Traditional credit model
The traditional MFI credit model is described in Fig. 1. MFIs receive funds from either or all available sources: private equity, financial institutions, and personal donors (Ledgerwood 1998). According to their policy, organizational structure, and partner organizations, the MFIs then distribute and channelize the fund to the poor people (Ledgerwood 1998). Based on the character of institutional structure (i.e., formal, semiformal, and informal), MFIs' objectives and operations may differ (Ledgerwood 1998; McGuire and Conroy 2000). Recently, MFIs have been cooperating with other development agencies (i.e., government, NGOs, donors) to enhance operating efficiency (Adegbite et al. 2013; Austin 2004; McGuire and Conroy 2000). Thus, the partner organizations' characteristics may also impact MFIs' objectives and operations.
Among others, the traditional credit model was described by Stiglitz and Weiss (1981), Bester (1985), and Wang et al. (2018). Based on the theory of collateral and limited liability along with the presence of information asymmetry, Stiglitz and Weiss (1981) examined why banks cannot increase the collateral requirement (or decrease borrowers' debttoequity ratio) besides knowing that there is a relatively higher demand of loans out in the market? Here, as per the credit channel theory, asymmetric information in the credit market propagates the effect of the interest rate channel. Their findings revealed that the observationally identical borrowers may or may not receive loans from a bank. Even if the rejected borrowers are willing to pay a higher interest rate for their loans or increase their collaterals, this would only upsurge the riskiness of a bank's portfolio by either encouraging them to invest in risky projects or discouraging safe investors. Details of the model are given below.
Following the models of Adegbite et al. (2013), Austin (2004), Ledgerwood (1998), McGuire and Conroy (2000), Stiglitz and Weiss (1981), and Wang et al. (2018), this section will describe the traditional credit model of MFIs. Assume that a hypothetical credit market has asymmetric information. Let us consider that loan takers classify under either \(i = :1:low risk,2:high risk\). The total credit amount is presented as M. The expected return, probability of success of an investment/project, and probability of failure of an investment/project shall be given as \(\pi_{i} \left( r \right) for loan taker and p_{i} \left( r \right) for MFIs, P_{i } and 1  P_{i}\) respectively.
According to Stiglitz and Weiss (1981), \(P_{1} X_{1} = P_{2} X_{2}\) because of mean preserving spread of returns^{Footnote 5}\(\left( {X_{i} } \right)\). The MR indicates total profit/interest earned, where R indicates gross profit rate (we are calling this profit rate since Islamic MFIs cannot charge interest) asked by MFIs. Primarily, R is determined by the market by analyzing opportunity costs. Finally, we assume that a credit institution depositor a deposit rate as denoted with \(\left( s \right)\). Based on the above assumptions, the basic models of expected returns in a traditional financing system can be expressed in Eq. 1 and 2. They explain the expected return expected of a loan taker and an MFIs, respectively.
$${\text{The}}\;{\text{expected}}\;{\text{return}}\;{\text{of}}\;{\text{loan}}\;{\text{taker}}\;\pi_{i} \left( r \right) = P_{i} \left( {X_{i} + W  MR_{i} } \right) + \left( {1  P_{i} } \right)\left( {W  C} \right)  W$$
(1)
$${\text{The}}\;{\text{expected}}\;{\text{return}}\;{\text{of}}\;{\text{MFI }}\rho_{i} \left( r \right) = P_{i} MR_{i} + \left( {1  P_{i} } \right)C  M\left( {1 + s} \right)$$
(2)
We now assume that initial investment \(\left( W \right)\) is 0, collateral (\(C\)) is 0, and probability of success of a highrisk project \(\left( {P_{2} } \right)\) is higher than a lowrisk project (\(P_{1} ):\) \(P_{2} > P_{1}\). This is to be mentioned here that, unlike the traditional MFIs operation, this model contributes theoretically by proposing that a loan taker can receive MFIs loan even without any collateral and initial investment. Theoretically, this assumption may increase the default risk of that loan taker, yet our model illustrates a better probability of success for a highrisk project. After adjusting the assumptions of Islamic MFIs into the traditional credit market model, the modified equations would be as follows:
$$\pi_{i} \left( r \right) = P_{i} \left( {X_{i}  MR} \right)$$
(3)
$$\rho_{i} \left( r \right) = P_{i} MR  M\left( {1 + s} \right)$$
(4)
The modifications are based on the following assumptions:

a)
Initial wealth W is 0

b)
Collateral C is 0

c)
If a project fails, the return is negative but not more than the loan amount M.
Thus, the estimated loss is ≤ M. Here, Eqs. 3 and 4 completely ignores the probability of failure of any project based on the assumptions of W = 0 and C = 0. Theoretically, with higher collateral, the rate of interest/profit (R) is supposed to get lower (Berger et al. 2011). In the case of Islamic MFIs, this paper proposes a new scheme based on blockchain to indicate the rate of R.
Blockchain embedded model
The underlying condition for getting approval of credit from a blockchainembedded system requires receiving a decentralized consensus by providing a minimum degree of information for verification approval of credit (Cong and He 2018). Therefore, a modified model for a blockchain embedded credit system is provided in this subsection. For illustration, a lowrisk investment from MFIs will be examined using a blockchainembedded model, and a highrisk investment is presented with a traditional model. Finally, Eq. 5 describes the expected return of a lowrisk loan taker through a blockchain model.
$$\begin{aligned} \pi_{1} \left( r \right) & = P_{1} \left( {X_{1} + W  MR_{1} } \right) + \left( {1  P_{1} } \right)\left( {W  D} \right)  W  f \\ & = P_{1} X_{1}  P_{1} MR_{1}  \left( {1  P_{1} } \right)D  f \\ \end{aligned}$$
(5)
Here, \(f\) is the usage fee of blockchain technology for the loan taker, and D is the default loss of a loan receiver. As shown in Eq. 5, M is the compensation the lender would make in a traditional model. However, in the case of Islamic MFIs, W is 0; hence, the modified model would be:
$$\begin{aligned} \pi_{1} \left( r \right) & = P_{1} \left( {X_{1}  MR_{1} } \right) + \left( {1  P_{1} } \right)\left( D \right)  f \\ & = P_{1} X_{1}  \left( {1  P_{1} } \right)D  P_{1} MR_{1}  f \\ & = P_{1} X_{1}  \left( {1  P_{1} } \right)D  (1  P_{1} )\left( {1  a} \right)M  M\left( {1 + s} \right)  f \\ \end{aligned}$$
(6)
Here, the equation introduces a category in a total investment of credit by all MFIs with (\(a)\) and \(\left( {1  a} \right)\) is the portion of MFIs credit distributed by the government as we assumed that in any country, collaborative work between MFIs and the government could only work appropriately (Sulkowski 2018; Treleaven et al. 2017). Furthermore, without involving government in the blockchain embedded MFIs operation, the legal framework would not work prospectively. Thus, the expected return of MFIs from investments into a lowrisk investment would be:
$$\rho_{1} \left( r \right) = P_{1} MR + \left( {1  P_{1} } \right)aM  M\left( {1 + s} \right)  g$$
(7)
Here, g is the cost involved by MFIs for using blockchain technology (Janssen et al. 2020). Figure 2 depicts a blockchainembedded credit system for Islamic MFIs. Figure 2 presents that during a transactions in the blockchainembedded credit system, all the stakeholders: donors; financial institutions; Islamic MFIs; credit receiver and private equity providers, will be notified and through distributive ledger system of blockchain, the Islamic MFIs require no additional document to be produced and submitted to the creditors, loan receivers, government agencies and whatsoever. Nevertheless, distributive ledger automatically updates all stakeholders on the updates of any loan repayments and failures.
For the blockchain service provider, the return can be calculated using \(\delta_{i} \left( r \right)\):
$$\delta_{i} \left( r \right) = f + g  \left( {1  P_{i} } \right)M$$
(8)
Now, if the expected return from a highrisk firm is intended with a lowrisk firm (assuming that \(R_{1}\) will equal here), its expected return would be:
$$\begin{aligned} \pi_{2} \left( r \right) & = X_{0}  \left( {1  P_{2} } \right)D  P_{2} MR_{1}  f \\ & = P_{1} X_{1}  \left( {1  P_{2} } \right)D  \frac{{P_{2} }}{{P_{1} }}\left( {M\left( {1 + s} \right)  \left( {1  P_{1} } \right)\left( {1  a} \right)M  f} \right)  f \\ \end{aligned}$$
(9)
Blockchain in Islamic MFI financing—does the adoption improve social welfare?
Subject to the ability of this blockchainembedded model to distinguish between a lowrisk firm and a highrisk firm, the return from a lowrisk firm would be positive (Eq. 10). However, if a highrisk firm that takes a loan from the traditional system can be intended with its counterpart (lowrisk firm \(R_{1}\)), the return of a highrisk firm would be lower than the benefit derived from the traditional manner (Eq. 11).
$$P_{1} X_{1}  \left( {1  P_{1} } \right)D  (1  P_{1} )\left( {1  a  b} \right)M  M\left( {1 + s} \right) > 0$$
(10)
$$P_{1} X_{1}  \left( {1  P_{2} } \right)D  \frac{{P_{2} }}{{P_{1} }}\left( {M\left( {1 + s} \right)  \left( {1  P_{1} } \right)\left( {1  a  b} \right)M  f} \right)  f < P_{1} X_{1}  M$$
(11)
Now, subject to the satisfaction parameter of default loss D:
$$\frac{{\left( {P_{2} \left( {1  P_{1} } \right)\left( {1  a  b} \right) + P_{1}  P_{2} } \right)M + \left( {P_{2}  P_{1} } \right)f  P_{2} s}}{{P_{1} \left( {1  P_{1} } \right)}} < D < \frac{{P_{1} X_{1}  (1  P_{1} )\left( {1  a  b} \right)M  M\left( {1 + s} \right)}}{{\left( {1  P_{1} } \right)}}$$
(12)
Thus, the expected outcome of this proposed blockchain embedded Islamic MFI model would explain a reasonable default cost, D, within which all lowrisk firms would prefer to switch their business to a blockchainembedded credit market. In contrast, highrisk firms will continue in the traditional market. Thus, the credit rationing problem (highrisk vs. lowrisk firms) in any credit market (Bester 1985) has been mitigated through a blockchain model.
Finally, since the highrisk firms will continue borrowing from the local market, the total social wealth will maximize with an assumption that lowrisk firms will adopt blockchain technology:
$$\Delta W = P_{1} X_{1}  \left( {1  P_{1} } \right)D  M\left( {1 + s} \right)$$
(13)
$$D < \frac{{P_{1} X_{1}  M\left( {1 + s} \right) }}{{\left( {1  P_{1} } \right)}}\quad ({\text{when}}\;\Delta W > 0)$$
(14)