The impact of contract enforcement efficiency on debt maturity structure using mathematical derivation and comparative static analysis

Abstract

In recent years, the systematic use of advanced mathematical methods to express, study and argue economic theories has become an important branch and research hotspot in economics. The choice of debt maturity structure is an interesting economics issue in corporate finance. The signing of debt financing contract is not only the enterprises’ independent behavior, but also will be affected by the contract environment. Differences in the contractual environment will inevitably lead to differences in the efficiency of contract enforcement between borrowers and lenders in debt contracts. This paper introduces the contract enforcement efficiency factor on the basis of the standard structure model constructed by Holmstrom and Tirole, and then theoretically investigates the influence mechanism of contract enforcement efficiency on debt maturity structure based on the mediating role of contract enforcement efficiency on the investors’ liquidation claims. By transforming the financing contract into a linear programming problem, and then solving the optimization model, mathematical derivation and static comparative analysis, the research shows that as the efficiency of contract enforcement increases, the amount of short-term debt and long-term debt increases, while the direction change in debt maturity structure is uncertain and the specific relationship between them is closely related to the magnitude of all exogenous variables.

Keywords

Contract enforcement efficiency debt maturity structure claims on liquidation value

1. Introduction

There are two types of exogenous financing for companies: equity financing and debt financing. In the case of debt financing, the entrepreneur must consider not only the amount of debt but also the maturity of debt, thus making the choice of debt maturity structure a necessary part of the firm’s debt financing decision [1]. In addition to affecting the company’s investment and financing costs, corporate governance and liquidity risk, such type of choice will be directly related to the distribution of benefits between creditors and debtors. It is therefore important to explore the various factors that influence the structure of debt maturity.

Since the 1990s, there has been a great deal of academic research on debt maturity structure in terms of internal corporate factors and some classical theories were developed [2, 3, 4]. However, the signing of financial contract is not only an autonomous act of the company, but also largely related to the contractual environment in which it lives.

The direct impact of the contractual environment on corporate debt financing is mainly at the insolvency and liquidation stage. When the company is unable to repay its debts and the contract cannot be executed successfully, it will be liquidated by the court. In these cases, the efficiency of the contractual environment will impact the actual liquidation value of the firm [5, 6].

Diamond argues that creditors tend to enforce short-term debt contracts with poor legal enforcement, while market-oriented reforms and improved equity regimes help both financing parties to choose the most advantageous debt maturity [7]. Childs et al. confirm that a plurality of financing sources motivates firms to borrow more short-term debt [8]. Aeberhardt et al. have shown that the legal system, which is an important element of the contractual environment, can have a significant impact on contractual frictions [9]. In addition, a number of empirical studies have demonstrated the impact of contract enforcement efficiency on debt maturity structure [10, 11, 12, 13].

Based on the enterprise’s liquidity needs perspective, Holmstrom and Tirole investigate the influence of the enterprise’s own factors on debt maturity by theoretical analysis, such as the amount of capital held by the firm itself, debt ratios, and agency costs [14, 15]. This model assumes that the firm is subject to a liquidity shock in its operating process and that if it is unable to raise the funds to meet this risk, it will face liquidation and investors can get 100% of the liquidation value. In reality, however, the enforcement of such liquidation rights for investors is often different depending on the contractual environment in which the firm is located. In particular, when the efficiency of contract enforcement is relatively low, the liquidation rights of financial institutions such as banks as investors are bound to suffer. Bolton and Rosenthal point out that difference in contract enforcement efficiency will change the creditor’s claims on income [16, 17]. The similar assumption is applied by Acemoglu et al. and Holmstrom and Tirole in their analysis [18, 19].

A review of the literature shows that there are relatively few academic studies addressing the relationship between the debt maturity structure of corporate and the contractual environment. Moreover, the only studies available are basically at the stage of empirical research, without analyzing the potential action mechanism. Therefore, the paper attempts to explore the issue theoretically by building a theoretical model with the help of linear programming analysis.

Based on the reality, this paper further expands the theoretical hypothesis of Holmstrom and Tirole and reconstructs the framework [14, 15]. And then through theoretical research and mathematical derivation, discusses the specific influence mechanism of contract enforcement efficiency on debt maturity structure under the condition that external investors’ liquidation rights are impaired.

2. Assumption

Consider an economy that lasts three periods and in which the stakeholders involved are one entrepreneur and lots of outside investors. The entrepreneur does not have sufficient funds to successfully complete the project, so he has to resort to external financing. In the contest of the variable-investment framework, this paper introduces contract enforcement efficiency into the standard debt maturity model. The model we consider is a transformed version of Holmstrom and Triole (2000) where we introduce a new contract enforcement efficiency parameter. Following the original model:

(1)
There are three periods, $t=$ 0, 1, 2. At date 0, the entrepreneur has a project requiring variable investment $I$ . He initially has “assets” $A$ and needs to borrow $I-A$ from investors.
(2)
At date 1, the investment yields deterministic and verifiable income $r I$ ; At the same time, the firm meets a new investment chance requiring an amount $\rho I$ ,1
¹
At date 1 the borrower, experiences a liquidity shock that needs to be withstood in order for the firm to continue and possibly succeed. A simple interpretation of this liquidity shock is as a reinvestment need (an investment cost overrun), but it can be equivalently thought of as being a new investment opportunity or else a shortfall in earnings at the intermediate stage, in which case a new external cash infusion is needed in order to cover operating expenses.

where $\rho$ is ex ante unknown and has cumulative distribution function $F(\rho)$ with density $f(\rho)$ on $\rho\in[0,\infty)$ . The realization of $\rho$ is learned at date 1.
(3)
If the firm reinvests $\rho I$ , then it yields, at date 2, $R I$ with probability $p$ and 0 with probability $1-p$ . If the entrepreneur would not get enough money for reinvestment, he will face bankruptcy liquidation, and liquidation value $L I$ belongs to outside investors.2
²
L means the residual value per unit of investment if the firm is liquidated at date 1. The salvage value is a monetary value that can be transferred to the lenders if the project is abandoned. It is worth noting that the upper-bound on L equal to the amount of liquidity available in the firm in the original model.

It should be emphasized that influenced by contract enforcement efficiency, claims on liquidation value of outside investors cannot be fully enforced. That is, the investors receive the liquidation value with probability $e$ at date 1. Where $e$ represents the level of contract enforcement efficiency, in fact increasing of $e$ shows that contract enforcement efficiency improves.3
³
About the contract enforcement efficiency $e$ here, the paper followed the assumptions of Acemoglu and Johnson in 2005 in the basic model in an imperfect-enforcement environment, where only a fraction of the investors’ nominal claim on the final profit is actually returned to investors. That is, the efficiency of contract enforcement affects the investor’s claim on the final profit. In this paper, as the efficiency of contract implementation e increases, investors will be able to obtain a greater probability of liquidation value.

(4)
The probability of success $p$ is affected by the effort degree of the entrepreneur which is unobservable. Behaving yields probability $p=p_{H}$ of success, and misbehaving results in probability $p=p_{L}<p_{H}$ of success and private benefit $BI>0$ . Let $\Delta p=p_{H}-p_{L}$ .

The project has positive NPV if the entrepreneur behaves, at this time,

$\displaystyle\mathop{\max}\limits_{\tilde{\rho}}\left.{\left\{{r+F(\tilde{\rho% })p_{H}R+\left[{1-F(\tilde{\rho})}\right]eL-1-\int_{0}^{\tilde{\rho}}{\rho f(% \rho)d\rho}}\right.}\right\}>0;$

But negative NPV if the entrepreneur does not behave, namely

$\displaystyle\mathop{\max}\limits_{\tilde{\rho}}\left.{\left\{{r+F(\tilde{\rho% })p_{L}R+\left[{1-F(\tilde{\rho})}\right]eL-1-\int_{0}^{\tilde{\rho}}{\rho f(% \rho)d\rho+B}}\right.}\right\}<0,$

even if it includes the borrower’s private benefit.

Further assumptions are as follows:

H1: Investors behave competitively in the sense that the loan, if any, makes zero profit. H2: Both the entrepreneur and investors are risk neutral and the riskless rate is taken to be 0. H3: The entrepreneur is protected by limited liability.

We summarize the timing in Fig. 1.

Figure 1.
Game timing.

3. Contract setting

The research process in this paper is illustrated in Fig. 2.

Figure 2.

Road map for the analysis process.

Firstly, the form of the contract between the entrepreneur and external investors is set, and the optimal model is given based on the contract, which leads to the definition of the optimal debt maturity structure. Secondly, the optimal model is solved to obtain $R_{b}^{\ast}$ , $I^{\ast}$ , and $\rho^{c\ast}$ in turn, which then gives expressions for the firm’s long-term debt $L D$ , short-term debt $S D$ and debt maturity structure $D M$ in equilibrium. Finally, the comparative analysis examines the impact of contract enforcement efficiency on debt maturity structure $D M$ .

The contract signed by external investors and the entrepreneur specifies that investment level is $I$ ; Only if $\rho\leqslant\rho^{c}$ , the firm reinvests, which means $\rho^{c}$ is a cutoff of reinvestment; The entrepreneur gets nothing at date 1 and $r I$ all belong to outside investors; If the project is successful, the entrepreneur and investors get $R_{b}$ and $RI-R_{b}$ respectively; If the project fails, both of them get 0; The outside investors can only get liquidation value $L I$ with probability $e$ .

For the purpose of modeling, it is supposed that the financing contract takes the following state-contingent form:

$\displaystyle\{I;\rho^{c};(0,R_{b},0);(rI,RI-R_{b},0)\}.$

To ensure that an equilibrium solution exists, the following conditions need to be satisfied about the parameters:

$\displaystyle\mathop{\max}\limits_{\rho^{c}}\left\{{\frac{F(\rho^{c})({p_{H}R-% eL})]-\left[1+\int_{0}^{\rho^{c}}{\rho f(\rho)}d\rho-eL-r\right]}{F(\rho^{c})p% _{H}}}\right\}<\frac{B}{\Delta p}$ (1) $\displaystyle 0<\mathop{\min}\limits_{\rho^{c}}\left\{{\frac{F(\rho^{c})({p_{H% }R-eL})]-\left[1+\int_{0}^{\rho^{c}}{\rho f(\rho)}d\rho-eL-r\right]}{F(\rho^{c% })p_{H}}}\right\}$ (2)

According to the contract, the probability of reinvestment is $\Pr ob\{\rho\leqslant\rho^{c}\}=F(\rho^{c})$ .

The revenue structure of the entrepreneur and investors from the above contract shown in Fig. 3.

Figure 3.

Revenue structure.

So the optimization problem becomes:

$\displaystyle\left\{{\begin{array}[]{l}\mathop{\max}\limits_{R_{b},\rho^{c},I}% F(\rho^{c})p_{H}R_{b}-A\\ s.t.(a1)F(\rho^{c})p_{H}R_{b}\geqslant F(\rho^{c})(p_{L}R_{b}+BI)\\ (b1)rI+[{p_{H}({RI-R_{b}})-eLI}]+eLI\\ \geqslant I+\int_{0}^{\rho^{c}}{\rho If(\rho)d\rho}-A\\ \end{array}}\right.$ (3)

where:

•

The objective function is the entrepreneur’s utility, which can be expressed as $U_{b}(\rho^{c})$ ;

•

(a1) is the entrepreneur’s incentive-compatibility constraint, and it could be simplified as $R_{b}\geqslant{BI}/{\Delta p}$ .

•

(b1) is the investors’ individual-rationality constraint, and it could be simplified as:

$\displaystyle R_{b}\leqslant\frac{F(\rho^{c})({p_{H}R-eL})]-\left[1+\int_{0}^{% \rho^{c}}{\rho f(\rho)}d\rho-eL-r\right]}{F(\rho^{c})p_{H}}$ (4) $\displaystyle\quad{}+\frac{A}{F(\rho^{c})p_{H}}$

In equilibrium, competitive lenders make no profit on the contract that is most advantageous for the borrower, that means the condition Eq. (3) should be an equality. So the optimal model Eq. (3) will be simplified as:

$\displaystyle\left\{{\begin{array}[]{l}\mathop{\max}\limits_{R_{b},\rho^{c},I}% rI+F(\rho^{c})p_{H}({R-eL})I+eLI\\ -\left[I+\int_{0}^{\rho^{c}}{\rho If(\rho)d\rho}\right]\\ s.t.(a2)R_{b}\geqslant BI/\Delta\\ (b2)rI+[{p_{H}({RI-R_{b}})-eLI}]+eLI\\ \geqslant I+\int_{0}^{\rho^{c}}{\rho If(\rho)d\rho}-A\\ \end{array}}\right.$ (5)

4. Optimal contract

Now, the model Eq. (5) could be solved in three steps.

Step 1: Solve $R_{b}^{\ast}$ for a given $\rho^{c}$ and $I$ .

The “feasible contract set” of Eq. (3) can be illustrated by Fig. 4, and it is constituted by the shaded area OEF.

Figure 4.

The solution for the model. ${}^{4}$

First of all, it is easy to get that the intercept of the line (EF) is $A\mathord{\left/{\vphantom{A{[F(\rho^{c})p_{H}e]}}}\right.\kern-1.2pt}{[F(\rho% ^{c})p_{H}e]}$ and the slope is:

$\displaystyle k_{1}=\frac{F(\rho^{c})({p_{H}R-eL})-\left[1+\int_{0}^{\rho^{c}}% {\rho f(\rho)d\rho-eL-r}\right]}{F(\rho^{c})p_{H}}.$

From Eqs (1) and (2), it can be seen that:

$\displaystyle 0<k_{1}<\frac{B}{\Delta P}.$

Therefore, the “feasible contract set” is not an empty set if $A>0$ . The objective function indicates that it is the bigger the better for $I$ . So the point F constitutes the optimal contract, that means: $R_{b}^{\ast}=\frac{BI}{\Delta P}$ .

Step 2: Find the optimal $I^{\ast}$ for a given $\rho^{c}$ .

Now, we need to introduce two concepts: the expected profit per unit of investment $m(\rho^{c})$ and the multiplier $k(\rho^{c})$ which denotes the financing capacity. Where:

$\displaystyle m(\rho^{c})=F(\rho^{c})({p_{H}R-eL})-\left[1+\int_{0}^{\rho^{c}}% {\rho f(\rho)d\rho-eL-r}\right]$ (6) $\displaystyle k(\rho^{c})=\frac{1}{1+\int_{0}^{\rho^{c}}{\rho f(\rho)d\rho}-eL% -r-F(\rho^{c})[{p_{H}\left({R-B\mathord{\left/{\vphantom{B{\Delta p}}}\right.% \kern-1.2pt}{\Delta p}}\right)-eL}]}$ (7)

Actually, take $R_{b}^{\ast}=\frac{BI}{\Delta P}$ into the Eq. (5) and it can be further simplified as that:

$\displaystyle\left\{{\begin{array}[]{l}\mathop{\max}\limits_{R_{b},\rho^{c},I}% m(\rho^{c})I\\ s.t.(b3)rI+[{p_{H}({RI-R_{b}})-eLI}]+eLI\\ \geqslant I+\int_{0}^{\rho^{c}}{\rho If(\rho)d\rho}-A\\ \end{array}}\right.$ (8)

As investors behave competitively in the sense that the loan, if any, makes zero profit, (b3) in Eq. (6) holds with equality. This implies:

$\displaystyle I^{\ast}(\rho^{c})=k(\rho^{c})A;U_{b}^{\ast}(\rho^{c})=m(\rho^{c% })k(\rho^{c})A.$

From Eq. (1), it can be shown that, for any $\rho^{c}$ , $k(\rho^{c})>0$ .

Step 3: Consider the optimal $\rho^{c\ast}$ .

Here, three definitions need to be presented: the expected unit cost of effective investment $c(\rho^{c})$ , the expected pledgeable income $\rho_{0}$ and the expected income $\rho_{1}$ . Where

$\displaystyle c(\rho^{c})=\frac{1+\int_{0}^{\rho^{c}}{\rho f(\rho)d\rho-eL-r}}% {F(\rho^{c})},$ $\displaystyle\rho_{0}=p_{H}(R-B\mathord{\left/{\vphantom{B{\Delta p}}}\right.% \kern-1.2pt}{\Delta p})-eL,$ $\displaystyle\rho_{1}=p_{H}R-eL.$

With respect to the optimal reinvestment threshold, there is an inference.

Inference 1: If the threshold liquidity shock $\rho_{c}^{\ast}$ is equal to $c(\rho^{c})$ , the entrepreneur’s welfare reaches its maximum, and $\rho_{c}^{\ast}$ lies between $\rho_{0}$ and $\rho_{1}$ , that is:

$\displaystyle c(\rho^{c\ast})=\rho^{c\ast};\rho_{0}<\rho^{c\ast}<\rho_{1}.$

Proof: In fact

$\displaystyle U_{b}(\rho^{c})=\frac{F(\rho^{c})\rho_{1}-\left[(1+\int_{0}^{% \rho^{c}}{\rho f(\rho)d\rho)-eL-r}\right]}{1+\int_{0}^{\rho^{c}}{\rho f(\rho)d% \rho}-eL-r-F(\rho^{c})\rho_{0}}A=\frac{\rho_{1}-c(\rho^{c})}{c(\rho^{c})-\rho_% {0}}A,$

So:

$\displaystyle\mathop{\max}\limits_{\rho^{c}}U_{b}(\rho^{c})\Leftrightarrow% \mathop{\min}\limits_{\rho^{c}}c(\rho^{c}),$

The first-order condition is:

$\displaystyle\rho^{c}f(\rho^{c})F(\rho^{c})-f(\rho^{c})\left({1+\int_{0}^{\rho% ^{c}}{\rho f(\rho)d\rho}-eL-r}\right)=0,$

That is:

$\displaystyle\int_{0}^{\rho^{c\ast}}{F(\rho)d\rho}=1-eL-r$ (9)

Then from Eq. (9), it implies:

$\displaystyle c(\rho^{c\ast})=\frac{1+\int_{0}^{\rho^{c\ast}}{\rho f(\rho)d% \rho-eL-r}}{F(\rho^{c\ast})}=\frac{1-eL-r+\rho^{c\ast}F(\rho^{c\ast})-\int_{0}% ^{\rho^{c\ast}}{F(\rho)d\rho}}{F(\rho^{c\ast})}=\rho^{c\ast},$

That means:

$\displaystyle\rho^{c\ast}=c(\rho^{c\ast})=\frac{1+\int_{0}^{\rho^{c\ast}}{\rho f% (\rho)d\rho-eL-r}}{{\kern 1.0pt}F(\rho^{c\ast})}$ (10)

Because of:

$\displaystyle U_{b}^{\ast}=\frac{\rho_{1}-\rho^{c\ast}}{\rho^{c\ast}-\rho_{0}}A,$

It may be achieved that:

$\displaystyle\rho_{0}<\rho^{c\ast}<\rho_{1}.$

According to the optimal contract, the optimal debt maturity structure can be defined. In inference 1, $\rho_{0}$ represents the expected pledgeable income per unit of investment when the project is ongoing, that is, when the entrepreneurial incentive compatibility constraint is satisfied, the highest income allocated to the external investor guaranteed by the unit investment in case the project is sustained and successful.

Definition 1: The debt maturity structure can be measured by the ratio of long-term debt to total debt, namely:

$\displaystyle DM=LD/(LD+SD).$

where short-term debt is:

$\displaystyle SD=(r+\rho_{0}-\rho^{c\ast})I^{\ast},$

And long-term debt is:

$\displaystyle LD=RI^{\ast}-R_{b}^{\ast}-\rho_{0}I^{\ast}.$

5. Comparative analysis

Comparative static analysis is an analytical method used in static economic models. This method can be used to analyze the specific effects of changes in exogenous variables on endogenous variables. In the theoretical model, contract enforcement efficiency is clearly an exogenous variable, and the ultimate goal of this paper is to explore the impact of contract enforcement efficiency on the optimal debt maturity structure in equilibrium. In order to achieve this purpose, the impact of contract enforcement efficiency on financing contract must be considered. For this question, two inferences can be drawn as follows.

Inference 2: The “first-best cutoff” of reinvestment $\rho^{c\ast}$ decreases as the contract enforcement efficiency becomes better, that is $\partial\rho^{c\ast}/\partial e<0$ .

Proof: From Eq. (9), it implies:

$\displaystyle\rho^{c\ast}F(\rho^{c\ast})=1+\int_{0}^{\rho^{c\ast}}{\rho f(\rho% )d\rho-eL-r},$

For the above formula, taking the partial derivative on both sides, it can be shown that:

$\displaystyle\rho^{c\ast}f(\rho^{c\ast})\frac{\partial\rho^{c\ast}}{\partial e% }+F(\rho^{c\ast})\frac{\partial\rho^{c\ast}}{\partial e}=\rho^{c\ast}f(\rho^{c% \ast})\frac{\partial\rho^{c\ast}}{\partial e}-L,$

That is:

$\displaystyle\frac{\partial\rho^{c\ast}}{\partial e}=-\frac{L}{F(\rho^{c\ast})% }<-L<0$ (11)

For this inference, it is worth noting that $\partial\rho^{c\ast}/\partial e<0$ is indeed strongly connected to $\partial\rho^{c\ast}/\partial L<0$ which is immediate from the description of the problem. In fact, it could be proved from the original model built by Holmstrom and Tirole in 2000 that $\partial\rho^{c\ast}/\partial e<0$ .

Inference 2 shows that when the contract enforcement efficiency becomes better, the “first-best cutoff” of reinvestment $\rho^{c\ast}$ decreases, and therefore the optimal probability of project continuation $F(\rho^{c})$ decreases. The reason for this is that the improvement in contract enforcement efficiency has reduced the incentives for outside investors to reinvest. On the one hand, more efficient contract enforcement increases the probability of obtaining liquidation value for external investors at date 1. On the other hand, as e increases, the expected pledgeable income $\rho_{0}$ of external investors will decrease as the project continues. For these two reasons, with improved contract enforcement efficiency, outside investors are less likely to reinvest in the project, i.e., the optimal probability of reinvestment decreases.

Inference 3: The optimum investment scale $I^{\ast}$ , is increasing with the improvement of contract enforcement efficiency, that is $\partial I^{\ast}/\partial e>0$ .

Proof: To simplify the proof process, set:

$\displaystyle q(\rho^{c\ast})=1+\int_{0}^{\rho^{c\ast}}{\rho f(\rho)d\rho}-eL-% r-F(\rho^{c\ast})[{p_{H}({R-B\mathord{\left/{\vphantom{B{\Delta p}}}\right.% \kern-1.2pt}{\Delta p}})-eL}],$

And then:

$\displaystyle\frac{\partial q(\rho^{c\ast})}{\partial e}=f(\rho^{c\ast})(\rho{% }^{c\ast}-\rho_{0})\frac{\partial\rho^{c\ast}}{\partial e}-[1-F(\rho^{c\ast})]L$

Equation (11) shows that $\partial\rho^{c\ast}/\partial e<0$ , and since $\rho^{c\ast}>\rho_{0}$ , $0<F(\rho^{c\ast})<1$ , it follows that:

$\displaystyle\frac{\partial q(\rho^{c\ast})}{\partial e}<0$

And because $I^{\ast}(\rho^{c\ast})=k(\rho^{c\ast})A=A/q(\rho^{c\ast})$ , easy to draw that:

$\displaystyle\frac{\partial I^{\ast}}{\partial e}=-\frac{A}{[q(\rho^{c\ast})]^% {2}}\cdot\frac{\partial q(\rho^{c\ast})}{\partial e}>0$ (12)

That means $I^{\ast}$ is increasing with the increase of $e$ .

In this paper, the contract enforcement efficiency means the probability that outside investors can receive the liquidation value. That is, when the contract enforcement efficiency is relatively high, external investors can be able to obtain the project liquidation value with a higher probability. In other words, as e increases, the interests of external investors will be protected. Thus by economic intuition, the amount of investment or the size of the project $I^{\ast}$ will increase at this time. Inference 3 coincidentally proves the above economic conjecture.

Next, it will be firstly analyzed for the relationship between short-term debt, long-term debt and contract enforcement efficiency respectively; and then the relationship between debt maturity structure and contract enforcement efficiency can be studied.

Proposition 1: The short-term debt is increasing with the improvement of contract enforcement efficiency, that is $\partial SD/\partial e>0$ .

Proof: The short-term debt is:

$\displaystyle SD=(r+\rho_{0}-\rho^{c\ast})I^{\ast},$

where:

$\displaystyle\rho_{0}=p_{H}(R-B/\Delta p)-eL,\text{ so:}$ $\displaystyle\frac{\partial SD}{\partial e}=(r+\rho_{0}-\rho^{c\ast})\frac{% \partial I^{\ast}}{\partial e}+\left({-L-\frac{\partial\rho^{c\ast}}{\partial e% }}\right)I^{\ast},$

From Eq. (11), it implies:

$\displaystyle{\partial\rho^{c\ast}}\mathord{\left/{\vphantom{{\partial\rho^{c% \ast}}{\partial e}}}\right.\kern-1.2pt}{\partial e}={-L}\mathord{\left/{% \vphantom{{-L}{F(\rho^{c\ast})}}}\right.\kern-1.2pt}{F(\rho^{c\ast})},$

And then:

$\displaystyle\frac{\partial SD}{\partial e}=(r+\rho_{0}-\rho^{c\ast})\frac{% \partial I^{\ast}}{\partial e}+\frac{L[1-F(\rho^{c\ast})]}{F(\rho^{c\ast})}I^{% \ast},$

where: $0<F(\rho^{c\ast})<1$ , $\partial I^{\ast}/\partial e>0$ , and the short-term debt is positive means $r+\rho_{0}-\rho^{c\ast}>0$ , so $\frac{\partial SD}{\partial e}>0$ .

Proposition 2: The long-term debt is increasing with the improvement of contract enforcement efficiency, that is $\partial LD/\partial e>0$ .

Proof: The long-term debt is $LD=RI^{\ast}-R_{b}^{\ast}-\rho_{0}I^{\ast}$ , where $\rho_{0}=p_{H}(R-B/\Delta p)-eL$ and $R_{b}^{\ast}=BI/\Delta p$ , now:

$\displaystyle LD=RI^{\ast}-\frac{BI^{\ast}}{\Delta p}-\rho_{0}I^{\ast}=\left[{% (1-p_{H})\left({R-\frac{B}{\Delta p}}\right)+eL}\right]I^{\ast},$

So:

$\displaystyle\frac{\partial LD}{\partial e}=LI^{\ast}+\left[{(1-p_{H})\left({R% -\frac{B}{\Delta p}}\right)+eL}\right]-\frac{\partial I^{\ast}}{\partial e}^{% \ast},$

And because $\partial I^{\ast}/\partial e>0$ (getting from inference 3) and the long-term debt is positive means $(1-p_{H})(R-B/\Delta p)+eL>0$ , it implies $\frac{\partial LD}{\partial e}>0$ .

From proposition 1 and proposition 2, we can obtain that both the short-term debt and long-term debt are increasing with the improvement of contract enforcement efficiency e. Actually, considering a deeper connection, the rise in SD and LD are both consequences of I-A increasing as the IR constraint of the firm relaxes.

Proposition 3: The changing direction in debt maturity structure is uncertain while the contract enforcement efficiency changes, the specific relationship between them is as follows:

(1)

If $X_{1}>X_{2}$ , the debt maturity structure is increasing with the improvement of contract enforcement efficiency, that is $\partial DM/\partial e>0$ ;

(2)

If $X_{1}<X_{2}$ , the debt maturity structure is decreasing with the improvement of contract enforcement efficiency, that is $\partial DM/\partial e<0$ ;

(3)

If $X_{1}=X_{2}$ , the debt maturity structure has nothing to do with contract enforcement efficiency, that is $\partial DM/\partial e=0.$ .

where:

$\displaystyle X_{1}=R+r-B/\Delta p-\rho^{c\ast}>0,$ $\displaystyle X_{2}={[{(1-p_{H})(R-B/\Delta p)+eL}]}\mathord{\left/{\vphantom{% {\left[{(1-p_{H})(R-B/\Delta p)+eL}\right]}{F(\rho^{c\ast})}}}\right.\kern-1.2% pt}{F(\rho^{c\ast})}>0.$

Proof: The debt maturity structure:

$\displaystyle DM=\frac{LD}{LD+SD}=\frac{(1-p_{H})(R-B/\Delta p)+eL}{R+r-B/% \Delta p-\rho^{c\ast}},$

And then:

$\displaystyle\frac{\partial DM}{\partial e}=\frac{LX_{1}-X_{2}F(\rho^{c\ast})(% {-\partial\rho^{c\ast}}\mathord{\left/{\vphantom{{-\partial\rho^{c\ast}}{% \partial e}}}\right.\kern-1.2pt}{\partial e})}{(X_{1})^{2}},$

From Eq. (11) shows that ${\partial\rho^{c\ast}}\mathord{\left/{\vphantom{{\partial\rho^{c\ast}}{% \partial e}}}\right.\kern-1.2pt}{\partial e}={-L}\mathord{\left/{\vphantom{{-L% }{F(\rho^{c\ast})}}}\right.\kern-1.2pt}{F(\rho^{c\ast})}$ , so:

$\displaystyle\frac{\partial DM}{\partial e}=\frac{L(X_{1}-X_{2})}{(X_{1})^{2}}.$

This indicates the proposition holds true.

Proposition 3 shows that the changing direction in debt maturity structure is uncertain while the contract enforcement efficiency changes, and that this relationship is related to the magnitude of each exogenous parameter in the theoretical model. In proposition 3, $X_{1}$ is the total amount of debt per unit of investment, and $X_{2}$ represents the ratio of the amount of long-term debt per unit of investment to the probability of project continuation. That is, the debt maturity structure increases as the institutional environment improves when the total debt per unit of investment is relatively large, the long-term debt is relatively small, and the probability of project persistence is high; and vice versa.

Proposition 1, 2 and 3 are the most important conclusions for this paper. They show the concrete system of the impact of contract enforcement efficiency on debt maturity structure.

6. Conclusions

Signing of financing contract is not just kind of the enterprise’s independent behavior; it will be led by the contract enforcement efficiency to a great extent. This paper analyzes the impacting process and influencing result of contract enforcement efficiency on debt maturity structure by constructing theoretical model. By condensing the financing contract into a linear programming problem, using mathematical derivation and static comparative analysis, and finally get the following conclusions.

(1)

The short-term debt will increase as the contract enforcement efficiency increases;

(2)

The long-term debt will increase as the contract enforcement efficiency increases;

(3)

The direction of change in debt maturity structure is uncertain with respect to the contract enforcement efficiency, and the specific relationship between them is closely related to the exogenous variables.

In addition, the study also proves that the “first-best cutoff” of reinvestment will decrease and the optimal investment scale will increase with the improvement of the contract enforcement efficiency.

Footnotes

Acknowledgments

The authors are grateful to the anonymous referee for a careful checking of the details and for helpful comments that improved this paper. The authors acknowledge the financial support of three projects: (1) Philosophy and Social Science Fund of Yunnan Province, China, project No. YB2020023, No. YB2019014; (2) National Natural Science Foundation of China, project No. 71964018, No. 71763034.

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