ISSN (Print) 0473-453X

Discussion Paper No. 1202

ISSN (Online) 2435-0982

FINANCIAL DEVELOPMENT AND
MINIMUM CAPITAL REQUIREMENTS IN
MACROECONOMIC ANALYSIS

Miho Sunaga

March 2023

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Financial Development and Minimum Capital Requirements in
Macroeconomic Analysis∗
Miho Sunaga†
February 2023

Abstract
We develop a macroeconomic model with a moral hazard problem between ﬁnancial intermediaries
and households, which causes ineﬃcient resource allocation, to make us reconsider the ﬁnancial
regulation according to ﬁnancial development, and individual and aggregate economic activities inthe short and long runs. First, we show that in an economy where ﬁnancial market has not developed,
raising minimum capital requirements improves resource allocation and welfare in the long run, while
it reduces welfare in an economy where ﬁnancial market has developed. Second, our study reveals thatan economy with a minimum capital adequacy ratio of 8% has a larger drop in aggregate net worth,consumption, and output when a negative capital quality shock occurs. However, during the ﬁnancial
crisis, the economy recovers faster than an economy with a higher minimum capital ratio (about 10%).
These results indicate that tighter bank requirements temporally mitigate crises in economies with adeveloped ﬁnancial market; however, they do not promote their activity in the long run.
Keywords: Minimum Capital Requirements; Financial Development; Macroeconomic Analyses
JEL Classiﬁcation Codes: E44, G21, G28, O11, O16
Declarations of interest: none
∗ I am grateful to Koichi Futagami and Nobuhiro Kiyotaki for their thoughtful comments and suggestions since 2013 when this project started. I would
like to thank Kosuke Aoki, Ryo Horii, and Tatsuro Iwaisako as well as several seminar participants for their helpful comments. I acknowledge the ﬁnancial
support from the Grant-in-Aid for JSPS Fellows (Grant No. 20K22120).
† Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. E-mail: sunaga@iser.osaka-u.ac.jp;
miho.sunaga08@gmail.com

1

Introduction

After the 2008 ﬁnancial crisis, banks worldwide began to implement tighter capital adequacy
requirements under international banking standards such as Basel III. The Basel Committee sets
the minimum capital requirements for banks regardless of how individual ﬁnancial markets havedeveloped across the world. However, banks operate in ﬁnancial markets, and the development ofﬁnancial markets varies across countries.1 In this study, we are interested in the following researchquestion: Is it desirable for countries to have tighter capital adequacy requirements irrespective of
their ﬁnancial development? To determine the desirability of a policy, we focus on three components:
If this policy will mitigate ineﬃciency caused by indirect and direct ﬁnancing in the long run, howwell an economy recovers from a crisis, and the magnitude of the drop in aggregate economic
variables. The last two components are analyses in the short runs. To approach these issues, weconstruct a macroeconomic model that explicitly represents the development of a ﬁnancial market
by reassessing these tighter minimum capital requirements. Using this model, we demonstrate howminimum capital requirements can improve welfare in an economy, depending on the developmentof its ﬁnancial market.
Our macroeconomic model involves indirect ﬁnance (where households invest their assets throughﬁnancial intermediaries) and direct ﬁnance (where households directly manage capital in the market). However, households must pay management costs due to their inexperience in managing
capital compared to banks. Meanwhile, there is a moral hazard problem between depositors (households) and ﬁnancial intermediaries in an economy, which causes monitoring costs. Thus, indirect
and direct ﬁnancing are responsible for resource allocation ineﬃciency in this model. An economy’s eﬃciency (or ineﬃciency) is determined by the degree to which investors are protected by law inthe country’ s ﬁnancial markets or legal systems. This also determines how its ﬁnancial market will

ˇ 𝑎k
1Based on the World Bank’ s global ﬁnancial development database, 𝐶ih´ et al. (2013) show that the degree ofﬁnancial development diﬀers across countries. The measures are divided into 4: depth, access, eﬃciency, and stability.
For example, in notable literature on ﬁnancial development and growth, the ratio of private credit to GDP is used tomeasure depth (Levine (2005)). Access is determined by the percentage of accounts per thousand adults in commercial
banks. Eﬃciency and stability are measured by lending-deposit spread and Z-score. This study focuses on access andeﬃciency because these measures are particularly relevant to ﬁnancial crises.

2develop. Our model allows analysis of the minimum capital requirements to remove ineﬃciency.
Gertler and Kiyotaki (GK 2010) and the following studies based on GK (2010): Gerler, Kiyotaki,
Queralto (GKQ 2012), Gertler and Kiyotaki (GK 2015), Aoki et al. (2016), Gertler, Kiyotaki,
and Prestipino (GKP 2016, 2020 (a), 2020 (b), 2020 (c)) also address the moral hazard problem
and the role of ﬁnancial intermediaries in a macroeconomic model. Other related studies onmacroprudential policies are GKQ (2012) and GKP (2020 (b), 2020 (c)). However, these studies
aim to replicate the U.S. economy during and after the recent ﬁnancial crisis. Thus, direct ﬁnanceis the main cause of ineﬃciency in their model. Regulations and policies on bank activities under
the incentive compatibility constraint cannot improve welfare in the long run.2 Compared to theliterature, one of the key features of this model is that resource allocation ineﬃciency results fromdirect and indirect ﬁnancing.3 This study presents the optimal requirements for improving welfareof an economy by reducing ineﬃcient resource allocation in the long run. We consider an economy
where agents have to choose between the marginal costs of indirect and direct ﬁnance, both ofwhich depend on the development of its ﬁnancial market. Our model can be used by regulators toadjust the capital requirements accordingly to ensure that the banking sector remains resilient toany future economic downturns and that the economy can recover from a crisis and return to its
pre-crisis growth rate. Therefore, this study contributes to the existing literature on optimal policies
and ﬁnancial intermediaries within macroeconomic analysis following GK (2010).
This study is relevant to Townsend (1979), and Holmström and Tirole (1997). Like them, we also
include the moral hazard problem between investors and depositors and monitoring activities in ourmodel. Furthermore, this study models depositors’ monitoring cost and investor protection by legal
systems, based on Townsend’ s (1979) costly state veriﬁcation model and the empirical results of
La Porta et al. (1997). As in this study and GK (2010), Holmström and Tirole (1997) examine themoral hazard problem of investors diverting a portion of their funds for private beneﬁt. Moreover,2GK (2010) and their subsequent studies demonstrate how asset price shocks contributed to the recent ﬁnancial
crisis in the U.S. They describe how these shocks cause bank activity ineﬃciency in the short run. Further, they show
how these shocks spread across the entire economy. Moreover, GKQ (2012) and GKP (2020 (b), 2020 (c)) suggest
optimal policies for such a crisis.
3Focusing on ”access” and ”eﬃciency” of ﬁnancial development, the marginal costs of direct and indirect ﬁnancing
are calibrated in this model.

3

in this study and Holmström and Tirole (1997), the monitoring activities as agents to resolve it
are incorporated. This study analyzes ineﬃciency due to indirect ﬁnance and direct ﬁnance, whichallows us to analyze monitoring activities, requirements, and policies to improve welfare. Therefore,
this devise in this model gives us richer understanding of the eﬀects of raising minimum capital
adequacy ratio on an economy.
In this study, our model ﬁnds that in the long run, raising minimum capital requirements in
can increase ineﬃciency and reduce welfare in an economy with a developed ﬁnancial market.
Conversely, the same policy can reduce ineﬃciency and improve welfare in an economy with an
undeveloped ﬁnancial market. These results can be interpreted as follows: In an economy witha developed ﬁnancial market, welfare losses caused by ineﬃciency due to indirect ﬁnance aresmaller than those caused by direct ﬁnance. Thus, tighter regulations on ﬁnancial intermediaries
will reduce the welfare of the agents in the economy because it will lead to more ineﬃcient direct
ﬁnance and suppress their activities. However, in an economy where the ﬁnancial market has notdeveloped, welfare losses caused by ineﬃciency due to indirect ﬁnance are larger than those causedby direct ﬁnance. Thus, ﬁnancial intermediaries are suppressed by tight requirements, which shifts
direct ﬁnance to indirect ﬁnance and improves the welfare of agents. Moreover, our study reveals
that an economy with a minimum capital adequacy ratio of 8 % has a larger drop in aggregate
net worth, consumption, and output when its ﬁnancial market has developed. However, when anegative capital quality shock occurs during ﬁnancial crises, the economy recovers faster than an
economy with a higher minimum capital ratio (about 10%). Thus, regulators should set minimum
capital requirements based on the structure of ﬁnancial markets, the causes of ineﬃciency, andthe development of the ﬁnancial market. Given these ﬁndings, we should reconsider Basel III’ sminimum capital requirements.
The remainder of this paper is organized as follows. Section 2 presents the baseline model, witha moral hazard problem between ﬁnancial intermediaries and households, which causes ineﬃcient
resource allocation. Section 3 describes the dynamic system of an economy. Section 4 analyticallyinvestigates how minimum capital requirements aﬀect the steady state economy in countries with4

developed ﬁnancial markets than those with undeveloped ﬁnancial markets. Section 5 numericallyanalyzes how minimum capital requirements aﬀect each country with diﬀerent degrees of ﬁnancial
development in the long and short runs. Finally, Section 6 presents our concluding remarks.

2

Model

We consider a closed economy in which time is discrete. There is a representative household witha continuum of members of measure unity. The members consist of 𝑓 bankers and 1 − 𝑓 workers,
where 𝑓 is constant over time. Each banker manages intermediary ﬁnancial ﬁrms, while the workers
supply labor to the ﬁnal goods production sector. The following subsections elaborate on this ﬁnal
goods production sector, households, and ﬁnancial intermediaries’ behavior.

2.1

Final Goods Production Sector

With capital and labor inputs, each ﬁrm (non–ﬁnancial ﬁrm) in the ﬁnal goods production sector
produces output using identical constant returns to scale Cobb–Douglas production function:
𝑌𝑡 = 𝐴𝐾𝑡𝛼 𝐿 𝑡1−𝛼 ,(1)with 0 < 𝛼 < 1. 𝑌𝑡 , 𝐾𝑡 , and 𝐿 𝑡 denote the output, aggregate capital, and labor in period 𝑡,
respectively. 𝐴 is a productivity parameter that is constant over time. We assume that 𝐴 ≥ 1.
Let the ﬁnal goods be the numeraire. Perfect competition prevails in this sector. The proﬁt
maximization conditions are as follows.
𝛼𝑌𝑡
,
𝐾𝑡

(2)

(1 − 𝛼)𝑌𝑡
,
𝐿𝑡

(3)

𝑍𝑡 =

𝑤𝑡 =

where 𝑍𝑡 is the rental price in period 𝑡, and 𝑤 𝑡 is the wage rate in period 𝑡.

5

2.2 Capital Market
The law of motion for capital is given by

𝐾𝑡+1 = 𝐼𝑡 + (1 − 𝛿)𝐾𝑡 .

(4)

𝐼𝑡 denotes the aggregate investment in period 𝑡, where 𝛿 ∈ [0, 1) is the depreciation rate.
Capital is either mediated by banks (ﬁnancial intermediate ﬁrms) or directly held by households.
𝐾𝑡𝑏 + 𝐾𝑡ℎ = 𝐾𝑡 ,(5)

where 𝐾𝑡 is the aggregate capital in period 𝑡, 𝐾𝑡ℎ is the aggregate capital holdings of households inperiod 𝑡, and 𝐾𝑡𝑏 is the aggregate capital holdings of banks in period 𝑡. We assume that households
have a disadvantage in ﬁnancing businesses relative to banks. Speciﬁcally, to manage capital in thecapital market, households require to pay extra management costs, while banks do not4:

𝑓

𝑘



𝐾𝑡ℎ ; 𝐾𝑡



 
𝜒 𝐾𝑡ℎ
𝐾𝑡ℎ ,
=
2 𝐾𝑡

(6)

where 𝜒 > 0 is a parameter reﬂecting the disadvantage of households in the ﬁnancing business
relative to banks, which is constant over time. Moreover, we assume that management costs areproportional to the rate of investment per unit of aggregate capital, rather than its level.5 Since
aggregate capital consists of the capital holdings of households and banks, and households knowthat their assets include banks’ capital as future dividends, they take care of both the level of theirinvestment and aggregate capital. The aggregate capital given to households is 𝐾𝑡 . Moreover,households are lenders in the capital market.6 We consider a moral hazard problem in which,
4GKQ(2012), GK (2015), Aoki et al. (2018), and GKP (2016, 2020(a), 2020(b), 2020(c)) make the same assumption.
This cost function (6) is an application of the cost of adjustment, as described in Uzawa (1969). GK(2010), GKQ(2012),
Aoki et al. (2016), and GKP(2020 (a)) introduce the cost of adjustment as well as the management cost in macroeconomic
models with capital goods producers.
5Turnovsky (1996) makes the same assumption and adopts the same function as Equation (6). He uses the quadratic
convex adjustment cost function, which depends on the aggregate capital.
6We assume that the cost is inﬁnity when households are borrowers in the capital market.

6instead of investing in capital, bankers divert it . Moreover, we assume that households incur
monitoring costs in the spirit of the costly state veriﬁcation model as Townsend (1979) outlined.7
We assume that the cost function is linear in the deposits.
𝑓 𝑑 (𝐷 𝑡 ) = 𝜃𝐷 𝑡 ,(7)

where 𝐷 𝑡 is the household deposits and 𝜃 ∈ (0, 1). In this model, the marginal monitoring cost𝜃 decreases as the ﬁnancial market develops. Thus, low (or high) 𝜃 reﬂects a country whose
ﬁnancial market has developed (or undeveloped).8 Due to the moral hazard problem, households
do not deposit all their assets. They lend capital in the capital market even if they incur extra
capital management costs. Thus, they must choose between the marginal costs of indirect and direct
ﬁnance, which cause ineﬃcient resource allocation and welfare losses. Figure 1 depicts the capital
market of the economy. Direct ﬁnance is ineﬃcient due to the management costs, while indirect
ﬁnance is ineﬃcient due to the moral hazard problem, which diminishes as the ﬁnancial market
develops. In Section 4, we analyze the optimal regulation that alleviates ineﬃciency and improves
an economy’ s welfare according to the development of its ﬁnancial market.

2.3

Households’ Optimizing Behavior

Each household has many members, including 1 − 𝑓 workers and 𝑓 bankers. There is a probability
that banks will return dividends to households with a probability of (1 − 𝜎) 𝑓 every time they retire
their banking business. (1 − 𝜎) 𝑓 workers become new bankers in each period. Therefore, theratio of workers to bankers is constant. Thus, the total population is 1 in every period as shownin Figure 2.9 When a banker becomes a worker, he/she brings the net worth of banking to the

7We describe the moral hazard problem in detail in subsection 2.4.
8The degree of ﬁnancial development 𝜃 can be captured by legal protection of the capital market. A large 𝜃
captures the poor legal protection of the capital market. Based on data from 49 countries, La porta et al. (1997) foundthat countries with poorer investor protection, measured by both legal rules and law enforcement, have narrower andundeveloped capital markets.
9This set up is based on Gertler and Karadi (2011), and GK (2010), GKQ (2012), Aoki et al. (2016), GKP (2020(a),2020(b), 2020(c)) make the same assumption.

7

Figure 1: Direct Finance and Indirect Finance

Figure 2: Each member of households

8Figure 3: Structure of the economy

household. When a worker becomes a banker, the representative household provides a proportionof the total assets of the entire economy to the new banker as start-up funds. Let 𝜁 ∈ (0, 1) denotethe proportion of total assets of the entire economy provided to new bankers.10 Workers supply theﬁnal goods sector with one unit of labor inelastically. A representative household uses wage andasset earnings for savings and consumption. There are two ways of saving: deposit of assets withbanks or manage assets in the capital market. The structure of the economy is shown in Figure 3.
The utility maximization problem of the representative household can be written as follows:

max 𝑈𝑡 =𝐶𝑡 ,𝐾𝑡ℎ ,𝐷 𝑡∞Õ𝛽𝑖 ln 𝐶𝑡+𝑖 ,

(8)

𝑖=0

subject to a budget constraint
𝐶𝑡 + 𝐾𝑡ℎ + 𝐷 𝑡 + 𝑓 𝑘

𝐾𝑡ℎ ; 𝐾𝑡where
Π𝑡 = (1 − 𝜎)ℎ+ 𝑓 𝑑 (𝐷 𝑡 ) = 𝑤 𝑡 𝐿 + Π𝑡 + 𝑅𝑡𝑘 𝐾𝑡−1
+ 𝑅𝑡 𝐷 𝑡−1 ,𝑏
𝑅𝑡𝑘 𝐾𝑡−1− 𝑅𝑡 𝐷 𝑡−1− (1 − 𝜎) 𝑓 𝜁 𝐾𝑡 ,

(9)

(10)

where 𝐶𝑡 is consumption in period 𝑡, 𝐾𝑡ℎ is the capital investment by workers in period 𝑡, 𝐷 𝑡 is
10GK (2010), Aoki et al. (2016) and GKP (2020 (a)) make the same assumption.

9the deposits in period 𝑡, and 𝛽 ∈ (0, 1) denotes the discount factor. The wage income is 𝑤 𝑡 𝐿,
where 𝐿 𝑡 = 𝐿 = 1 − 𝑓 . 𝑅𝑡 represents the gross rate of return on the deposits. 𝑅𝑡𝑘 is the grossinterest rate in the capital market, where 𝑅𝑡𝑘 = 1 + 𝑍𝑡 − 𝛿. This implies that the rental price equals
the sum of the interest rate and depreciation cost such that 𝑍𝑡 = 𝑅𝑡𝑘 − 1 + 𝛿. The household
solves the above optimization problem, given the net funds transferred to households Π𝑡 . The ﬁrst
term on the right-hand side of Equation (10) is the net worth of retired bankers who return to thehousehold as dividends. The probability of retired bankers returning to the household as dividends
is 1 − 𝜎 ∈ (0, 1). 𝐾𝑡𝑏 denotes the capital investment by bankers in period 𝑡.11 The household
provides a proportion of total assets, 𝜁𝑡 𝐾𝑡 , to new bankers, (1 − 𝜎) 𝑓 , in each period.
The ﬁrst-order conditions for the utility maximization of consumption, capital investment ofhouseholds, and deposits yield the following conditions:

1 = Λ𝑡,𝑡+1
1 = Λ𝑡,𝑡+1

𝑅𝑡+1
;
1+𝜃
𝑘
𝑅𝑡+1

1 + 𝑓 0 𝐾𝑡ℎ ; 𝐾𝑡

(11)
,

(12)

where
Λ𝑡,𝑡+1 ≡
and
𝑓

0




𝐾𝑡ℎ
=𝜒
.
𝐾𝑡

𝐷 𝑇 + 𝐾𝑇ℎ = 0.

𝐾𝑡ℎ ; 𝐾𝑡

The transversality condition is lim𝑇→∞ Λ0,𝑇

𝛽𝐶𝑡
,
𝐶𝑡+1





Equations (11) and (12) provide the no-arbitrage condition in which the rate of return of capitalmanagement equals the rate of return of deposits as follows:𝑘Λ𝑡,𝑡+1 𝑅𝑡+1
− 𝑓 0 𝐾𝑡ℎ ; 𝐾𝑡 = Λ𝑡,𝑡+1 𝑅𝑡+1 − 𝜃.

(13)

11The details of aggregating each bankers’ net worth, capital investment, and deposit market clearing condition are
given in subsection 2.5.

10

2.4 Bankers’ Optimizing Behavior (Financial intermediate ﬁrms)
Each banker maximizes the discounted sum of the expected value of its own net worth 𝑉𝑡 . The
problem of a banker who begins the banking business in period 𝑡 and exits the bank at the end of
period 𝜏 and brings net worth back to the household is
max 𝑉𝑡 = 𝐸 𝑡
𝑛𝜏

" ∞
Õ

#
Λ𝑡,𝜏 𝜎 (𝜏−1)−𝑡 (1 − 𝜎)𝑛𝜏 ,

(14)

𝜏=𝑡+1

subject to the balance sheet condition:
𝑘 𝑖𝑏 = 𝑛𝑖 + 𝑑𝑖 , for 𝑡 ≤ 𝑖 ≤ 𝜏,(15)

the evolution of a continuing banker’s net worth:𝑛 𝑗 = 𝑅 𝑘𝑗 𝑘 𝑏𝑗−1 − 𝑅 𝑗 𝑑 𝑗−1 , for 𝑡 + 1 ≤ 𝑗 ≤ 𝜏,

(16)

the incentive compatibility constraint (17), and the minimum capital requirement (18), where 𝑛𝑡
denotes the net worth of each banker in period 𝑡, 𝑑𝑡 denotes the funds from households’ deposits ofeach banker in period 𝑡, and 𝑘 𝑡𝑏 represents the investment of each banker in period 𝑡. In the objectivefunction, Λ𝑡,𝜏 represents the stochastic discount factor of the representative household such that𝜏−𝑡Λ𝑡,𝜏 = 𝛽 𝐶 𝜏𝐶𝑡 , 𝜎 (𝜏−1)−𝑡 (1 − 𝜎) is the probability of the banker retiring at time 𝜏, and 𝑛𝜏 is the net
worth of the banker retiring at time 𝜏. Equations (15) and (16) are the constraints of the ﬂow offunds of a banker who continues banking business from 𝑡 to 𝜏. The initial conditions for the banker
who begins doing business at time 𝑡 are as follows: 𝑛𝑡 = 𝑘 𝑡𝑏 , 𝑑𝑡 = 0, and 𝑛𝑡 = 𝜁 𝐾𝑡 , where 𝜁 isconstant and very small for 0 ≤ 𝑡 < ∞.
We consider the following moral hazard problem in the spirit of Holmström and Tirole (1997) andGK (2010).12 After bankers collect deposits from households, they can leave the bank and divert
12GKQ (2012), GKP (2020 (a), 2020 (b), and 2020 (c)), and other related pieces of literature following GK (2010)that consider the moral hazard problem. We assume that the diversion of funds depends on households’ monitoringin the spirit of costly state veriﬁcation, similar to Townsend (1979) but contrasting with these studies. Moreover, eachbanker in this model faces not only the incentive compatibility constraint but also the minimum capital requirement.

11

a proportion of the funds for their private beneﬁt. We assume each banker can divert a fractionof the funds because of depositors’ protection and household monitoring. Thus, the incentive
compatibility constraint can be written as
𝑉𝑡 ≥ Θ𝑘 𝑡𝑏 = Θ𝑛𝑡 + Θ𝑑𝑡 ,(17)

where Θ ≡ 𝑚𝜃 and 0 < 𝑚 < 1. As we mentioned, in this model, as the ﬁnancial market develops, 𝜃decreases, and thus Θ decreases. The left-hand side of Equation (17) is the value of the investmentof bankers’ funds, whereas the right-hand side is the value of diverting these funds. The last equalityof (17) is given by Equation (15). Constraint (17) implies depositors’ monitoring activities andﬁnancial market developments determine how bankers divert funds. Both the continuing and newbankers are subjected to (17).
We introduce the minimum capital requirements into this model.13 This requirement forces
bankers to keep the ratio of their net worth to risky assets, that is, a larger capital adequacy ratio
than the minimum 𝜅. Thus, 𝜅 is a key policy parameter in this model, which ranges from 0 to 1. Let
𝜅𝑡 denote the capital adequacy ratio, such that 𝜅𝑡 ≡ 𝑘𝑛𝑏𝑡 . Formally, the minimum capital requirements𝑡in this model are described as follows:𝜅𝑡 ≥ 𝜅.

(18)

Generally, from (14), the value of the banker who begins his or her business in period 𝑡 satisﬁes
the following Bellman equation:
𝑉𝑡 = 𝐸 𝑡 Λ𝑡,𝑡+1 (1 − 𝜎)𝑛𝑡+1 + Λ𝑡,𝑡+1 𝜎𝑉𝑡+1 .

(19)

13Holmström and Tirole (1997) introduce the ﬁnancial intermediaries instead of other ﬁnancial regulations to monitor
the moral hazard problem between depositors and investors who divert a proportion of funds for private beneﬁt. We
introduce these capital requirements because this model has both a moral hazard problem and a problem causedby households’ ineﬃcient investment. GKP (2020 (b), 2020(c)) proposes a state-dependent time-varying capitalrequirement as a macroprudential policy to prevent bank runs. Instead of considering time-varying requirements andbank runs, we introduce the minimum capital requirement as a constant policy parameter to demonstrate the optimal
policy to improve ineﬃcient resource allocation in an economy in the long run.

12

We guess the following linear value function to solve the decision problem:
𝑉𝑡 = 𝜄𝑡 𝑘 𝑡𝑏 − 𝜈𝑡 𝑑𝑡 ,(20)

where 𝜄𝑡 > 0 and 𝜈𝑡 > 0.
Let 𝜇𝑡 be deﬁned such that 𝜇𝑡 ≡ 𝜄𝑡 − 𝜈𝑡 . Assume that 𝜇𝑡 < Θ.14 The incentive compatibility
constraint (17) with Equations (20) and (15) can be rewritten as follows:
𝑉𝑡 = 𝜇𝑡 𝑘 𝑡𝑏 + 𝜈𝑡 𝑛𝑡 ≥ Θ𝑘 𝑡𝑏 .

(21)

We assume that Equation (21) is binding. Then, the minimum capital requirements (18) and thedeﬁnition of 𝜅𝑡 yield𝜅𝑡 =Θ − 𝜇𝑡
≥ 𝜅.
𝜈𝑡

(22)

As a result, 𝜅˜𝑡 is deﬁned as the endogenous capital adequacy ratio that satisﬁes the incentive
compatibility constraint, such that𝜅˜𝑡 ≡

Θ − 𝜇𝑡.
𝜈𝑡Using the above deﬁnition of 𝜅˜𝑡 , (22) can be rewritten as
𝜅𝑡 = max ( 𝜅˜𝑡 , 𝜅) .

(22’)

After combining the conjectured value function (21), the Bellman equation (19), and constraints on
the ﬂow funds of a continuing banker (15) and (16), we verify that the value function is linear in 𝑘 𝑡𝑏
and 𝑛𝑡 such that 𝑉𝑡 = 𝜇𝑡 𝑘 𝑡𝑏 + 𝜈𝑡 𝑛𝑡 with 15hi𝑘𝜇𝑡 = 𝐸 𝑡 Λ𝑡,𝑡+1 Ω𝑡+1 𝑅𝑡+1
− 𝑅𝑡+1 ,(23)

14The incentive compatibility constraint (17) can be written as 𝜈𝑡 𝑛𝑡 ≥ (Θ − 𝜇𝑡 ) 𝑘 𝑡𝑏 . Because 𝜈𝑡 > 0, 𝑛𝑡 > 0, and𝑏𝑘 𝑡 > 0, (Θ − 𝜇𝑡 ) > 0 allows us to analyze an economy in which the incentive compatibility constraint is binding.
15See Appendix A.

13



𝜈𝑡 = 𝐸 𝑡 Λ𝑡,𝑡+1 Ω𝑡+1 𝑅𝑡+1 ,

(24)

where
Ω𝑡+1 ≡ (1 − 𝜎) +𝜎𝜇𝑡+1
+ 𝜎𝜈𝑡+1 ,
𝜅𝑡+1

(25)

and
𝜅𝑡+1 = max ( 𝜅˜𝑡+1 , 𝜅) .

(22’)

Λ𝑡,𝑡+1 Ω𝑡+1 represents the stochastic discount factor of the banker. Ω𝑡+1 is the marginal value of net
worth, which is the weighted average of the marginal values for existing and continuing banks. The
shadow value of a banker’s capital holdings is 𝜇𝑡 . The shadow value of the banker’s net worth is 𝜈𝑡 .

2.5

Aggregation and Equilibrium

In period 𝑡, there are 𝜎 𝑓 continuing bankers and (1−𝜎) 𝑓 new bankers. Each 𝜎 of the 𝜎 𝑓 continuingbankers who continue their business before 𝑡 − 1 has capital holdings 𝑘 𝑡𝑏 , deposits 𝑑𝑡 , and the net
𝑏 −𝑅 𝑑worth 𝑅𝑡𝑘 𝑘 𝑡−1
𝑡 𝑡−1 in period 𝑡. Each 1 − 𝜎 out of the 𝜎 𝑓 bankers who begin their business inperiod 𝑡 − 1 have capital holdings 𝑘 𝑡𝑏 , deposits 𝑑𝑡 , and the net worth 𝑅𝑡𝑘 𝜁 𝐾𝑡−1 in period 𝑡. Each newbanker has capital holdings 𝜁 𝐾𝑡 and net worth 𝜁 𝐾𝑡 in period 𝑡. Thus, the aggregate capital holdings
and aggregate net worth of bankers 𝐾𝑡𝑏 and 𝑁𝑡 are as follows:
𝐾𝑡𝑏 = 𝜎 𝑓 𝑘 𝑡𝑏 + (1 − 𝜎) 𝑓 𝜁 𝐾𝑡 ;(26)𝑏
𝑁𝑡 = 𝜎𝜎 𝑓 (𝑅𝑡𝑘 𝑘 𝑡−1− 𝑅𝑡 𝑑𝑡−1 ) + (1 − 𝜎)𝜎 𝑓 𝑅𝑡𝑘 𝜁 𝐾𝑡−1 + (1 − 𝜎) 𝑓 𝜁 𝐾𝑡 .

The deposit market clearing condition is

𝐷 𝑡 = 𝜎 𝑓 𝑑𝑡 .

14

(27)

Using Equations (26) and (27), the aggregate net worth 𝑁𝑡 can be rewritten as follows:𝑏
𝑁𝑡 = 𝜎 𝑅𝑡𝑘 𝐾𝑡−1
− 𝑅𝑡 𝐷 𝑡−1 + (1 − 𝜎) 𝑓 𝜁 𝐾𝑡 .

(28)

The goods market clearing condition is as follows:

𝑌𝑡 = 𝐶𝑡 + 𝐼𝑡 + 𝑓𝑘

𝐾𝑡ℎ ; 𝐾𝑡+ 𝑓 𝑑 (𝐷 𝑡 ) .

(29)

Because the sum of households’ total assets and bankers’ total assets is equal to the aggregate
capital, we obtain
𝐾𝑡ℎ + 𝐷 𝑡 + 𝑁𝑡 = 𝐾𝑡 .

(30)

𝐷 𝑡 + 𝑁𝑡 = 𝐾𝑡𝑏 .

(31)

From (5), (30) can be rewritten as

2.6

Summary of the Key Ingredients of This ModelIn this model, Θ and 𝜒 are key parameters which reﬂect how the ﬁnancial market develops. In
particular, Θ reﬂects a parameter that causes ineﬃcient resource allocation through direct ﬁnancing,while 𝜒 reﬂects a parameter that causes ineﬃcient resource allocation through indirect ﬁnancing, 𝜃determines how bankers can divert the funds: Θ in (17). In this model, we obtain the no-arbitrage
condition (13), which is the key equation, from the ﬁrst order conditions of utility maximization
problem. (13) includes the spread determined by the market mechanism. By adjusting the optimum
household behavior and the market, the key parameters of ﬁnancial development in this model:
Θ and 𝜒 play a critical role in determining resource ineﬃciency caused by direct and indirect
ﬁnancing. As a result of these mechanisms, raising minimum capital requirements (the policy inthis model) might work to improve welfare in the long run and to mitigate a crisis in the short run.

15

3

Dynamic System

To describe the dynamic system of the economy, we deﬁne two variables, 𝜂𝑡 and 𝑥𝑡 , as follows:𝐾𝑏

Let 𝜂𝑡 be the ratio of banks’ capital holdings to aggregate capital as 𝜂𝑡 ≡ 𝐾𝑡𝑡 . Then, the ratio of𝐾ℎhouseholds’ capital holdings to aggregate capital is 1 − 𝜂𝑡 ≡ 𝐾𝑡𝑡 . In an economy with large (small)𝜂𝑡 , more capital is managed by direct (indirect) ﬁnancing. Let 𝑥𝑡 denote the ratio of consumption toaggregate capital as 𝑥𝑡 ≡ 𝐶𝐾𝑡𝑡 . We describe the dynamic system of the model with 𝜂𝑡 , 𝑥𝑡 , and 𝐾𝑡 with
𝜅𝑡 , where 𝜅𝑡 = 𝑘𝑛𝑏𝑡 is the capital adequacy ratio, which must be larger than the minimum 𝜅 because𝑡of the minimum capital requirements, as deﬁned in Section 2.16
Combining the Euler equation (12) with management costs (6), the ﬂow constraints of continuingbankers (15) and (16), aggregation (26), (28), and (31), and deposit market clearing (27) yields
𝜎 𝑓 𝜁 [1 + 𝜒 (1 − 𝜂𝑡−1 )]𝜅𝑡 = 𝛽𝑥
,
𝑡 −1
[𝜂
]
−
(1
−
𝜎)𝜁
𝑓
𝑡
𝑥𝑡

(32)

where 𝜅𝑡 = max ( 𝜅˜𝑡 , 𝜅).17
From the bankers’ value functions (19) and (21), as well as aggregation (26), we obtain



𝑥𝑡−1
𝜂𝑡−1 − (1 − 𝜎) 𝑓 𝜁[(1 − 𝜎) 𝜅˜𝑡 + 𝜎Θ] ,Θ=𝛽𝜂𝑡 − (1 − 𝜎) 𝑓 𝜁
𝑥𝑡

(33)

where 𝜅˜𝑡 satisﬁes both bankers’ and workers’ optimization conditions (11)–(13) and (22)–(25).18
Using the goods production function (1), the proﬁt maximization conditions (2), and the households’ utility maximization conditions (12) with management costs (6), we obtain191 1− 𝛼𝛼𝐾𝑡 = (1 − 𝑓 )  1+𝜒(1−𝜂 ), Λ𝑡 −1,𝑡𝑡 −1 − (1 − 𝛿) 



(34)

16The aggregate ﬂow variables in period 𝑡 are 𝑌𝑡 , 𝑍𝑡 , 𝑤 𝑡 , 𝑅𝑡𝑘 , 𝐼𝑡 , 𝐶𝑡 , and 𝑅𝑡 . The aggregate stock variables are 𝐾𝑡 , 𝐾𝑡𝑏 ,ℎ
𝐾𝑡 , 𝐷 𝑡 , and 𝑁𝑡 . All these variables are described by 𝜂𝑡 , 𝑥 𝑡 , and 𝐾𝑡 with 𝜅 𝑡 , subject to minimum capital requirements.
17See Appendix B.
18See Appendix C.
19See Appendix D.

16where Λ𝑡−1,𝑡 = 𝛽𝑥𝑥𝑡𝑡−1 𝐾𝐾𝑡 −1.
𝑡
From the goods market clearing condition (29), the law of motion for capital (4), the production
function (1), the Euler equation (12) with (6) and (7), the ﬂow constraints for banks (15) and (16),as well as the aggregations (26)–(28), and (31), we obtain20
"

𝑥𝑡 =

#

𝐴[1 + 𝜒(1 − 𝜂𝑡−1 )] 𝐾𝑡+1 (1 − 𝛿)( 𝐴 − 𝛼) 𝜒
1 + 𝜒 (1 − 𝜂𝑡−1 )−𝜃 (1−𝜎) 𝑓 𝜁−𝜃𝜂𝑡 ,−−− (1 − 𝜂𝑡 ) 2 −𝜃𝜎 𝑓 𝜁𝛼Λ𝑡−1,𝑡𝐾𝑡
𝛼2𝛽 𝑥𝑥𝑡 −1𝑡(35)where Λ𝑡−1,𝑡 = 𝛽𝑥𝑥𝑡𝑡−1 𝐾𝐾𝑡 −1.
𝑡Using the deﬁnitions of 𝜂𝑡 , 𝑥𝑡 , and 𝑅𝑡𝑘 , combining the households’ budget constraint (9) with
(10), the goods market clearing condition (29) with (30), the aggregate net worth (28) and the lawof motion for capital (4) with (5), the production function (1) with the non-ﬁnancial ﬁrms’ proﬁt
maximization conditions (2) and (3) yields21
"



1− 𝑓
𝐾𝑡+1 − 𝐾𝑡 = (1 − 𝛿) + 𝛼
𝐾𝑡 1−𝛼 #
(𝐾𝑡 − 𝐾𝑡−1 ) .

(36)

Equations (32), (33), (34), (35), (36), and (22’) constitute the dynamic system of 𝑥𝑡 , 𝜂𝑡 , and 𝐾𝑡 with𝜅𝑡 that describes the economy.
Let Φ𝑥 and Φ𝐾 be the functions of the variables in periods 𝑡 − 1 and 𝑡, respectively. Let Ψ𝜂 ,
Ψ𝑥 , and Ψ𝐾 be the dynamics of the following variables: 𝜂, 𝑥, and 𝐾, which are described byvariables in period 𝑡 − 1, respectively. Equations (32) and (33) with (22’) yield the dynamics
of 𝑥𝑡 , 𝑥𝑡−1 , 𝜂𝑡 , and 𝜂𝑡−1 : 𝑥𝑡 = Φ𝑥 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝜂𝑡 ). Plugging Φ𝑥 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝜂𝑡 ) into (34) yields
𝐾𝑡 = Φ𝐾 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝐾𝑡−1 , 𝜂𝑡 ). By combining (35) and (36), and using Φ𝑥 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝜂𝑡 ) andΦ𝐾 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝐾𝑡−1 , 𝜂𝑡 ), we obtain the dynamics of 𝜂𝑡 as follows:𝜂𝑡 = Ψ𝜂 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝐾𝑡−1 ) .
Then, the dynamics of 𝑥𝑡 and 𝐾𝑡 are derived from 𝑥𝑡 = Φ𝑥 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝜂𝑡 ), 𝐾𝑡 = Φ𝐾 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝐾𝑡−1 , 𝜂𝑡 ),20See Appendix E.
21See Appendix F.

17and 𝜂𝑡 = Ψ𝜂 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝐾𝑡−1 ) as follows:
𝑥𝑡 = Φ𝑥 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝜂𝑡 ) = Φ𝑥 𝑥𝑡−1 , 𝜂𝑡−1 , Ψ𝜂 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝐾𝑡−1 ) ≡ Ψ𝑥 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝐾𝑡−1 ) .

𝐾𝑡 = Φ𝐾 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝐾𝑡−1 , 𝜂𝑡 ) = Φ𝐾 𝑥𝑡−1 , 𝜂𝑡−1 , Ψ𝜂 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝐾𝑡−1 ) ≡ Ψ𝐾 (𝑥𝑡−1 , 𝜂𝑡−1 , 𝐾𝑡−1 ) .
After 𝜂𝑡−1 , 𝑥𝑡−1 , 𝜂𝑡 , and 𝑥𝑡 are determined by the above system, 𝜅𝑡 is determined by (32) with (22’)
and (33).

4

Steady State Analyses

This section examines the optimal minimum capital requirements for steady-state economies. Minimum capital requirements in this model play the role of improving ineﬃcient resource allocation
due to the marginal costs of both direct and indirect ﬁnance in an economy.22 Let 𝑦 ∗ denote thelevel of variable 𝑦 in the steady-state. First, we compute the following four variables by evaluating
Equations (32), (33), (34), and (35) as well as (22’) at the steady-state: the capital adequacy ratio𝜅 ∗ , the ratio of households’ (banks’) management capital 1 − 𝜂∗ (𝜂∗ ), the ratio of consumption toaggregate capital 𝑥 ∗ , and aggregate capital 𝐾 ∗ in the following subsection.23

4.1

Aggregate Variables in the Steady StateIn the steady state, where 𝜂𝑡−1 = 𝜂𝑡 = 𝜂∗ , 𝑥𝑡−1 = 𝑥𝑡 = 𝑥 ∗ , and 𝜅𝑡 = 𝜅 ∗ , evaluating Equations (22’)
and (33) at the steady state yields
𝜅 ∗ = max ( 𝜅˜∗ , 𝜅) ;(22’ ∗)

Θ(1 − 𝛽𝜎),𝛽(1 − 𝜎)(33 ∗)

𝜅˜∗ =22In the steady-state, ineﬃciency of the model is caused by household capital investment and banks’ moral hazard
problem. Thus, the analysis of optimal capital requirements in the steady-state economy is meaningful for consideringa policy for improving the welfare of agents in an economy in the long run.
23(36) is satisﬁed in the steady-state where 𝐾𝑡 −1 = 𝐾𝑡 = 𝐾𝑡+1 = 𝐾 ∗ .

18where Θ = 𝑚𝜃 < 1 with parameters 𝛽, 𝜎, Θ ∈ (0, 1) and policy 𝜅 ∈ [0, 1]. Let Θ𝑚𝑎𝑥 be such that
Θ𝑚𝑎𝑥 ≡ 𝛽(1−𝜎)1−𝛽𝜎 and Θ𝑚𝑎𝑥 < 1 from 𝛽, 𝜎 ∈ (0, 1). We then assume that Θ ≤ Θ𝑚𝑎𝑥 , allowing us toanalyze an economy in which 𝜅˜∗ ≤ 1. The development of the ﬁnancial market is represented by𝜃 and Θ = 𝑚𝜃 < 𝜃. In this and the following sections, we focus on Θ as a parameter of ﬁnancial
development. Equation (33 ∗) shows the endogenous capital adequacy ratio, which increases as theﬁnancial market develops.
In the steady state, 𝜂𝑡−1 = 𝜂𝑡 = 𝜂∗ and 𝑥𝑡 = 𝑥𝑡−1 = 𝑥 ∗ . Using (22’ ∗) and (33 ∗), evaluating
Equation (32) in the steady state gives us𝜂∗ =𝑓 𝜁 [𝜎(1 + 𝜒) + 𝜅 ∗ 𝛽(1 − 𝜎)],𝜅∗ 𝛽 + 𝜎 𝑓 𝜁 𝜒(32 ∗)

where 𝜅 ∗ is given by (22’ ∗) and (33 ∗) with parameters 𝑓 , 𝜁, 𝛽, 𝜎 ∈ (0, 1), and 𝜒 > 0. From𝑓 𝜁𝜎∗Equation (32 ∗), 𝜂∗ ≤ 1 corresponds to 𝛽[1−(1−𝜎)𝑓 𝜁] ≤ 𝜅 , the following lemma gives the policy

parameter restriction of optimal requirements for 𝜂∗ ∈ (0, 1].
𝑓 𝜁𝜎Lemma 1 Let 𝜅 𝑚𝑖𝑛 be such that 𝜅 𝑚𝑖𝑛 = 𝛽[1−(1−𝜎)𝑓 𝜁] with parameters 𝑓 , 𝜁, 𝜎, 𝛽 ∈ (0, 1). Suppose𝜅 𝑚𝑖𝑛 is less than one. Then, the optimal minimum capital requirement must be 𝜅 such that 𝜅 ≥ 𝜅 𝑚𝑖𝑛 .
𝑓 𝜁 𝜎(1−𝜎)Let Θ𝑚𝑖𝑛 be arranged so that Θ𝑚𝑖𝑛 ≡ [1−(1−𝜎)𝑓 𝜁] (1−𝛽𝜎) . Then, from Equations (32 ∗) and (33 ∗),Θ𝑚𝑖𝑛 ≤ Θ correspond to 𝜂∗ ≤ 1 when 𝜅 ∗ = 𝜅˜∗ . We suppose that Θ𝑚𝑖𝑛 ≤ Θ ≤ Θmax to analyze
the economy in which the endogenous capital adequacy ratio and the proportion of direct (indirect)
ﬁnance in the economy are less than 1, respectively.
In the steady state, 𝜂𝑡−1 = 𝜂𝑡 = 𝜂∗ , 𝑥𝑡 = 𝑥𝑡−1 = 𝑥 ∗ , 𝐾𝑡−1 = 𝐾𝑡 = 𝐾𝑡+1 = 𝐾 ∗ ; thus, Λ𝑡−1,𝑡 = Λ𝑡,𝑡+1 =𝛽. Evaluating Equations (34) and (35) in steady-state yields
𝛼𝛽
𝐾 = (1 − 𝑓 )1 + 𝜒 (1 − 𝜂∗ ) − 𝛽(1 − 𝛿)∗𝑥∗ =

 1−1 𝛼,(1 − 𝛿)( 𝐴 − 𝛼) 𝜒
Θ 𝐷∗
𝐴[1 + 𝜒(1 − 𝜂∗ )]−1−− (1 − 𝜂∗ ) 2 −
,
𝛼𝛽
𝛼
2
𝑚 𝐾∗

19

(34 ∗)(35 ∗)

where𝐷∗ 𝜎 𝑓 𝜁[1 + 𝜒(1 − 𝜂∗ )] + (1 − 𝜎) 𝑓 𝜁 + 𝜂∗ ,
=
∗
𝐾
𝛽

(37)

and 𝜂∗ are given by (32 ∗), (22’ ∗), and (33 ∗) with parameters 𝑓 , 𝛼, 𝛽, 𝛿, 𝜎, 𝜁 ∈ (0, 1), 𝜒 > 0,
Θ𝐴 ≥ 1, and 𝑚= 𝜃, where Θ ∈ [Θ𝑚𝑖𝑛 , Θ𝑚𝑎𝑥 ], 𝑚 ∈ (0, 1), and 𝜃 ∈ (0, Θ𝑚𝑎𝑥𝑚 ].24

Next, we describe additional aggregate variables of the real economy in the steady state. Aggregate
output 𝑌 ∗ , and aggregate consumption 𝐶 ∗ are obtained from Equations (1), (4), (6), (7), (29), (22’∗), (32 ∗), (33 ∗), (34∗), and (37) as follows:𝑌 ∗ = 𝐴𝐾 ∗𝛼 𝐿 1−𝛼 = 𝐴𝐾 ∗𝛼 (1 − 𝑓 ) 1−𝛼 ,𝐶 ∗ = 𝑌 ∗ − 𝛿𝐾 ∗ −𝜒Θ(1 − 𝜂∗ ) 2 𝐾 ∗ − 𝐷 ∗ ,
2
𝑚

(1 ∗)

(38)

where 𝜂∗ , 𝐾 ∗ , and 𝐷 ∗ are given by Equations (32 ∗), (33 ∗), (34 ∗), and (37), and (22’ ∗) with
Θ
𝛼, 𝑓 , 𝛿, 𝑚 ∈ (0, 1), 𝜒 > 0, 𝐴 ≥ 1, and 𝑚= 𝜃, where Θ ∈ [Θ𝑚𝑖𝑛 , Θ𝑚𝑎𝑥 ]. We can evaluate the welfare

of all agents in the steady-state economy by using the aggregate consumption determined by (38)since the representative households consist of workers and bankers, and the utility function depends
on aggregate consumption.25

4.2

Eﬀects of Tighter Minimum Capital Requirements on the Aggregate
Economic Activity in the Long Run

This subsection examines how minimum capital requirements relating to ﬁnancial development in
the long-run economy before analyzing the optimal capital requirements. Equations (32 ∗) – (35 ∗),(37), (38), and (22’ ∗) show that the minimum capital adequacy ratio 𝜅 aﬀects aggregate variables
only through the ratio of direct ﬁnance to indirect ﬁnance 𝜂∗ when the steady-state adequacy ratio
sticks to the minimum. We determined that in an unregulated economy the steady-state capital
24Equation (37) is obtained by evaluating Equation (E1’) at the steady state in Appendix E.
25Van den Heuvel (2008) examines household consumption and bank deposits to determine the welfare of the bankcapital requirements.

20adequacy ratio is endogenous, such that 𝜅 ∗ = 𝜅˜∗ . By contrast, the steady-state capital adequacy ratio
in a regulated economy sticks to the minimum requirement: 𝜅 ∗ = 𝜅 from (22’ ∗) and (33 ∗). Then,
we obtain the relationship between ﬁnancial development and minimum capital requirements in thefollowing proposition:𝛽(1−𝜎)
Proposition 1 Let Θ be such that Θ ≡ (1−𝛽𝜎)𝜅. If the ﬁnancial market has developed enough suchthat Θ ≥ Θ, then the economy is regulated in the steady state, where the bankers’ capital adequacy
ratio is equal to the minimum such that 𝜅 ∗ = 𝜅. If the ﬁnancial market has not developed enoughsuch that Θ < Θ, then the economy is unregulated in the steady state, where the bankers’ capital
adequacy ratio is larger than the minimum such that 𝜅 ∗ = 𝜅˜∗ > 𝜅.
𝛽(1−𝜎)
Proof. Let Θ be such that Θ ≡ (1−𝛽𝜎)𝜅. Then, from (22’ ∗) and (33 ∗), Θ ⋛ Θ correspond to 𝜅 ⋛ 𝜅˜∗ .

Proposition 1 provides the threshold between the unregulated and regulated economy Θ. It
depends on the discount factor 𝛽 ∈ (0, 1), the ratio of continuing bankers in the banking sector𝜎 ∈ (0, 1), and the minimum requirements 𝜅. To interpret Proposition 1, Figure 4 exhibits ﬁnancial
markets in the regulated and unregulated economy range between in the range of Θ𝑚𝑖𝑛 < Θ ≤ Θ ≤Θmax and the endogenous bank capital adequacy ratio, which takes the value of 𝜅 𝑚𝑖𝑛 to 1.26 The
solid line in Figure 4 represents the steady-state capital adequacy ratio, which is determined by (22’∗∗∗) with (33 ∗). Because (1−𝛽𝜎)
𝛽(1−𝜎) > 1, the slope 𝜅˜ is steeper than the slope of 𝜅 = Θ. In Figure 4, thedotted area indicates an economy with low minimum requirements and a ﬁnancial market that hasdeveloped enough that Θ ≤ Θ is the regulated economy. In contrast, an economy without a large Θthat Θ < Θ is an unregulated economy.
From Proposition1, Equations (22’ ∗), (32 ∗)-(35∗), (1∗), (37), and (38), all the following steadystate variables: the ratio of capital holdings by banks to the aggregate capital 𝜂∗ , aggregate capital 𝐾 ∗ ,aggregate output 𝑌 ∗ , and aggregate consumption 𝐶 ∗ depend on how the ﬁnancial market develops
Θ, but not on the minimum requirement 𝜅 in an unregulated economy where ﬁnancial market has
not developed. Meanwhile, in a regulated economy where ﬁnancial market has developed, these(1− 𝜎)
26Lemma 1ensures that Θ𝑚𝑖𝑛 < Θ. Because 𝜅 < 1 and 𝛽(1−𝛽𝜎) = Θ𝑚𝑎𝑥 , Θ < Θ𝑚𝑎𝑥 , and Θ < 𝜅 are ensured.

21

Figure 4: Relationship between ﬁnancial development and minimum capital requirements

variables, except for aggregate consumption (welfare), depend on 𝜅 but not on Θ. Also, in a
regulated economy, welfare is aﬀected by both the minimum capital requirements 𝜅 and ﬁnancial
development Θ of the economy.
Now, we examine the optimal minimum capital requirements to improve ineﬃcient resource
allocation and welfare. As we have argued before, the economic variables in only the regulated
economy are aﬀected by raising (reducing) the minimum capital requirements. For an unregulated
economy, because the threshold Θ is aﬀected by raising the minimum capital requirements, if a large
capital adequacy ratio improves welfare, such a tight requirement raises the threshold, moves the
economy from unregulated to regulated, and thus improves welfare. We ﬁrst consider a regulated
economy where ﬁnancial market is well-developed and analyze the optimal capital requirements.
Consider an economy in which Θ ≤ Θ and 𝜅 ∗ = 𝜅. We examine how raising the minimum
capital requirements aﬀects the ratio of direct to indirect ﬁnance 𝜂∗ in the regulated economy. From
Equation (32 ∗), diﬀerentiating 𝜂∗ with respect to 𝜅 yields− 𝑓 𝜁 𝛽𝜎𝑑𝜂∗{[1 − (1 − 𝜎) 𝑓 𝜁] 𝜒 + 1} < 0.
=𝑑𝜅(𝜅𝛽 + 𝜎 𝑓 𝜁 𝜒) 2

(39)

The inequality (39) is obtained by (1 − 𝜎) 𝑓 𝜁 < 1 from 𝛽, 𝜎, 𝑓 , 𝜁 ∈ (0, 1), 𝜒 > 0, and 𝜅 ∈ [𝜅 𝑚𝑖𝑛 , 1].

22

Ineﬃciency in this economy is caused by households’ extra management cost of capital holdings
and monitoring costs for banks’ moral hazard problems in terms of goods. Since the marginal
management cost is 𝜒(1 − 𝜂∗ ), the inequality in (39) shows that raising the minimum capitalrequirements enhances indirect ﬁnance more than direct ﬁnance, and thus, its policy causes more
ineﬃcient resource allocation stemming from the former causes in the economy. From (34 ∗) and(39), diﬀerentiating 𝐾 ∗ with respect to 𝜅, we obtain𝑑𝐾 ∗ 𝑑𝐾 ∗ 𝑑𝜂∗ 𝜒 [1 + 𝜒(1 − 𝜂∗ ) − 𝛽(1 − 𝜎)] ∗ 𝑑𝜂∗
=
=
𝐾
< 0,
𝑑𝜅
𝑑𝜂∗ 𝑑𝜅
𝛼𝛽(1 − 𝛼)𝑑𝜅(40)where 1 + 𝜒(1 − 𝜂∗ ) − 𝛽(1 − 𝜎) > 0 from 𝜂∗ ≤ 1, 𝛽(1 − 𝜎) < 1, and 𝛼𝛽(1 − 𝛼) > 0. From (1 ∗)and (40), diﬀerentiating 𝑌 ∗ with respect to 𝜅, we obtain𝑌 ∗ 𝑑𝐾 ∗𝑑𝑌 ∗ 𝑑𝑌 ∗ 𝑑𝐾 ∗
=
=
𝛼
< 0.
𝑑𝜅
𝑑𝐾 ∗ 𝑑𝜅
𝐾 ∗ 𝑑𝜅

(41)

The inequalities in (40) and (41) show that raising the minimum capital requirements decreases
aggregate capital and, thus, output in an economy where ﬁnancial market has developed. Therefore,
the results from the comparative statistics imply that the regulator should not raise the minimum
capital requirement to attain a large output in an economy where ﬁnancial market has developed
such that Θ ≤ Θ. However, in this economy, households face a trade-oﬀ between the marginal
costs of direct and indirect ﬁnance, both of which cause ineﬃciency. Even if an economy attains alarge output by raising the minimum capital requirements, if the increase in large output is allocatedineﬃciently, this policy causes welfare losses. Conversely, even if the output is smaller, raising theminimum capital adequacy ratio can improve welfare through more eﬃcient resource allocation.
Therefore, we examine how raising the minimum capital requirement aﬀects welfare in a regulated
economy.

23

From (37) and (38), diﬀerentiating 𝐶 ∗ and 𝐷 ∗ with respect to 𝜅 yields𝑑𝐷 ∗𝛽 − 𝜎𝑓𝜁𝜒𝑑𝜂∗ 𝐷 ∗ 𝑑𝐾 ∗
=
𝐾∗
+
< 0;
𝑑𝜅
𝛽
𝑑𝜅 𝐾 ∗ 𝑑𝜅
| {z }

(42)

>0

 ∗

∗
∗
𝑑𝑌
𝜒
Θ 𝑑𝐷 ∗𝑑𝐶 ∗∗ 2 𝑑𝐾
∗ 𝑑𝜂
=
−
𝛿
−
(1
−
𝜂
)
+
𝜒(1
−
𝜂
)
−
,
𝑑𝜅
𝑑𝐾 ∗
2
𝑑𝜅
𝑑𝜅 𝑚 𝑑𝜅|{z} | {z }(43)(∗∗)<0

(∗)<0
∗

𝑑𝜂
𝑑𝑌
∗
∗
𝑘∗
where 𝑑𝐾𝑑𝜅 < 0 from (40), 𝑑𝜅 < 0 from (39), 𝜒(1 − 𝜂 ) > 0, and 𝑑𝐾 ∗ − 𝛿 = 𝑍 − 𝛿 = 𝑅 − 1 =
1+𝜒(1−𝜂∗ )−𝛽∗ = 𝑅 𝑘∗ − 1 + 𝛿. For the ﬁrst term on the.
Fromequations
(2),
(6),
(12),
and
𝑍
𝛽
∗

∗

right-hand side of (42), we assume that 𝛽 − 𝜎 𝑓 𝜁 𝜒 > 0. For the term (∗) on the right-hand side of 𝑑𝑌 ∗
𝜒
∗ 2 > 0 is obtained from 1 − 𝜂 ∗ ≤ 1.27 The (∗) in (43) represents how(43), 𝑑𝐾∗ − 𝛿 − 2 (1 − 𝜂 )raising minimum capital requirements causes ineﬃciency through direct ﬁnance with management
costs, while the (∗∗) in (43) represents how raising minimum capital requirements causes ineﬃciency through indirect ﬁnance with monitoring costs. The eﬀect of raising the minimum capitalrequirement on welfare depends on the magnitude of (∗) and (∗∗) in (43). When the magnitude∗of (∗) in (43) is larger than that of (∗∗) in (43), 𝑑𝐶𝑑𝜅 < 0, and vice versa. For the latter case, tightcapital requirements help monitor activities and improve welfare. The eﬀects (∗) and (∗∗) are larger
depending on how the ﬁnancial market develops. The following proposition summarizes the eﬀects
of raising minimum capital requirements on welfare according to the development of the ﬁnancial
market:Proposition 2 Consider an economy in which Θ𝑚𝑖𝑛 ≤ Θ ≤ Θ. Let Θ be such that

Θ≡

−

n


∗
𝜒
𝑑𝑌 ∗
∗ ) 2 𝑑𝐾 ∗ + 𝜒(1 − 𝜂 ∗ ) 𝑑𝜂
−
𝛿
−
(1
−
𝜂
∗
𝑑𝐾
2
𝑑𝜅
𝑑𝜅
−1 𝑑𝐷 ∗
𝑚 𝑑𝜅

o
.

∗

Suppose that Θ < Θ. Then, 𝑑𝐶𝑑𝜅 ⋛ 0 corresponds to Θ ⋛ Θ.
27The derivation of the sign of 𝑑𝑌 ∗

𝜒
∗ 2 in (43) in detail is given in Appendix G.
𝑑𝐾 ∗ − 𝛿 − 2 (1 − 𝜂 )24∗Proof. From (43), we obtain 𝑑𝐶𝑑𝜅 ⋛ 0 as follows:

Θ⋛

−

n


∗
𝜒
𝑑𝑌 ∗
∗ 2 𝑑𝐾 ∗ + 𝜒(1 − 𝜂 ∗ ) 𝑑𝜂𝑑𝐾 ∗ − 𝛿 − 2 (1 − 𝜂 )𝑑𝜅𝑑𝜅−1 𝑑𝐷 ∗
𝑚 𝑑𝜅

o
.

(43’)

Proposition 2 implies that when the ﬁnancial market has developed suﬃciently such that Θ < Θ,raising the minimum capital requirements causes welfare loss. However, when the ﬁnancial market
has not developed enough such that Θ > Θ, raising the minimum capital requirements improves
welfare. Therefore, lemma 1 and Proposition 2 suggests that the regulator should reduce theminimum capital adequacy ratio to 𝜅 𝑚𝑖𝑛 in an economy where ﬁnancial market has developed suchthat Θ < Θ. However, to be a regulated economy, its policy must keep 𝜅 ≥ 𝜅˜∗ , and if 𝜅˜∗ > 𝜅 𝑚𝑖𝑛 , theregulator cannot set the minimum to 𝜅 𝑚𝑖𝑛 under minimum capital requirements (22’ ∗). Therefore,
in an economy with well-developed ﬁnancial markets, the regulator should set the following new
requirement such that 𝜅 = 𝜅 𝑚𝑖𝑛 for each bank, instead of the minimum capital requirements toimprove welfare. From (41), the economy under this rule can attain a large output. When Θ = Θ,
𝑑𝐶 ∗
𝑑𝜅

∗

= 0, the optimal minimum capital requirements that satisfy 𝑑𝐶𝑑𝜅 = 0 can attain maximum

consumption and welfare.28 Meanwhile, the regulator should raise the minimum capital adequacy
ratio in an economy where ﬁnancial market has not developed such that Θ < Θ to improve welfare,although raising the minimum capital requirements reduces output. So far, we consider a regulated
economy where ﬁnancial market has well developed such that Θ ∈ [Θ𝑚𝑖𝑛 , Θ]. However, the resultof Proposition 2 implies that for an economy where ﬁnancial market has not developed, larger
welfare can be attained as the capital adequacy ratio is higher. Thus, for an unregulated economy
where Θ ∈ [Θ, Θ𝑚𝑎𝑥 ], the regulator can improve welfare in the economy by raising the minimum
capital adequacy ratio, which shifts the threshold between regulated and unregulated economies.
The following proposition summarizes the optimal minimum capital requirement in an economy
based on how its ﬁnancial market has developed.

28We show it numerically in the next section.

25

Proposition 3 Consider an economy in which the ﬁnancial market has developed Θ𝑚𝑖𝑛 ≤ Θ ≤Θ𝑚𝑎𝑥 . Let 𝜅 𝑜 𝑝 be the optimal capital requirement such that 𝜅 ∈ [𝜅 𝑚𝑖𝑛 , 1]. Then,




𝜅 𝑚𝑖𝑛






𝜅 𝑜 𝑝 = 𝜅 ∗𝑜 𝑝







1


𝑓 𝑜𝑟 Θ𝑚𝑖𝑛 ≤ Θ < Θ
𝑓 𝑜𝑟 Θ = Θ
𝑓 𝑜𝑟 Θ < Θ ≤ Θ𝑚𝑎𝑥 ,∗where 𝜅 ∗𝑜 𝑝 is 𝜅, such that Θ = Θ, where 𝑑𝐶𝑑𝜅 = 0.
Propositions 2 and 3 imply that the regulator should raise the minimum capital adequacy ratio foran economy where ﬁnancial market has not developed, whereas it should be reduced for an economy
where ﬁnancial market has developed. We now interpret the intuition behind the optimal minimum
capital requirement described by Propositions 2 and 3. In an economy where ﬁnancial market hasdeveloped, the economy is not aﬀected much by the problems caused by undeveloped ﬁnancial
markets through indirect ﬁnance but is aﬀected much by households’ ineﬃcient investment throughdirect ﬁnance. Thus, in this economy, reducing minimum capital requirements promotes investment
through more eﬃcient indirect ﬁnancing through ﬁnancial intermediaries. In such an economy, if
the regulator raises the minimum capital requirements, its policy hinders a large output and leads
to less consumption, thus reducing welfare. Meanwhile, in an economy where ﬁnancial market has
not developed, the economy is greatly aﬀected by the problems caused by undeveloped ﬁnancial
markets through indirect ﬁnance with moral hazard problems. Thus, in this economy, raisingminimum capital requirements, such as Basel III, promotes investment by more direct ﬁnance but
suppresses indirect ﬁnance through ﬁnancial intermediaries, and then, the policy improves welfare.
Therefore, Propositions 2 and 3 show that optimal capital requirements should be set according to
the development of the ﬁnancial market.

26

5

Numerical Analyses in the Long and Short Runs

This section numerically analyzes the eﬀects of tighter minimum capital requirements on theaggregate economic activity in the long and short runs. In the long run analyses, we investigate how
this policy might work to mitigate the ineﬃcient resource allocation caused by distortions of the
ﬁnancial market through indirect and direct ﬁnancing and promote the aggregate economic activity,
according to the degree of ﬁnancial development. In the short run analyses, focusing on economies
with developed ﬁnancial markets, such as the U.S. economy, we examine whether this policy might
mitigate a ﬁnancial crisis regarding the speed of recovery and the magnitude of a drop in aggregate
economic variables. We ﬁrst start with the calibration.
Table 1: Parameter Values
Parameter
Households (workers and bankers)𝛽𝜁𝑓𝜎
Production Sector
𝛼
𝛿
𝐴

Value

Description

Source0.99
0.002890.2
0.968

Discount factor (impatience)
Transfer to entering bankers (Startup net worth)
Ratio of bankers to workers
Proportion of continuing bankers

Literature
Literature: GKQ (2012)
Literature: GK (2010)
Literature: GKQ (2012)

0.33
0.025
1

Capital share
Depreciation rate
Productivity

Literature
Literature
Literature: GK (2010)

There are eleven parameters for which we need to assign values. Four are standard preference
and technology parameters. These include the discount factor 𝛽, the capital share parameter 𝛼, thedepreciation rate 𝛿 and the productivity of the production of the ﬁnal goods 𝐴. We use reasonably
conventional values for these parameters as reported in Table 1. The other three parameters reported
in Table 1 are speciﬁc to the models based on GK (2010). The proportion of total assets to newbankers 𝜁 is minimal (0.00289) as in GKQ (2012) and the ratio of bankers to workers 𝑓 is 0.2 asin GK (2010). The proportion of continuing bankers 𝜎 is 0.968, implying that bankers survive foreight years on average.
The other four parameters are speciﬁc to this model, including the key three parameters in thismodel: ﬁnancial market development Θ and 𝜒, and minimum capital requirement (a policy) 𝜅. We𝑏consider the steady state leverage ratio before a crisis 𝐾𝑁 is as follows: 12.5 when the minimum
capital adequacy ratio 𝜅 = 0.08; 10.5 when 𝜅 = 0.095; and 10 when 𝜅 = 0.10. We choose 𝜒 and27

Θ to hit the following two targets: an average credit spread of one hundred basis points per year
and an economy-wide leverage ratio of four, as in GK (2010), and the ratio of total housing stock
to aggregate capital: 𝐾 𝐻 /𝐾 = 1/3 as in the U.S economy in 2008.29 We ﬁnally set the monitoring
costs 𝜃 as half of asset diversion.30 Table 2 summarizes the above calibrated parameters.
Table 2: Calibration

Parameter𝜒ΘDescription
Management cost:
Financial development (access)Asset diversion:
Financial development (eﬃciency)

𝜅Minimum capital adequacy ratio𝜃

Monitoring costs

Target
The ratio of total housing stock to aggregate
capital: 𝐾 𝐻 /𝐾 = 1/3
1% annual external ﬁnance premium:𝐸 [𝑅 𝐾 ] − 𝑅 = (1.01) 1/4 − 1(Literature: GK (2010))
Leverage 𝐾 𝑏 /𝑁 = {12.5, 10.5, 10}
A policy {0.08, 0.095, 0.10}
Leverage 𝐾 𝑏 /𝑁 = {12.5, 10.5, 10}
𝜃 = 2Θ

5.1 Financial Development and Minimum Capital Requirements in the LongRun Economy
This subsection presents numerical analyses of our main results in Propositions 2 and 3 outlinedin the previous section. We consider three economies, corresponding to Proposition 3: a welldeveloped ﬁnancial market, a developed ﬁnancial market, and an undeveloped ﬁnancial market asshown in Table 3 and Figure 5.
Figure 5 summarizes the results of Proposition 2 and shows how raising minimum capital requirements from 0.08 to 0.10, as recommended by Basel requirements, aﬀects economies whose ﬁnancial
market has developed shown in Table 3. A tight policy requirement reduces welfare in an economy
with a developed ﬁnancial market (the upper two lines in Figure 5). By contrast, a tight policy
improves welfare in an economy with an undeveloped ﬁnancial market (the lowest line in Figure
29GKQ (2012) noted that the total housing stock accounted for about a third of the overall capital stock before thebeginning of the 2008 ﬁnancial crisis. As a result, a leverage ratio of 4 is reasonable as described by GKQ(2012).
30As long as we consider the monitoring cost function is a linear function of deposits in Equation (7), the othersettings of 𝜃 do not have a signiﬁcant eﬀect on the numerical results shown in the following subsections. There may
be a diﬀerence in illustration if monitoring costs are used in other shapes ; however, the substantial elements for themechanism of this model is depositors’ monitoring activities, that is, the existence of monitoring costs, not the shapesof the cost functions.

28

Table 3: Financial Development and Minimum Capital Requirements in the Long Run Economy

Classiﬁcation
a well-developed ﬁnancial
marketa developed ﬁnancial market
an undeveloped ﬁnancial
marketFinancial development (access) 𝜒,
Financial development (eﬃciency) Θ
0.2580,0.08410.2597,0.08410.2597,0.3000Description
Low 𝜒 and Low Θ
High 𝜒 and Low Θ
High 𝜒 and High Θ

Figure 5: Eﬀects of raising minimum capital requirements in the steady state economy in the long run
according to the degree of ﬁnancial development5). In Figure 5, Θ, is the diﬀerence between the upper and lowest lines, which causes ineﬃcient
resource allocation from indirect ﬁnancing. Thus, Figure 5 demonstrates that the minimum capitalrequirements should be set according to the degree of ﬁnancial development, especially measured
by eﬃciency, to improve welfare in the long run. Moreover, in an economy where 𝜒 = 0.2597
shown in the lowest and middle lines in Figure 5, the optimal minimum capital requirement 𝜅 ∗𝑜 𝑝
is 8% when the degree of ﬁnancial development (access) is 0.2203, that is, Θ in Proposition 2 is0.2203. For this economy, the present minimum requirement, 0.08 is optimal, and to avoid welfare
losses, regulators should not change its minimum requirements.
29

5.2 Tighter Minimum Capital Requirements and Aggregate Economic Activity in a Country with a Financial Crisis
This subsection presents numerical results how raising minimum capital adequacy ratio might work
to mitigate the crisis. Figure 6 shows the impulse responses of the key aggregate economic variables
to a negative shock to capital quality in an economy whose ﬁnancial market has developed.31 We
suppose that the capital quality shock obeys a ﬁrst order autoregressive process. The dotted red (solidblue) line reports a case with minimum capital requirements is 0.08 (0.10). We set the initiating
disturbance as a ﬁve percent unanticipated decline in capital quality with an autoregressive factorof 0.66 for both economies with diﬀerent policies. We ﬁx the size of the shock as in GK (2010).
This negative capital quality shock captures an exogenous force that triggers a decline in the valueof assets, as explained by GK (2010) and GKQ (2012).
Despite its impact on ﬁnancial development in the short run, both values indicate high ﬁnancial
development. The dotted line illustrates a condition with ﬁnancial development 𝜒 (access: a frictionof direct ﬁnancing) and Θ (eﬃciency: a friction of indirect ﬁnancing) is {0.2580, 0.0841}, whilesolid line is {01974, 0.0633}.32 Both cases experience drops in expected spreads due to this negative
shock, and the path is nearly identical. However, this shock has diﬀerent eﬀects on the gross rateof deposits 𝑅, and the gross interest rate in the capital market 𝑅 𝑘 . 𝑅 in the red 8 % requirements
case has a larger drop than the blue 10% requirements case.This makes a diﬀerence in responses ofthe variables in the ﬁnancial and real sectors between the 8% and 10% requirements cases. Thus,
Figure 6 indicates that ineﬃcient resource allocation is caused by both direct and indirect ﬁnancing
through households’ behaviors and interest rates in the market.

31All small letters in Figure 6 shows the aggregate variables. For example, 𝑦 in Figure 6 implies the aggregate
variable 𝑌 , as described in the previous sections.
𝜂∗< 0 even if higher 𝜅 might change 𝜒 and Θ.
32In both economies, 𝜒 is so small that we analytically obtain 𝑑𝑑𝜅
𝑑𝐶 ∗Moreover, we obtain 𝑑𝜅 < 0 even if higher 𝜅 might change 𝜒 and Θ for both economies in the long run, using theother parameters in Tables 1 and 2.

30

Figure 6: Eﬀects of raising minimum capital adequacy ratio on aggregate economic activities in an economy
where ﬁnancial market has developed31

6Discussion: Interpretation of Results

The eﬀects of ineﬃciency caused by indirect ﬁnance Θ are larger than those by direct ﬁnance 𝜒 inthe long run. As a result, this minimum capital requirements should be determined based on the
degree of ﬁnancial development (eﬃciency). However, in the short run, an increase in the minimum
capital ration makes the drop in aggregate output in a recession (in response to a capital quality
shock) smaller. Its policies (tighter requirements) slow down the recovery process in the U.S. (acountry with a developed ﬁnancial market).
Tighter regulation suppress banks’ activities when ﬁnancial markets are developed (Θ is low),which harms households wanting to deposit more assets. The households manage more capitalon the market (direct ﬁnancing), taking account of balance sheet (15). This causes resource
ineﬃciency due to 𝜒, and decreases their consumption in the steady state (in the long run). Further,a negative shock to capital quality results in a moderate response of consumption and output. Theseeﬀects imply that both considering the signiﬁcance of balance sheet condition and the diﬀerence
sources of ineﬃciency resource allocation: direct and indirect ﬁnancing is important for setting ofminimum capital requirements during a ﬁnancial crisis as well as in the long run.33 In a country
with an undeveloped ﬁnancial market (Θ is high), when households deposit more assets, theresource ineﬃciency caused by indirect ﬁnancing decreases their consumption. What can policies(requirements) do? Raising minimum capital requirements can mitigate the resource ineﬃciency
caused by indirect ﬁnancing, suppressing the banks’ activities.34

33This study insists that both the signiﬁcance of balance sheet condition and the importance of an imperfect ﬁnancial
market is important for considering a policy for a ﬁnancial crisis in a large literature based on GK (2010). This study’s
model incorporates the diﬀerence sources of ineﬃciency resource allocation: direct and indirect ﬁnancing.
34The other research subjects (policies and requirements) include: (capital) taxes and policies regarding the activities
of banks and depositors that might resolve the problems caused by such ineﬃciency. In this study, we focus on
the minimum capital requirements which is one of international standards regardless of the diﬀerence in ﬁnancial
development.

32

7Concluding Remarks

This study aims to determine the optimal minimum capital requirements for an economy based on
its ﬁnancial market development. We construct a macroeconomic model in which both indirect anddirect ﬁnance cause ineﬃciency due to moral hazard problems between ﬁnancial intermediaries
and households. We analytically and numerically show that raising minimum capital requirements
improves resource allocation and welfare in an economy where ﬁnancial market has not developed,
whereas it reduces welfare in an economy where ﬁnancial market has developed. Our model showsthat raising minimum capital requirements in an economy with undeveloped ﬁnancial markets
increases resource allocation and welfare. By contrast, it reduces welfare in an economy with
developed ﬁnancial markets. Our numerical analyses allow us to calculate each economy’ s optimal
minimum capital requirements based on how much its ﬁnancial market has developed. We also showthat raising the minimum capital requirements can mitigate the crisis; however, there is a choice ofits purposes between the high speed of recovery from a ﬁnancial crisis and a small magnitude of adrop in aggregate variables when the crisis occurs, through ineﬃciency resource allocation caused
both by indirect and direct ﬁnancing. Our ﬁndings imply that, when setting the minimum capitalrequirements, regulators should consider how the ﬁnancial market of each economy has developed.
For future research, there are some directions. First, this model may be extended to modelsof an open economy to analyzes the capital requirement, considering the international ﬁnancial
market. Second is the time–varying capital requirements. This will contribute to a large bodyof literature based on GK (2010).35 Third, the dynamic equilibrium model in which endogenous
ﬂuctuations in the asset market prices can be extended, as in Kiyotaki and Moore (1997), especially
focusing on more complex contracts between banks and depositors and on the assumption of rational
expectation. Finally, the results of this model suggest that we should adopt diﬀerent measures ofﬁnancial development based on their intended purposes, such as minimum capital requirements.
The implications of this ﬁnding could contribute to a large body of empirical research and calibration
regarding ﬁnancial development, economic growth, and economic crises.
35GKP (2020 (b), 2020 (c)) recently conducted analyzes on the time–varying capital requirements.

33

Appendix
A

Derivations of Equations (23), (24), and (25)Substituting Equation (21) into 𝑉𝑡 on the left-hand side of Equation (19) and 𝑉𝑡+1 on the right-handside of Equation (19), we obtain𝑏+ Λ𝑡,𝑡+1 𝜎𝜈𝑡+1 𝑛𝑡+1 ,𝜇𝑡 𝑘 𝑡𝑏 + 𝜈𝑡 𝑛𝑡 = 𝐸 𝑡 Λ𝑡,𝑡+1 (1 − 𝜎)𝑛𝑡+1 + Λ𝑡,𝑡+1 𝜎𝜇𝑡+1 𝑘 𝑡+1
where 𝜇𝑡 = 𝜄𝑡 − 𝜈𝑡 . Substituting the deﬁnition of 𝜅𝑡 into the above equation yields:
𝜇𝑡 𝑘 𝑡𝑏 + 𝜈𝑡 𝑛𝑡 = 𝐸 𝑡




𝑛𝑡+1
Λ𝑡,𝑡+1 (1 − 𝜎)𝑛𝑡+1 + Λ𝑡,𝑡+1 𝜎𝜇𝑡+1+ Λ𝑡,𝑡+1 𝜎𝜈𝑡+1 𝑛𝑡+1 ,
𝜅𝑡+1

(A1)

with
𝜅𝑡+1 = max (𝜅, 𝜅˜𝑡+1 ) .
Let Ω𝑡+1 be deﬁned as

(22’)

𝜎𝜇𝑡+1
+ 𝜎𝜈𝑡+1 .
𝜅𝑡+1

(25)



𝜇𝑡 𝑘 𝑡𝑏 + 𝜈𝑡 𝑛𝑡 = 𝐸 𝑡 Λ𝑡,𝑡+1 Ω𝑡+1 𝑛𝑡+1 ,(A1’)

Ω𝑡+1 ≡ (1 − 𝜎) +Then, (A1) can be rewritten as follows:with (25) and (22’).
Combining (15) with (16) for 𝑖 = 𝑡 and 𝑗 = 𝑡 + 1, we obtain

 

𝑘
𝑘
𝑛𝑡+1 = 𝑅𝑡+1
𝑘 𝑡𝑏 − 𝑅𝑡+1 𝑘 𝑡𝑏 − 𝑛𝑡 = 𝑅𝑡+1
− 𝑅𝑡+1 𝑘 𝑡𝑏 + 𝑅𝑡+1 𝑛𝑡 .

34

(A2)

Substituting (A2) into (A1’), we obtain
n
h

io
𝑘
𝜇𝑡 𝑘 𝑡𝑏 + 𝜈𝑡 𝑛𝑡 = 𝐸 𝑡 Λ𝑡,𝑡+1 Ω𝑡+1 𝑅𝑡+1
− 𝑅𝑡+1 𝑘 𝑡𝑏 + 𝑅𝑡+1 𝑛𝑡 ,

(A1”)

where
Ω𝑡+1 ≡ (1 − 𝜎) +𝜎𝜇𝑡+1
+ 𝜎𝜈𝑡+1 ,
𝜅𝑡+1

(24)

with
𝜅𝑡+1 = max (𝜅, 𝜅˜𝑡+1 ) .

(22’)

Because the coeﬃcients on the left-hand side of Equation (A1”) are equivalent to the coeﬃcientson the right-hand side of (A1”), if the guess is correct, we obtain the following equations:hi𝑘𝜇𝑡 = 𝐸 𝑡 Λ𝑡,𝑡+1 Ω𝑡+1 𝑅𝑡+1
− 𝑅𝑡+1 ,(23)𝜈𝑡 = 𝐸 𝑡 Λ𝑡,𝑡+1 Ω𝑡+1 𝑅𝑡+1 ,

(24)

where
Ω𝑡+1 ≡ (1 − 𝜎) +𝜎𝜇𝑡+1
+ 𝜎𝜈𝑡+1 ,
𝜅𝑡+1

(25)

and
𝜅𝑡+1 = max ( 𝜅˜𝑡+1 , 𝜅) .

B

(22’)

Derivation of Equation (32)

Using (26), Equation (28) can be rewritten as follows:𝑏
𝑁𝑡 = 𝜎𝜎 𝑓 𝑅𝑡𝑘 𝑘 𝑡−1− 𝑅𝑡 𝑑𝑡−1 + (1 − 𝜎)𝜎 𝑓 𝑅𝑡𝑘 𝜁 𝐾𝑡−1 + (1 − 𝜎) 𝑓 𝜁 𝐾𝑡 ..

35

(B1)

Substituting (16) into (B1), we obtain
𝑁𝑡 = 𝜎𝜎 𝑓 𝑛𝑡 + (1 − 𝜎)𝜎 𝑓 𝑅𝑡𝑘 𝜁 𝐾𝑡−1 + (1 − 𝜎) 𝑓 𝜁 𝐾𝑡 ..

(B1’)

Combining Equation (31) with Equations (26) and (27) yields
𝑁𝑡 = 𝐾𝑡𝑏 − 𝐷 𝑡 = 𝜎 𝑓 𝑘 𝑡𝑏 + (1 − 𝜎) 𝑓 𝜁 𝐾𝑡 − 𝜎 𝑓 𝑑𝑡 = 𝜎 𝑓𝑘 𝑡𝑏 − 𝑑𝑡+ (1 − 𝜎) 𝑓 𝜁 𝐾𝑡 .

Substituting (15) into the above equation, we obtain

𝑁𝑡 = 𝜎 𝑓 𝑛𝑡 + (1 − 𝜎) 𝑓 𝜁 𝐾𝑡 .

(B2)

Substituting Equation (B2) into the left-hand side of Equation (B1’), we obtain𝑛𝑡 = 𝑅𝑡𝑘 𝜁 𝐾𝑡−1 .

(B3)

By dividing both sides of Equation (B3) by 𝑘 𝑡𝑏 and using the deﬁnition of 𝜅𝑡 , we obtain
𝑘 𝑡𝑏
𝐾𝑡−1
= 𝑅𝑡𝑘 𝜁
,
𝜅𝑡
𝐾𝑡
𝐾𝑡

(B3’)

with 𝜅𝑡 = max ( 𝜅˜𝑡 , 𝜅) (22’). Equation (26) and the deﬁnition of 𝜂𝑡 yields
𝑘 𝑡𝑏 𝜂𝑡 − (1 − 𝜎) 𝑓 𝜁
=
.
𝐾𝑡
𝜎𝑓

(B4)

Combining Equation (B3’) with Equation (B4) yields

𝜅𝑡 =𝜎 𝑓 𝜁 𝑅𝑡𝑘 𝐾𝐾𝑡 −1𝑡[𝜂𝑡 − (1 − 𝜎)𝜁 𝑓 ]

36

,

(B5)

with 𝜅𝑡 = max ( 𝜅˜𝑡 , 𝜅) (22’). Taking one lag of Equation (12) with Equation (6) and the deﬁnition of𝜂𝑡 yields
𝑅𝑡𝑘 =𝐶

where Λ𝑡−1,𝑡 =𝛽 𝐾𝑡 −1 𝐾
𝑡 −1
𝐶𝑡
𝐾𝑡

𝑡 −1

𝐾𝑡



ℎ ;𝐾
1 + 𝑓 0 𝐾𝑡−1𝑡−1
Λ𝑡−1,𝑡=1 + 𝜒 (1 − 𝜂𝑡−1 ),.
Λ𝑡−1,𝑡(B6)= 𝛽𝑥𝑥𝑡𝑡−1 𝐾𝐾𝑡 −1, where 𝜅𝑡 = max ( 𝜅˜𝑡 , 𝜅), (22’). Combining (B5) with (B6),𝑡we obtain:
𝜎 𝑓 𝜁 [1 + 𝜒 (1 − 𝜂𝑡−1 )]𝜅𝑡 = 𝛽𝑥,𝑡 −1𝑥 𝑡 [𝜂𝑡 − (1 − 𝜎)𝜁 𝑓 ](32)

with 𝜅𝑡 = max ( 𝜅˜𝑡 , 𝜅) (22’).

C

Derivation of Equation (33)Combining the Bellman equation (19) and the binding incentive compatibility constraint (21), we
obtain𝑏Θ𝑘 𝑡𝑏 = 𝐸 𝑡 Λ𝑡,𝑡+1 (1 − 𝜎)𝑛𝑡+1 + Λ𝑡,𝑡+1 𝜎Θ𝑘 𝑡+1.
Taking one lag of the above equation and dividing both sides of it by 𝑘 𝑡𝑏 yields
Θ

𝑏
𝑘 𝑡−1

𝑘 𝑡𝑏

= Λ𝑡−1,𝑡 (1 − 𝜎)𝑛𝑡
+ Λ𝑡−1,𝑡 𝜎Θ.
𝑘 𝑡𝑏Using (B4) and the deﬁnitions of 𝜅˜𝑡 , 𝑥𝑡 , and Λ𝑡−1,𝑡 , the above equation can be rewritten as follows:



𝑥𝑡−1
𝜂𝑡−1 − (1 − 𝜎) 𝑓 𝜁[(1 − 𝜎)𝜅𝑡 + 𝜎Θ] .
Θ=𝛽𝜂𝑡 − (1 − 𝜎) 𝑓 𝜁
𝑥𝑡

D

(33)

Derivation of Equation (34)

Combining Equations (1) and (2) with 𝑍𝑡 = 𝑅𝑡𝑘 − 1 + 𝛿 and 𝐿 = 1 − 𝑓 yields
𝛼𝑌𝑡
𝑅𝑡𝑘 = (1 − 𝛿) += (1 − 𝛿) + 𝛼
𝐾𝑡
37



1− 𝑓
𝐾𝑡

 1−𝛼
.

(D1)

From Equations (12) and (6), we obtain𝑅𝑡𝑘 =1 + 𝜒(1 − 𝜂𝑡−1 ),Λ𝑡−1,𝑡(12’)

where Λ𝑡−1,𝑡 = 𝛽𝑥𝑥𝑡𝑡−1 𝐾𝐾𝑡 −1. Plugging Equation (12’) into Equation (D1), we obtain𝑡1

 1+𝜒(1−𝜂𝑡 −1 ) − (1 − 𝛿)  1− 𝛼 Λ𝑡 −1,𝑡

1− 𝑓

= 
.

𝐾𝑡
𝛼





(D2)

(D2) can be rewritten as follows:1 1− 𝛼𝛼,𝐾𝑡 = (1 − 𝑓 )  1+𝜒(1−𝜂 ) Λ𝑡 −1,𝑡𝑡 −1 − (1 − 𝛿) 



(34)

where Λ𝑡−1,𝑡 = 𝛽𝑥𝑥𝑡𝑡−1 𝐾𝐾𝑡 −1.
𝑡E

Derivation of Equation (35)Dividing the goods market clearing condition (29) by 𝐾𝑡 , we obtain
𝑓 𝑘 𝐾𝑡ℎ ; 𝐾𝑡
𝑌𝑡𝐶𝑡𝐼𝑡𝑓 𝑑 (𝐷 𝑡 )=+++𝐾𝑡 𝐾𝑡 𝐾𝑡
𝐾𝑡
𝐾𝑡
𝐼𝑡
𝜒
𝐷𝑡
= 𝑥𝑡 ++ (1 − 𝜂𝑡 ) 2 + 𝜃 .
𝐾𝑡 2
𝐾𝑡

(29’)

The last equality is given by the deﬁnitions of 𝑥𝑡 and 𝜂𝑡 as well as Equations (6) and (7). Dividing
both sides of (31) with the deﬁnition of 𝜂𝑡 yields
𝐷 𝑡 𝑁𝑡=+ 𝜂𝑡 .
𝐾𝑡
𝐾𝑡

38

(E1)

Combining (B2) and (B3), we obtain𝑁𝑡
𝐾𝑡−1= 𝜎 𝑓 𝑅𝑡𝑘 𝜁+ (1 − 𝜎) 𝑓 𝜁 .
𝐾𝑡
𝐾𝑡
Using (B6) and the deﬁnition of 𝑥𝑡 , we can rewrite the above equation as follows:#"𝑁𝑡
1 + 𝜒 (1 − 𝜂𝑡−1 )+ (1 − 𝜎) 𝑓 𝜁 .
= 𝜎𝑓𝜁𝐾𝑡
𝛽 𝑥𝑥𝑡 −1𝑡

(E2)

Using (E2), Equation (E1) can be rewritten as#"𝐷𝑡
1 + 𝜒 (1 − 𝜂𝑡−1 )= 𝜎𝑓𝜁+ (1 − 𝜎) 𝑓 𝜁 + 𝜂𝑡 .
𝐾𝑡
𝛽 𝑥𝑥𝑡 −1𝑡

(E1’)

Substituting the law of motion for capital (4) and (E1’) into Equation (29’), we obtain
#
"
(1
)
𝑌𝑡
𝐾𝑡+1
𝜒
1
+
𝜒
−
𝜂
𝑡−1
+ 𝜃 (1 − 𝜎) 𝑓 𝜁 + 𝜃𝜂𝑡 .
= 𝑥𝑡 +− (1 − 𝛿) + (1 − 𝜂𝑡 ) 2 + 𝜃𝜎 𝑓 𝜁
𝑥 𝑡 −1𝐾𝑡
𝐾𝑡
2
𝛽 𝑥𝑡
Equation (1) yields

  1−𝛼
𝑌𝑡
𝐿
.
=𝐴
𝐾𝑡
𝐾𝑡

(E3)

(1’)

Substituting Equation (1’) into the left-hand side of (E3), we obtain
"
#

 1−𝛼
(1
)
𝐾𝑡+1
1− 𝑓
𝜒
1
+
𝜒
−
𝜂
𝑡−1
𝑥𝑡 = −+ (1 − 𝛿) + 𝐴− (1 − 𝜂𝑡 ) 2 − 𝜃𝜎 𝑓 𝜁− 𝜃 (1 − 𝜎) 𝑓 𝜁 − 𝜃𝜂𝑡 ..
𝐾𝑡
𝐾𝑡
2
𝛽 𝑥𝑥𝑡 −1𝑡
(E4)Note that 𝐿 = 1 − 𝑓 . Plugging (D2) into (E4) yields
"
#
(1
)
1
+
𝜒
−
𝜂
𝐴[1 + 𝜒(1 − 𝜂𝑡−1 )] 𝐾𝑡+1 (1 − 𝛿)( 𝐴 − 𝛼) 𝜒
𝑡−1
𝑥𝑡 =
−
−
− (1 − 𝜂𝑡 ) 2 −𝜃𝜎 𝑓 𝜁−𝜃 (1−𝜎) 𝑓 𝜁−𝜃𝜂𝑡 ..
𝛼Λ𝑡−1,𝑡𝐾𝑡
𝛼2𝛽 𝑥𝑥𝑡 −1
𝑡
(35)

39

F

Derivation of Equation (36)Substituting (10) into Equation (9) yields
𝐶𝑡 +𝐷 𝑡 +𝐾𝑡ℎ + 𝑓 𝑘

𝐾𝑡ℎ ; 𝐾𝑡𝑑+ 𝑓 (𝐷 𝑡 ) = 𝑤 𝑡 𝐿+(1−𝜎)





𝑏
ℎ
𝑅𝑡𝑘 𝐾𝑡−1− 𝑅𝑡 𝐷 𝑡−1 −(1−𝜎) 𝑓 𝜁 𝐾𝑡 +𝑅𝑡𝑘 𝐾𝑡−1
+𝑅𝑡 𝐷 𝑡−1 .

(9’)
Substituting (28) and (30) into (9’), we obtain:

𝐶𝑡 + 𝐾𝑡 + 𝑓𝑘

𝐾𝑡ℎ ; 𝐾𝑡𝑏ℎ+ 𝑓 𝑑 (𝐷 𝑡 ) = 𝑤 𝑡 𝐿 + 𝑅𝑡𝑘 𝐾𝑡−1
+ 𝑅𝑡𝑘 𝐾𝑡−1.

(F1)

Combining (29) and (5) with (F1) yield
𝐾𝑡 + 𝑌𝑡 − 𝐼𝑡 = 𝑤 𝑡 𝐿 + 𝑅𝑡𝑘 𝐾𝑡−1 .
From (2) and (3),𝑤 𝑡 𝐿 = 𝑌𝑡 − 𝑍𝑡 𝐾𝑡 = 𝑌𝑡 −



𝑅𝑡𝑘 − 1 + 𝛿

(F2)



𝐾𝑡 .

(F3)

Substituting (F3) and (4) into (F2), we obtain
𝐾𝑡+1 − 𝐾𝑡 = 𝑅𝑡𝑘 (𝐾𝑡 − 𝐾𝑡−1 ) .

(F4)Substituting (D1) into (F4), we obtain
"



1− 𝑓
𝐾𝑡+1 − 𝐾𝑡 = (1 − 𝛿) + 𝛼
𝐾𝑡40 1−𝛼 #
(𝐾𝑡 − 𝐾𝑡−1 ) .

(36)

G

The sign of 𝑑𝑌 ∗


𝜒
∗ 2
−
𝛿
−
(1
−
𝜂
)
in (43)𝑑𝐾 ∗2Now, we examine the sign of𝑍 ∗ = 𝑅 𝑘∗ − 1 + 𝛿, we obtain

 𝑑𝑌 ∗𝑑𝐾 ∗− 𝛿 − 𝜒2 (1 − 𝜂∗ ) 2 in (43). From Equations (2) and (12), and




 𝜒
𝑑𝑌 ∗
1 + 𝜒(1 − 𝜂∗ ) − 𝛽 𝜒
𝜒
𝜒
∗ 2
∗
∗ 2
𝑘∗
∗ 2
−𝛿− (1−𝜂 ) = 𝑍 −𝛿− (1−𝜂 ) = 𝑅 − 1 − (1−𝜂 ) =− (1−𝜂∗ ) 2 .
∗
𝑑𝐾
2
2
2
𝛽
2
(G1)
Let Γ(𝜂∗ ) on the right-hand side of Equation (G1). Taking the derivative of Γ(𝜂∗ ), we obtain𝑑Γ(𝜂∗ )
𝜒 [1 − 𝛽(1 − 𝜂∗ )]
=−
< 0,
𝑑𝜂∗
𝛽
where 𝛽 < 1 and 1 − 𝜂∗ ≤ 1. By computing Γ(1), we obtain:
1−𝛽Γ(1) =
> 0.
𝛽

(G2)

Because Γ(𝜂∗ ) is a decreasing function of 𝜂∗ and its maximum value is positive from (G2) and 𝑑𝑌 ∗
𝜒
∗ 2 > 0𝜂∗ ∈ [0, 1], we obtain Γ(𝜂∗ ) > 0 for all 𝜂∗ ∈ [0, 1], which implies that 𝑑𝐾∗ − 𝛿 − 2 (1 − 𝜂 )for all 𝜂∗ ∈ [0, 1].

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