An agent has one unit of input, but has no access to productive production project.
An agent can allocate his endowment into different assets: deposits (α) and se-curities (1−α). If he invests its endowment in securities, he receives uncertain incomes, while deposits give him certain amounts of repayments.
We are interested in equilibrium in which both loans and securities appear.
The other type e2will choose loan finance in equilibrium. An agent faces uncer-tainty if he buys securities. Thus the agent’s optimization problem is a standard portfolio selection problem in financial economics, and can be written as:
αmax,c1,c2
Pu(c˜ 1) + (1 − ˜P)u(c2)
s.t. c1=αRD+ (1 −α)RM = RU+α(RD− RM) c2=αRD
where RD is the deposit rate and RM is the rate of gross returns an agent expects to obtain from the market (called market rate). When there are a number of se-curities, RM stands for the expected returns an agent expects to obtain from the
market average portfolio he invests in, and ˜P is the probability with which an agent believes his portfolio will succeed to obtain the return RM. An agent uses ˜P to calculate his expected utility.
We assume that commercial banks and investment banks are run by risk neu-tral agents whose goals is to maximize their expected profits and then consume what they earn. Similarly, the entrepreneurs are also risk neutral and consume the profits they earn from running investment projects. Thus, not like in the usual general equilibrium framework, the utility-maximizer in this model does not have distributed profits from profit earners in their budget constraint.
The first order necessary condition (with respect toα):
H≡ ˜Pu′(c1)(RD− RM) + (1 − ˜P)u′(c2)RD= 0.
The second order sufficient condition is
Hα = ˜P(RD− RM)2u′′(c1) + (1 − ˜P)(RD)2u′′(c2) < 0.
By the implicit function theorem a solution α∗(RD, RM) exists and satisfies the following properties:
and RRA(c) = −cuu′′′(c)(c) is Arrow’s relative risk aversion coefficient of the utility
−) and an individual fund supply for securities β∗(RD
In this section we illustrate how to solve the equilibrium deposits rate (RD) and equilibrium market rate (RM) for our general equilibrium model. Funds markets operate in two forms in our economy: securities and loans. The market clearings for these two forms of funds are
n· respectively. Combining (10) and (11), we have the market clearing for the whole of funds markets:
n· Z e¯
e(R˜ D)
f(e)de = m. (12)
First, we can use (12) to solve RD as a function of m/n and other relevant parameters. To abuse the math expression we denote the solution by RD∗(m/n).
Since the total demand of funds is decreasing in RD, an increase in m/n results in an decrease in RD. The result is quite intuitive. When on average an entrepreneur has more funds available to him, the cost of funds drops. We can use RD as an index of the cost of funds.
Then we substitute RD∗(m/n) into (10) to solve RM. Note that ˜P is a func-tion of RM and is increasing in RM due to the fact that entrepreneurs with higher skill levels of management have higher probabilities of success. When RM in-creases, only those with higher management skill are able to obtain market fi-nance and, thus, the average probability of success of all market-fifi-nanced projects increases with RM, i.e., ˜P′(RM) > 0. From the above analysis we also know that α∗(RD, RM, ˜P) is decreasing in both RM and ˜P, and the supply of funds in the form of securities (1−α∗) is increasing in RM. When RM approaches zero, the supply of funds in the form of securities also approaches zero. The demand of funds in the form of securities (n·Ree¯∗(RM) f(e)de) is decreasing in RM and when RM is very high, the quantity demanded for funds is zero. When RM is very low (given loan rates), most of entrepreneurs would choose market finance. Figure 5 shows that how the market rate (RM) is determined and its relation to the deposit rate (RD). As RD increase, the household diverts more funds to deposits and less funds are available to market finance, resulting in the supply of funds in the form of securities shifts to the left and, thus, the market rate RM is pushed up.
One justification for the use of a general equilibrium model is to explore how the primitive parameters of the economy, such as intermediation technology (c,
LHS of (10) RHS(RD) RHS(RD′)
RD′ > RD
RM∗(RD) RM∗(RD′)
Funds RM
Figure 5: Determination of RM
θ, andφ), the populations of households and entrepreneurs (m, and n), and the risk attitude of households, affect the financial structure of the economy. In our economy two critical value of management skills, ˜e(RD) and e∗(RM), determines the ratio of market finance to loan finance. They also play an important role in understanding how economic primitive elements, such as technologies and popu-lations of households and entrepreneurs, affect the prices of funds and the financial structure.
Using our specific form of production technologies P(S, e) = 1 − AS/e, and intermediation technology (c, θ, andφ), we are able to derive:
˜
e= 8A(c + RD), and e∗= 16A 3
RM
θ +φ. (13) To simplify the analysis, we assume that e is uniformly distributed over[e, ¯e] such that f(e) = 1/( ¯e − e). From the market clearing of the whole funds market (12)
one can derive
RD= 1
8A[ ¯e − (m/n)∆e] − c, (14)
where∆e= ¯e − e. Notice that the underwriting technology has no impact at all on the determination of the deposit rate.
The market clearing of the funds market in the form of securities can be writ-ten as
We start with examining how changes in loan technology affect both loan and market rates and the financial structure. From (14) we know that the screening cost (c) is negatively related to the loan rate. The bank’s pricing practices always take the screening cost into account. When c decreases, the bank adjusts loan rates accordingly. This adjustment feeds back to the fund market and increases the demand for funds through loans. The bank then adjusts up the deposit rate in order to attract more deposits to meet their loan business.
An increase in RD drives households to shift their funds from markets to de-posits and, thus, reduces the funds available to market finance and results in a rise in the market rate.
From (14) we know that in respond to a change in c, the equilibrium deposit rate changes by the same magnitude as the change in c, but in the completely
opposite direction. Consequently, an improvement in c pushes up the deposit rate, which in turns increases RM (see Figure 5). The intuition is that when RD increases, the household reallocates their funds to deposits and reduces the supply of funds through buying securities.
From (13) we can see that ˜e remains unchanged when c decreases; e∗ in-creases due to an increase in RM caused by an increases in RD. A decrease in c is offset by its general-equilibrium feedback effect on RD, an equal amount of changes in RDbut in the opposite direction. As a result, a change in c does not af-fect ˜e, and thus does not affect the fund availability for production projects. Other things being identical, the economy becomes more loan-financed when facing an improvement in the loan technology.
We summarize the effects of an improvement in the loan technology in the following proposition.
Proposition 5 In response to an improvement in the loan technology (a de-crease in c),
[1] both equilibrium deposit and market rates (RDand RM) increase, and
[2] the economy becomes more loan-financed, while the total number of
exter-nally financed projects remains unchanged (i.e., e∗increases while ˜e remains unchanged).
Underwriting Technology Responding to an improvement in underwriting
Funds
RM θ ↑ orφ ↓ ⇒ RM ↑
Figure 6: Underwriting Technology and RM
technology (an increase in the convincing power θ and/or a decrease in under-writing feeφ), the demand for funds of market finance shifts to the right and the market rate increases (as shown in Figure 6 and equation (15).) From (14) we know that the underwriting technology has no impact on the determination of the deposit rate. This is the total demand for funds is determined by the lower bound of management skill levels ( ˜e) which is a function of RDand is independent of RM. The market rate matters when an entrepreneur chooses between loan finance and market finance. In our setup all those who consider market finance always have loan finance available. As a result, an improvement in underwriting technology does not affect total number of projects which are financed.
Concerning the change of e∗, we need more structure of the economy such as the risk attitude of households to pin down the direction of changes in e∗whenθ
andφ change. In (13) one can see that the relative magnitude of the changes of RM to that ofθ (orφ) determines the direction of changes in e∗. From (15) orFigure 5 we know that the slope of the supply of funds in the form of securities determines the magnitude of the change in RM. The risk aversion coefficient plays an impor-tant role for determining the slope of the supply curve. The implication is that the financial structure (defined by the shares of of market finance and loan finance of the economy) is determined by not only the financial technology but also the household’s risk attitude. The effect of changes in the underwriting technology can be summarized as:
Proposition 6 When the underwriting technology improves (θ orφ decreases), [1] the market rate increases, while the deposit rate remains unchanged, and
[2] the total number of externally financed projects remains unchanged (i.e., ˜e remains unchanged), and the number of market-financed projects increases or decreases, depending on the magnitude of changes in the market rates.
Populations of Entrepreneurs and Households
It is clear that an increase in m/n improves the availability of funds. Equation (14) indicates the deposit rate falls. From the above analysis (see Figure 5) we know that the market rate is positively related to the loan rate, implying that an increase in m/n makes both RDand RM decrease.
From (13) we know that a drop in RDcauses a decrease in ˜e. The total number of externally financed projects increases. Similarly, a drop in RMcauses a decrease
in e∗, the number of market-financed projects increases. However, we are unable to figure out which one has a greater drop. The change of the financial structure is indeterminate.
Proposition 7 When the availability of funds improves,
[1] both deposit and loan rates drop,
[2] the total number of externally financed projects increases,
[3] the number of market-financed projects increases, while the change in the number of loan-financed projects is indeterminate.
Financial Fragility
Boyd and de Nicol´o (2005) argue that there exists a fundamental risk-incentive mechanism that operates in exactly opposite direction, causing banks to become more risky as their market become more concentrated. Chiang (2008) modifies their model by incorporating market finance into a partial equilibrium framework and show that the presence of market finance provides alternatives for firms to finance their projects. The market alternative substitutes expensive loans with security finance of lower rates, which in turn keep firms from choosing high risk projects.
In our general equilibrium framework all properties about financial fragility remain intact. Market-financed projects have greater probabilities of success than
loan-financed projects. Thus the appearance of market finances help to alleviate the fragility problems caused by more concentrated loan markets.
Moreover, we are able to analyze how changes in financial technologies and the availability of funds affects loan and market rates by taking into account gen-eral equilibrium feedback effect. It is well perceived that technology improve-ments, both in the real and the financial sectors benefits the economy. However, our analysis shows that improvements in the loan and underwriting technologies raise both loan and market rates and induce entrepreneurs to choose more risky projects and causes more serious financial fragility problem. An improvement in financial technologies increases demands for funds due to the lower costs of ob-taining funds; however, the fund supply is fixed and, thus, the cost of funds itself goes up and dominates the effect of lower funds-obtaining costs.
By adding general equilibrium feedback effects to the analysis, we are able to examine the fragility problem more extensively and convincingly.
6 Conclusions
In Chiang (2008) we show that when an entrepreneur facing a higher loan rate due to the loan-lender’s monopoly power, the presence of market finance provides an alternative of external finance with a lower rate. The lower borrowing rate induces the firm chooses less risk projects and thus reduce financial fragility.
Chiang (2008)’s analysis is conducted in a partial equilibrium framework. In this paper we extend the analysis into a general equilibrium framework to explore
Table 1: General Equilibrium Effects of Changes in Model Parameters
RD RM e˜ e∗
c↓ ↑ ↑ unchanged ↑
θ ↑/φ ↓ unchanged ↑ unchanged depends
m/n ↑ ↓ ↓ ↓ ↓
how financial (loan and underwriting) technologies play their roles in determining the costs of different forms of external finance. We then go further and analyze the issue of financial fragility with general equilibrium feedback effects which is beyond the scope of Chiang (2008).
Table 1summarizes our general equilibrium effects of changes in intermedia-tion technologies and the availability of funds. Improvements in the technologies of the financial sector reduce the cost of obtaining funds. Such a cost reduction in obtaining funds increases the demand for funds. When the available funds are fixed, a higher demand means higher costs of funds. This general equilibrium feedback effect dominates the first round effect of technology improvements and both deposit and security (market) rates (RDand RM) increase. A reduction in the screening cost (c) is offset by the resulting increase in deposit rate (in our setup) and the loan rate remain unchanged. As a result, the total number of externally financed projects does not change. In contrast, those entrepreneurs issuing secu-rities to finance their projects face higher market rates and chooses projects with lower probability of success. The economy faces a more fragile financial sector.
We use a specific form of production technology P(S, e) = 1 − AS/e to sim-plify the analysis of firms’ finance behaviors. The simplification allows us to extract a clear picture of how these parameters affect the financial structure. We do not mean that the conclusion from the simplification is generally applicable.
Instead, it shows a simple but clear mechanism through which economic primi-tives work to influence the financial structure. The full understanding of simple mechanisms makes us better equipped to handle more complex mechanisms that work in the real world.
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