Discussion Papers In Economics
And Business

Finance, Technology and Inequality
in Economic Development
Ryo Horii Ryoji Ohdoi Kazuhiro Yamamoto

Discussion Paper 05-08-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANFinance, Technology and Inequality
in Economic Development
Ryo Horii Ryoji Ohdoi Kazuhiro Yamamoto

Discussion Paper 05-08-Rev.
August 2005
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANFinance, Technology and Inequality
in Economic Development∗
Ryo Horii† Ryoji Ohdoi‡ Kazuhiro Yamamoto§
Graduate School of Economics, Osaka University
August 1, 2005
Abstract
This paper presents an overlapping generations model with technology
choice and credit market imperfections, in order to investigate a possible sourceof underdevelopment. The model shows that a better financial infrastructurethat provides stronger enforcement of contracts facilitates the development offinancial markets, which, in turn, enables firms to switch to more productive
and capital-intensive technologies, thereby promoting economic development.
In the presence of credit rationing, however, this technological switch widens
inequality. Therefore, risk-averse agents would not be willing to improve thefinancial infrastructure to the level at which the technological switch occurs,
resulting in a development trap. A remedy is to facilitate small firms’ adoptionof the currently used technology rather than the new one.
JEL Classification Numbers: O14, O16.
Key words: Enforcement; Technological Switch; Income Distribution; Credit
Rationing; Development Trap; Institutions.
∗

We are grateful for the comments from Koichi Futagami, Akiomi Kitagawa, Nobuhiro Kiyotaki,

Makoto Saito, Masaya Sakuragawa, Akihisa Shibata, Ping Wang, and the seminar participants at
Kansai Macroeconomics Workshop, Public Economic Theory 04 Peking, and Tokyo University.
This study was partly supported by JSPS Grant-in-Aid for Scientific Research (No.16730097). Allremaining errors are our own.
†

Correspondence: Graduate school of economics, Osaka University, 1-7, Machikaneyama, Toy-

onaka, Osaka, JAPAN; E-mail: horii@econ.osaka-u.ac.jp‡

E-mail: bg018or@srv.econ.osaka-u.ac.jp§

E-mail: yamamoto@econ.osaka-u.ac.jp

1

Introduction

Although earlier studies point to the accumulation of physical and human capital as
the main determinant of economic development, recent empirical analysis increasingly suggests that there remains a large gap in per capita income between rich andpoor countries even after controlling for diﬀerences in those factors.1 The search formissing determinants has been conducted from several directions. First, there is aconvincing argument that the unexplained gap implies the existence of some barrierthat prevents poor countries from adopting and eﬃciently using better technologies(e.g., Parente and Prescott, 2000, 2005). Second, as surveyed extensively by Levine(1997, 2005), a line of research supports the view that a country’s financial system
plays an important role in economic development.2 Developed financial markets contribute to distributing productive factors more eﬃciently across production units sothat more output can be produced from given aggregate inputs. Third, institutions
and government policies also appear to be important factors since they determine
the economic environment of individuals and firms and aﬀect how they behave. Halland Jones (1999) call those factors “social infrastructure,” which, they argue, causesa large portion of the observed international per capita income diﬀerences.
These alternative explanations are not necessarily in conflict with each other,
but they are all integral aspects of the process of economic development. Historicalevidence shows that the timing of technology adoption is critically dependent on
the development of financial markets. Sylla (2002) documents that, in most developed economies, ‘financial revolutions’ preceded major technological changes –theIndustrial Revolution in the case of England– and subsequent rapid economic development. 3 Financial markets, in turn, are aﬀected by the social infrastructure. La
1For example, see Mankiw, Romer, and Weil (1992) and Islam (1995), respectively, for earlier

and more recent views.
2

Christopoulos and Tsionas (2004) argue that the causality runs from finance to economic de-velopment and not vice versa.
3

Sylla (2002) reports “The Dutch financial revolution had occurred by the first decades of theseventeenth century, before the Dutch Golden Age... The British financial revolution in the late1Porta et al. (1997, 1998) and Levine, Loayza, and Beck (2000) show evidence thatthe development of financial markets is strongly influenced by legal and accounting
systems–i.e., by creditor rights, contract enforcement, and accounting standards,
all of which are parts of the social infrastructure.
This paper develops an overlapping generations model with technology choiceand credit market imperfections, which are used to investigate a possible source ofunderdevelopment. Consistent with the observations above, the model shows thatthe enforceability of contracts (a part of social infrastructure) determines the development of financial markets, which, in turn, aﬀects the choice and utilizationof technologies and, hence, economic performance. More specifically, stronger enforcement of financial contracts facilitates the productive use of capital through two
mechanisms. First, it makes credit accessible to an increased number of agents and,thus, raises the aggregate capital-labor ratio imputed to a given technology. Second,
while the agents must rely on labor intensive technologies when obtaining credit isdiﬃcult, improved enforcement of financial contracts enables them to choose fromwider set of technologies, including those with higher capital intensities and higher
productivities. Both imply that stronger contract enforcement, or, more broadly, abetter financial infrastructure, improves productivity and, therefore, raises the average income.
In this framework, the issue of economic development, or, equivalently, that ofincome diﬀerences, can be restated in terms of the diﬀerences in the financial infrastructure that determines the enforceability of contracts. A fundamental questionis why some countries are stuck with poor enforcement even though it results inprimitive financial markets and unproductive technologies. From the point of viewof political economy, socially optimal reforms are not always adopted because of
seventeenth and early eighteenth centuries, before the English industrial revolution. The U.S.
financial revolution occurred ..., before the U.S. economy accelerated its growth in the ‘statisticaldark age’ of the early nineteenth century”. He also notes “In the early Meiji era of the 1870s and1880s, Japan had a financial revolution ... Once their financial revolution was in place, the Japanese
were oﬀ and running.” See also Dickson (1967) for similar arguments.

2

conflicts between interest groups.4 While such conflicts may have played importantroles in a number of cases, it theoretically remains open whether it is in fact optimal to upgrade an economy’s financial infrastructure. While stronger enforcementof financial contracts improves the per capita income, as explained above, it alsochanges the distribution of income among agents in either desirable or undesirable
directions. In this paper, we show that, in certain situations, people are reluctant toimprove their financial infrastructure because they know that such an improvement
would undermine their welfare by widening income inequality.
The intuition behind this mechanism is as follows. When enforcement of financial
contracts is imperfect, credit rationing may be required to give borrowers the incentive to repay rather than to default. As long as the same technology is employed,
stronger enforcement weakens the necessity of credit rationing, thereby reducing theinequality. However, if improved contract enforcement triggers a technological switch
and the new technology requires a larger scale of operation, as is often the case, then
the number of agents who obtain credit may decline. As a result, the surplus fromthe better technology is monopolized by a small number of agents, and, therefore,
inequality rises.5
In a circumstance in which the means of redistribution are limited, the foreseeable rise in inequality gives a strong disincentive for risk-averse people to achieve
economic development through improving their institutions for enforcing financial
contracts. In this paper, we demonstrate that stronger enforcement actually worsens the (Benthamian) social welfare, if the minimum size of investment required
4

Rajan and Zingales (2003) suggest that industrial incumbents may want to leave financial

infrastructure underdeveloped in order to limit competition. Erosa and Hidalgo-Cabrillana (2005)
argues that if entrepreneurs, as a class, have the power to impede financial development, they would
do since it allows them to extract rents from the factor servicing they hire. See Drazen (2000) for
general discussions about the conflicting interests in economic reforms.
5

In England, for example, the technological switch from cottage technologies to manufacturing

technologies occurred in the textile industry and many others between 1759 and 1801, during which
the nominal Gini index was raised from 52.2 to 59.3 (Hobsbawm 1968; Lindert 1999). See also
Lindert and Williamson (1985).

3

for adoption of the currently used technology is within a certain range.6 Given theconsiderable variability across countries in both social and regulatory circumstancesthat aﬀect the minimum requirement, our finding provides one possible reason whysome economies face diﬃculties in achieving industrialization due to opposition fromtheir residents, while other economies successfully industrialize without major conflicts.7 Another implication of the model is that it may not be a good idea to force
developing countries to adopt ‘better’ institutions when the non-adoption of suchinstitutions is caused by welfare concerns rather than political reasons. If this is thecase, we show that an appropriate remedy is to facilitate adoption by small firms ofthe currently used technology, not the new one, until credit rationing is resolved.
In the literature, a number of theoretical works explain the rise of income inequality at early stages of economic development using models with credit market
imperfections. Banerjee and Newman (1998) and Greenwood and Jovanovic (1990)
derive the Kuznets curve as a result of the technological shift.8 If agents can only
gradually shift to the new sector where they receive a higher income, the distribution of income within the whole economy widens during the period when agentsare divided between the new and old sectors. This inequality, however, is always
welfare-improving as those who moved to the new sector are better oﬀ because they
voluntarily chose to do so while those who remained in the old sector are never worse
oﬀ because reduced labor supply in the old sector increases their wage. Aghion and
Bolton (1997) also shows that the rise in inequality in the early phases of development is beneficial to the poor since accumulation of capital by the rich makes
lending terms more favorable to poor borrowers. Thus, in these models, the fore6

Galor and Zeira (1993) and Matsuyama (2000) have already shown that the minimum invest-

ment requirement gives rise to inequality when the credit market is imperfect. Our new point isthat the degree of inequality after the technological switch is critically dependent on indivisibility
before the switch, mainly because the latter aﬀects the timing of the switch.
7

For example, as reported by Mokyr (1990), resistance to new technologies delayed their adoption

in 19th Century Continental Europe, where the old urban guilds restricted new entrants. In Britain,
where such guilds were not common, new technologies were adopted with little resistance.
8

See also Barro (2000, p. 9) for a survey of related work.

4

seeable rise in inequality does not retard economic development since its benefit isevery individual. By contrast, inequality in our model can be welfare-reducing since
it is caused by a particular sort of crowding-out eﬀect: once improvement in thefinancial infrastructure allows adoption of the new technology, those who are notgiven an entrepreneurial chance to start projects with that technology are no longer
even able to start projects with the old technology because demand for productive
factors by the entrepreneurs with the new technology crowds the old out of factor
markets. Thus, people resist the technological shift, thereby causing a development
trap.
The impact of financial markets on technological choice is also stressed by SaintPaul (1992), Castro, Clementi and MacDonald (2005), and Bencivenga, Smith andStarr (1995). Saint-Paul (1992) shows that, without a well-functioning financial
market, risk-averse agents may choose less specialized and less productive technologies. Castro et al. (2005) demonstrate that stronger investor protection facilitates
economic development given that the technology for producing investment goods
involves higher idiosyncratic risk than the technology for producing consumption
goods. In contrast, Bencivenga et al. (1995) show that a technological switch led bya better financial infrastructure may reduce the growth rate if the new technology
requires a longer length of periods for which investments must be committed. Thosestudies are closely related to ours in motivation, but our primary focus is to examine
the eﬀects of financial development on inequality and welfare when technologies are
diﬀerent in capital intensities and the sizes of minimum investments.
The rest of this paper is organized as follows. In Section 2, a model of technology
choice and credit rationing is constructed, in which we demonstrate how inequality
arises in equilibrium even when agents are almost homogenous ex ante. Section 3
contains a clarification of how enforcement of financial contracts aﬀects the choice oftechnology in equilibrium and when credit rationing occurs. Section 4 is a long-term
analysis, in which the eﬀects of improved enforcement on aggregate consumption,
inequality, and the welfare of agents are investigated. Using the results obtained,
Section 5 presents a discussion of the possibility of a development trap and eﬀective

5

remedies. The paper is concluded in Section 6. The appendix contains the proofs forselected propositions, while proofs of the other propositions are available from thecorresponding author upon request.

2

The Model

In this section, a model of technology choice and credit rationing is presented in order
to explain why credit rationing occurs when the enforceability of financial contracts
is limited and characterize the equilibrium distribution of income among agents. One
problem in deriving equilibrium is that the demand functions of production factors
may become discontinuous if individuals change the choice of technology all at oncein response to a marginal change in factor prices. We overcome this problem by first
introducing small ex ante heterogeneity among agents and then considering the limit
in which ex ante heterogeneity almost vanishes.

2.1

Economic Environments

An overlapping generations economy is considered, in which time is discrete and extends from zero to infinity (t = 0, 1, 2, . . . ). In each period, there are two generationsof two-period-lived agents. All generations are identical in size, and each contains acontinuum of agents with unit mass, indexed by i ∈ [0, 1]. In the first period of his
life, each agent born in period t supplies one unit of labor inelastically to the competitive labor market and receives wt units of consumption goods as wage income at
the end of that period. He (or she) is also endowed with a small exogenous income,²it , which is chosen randomly from a uniform distribution between 0 and ² > 0.9
The agent maximizes his utility, which depends only on the amount of consumption in the second period, ci t+1 . In order to finance this consumption, he makes useof his wealth, wt + ²it , in either of two alternative ways. First, he may save his entire
9²it gives a small perturbation to the realized incomes of otherwise identical agents. Later, weshow that this term is necessary for establishing the existence of equilibrium and that it can bearbitrarily small for that end. The limiting case where ² → 0 is considered at the end of this section.

6

wealth and consume ci t+1 = r(wt + ²it ) in the second period. Interest rate r ≥ 1 isconstant either because the economy considered is a small open economy or because
there is a storage technology that yields the gross rate of return r. In the latter
case, we assume that the demand for capital never exceeds the amount of aggregate
savings.10
His second option is to become an entrepreneur and start a project. Each agent
can run at most one project, and a project cannot be shared by multiple entrepreneurs
due to information and enforcement problems among them. When starting a project,an agent can choose from several types of technology, the set of which is finite
and denoted by J . Every technology produces a homogeneous consumption goodfrom capital and labor with constant returns to scale. Specifically, if agent i adopts
technology j, where j ∈ J , his project produces the consumption good according to kif ki t+1 ≥ Ij ,i t+1 fj (`i t+1 /ki t+1 ),(1)
yi t+1 =
 0
<I ,
if k
i t+1

j

where fj (·) is the per unit capital production function of technology j,`i t+1 , andki t+1 are the amounts of labor and capital inputs, respectively. Equation (1) showsthat exploiting the potential of each technology requires at least a certain amountof investment. The minimal required amount of capital, denoted by Ij ≥ 0, diﬀers
across technologies depending on technical aspects (e.g., the scope of scale economy
for that technology) and various barriers to the adoption of those technologies, which
may be specific to each economy. Capital depreciates completely within one period
and fj (·) satisfies the standard Inada conditions for all j ∈ J .
Capital to be used at period t+1 is obtained by converting the consumption good10

The open-economy assumption enables us to focus on the role of financial markets in determin-

ing the demand for capital and its composition rather than the supply of capital, which is givenby the amount of savings in the closed-economy setting. Consistent with our assumption, recent
empirical studies suggest that financial markets promote economic development not by enhancing
overall capital accumulation but by eﬃciently allocating capital across sectors (e.g., Wurgler 2000).
The assumption of storage technology is more suitable for low income countries, where inventories
are the principal substitutes for investment (see discussions by Bencivenga and Smith 1993, Section
5).

7

at the end of period t on a one-to-one basis. Thus, if the wealth of agent i at the endof his first period is larger than Ij , then he can manage the investment for the project,
adopting technology j using his own funds.11 If the agent’s wealth, wt + ²it , falls shortof the minimum required amount of capital, Ij , he must finance the gap by borrowing
from the competitive financial intermediaries, which we call banks. Banks can borrowfrom the international credit market at the constant world interest rate r, while
agents cannot do so because of the issue of limited enforcement, as explained below.
In order to obtain the loan that is needed to finance the investment, ki t+1 − wt − ²it ,the agent applies to banks by announcing the plan of his project, consisting of a
triple (j, ki t+1 , wt + ²it ). The choice of technology, j, the size of investment, ki t+1 ,and the amount of his own fund, wt + ²it , are verifiable and, thus, contractable. If
the agent is approached by several banks, he chooses a loan contract from the bankthat oﬀers the lowest gross interest rate, denoted by rit . If the agent is denied theloan at any interest rate–i.e., if he is credit rationed– he gives up becoming anentrepreneur and lends his entire wealth to the credit market.
At period t + 1, an entrepreneur (an agent who has successfully obtained credit
or has managed his investment fully using his own funds) hires a positive numberof workers, denoted by `i t+1 , at the market wage rate wt+1 . The revenue from his
project at the end of this period isρi t+1 = yi t+1 − wt+1 `i t+1 ,(2)

where yi t+1 is given by (1). The entrepreneur is obliged to repay the loan fromthis revenue, but he has an option to default at a certain cost. We assume thatthe cost of default is proportional to the revenue from the project, λρi t+1 , whereλ ∈ (0, 1) is a parameter representing the quality of enforcing financial contracts.
This is equivalent to assuming that only 100λ percent of the cash flow from any
project can be collateralized. If he defaults, his end of period consumption becomes
ci t+1 = (1 − λ)ρi t+1 ; otherwise, he repays the loan and consumes ci t+1 = ρi t+1 −11

Whenever an agent starts a project, he invests all of his wealth in the project since it is always

optimal to do so.

8ri t+1 (ki t+1 − wt − ²it ) units of the good.

2.2

Behaviors of Households and BanksThis subsection examines the rational behaviors of generation-t households (whobecome entrepreneurs at period t+1 if they obtain credit) and banks, taking as given
the market wage rate wt+1 . The decision processes are sequential and, therefore,
can be solved backward from the final decision: the decision at period t + 1 ofan entrepreneur who has already chosen technology j and the amount of capital
ki t+1 ≥ Ij . The remaining variable to be determined is the number of workers to hire.
Whether to default or not, the entrepreneur’s objective at this stage is to maximize
ρi t+1 with respect to `i t+1 . Diﬀerentiating (2) by `i t+1 yields fj0 (`i t+1 /ki t+1 ) = wt+1 ,which shows that there is a unique solution to this maximization problem,`i t+1 = èj (wt+1 )ki t+1 ,ρi t+1 = ρej (wt+1 )ki t+1 ,−1where èj (w) ≡ f 0 (w),jρej (w) ≡ fj ( èj (w)) − w èj (w).

(3)

Out of the maximized revenue ρej (wt+1 )ki t+1 , he repays the loan when it is in his
interest to do so. That is, the loan will be paid back if and only if

ρj (wt+1 )ki t+1 .
(eρj (wt+1 ) − ri t+1 ) ki t+1 + ri t+1 (wt + ²it ) ≥ (1 − λ)e(4)

Now let us focus on the decisions made by banks. Banks oﬀer loans to potential
entrepreneurs if and only if entrepreneurs are expected to repay and banks can earninterest at least as large as the market interest rate r. As long as repayment is
expected, competition among banks brings the interest rate down to r. Banks are
assured of the repayment if a prospective entrepreneur’s planned project, summarizedby (j, ki t+1 , wt + ²it ), satisfies condition (4) at interest rate ri t+1 = r as well as theminimum requirement for the size of investment. These conditions are summarized
in terms of the size of investment for the proposed project:¸·wt + ²it Ij ,if λe
ρj (wt+1 ) < r.
1 − λeρj (wt+1 )/r
ki t+1 ∈[Ij , ∞) if λe
ρj (wt+1 ) ≥ r,
9(5)

If the plan fails to satisfy (5), the project cannot obtain credit at any interest rate.12
We then go back to the decision of a prospective entrepreneur at the end of his
first period. He chooses technology j and the size of investment ki t+1 to maximize
his second-period consumption, (eρj (wt+1 ) − r) ki t+1 + r(wt + ²it ) if the plan satisfies (5),ci t+1 = r(w + ² ) otherwise.
t
it

(6)

Let us first examine whether it is feasible to adopt technology j. Note that theρj (wt+1 ) < r, or, equivalently,
equilibrium wage, wt+1 , must satisfy λe
wt+1 > max ρe−1
j (r/λ) ≡ w(λ),
j∈J

(7)

because violation of (7) will result in infinite amounts of investment and, therefore,in excess labor demand.13 Under (7), the amount of investment must be within thefinite interval, as designated by the first line of (5), which implies that technology
j can be adopted only when this interval is not an empty set. This condition isequivalent toµ¶λe
ρj (wt+1 )Ij ≡ ηj (wt+1 ; λ, Ij ),wt + ²it ≥ 1 −r(8)

where function ηj (·) represents the minimum amount of own funds required to borrowfrom banks to start a project with technology j. Since function ηj (wt+1 ; λ, Ij ) is
positive and upward-sloping under (7), condition (8) can be inverted to yield12

wt+1 ≤ ρe−1
j [(r/λ) (1 − (wt + ²it )/Ij )] ≡ ωjb (wt + ²it ; λ, Ij ).

(9)Note that a higher interest rate makes condition (4) more strict and gives borrowers more

incentive to default. Thus, banks cannot make (zero) profit by oﬀering a loan for projects that do
not satisfy (5) with an interest rate higher than r.
13

Suppose that λe
ρj (wt+1 ) ≥ r for some j ∈ J . Then, from the second line of (5), entrepreneurs

can obtain an infinite payoﬀ by investing an infinite amount of capital and hiring an unbounded
number of workers, which, necessarily, results in excess demand in the labor market. Thus, inequilibrium, wt+1 must be in a range where λe
ρj (wt+1 ) ≥ r does not hold for any j, which isguaranteed under (7).

10

We call (9), or, equivalently, (8), the borrowing constraint for technology j. Anytechnology is adoptable for agent i if and only if (9) is satisfied.
An agent also examines whether it is desirable to become an entrepreneur (i.e., tochoose some j and set ki t+1 > 0) rather than simply save his entire wealth (ki t+1 = 0).
From (6), it is optimal to invest a positive amount of capital in technology j only
when ρej (wt+1 ) ≥ r. This condition is equivalent to

wt+1 ≤ ρe−1
j (r) ≡ ω jp ,(10)which we call the profitability constraint. The constant ω jp represents the level ofwage at which a project with technology j breaks even.
Combining (9) and (10), we see that technology j satisfies both the borrowing
and profitability conditions if and only if
wt+1 ≤ min{ω jp , ωjb (wt + ²it ; λ, Ij )} ≡ φj (wt + ²it ; λ, Ij ).

(11)

Function φj (wt + ²it ; λ, Ij ) gives the upper bound for wt+1 , at which the adoptionof technology j is both feasible and desirable for an agent with a given amount ofown funds. Accordingly, we find that there is at least one technology that is both
adoptable and profitable for agent i if and only if
wt+1 ≤ max φj (wt + ²it ; λ, Ij ) ≡ θ(wt + ²it ; λ, I),
j∈J

(12)

where I ≡ {Ij }j∈J represents the list of minimum requirements. Note that the valueof function θ(·) on the right-hand side diﬀers among agents due to a random term
²it . According to the relative magnitude between wt+1 and the individual value of
θ(·), the optimal behavior of each agent can be categorized into three cases.
First, if wt+1 < θ(wt +²it ; λ, I) for agent i, then there must be some technology that
he can adopt whose rate of return ρej (wt+1 ) exceeds r.14 He becomes an entrepreneur,
14

In this case, (11) and (12) imply that there must be some j such that wt+1 < θ(wt + ²it ; λ, I) =min{ω jp , ωjb (wt + ²it ; λ, Ij )}. It follows that wt+1 < ωjb (wt + ²it ; λ, Ij ) and wt+1 < ω jp ; that is,
technology j is adoptable, and its rate of return is strictly larger than r.

11adopting that technology,15 and invests
ki t+1 =wt + ²itwt + ²it=Ij
1 − λeρj (wt+1 )/r
ηj (wt+1 ; λ, Ij )(13)units of capital because (6) implies it is optimal to invest as much as permittedby (5). Note that (wt + ²it )/ηj (wt+1 ; λ, Ij ) in equation (13) represents the ratio of
actual own funds to the amount required to obtain the credit and, therefore, must
be above 1. From (3) and (6), consumption of the entrepreneur and the individuallabor demand from this project are
wt + ²itIj èj (wt+1 ),ηj (wt+1 ; λ, Ij )wt + ²itρj (wt+1 ) − r)
ci t+1 = r (wt + ²it ) + (eIj .
ηj (wt+1 ; λ, Ij )`i t+1 = ki t+1 èj (wt+1 ) =

(14)
(15)

The second term in (15) is the extra income, or rent, obtained by becoming anentrepreneur. Second, if wt+1 > θ(wt +²it ; λ, I), the rate of return from any adoptable
technology falls short of r. Then, it is best for the agent to save his entire wealth(ki t+1 = `i t+1 = 0) and receive ci t+1 = r (wt + ²it ). Finally, if wt+1 = θ(wt + ²it ; λ, I),he may act similarly to the first case or may be indiﬀerent as to which option tochoose.16 If he is indiﬀerent, investment ki t+1 can be anywhere between the minimum
amount Ij and (13); the labor demand is `i t+1 = ki t+1 èj (wt+1 ), and any choice resultsin ci t+1 = r (wt + ²it ).

2.3

Inequality in Equilibrium

The analysis above implicitly gives individual labor demand as a function of themarket wage rate, wt+1 . This subsection establishes the existence of an equilibrium
15

In principle, there may be two or more technologies whose rate of return is larger than r. In

that case, the entrepreneur chooses the most profitable technology. In the limiting case of ² → 0
which we will consider later, however, there is generically only one technology that satisfies thiscondition and, therefore, this possibility can be ignored.
16

When wt+1 = θ(wt + ²it ; λ, I), there must be some j such that min{ω jp , ωjb (wt + ²it ; λ, Ij )} =wt+1 . This means that either the borrowing or profitability condition is binding. It is possible thatω jp > wt+1 and the agent strictly prefers to start a project, in which case the borrowing constraint
should be binding. Otherwise, the profitability condition is binding, and he is indiﬀerent as towhether to start a project or not.

12

wage rate at which the aggregate supply of and demand for labor are equalized. Wethen examine the extent of inequality arising in equilibrium.
Aggregate labor demand is obtained by summing the decisions of individual
agents,LD
t+1 (wt+1 ; wt ) ≡Z 1`i t+1 di,

(16)

0

which explicitly shows its dependence on wt+1 and wt . Recall that θ(wt + ²it ; λ, I)
is the threshold level of wt+1 for starting a project. Individual labor demand may
diﬀer across agents because they have heterogeneous amounts of own funds, wt + ²it ,and, therefore, the threshold θ(wt + ²it ; λ, I) may vary across agents. A rise in wt+1
decreases the number of agents whose θ(wt + ²it ; λ, I) is above wt+1 , which causes
aggregate labor demand (16) to decrease in wt+1 . Intuitively, an increase in wt+1
strengthens both the borrowing constraint (9) and the profitability constraint (10)and thus reduces the number of agents who are both able and willing to start projects.
A subtle point is that LD
t+1 (wt+1 ; wt ) may be set-valued at certain levels of wt+1 ; thatis, the demand curve of labor may have flat segments at which a mass of agents are
indiﬀerent as to whether to save or starts a project.17 If the set of labor demand ata certain wage level, LD
t+1 (wt+1 ; wt ), includes the level of labor supply, 1, then wt+1
is an equilibrium wage level. The following proposition establishes the existence ofan equilibrium wage level.
Proposition 1 Suppose that Ij > èj (ω jp )−1 for all j ∈ J and that there are, at
most, a finite number of intersections between functions ρej (w) and ρej 0 (w) for any

j 6= j 0 . Then, for any wt > 0, there is an equilibrium level of wt+1 ∈ [θ t , θ t ] withwhich 1 ∈ LD
t+1 (wt+1 ; wt ) holds, where θ t ≡ θ(wt ; λ, I) and θ t ≡ θ(wt + ²; λ, I).
Proof: in Appendix

Condition Ij > èj (ω jp )−1 means that the minimum size of each project is suchthat it requires hiring more than one worker, which we reasonably assume to besatisfied. Note that, on the one hand, wt+1 < θt cannot be an equilibrium since,in that case, all agents start projects and therefore aggregate labor demand would
17

Another possibility is that they are indiﬀerent as to which technology to adopt.

13

Figure 1: An Example of the θ Curve. It depicts a case of two technologies, J = {A, M }.
From (11) and (12), it is easily confirmed that function θ(·) is weakly upward-sloping and bounded
below by maxj∈J φj (0, Ij ) = maxj∈J ρe−1 (λ/r) = w(λ) and above by maxj∈J ω jp . The θ curveconsists of several flat segments and upward-sloping segments: each upward-sloping segment corre-

sponds to the borrowing constraint for a certain technology, (9), while each flat segment corresponds
to the profitability constraints, (10).

exceed the labor supply. On the other hand, if wt+1 > θ t , then no agent would starta project and, therefore, the aggregate labor demand would become zero. Moreover,
we prove in the Appendix that aggregate labor demand is continuous with respect
to wt+1 , relying upon heterogeneity in the amounts of own funds among prospective
entrepreneurs.18 It follows that there must be a level of wt+1 between θt and θ t at
which aggregate labor demand coincides with the supply.
In the remainder of this section, we present an intuitive explanation of how andwhen a significant income inequality arises among old agents in equilibrium.19 Figure
1 depicts a typical shape of function θ(·) against the amount of own funds, wt +18

Since LDt+1 (·) is a correspondence, the notion of continuity is diﬀerent from that for a function.

Precisely, we show in the Appendix that LD
t+1 (·) is convex-valued, non-empty, and upper hemi
continuous, which implies that the graph of labor demand in (LDt+1 , wt+1 ) space is jointed.
19

We concentrate on the income of the old agents by two reasons. First, the income of young

agents is uniformly distributed between [wt , wt + ²], the inequality among whom is negligible when ²
is suﬃciently small. Second, in our model, the welfare of agents is determined solely by the amountof consumption (= income) in their old age.

14

²it , which we call the θ curve. Proposition 1 implies that what is important indetermining the equilibrium is the shape of the θ curve only on a short interval[wt , wt + ²]. One possibility is that the curve is entirely flat in that interval. In thiscase, the labor demand curve is completely elastic (flat) at wt+1 = θt = θt , and theequilibrium wage is uniquely determined at this level. In equilibrium, the rate of
return on the most profitable technology among all feasible technologies is equal tothe exogenous interest rate.20 All agents are indiﬀerent as to whether to become
entrepreneurs or save their wealth. Either way, they obtain ct+1 = r(wt + ²it ). Given
that the magnitude of random income ²it is marginal, the inequality of consumption
in the second period is also marginal.
We have a diﬀerent distributional consequence, however, when the θ curve isupward-sloping in interval [wt , wt + ²]. The labor demand curve is downward-sloping
for wt+1 ∈ [θt , θt ] because the level of wt+1 aﬀects the number of entrepreneurs
who can obtain credit. In this case, the profitability constraint is not (generically)binding, and, therefore, the rate of return from starting a project is strictly higher
than r. This means that every agent strictly prefers to start a project, but theoverall labor demand would exceed the aggregate supply if all agents start projects.
Thus, the equilibrium wage wt+1 must be between θ t and θt so that some agents(whose θ(wt + ²it ; λ, I) is below wt+1 ) do not satisfy the borrowing constraint. In
other words, some agents must be rationed from the credit market. Consumption of
those credit-rationed agents is lower than entrepreneurs.
The income inequality that emerged in this way would not be removed even whenthe ex ante heterogeneity, represented by ², is infinitesimally small. The following
proposition explicitly makes this point.
Proposition 2 Suppose that the assumptions in Proposition 1 hold and the ex anteheterogeneity is vanishingly small (² → 0). Then, equilibrium at the limit is charac20

To understand this, note that the flat segments of the θ curve correspond to ω jp for some j.

In equilibrium, wt+1 = θt = θ t = ω jp means ρej (wt+1 ) = r from the definition of ω jp . For any
technology that is more profitable than j, that is, for all j 0 such that ω j 0 p > ω jp , definitions (11)and (12) imply that ωj 0 b (wt + ²it ; λ, Ij 0 ) < θ(wt + ²it ; λ, I) = ω jp = wt+1 for almost all agents and,
therefore, such a technology cannot be adopted because of the borrowing constraint.

15

terized as follows:(i) the equilibrium wage is determined by wt+1 = θ(wt ; λ, I).
(ii) the technology used in equilibrium is in argmaxj∈J φj (wt ; λ, Ij ).
(iii) If j ∗ = argmaxj∈J φj (wt ; λ, Ij ) and ωj ∗ b (wt ; λ, Ij ∗ ) < ω j ∗ p , then the economy specializes in technology j ∗ , and credit rationing occurs. The number of entrepreneurs
and their consumption are given by³

nt+1 = Ij ∗ èj ∗ (ωj ∗ b (wt ; λ, Ij ∗ ))

´−1

< 1,

ci t+1 = rwt + r((1 − λ)Ij ∗ − wt )/λ > rwt .

(17)
(18)

Proof: in Appendix.
(17) and (18) clearly show that, even at the limit where almost no ex ante heterogeneity exists, a significant number of agents (i.e., 1 − nt+1 of them) are credit
rationed, and their consumption rwt is significantly lower than those lucky enoughto obtain credit. In the remainder of the paper, we focus on this limiting case inorder to examine the degree of inequality arising from credit rationing rather thanthat arising from ex ante heterogeneity among agents.21

3

Technology Choice and Credit Regime

The objective of this paper is to clarify how the economy’s financial infrastructure,
represented by the enforceability of financial contracts λ, aﬀects the equilibrium
distribution of income and the welfare of agents. From (17) and (18) in Proposition
2, we can see that, other things being equal, a better financial infrastructure (largerλ) results in more equal income distribution (i.e., in larger nt+1 and smaller ct+1 )by easing credit rationing. However, Proposition 2 also implies that there are othermechanisms through which λ aﬀects income distribution in direct or indirect ways.
21

Note that, however, we could not start the analysis of the model without the random term ²it

since, in that case, Proposition 1 does hold and there may be no equilibrium. Fortunately, the proofof Proposition 1 in Appendix is valid regardless of how small ² > 0 is. Therefore, at the limit, we
can rely on Proposition 1 without actually considering the random term.

16

First, as property (i) says, λ aﬀects the dynamics of equilibrium wage wt+1 . Second,
property (ii) shows that λ is a determinant of the equilibrium choice of technology by
aﬀecting the highest level of wage that can be oﬀered by entrepreneurs using each typeof technology. Third, property (iii) implies that λ determines credit regime; that is,credit rationing occurs only when the borrowing constraint, which obviously dependson λ as represented by ωj ∗ b (wt ; λ, Ij ∗ ), is tighter than the profitability constraint, ω j ∗ p .
From among these issues, this section examines those of technology choice and credit
regime, leaving the analysis of dynamic issues for the next section.

3.1

Credit Regime

Let us start by considering whether or not credit rationing occurs in equilibrium,
given the choice of technology j ∗ . As shown by property (iii) of Proposition 2, credit
rationing occurs if borrowing constraint (9) is stronger than profitability constraint(10). A comparison of these conditions, where ²it → 0, givesωj ∗ b (wt ; λ, Ij ∗ ) Q ω j ∗ p ⇔ wt Q (1 − λ)Ij ∗ .

(19)(19) shows that there is a threshold level of the amount of own funds below which
entrepreneurs are constrained by the borrowing limit. This condition can also be
written as λ Q 1−wt /Ij ∗ . That is, given the amount of own funds wt , credit rationing
occurs if and only if the enforceability of financial contract is below 1 − wt /Ij ∗ . This
explains one mechanism through which stronger enforcement (a larger λ) leads tomore equality. Note that, however, the threshold level 1 − wt /Ij ∗ depends on theequilibrium choice of technology j ∗ . Therefore, we need to clarify how λ aﬀects thetechnology choice, which is to be examined below.

3.2

Technology Choice

The technology used in equilibrium changes, or a technological switch occurs, when
the technology with the largest φj (wt ; λ, Ij ) is overtaken by another technology (recall
property (ii) of Proposition 2). The number of usable technologies is potentially large,
and it is not unusual for any economy to experience a technological switch several
17

times during the process of economic development. Nonetheless, to see how income
distribution is aﬀected at each instance of a technological switch, it generically suﬃces
to focus on the two technologies involved, i.e., that with the largest φj (wt ; λ, Ij ) andthat with the second largest one. To examine the role played by the diﬀerence incapital intensity, let us suppose that these two are Cobb-Douglas technologies,
fj (`/k) = Aj (`/k)1−αj

where αA < αM .

(20)

The one with a lower share of capital is called technology A (e.g., agriculture), while
the other technology is called M (e.g., manufacturing). Substituting (20) into (3)and then into (10) and (9) gives the profitability and borrowing constraints for eachtechnology:
αb +1

wt+1 ≤ (1 − αj ) (αj /r)αbj Aj j

≡ ω jp ,wt+1 ≤ ω jp (λIj /(Ij − wt ))αbj ≡ ωjb (wt ; λ, Ij ),

(21)
(22)

where αbj ≡ αj /(1 − αj ) > 0. The value of φj (wt ; λ, Ij ) is given by the smaller of ω jpand ωjb (wt ; λ, Ij ) in (21) and (22).

While our concern is when a marginal increase in λ causes a technological switch,

it is insightful to see how the pattern of technological choice is aﬀected by largechanges in λ. In particular, when the economy’s financial infrastructure is quite primitive (λ → 0), then the borrowing constraint becomes very tight (ωjb (wt ; λ, Ij ) → 0
for every j), which means that φj (wt ; λ, Ij ) is determined by ωjb (wt ; λ, Ij ). In addition, (22) shows that the higher the capital intensity is, the more rapidly ωjb (wt ; λ, Ij )converges to 0. This means that, with a suﬃciently low λ, the economy specializes
in labor-intensive technology. Intuitively, if the enforcement of financial contracts isweak, only a small number of agents obtain funds due to tight credit rationing. In
that situation, entrepreneurs who have successfully obtained funds can hire a large
number of workers at a low wage level, for which case the labor-intensive technology
is more profitable. Conversely, when the enforcement of financial contracts is nearly
perfect (λ → 1), the borrowing constraint becomes weaker than the profitability constraint (ωjb (wt ; λ, Ij ) > ω jp for every j), which means φj (wt ; λ, Ij ) is determined by
ω jp . Hence, the economy specializes in the technology with the highest ω jp , which
18Figure 2: Technology choice and credit regime.The economy specializes in technology A inregions Ab and Ap and in technology M in M b and M p. The borrowing constraint is binding inregions Ab and M b, while the profitability constraint is binding in Ap and M p. This result is valid
within a range of (λ, wt ), in which technologies A and M have higher φj (wt ; λ, Ij )’s than others. The
parameters are αA = .20, αM = .45, r = 2.0, ω Ap = 1.20, and ω M p = 2.25 (these values are used
for all following figures). The figure shows the case of IA < IM (specifically, IA = 1.8, IM = 3.5).
Region Ap disappears if IA > IM .

is largely determined by the TFP of that technology Aj . Intuitively, when credit
is easy to obtain, many entrepreneurs start businesses, and wages go up. In thatcircumstance, entrepreneurs choose a more productive technology using which they
can break even with a higher wage level. Thus, when development in financial infrastructure triggers a technological change, the new technology tends to have higher
capital intensity and higher productivity.22
Reflecting this observation, we focus on the case in which TFP Aj is such thattechnology M can break even at a higher wage level. Specifically, ω M p ≥ ω Ap /(1 −αM ) is assumed, which implies ω M p > ω Ap since αM > 0.23 In this setting, φM (·)
22

If the productivity of the new technology is lower, it is not adopted. If its capital intensity

is lower, it should be adopted before the development of financial infrastructure. Within a large
collection of potentially usable technologies, there are of course some technologies that are highly
capital intensive but not very productive. Our argument is that, even though such technologies
exist, they are unlikely to be chosen in equilibrium at any stage of the development of financial
markets.
23

As shown by (21), the value of ω jp is determined by TFP Aj , and this is the only way Aj aﬀects

19and φA (·) can be compared by substituting (21) and (22) into (11). Since ω M p >ω Ap ≥ φA (·), the value of φM (·) becomes larger than φA (·) when either ωM b (·) ≥ ω Ap
or ωM b (·) ≥ ωAb (·) holds. From (21) and (22), solving ωM b (w; λ, IM ) ≥ ω Ap gives
λ≥

µ

ω Ap
ωM p

¶1/bαM µ

w
1−
IM

¶

≡ Λ1 (w; I),(23)

whereas ωM b (w; λ, IM ) ≥ ωAb (w; λ, IA ) holds if and only if w < IA and
"

ω Ap
λ≥
ωM p

µ

w
1−
IM

¶αbM µw
1−
IA¶−bαA #1/(bαM −bαA )≡ Λ2 (w; I).

(24)

For convenience, define Λ2 (w; I) = ∞ when w ≥ IA . From (23) and (24), it turns
out thatφM (wt ; λ, IM ) Q φA (wt ; λ, IA ) ⇔λ Q min{Λ1 (wt ; I), Λ2 (wt ; I)} ≡ Λ(wt ; I).

(25)

The representative shape of function Λ(wt ; I) is calculated numerically in Figure 2.
(25) shows the way in which the choice of technology in equilibrium depends on theenforceability of financial contracts. Since technology M is both more productive
and more capital intensive compared to A, there is a threshold level Λ(wt ; I) in theenforceability of a financial contract at which a technological switch from A to M
occurs.
Figure 2 also shows how the (λ, wt ) space is separated by condition (19) in addition to (25). Observe that there is no simple relationship between the degree ofcontract enforcement and the existence of credit rationing. Specifically, economies
in region M b experience credit rationing even though they have a better financial
infrastructure than those in region Ap, where no such rationing occurs. While it
may appear strange, this is not particularly at odds with reality. Credit rationing is
not necessarily most prevalent at the initial stage of economic development when thefinancial infrastructure is weak. Our model shows that such non-monotonic behavior
equilibrium. Therefore, even though Aj is an underlying parameter, we can specify the value of ω jp
as a parameter, and then Aj is determined accordingly by (21). We assume an inequality slightly
stronger than ω M p > ω Ap in order to reduce the number of cases to be analyzed without aﬀecting
the main finding.

20

arises because the degree of enforcement λ not only aﬀects the diﬃculty of obtaining
credit for a given technology but is also a determinant of the economy’s technology
specialization.

4Long-term Analysis

The previous section has shown that stronger enforcement of financial contracts
enables the economy to adopt more productive technologies but, at the same time,
may cause more inequality. As can be observed from Figure 2, the level of λ at whicha technological switch occurs as well as the degree of inequality caused by the switch
depends on the amount of own funds, wt , which is taken as given in the previous
section. However, since wt is determined by the previous period’s equilibrium inthe labor market and since the latter is critically dependent on λ, as we see in
Propositions 1 and 2, an examination of the long-term consequence of a stronger
enforcement on income distribution must take into account its eﬀects on the dynamics
of wages. The first half of this section investigates the dynamics of wages underdiﬀerent levels of enforcement. Then, in the second half, we examine how the level ofenforcement aﬀects income distribution and the welfare of agents in the long run.24

4.1

Dynamics of Wages

To be consistent with historical experiences that financial markets aﬀected economic
performance, it is reasonable to consider situations in which the size of minimum
investment is such that entrepreneurs must borrow part of their investments to start
projects: specifically, Ij > ω jp , where ω jp is the upper bound of the market wage
when the economy specializes in technology j. For notational simplicity, we retain
the two-technology Cobb-Douglas setting introduced in the previous section.
24

We use the term ‘long-term’ or ‘long-run’ to mean the stationary state of the economy for aconstant value of λ. The consequences when the financial infrastructure is gradually improved (i.e.,
when λ increases gradually) are explained in the next section, where the implications of having
more than three technologies are also discussed.

21

(i) IA ≤ ζ(IM )(ii) IA > ζ(IM )Figure 3: Patterns of long-term dynamics for diﬀerent sizes of minimum investments.
The thick locus represents the steady-state value(s) of wt for each λ. Arrows indicate the directionof the movement of wt below and above the thick curve. Minimum sizes of investments are IA = 1.8,
IM = 3 (panel i); IA = 1.5, IM = 4 (panel ii). The other parameters are the same as in Figure 2.

From property (i) of Proposition 2, wt evolves over generations according to
wt+1 = θ(wt ; λ, I). Using (19) and (25), the pattern of movements of wt , given λ andother parameters, is summarized as follows. When the (λ, wt ) pair is in region jp of
Figure 2, where j ∈ {A, M }, the equilibrium wage is determined by wt+1 = ω jp . In
region jb, it is given by wt+1 = ωjb (wt ; λ, Ij ), which, from (22), implies that
wt+1 R wt ⇔ λ R (1 − wt /Ij )(wt /ω jp )1/bαj ≡ λ∗jb (wt ; Ij ),(26)

where λ∗jb (wt ; Ij ) gives the level of enforcement at which wt becomes stationary.
Figure 3 depicts representative patterns of the evolution of wt , numerically calculated under two diﬀerent sets of parameters. A number of properties can be observed.
First, there is at least one steady state for each level of λ. Some steady states are
unstable, but the lowest steady state (i.e., the steady state with the lowest wt ) is always stable. Second, in the long run, the economy specializes in the labor-intensive
technology if enforcement of financial contracts is poor, whereas it specializes in
the capital-intensive technology if enforcement is stronger than a certain threshold.
Third, the steady-state income of young agents and, therefore, the amounts of theirown funds grow with the enforcement of contracts. This property holds even at thethreshold level of λ, and the technology switch does not aﬀect wages in a drastic
22

way. Fourth, the precise pattern of the evolution depends on the minimum size ofinvestment. The locus of the steady state transits region Ap, as shown by panel (i),only if IA is smaller than a certain threshold ζ(IM ), which is defined by¡¢−1.
ζ(IM ) ≡ ω Ap 1 − (ω Ap /ω M p )1/bαM (1 − ω Ap /IM )(27)

Intuitively, credit rationing in an economy specialized in technology A cannot be
resolved until that technology is abandoned if the size of the project is large. The
following proposition establishes that those four properties are fairly robust.
Proposition 3 Let j ∗ (λ) and w∗ (λ) denote the choice of technology and wage rate
at the lowest steady state. Suppose that increases in λ do not cause j ∗ (λ) ∈ {A, M }to switch more than twice.25 Then, the following properties hold:a. For any λ ∈ (0, 1), there exists w∗ (λ) ∈ (0, ω M p ]. In addition, if wT ≤ w∗ (λ) forsome T , then wt converges to w∗ (λ);
b. There exists λsw ∈ (0, 1) such that j ∗ (λ) = A if λ < λsw and j ∗ (λ) = M if
λ > λsw ;c. w∗ (λ) is weakly increasing in λ; in addition, it is continuous at λ = λsw ;d. w∗ (λsw ) < ω Ap whenever IA > ζ(IM ).
Proof: Available upon request.

4.2

Long-term Eﬀects of Stronger Enforcement

Relying upon the above result, we now examine the impacts of stronger enforcement
on income distribution among agents. Those considered are old agents since theutility of agents depends only on the amount of consumption when old. In region
jp (i.e., in region Ap or M p), no credit rationing occurs, and all old agents receive
25

By this, we ignore the possibility that the economy switches back to the old, labor-intensive

technology as a result of improvements in the financial infrastructure because there is little evidence supporting such a possibility and also because it is also theoretically very unlikely to occur.
Specifically, for this condition to be violated, functions λ∗Ab (w; IA ) and λ∗M b (w; IM ) have multiple
points of intersection in the range of w ∈ (0, ω Ap ), which we numerically found to occur only for avery narrow range of parameters.

23

(i) IA ≤ ζ(IM )(ii) IA > ζ(IM )Figure 4: Aggregate consumption and the Gini coeﬃcient at the lowest steady state.
Parameters: IA = 1.5, IM = 10 (panel i); IA = 8.5, IM = 10 (panel ii).

rwt . In region jb, substituting the equilibrium wage for (17) and eliminating Aj andωjb (wt ; λ, Ij ) by (21) and (22) show thatαbj ω jpN (j, wt , λ) ≡
rIj

µ

λIj
Ij − w t¶1+bαj(28)

Agents start projects and earn rwt + r((1 − λ)Ij − wt )/λ units of goods at the endof their second period, whereas 1 − N (j, wt , λ) agents are rationed from the credit
market and end up consuming rwt in a credit-constrained economy. The dependence
of income distribution on λ can be illustrated by focusing on aggregate consumption,
denoted by C, and the Gini coeﬃcient, G. Given technology j, current wage wt , andenforcement λ, they are expressed as26
C(j, wt , λ) = rwt + (r/λ)N (j, wt , λ) max [(1 − λ)Ij − wt , 0] ,(29)

G(j, wt , λ) = (1 − N (j, wt , λ))(1 − rwt /C(j, wt , λ)).

(30)

Their steady-state values for a given level of λ are obtained by substituting j = j ∗ (λ)
and w = w∗ (λ); namely, C ∗ (λ) ≡ C ∗ (j ∗ (λ), w∗ (λ), λ) and G∗ (λ) ≡ G(j ∗ (λ), w∗ (λ), λ).
In Figure 4, we calculated the representative shapes of C ∗ (λ) and G∗ (λ) forthe cases of IA > ζ(IM ) and IA ≤ ζ(IM ), respectively. In either case, a stronger
26

In regions Ap and M p, where credit rationing is absent, condition (19) shows that (1−λ)Ij −wt

is negative. In that case, (29) shows that C(j, wt , λ) = rwt .

24

enforcement increases aggregate consumption while reducing inequality, except at the
point at which a technology switch occurs. With strengthened enforcement, a larger
number of agents become able to start projects both through the weakened borrowing
constraint and their increased own funds (recall that w∗ (λ) is increasing). Then, theimproved allocation of resources increases the aggregate income of agents and, atthe same time, reduces inequality among agents. Thus, policies that strengthen
the enforcement of contracts unambiguously improve welfare as long as they do notcause a technological switch. The following proposition formally establishes those
properties.
Proposition 4 a. C ∗ (λ) is weakly increasing in λ for all λ;b. G∗ (λ) is weakly decreasing in λ for all λ except at λ = λsw .
Proof: Available upon request.
The remaining task is to examine the welfare consequences of stronger enforcement when it causes a technological switch. Figure 4 shows that the degree of
inequality increases discretely when λ reaches the threshold level λsw . Since aggregate consumption also increases discretely at the same time, the welfare eﬀect of atechnological switch is, in general, ambiguous. In the following, we explicitly examine the distributions of income before and after a technological switch and show thatthe minimum investment size plays an important role in determining whether thetechnological switch is desirable or not in terms of welfare.
First, consider the case in which IA ≤ ζ(IM ) (see panel (i) of Figure 3). Whenλ is slightly below the threshold λsw , the economy is in region Ap, with all old
agents receiving rω Ap . Credit rationing occurs when λ is slightly above the criticallevel, since the economy is now in region M b. This creates income inequality, but,
nonetheless, all agents receive at least rω Ap , which can be understood by noting thatthe steady-state level of wt is weakly increasing in λ and that all old agents receive atleast rwt regardless of their entrepreneurial opportunities. Technically, the (steadystate) income distribution with λ > λsw first-order dominates the distribution withλ < λsw , and, therefore, the overall welfare of agents is improving regardless of the
choice of criteria unless the disutility caused by envy, which is ignored in our analysis,25

Figure 5: Welfare consequence of the technological switch. The technological switch canworsen welfare if the (IA , IM ) pair is within the dark-gray area between the two thick curves. Theχ curve locates between the ζ curve and the 45-degree line when IM > I M , while the two curves
overlap for IM ≤ I M . The curves are calculated numerically under the same parameters as in
Figure 2 (αA = .20, αM = .45, r = 2.0, ω Ap = 1.20, and ω M p = 2.25).

is taken into account.
Next, let us consider the case in which IA > ζ(IM ), for which a comparison must
be made between income distribution in region Ab and that in region M b. In thiscase, the above argument cannot be applied because some old agents earn an extra
rent over rω Ap in region Ab. Specifically, if the number of agents who can get credit
in region M b is lower than that in region Ab, then the distribution of income withλ > λsw would not first-order dominate one with λ < λsw . The following propositionshows when this is the case.
Proposition 5 When a slight increase in λ causes a technological switch, the numberof old agents who earn more than rwt in the lowest steady state decreases if and only if
IA ∈ (ζ(IM ), χ(IM )), where χ(IM ) is a continuous function defined for IM ≥ I M , sat¡¢
αM /b
αA − 1) / (ω M p /ω Ap )1/bαM − 1 ,isfying ζ(IM ) < χ(IM ) < IM for IM > I M ≡ ω Ap (band χ(IM ) = ζ(IM ) for IM ≤ I M .

Proof: in Appendix.
Figure 5 shows representative shapes of functions ζ(IM ) and χ(IM ), by which
the (IA , IM ) space is divided into three areas. The technological switch, induced by26

a stronger enforcement of contracts, is welfare-improving when the minimal size ofinvestment for the labor-intensive technology is either suﬃciently small (IA ≤ ζ(IM ))or suﬃciently large (IA ≥ χ(IM )). In those cases, a technological switch enables moreagents to obtain rents from entrepreneurial opportunities, while the remaining agents
earn at least as much as the pre-switch levels (recall that rw(λ) is weakly increasingin λ). In addition, Proposition 4 implies that the aggregate amount of rents also
increases. Therefore, the technological switch is desirable under standard welfare
criteria.
When the size of investment is in an intermediate range, IA ∈ (ζ(IM ), χ(IM )),by contrast, the number of rent-earners is discretely reduced by the technologicalswitch even though it increases the amount of rent received by each entrepreneur.
The reduced probability of obtaining credits (and therefore rents) lowers the expectedutility of agents if they are highly risk-averse.27 This result can be explained as acertain type of crowding-out eﬀect. Once enforcement is strengthened to a criticalstandard, banks can expect some agents (those with relatively large ²it ) to repay thedebt when banks lend them enough funds to adopt the capital-intensive technology.
Agents who are in the fortunate position to be able to borrow enough for the newtechnology employ workers by paying a wage rate marginally higher than any that
could be oﬀered by entrepreneurs using the old, labor-intensive technology within
their respective borrowing constraints. In addition, they strictly prefer to do so
because the new technology yields higher returns to their projects than the oldtechnology. This implies, however, that agents without an entrepreneurial chance to27

We measure the expected utility before the random income is realized when all agents share an

equal chance of obtaining credit. The technological switch caused by a marginal increase in λ aﬀects
the steady-state income distribution in three respects. First, the income of credit-rationed agents(i.e., savers) increases marginally. Second, the number of entrepreneurs falls discretely. Third, theamount of income received by each entrepreneur increases discretely. The first eﬀect is a marginal
one and is, therefore, dominated by the second and the third ones, unless risk aversion is infinite.
The change of probability (the second eﬀect) linearly aﬀects the expected utility, while the thirdis subject to decreasing marginal utility. Therefore, when the degree of risk aversion is suﬃciently
high (but not infinite), the second eﬀect dominates.

27

start projects with the new technology are no longer able even to start projects withthe old technology since they are, in eﬀect, crowded out from the factor markets.
Thus, even when nobody wants a technological switch ex ante, i.e., before their young
period incomes are revealed, they cannot stay with the old technology if λ reaches
the threshold.

5

Discussion: Possibility of a Development Trap

Before successful industrialized countries achieved current levels of prosperity, history
witnessed a number of major technological changes. While, for notational simplicity,
the analysis in the previous section focused on a two-technology case, it can easily
be fitted into a multi-technology context. The result is qualitatively the same as
the two-technology case: whenever a marginal increase in λ causes a technologicalswitch, welfare may either increase or decrease discretely depending on the size ofinvestment required for the pre-switch technology as well as other technological andinstitutional properties.28 The size of the minimum investment required for a given
technology, in turn, depends not only on the technical features of that technology but
also on institutions, such as regulations and the forms of organizations, which varyconsiderably across countries. Therefore, even when the set of available technologies
are the same across countries, countries may or may not face deteriorating welfare
at diﬀerent stages in the development of their financial infrastructure.
Such a diﬀerence in the welfare eﬀects gives people diﬀerent degrees of incentive(or disincentive) to improve their financial infrastructure. Let us illustrate howdevelopment traps might emerge in some economies but not in others. Recall thatparameter λ represents the enforceability of financial contracts, which relies on theeconomy’s legal and accounting systems. Since such systems are quite complex inevery economy, their reforms take considerable time and have, in general, a gradual28

Proposition 2 gives the choice of technology under given λ, from which we can identify eachtime that a marginal increase in λ causes a technological switch. Once we know the pre-switch andpost-switch technologies, Proposition 5 gives the welfare eﬀect of the switch.

28

nature.29 In the context of our model, these properties imply that a reform that aims
to improve the enforceability of a financial contract (i.e., to increase λt+1 relative toλt ) must be prepared at least one period in advance before it takes eﬀect and that
there is a small upper bound for the size of improvement that is feasible in one period(e.g., λt+1 − λt must be lower than some constant).30 This means that, unless theinitial wealth w0 is too high, we can approximate the evolution of the economy as
being always at the lowest steady state (recall Proposition 3). Suppose that thecurrent state of an economy is such that a small improvement in λ is expected tocause a technological switch. Then, the analysis in the previous section implies that,even without the costs associated with a reform,31 society as a whole is reluctant toimprove its financial infrastructure when there is a considerable (but not too huge)
fixed cost in adopting the currently used technology, since, in that case, it expects
the reform to result in a welfare-reducing technological switch.32 Thus, even though
the continual upgrading of such systems is an integral part of achieving economic
29

Actually, Li, Squire, and Zou (1998) confirm that the degree of credit market imperfections candiﬀer markedly across countries but change only slowly within countries.
30

Without this restriction, any reform would aim to improve λ to one, since it is obviously thebest: it means that 100 percent of the flow of future profits from any project can be collateralized,
under which the most profitable technology can be adopted without borrowing limits. However,even equipped with state-of-the-art tools, modern financial intermediaries cannot monitor lenders
perfectly and, therefore, do not usually lend too much. This implies the existence of a technologicallimit in the range of λ. In addition, the observed dependence of the current performance of thefinancial market on an economy’s colonial and legal origins (see La Porta et al., 1998) implies thatthe size of improvement within a given period is not very large.
31

Ando and Yanagawa (2004) consider a model in which a country has to hire workers to raise λt .

The inclusion of such a cost to our model would further increase the possibility of a development
trap.
32

Recall that there is no heterogeneity among young agents until the random income is realized

at the end of their first period. Note also that the policy does not aﬀect the utility of old agents
because it does not take eﬀect during their lives. By this setting, we consider an ideal situation
with no conflict in choosing whether or not to implement the reform. In this respect, our study iscomplementary to the literature of political economy, in which the importance of conflicts between
diﬀerent parties is intensively studied (see Drazen, 2000).

29

development, the necessary reforms may not be implemented when the adoption of
the currently used technology requires a relatively large amount of fixed investment,
possibly due to some barriers for entry. Welfare losses can be prevented, but only at
the cost of causing a development trap.
Some historical events are consistent with this observation. Mokyr (1990) argues
that, in the 19th Century, the old guilds of Continental Europe, which restricted
the entry of new firms into the market, became a fetter on technological progress.
He documents workers’ riots against spinning machines imported from Britain andconcludes that this resistance was one reason that the Industrial Revolution started
in Britain rather than on the Continent. Unions can also be an obstacle to theadoption of new technologies, particularly, when they monopolize the currently usedtechnology. Wolcot (1994) reports that the textile industry in India is an example.
One way implied by our theory to escape from this development trap is to facilitate the adoption of the currently used technology by small businesses so thatthe minimum size of investment is reduced. Such a policy is beneficial to people
even without considering a technological switch, since it weakens credit rationing
and thereby distributes the rent obtained by entrepreneurs to workers (in fact, it
can be confirmed that the income of credit-rationed agents, rw∗ (λ), increases as IA
falls). Moreover, if the minimum size can be reduced to a level below a threshold(specifically, when IA < ζ(IM )), credit rationing disappears, and technological switch
becomes Pareto improving.33 That is, even in a case in which the long-term object
is to promote the adoption of the new technology, the immediate policy should be toremove barriers to the adoption of the currently used technology rather than the newtechnology. In the above example of Continental Europe, the introduction of newtechnologies was retarded until the entry costs to guilds fell substantially, when resistance eventually subsided. A more contemporary issue is the cost of entry associatedwith regulation. Djankov et al. (2002) reports that the oﬃcial cost of following the33

Note that the threshold level of λ, given by Λ(wt ; I) in (25), gets larger when IA is smaller.

Thus, when IA is small, the credit rationing immediately after switching to technology M is not asstrong as when IA is large. This is another reason that the switch becomes welfare improving.

30

procedures required to start up a simple firm averages 46 percent of annual per capita
GDP in the world, with this number being systematically high in low-income countries. Our result implies that simplifications of procedures in low-income countries
would be essential to get out of the development trap.
Another eﬀective policy implied by our analysis is to protect the firms with thecurrently used technology, or even to deter the adoption of new technology, so thatthe technological switch does not occur until λt becomes large enough to resolve
credit rationing. Specifically, equation (19) shows that credit rationing is resolved
if the adoption of technology M is somehow deterred until λt reaches 1 − ω Ab /IA .
Once this is accomplished, no party earns rents or loses from the adoption of the
new technology. Although these kinds of policies are often criticized as protecting
those with vested interests in the status quo and often also regarded as an obstaclein the way of economic development,34 our analysis has shown that, under certain
conditions, they are eﬀective for avoiding welfare losses due to technological switchand, thereby, the development trap. In the process of economic development afterWorld War II, Okazaki (1996) reports that the Japanese government subsidized smallscale firms with primitive technologies while continually strengthening its financial
infrastructure. Our model confirms that this combination of policies was eﬀective in
fostering economic development while keeping down social conflicts.

6

Concluding Remarks

We have presented a model in which improvements in the enforcement of credit contracts facilitate economic development. With the provision of improved enforcement,
the economy switches from a labor-intensive technology to a more productive andcapital-intensive technology, following which both the average income and the degreeof inequality increase discretely. Our model shows that, in an economy where thecredit market is imperfect, the scale of fixed investments has important meanings.
If the scale of the fixed investments in the currently used technology is within a34See, for example, Canton, de Groot, and Nahuis (2002).

31

certain range and economic agents are suﬃciently risk-averse, industrialization worsens the welfare of agents due to the concentration of wealth on a small number ofentrepreneurs. This creates the possibility of a development trap, in which society
as a whole is reluctant to provide improved legal and accounting systems for fear of
causing a welfare-reducing technological switch.
The theory implies that a way to escape from the development trap is to facilitate
the adoption of the currently used (not new) technology by small businesses. It evenlegitimizes the protection of existing firms against new technologies for some time
while promoting the development of financial markets. Such policies will graduallyredistribute the rent received by existing entrepreneurs to the broader populationand thus mitigate the welfare loss and opposition associated with the technologicalswitch. There is a caveat, however. If the indivisibility of current technology is too
high, or entry is too restricted, then this kind of protection is not only unnecessary
but also causes a development trap.35 Therefore, policymakers should cautiously
examine whether the demand from some parties to protect existing sectors is simply
based on their vested interests or the protection is truly required to avoid economywide welfare losses.

35

If IA is originally higher than χ(IM ), policies that reduce IA cause an unnecessary trap if IA ispushed to below χ(IM ).

32

Appendix
Proof of Proposition 1
Here, we establish the equilibrium market wage at t + 1, denoted simply by w, taking
as given the predetermined market wage of the previous period wt . As explained inD
the text, min LD
t+1 (w; wt ) > 1 for all w ∈ (w(λ)θ t ) and Lt+1 (w; wt ) = {0} for all

w > θ t . Thus, the intermediate value theorem implies the existence of w ∈ [θt , θt ]Dsuch that LD
t+1 (w; wt ) 3 1 if the graph of Lt+1 (w; wt ) is jointed for all [Θ, Θ], where

Θ and Θ are arbitrary constants with Θ ∈ (w(λ), θ t ) and Θ > θ (note that we canalways choose such constants since θ is finite and θt > w(λ) from wt > 0). For this,
it is suﬃcient to show that LD
t+1 (w; wt ) is convex-valued, non-empty, and upper hemi
continuous (hereafter u.h.c.) for all w ∈ [Θ, Θ].
From the definition that LD
t+1 (w; wt ) is an aggregation of individual labor demand,
its non-emptiness and convexity are obvious; there is a continuum of agents, and theset of most preferred actions of each agent is non-empty and does not depend on theaction of others given w. It is also easy to see that set LD
t+1 (w; wt ) is compact. For
any w ∈ [Θ, Θ], condition (7) implies that the size of investment of each project isfinite. Then, there exists a finite upper bound in labor demand L in LD
t+1 (w; wt ),from (3). Since LD
t+1 (w; wt ) is closed by construction, it is compact.
According to the definition of Stokey and Lucas (1989, p. 56), a compact-valued∗
correspondence LD
t+1 : [Θ, Θ] → [0, L] is u.h.c. at w , if, for every sequence {wn }such that wn ∈ [Θ, Θ] and wn → w∗ , and for every sequence {Ln } such that Ln ∈∗LD
t+1 (wn ; wt ), there exists a convergent subsequence of {Ln } whose limit point L is in∗D∗LD
t+1 (w ; wt ). To show that Lt+1 (·; wt ) is u.h.c., fix w ∈ [Θ, Θ] and pick any arbitrary

sequences {wn } and {Ln } such that wn ∈ [Θ, Θ], wn → w∗ , and Ln ∈ LD
t+1 (wn ; wt ).
S∞
If n=1 wn is finite, there must be some N > 0 such that wn = w∗ for all n ≥ N .

∗D∗
Then, since Ln ∈ LD
t+1 (w ; wt ) for all n ≥ N and Lt+1 (w ; wt ) is compact, there is a∗convergent subsequence of {Ln } whose limit point L∗ is in LD
t+1 (w ; wt ). Therefore,

the conditions for u.h.c. are satisfied.
Sej (w)’s intersect
The remaining case is that ∞n=1 wn is infinite. Note that, since ρ33

with each other only a finite number of times, we can choose a subsequence of {wn }so that each element in set {eρj (wn )}j∈J ∪ r has a distinct value, i.e., so that there isalways a strict ordering of profitability among technologies as well as between investment and saving. Since the set of technologies J is finite, the number of patterns in
the ordering of profitability that possibly appear in that subsequence is also finite.
Therefore, there is at least one pattern of the strict ordering of profitability thatappears infinite times in sequence {wn }, from which we can construct a subsequence
wnk → w∗ such thatρe1 (wnk ) < · · · < ρebj−1 (wnk ) < r < ρebj (wnk ) < · · · < ρeJ (wnk ) for all k,(31)

where each of the available technologies is numbered in ascending order of profitability. In this particular ordering, J is the number of technologies, and b
j is the index
of the least profitable technology above saving.

Let us derive LD
t+1 (wnk ; wt ) using (31). Note that, from (5) and (6), it is optimalfor each agent to invest in the most profitable technology (that with the largest
index) under his specific borrowing constraint wt + ²it ≥ ηj (wnk ; λ, Ij ) as long as

there is such a technology above b
j. More specifically, he adopts technology j if andonly if the amount of his own funds is within the range of wt + ²it ∈ Wj (wnk ), wherej by
Wj (wnk ) is defined recursively for j = J, J − 1, . . . , b[ηJ (wn ; λ, IJ ), ∞)for j = J;kWj (wnk ) ≡[ηj (wn ; λ, Ij ), ∞) \ Wj+1 (wn ) for j = J − 1, · · · , bj.
kkFrom (9) and (14), an entrepreneur with own funds wt + ²it demands è(wnk )(wt + ²it )j. Since/(1 − λeρj (wnk )/r) units of labor whenever wt + ²it ∈ Wj (wnk ) for some j ≥ b²it is distributed uniformly between 0 and ², the aggregate labor demand is given by
J Z
X
j=b
j

(wt + ²)el(wnk ) d²
et+1 (wn ; wt ),
≡L
k
1
−
λe
ρ
(w
)/r
²
j
nk
wt +²∈Ej (wnk ;wt )(32)

where set Ej (wnk ; wt ) ≡ {² ∈ [0, ²]| wt +² ∈ Wj (wnk )} represents the range of random
et+1 (wn ; wt ) is a function
incomes with which technology j is chosen. Note that Lkand, therefore, set LD
t+1 (wnk ; wt ) has only one element. The only choice of sequence34

et+1 (wn ; wt ) for all k. When viewed as a correspondence,{Lnk } is such that Lnk = LkEj (w; wt ) is well defined and continuous for all w ∈ [Θ, Θ]. From (32), function
et+1 (w; wt ) is also well defined and continuous for all w ∈ [Θ, Θ]. Thus, given thatL

et+1 (wn ; wt ) converges to L∗ ≡ L
et+1 (w∗ ; wt ).
wnk ∈ [Θ, Θ] converges to w∗ , Lnk = L
k∗The final task is to show that L∗ ∈ LD
t+1 (w ; wt ). Consider the relative prof-

itability of each technology when the market wage is given by w∗ . Since ρej (w) is
continuous, taking limit wnk → w∗ in (31) implies

ρe1 (w∗ ) ≤ · · · ≤ ρebj−1 (w∗ ) ≤ r ≤ ρebj (w∗ ) ≤ · · · ≤ ρeJ (w∗ ).

(33)

In the limit, we only have a weak ordering of profitability. Nonetheless, agentsj find it at least weakly optimal towith own funds ²it ∈ Wj (w∗ ) for some j ≥ bchoose technology j, while other agents find it at least weakly optimal to save.

∗
∗
∗
D
e
Thus, LDt+1 (w ; wt ) 3 Lt+1 (w ; wt ) = L . It establishes that Lt+1 (w; wt ) is u.h.c. atw = w∗ . Since w∗ ∈ [Θ, Θ] is arbitrary, the correspondence LD
t+1 (w; wt ) is u.h.c. for

all w ∈ [Θ, Θ]. This completes the proof.

¥

Proof of Proposition 2
Proportions (i) and (ii) are directly obtained by taking limit ² → 0 in Proposition 1. The following proves property (iii). Note that, from the continuity offunction ωjb (·) for all j, we can choose suﬃciently small ² > 0 such that θ(wt +²it ; λ, Ij ∗ ) = ωj ∗ b (wt + ²it ; λ, Ij ∗ ) for all ²it ∈ [0, ²]. Then, Proposition 1 says that
wt+1 ∈ [ωj ∗ b (wt ; λ, Ij ∗ ), ωj ∗ b (wt +²; λ, Ij ∗ )] or, equivalently, ηj ∗ (wt+1 ; λ, Ij ∗ ) ∈ [wt , wt +²]. Now, consider the limiting case in which the degree of heterogeneity ² is infinitesimally small (² → 0). In the limit, the previous relationships indicate wt+1 →ωj ∗ b (wt ; λ, Ij ∗ ) and ηj ∗ (wt+1 ; λ, Ij ∗ ) → wt . Applying these for (14) yields li t+1 →Ij ∗ èj ∗ (ωj ∗ b (wt ; λ, Ij ∗ )), which gives the limiting value of labor demand by each en-

trepreneur. Since the total labor demand should be 1 in equilibrium, it suggests´−1³, which isthat the number of entrepreneurs in the limit is Ij ∗ èj ∗ (ωj ∗ b (wt ; λ, Ij ∗ ))smaller than the number of agents, 1, under Ij > èj (ω jp )−1 , as assumed in Proposi-tion 1. Similarly, taking the limit in (15) and eliminating ωj ∗ b (·) by (9) show that35

the limiting value of consumption of entrepreneurs is rwt + r((1 − λ)Ij ∗ − wt )/λ,where the second term is positive since, in any borrowing-constrained equilibrium,
the amount of own funds wt falls short of the portion of investment that cannot becollateralized, (1 − λ)Ij ∗ .

¥Proof of Proposition 5
It is suﬃcient to consider only the case of IA > ζ(IM ) since, as explained in thetext, we know that the number of rent earners before the technological switch isalready zero whenever IA ≤ ζ(IM ). Let wsw ≡ w∗ (λsw ) denote the steady-state
wage at the threshold. Using (28) and the continuity of w∗ (λ) at the threshold,the number of entrepreneurs at levels of enforcement slightly below and above thethreshold is given by N (A, wsw , λsw ) and N (M, wsw , λsw ), respectively. Let Q denote
the ratio N (M, wsw , λsw )/N (A, wsw , λsw ). Our concern is whether Q ≥ 1 or Q < 1.
Note that, from (26), the continuity of the steady state at the threshold means thatλ∗Ab (wsw ; IA ) = λsw = λ∗M b (wsw ; IM ). Using this relationship and (28), the expression
for Q can be simplified asQ=

αbM IA − wsw
N ss (M, wsw ).
=N ss (A, wsw )αbA IM − wsw(34)

From assumptions Ij > ω jp and ω M p > ω Ap , it follows that both IA − wsw andIM − wsw are positive, guaranteeing Q > 0. Moreover, it is implied that Q > 1
bA < αbM ).
whenever IA ≥ IM (recall that α

Let us examine how Q responds to changes in IA when IA < IM . Diﬀerentiating(34) with respect to IA gives

·
¸
α
bM
dwsw
dQ
sw
=
(IM − w ) − (IM − IA ),dIAαbA (IM − wsw )2
dIA

(35)

the sign of which depends on that of dwsw /dIA . Note that function λ∗Ab (w; IA ) andfunction Λ(w; IA , IM ) intersect at the point (λsw , wsw ), as shown by panel b of Figure
3. Equations (26) and (25) show that, with an increase in IA , λ∗Ab (w; IA ) shifts
to the right, whereas Λ(w; IA , IM ) shifts to the left, pushing the intersecting point
downward. This means that dwsw /dIA < 0, and, therefore, dQ/dIA > 0 from (35).
36

We confirmed that Q > 1 when IA = IM and that it gradually decreases as IA falls
for all IA > ζ(IM ). If limIA →ζ(IM ) Q < 1, the intermediate value theorem shows that
there exists a value of IA below which Q < 1 holds. We now calculate the limiting
value. From the definition ζ(IM ) and the continuity of wsw with respect to IA , observethat wsw → ω Ap when IA → ζ(IM ) (i.e., the point of technological switch approaches
region Ap, where w = ω Ap ). Substituting it into Q = N ss (M, wsw )/N ss (A, wsw ) andusing the definition of ζ(IM ) in (27) show
αbM
lim Q =
IA →ζ(IM )
α
bA

µ

ω Ap
ωM p

¶1/bαM

ζ(IM )
Q 1 ⇔ IM R I.
IM

If IM > I M , there exists χ(IM ) ∈ (ζ(IM ), IM ) such that Q < 1 for IA ∈ (ζ(IM ), χ(IM )).
When IM ≤ I M , we can χ(IM ) should be equal to ζ(IM ) so that (ζ(IM ), χ(IM )) is anempty set. Finally, since Q is continuous with respect to IM from (34), the implicit
function theorem guarantees that the value of IA at which Q = 1 changes continuously with respect to IM > I M and approaches ζ(IM ) as IM → I M . This implies the¥continuity of function χ(IM ).

37

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