, the goal of this paper is to quantify the
importance of financial frictions and entry costs in cross-country differences with respect to
income per capita and TFP.
There are a number of studies that have examined the effect of financial frictions and/or
the effect of entry costs on cross-country differences. This paper, however, investigates
whether there is any interaction between entry costs and financial frictions and, if so, how
such interaction may contribute to cross-country income and TFP differences. Intuitively,
underdeveloped financial markets may amplify the effect of entry costs as entrepreneurs
cannot borrow enough to overcome such costs. In contrast, a better developed financial
market may have less impact on how entry costs affect output and TFP. Understanding such
interactions can guide policymakers in improving overall TFP and output. For instance, if
the interaction is important, simplifying the entry process and improving the financial market
conditions will have a much greater impact on economic development than addressing just
one of the two factors.
1
See, for example, Klenow and Rodriguez-Clare (1997); Prescott (1998); Hall and Jones (1999). One
exception is Manuelli and Seshadri (2010).
1
To explore this issue, this paper develops a model that incorporates both financial frictions
and entry costs and then uses the calibrated model to analyze the quantitative importance of
the interaction between the two frictions. We find that financial frictions amplify the effect
of entry costs on economic development. Moreover, the interaction accounts for a substantial
portion of the differences in cross-country income and TFP.
The model builds on the industry model studied by Hopenhayn (1992) and Hopenhayn
and Rogerson (1993). In the model, establishments have different levels of productivity
that evolve over time. The technology is subject to decreasing returns to scale with fixed
production costs. Although an establishment finances capital and the fixed production cost
from the financial market, the financial market is imperfect, and an establishment can only
borrow up to a fraction of its expected discounted life-time profits. Furthermore, existing establishments may exit if they experience lower productivity. In contrast, new establishments
can enter after paying an upfront entry cost that can be financed from the financial market
subject to a borrowing constraint similar to the one faced by existing establishments.
The model is calibrated to match the establishment level statistics in the U.S. economy,
assuming a perfect financial market for the U.S. The calibrated model is then used to analyze
the cross-country differences in income per capita and TFP. To perform the analysis, we
simulate the model to jointly match the ratio of entry cost to GDP per worker and the ratio
of debt to GDP for a large set of countries. The linear regression of the data on the model
prediction shows that entry costs and financial frictions together can account for 55% and
46% of the cross-country variation in output and TFP, respectively, as measured by R2 . In
addition, the two frictions reduce output to 9% and TFP to 25% of the U.S. level in low
income countries. More importantly, a substantial portion of the decline in output and TFP
is generated by the interaction between the two frictions. This finding suggests that to fully
take advantage of the reduction in entry costs, it is better to improve the conditions of entry
and the financial market simultaneously.
2
The intuition for the results consists of three parts. First, higher entry costs protect
existing establishments. Hence, establishments with lower productivity can survive, and
output and TFP decrease. Second, financial frictions limit borrowing and lead to lower
capital to output ratio. Moreover, financial frictions distort the allocation of capital and labor
towards establishments with more capital stocks. This drives down TFP. Third, financial
frictions amplify the effect of entry costs on output and TFP. To understand this, note that
when there are frictions in the financial market, some of the profitable entrants may not
be able to open their businesses as they cannot finance the required upfront entry cost.
This raises the effective entry cost. The effect is equivalent to an increase in the entry cost.
Hence, output and TFP decline. Furthermore, as financial market conditions deteriorate,
the amplification effect increases, as does the decrease in output and TFP.
Entry costs and financial frictions affect output and TFP through the effects on the capital
to output ratio and distribution of establishments. To evaluate the model prediction on these
dimensions, we compare the capital to output ratio, entry rate, average establishment size,
and variance of establishment size from the model to the cross-country data. We find that
the model is broadly consistent with the cross-country data and a version of the model with
capital adjustment costs fits the data better.
This paper is related to the broad literature that investigates the effects of financial
frictions on economic development. Levine (2005) conducts a comprehensive literature review
in this area. Many of these works cited by Levine (2005), both theoretically and empirically,
show that poor financial development leads to low economic development. However, a recent
quantitative paper by Chakraborty and Lahiri (2007) finds that financial frictions modeled
as intermediation costs of capital in a standard neoclassical model cannot account for the
large income differences between rich and poor countries. Banerjee and Dufflo (2005) point
out that financial frictions may affect income through borrowing constraints and also provide
evidence for the prevalence of borrowing constraints in poor countries. Following the idea of
3
modeling financial frictions as borrowing constraints, Amaral and Quintin (2010) and Buera
et al. (2011) show that financial frictions can generate sizable differences in output and TFP
across countries, and D’Erasmo and Boedo (2012) find that the financial market structure
and the costs of informality have important quantitative implications for cross-country TFP
differences.
This paper is also related to several other papers in the literature that emphasize the importance of entry costs on cross-country income and TFP differences. Barseghyan and DiCecio (2011) quantify the effect of entry costs on economic development, and Moscoso Boedo
and Mukoyama (2012) analyze the effects of entry costs and firing costs on cross-country differences in income and productivity. We view this paper as a complement to these works of
entry costs and financial frictions on cross-country income and TFP differences. We develop
an industry model that incorporates both entry costs and financial frictions, allowing us to
investigate how the interaction between entry costs and financial frictions affects income and
TFP. Moreover, the simulations based on the calibrated model show that the interaction is
quantitatively important. Our findings are consistent with Midrigan and Xu (2013) which
find that financial frictions generate small losses from misallocation, but potentially sizable
losses from the inefficiently low level of entry because traditional producers cannot borrow
to overcome the barriers to entry to enter the modern sector.
This paper is connected to the literature that studies the relationship between various policies and the cross-country income and TFP differences. For instance, Parente and
Prescott (1999) and Herrendorf and Teixeira (2011) examine the role of monopoly rights
in blocking the use of more efficient technologies, while Lagos (2006) examines how labor
market institutions affect TFP. Erosa and Cabrillana (2008) investigate the role of poor
contract enforcement in explaining the use of inefficient technologies and low TFP in poor
countries, and Guner et al. (2008) and Restuccia and Rogerson (2008) study the effect of
size-dependent policies on macroeconomic aggregates.
4
The rest of the paper is organized as follows. Section 2 presents the model and defines the
steady state equilibrium. Section 3 describes the calibration strategy. Section 4 assesses the
quantitative implication of the calibrated model and the robustness of the results. Section
5 concludes the paper.
2
The Economy
We consider a discrete-time model with heterogeneity in establishment level productivity.
The model can be best described as embedding borrowing constraints in the industry model
studied by Hopenhayn and Rogerson (1993). In the model economy, an establishment must
finance capital and the fixed production cost ahead of production by borrowing from an
imperfect financial market. More importantly, there are many potential entrants who can
enter after paying a front-loaded set-up cost upon entry, and this cost must also be financed.
The details follow.
2.1
Production
2.1.1
Technology
The production unit is the establishment. There is a continuum of existing establishments
that differ in their productivity z. Each of these establishments hires labor, invests in capital,
and produces according to the following production function:
y = zk α hγ ,
(2.1)
where the individual establishment productivity z changes over time. Specifically, z is the
same as the last period value with probability λ, and evolves according to the distribution
F (z) with probability 1 − λ. The parameter λ controls the persistence of the idiosyncratic
5
productivity shock. The establishment’s production technology is assumed to be decreasing
returns to scale, i.e., α + γ < 1. To stay in operation, each establishment must pay a fixed
production cost f every period, measured in the unit of output. As in Hopenhayn and Rogerson (1993), the fixed production cost generates an explicit exit and prevents establishments
from staying in the economy while not producing. Capital is owned by establishments. We
use k−1 to denote the establishment’s capital holdings during the last period and k to denote
the optimal choice of capital in the current period. The capital good is homogenous across
establishments and can be freely traded in the market. Hence, if k > (1 − δ)k−1 , establishments expand and raise capital, and otherwise, establishments downsize and sell capital in
the market.
2.1.2
Financial Market
The financial market consists of many competitive intermediaries who receive deposits and
lend to establishments at a constant rate r. We assume that borrowing and lending are
within the same period and that establishments cannot default on the debt. Thus, the zeroprofit condition for the intermediaries implies that the interest rate paid on the deposit is
also r.
An establishment’s ability to borrow is limited by its prospect and current capital holdings. For simplicity, we assume that an establishment can borrow up to a fraction η of its
discounted life-time profits. Each establishment finances capital and fixed production cost,
and the borrowing constraint is described as follows:
k − (1 − δ)k−1 + f ≤ ηv(z, k−1 ),
(2.2)
where v(z, k−1 ) represents the value of an establishment, and the establishment’s state at
the beginning of a period is summarized by (z, k−1 ). The credit constraint imposes an
6
upper bound on the current period capital usage. As demonstrated later in this section,
v(z, k−1 ) is increasing in both z and k−1 . Hence, the credit constraint captures the idea
that establishments with more collateral and better productivity can borrow more from
the financial market.2 The development of financial markets differs across countries due to
differences in contract enforcement. For simplicity, we use η to capture the degree of financial
development in different countries with the interpretation that a larger η represents a better
financial market.
The timing of decisions within a period is as follows. At the beginning of a period, the
productivity z realizes. After seeing the new productivity, an establishment with capital
holding k−1 can choose to stay in operation if the continuation value is larger than the value
of its non-depreciated capital, or exit otherwise. If the establishment decides to stay, it
chooses how much labor to hire, how much capital to use, and therefore, how much money
to borrow from the financial market, taking into account the borrowing constraint. At the
end of the period, production takes place, and the establishment repays the debt. If the
establishment decides to exit, it sells its capital and exits the market.
Subject to the borrowing constraint (2.2) and the non-negativity constraint on capital,
an establishment’s value v(z, k−1 ) measured immediately after the realization of productivity
is given by:
v(z, k−1 ) = max zk α hγ − wh − (1 + r)(k − (1 − δ)k−1 + f ) + βλ max[v(z, k), (1 + r)(1 − δ)k]
k,h
Z
+β(1 − λ) max[v(z 0 , k), (1 + r)(1 − δ)k]dF (z 0 )
(2.3)
An establishment’s value consists of its current period profit and the next period’s value,
which reflects the evolution of productivity and the establishment’s staying or exiting decision
as indicated by the maximization operator nested on the right-hand side. When exiting, an
2
Buera et al. (2011) and Amaral and Quintin (2010) derive the borrowing constraint endogenously through
contractual enforcement and their borrowing constraints have the same properties.
7
establishment can sell its non-depreciated capital (1 − δ)k and earn interest on the proceeds,
as described by the second term inside the maximization operator. As there is no distortion
α
1
in the labor market, the first order condition implies h = ( zkw γ ) 1−γ .
The value function (2.3) implicitly defines three types of establishments: expanding,
downsizing, and exiting establishments. The expanding establishments raise capital and
the fixed production cost through borrowing from the financial market. The downsizing
establishments sell the extra capital and deposit the proceeds and also borrow from the
financial market to pay for the fixed production cost.3 The exiting establishments sell their
capital and earn interest.
If the financial market is perfect, equation (2.2) will not bind. In this case, regardless of
their capital holdings, establishments can always borrow the optimal amount of capital as
determined only by the productivity. Hence, all the decision rules, including usage of capital
and labor and the stay/exit decision, only depend on productivity. However, if the financial
market is imperfect, the decision rules will depend on both z and k−1 . In particular, the
capital usage could be either less or greater than the level in the perfect financial market
for an establishment with the same state (z, k−1 ). On the one hand, equation (2.2) sets an
upper bound for k and may force some establishments to operate on a smaller scale; on the
other hand, establishments with larger k−1 and smaller z have some probability of drawing
a better productivity in the future and optimally choosing to hold more capital.
When the financial market is perfect, the stay/exit decision does not depend on capital
holdings, as it is characterized by a cutoff rule for z. In contrast, when the financial market
is imperfect, an establishment with more capital holdings can operate on a larger scale, as
more capital holdings imply not only that the establishment has more capital to begin with
3
As the borrowing and lending rates are the same, this is equivalent to the following: the downsizing
establishment sells the extra capital and uses the proceeds to pay for the fixed production cost. If the
fixed production cost is large, the establishment finances the difference, and if the proceeds are large, the
establishment deposits the extra proceeds.
8
but also that the establishment can borrow more from the financial market because v is an
increasing function k−1 , as established in the following lemma.
Lemma 1 (i) v(z, k−1 ) is increasing in z and k−1 ;
(ii) v(z, k−1 ) − (1 + r)(1 − δ)k−1 is increasing in k−1 .
Proof : See appendix.
As v is increasing in z, the decision to stay or exit is characterized by a cutoff rule for z
at a given value of k−1 . In particular, the rule is to exit if z is below the cutoff value and
to stay otherwise. Lemma 1 (ii) proves that an increase in k−1 leads to a greater increase
in v. It follows that the decision to stay or exit is also characterized by a cutoff rule for
k−1 at a given value of z. Moreover, the monotonicity proven in lemma 1 also implies the
monotonicity of the cutoff values. Specifically, the cutoff value of k−1 becomes smaller as z
increases and the cutoff value of z becomes smaller as k−1 increases.
2.1.3
Entry
There is a continuum of an infinite amount of ex ante identical establishments that can enter
each period after paying the entry cost fe , measured in the unit of output. To pay the entry
cost, the entrant can borrow from the financial market at the rate r up to the fraction η
of its value of entry. The debt is, again, within the period and must be repaid at the end
of the period. Once the entry cost is paid, each establishment receives a productivity draw
z from the same distribution as the existing establishments F (z). The productivity draws
are i.i.d across entering establishments. After the productivity draw is realized, the entering
establishment decides to stay or exit. If the establishment chooses to stay, it then decides
how much to borrow and how much to produce. The borrowing constraint for entrants
choosing to stay is as follows:
k + f + fe ≤ ηv(z, 0).
9
(2.4)
Perfect Financial Market
As in Hopenhayn and Rogerson (1993), as there is an infinite amount of potential entrants
in each period, the value of entry for an entering establishment should not exceed the entry
cost in the equilibrium when the financial market is perfect. In this economy, the entrant,
after paying the entry cost, is in the same position as the existing establishment with the
same productivity and zero capital. Hence, when making the stay/exit decision, the entrant
will compare the value of staying v(z, 0) with the value of exiting 0. Thus, the free entry
condition can be described as follows:
Z
max(v(z, 0), 0)dF (z) − rfe ≤ fe
(2.5)
where fe on the right-hand side denotes the entry cost and rfe on the left-hand side is the
interest payment on the entry cost. The integral is taken over all the possible productivity
draws. For future reference, note that the left-hand side of (2.5) denotes the value of entry
for a new establishment.
If there is no financial friction, the free-entry condition (2.5) must hold in the steady
state equilibrium. In this paper, we will focus on the steady state equilibrium with entry
and exit.4 As proven in Hopenhayn (1992), if a steady state equilibrium with entry and
exit exists, it is a unique equilibrium and (2.5) will hold with equality. Otherwise, more
establishments will enter and produce, which drives down the value of entry until it is no
longer profitable for more establishments to enter.
Imperfect Financial Market
When there are financial frictions, the free-entry condition may not hold in the steady
state equilibrium. To see this, note that as for an existing establishment, a new establishment
can only borrow up to η fraction of its value of entry. Hence, if fe is greater than the
4
This equilibrium exists for all the simulations in section 5.
10
borrowing limit for a potential entrant, no establishment can pay the up-front cost to enter.
In such cases, a steady state equilibrium with entry and exit cannot exist even when the freeentry condition holds. This implies that fe must be less than or equal to the borrowing limit
for a new establishment in the steady state equilibrium. As the left-hand side of equation
(2.5) is the value of entry for a new establishment, the borrowing constraint is as follows:
Z
max(v(z, 0), 0)dF (z) − fe r} ≥ fe
η{
(2.6)
Simple manipulation gives
Z
max(v(z, 0), 0)dF (z) − fe r ≥
fe
η
(2.7)
If η ≥ 1, there is no contradiction between equations(2.5) and (2.7), and therefore, the
free-entry condition will hold with equality in the equilibrium. This implies that if the friction
in the financial market is moderate, all the profitable new establishments can borrow fe , and
the entry decision is not distorted. However, if η < 1, the entry decision is distorted, and
the free-entry condition cannot hold in equilibrium. In such cases, the entrant’s borrowing
constraint (2.7) binds. Otherwise, more establishments can acquire the up-front cost fe from
the financial market, and it is also profitable for these establishments to enter because the
value of entry is greater than the entry cost. This drives up the labor demand and wage
rate, therefore driving down the value of entry until equation (2.7) holds with equality.
For future reference, note that from equations (2.5) and (2.7), it is evident that when
η ≥ 1, the value of entry equals to the entry cost fe , but when η < 1, the value of entry equals
to
fe
.
η
Thus, when η < 1,
fe
η
entry decisions according to
can be viewed as the effective entry cost, as the entrants make
fe
η
instead of fe , and therefore output and TFP also adjust
according the effective entry cost. Hence, as long as η is small, even if the entry cost is
not substantial, the effective entry cost could still be significantly large. This implies that
11
financial frictions interact with entry costs and amplify the effects of entry costs on crosscountry incomes and TFP differences by boosting the effective entry cost. Moreover, the
magnitude of the amplification effect depends on the severity of the friction.
2.2
Household
There is an infinitely lived representative household who inelastically supplies one unit of
labor each period and values a single consumption good c according to the utility function:
∞
X
β t log(ct ),
t=0
where 0 < β < 1 is the discount factor. The household can deposit its savings a and
earn interest at rate r from the financial intermediaries. The household also owns all the
establishments in the economy. Let W (a) denote the value function of the household. The
problem of the representative household is given by:
W (a) = max
log(c) + βW (a0 )
0
c,a
s.t.
c + a0 = w + a(1 + r) + Π,
(2.8)
where w is the wage rate and Π is the total profits generated by the production sector.
A simple manipulation of the first order condition implies that if a stationary equilibrium
exists, r = β1 .
2.3
Definition of the Steady State Equilibrium
A steady state competitive equilibrium is composed of: prices w and r, value functions W (a)
and v(z, k−1 ), a measure of productive establishments µ(z, k−1 ), total profit Π, a mass of
entry M , policy functions c(a), a0 (a), h(z, k−1 ), k(z, k−1 ), and the stay/exit decision x(z, k−1 )
12
with the convention that x(z, k−1 ) = 1 corresponds to stay and x(z, k−1 ) = 0 corresponds to
exit, such that:
(i) Given prices, all agents solve their maximization problems.
(ii) r = 1/β.
(iii) If η ≥ 1, (2.5) holds with equality and if η < 1, (2.7) holds with equality.
(iv) µ is time-invariant.
(v) Labor, good, and credit market clear:
Z
1=
h(z, k−1 )dµ(z, k−1 ),
(2.9)
Z
c(a) + δK = Y −
f dµ(z, k−1 ) − M fe ,
(2.10)
Z
a = δK +
where K and Y are defined as K =
R
f dµ(z, k−1 ) + M fe ,
k(z, k−1 )dµ(z, k−1 ) and Y =
(2.11)
R
zk(z, k−1 )α h(z, k−1 )γ dµ(z, k−1 ).
(vi) Profit Π is as follows:
Z
Π = Y − w − δK(1 + r) − (1 + r)
f dµ(z, k−1 ) − M fe (1 + r).
(2.12)
The labor market and good market clearing conditions (2.9) and (2.10) are standard. To
understand the credit market clearing condition (2.11), note that the deposits received by the
financial intermediary come from three sources: household savings, capital sold by downsizing
and exiting establishments. The lending by the financial intermediary is applied to three
sources: capital raised by expanding establishments, fixed production costs and entry costs.
In the steady state equilibrium, the economy-wide capital stock equalize across periods, and
new capital must be raised only to replace the depreciated capital. Hence, the credit market
clearing can be described as in equation (2.11). The total profit received by the household
13
is the aggregation of current period profits net of the entry costs. Equation (2.12) can then
be derived using the labor market and the credit market clearing conditions.
3
Calibration
This section calibrates the parameters to match observations in the steady state to data in
the U.S. economy. For this purpose, the U.S. economy is treated as an economy without
distortion in the financial market.5 We assume that one period in the model corresponds to
one year in the data, and therefore, we target the steady state interest rate r to be 4% per
year. This implies that β = 0.96. We follow the literature and set the returns to scale in
the establishment level to be 0.8, and we set the capital share to be one-third of the returns
to scale and the labor share to be two-thirds of the returns to scale.6 This indicates that
α = 0.27 and γ = 0.53. To calibrate the depreciation rate, we follow Guner et al. (2008) and
target the capital to output ratio in the U.S. business sector to be 2.3 and the implied δ to
be 0.08.
To calibrate the persistence of the productivity process λ, we target the entry/exit rates
of U.S. establishments. The most recent value is approximately 10% as reported by the
U.S Census Business Dynamics Statistics (BDS).7 We assume a lognormal distribution F (z)
with support [0, zmax ] for the productivity process. In an economy without financial frictions,
all establishments operate at their optimal scale and the establishment employment level is
uniquely determined by z for any given price. Hence zmax can be inferred from the maximum
employment level of establishments in the steady state equilibrium, which we assume to be
5
The financial market in the U.S. is certainly not perfect. Hence, the quantitative results in section 4
should be interpreted as the effects of financial frictions on income and TFP relative to the U.S.
6
The return to scale parameter is found to be between 0.8 and 0.9. See, for example, Basu (1996),
Veracierto (2001), Chang (2000), Atkeson and Kehoe (2005), and Guner et al. (2008). Section 4.4.2 discusses
the results for the return to scale of 0.9.
7
The data can be found in the table “Economy Wide” at the following webpage: http://www.census.
gov/ces/dataproducts/bds/data_estab.html.
14
1500 as in Moscoso Boedo and Mukoyama (2012).
The parameters that remain to be assigned are the entry cost fe , the fixed production cost
f , the mean φ and the variance σ of the distribution F . We follow Hopenhayn and Rogerson
(1993) to normalize the wage rate to be one and calibrate the four parameters jointly to
match the ratio of entry cost to GDP per worker, the average establishment size, and the
share of the total number of establishments at different sizes in the U.S. economy.8 The World
Development Indicators (WDI) dataset by the World Bank provides entry costs in terms of
GDP per capita for a large set of countries.9 Since we abstract from the complexity of the
household sector and assume the representative household supplies all its labor endowment,
the appropriate counterpart of output in the model is GDP per worker in the data. Hence we
use data on GDP per capita and GDP per worker from the Penn World Table 7.1 (PWT7.1)
to convert the ratio of entry cost to GDP per capita to the ratio of entry cost to GDP per
worker. The most recent value of the ratio of entry cost to GDP per worker for the U.S. is
0.71%. This number is used to calibrate the entry cost fe . The establishment level statistics
are borrowed from the 2007 U.S. Economic Census, which summarizes the establishment
level distributional statistics by size.10 Specifically, the establishment level targets include
10 moments: the average establishment size and nine statistics related to the distribution of
the share of establishments by size.
The calibrated parameters are reported in table 1, while table 2 lists the targets and the
corresponding statistics generated by the model.11 Overall, the calibrated model matches
8
As noted by Hopenhayn and Rogerson (1993), one cannot disentangle the effect of a high wage from a
high mean of the productivity shock on the establishment’s objective function. This problem is dealt with
by normalizing the wage rate in the calibrated model to be one and choosing the remaining parameters to
be consistent with the data.
9
The data can be found at the following webpage: http://databank.worldbank.org/data/views/
variableSelection/selectvariables.aspx?source=world-development-indicators.
10
The establishment statistics can be found in the table “U.S. & States, Totals” at the following webpage:
http://www.census.gov/econ/susb/data/susb2007.html.
11
Note that the fixed production cost is 42 times that of the entry cost, thus, implies that f is 30% of
GDP per worker in the U.S. and is a small fraction of the total GDP.
15
the data well.
4
Quantitative Analysis
This section uses the calibrated model to assess the effects of entry costs, financial frictions,
and the interaction between these two on the cross-country income and TFP differences.
The strategy is to compare the steady state equilibrium in economies that differ in the entry
cost and the ability to acquire external finance. Data on entry costs have been discussed in
section 3. A common measure for a country’s level of financial development in the empirical
literature is the ratio of external credit to GDP ratio, which has been found to be negatively
correlated with economic development.12 Research works on assessing the effects of financial
frictions on cross-country income and TFP differences, such as Amaral and Quintin (2010),
Buera et al. (2011), and Midrigan and Xu (2013), use this measure to pin down the crosscountry variation in financial development. We follow the literature and adjusting η to match
the debt to GDP ratio in the model to the credit to the private sector as a fraction of GDP
in the data.13 The data also comes from the WDI dataset.
4.1
Aggregate Effects of Entry Costs and Financial Frictions
In this section, we adjust fe and η jointly to match the observed entry cost to GDP ratio and
the external credit to GDP ratio in the data for all available countries. Figure 1 plots the
log values of output and TFP from the model against the data. Each circle represents one
country. Data on GDP per worker comes from the PWT7.1. TFP in the data is calculated
12
See Levine (2005) for a comprehensive literature review.
The debt to GDP ratios are normalized by the perfect financial market levels in the model and by the
U.S. levels in the data.
13
16
following Hall and Jones (1999):
TFP =
Y
,
K α H (1−α)
where Y is the aggregate output, K is the aggregate capital, calculated using the perpetual
inventory method with investment data from PWT7.1, and H is the aggregate labor adjusted
for human capital, calculated using educational attainment data from Barro and Lee (2012).
The reported GDP and TFP series are normalized by the perfect financial market levels in
the model and by the U.S. levels in the data.
Output per worker and TFP from the model are highly correlated with the data. The
correlation coefficient is 0.74 for the output and 0.68 for TFP. Figure 1 also plots the regression line that regresses data on model values. If the model accounts perfectly for the
data, the slope and the R2 would both be one. Hence the model fits the data better if both
the slope and R2 are closer to one. The regression for the output has a R2 of 0.55 and a
slope of 1.08. The regression for TFP has a R2 of 0.46 and a slope of 0.97.14 Moreover,
both slopes are significantly different from zero at 1% level. Therefore, we conclude that the
model explains 55% of the cross-country variation in output and 46% of the cross-country
variation in TFP as measured by the R2 .15
4.2
Understanding the Contributions of the Two Frictions
To illustrate the importance of the two frictions and their interaction, we simulate the model
outcomes for three groups of countries: high income countries (HIC), middle income countries
(MIC), and low income countries (LIC), where the definitions of the groups follow the Atlas
method from the World Bank. For the analysis, we calibrate fe and η jointly to match the
14
The correlation coefficient and the regression results are very similar for TFP computed without human
capital adjustment.
15
The simulations with entry cost data from Djankov et al. (2002) deliver similar results.
17
observed average values of the entry cost to GDP ratio and the debt to GDP ratio for each
group. The simulation results are reported in the first column of table 3. For comparison
purposes, we normalize the U.S. values to be one.
The second column of table 3 reports the results of the model where fe is set to each
group’s own value and η is set to the U.S. value (fei , η us , where i = HIC, M IC, LIC). Hence
the second column shows the sole effects of the entry cost. As reported, both output and
TFP decrease with the entry cost. The intuition goes back to Hopenhayn and Rogerson
(1993). When there is no borrowing constraint, the free-entry condition must always be
satisfied in the steady state equilibrium. Thus, a higher entry cost necessarily leads to a
higher expected value of entry through a lower wage rate. This implies a larger v at each
state, and therefore, an establishment with a smaller productivity can survive, and the cutoff
value for z decreases. This leads to lower TFP and output. In particular, the cutoff values for
z are 91%, 90%, and 79% of the U.S. value for the simulated high income, middle income, and
low income countries. These cutoff values are independent of establishments’ capital stocks
since the experiments performed in the second column assume perfect financial markets.
The entry cost in low income countries alone can generate a drop of 38% in output
and a drop of 29% in TFP. The quantitative effects generated are similar to that found in
Barseghyan and DiCecio (2011) and Moscoso Boedo and Mukoyama (2012). As entry costs
do not affect capital accumulation, the capital to output ratio remains constant. Hence, the
reduction in output with entry costs is due to the reduction in TFP.
The third column presents the effects of financial frictions by holding η at each group’s
value and setting fe to be the same as the U.S. value (feus , η i , where i = HIC, M IC, LIC).
Observing that the product of the values from the second and the third column is approximately the same as that in the first column, one may be tempted to conclude that there is
no interaction between the two frictions. However, as equation (2.7) illustrates, when there
are financial frictions there will be interaction between the two frictions as long as the entry
18
cost is not zero. In the third column, the entry cost is set to the U.S. value and therefore,
the effective entry cost is
feus
.
ηi
To isolate the effect of the interaction, we decompose the
effect of financial frictions into two parts: the effect of financial frictions on capital accumulation as reported in the fourth column, and the effect of financial frictions on business
entry as reported in the fifth column. We interpret the latter effect as the contribution of
the interaction to the cross-country income and TFP differences. Specifically, the fourth
column reports the results when the borrowing constraint is imposed only on the financing
of capital and the fixed production cost (feus , η i on K, where i = HIC, M IC, LIC) and
the fifth column reports the results when the borrowing constraint is imposed only on the
financing of entry costs (feus , η i on fe , where i = HIC, M IC, LIC).
We now discuss the simulation results in the fourth column. In this case the free-entry
condition (2.5) holds with equality since the borrowing constraint is not imposed on the
financing of entry costs. As expected, tighter borrowing constraints decrease output and
TFP. The intuition is standard. First, financial frictions restrict borrowing and lending and
drive down the capital to output ratio and hence output. It is also worth noting that the
capital share and labor share do not change with financial development despite the sharp
decrease in the capital to output ratio. This is because the capital and labor markets are
both competitive.
Second, tighter borrowing constraints distort the allocation of capital and labor. Specifically, establishments with high productivity, but small capital stocks may not raise enough
capital and have to operate on smaller scales while establishments with low productivity but
large capital stocks can operate on larger scales. Furthermore, establishments with large capital stocks and low productivity may survive, and establishments with small capital stocks
and high productivity may not survive. This misallocation drives down TFP and therefore
also contributes to the decline in output. However, such an effect only accounts for a small
portion of the decline in output, as the change in TFP is small. Although this is consistent
19
with Midrigan and Xu (2013) and Gilchrist et al. (2013), this differs from the findings of
Amaral and Quintin (2010) and Buera et al. (2011). To understand the difference, note
that Amaral and Quintin (2010) have a three-period overlapping generations model in which
the entrepreneur can only save for one period and cannot overcome borrowing constraints
through self-financing over time. As a result, the misallocation effect of financial frictions
on TFP is large. Buera et al. (2011) generate a greater effect on TFP through an industry
model with risk-averse entrepreneurs and the misallocation of capital and talent. In contrast,
we did not model the misallocation of talent and show that the misallocation of capital with
risk neutral establishments cannot generate a large quantitative effect of financial frictions
on TFP. This abstraction simplifies the analysis and does not undermine our results, as
our main focus is on how the interaction between entry costs and financial market frictions
affects the cross-country income and TFP differences.
The rest of this section discusses the interaction between entry costs and financial frictions. As reported in the fifth column, the interaction between the two frictions decreases
output and TFP significantly. To understand this, note that η is less than one for all three
groups. Hence, the free-entry condition (2.5) cannot be satisfied in the case reported in
the fifth column; thus (2.7) holds with equality. In such cases, the equilibrium wage rate
adjusts according to the effective entry cost
the effective entry cost
feus
ηi
feus
,
ηi
so do output and TFP. Because η i < 1,
is greater than the entry cost in the U.S. feus . Hence output and
TFP decrease as reported in the fifth column and the effect is equivalent to an increase in
the entry cost alone since the borrowing constraint on capital accumulation is eliminated
in this case. This implies that financial frictions interact with entry costs and amplify the
effect of entry costs on output and TFP. Furthermore, as the financial market conditions
deteriorate, such amplification effects increase, as a smaller η leads to greater effective entry
cost. This is demonstrated by the larger decline of output and TFP in the fifth column as
the income level decreases. Moreover, the interaction between the two frictions decreases
20
TFP substantially while having no effect on the capital output ratio. This is because the
interaction only raises the effective entry cost and has no impact on capital accumulation.
Hence, the decline of output in the fifth column is completely driven by the decline in TFP.
We now turn to the quantitative magnitude of each factor. Based on the first column,
when both frictions are included, output per worker in the simulated low income country is
9% of the U.S. level. As in the data, the drop in TFP accounts for a substantial portion
of the drop in output. In particular, TFP in the low income country is 25% of the U.S.
level. The entry cost alone can bring down TFP (output per worker) to 71% (62%) and the
financial friction on capital accumulation alone can bring down TFP (output per worker) to
92% (51%). Hence, if we eliminate the interaction between entry costs and financial frictions,
TFP (output per worker) in the simulated low income country would be 65% (32%) of the
U.S. level, as measured by the product of values in the second column and the fourth column.
The fifth column shows that the interaction can bring down TFP (output per worker) to
39% (27%) in the low income country and therefore accounts for more than half of the
drop in TFP (output per worker). Moreover, the interaction accounts for a larger portion
of the decline in TFP than in output. Because the interaction reduces TFP but not the
capital to output ratio. Similarly, the interaction accounts for a substantial portion of the
decline in output and TFP for the simulated high and middle income countries. Hence, when
analyzing the effects of financial frictions and entry costs on output and TFP, it is important
to explicitly model business entry and to explore the interaction between financial frictions
and entry costs. This also suggests that policymakers should remove barriers to entry and
mitigate financial frictions to achieve the greatest improvement in economic development.
In summary, while the entry cost and the financial friction in the model generate large
declines in output and TFP, a substantial portion of the decline is accounted for by the
interaction between the two. Hence, such interaction cannot be ignored when analyzing the
cross-country income and TFP differences.
21
4.3
Model Evaluation
Section 4.2 has shown that the two frictions affect output and productivity through the effects on the capital to output ratio and distribution of establishments. One natural question
is whether the model’s predictions on these dimensions are consistent with the cross-country
data. To address this question, figure 2 compares the capital to output ratio, entry rate,
average establishment size as measured by the number of employees, and variance of establishment size from the model to the data. Data on capital to output ratios are constructed
when we compute TFP.16 Data on business entry rates are borrowed from Djankov et al.
(2010). Tybout (2000) and Alfaro et al. (2009) both study firm size distributions across
countries. But the former studies, both formal and informal firms, while the latter focuses
only on plants in the formal sector. Since we focus on the formal sector, we compare our
model predictions with data constructed by Alfaro et al. (2009). In figure 2, we compare
the log values of the average establishment size and variance of establishment size from the
model to the data in Alfaro et al. (2009).
Table 4 reports the correlation coefficients and the regression results for the above variables. Figure 2 and table 4 show that the model is consistent with the data in terms of the
capital to output ratio, average establishment size, and variance of establishment size. This
is demonstrated by the positive correlation coefficients, the positiveness and the significance
of the slopes for the regression lines and the associated R2 .17 The model misses the crosscountry variation in the entry rate. The next section explores the implications of the model
with capital adjustment costs. The model’s explanatory power for the entry rate improves
substantially with capital adjustment costs.
16
The values are normalized by the U.S. values in both the model and the data.
The R2 for the variance of establishment size is small. This may be due to the fact that there is no
selection of establishments at the point of entry. D’Erasmo and Boedo (2012) model selection upon entry.
They assume that an establishment receives the productivity draw before entering the formal sector and find
that the implied variance is close to the data.
17
22
4.4
4.4.1
Discussion
Capital Adjustment Cost
So far, we have assumed establishments can adjust their capital stock freely. However,
there are often costs associated with changing the level of production, such as inventory
costs, machine set-up costs, and hiring and lay-off costs. This section explores the models
implications with capital adjustment costs. Specifically, establishments are subject to the
capital adjustment cost Ω(k, k−1 ) and are recovering only the scrap value of capital when
exiting. Cooper and Haltiwanger (2006) estimate the capital adjustment cost function using plant level data and find that both convex and non-convex capital adjustment costs
are important in explaining the investment behavior. We follow their estimates and set
−1 2
) k. The scrap value of capital is a fraction of the
Ω(k, k−1 ) = 0.039k + 0.0245( k−(1−δ)k
k
original capital stock and this fraction is set to be 40% following the estimate by Ramey and
Shapiro (2001).
We simulate the model with capital adjustment costs for each country as in section 4.1.
In the simulation, the capital adjustment cost is also subject to the borrowing constraint.
The simulation results are reported in figure 3 and table 5. Including capital adjustment
costs improves significantly the fit of the model for the entry rate with no major changes in
the other variables. The correlation coefficient for the entry rate between the model and the
data is more than doubled, and the regression coefficient of the data on the model is now
significantly different from zero and R2 increases from the benchmark case.
4.4.2
Technology Parameters
We first examine the effect of a larger capital share. We set the capital share to be 0.3
and recalibrate the model. The quantitative effects are slightly larger for output and almost unchanged for TFP. We set the returns to scale parameter to be 0.8 in the benchmark
23
calibration. Research of the literature normally indicates a value between 0.8 and 0.9. Recalibrating the model with the upper bound of 0.9 generates smaller but still sizable effects.
In particular, entry costs and financial frictions together can bring down TFP to 53% and
output to 19% in the simulated low income country. Importantly, a substantial portion of
the quantitative effect is, again, derived from the interaction for the alternative values of the
capital share and returns to scale parameter.
4.4.3
Internal and External Financing
In this paper, we did not explicitly allow establishments to finance capital using retained
earnings. However, excluding self-financing is not an issue for incumbents in our model.
Because in our formulation, the collateral of borrowing is the value of the establishment,
which includes present and future net profits. Since an establishment’s value increases in
profits, an establishment with higher profits or cash flow is less financially constrained. This
is similar to what one would obtain by specifically modeling self-financing.
For new entrants, as we did not model the entrepreneurial sector, we assume that establishments have zero wealth before entry and must finance the entire entry cost. In reality,
entrepreneurs use some of their assets to start businesses. However, external financing is still
important for start-ups. Berger and Udell (1998) use data from National Survey of Small
Business Finances (NSSBF) to analyze financial structure of small businesses. They find
that the upper-bound of internal financing (principal owner and other equity) is 56.8% for
small business start-ups in the U.S. Cassar (2004) finds that 40% of the start-up funds in
Australia comes from outside sources. Huyghebaert and Van de Gucht (2002) report, for a
sample of 244 manufacturing start-ups in Belgium, 77.75% of their initial funding is from
external debts (bank debt and trade credit).
To illustrate the model’s prediction when entrepreneurs finance part of the entry cost
through their savings, we recalibrate the model assuming that only half of the entry cost is
24
financed by debt and the other half is paid through the household savings. The partition is
in the middle range as reported by the referenced studies. The simulation results show that
all the qualitative predictions are the same as in the benchmark case, and the quantitative
results are smaller but close to the benchmark case. For example, output and TFP in the
simulated low income country are now 10% and 28%, respectively. The entry cost alone can
reduce TFP (output) to 71% (62%) and the borrowing constraint on capital accumulation
alone can reduce TFP (output) to 91% (45%), and the interaction can reduce TFP (output)
to 43% (32%).
4.4.4
Informal Sector
There is a large informal sector in many poor countries. This paper exclusively focuses on
the formal sector not because we think the informal sector is not important, but because we
view the analysis as a benchmark in assessing the effects of the interaction between entry
costs and financial frictions on output and TFP. In this paper, financial frictions amplify
the effect of entry costs by increasing the effective entry cost. This channel will still operate
if the informal sector is modeled. D’Erasmo and Boedo (2012) take the existence of the
informal sector seriously and explore how the financial market structure and the costs of
informality, including the entry cost, affect cross-country TFP differences. However, they do
not focus on the interaction between entry costs and financial frictions. They find that the
model with the informal sector generates a larger decline in TFP than the model without the
informal sector. Because in countries with large entry costs, a large fraction of total output
is produced by the informal sector, which consists of low productivity firms compared to the
formal sector. This intuition is also valid in our model. Moreover, because the borrowing
constraint on business entry raises the effective entry cost, an even larger share of the output
would be produced by the low productivity informal firms if we were to include an informal
sector. Hence TFP and output would be even lower.
25
5
Conclusion
This paper analyzed how the interaction between entry costs and financial frictions affect
cross-country income and TFP differences. To perform such an analysis, we developed
a model that incorporates both entry costs and financial frictions. In the model, entry,
production, and exit decisions are all endogenous. To raise capital, establishments can
borrow from the financial market. To enter the market, new establishments must borrow
from the financial market to pay an upfront entry cost. Because the financial market is
imperfect, each establishment can only borrow up to a fraction of its expected discounted
lifetime profits.
The model is calibrated to match the establishment level statistics in the U.S. economy,
assuming a perfect financial market for the U.S. We simulate the model to jointly match the
entry cost to GDP per worker ratio and the debt to GDP ratio for a large set of countries.
The regression of the data on the model results show that the model accounts for 55%
and 46% of the cross-country variation in output and TFP, respectively, as measured by
R2 . Moreover, a large portion of the model’s explanatory power comes from the interaction
between entry costs and financial frictions. The main mechanism is that financial frictions
amplify the effect of entry cost by boosting the effective entry cost. This finding implies that
financial frictions and business entry costs must be addressed together to improve overall
productivity.
We assume that all new establishments pay the same entry cost, which might not be
true in reality. As Buera et al. (2011) has shown, allowing entry costs to vary across sectors
can generate significant quantitative effects on income and TFP. Similarly, the interaction
between financial frictions and sectoral or industrial entry costs may be worth studying. We
leave this for future research.
26
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30
Table 1: Parameter Values
Parameter
r
Value
4%
β
α
γ
δ
λ
zmax
0.96 0.27 0.53 0.08 0.9
5.48
fe
f
φ
σ
0.013 0.55 -5.78 1.25
Notes: The table reports calibrated parameter values.
Table 2: Targets
Statistics
Data
Model
Entry cost (% of GDP)
0.71%
0.71%
Average establishment Size 15.65
15.18
% of establishments with
1-4 employees
54.45% 59.27%
5-9 employees
18.92% 20.30%
10-19 employees
12.72% 9.81%
20-49 employees
8.63%
6.14%
50-99 employees
2.94%
2.20%
100-249 employees
1.67%
1.42%
250-499 employees
0.42%
0.51%
500-999 employees
0.16%
0.27%
1000+ employees
0.09%
0.09%
Notes: The table compares the targeted moments and the corresponding statistics implied by the
calibration.
31
Figure 1: The Effects of the Two Frictions
GDP per Worker
TFP
Notes: The figure plots GDP per worker and TFP from the data against the model’s predictions.
Each circle represents one country. The predicted values are obtained from simulations with
individual country’s values of entry cost to GDP ratio and debt to GDP ratio.
32
Table 3: Decomposition of the Contributions of the Two Frictions
Case 1
Case 2
Case 3
Case 4
Case 5
(fei , η i ) (fei , η us ) (feus , η i ) (feus , η i on K) (feus , η i on fe )
Y
HIC
0.62
0.84
0.72
0.97
0.77
MIC
0.23
0.82
0.28
0.75
0.38
LIC
0.09
0.62
0.14
0.51
0.27
HIC
0.71
0.88
0.79
0.99
0.83
MIC
0.41
0.86
0.47
0.96
0.49
LIC
0.25
0.71
0.36
0.92
0.39
HIC
0.96
1
0.96
0.96
1
MIC
0.51
1
0.51
0.51
1
LIC
0.22
1
0.22
0.22
1
TFP
K
Y
Notes: The table reports output, TFP and capital-output ratio in the average high income countries
(HIC), middle income countries (MIC), and low income countries (LIC). The values are all relative
to the corresponding values in the U.S. Case 1 reports the benchmark results with each group’s
levels of entry cost and financial frictions. Case 2 reports the results with each group’s level of
entry cost but without financial frictions. Case 3 reports the results with U.S. level entry cost but
with each group’s levels of financial frictions. Case 4 reports the results with U.S. level entry cost
but with financial frictions on capital accumulation only. Case 5 reports the results with U.S. level
entry cost but with financial frictions on entry only.
33
Figure 2: Model and Data Comparison
capital to output Ratio
Entry Rate
Average Size
Variance of Size
Notes: The figure plots capital-output ratio, entry rate, average establishment size, and the variance
of establishment size from the data against the model’a predictions. Each circle represents one
country. The predicted values are obtained from simulations with individual country’s values of
entry cost to GDP ratio and debt to GDP ratio.
34
Table 4: Model and Data Comparison
Variable
Correlation
R2
Slope
t-Statistics on Slope
Output
0.74
0.55
1.08
13.41
TFP
0.68
0.46
0.97
9.88
Capital to Output Ratio
0.41
0.17
0.36
4.86
Entry Rate
0.15
0.02
2.38
1.07
Average Establishment Size
0.66
0.44
0.42
7.25
Variance of Establishment Size
0.30
0.09
0.16
2.57
Notes: The table reports the correlation between the data and the model’s predictions on output,
TFP, capital-output ratio, entry rate, average establishment size, and the variance of establishment
size. The predicted values are obtained from simulations with individual country’s values of entry
cost to GDP ratio and debt to GDP ratio. The R2 , slope and t-Statistics reported are from the
fitted OLS regression in figure 1 and figure 2.
35
Figure 3: Model Implications with Capital Adjustment Costs
GDP per Worker
TFP
capital to output Ratio
Entry Rate
Average Size
Variance of Size
Notes: The figure plots output, TFP, capital-output ratio, entry rate, average establishment size,
and the variance of establishment size from the data against the predictions from the model with
capital adjustment costs. Each circle represents one country. The predicted values are obtained
from simulations with individual country’s values of entry cost to GDP ratio and debt to GDP
36
ratio.
Table 5: Model and Data Comparison with Capital Adjustment Costs
Variable
Correlation
R2
Slope
t-Statistics on Slope
Output
0.74
0.55
1.07
13.42
TFP
0.68
0.46
0.97
10.02
Capital to Output Ratio
0.41
0.17
0.37
4.88
Entry Rate
0.33
0.11
1.30
2.48
Average Establishment Size
0.67
0.45
0.42
7.37
Variance of Establishment Size
0.31
0.09
0.17
2.64
Notes: The table reports the correlation between data and predictions on output, TFP, capitaloutput ratio, entry rate, average establishment size, and the variance of establishment size from
the model with capital adjustment costs. The predicted values are obtained from simulations with
individual country’s values of entry cost to GDP ratio and debt to GDP ratio. The R2 , slope and
t-Statistics reported are from the fitted OLS regression in figure 3.
37
6
Appendix
Proof of Lemma 1.
(i): Since the per period profits and the choice set for k are both increasing in z and k−1 ,
standard dynamic programming argument can easily show that v(z, k−1 ) is increasing in z
and k−1 .
(ii) Let g(z, k−1 ) = v(z, k−1 ) − (1 + r)(1 − δ)k−1 . g(z, k−1 ) is then defined by:
g(z, k−1 ) = max zk α hγ − wh − (1 + r)(k + f )
k,h
Z
+β(1 − λ) max[g(z 0 , k), 0]dF (z 0 ) + βλ max[g(z, k), 0]
s.t.
k + f ≤ ηg(z, k−1 ) + (η(1 + r) + 1)(1 − δ)k−1
(6.1)
Since the per period payoff and the choice set for k are increasing in k−1 , it is easy to show
that g(z, s) is increasing in k−1 by applying the standard dynamic programming analysis to
the above problem.
38