The optimal financing of a conglomerate firm with hidden information and costly state verification

Abstract

This manuscript addresses the issue, particularly interesting for a conglomerate firm, of the choice of the optimal financing method (namely, the most efficient one) between the joint one and the separate one. In particular, the authors identify the properties of the optimal financing contract for three investment projects under the assumptions of the literature on Costly State Verification (CSV), namely, uncorrelated returns, hidden information (the return of a single project is a borrower’s private information), lender performing sequential audit and residual claimant borrower. The authors’ research method consists of solving the optimization problem of the borrower’s expected utility subject to appropriate incentive constraints and the lender’s participation constraint. The novelty of this contribution is the demonstration that joint financing with return pooling between the high and low states is more efficient than separate financing, as it implies a lower expected audit cost for the lender and, if the investment cost is not too high, also less credit rationing for the borrower. Joint financing with return pooling between the intermediate and low states, instead, is found to be less efficient than separate financing in terms of expected audit cost and, in the presence of sufficiently high investment cost, also credit rationing.

Appendices

Appendix

Proof of Proposition 2

2.1 Maximum punishment for false report

Note that punishment repayments are found only on the second member of the incentive constraints (namely in the constraints (2), (3), (4), (5), (6) and (7)) and are completely absent both in the objective function and in the participation constraint (Eq. (1)).

Therefore, it is possible to increase the generic punishment repayment by simultaneously reducing the relative audit probabilities in each of the incentive constraints. In this way, the generic constraint becomes slacker, and it is possible to reduce ${R}_{3}$ leaving $PC$ and $E{\Pi }_{B}$ unchanged.

More precisely, the authors proceed as follows:

(i) Increase ${R}_{0|\text{HL}.}$ and reduce ${m}_{0}$ while keeping ${m}_{0}{m}_{0,H}\left(1-{m}_{0,HL}\right){R}_{0|\text{HL}.}$ constant in (6) and (7). Doing so, these constraints become slacker because the term $\left(1-{m}_{0}\right){R}_{0|.}$ increases. So, the expected audit cost ${m}_{0}c$ and ${R}_{3}$ may be reduced. Then, set ${R}_{0|\text{HL}.}=H+2L.$ The second members of the (6) and (7) are increasing in ${m}_{0,H}{m}_{0,HL}$, but since this term is absent in the objective function and in the participation constraint, it may be set equal to 1 to slacken the (6) and (7) as much as possible.

A similar argument may be applied to ${R}_{0|LLH}$ and to the changes in ${m}_{0}$ making ${m}_{0}{m}_{0,L}{m}_{0,LL}{R}_{0|LLH}$ constant in the (7). Even in this case, it si possible to set ${R}_{0|LLH}=H+2L$ to slacken the (7). However, here it is impossible to set ${m}_{0,L}{m}_{0,LL}=1$ because this term appears both in the objective function and in the participation constraint. Again, in the (7) it is possible to increase ${R}_{0|LHL}$ and reduce ${m}_{0}$ while keeping ${m}_{0}{m}_{0,L}{m}_{0,LH}{R}_{0|LHL}$ constant and, at the same time, raise $\left(1-{m}_{0}\right){R}_{0|.}$. Doing so allows the reduction of the expected audit cost ${m}_{0}c$ in the (1) and, consequently, the decrease in ${\text{R}}_{3}$. So, set ${R}_{0|LHL.}=H+2L$. The second member of the (7) is increasing in ${m}_{0,LH}$ and, since this term appears neither in the objective function nor in the participation constaint, it may be set to 1 to slacken the (7).

(ii) Increase ${R}_{1|HL}$ and reduce at the same time ${m}_{1}$ while keeping ${m}_{1}{m}_{1,H}{R}_{1|HL}$ constant in the (5). Doing so slackens the same constraint because of the growth of the term $\left(1-{m}_{1}\right){R}_{1|.}$ allowing also to reduce the expected audit cost ${m}_{1}c$ and ${R}_{3}$ into the (1). So, set ${R}_{1|HL.}=2H+L.$ The second member of the (5) is increasing in ${m}_{1,H}$, but since this term does appear neither in the objective function nor in the participation constraint, it may be set equal to 1 to slacken the (5) as much as possible.

A similar argument may be applied to ${R}_{1|LH}$ and to the changes of ${m}_{1}$ while keeping ${m}_{1}{m}_{1,L}{R}_{1,LH}$ constant in the (5). Again, set ${R}_{1|LH}=2H+L$ to slacken the (5). However, in this case, it is impossible to set ${m}_{1,L}=1$ because this term appears both in the objective function and in the participation constaint. In any case, in the (1) the reduction of ${m}_{1}$ allows to decrease ${R}_{3}$.

It is also possible to raise ${R}_{1|H.}$ and reduce at the same time ${m}_{1}$ keeping ${m}_{1}\left(1-{m}_{1,H}\right){R}_{1|H.}$ constant into the (3) and allow to reduce ${R}_{3}$ in the (1) thanks to a decrease of ${m}_{1}c$. Set ${R}_{1|H.}=2H+L$ and ${m}_{1,H}=1$ to slacken the (3) as much as possible. Doing so is possible because of the absence of ${m}_{1,H}$ both in the objective function and in the participation constraint.

(iii) Increase ${R}_{0|\text{HH}.}$ reducing at the same time ${m}_{0}$ and keeping ${m}_{0}{m}_{0,H}\left(1-{m}_{0,HH}\right){R}_{0|\text{ HH}.}$ constant in the (4) and (6). Doing so raises $\left(1-{m}_{0}\right){R}_{0|.}$ and then slackens these constraints. In the (1), then, it is possible to decrease ${m}_{0}c$ and, consequently, also ${R}_{3}$. So, set ${R}_{0|\text{HH}.}=2H+L.$ The second members of the (4) and (6) are increasing in ${m}_{0,H}{m}_{0,HH}$ but, since these terms appears neither in the objective function nor in the participation constraint, they may be set equal to 1 to slacken the (4) and (6) as much as possible. Moreover, the reduction of ${m}_{0}$ in the (1) allows to reduce ${R}_{3}$ in the same constaint.

(iv) Increase ${R}_{0|\text{HHL}}$ and reduce ${m}_{0}$ while keeping ${m}_{0}{m}_{0,H}{m}_{0,HH}{R}_{0|\text{HHL}}$ constant in the (6). Doing so slackens the same constaint because of the growth of $\left(1-{m}_{0}\right){R}_{0|.}$ and lower ${m}_{0}c$ in the (1), allowing to reduce ${R}_{3}$ in the same expression. So, set ${R}_{0|\text{HHL}}=2H+L$. The second member of the (6) is increasing in ${m}_{0,H}{m}_{0,HH}$, but, since this term appears neither in the objective function and in the participation constraint, it may be set equal to 1 to slacken the (6) as much as possible.

Decreaing ${m}_{0}$ into the (1) allows to reduce ${R}_{3}$.

Again, raise ${R}_{0|\text{LHL}}$ and reduce ${m}_{0}$ while keeping ${m}_{0}{m}_{0,L}{ m}_{0,LH}{R}_{0|\text{LHH}}$ constant in the (6). Doing so slackens the second member of this constraint because of the reduction of $\left(1-{m}_{0}\right){R}_{0|.}$ and decreases ${m}_{0}c$ in the (1), allowing a reduction of ${R}_{3}$. So, set ${R}_{0|\text{LHH}}=2H+L$. Note also that ${m}_{0,LH}$ appears neither in the objective function and in the participation constraint and then this term may be set equal to 1 to slacken the (6) as much as possible.

(v) Increase ${R}_{1|\text{HH}.}$ and reduce ${m}_{0}$ while keeping ${m}_{1}{m}_{1,H}{R}_{1|\text{HH}}$ constant in the (3). Doing so slackens the same constraint because of the growth of $\left(1-{m}_{1}\right){R}_{1|.}$ and decreases ${m}_{1}c$ allowing a reduction of ${R}_{3}$ in the (1). So, set ${R}_{1|\text{HH}.}=2H+L.$ The second member of the (3) is increasing in ${m}_{1,H}$ and, since this term appears neither in the objective function nor in the participation constraint, it may be set equal to 1 to slacken the (3) as much as possible.

(vi) Increase ${R}_{0|\text{H}.}$ and reduce ${m}_{0}$ while keeping ${m}_{0}\left(1-{m}_{0,H}\right){R}_{0|\text{H}.}$ constant in the (4) and (6). Doing so slackens these constraints because of the growth of the term $\left(1-{m}_{0}\right){R}_{0|.}$ and allows a reduction of ${m}_{0}c$ in the (1) and, consequently, also of ${R}_{3}$. Then, set ${R}_{0|\text{H}.}=H+2L.$ The second members of the (4) and (6) are increasing in ${m}_{0,H}$ and, since this term is absent both in the participation constraint and in the incentive constraint, it may be set equal to 1 to slacken the (4) and (6) as much as possible.

(vii) Increase ${R}_{0|HHH}$ and reduce ${m}_{0}$ while keeping ${m}_{0}{m}_{0,H}{m}_{0,HH}{R}_{0|\text{ HHH}}$ constant in the (6). Doing so slackens the same constraint because of the growth of the term $\left(1-{m}_{0}\right){R}_{0|.}$ and lower ${m}_{0}c$ in the (1) allowing, therefore, a reduction of ${R}_{3}$. So, set ${R}_{0|\text{HHH}}=3H.$ The second member of the (6) is increasing in ${m}_{0,H}{m}_{0,HH}$ and, since it appears neither in the objective function nor in the participation constraint, this term may be set equal to 1 to slacken the (6) as much as possible.

A similar argument holds for ${R}_{2|\text{H}.}$. Increase this term and reduce at the same time ${m}_{2}$ while keeping ${m}_{2}{R}_{2|\text{H}}$ constant in the (2). Doing so slackens the same constraint because of the growth of $\left(1-{m}_{2}\right){R}_{2|.}$, lowers ${m}_{2}c$ and, consequently, allows also to reduce ${R}_{3}$ in the same constraint. Then, set ${R}_{2|\text{H }}=3H$. However, it is impossible to set ${m}_{2}=1$, because this term appears both in the objective function and in the participation constraint.

Following all of these modifications, the participation constraint becomes binding, as the punishment repayments are set at their maximum values, and the corresponding expected audit cost and repayment ${R}_{3}$ have been reduced as much as possible.

So, the initial optimization problem of the borrower’s expected utility becomes:

$$\begin{aligned} {}&\text{max} {p}^{3}\left(3H-{R}_{3}\right)+3{p}^{2}\left(1-p\right)\left[2H+L-{m}_{2}{R}_{2|\text{L}. }-\left(1-{m}_{2}\right){R}_{2|. }\right]\\ &\quad +3p{\left(1-p\right)}^{2}\left\{2L+H-{m}_{1}\left[{m}_{1,L}{R}_{1|\text{ LL}.}+\left(1-{m}_{1,L}\right){R}_{1|\text{L}. }\right]-\left(1-{m}_{1}\right){R}_{1|. }\right\}\\ &\quad +{\left(1-p\right)}^{3}\left\{3L-{m}_{0}\left[{m}_{0,L}\left({m}_{0,LL}{R}_{0|\text{LLL}}+\left(1-{m}_{0,LL}\right){R}_{0|\text{LL}.}\right)\right.\right.\\ &\quad+\left.\left.\left(1-{m}_{0,L}\right){R}_{0|\text{L}.}\right]-\left(1-{m}_{0}\right){R}_{0|.}\right\} \end{aligned} $$

subject to:

$$\begin{aligned}{p}^{3}{R}_{3}&+3{p}^{2}\left(1-p\right)\left\{{m}_{2}\left({R}_{2|\text{L }}-c\right)+\left(1-{m}_{2}\right){R}_{2|. }\right\}\\ &+3p{\left(1-p\right)}^{2}\left\{{m}_{1}\left[{m}_{1,L}\left({R}_{1|\text{LL}}-c\right)+\left(1-{m}_{1,L}\right){R}_{1|\text{L}.}-c\right]+\left(1-{m}_{1}\right){R}_{1|.}\right\}\\ &+{\left(1-p\right)}^{3}\left\{{m}_{0}\left[{m}_{0,L}\left({m}_{0,LL}\left({R}_{0|\text{LLL}}-c\right)+\left(1-{m}_{0,LL}\right){R}_{0|\text{LL}.}-c\right)\right.\right. \\ &+\left.\left.\left(1-{m}_{0,L}\right){R}_{0|\text{L}.}-c\right]+\left(1-{m}_{0}\right){R}_{0|.}\right\}=3I \end{aligned}$$

(12)

$${R}_{3}\le \left(1-{m}_{2}\right){R}_{2|. }+{m}_{2}3H$$

(13)

$${R}_{3}\le \left(1-{m}_{1}\right){R}_{1|. }+{m}_{1}\left(2H+L\right)$$

(14)

$${R}_{3}\le \left(1-{m}_{0}\right){R}_{0|. }+{m}_{0}3H$$

(15)

$$\begin{aligned}\left(1-{m}_{2}\right){R}_{2|.}&+{m}_{2}{R}_{2|\text{L}}\le \left(1-{m}_{1}\right){R}_{1|.}\\&+{m}_{1}\left\{{0,5} \left(2H+L\right)+{0,5} \left[{m}_{1,L}\left(2H+L\right)+\left(1-{m}_{1,L}\right){R}_{1|\text{L}.}\right]\right\}\end{aligned}$$

(16)

$$\begin{aligned}\left(1-{m}_{2}\right){R}_{2|.}&+{m}_{2}{R}_{2|\text{L}}\le \left(1-{m}_{0}\right){R}_{0|.}\\&+{m}_{0}\left\{{0,5}(2H+L)+{0,5}\left[{m}_{0,L}(2H+L)+\left(1-{m}_{0,L}\right){R}_{0|\text{L}.}\right]\right\}\end{aligned}$$

(17)

$$\begin{aligned} \left(1-{m}_{1}\right){R}_{1|.}&+{m}_{1}\left[\left(1-{m}_{1,L}\right){R}_{1|\text{L}.}+{m}_{1,L}{R}_{1|\text{LL}.}\right]\le \left(1-{m}_{0}\right){R}_{0|.}\\ &+{m}_{0}\left\{{0,5}\left(H+2L\right) +{0,5}\left[{ m}_{0,L}\left[{0,5}\left(\left(1-{m}_{0,LL}\right){R}_{0|LL.}\right.\right.\right.\right.\\ &\left.\left.\left.\left.+{m}_{0,LL}\left(H+2L\right)\right)+{0,5}\left(H+2L\right)\right]+\left(1-{ m}_{0,L}\right){R}_{0|\text{L}.}\right]\right\} \end{aligned}$$

(18)

2.2 Lower true state returns

Since ${R}_{0|\text{L}.}<3L$, then it is possible to reduce ${R}_{3}$ and increase ${R}_{0|\text{L}.}$ keeping ${p}^{3}{R}_{3}+{\left(1-p\right)}^{3}{m}_{0}\left(1-{m}_{0,L}\right){R}_{0|\text{L}.}$ into the (12) constant until ${R}_{0|\text{L}.}=3L$. Doing so, the (13), (14), (15) and (18) become slacker, allowing to decrease ${m}_{0}$, while the objective function and the participation constraint remain unchanged.

Replicating the same argument for ${R}_{0|.}$ and ${R}_{0|\text{LL}.}$ it results ${R}_{0|.}={R}_{0|\text{L}.}={R}_{0|\text{LL}.}=3L$. The return ${R}_{0|\text{LLL}}$ only appears in the objective function and in the participation constraint. In the (12), by replacing ${R}_{0|.}={R}_{0|\text{L}.}={R}_{0|\text{LL}.}=3L$, by lowering ${R}_{3}$ and by increasing ${R}_{0|\text{LLL}}$ keeping, at the same time, ${p}^{3}{R}_{3}+{\left(1-p\right)}^{3}{m}_{0}{m}_{0,L}{m}_{0,LL}{R}_{0|\text{LLL}}$ constant, the (13), (14) and (15) become more slack and the objective function and the participation constraint remain unchanged. Then, also ${R}_{0|\text{LLL}}=3L$.

Moreover, it ever holds ${R}_{3}>3L>0$ because, if ${R}_{3}\le 3L$, the return of the three investment projects would be ever insufficient to repay the investment cost.

The new reduced form of the problem becomes:

$$\begin{aligned}& \text{max} {p}^{3}\left(3H-{R}_{3}\right)+3{p}^{2}\left(1-p\right)\left[2H+L-{m}_{2}{R}_{2|\text{L}. }-\left(1-{m}_{2}\right){R}_{2|. }\right]+3p{\left(1-p\right)}^{2}\\&\quad \times\left\{2L+H-{m}_{1}\left[{m}_{1,L}{R}_{1|\text{ LL}.}+\left(1-{m}_{1,L}\right){R}_{1|\text{L}. }\right]-\left(1-{m}_{1}\right){R}_{1|.}\right\}\end{aligned}$$

subject to:

$$\begin{aligned}{p}^{3}{R}_{3}&+3{p}^{2}\left(1-p\right)\left\{{m}_{2}\left({R}_{2|\text{L }}-c\right)+\left(1-{m}_{2}\right){R}_{2|. }\right\}\\&+3p{\left(1-p\right)}^{2}\left\{{m}_{1}\left[{m}_{1,L}\left({R}_{1|\text{LL}}-c\right)+\left(1-{m}_{1,L}\right){R}_{1|\text{L}.}-c\right]+\left(1-{m}_{1}\right){R}_{1|.}\right\}\\&+{\left(1-p\right)}^{3}\left\{3L-{m}_{0}{m}_{0,L}\left(1+{m}_{0,LL}\right)c-{m}_{0}c\right\}=3I\end{aligned}$$

(19)

$${R}_{3}\le \left(1-{m}_{2}\right){R}_{2|. }+{m}_{2}3H$$

(20)

$${R}_{3}\le \left(1-{m}_{1}\right){R}_{1|. }+{m}_{1}\left(2H+L\right)$$

(21)

$${R}_{3}\le \left(1-{m}_{0}\right)3L+{m}_{0}3H$$

(22)

$$\begin{aligned}\left(1-{m}_{2}\right){R}_{2|.}&+{m}_{2}{R}_{2|\text{L}}\le \left(1-{m}_{1}\right){R}_{1|.}\\&+{m}_{1}\left\{{0,5} \left(2H+L\right)+{0,5} \left[{m}_{1,L}\left(2H+L\right)+\left(1-{m}_{1,L}\right){R}_{1|\text{L}.}\right]\right\}\end{aligned}$$

(23)

$$\begin{aligned} \left(1-{m}_{2}\right){R}_{2|.}&+{m}_{2}{R}_{2|\text{L}}\le \left(1-{m}_{0}\right)3L\\ &+{m}_{0}\left\{{0,5}(2H+L)+{0,5}\left[{m}_{0,L}(2H+L)+\left(1-{m}_{0,L}\right)3L\right]\right\} \end{aligned} $$

(24)

$$\begin{aligned}\left(1-{m}_{1}\right){R}_{1|.}&+{m}_{1}\left[\left(1-{m}_{1,L}\right){R}_{1|\text{L}.}+{m}_{1,L}{R}_{1|\text{LL}.}\right]\\ &\le \left(1-{m}_{0}\right)3L+{m}_{0}\left\{{0,5}\left(H+2L\right) +{0,5}\left[{ m}_{0,L}\right.\right.\\ &\left[{0,5}\left(\left(1-{m}_{0,LL}\right)3L+{m}_{0,LL}\left(H+2L\right)\right)\right.\\ &\left.\left.\left.+\,{0,5}\left(H+2L\right)\right]+\left(1-{ m}_{0,L}\right)3L\right]\right\} \end{aligned}$$

(25)

${{\varvec{m}}}_{0}>0$

If it were ${m}_{0}=0$, from (22), from (24) and from (25) one would deduce ${R}_{3}$, ${R}_{2|.}$, ${R}_{2|\text{L}}$, ${R}_{1|.}$, ${R}_{1|\text{L}.}$, ${R}_{1|\text{LL}.}\le 3L$ and the participation constraint (19) would no longer hold, because it would result:

$$3L-3{p}^{2}\left(1-p\right){m}_{2}c-3p{\left(1-p\right)}^{2}{m}_{1}c\left({m}_{1,L}c-1\right)<3I$$

So, it must necessarily hold ${m}_{0}>0$.

Also note that (24) and (25) must necessarily be binding. Otherwise, it would be possible to further reduce ${m}_{0}$ by slackening the participation constraint and allowing an additional reduction of ${R}_{3}$.

Proof of Proposition 3

3.1 Optimal values of $\widehat{{R_{3} }} \equiv R_{3}$,$\widehat{{R_{2|.} }} \equiv R_{2|.} ,\widehat{{R_{1|.} }} \equiv R_{1|.}$, $\widehat{{m_{2} }} \equiv m_{2} ,$ $\widehat{{m_{1} }} \equiv m_{1}$ $e$ $\widehat{{m_{1,L} }} \equiv m_{1,L}$

From the Eqs. (24) and (25) are obtained, respectively:

$$\left(1-{m}_{2}\right){R}_{2|.}+{m}_{2}{R}_{2|\text{L}}= 3L+(H-L){m}_{0}(1+{m}_{0,L})$$

and:

$$\begin{aligned}\left(1-{m}_{1}\right){R}_{1|.}&+{m}_{1}\left[\left(1-{m}_{1,L}\right){R}_{1|\text{L}.}+{m}_{1,L}{R}_{1|\text{LL}.}\right]\\ &=3L+{0,5} \left(H-L\right) {m}_{0}[1 +{0,5}{ m}_{0,L}(1+{m}_{0,LL})]\end{aligned}$$

Since the authors have just deduced the value of $\left(1-{m}_{2}\right){R}_{2|.}+{m}_{2}{R}_{2|L}$ from the constraint (24), the constraint (23) becomes negligible.

By replacing the two expressions above into the objective function and the participation constraint (19), the optimization problem becomes:

$$\begin{aligned}\text{max} {p}^{3}\left(3H-{R}_{3}\right)&+3{p}^{2}\left(1-p\right)(H-L)\left[2-{m}_{0}\left(1+{m}_{0,L}\right)\right]\\&+3p{\left(1-p\right)}^{2}(H-L)\left\{1-{m}_{0}[1 +{0,5}{ m}_{0,L}(1+{m}_{0,LL})]\right\}\end{aligned}$$

subject to:

$$\begin{aligned} {p}^{3}{R}_{3}&+3{p}^{2}\left(1-p\right)\left\{3L+(H-L){m}_{0}(1+{m}_{0,L})-{m}_{2}c\right\}\\ &+3p{\left(1-p\right)}^{2}\left\{3L+{0,5}\left(H-L\right){m}_{0}[1 +{0,5}{ m}_{0,L}(1+{m}_{0,LL})]-{m}_{1}\left(1+{m}_{1,L}\right)c\right\}\\ &+{\left(1-p\right)}^{3}\left\{3L-{m}_{0}\left[1+{m}_{0,L}\left(1+{m}_{0,LL}\right)\right]c\right\}=3I \end{aligned}$$

(26)

$${R}_{3}\le \left(1-{m}_{2}\right){R}_{2|. }+{m}_{2}3H$$

(27)

$${R}_{3}\le \left(1-{m}_{1}\right){R}_{1|. }+{m}_{1}\left(2H+L\right)$$

(28)

$${R}_{3}\le \left(1-{m}_{0}\right)3L+{m}_{0}3H$$

(29)

Note that it is possible to slacken the (27) and (28) leaving the constraints (26) and (29) and the borrower's expected utility unchanged by setting ${R}_{2|.}$ and ${R}_{1}$ to their corresponding maximum values:

$$\widehat{{R}_{2|. }}=2H+L$$

and:

$$\widehat{{R}_{1}}=H+2L$$

Note also that the borrower’s expected utility may be increased complying with the constraints (26), (27), (28) and (29) only by simultaneously reducing ${R}_{3}$, ${m}_{2}$, ${m}_{1}$ e ${m}_{1,L}$.

More precisely, these four values may be reduced until (27) and (28) become binding and ${m}_{1,L}$ goes to zero ($\widehat{{m}_{1,L}}=0$).

From the binding (27):

$${R}_{3}=\left(1-{m}_{2}\right){R}_{2|. }+{m}_{2}3H$$

and from the binding (28):

$${R}_{3}=\left(1-{m}_{1}\right){R}_{1|. }+{m}_{1}\left(2H+L\right)$$

it is deduced:

$${R}_{3}=\left(1-{m}_{2}\right)(2H+L)+{m}_{2}3H=\left(1-{m}_{1}\right)(H+2L)+{m}_{1}\left(2H+L\right)$$

The values of ${m}_{1}$ and ${m}_{2}$ solving this equation are $\widehat{{m}_{1}}=1$ and $\widehat{{m}_{2}}=0$ and they imply $\widehat{{R}_{3}}=2H+L$.

3.2 Optimal value of ${\widehat{{{\varvec{m}}}_{0}}\equiv {\varvec{m}}}_{0}$ and borrower’s expected utility optimal value

By replacing $\widehat{{R}_{3}}=2H+L$ in the (26) it is obtained:

$$\widehat{{m}_{0}}=\frac{{D}_{1}}{{D}_{2}}$$

with:

$${D}_{1}=3I-{p}^{3}\left(2H+L\right)-\left[3{p}^{2}\left(1-p\right)+3p{\left(1-p\right)}^{2}+{\left(1-p\right)}^{3}\right]3L$$

and:

$$\begin{aligned}{D}_{2}&=3{p}^{2}\left(1-p\right)(H-L)(1+\widehat{{m}_{0,L}})+3p{\left(1-p\right)}^{2}\\&\quad\times\left\{{0,5}\left(H-L\right)[1 +{0,5}\widehat{{m}_{0,L}}(1+\widehat{{m}_{0,LL}})]-c\right\}-{\left(1-p\right)}^{3}\left[1+\widehat{{m}_{0,L}}(1+\widehat{{m}_{0,LL}}\right]c\end{aligned}$$

and by replacing $\widehat{{R}_{3}}$ and $\widehat{{m}_{0}}$ in the objective function it is deduced:

$$\begin{aligned}\widehat{{E\Pi }_{B}}&={p}^{3}\left(H-L\right)+6{p}^{2}\left(1-p\right)\left(H-L\right)+3p{\left(1-p\right)}^{2}\left(H-L\right)\\&-\frac{{D}_{1}\left\{(1+\widehat{{m}_{0,L}})+{0,5}[1 +{0,5}\widehat{{m}_{0,L}}(1+\widehat{{m}_{0,LL}}]\right\}}{{D}_{2}}\end{aligned}$$

Proof of Proposition 4

4.1 Optimal value of ${{\varvec{m}}}_{0}^{\boldsymbol{*}}\equiv {{\varvec{m}}}_{0}$

The hypothesis that ${PF}_{1}$ holds, namely that the contract provides for the return pooling from three successes and from two successes and one fail, implies ${m}_{2}=0$; ${m}_{1}={m}_{1,L}=1$; ${R}_{3}=$ ${R}_{2|.}=$ ${R}_{2|\text{L}}\le 2H+L$ and ${R}_{1|.}=$ ${R}_{1|\text{L}.}={R}_{1|\text{LL}.}\le H+2L$.

By replacing these values ​​in the objective function and into the constraints (19), (20), (21), (22), (23), (24) and (25) the reduced form of the optimization problem under the hypothesis that Condition 2 holds is obtained:

$$\text{max} {p}^{3}\left(3H-{R}_{2|. }\right)+3{p}^{2}\left(1-p\right)(2H+L-{R}_{2|. })+3p{\left(1-p\right)}^{2}(2L+H-{R}_{1|\text{ L}.})$$

subject to:

$$\begin{aligned}{p}^{3}{R}_{2|. }&+3{p}^{2}\left(1-p\right){R}_{2|. }+3p{\left(1-p\right)}^{2}({R}_{1|\text{L}}-2c)+{\left(1-p\right)}^{3}\\&\times\left\{3L-{m}_{0}{m}_{0,L}\left(1+{m}_{0,LL}\right)c-{m}_{0}c\right\}=3I\end{aligned}$$

(30)

$${R}_{2|. }\le \left(1-{m}_{0}\right)3L+{m}_{0}3H$$

(31)

$${R}_{2|.}= \left(1-{m}_{0}\right)3L+{m}_{0}\left\{{0,5}(2H+L)+{0,5}\left[{m}_{0,L}(2H+L)+\left(1-{m}_{0,L}\right)3L\right]\right\}$$

(32)

$$ \begin{aligned} R_{{1{\text{L}}.|}} & = \left( {1 - m_{0} } \right)3L + m_{0} \left\{ 0,5\left( {H + 2L} \right) + 0,5\left[ m_{{0,L}}\right. \right. \\ & \quad \left. \left. \left[ 0,5\left( {\left( {1 - m_{{0,LL}} } \right)3L + m_{{0,LL}} \left( {H + 2L} \right)} \right) + 0,5\left( {H + 2L} \right) \right]+ \left( {1 - m_{{0,L}} } \right)3L \right] \right\} \end{aligned} $$

(33)

From the (30) it is deduced:

$$\begin{aligned} {}&{m}_{0}[{m}_{0,L}\left(1+{m}_{0,LL}\right)+1]\\&=\frac{\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]{R}_{2|. }+3p{\left(1-p\right)}^{2}{R}_{1|\text{L}}-3\left[I-{\left(1-p\right)}^{3}L\right]-6p{\left(1-p\right)}^{2}c}{{\left(1-p\right)}^{3}c}\end{aligned}$$

(34)

while from the (32) and from the (33) are obtained, respectively:

$${R}_{2|.}={m}_{0}\left(H-L\right)\left(1+{m}_{0,L}\right)+3L$$

(35)

and:

$${R}_{1|\text{L}.}={m}_{0}\left[1 +{0,5}{m}_{0,L}\left(1+{m}_{0,LL}\right) \right]{0,5}\left(H-L\right)+3L$$

(36)

By replacing the (35) and (36) in the (34), it is deduced:

$${m}_{0}^{*}=\frac{{D}_{3}}{{D}_{4}}$$

(37)

where ${m}_{0}^{*}$ is ever higher than zero for the constraint (31),

$${D}_{3}=-3\left[I-{\left(1-p\right)}^{3}L\right]-6p{\left(1-p\right)}^{2}c+\left[{p}^{3}+3{p}^{2}\left(1-p\right)+3p{\left(1-p\right)}^{2}\right]3L$$

and:

$$\begin{aligned} {D}_{4}&=\left[{m}_{0,L}^{*}\left(1+{m}_{0,LL}^{*}\right)+1\right]{\left(1-p\right)}^{3}c-\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(H-L\right)\left(1+{m}_{0,L}^{*}\right)\\ &\quad -3p{\left(1-p\right)}^{2}\left[1 +{0,5}{m}_{0,L}^{*}\left(1+{m}_{0,LL}^{*}\right) \right]{0,5}\left(H-L\right) \end{aligned} $$

By setting ${m}_{0,L}^{*}={m}_{0,LL}^{*}=1$, it is deduced:

$${m}_{0}^{*}=\frac{-3\left[I-{\left(1-p\right)}^{3}L\right]-6p{\left(1-p\right)}^{2}c+\left[{p}^{3}+3{p}^{2}\left(1-p\right)+3p{\left(1-p\right)}^{2}\right]3L}{3{\left(1-p\right)}^{3}c-2\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(H-L\right)-3p{\left(1-p\right)}^{2}\left(H-L\right)}$$

and using the binding ${PF}_{1}$:

$$\begin{aligned}& \left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(2H+L\right)+3p{\left(1-p\right)}^{2}\left(H+2L\right)-3{\left(1-p\right)}^{3}c\\ &\quad =3\left[I-{\left(1-p\right)}^{3}L\right]+6p{\left(1-p\right)}^{2}c\end{aligned}$$

(38)

this expression becomes:

$${m}_{0}^{*}=\frac{\begin{aligned}\left[ -{p}^{3}-3{p}^{2}\left(1-p\right)\right]\left(2H+L\right)-3p{\left(1-p\right)}^{2}\left(H+2L\right)\\ +3{\left(1-p\right)}^{3}c+\left[{p}^{3}+3{p}^{2}\left(1-p\right)+3p{\left(1-p\right)}^{2}\right]\end{aligned} 3L}{3{\left(1-p\right)}^{3}c-2\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(H-L\right)-3p{\left(1-p\right)}^{2}\left(H-L\right)}$$

from which, by adding the terms in the numerator, it is obtained:

$${m}_{0}^{*}=\frac{2\left[-{p}^{3}-3{p}^{2}\left(1-p\right)\right]\left(H-L\right)-3p{\left(1-p\right)}^{2}\left(H-L\right)+3{\left(1-p\right)}^{3}c}{3{\left(1-p\right)}^{3}c-2\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(H-L\right)-3p{\left(1-p\right)}^{2}\left(H-L\right)}=1$$

Setting ${m}_{0,L}^{*}={m}_{0,LL}^{*}=0$ in the (37), instead, it is deduced:

$${m}_{0}^{*}=\frac{-3\left[I-{\left(1-p\right)}^{3}L\right]-6p{\left(1-p\right)}^{2}c+\left[{p}^{3}+3{p}^{2}\left(1-p\right)+3p{\left(1-p\right)}^{2}\right]3L}{{\left(1-p\right)}^{3}c-\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(H-L\right)-3p{\left(1-p\right)}^{2}{0,5}\left(H-L\right)}$$

(39)

The maximum value of the (39) is computed by the (38) and it is equal to:

$$-\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]2\left(H-L\right)-3p{\left(1-p\right)}^{2}\left(H-L\right)+3{\left(1-p\right)}^{3}c$$

and by replacing this value into the (39) it is obtained:

$${m}_{0}^{*}=\frac{-\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]2\left(H-L\right)-3p{\left(1-p\right)}^{2}\left(H-L\right)+3{\left(1-p\right)}^{3}c}{{\left(1-p\right)}^{3}c-\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(H-L\right)-3p{\left(1-p\right)}^{2}{0,5}\left(H-L\right)}$$

The difference between the numerator and the denominator of the previous expression is:

$$2{\left(1-p\right)}^{3}c-\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(H-L\right)-\text{1,5}p{\left(1-p\right)}^{2}\left(H-L\right)$$

and one may easily prove that this term is negative. From the (38), in fact, it is deduced:

$$\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]2\left(H-L\right)+3p{\left(1-p\right)}^{2}\left(H-L\right)-3{\left(1-p\right)}^{3}c=3(I-L)+6p{\left(1-p\right)}^{2}c$$

from which:

$$\begin{aligned} & 2{\left(1-p\right)}^{3}c-\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(H-L\right)-\text{1,5}p{\left(1-p\right)}^{2}\left(H-L\right)\\ &\quad =-{\left(1-p\right)}^{3}c-3(I-L)-6p{\left(1-p\right)}^{2}c\\&\qquad+\left[{p}^{3}+3{p}^{2}\left(1-p\right)+3p{\left(1-p\right)}^{2}\right]\left(H+2L\right)-3p{\left(1-p\right)}^{2}{0,5}\left(H-L\right)\end{aligned}$$

For ${PF}_{1}$, this expression should necessarily be less than zero. So, ${m}_{0}^{*}<1$.

4.2 Optimal value of ${{\varvec{R}}}_{2|.}^{\boldsymbol{*}}\equiv {{\varvec{R}}}_{2|.}$

By replacing the (35) into the (30) it is obtained:

$${R}_{2|.}^{*}=3L+\frac{\left(H-L\right)\left(1+{m}_{0,L}\right){D}_{3}}{{D}_{4}}$$

where, for ${m}_{0,L}^{*}={m}_{0,LL}^{*}=1$ and ${PF}_{1}$ binding, as already proved in the Sect. 4.1 of the Appendix, it is deduced:

$${m}_{0}^{*}=\frac{{D}_{3}}{{D}_{4}}=1$$

and consequently, ${R}_{2|.}^{*}=2\left(H-L\right)+3L=2H+L$.

For ${m}_{0,L}^{*}={m}_{0,LL}^{*}=0$, instead, as already proved in the Sect. 4.1 of the Appendix, holds:

$${m}_{0}^{*}=\frac{{D}_{3}}{{D}_{4}}<1$$

and then ${R}_{2|.}^{*}<2H+L$.

4.3 Optimal value of ${{\varvec{R}}}_{1|{\varvec{L}}.}^{\boldsymbol{*}}\equiv {{\varvec{R}}}_{1|{\varvec{L}}.}$

By replacing the (36) in the (30), it is obtained:

$${R}_{1|\text{L}.}^{*}=\frac{{D}_{3} \left\{\left[1 +{0,5 }{m}_{0,L}^{*}\left(1+{m}_{0,LL}^{*}\right)\right]{0,5}\left(H-L\right)+3L\right\}}{{D}_{4}}$$

where, for ${m}_{0,L}^{*}={m}_{0,LL}^{*}=1$ and ${PF}_{1}$ binding, as proved in the Sect. 4.1 of the Appendix, it is deduced:

$${m}_{0}^{*}=\frac{{D}_{3} }{{D}_{4}}=1$$

and, consequently, ${R}_{1|\text{L}.}^{*}=\left(H-L\right)+3L=H+2L$.

For ${m}_{0,L}^{*}={m}_{0,LL}^{*}=0$, instead, as already proved in the Sect. 4.1 of the Appendix, holds:

$${m}_{0}^{*}=\frac{{D}_{3} }{{D}_{4}}<1$$

and then ${R}_{1|\text{L}.}^{*}<H+2L$.

4.4 The optimal borrower’s expected utility

By replacing ${R}_{1|\text{L}.}^{*}$ and ${R}_{2|.}^{*}$ in the objective function, it is obtained:

$$\begin{aligned}{E\Pi }_{B}^{*}&={p}^{3}3H+3{p}^{2}\left(1-p\right)\left(2H+L\right)+3p{\left(1-p\right)}^{2}\left(H+2L\right)-\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]3L\\&-\frac{{D}_{3} \left\{\left[1 +{0,5}{m}_{0,L}^{*}\left(1+{m}_{0,LL}^{*}\right) \right]{0,5}\left(H-L\right)+3L+\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(H-L\right)\left(1+{m}_{0,L}^{*}\right)\right\}}{{D}_{4}}\end{aligned}$$

that may rewritten as:

$$\begin{aligned}{E\Pi }_{B}^{*}&={p}^{3}3H+3{p}^{2}\left(1-p\right)\left(2H+L\right)+3p{\left(1-p\right)}^{2}\left(H+2L\right)-\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]3L\\&-\frac{{D}_{3}\left\{\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(H-L\right)\left(1+{m}_{0,L}^{*}\right)-3p{\left(1-p\right)}^{2}\left\{\left[1 +{0,5}{m}_{0,L}^{*}\left(1+{m}_{0,LL}^{*}\right) \right]{0,5}\left(H-L\right)+3L\right\}\right\}}{{D}_{4}}\end{aligned}$$

Since, as proved in the Sect. 4.1 of the Appendix, with ${m}_{0,L}^{*}={m}_{0,LL}^{*}=1$ and ${PF}_{1}$ binding holds:

$${m}_{0}^{*}=\frac{{D}_{3}}{{D}_{4}}=1$$

it derives:

$$\begin{aligned}{E\Pi }_{B}^{*}&={p}^{3}3H+3{p}^{2}\left(1-p\right)\left(2H+L\right)+3p{\left(1-p\right)}^{2}\left(H+2L\right)-\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\\ & \times 2\left(H-L\right)-3p{\left(1-p\right)}^{2}[\left(H-L\right)+3L]-\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]3L={p}^{3}(H-L)\end{aligned}$$

Moreover, since, as proved in the Sect. 4.1 of the Appendix, ${m}_{0,L}^{*}={m}_{0,LL}^{*}=0$ implies:

$${m}_{0}^{*}=\frac{{D}_{3}}{{D}_{4}}<1$$

then ${E\Pi }_{B}^{*}>{p}^{3}(H-L)$.

Proof of Proposition 5

5.1 Optimal value of $\widetilde{{{\varvec{R}}}_{1|{\varvec{L}}{\varvec{L}}.}}\equiv {{\varvec{R}}}_{1|{\varvec{L}}{\varvec{L}}.}$

The hypothesis that ${PF}_{2}$ holds, namely that the contract provides for the return pooling from three successes, from two successes and one fail and from one success and two fails, implies ${m}_{2}={m}_{1}={m}_{1,L}=0$; $\widetilde{{m}_{0}}=\widetilde{{m}_{0,L}}=\widetilde{{m}_{0,LL}}\le 1$ and ${R}_{3}=$ ${R}_{2|.}=$ ${R}_{2|\text{L}}=$ ${R}_{1|.}=$ ${R}_{1|\text{L}.}={R}_{1|\text{LL}.}\le H+2L$.

By replacing these values into the objective function and into the constraints (19), (20), (21), (22), (23), (24) and (25), the reduced form of the optimization problem under the hypothesis that Condition 3 holds is derived:

$$\text{max} {p}^{3}\left(3H-{R}_{1|\text{LL}.}\right)+3{p}^{2}\left(1-p\right)\left[(2H+L)-{R}_{1|\text{LL}.}\right]+3p{\left(1-p\right)}^{2}[(H+2L)-{R}_{1|\text{LL}.}]$$

subject to:

$$\begin{aligned}{p}^{3}{R}_{1|\text{LL}.}&+3{p}^{2}\left(1-p\right){R}_{1|\text{LL}.}+3p{\left(1-p\right)}^{2}{R}_{1|\text{LL}.}+{\left(1-p\right)}^{3}\\&\times\left\{3L-{m}_{0}{m}_{0,L}\left(1+{m}_{0,LL}\right)c-{m}_{0}c\right\}=3I\end{aligned}$$

(40)

$${R}_{1|\text{LL}.}\le \left(1-{m}_{0}\right)3L+{m}_{0}3H$$

(41)

$${R}_{1|\text{LL}.}= \left(1-{m}_{0}\right)3L+{m}_{0}\left\{{0,5}(2H+L)+{0,5}\left[{m}_{0,L}(2H+L)+\left(1-{m}_{0,L}\right)3L\right]\right\}$$

(42)

$$\begin{aligned} {R}_{1|\text{LL}.}&=\left(1-{m}_{0}\right)3L+{m}_{0}\left\{{0,5}(H+2L)+{0,5}\left[{m}_{0,L}\left[{0,5}\left((1-{m}_{0,LL})3L\right.\right.\right.\right.\\ &\quad \left.\left.\left.\left. +{m}_{0,LL}\left(H+2L\right)\right)+{0,5}\left(H+2L\right)\right]+\left(1-{m}_{0,L}\right)3L\right]\right\}\end{aligned}$$

(43)

From the (40) and from the (43) are deduced, respectively:

$$\widetilde{{R}_{1|\text{LL}}}=\frac{3\left[I-L{\left(1-p\right)}^{3}\right]+{\left(1-p\right)}^{3}\left\{\widetilde{{m}_{0}}\left[\widetilde{{m}_{0,L}}\left(1+\widetilde{{m}_{0,LL}}\right)+1\right]c\right\}}{{p}^{3}+3p(1-p)}$$

(44)

and:

$$\widetilde{{m}_{0}}=\frac{2(\widetilde{{R}_{1|\text{LL}}}-3L)}{\left[1 +{0,5} \widetilde{{m}_{0,L}}\left(1+\widetilde{{m}_{0,LL}}\right)\right](H-L)}\le 1$$

(45)

By replacing the (45) into the (44) it is obtained:

$$\widetilde{{R}_{1|\text{LL}}}=\frac{3\left[I-L{\left(1-p\right)}^{3}\right]}{{p}^{3}+3p(1-p)}+\frac{3\left(I-L\right){\left(1-p\right)}^{3}\left[\widetilde{{m}_{0,L}}\left(1+\widetilde{{m}_{0,LL}}\right)+1\right]c}{{D}_{5}}$$

(46)

with:

$$\begin{aligned}{D}_{5}&=\left[{p}^{3}+3p(1-p)\right]{0,5}\left(H-L\right)+\widetilde{{m}_{0,L}}\left(1+\widetilde{{m}_{0,LL}}\right)\\ &\quad\times\left\{\left[{p}^{3}+3p(1-p)\right]{0,25}\left(H-L\right)-{\left(1-p\right)}^{3}c\right\}-{\left(1-p\right)}^{3}c\end{aligned}$$

By setting $\widetilde{{m}_{0,L}}=\widetilde{{m}_{0,LL}}=1$ and assuming that ${PF}_{2}$ is binding, the previous expression becomes:

$$\widetilde{{R}_{1|\text{LL}}}=\frac{3\left[I-L{\left(1-p\right)}^{3}\right]+{\left(1-p\right)}^{3}c}{{p}^{3}+3p(1-p)}$$

(47)

and may be easily proved that this expression is equal to $H+2L$. In fact, the binding ${PF}_{2}$:

$${p}^{3}\left(H+2L\right)+3{p}^{2}\left(1-p\right)\left(H+2L\right)+3p{\left(1-p\right)}^{2}\left(H+2L\right)+{\left(1-p\right)}^{3}3L=3\left[I+{\left(1-p\right)}^{3}c\right]$$

may be rewritten as:

$$H+2L=\frac{3\left[I+{\left(1-p\right)}^{3}c\right]-{\left(1-p\right)}^{3}3L}{\left[{p}^{3}+3p(1-p)\right]}$$

from which:

$$\left[{p}^{3}+3p(1-p)\right](H-L)+3L=3\left[I+{\left(1-p\right)}^{3}c\right]$$

(48)

Finally, by replacing the (48) in the (47), it is deduced $\widetilde{{R}_{1|\text{LL}}}=H+2L$.

5.2 Optimal value of $\widetilde{{{\varvec{m}}}_{0}}\equiv$ ${{\varvec{m}}}_{0}$

By replacing the (45) in the (46) it is obtained:

$$\widetilde{{m}_{0}}=\frac{3\left(I-L\right)}{{D}_{5}}$$

(49)

The lowest value of $\widetilde{{m}_{0}}$ is obtained by setting $\widetilde{{m}_{0,L}}=\widetilde{{m}_{0,LL}}=1$ (namely setting $\widetilde{{m}_{0,L}}\left(1+\widetilde{{m}_{0,LL}}\right)=2$) in the (49):

$$\frac{3\left(I-L\right)}{\left[{p}^{3}+3p(1-p)\right]\left(H-L\right)-3{\left(1-p\right)}^{3}c}$$

(50)

where $\left[{p}^{3}+3p(1-p)\right]\left(H-L\right)$ is the top state and intermediate state expected return net of the high state and intermediate state expected loss:

$$\left[{p}^{3}+3p(1-p)\right]\left(H-L\right)=\left[{p}^{3}+3p\left(1-p\right)\right]\left(H+2L\right)-\left[{p}^{3}+3p\left(1-p\right)\right]3L$$

The denominator of the (50) is higher than 0. In fact, the ${PF}_{2}$:

$$\left[{p}^{3}+3p(1-p)\right](H+2L)\ge 3\left[I+{\left(1-p\right)}^{3}c\right]-{\left(1-p\right)}^{3}3L$$

may be rewritten as:

$$\left[{p}^{3}+3p\left(1-p\right)\right]\left(H-L\right)+3L\ge 3I+3{\left(1-p\right)}^{3}c$$

(51)

This expression represents a necessary condition to finance three investment projects with the pooling between the top and the intermediate states: the expected return net of the expected loss in the top and intermediate states and the return in the low state have to be at least equal to the investment cost and the expected audit cost.

Since in the (51) $3\left(I-L\right)>0$, then $\left[{p}^{3}+3p(1-p)\right]\left(H-L\right)-3{\left(1-p\right)}^{3}c+3L>0$.

When $\widetilde{{m}_{0,L}}=\widetilde{{m}_{0,LL}}=1$ and ${PF}_{2}$ is binding, the (50) is equal to 1.

The highest value of $\widetilde{{m}_{0}}$, instead, is computed setting $\widetilde{{m}_{0,L}}=\widetilde{{m}_{0,LL}}=0$ in the (50) (namely by setting $\widetilde{{m}_{0,L}}\left(1+\widetilde{{m}_{0,LL}}\right)=0$):

$$\frac{3\left(I-L\right)}{\left[{p}^{3}+3p(1-p)\right]{0,5}\left(H-L\right)-{\left(1-p\right)}^{3}c}$$

(52)

It is possible to prove that the denominator of this expression is positive. From the (51), in fact, it may be deduced:

$$\left[{p}^{3}+3p\left(1-p\right)\right]{0,5}\left(H-L\right)-{\left(1-p\right)}^{3}c\ge {0,5}[3\left(I-L\right)+{\left(1-p\right)}^{3}c]$$

where ${0,5}[3\left(I-L\right)+{\left(1-p\right)}^{3}c]$ is higher than zero because $I>L$.

It may also be proved that the (52) must necessarily be less than 1 because, otherwise, the condition (51) would be violated.

To this end, consider the difference between the numerator and the denominator of the (52):

$$3\left(I-L\right)-\left[{p}^{3}+3p(1-p)\right]{0,5}\left(H-L\right)-{\left(1-p\right)}^{3}c$$

(53)

The maximum value of $3\left(I-L\right)$ may be calculated assuming that the (51) is binding:

$$3\left(I-L\right)=\left[{p}^{3}+3p\left(1-p\right)\right]\left(H-L\right)-3{\left(1-p\right)}^{3}c$$

The (53), then, becomes:

$$\left[{p}^{3}+3p\left(1-p\right)\right]\left(H-L\right)-4{\left(1-p\right)}^{3}c$$

and, since according to the (51):

$$\left[{p}^{3}+3p\left(1-p\right)\right]\left(H-L\right)-3{\left(1-p\right)}^{3}c=3\left(I-L\right)$$

results:

$$\left[{p}^{3}+3p\left(1-p\right)\right]\left(H-L\right)-4{\left(1-p\right)}^{3}c<3(I-L)$$

then:

$$\left[{p}^{3}+3p\left(1-p\right)\right]\left(H-L\right)+3L<3I+4{\left(1-p\right)}^{3}c$$

The investment project, therefore, is unviable because the expected return net of the expected loss in the top, intermediate low states are less than the investment cost and the expected audit cost.

This result has been obtained by assuming binding ${PF}_{2}$. When ${PF}_{2}$ is slack, the minuend of (52), that is $3\left(I-L\right)$, is smaller than that of the case in which ${PF}_{2}$ is binding and so this result remains valid.

5.3 Optimal borrower’s expected audit cost ${{\widetilde{{\varvec{E}}{\varvec{\Pi}}}}_{{\varvec{B}}}\equiv {\varvec{E}}{\varvec{\Pi}}}_{{\varvec{B}}}$

By replacing the (46) in the objective function, it is obtained the borrower’s utility with ${PF}_{2}$:

$${\widetilde{E\Pi }}_{B}=\frac{3\left[I-L{\left(1-p\right)}^{3}\right]}{{p}^{3}+3p(1-p)}+\frac{3\left(I-L\right){\left(1-p\right)}^{3}\left[{p}^{3}+3p(1-p)\right]\left[\widetilde{{m}_{0,L}}\left(1+\widetilde{{m}_{0,LL}}\right)+1\right]c}{{D}_{5}}$$

(54)

while by etting $\widetilde{{m}_{0,L}}=\widetilde{{m}_{0,LL}}=1$ in the previous it is deduced:

$$\begin{aligned}{\widetilde{E\Pi }}_{B}&={p}^{3}3H+3{p}^{2}\left(1-p\right)\left(2H+L\right)+3p{\left(1-p\right)}^{2}\left(H+2L\right)-3\left[I-L{\left(1-p\right)}^{3}\right]\\&\quad-\frac{9\left(I-L\right){\left(1-p\right)}^{3}c}{\left[{p}^{3}+3p(1-p)\right]\left(H-L\right)-3{\left(1-p\right)}^{3}c}\end{aligned}$$

and assuming that ${PF}_{2}$ is binding, it is obtained:

$$\begin{aligned}{\widetilde{E\Pi }}_{B}&={p}^{3}3H+3{p}^{2}\left(1-p\right)\left(2H+L\right)+3p{\left(1-p\right)}^{2}\left(H+2L\right) \\&\quad-3\left[I-L{\left(1-p\right)}^{3}+{\left(1-p\right)}^{3}c\right]\end{aligned}$$

Proof of Proposition 6

6.1 $3 {{\varvec{m}}}_{0}^{{\varvec{s}}{\varvec{i}}{\varvec{n}}}\left(1-{\varvec{p}}\right){\varvec{c}}<{{\varvec{m}}}_{0}^{*}\left[3{\left(1-{\varvec{p}}\right)}^{3}+6{\varvec{p}}{\left(1-{\varvec{p}}\right)}^{2}\right]{\varvec{c}}$

Consider the difference between the expected audit cost with separate financing and the expected audit cost of the joint financing with ${PF}_{1}$:

$$3 {m}_{0}^{sin}\left(1-p\right)c-{m}_{0}^{*}\left[3{\left(1-p\right)}^{3}c+6p{\left(1-p\right)}^{2}\right]c$$

The maximum value of ${m}_{0}^{*}\left[3{\left(1-p\right)}^{3}+6p{\left(1-p\right)}^{2}\right]c$ is $3{\left(1-p\right)}^{3}c+6p{\left(1-p\right)}^{2}c$ and it is obtained by setting ${m}_{0,L}^{*}={m}_{0,LL}^{*}=1$ and assuming that ${PF}_{1}$ is binding.

The previous expression, under these two assumptions, becomes:

$$3 {m}_{0}^{sin}\left(1-p\right)c-\left[3{\left(1-p\right)}^{3}c+6p{\left(1-p\right)}^{2}\right]c$$

that is equal to:

$$\frac{3(I-L)\left(1-p\right)c}{p\left(H-L\right)-\left(1-p\right)c}-\left[3{\left(1-p\right)}^{3}c+6p{\left(1-p\right)}^{2}\right]c$$

from which:

$$\frac{3\left(I-L\right)\left(1-p\right)c-\left[p\left(H-L\right)-\left(1-p\right)c\right]\left[3{\left(1-p\right)}^{3}c+6p{\left(1-p\right)}^{2}\right]c}{p\left(H-L\right)-\left(1-p\right)c}$$

Since:

$$p\left(H-L\right)-\left(1-p\right)c>3(I-L)\left(1-p\right)c$$

then holds:

$$3\left(I-L\right)\left(1-p\right)c<\left[p\left(H-L\right)-\left(1-p\right)c\right]\left[3{\left(1-p\right)}^{3}c+6p{\left(1-p\right)}^{2}\right]c$$

from which:

$$\frac{3\left(I-L\right)\left(1-p\right)c-\left[p\left(H-L\right)-\left(1-p\right)c\right]\left[3{\left(1-p\right)}^{3}c+6p{\left(1-p\right)}^{2}\right]c}{p\left(H-L\right)-\left(1-p\right)c}<0$$

Since $3 {m}_{0}^{sin}\left(1-p\right)c<\left[3{\left(1-p\right)}^{3}c+6p{\left(1-p\right)}^{2}\right]c$ and ${m}_{0}^{*}\in [{0,1}]$, then $3 {m}_{0}^{sin}\left(1-p\right)c<{m}_{0}^{*}\left[3{\left(1-p\right)}^{3}c+6p{\left(1-p\right)}^{2}\right]c$.

6.2 Graphical representation of the credit rationing regions with ${{\varvec{P}}{\varvec{F}}}_{1}$

Consider $3 I{F}^{\mho }$:

$$3 I{F}^{\mho }=3pH+ 3\left(1-p\right)L-3\left(1-p\right)c$$

and ${PF}_{1}^{{\mho }}$:

$$\begin{aligned}{PF}_{1}^{\mho }&=\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(2H+L\right)+3p{\left(1-p\right)}^{2}\left(H+2L\right)\\ &-6p{\left(1-p\right)}^{2}c+{\left(1-p\right)}^{3}3L-3{\left(1-p\right)}^{3}c\end{aligned}$$

Note that:

$$3pH+ 3\left(1-p\right)L>\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(2H+L\right)+3p{\left(1-p\right)}^{2}\left(H+2L\right)+{\left(1-p\right)}^{3}3L$$

namely, the vertical intercept of $3 I{F}^{\mho }$ is higher than that one of ${PF}_{1}^{\mho }$. Also note that, for every $p\in [{0,1}]$ and $c>0$, the inequality:

$$-6p{\left(1-p\right)}^{2}c-3{\left(1-p\right)}^{3}c>-3\left(1-p\right)c$$

ever holds, i.e., the slope of $3 I{F}^{\mho }$ is lower than that of ${PF}_{1}^{\mho }$. In fact, by dividing the right-hand side and the left-hand side for $3\left(1-p\right)c$, the following equivalent inequality is obtained:

$$-2p\left(1-p\right)-{\left(1-p\right)}^{2}>-1$$

and the difference between the right-hand side and the left-hand side of the previous expression gives:

$${p}^{2}>0$$

Finally, from the resolution of the system including ${PF}_{1}^{\mho }$ and $3 I{F}^{\mho }$ may be deduced that the two corresponding lines have an intersection point given by:

$${c}^{*}=\frac{\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(2H+L\right)+3p{\left(1-p\right)}^{2}\left(H+2L\right)+{\left(1-p\right)}^{3}3L-3pH- 3\left(1-p\right)L}{-3\left[1-2p\left(1-p\right)-{\left(1-p\right)}^{2}\right](1-p)}$$

where ${c}^{*}>0$, because $\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(2H+L\right)+3p{\left(1-p\right)}^{2}\left(H+2L\right)+{\left(1-p\right)}^{3}3L-3pH- 3\left(1-p\right)L<0$ and $-3\left[1-2p\left(1-p\right)-{\left(1-p\right)}^{2}\right](1-p)<0$ for every $p\in [{0,1}]$.

Proof of Proposition 7

7.1 $3 {{\varvec{m}}}_{0}^{{\varvec{s}}{\varvec{i}}{\varvec{n}}}\left(1-{\varvec{p}}\right){\varvec{c}}>\widetilde{{{\varvec{m}}}_{0}}{\left(1-{\varvec{p}}\right)}^{3}{\varvec{c}}$

To prove $3 {m}_{0}^{sin}\left(1-p\right)c>\widetilde{{m}_{0}}{\left(1-p\right)}^{3}c$, is sufficient to demonstrate that the difference between the minimum value of $\widetilde{{m}_{0}}{\left(1-p\right)}^{3}c$ (namely that obtained with $\widetilde{{m}_{0,L}}\left(1+\widetilde{{m}_{0,LL}}\right)=2$) and $3 {m}_{0}^{sin}\left(1-p\right)c$ is less than zero. This difference is equal to:

$$\begin{aligned}{}&\widetilde{{m}_{0}}{\left(1-p\right)}^{3}c-3 {m}_{0}^{sin}\left(1-p\right)c=\frac{9\left(I-L\right){\left(1-p\right)}^{3}c}{\left[{p}^{3}+3p(1-p)\right]\left(H-L\right)-{\left(1-p\right)}^{3}c}\\&\quad-\frac{3\left(I-L\right)\left(1-p\right)c}{p\left(H-L\right)-\left(1-p\right)c}\end{aligned}$$

and may be rewritten as:

$$\widetilde{{m}_{0}}{\left(1-p\right)}^{3}c-3 {m}_{0}^{sin}\left(1-p\right)c=\frac{3\left(I-L\right)(1-p)c}{\frac{\left\{\left[{p}^{3}+3p(1-p)\right]\left(H-L\right)-{\left(1-p\right)}^{3}c\right\}}{3{\left(1-p\right)}^{2}}}-\frac{3\left(I-L\right)\left(1-p\right)c}{p\left(H-L\right)-\left(1-p\right)c}$$

Given the same numerator, for every $p\in \left[{0,1}\right]$, it is obtained:

$$\frac{\left\{\left[{p}^{3}+3p(1-p)\right]\left(H-L\right)-{\left(1-p\right)}^{3}c\right\}}{3{\left(1-p\right)}^{2}}-\left(1-p\right)c>0$$

Since the denominator of $\widetilde{{m}_{0}}{\left(1-p\right)}^{3}c$ is higher than the denominator of $3 {m}_{0}^{sin}\left(1-p\right)c$ and the numerator is the same, then $3 {m}_{0}^{sin}\left(1-p\right)c>\widetilde{{m}_{0}}{\left(1-p\right)}^{3}c$.

7.2 Graphical representation of the credit rationing regions with ${{\varvec{P}}{\varvec{F}}}_{2}$

Consider again $3 I{F}^{\mho }$ and ${PF}_{2}^{\mho }$ and note that:

$$3pH+ 3\left(1-p\right)L>\left[{p}^{3}+3{p}^{2}\left(1-p\right)+3p{\left(1-p\right)}^{2}\right]\left(H+2L\right)+{\left(1-p\right)}^{3}3L$$

namely the vertical intercept of $3 I{F}^{\mho }$ is higher than that one of ${PF}_{2}^{\mho }$.

Note also that, for every $p\in [{0,1}]$ and $c>0$, the inequality:

$$ - 3\left( {1 - \rho } \right)c < - 3\left( {1 - \rho } \right)^{3} c $$

ever holds, i.e., the slope of $3 I{F}^{\mho }$ is lower than that one of ${PF}_{2}^{\mho }$.

By solving the system formed by ${PF}_{2}^{\mho }$ and $3 I{F}^{\mho }$ is deduced that the two corresponding lines have an intersection point given by:

$${c}^{**}=\frac{\left[{p}^{3}+3{p}^{2}\left(1-p\right)+3p{\left(1-p\right)}^{2}\right]\left(H+2L\right)+{\left(1-p\right)}^{3}3L-3pH- 3\left(1-p\right)L}{-3\left[1-{\left(1-p\right)}^{2}\right]\left(1-p\right)}$$

where ${c}^{**}>0$, because $\left[{p}^{3}+3{p}^{2}\left(1-p\right)+3p{\left(1-p\right)}^{2}\right]\left(H+2L\right)+{\left(1-p\right)}^{3}3L-3pH- 3\left(1-p\right)L<0$ and $-3\left[1-{\left(1-p\right)}^{2}\right]\left(1-p\right)<0$.

Finally, note that the intercept of ${PF}_{1}^{\mho }$ is higher than that of ${PF}_{2}^{\mho }$:

$${PF}_{1}^{0}-{PF}_{2}^{0}=\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(2H+L\right)-\left[{p}^{3}+3{p}^{2}\left(1-p\right)\right]\left(H+2L\right)>0$$

and that the slope of ${PF}_{2}^{\mho }$ is higher than that of ${PF}_{1}^{\mho }$:

$$ - 3\left( {1 - p} \right)^{3} c - 6p\left( {1 - p} \right)^{2} c < - 3\left( {1 - p} \right)^{3} c $$