1. Introduction
When financial system is characterized by financial dualism that threatens financial liberalization, the control and efficiency of the monetary transmission channels must be perfectly apprehended and then integrated into the price stability policy frame. This question arises more with the development of the MFIs sector which carries out both micro financial intermediation and transformation. Theoretically and traditionally, four transmission channels should be distinguished: the interest rate channel, the asset price channel, the credit channel and the anticipation channel. And the effectiveness of each channel depend intrinsically on the economic structures, the dynamics of the financial architecture, the institutional constraints and the economic choices of agents in terms of consumption, savings and investment.
The results of Babatoundé (2015) on the monetary implications of financial dualism are quite conclusive on the need to re-examine the transmission channels of monetary shocks when financial dualism is persistent. In particular, compatible with the effectiveness of monetary policy, the persistence of financial dualism and the development of financial micro intermediation its corollary, allow strengthening the monetary creation power of the bank when the complementarity between the two institutions is implemented. Given the preference of agents for decentralized financial services, the demand for deposits from non-financial agents (NFA) in the MFIs sector stimulates that for macro-deposits with the banking system or the supply of decentralized credit. These two dynamics in the MFIs sector inaugurate a new mechanism which amplifies indirectly the monetary shocks. In this process, the credit interest rate differential is critical. Moreover, if the dynamics of decentralized deposits threaten the success of financial liberalization policies, it has the advantage of strengthening the effectiveness of monetary policy via the monetary multiplier and the lending interest rate.
As this analysis focused on the behavior of non-financial agents, the role of financial agents requires an in-depth study of the determinants of their behaviors which would explain this amplification mechanism of monetary shocks. To this end, in addition to the assumption of constitution of macro-deposits of the MFI with the bank, the complementarity between the two sub-sectors will be enhanced by the micro-refinancing of MFIs by the commercial bank. Beyond the traditional channels of monetary transmission, this paper aims to characterize the bank and MFI credit markets to deduce the implications on the effectiveness of monetary policy in an economy with segmented financial system.
With reference to Ary Tanimoune (2007), the two credit markets are complementary, justified by the purpose of the financing. Indeed, beyond rationing, decentralized credits are not likely to meet large-scale investment demands given the constraints of size, capacity and expertise; those demands are covered by banks. Also, given the different financial and transactional costs, non-financial agents do not apply for bank loans to ensure the financing of projects requiring small amounts; those demands are addressed to MFIs. Adera (1995) justifies the complementarity between “formal” and “informal” markets by the fact that an expansion of the banking sector is accompanied by an extension of the decentralized sector. However, the nature of the relationship between the two types of credits is not so clear-cut. For Montiel (1991), bank credit and decentralized credit are perfect substitutes for agents’ portfolio choice; this makes it possible to take particular account of the “spillover” effects of bank credit rationing. Through a study applied to Ghanaian regions, Awunyo and Abankwah (2012) show that the two types of credit are not necessarily perfect substitutes, but that one complements the other in financing producers.
That is why, within the framework of this study, the hypothesis of complementarity of the banking and decentralized markets (Adera 1995; Ary Tanimoune 2007), is retained, but the two types of credit are assumed to be imperfect substitutes (Montiel 1991; Awunyo and Abankwah 2012).[1] The rest of the paper is organized as follows. After this introduction, the second section states the basic assumptions and the analytical framework; sections 3 and 4 examine bank and decentralized credit markets by characterizing the supply and demand functions and then the resulting market equilibrium. Section 5 concludes the paper.
2. Basic assumptions and analytical framework
The economy is considered with three imperfectly substitutable assets (Fama 1980): money, bank deposits and decentralized deposits. Due to the lack of a developed and organized financial market, it is assumed that there are no financial securities (stocks and bonds) in the economy. In addition to money, agents therefore choose to hold their assets either in bank deposits or in decentralized deposits. Firms and households have a need for financing consumption and / or investment: there are two sources of funding for them (bank credit and microcredit). Banks’ resources consist mainly on deposits mobilized from agents (households, firms and MFIs) and refinancing from the central bank. Given a transformation ratio, they are recycled into bank loans including micro-refinancing for MFIs.
Two types of complementary relationships are thus highlighted between MFIs and banks: on the one hand, the micro-refinancing of MFIs by banks and, on the other hand, the macro account of MFI deposits in banks what we call macro-deposits. We assume two types of MFIs: it is either “in the bank” with the benefit of micro-refinancing or “out of the bank” with a macro account of deposits. As a result, at any time, it will be in one of these two scenarios.[2] The MFI may have a need for refunding and it obtains this from the bank as additional resources which can be recycled into credit; this assumption abolishes that of the deposit-based of credit in this sector, linking the bank’s assets and MFIs liabilities through the interest rate and in this case, the MFI is said to be “in bank”. The MFI may finally have an excess of resources because it has mobilized more deposits than supplying credit; in this case, there is an “urban bias” in reference to Wickramanayake (2004), insofar as deposits from MFIs serve as resources for banks;[3] with this macro-deposit mechanism, the reserve requirement ratio links the MFIs assets to the bank’s liabilities and in this case, MFI is said to be “out of the bank”.
Given these different assumptions, the aggregate and simplified balance sheets of the four agents is presented in the table below.
The accounting framework thus described makes it possible to establish the various links between the agents and the main balances that emerge.
Bank mobilizes deposits from NFA, macro-deposits from MFI which is “out of the bank” and receives refinancing from central bank. It uses these resources to provide loans for both NFA and MFIs whose are “in the bank”, but also to constitute required reserves or excess reserves Then the bank balance can be written as,
\[D_{B}^{NFA} + D_{B}^{MFI} + RF = L_{B}^{NFA} + L_{B}^{MFI} + RR + ER\]
The MFI can receive loan from banks in terms of micro-refinancing when needed, and collect deposits from NFA it uses these funds to meet the demand of NFA for decentralized loans and constitutes macro-deposits in bank Therefore the balance gives,
\[D_{M} + L_{B}^{MFI} = L_{M} + D_{B}^{MFI}\]
For NFAs, resources are allocated between money holding bank deposits and MFI deposits they receive bank loans and MFI loans Then their balance can be written as,
\[{B + D}_{M} + D_{B}^{NFA} = L_{M} + L_{B}^{NFA}\]
Assuming no government and autarky in the simplified model, this balance gives components and counterparties of the monetary aggregate in the economy.
Finally, the central bank has as liabilities, bank notes and requirements and/or excess reserves, ensuring in last resort, the function of bank refinancing. The required reserves ratio and the refinancing interest rate are the two main monetary instruments of the central bank. Its balance gives the form,
\[B + RR + ER = RF\]
Given these different assumptions, the aggregate and simplified balance sheets of the four agents is like what discussed in Ary Tanimoune (2007) or with more details in Babatoundé (2015). Such framework allows establishing the different links between financial and non-financial agents in Least Developed Countries (LDCs). The only costs remain the different credit and deposit interest rates on each segment of the financial system; this implies an interest rate differential that we assume to be a proxy for financial dualism measure (Agénor and Alper 2009).
By assumption, the NFAs choose to hold bank deposits or MFI deposits We note the relationship between these quantities; it measures the preference of the agents for MFI deposits (compared to bank deposits). The parameter is therefore such as, or Later, for reasons of simplification and interpretation, one or the other of the two indicators will be used.
\[\hspace{25mm} qD_{B}^{NFA} = D_{M} \hspace{25mm} [1]\]
Given the relationship between the two types of deposits, it is possible to link the parameter and the MFIs market share in the deposit side. Indeed,
\[D_{M} + D_{B}^{NFA} = (1 + q)D_{B}^{NFA}\]
Reversing the identity and multiplying by it becomes,
\[D_{M}/(1 + q)D_{B}^{NFA} = D_{M}/(D_{M} + D_{B}^{NFA})\]
The second member represents the share of MFIs in the deposits market that we note such that from (1),
\[\hspace{10mm} q/(1 + q) = p(q) \text{ or } p/(1 - p) = q(p) \hspace{10mm} [2]\]
In the following, for simplification and interpretation purposes, one or the other of these two indicators will be used.
3. The bank credit market: equilibrium and its implications
The representative bank chooses the deposits and credits levels that maximize its profit in each period given the constraints on funds and financial intermediation costs. Assuming complementarity between bank and MFI, we introduce two additional variables in traditional framework: the macro-deposits at interest rate and the micro-refinancing at interest rate The profit of the bank can be expressed as,
\[ \begin{align} \pi_{B}\left( . \right) = &i_{B}L_{B}^{NFA} + i_{BL}L_{B}^{MFI} + i_{m}ER - r_{B}D_{B}^{NFA} \\ &- r_{BD}D_{B}^{MFI} - \varsigma RF - C_{B}(.) \end{align} \]
represent respectively the lending interest rate, the money market interest rate and the interest rate that the bank pays on saving. (.) indicates the financial intermediation cost. Regard to this profit function and in accordance with the simplified balance sheet defined in the previous section, the budget constraint of the bank is given by,
\[D_{B}^{NFA} + D_{B}^{MFI} + RF = L_{B}^{NFA} + L_{B}^{MFI} + RR + ER \text{.}\]
Like Freixas and Rochet (2008), to simplify the analysis, it is made the assumption that the refinancing interest rate is equal to that of the interbank market that remunerates excess reserves. In contrast, for the purposes of monetary policy, lending interest rates concerning the supply of credit to NFAs and MFIs are supposed to be distinct, i.e. Also the deposits interest rates applied by both bank and MFI are assumed to be different, There are two main reasons that justify this double hypothesis. Given the enhancing of the money creation power of banks related to the MFI financial intermediation and financial transformation (Babatoundé 2018), a discriminating rates policy is critical for the monetary policies implementation. Furthermore, with such linkage, the bank can take advantage in trading-off given the substitutability between bank deposits and MFI deposits or the complementarity between bank credits and MFI credits.
Referring to the definition of we note the parameter of preference for MFI micro-refinancing; that is Moreover, when the MFI adopts a transformation ratio of deposits to credit, the expression of the macro-deposits that it can constitute is given by By substituting these expressions in the profit function of the bank, it becomes
\[ \begin{align} \hspace{2mm} \pi_{B}\left( . \right) = &{(i}_{B} + i_{BL}k)L_{B}^{NFA} - \left\lbrack r_{B} + r_{BD}(1 - \tau)q \right\rbrack \\ &\times D_{B}^{NFA} - \varsigma Z - C_{B}(.) \hspace{25mm} [3] \end{align} \]
where is the bank’s net position in the interbank market, derived from the budget constraint as,
\[ \begin{align} \hspace{8mm} Z = &RF - ER = (1 + k)L_{B}^{NFA} \\ &- (1 - \varphi^{RO})\left\lbrack 1 + (1 - \tau)q \right\rbrack D_{B}^{NFA} \hspace{8mm} [4] \end{align} \]
If central bank provides refinancing to bank with interest payment; in this case, represents an additional cost. In contrast, when the bank can invest through the interbank market with received interest payment; in this case, represents an additional revenue. Combining equations [3] and [4], we get the profit maximization program of the bank which can be written as,
\[ \begin{align} \hspace{0mm} \underset{(..)}{Max} \; \pi_{B}\left( . \right) = &- C_{B}\left( . \right) + \left\lbrack i_{B} + i_{BL}k - \varsigma(1 + k) \right\rbrack \\ &\times L_{B}^{NFA} + \biggl\lbrack \varsigma\left( 1 - \varphi^{RR} \right)\left\lbrack 1 + (1 - \tau)q \right\rbrack \\ &- \left\lbrack r_{B} + r_{BD}(1 - \tau)q \right\rbrack \biggr\rbrack D_{B}^{NFA} \hspace{15mm} [5] \end{align} \]
As result, the profit is therefore the sum of the intermediation margins on deposits and loans, net of intermediation costs. The specification of the intermediation function is critical for the characterization of the loan supply function and the demand for deposits (Klein 1971). It depends in fact on the structure of the banking system, the financial architecture, the monetary and financial regulation, the behavior of non-financial agents.[4]
More specifically, when both loans and deposits interest rates are not discriminated between NFAs and MFIs, that is to say and the bank’s profit represented by [5] is reduced to,
\[ \begin{align} \hspace{2mm} \underset{(..)}{Max} \; \pi_{B}\left( . \right) = &- C_{B}\left( . \right) + \left( 1 + k \right)(i_{B} - \varsigma)L_{B}^{NFA} \\ &+ \bigl\lbrack \left\lbrack 1 + (1 - \tau)q \right\rbrack\bigl\lbrack \left( 1 - \varphi^{RR} \right) \\ &\times \varsigma - r_{B} \bigr\rbrack \bigr\rbrack D_{B}^{NFA} \hspace{27mm} [6] \end{align} \]
Relation [6] is equivalent to an objective of profit maximization in the absence of MFI sector or when the financial system is unified. The difference here remains that the two intermediation margins are weighted respectively by for loans and for deposits with, and
Result 1
When micro-refinancing and / or macro-deposit mechanisms exist between banks and MFIs, the bank’s financial intermediation margins are higher than those prevailing in a integrated or compartmentalized system.
It is obvious that when the preferences for decentralized loans and decentralized deposits are zero, that is the bank’s profit is exactly the same as in an unified financial system. First, when the financial system is not integrated, an increase in the parameter implies better resources mobilization for banks under the hypothesis of macro-deposits. Second, a rise in the parameter strengthens the credit supply by the banks with significant financial revenues. The combination of these effects provides an explanation for the difference in financial intermediation margins from an integrated to a dualistic financial system.
The first-order conditions of the bank’s maximization program give the functions of both bank credit and bank deposits; since they are defined implicitly, we proceed from the bank constraint defined by [4] which can be written as,
\[ \begin{align} \hspace{5mm} \left( 1 + k \right)L_{B}^{NFA} + ER = &RF + (1 - \varphi^{RR}) \\ &\times \left\lbrack 1 + (1 - \tau)q \right\rbrack D_{B}^{NFA} \hspace{8mm} [7] \end{align} \]
Relation [7] is a representation of the bank’s simplified balance sheet where is the micro-refinancing share in the bank’s portfolio. In the following, like Bernanke and Blinder (1988), we assume that the bank makes up its portfolio for a proportion of loans granted to NFAs. Under this assumption and given the total bank deposits the equation [7] allows expressing the credit supply function to NFAs as,
\[\hspace{2mm} L_{B}^{NFA} = \mu\left( i_{m},i_{BL},\ i_{B} \right)\left\{ RF + (1 - \varphi^{RR})D_{B} \right\} \hspace{3mm} [8]\]
with and The ratio thus depends on the different returns of the three assets in the bank’s portfolio: the lending interest rates applied to NFAs and MFIs but also the interest rate in the interbank market. Likewise, it is possible to show that the ratio depends negatively on the loan interest rate applied by the MFIs. Let be this rate. We can derive,
\[\frac{\partial\mu}{\partial i_{BL}} = \frac{\partial\mu}{\partial i_{M}}\frac{\partial i_{M}}{\partial i_{BL}}\]
By assumption since the MFI has to report on its interest rate any change in the micro-refinancing interest rate; then implies Henceforth the following result holds.
Result 2
The bank loan supply is a decreasing function of the lending interest rate applied by the MFI.
Like Bernanke and Blinder (1988), we can express the total deposits as a function of the augmented monetary multiplier taking account for the micro financial intermediation of MFIs. Let be the proportion of money holding in the monetary aggregate, the relation between the monetary base and the monetary aggregate can be written as, This gives,
\[\hspace{24mm} D_{B} = (1 - b)\overline{\phi}H \hspace{24mm} [9]\]
where we adopt an endogenous form of the monetary multiplier in Babatoundé (2018) such that is the interest rate on MFI deposits and is the interest differential given as proxy of the level of financial dualism.
Equation [9] represents a supply function similar to what Bernanke and Blinder (1988) propose, when the very basic assumption of no real cash balances[5] is postulated with consequently, Substituting [9] in [8], the credit supply function to NFAs can be deduced as,
\[\hspace{0mm} L_{B}^{s} = \mu\left( i_{m},i_{BL},\ i_{B} \right)\left\{ RF + (1 - \varphi^{RR})(1 - b)\overline{\phi}H\ \right\} \hspace{1mm} [10]\]
Before determining the equilibrium of the bank credit market, it is useful to analyze the determinants of this credit supply function. Partial derivatives allow us to examine the different interactions and, in particular, the effect of monetary policy variables on the credit supply function of bank. Differentiating [10] with respect to the three interest rates, it comes
\[\hspace{5mm} \frac{\partial L_{B}^{s}}{\partial i_{B}} = \frac{T\partial\mu}{\partial i_{B}} > 0 \hspace{5mm}\]
\[\hspace{5mm} \frac{\partial L_{B}^{s}}{\partial i_{BL}} = \frac{T\partial\mu}{\partial i_{BL}} < 0 \hspace{5mm}\]
\[ \begin{align} \frac{\partial L_{B}^{s}}{\partial i_{m}} = &\frac{T\partial\mu}{\partial i_{m}} + \mu\left( i_{m},i_{BL},\ i_{B} \right) \\ &\times (1 - \varphi^{RR})(1 - b)H\frac{\partial\overline{\phi}}{\partial i_{m}} \lessgtr 0 \end{align} \]
where
Referring to the definition of the bank credit supply is an increasing function of the lending interest rate and a decreasing function of the interest rate at which the bank refinances the MFI. Furthermore, the effect of the interest rate in the interbank or money markets remains ambiguous, determined by the sensitivity of the money multiplier to the rate. For an integrated financial system, the effect remains undetermined as long as the interest elasticity of the multiplier is not low.[6] Moreover, since if the financial system displays dualism, a positive effect on the credit supply is not excluded, but it works without great efficiency in terms of monetary transmission channel.
Following Jagdish (2009), the loan supply appears as a better channel in implementing monetary policies, especially in LDCs. Taking the derivatives of the credit supply function with respect to the main monetary instruments, we find some traditional theoretical results. Indeed, from [10] it comes,
\[\hspace{5mm} \frac{\partial L_{B}^{s}}{\partial RF} = \mu\left( i_{m},i_{BL},\ i_{B} \right) > 0 \hspace{5mm}\]
\[ \begin{align} \frac{\partial L_{B}^{s}}{\partial\varphi^{RR}} = &- \left( 1 - b \right)Hμ\left( i_{m},i_{BL},\ i_{B} \right) \\ &\times \left\lbrack \overline{\phi} - (1 - \varphi^{RR})\frac{\partial\overline{\phi}}{\partial\varphi^{RR}} \right\rbrack < 0 \end{align} \]
\[\frac{\partial L_{B}^{s}}{\partial\overline{\phi}} = \mu\left( i_{m},i_{BL},\ i_{B} \right)\left( 1 - \varphi^{RR} \right)\left( 1 - b \right)H > 0\]
\[\frac{\partial L_{B}^{s}}{\partial H} = \mu\left( i_{m},i_{BL},\ i_{B} \right)\left( 1 - \varphi^{RR} \right)\left( 1 - b \right)\overline{\phi} > 0\]
The credit supply is an increasing function of the monetary base or the refinancing of the central bank whereas the requirement reserves ratio has a negative effect on supply. This confirms the monetary policy theories and also reflects the positive relation between the money multiplier and the bank credit supply. Beyond the volume of refinancing, the first derivative shows the indirect effect of the central bank interest rate policy. That is, By assumption, Then implies Consequently, the supply of bank credit is a decreasing function of the central bank interest rate.
Finally, given the determinants of the augmented monetary multiplier, it is possible to expand this analysis to the MFI sector. Indeed, Babatoundé (2018) shows that and On this basis, the partial derivatives of equation [10] give,
\[\frac{\partial L_{B}^{s}}{\partial q} = \mu\left( i_{m},i_{BL},\ i_{B} \right)\left( 1 - \varphi^{RR} \right)\left( 1 - b \right)H\frac{\partial\overline{\phi}}{\partial q} > 0\]
\[\frac{\partial L_{B}^{s}}{\partial r_{B}} = \mu\left( i_{m},i_{BL},\ i_{B} \right)\left( 1 - \varphi^{RR} \right)\left( 1 - b \right)H\frac{\partial\overline{\phi}}{\partial r_{B}} \lessgtr 0\]
\[\frac{\partial L_{B}^{s}}{\partial r_{M}} = \mu\left( i_{m},i_{BL},\ i_{B} \right)\left( 1 - \varphi^{RR} \right)\left( 1 - b \right)H\frac{\partial\overline{\phi}}{\partial r_{M}} > 0\]
From these derivatives, we conclude that the preference for MFIs deposits and the deposits interest rate have a positive effect on the bank credit supply. This mechanism derives from the complementarity between banks and MFIs, with notably, the direct effect of these two variables on macro-deposits. Moreover, the effect of the deposit interest rate on the loan supply is ambiguous: it depends on the volatility of the induced deposits. When these new deposits are highly volatile, the negative effect outweighs the positive effect and an increase in excess reserves will occur; in contrast, if induced deposits are not volatile, the positive effect may prevail.[7]
After this comparative static of the credit supply function, the market equilibrium supposes understanding the demand function. Since NFAs choose to demand for one of the two types of loan given their interest rates, the demand function for bank credit can be written as,
\[\hspace{18mm} L_{B}^{d} = L_{B}\left( y,\ i_{B},i_{M} \right) \hspace{18mm} [11]\]
with and The conditions and are evident; is derived from the transaction motive, where is for the income and represents the lending interest rate applied by MFIs. Further, the equilibrium condition of the bank credit market is obtained by equalizing supply and demand. That is,
\[ \begin{align} \hspace{2mm} L_{B}\left( y,\ i_{B},i_{M} \right) = &\mu\left( i_{m},i_{BL},\ i_{B} \right)\bigl\{ RF \\ &+ (1 - \varphi^{RR})(1 - b)\overline{\phi}H\ \bigr\} \hspace{4mm} [12] \end{align} \]
Resolving equation [12], we implicitly deduce the bank lending interest rate as a function of the variables and That is,
\[\hspace{5mm} i_{B} = \Omega(i_{m},i_{M},i_{BL},y,\varphi^{RR},b,\overline{\phi},RF,H) \hspace{5mm} [13]\]
with,
\[ \begin{align} \Omega_{1} = \frac{\partial i_{B}}{\partial i_{m}} = &\mathrm{\Delta}^{- 1}\biggl\lbrack T\frac{\partial\mu}{\partial i_{m}} + \mu\left( i_{m},i_{BL},\ i_{B} \right) \\ & \times (1 - \varphi^{RR})(1 - b)H\frac{\partial\overline{\phi}}{\partial i_{m}} \biggr\rbrack \lessgtr 0 \end{align} \]
\[\Omega_{2} = \frac{\partial i_{B}}{\partial i_{M}} = - \mathrm{\Delta}^{- 1}\frac{\partial L_{B}}{\partial i_{M}} > 0\]
\[\Omega_{3} = \frac{\partial i_{B}}{\partial i_{BL}} = \mathrm{\Delta}^{- 1}T\frac{\partial\mu}{\partial i_{BL}} > 0\]
\[\Omega_{4} = \frac{\partial i_{B}}{\partial y} = - \mathrm{\Delta}^{- 1}\frac{\partial L_{B}}{\partial y} > 0\]
\[ \begin{align} \Omega_{5} = \frac{\partial i_{B}}{\partial\varphi^{RR}} = &- \mathrm{\Delta}^{- 1}\left( 1 - b \right)Hμ\left( i_{m},i_{BL},\ i_{B} \right) \\ & \times \left\lbrack \overline{\phi} - (1 - \varphi^{RR}) \frac{\partial\overline{\phi}}{\partial\varphi^{RR}} \right\rbrack > 0 \end{align} \]
\[ \begin{align} \Omega_{6} = \frac{\partial i_{B}}{\partial b} = &- \mathrm{\Delta}^{- 1}\left( 1 - \varphi^{RR} \right)Hμ\left( i_{m},i_{BL},\ i_{B} \right) \\ &\times \left\lbrack \overline{\phi} - (1 - b)\frac{\partial\overline{\phi}}{\partial b} \right\rbrack > 0 \end{align} \]
\[ \begin{align} \Omega_{7} = &\frac{\partial i_{B}}{\partial\overline{\phi}} = \mathrm{\Delta}^{- 1}\left( 1 - \varphi^{RR} \right)\left( 1 - b \right) \\ &\times Hμ\left( i_{m},i_{BL},\ i_{B} \right) < 0 \end{align} \]
\[\Omega_{8} = \frac{\partial i_{B}}{\partial RF} = \mathrm{\Delta}^{- 1}\mu\left( i_{m},i_{BL},\ i_{B} \right) < 0\]
\[ \begin{align} \Omega_{9} = &\frac{\partial i_{B}}{\partial H} = \mathrm{\Delta}^{- 1}\left( 1 - \varphi^{RR} \right)\left( 1 - b \right) \\ &\times \overline{\phi}\mu\left( i_{m},i_{BL},\ i_{B} \right) < 0 \end{align} \]
where . These different partial derivatives can be interpreted easily.
First, the lending interest rate of MFIs the micro-refinancing interest rate by bank the income (y), the requirement reserves ratio and the preference for real cash balances affect positively the bank loan interest rate Any positive change in these variables leads to a rise of the interest rate, reflecting a tightening of financing conditions in the credit market.[8] In fact, given substitution and income effects, increase in and induces positive change in demand for credit. All things being equal, this causes a rise in bank loan interest rate; likewise, when and increase, the supply is reducing, pushing up the loan interest rate. Second, the monetary multiplier the refinancing and the monetary base have a negative effect on the bank lending interest rate. For any one of these three variables, positive change induces a fall in stressing a less constrained financing conditions in the market. Finally, the effect of the interest rate in the interbank or money market is ambiguous, determined by the sensitivity of the money multiplier to the interest rate changes. As highlighted previously with the supply function, this effect remains undetermined if financial system is unified, as long as the interest elasticity of the multiplier is not low[9] because In opposite, if the financial system is dualistic with an augmented money multiplier, it is possible that any variation in affects the lending interest rate in the same direction with implications in terms of the monetary policy effectiveness.
Result 3
The sensitivity of the bank loan interest rate to the monetary policy instrument depends on three elasticities: that of the refinancing to the instrument and those of both credit demand and credit supply to the loan interest rate.
Proof
allows checking for the effectiveness of monetary transmission through the bank lending interest rate channel. Indeed, it assumes that changes in the central bank interest rate (what we call monetary policy instrument) pass through two mechanisms: from monetary instruments to banks loan interest rate and from this to real variables such as investment and consumption of NFAs.
\[\frac{\partial i_{B}}{\partial\varsigma} = \frac{\partial i_{B}}{\partial RF}\frac{\partial RF}{\partial\varsigma}\]
Given by definition, implies Consequently, the analytical framework helps to confirm the first transmission mechanism of monetary policy, since all changes in passed on the bank lending interest rates in the same direction. At what extent does this passthrough take place? From it is possible to write explicitly
\[\frac{\partial i_{B}}{\partial\varsigma} = \frac{\frac{\partial RF}{\partial\varsigma}}{\left(\frac{\partial L_{B}}{\partial i_{B}} - T\frac{\partial\mu}{\partial i_{B}}\right)}\mu\left( i_{m},i_{BL},\ i_{B} \right)\]
Multiplying the two members of this equation by the elasticity of the bank lending interest rate to the monetary policy instrument is obtained as,
\[\varepsilon_{\varsigma}^{i_{B}} = \frac{\frac{\partial RF}{\partial\varsigma}}{\left(\frac{\partial L_{B}}{\partial i_{B}} - T\frac{\partial\mu}{\partial i_{B}}\right)}\frac{\varsigma}{i_{B}}\mu\left( i_{m},i_{BL},\ i_{B} \right)\]
\[\varepsilon_{\varsigma}^{i_{B}} = \frac{RF\left(\frac{\partial RF}{\partial\varsigma}\right)\left(\frac{\varsigma}{RF}\right)}{L_{B}\left(\frac{\partial L_{B}}{\partial i_{B}}\right)\left(\frac{i_{B}}{L_{B}}\right) - T\mu\left(\frac{\partial\mu}{\partial i_{B}}\right)\left(\frac{i_{B}}{\mu}\right)}\mu\left( i_{m},i_{BL},\ i_{B} \right)\]
At the equilibrium of the credit market, Then, substituting this equilibrium condition in the expression of elasticity above, we get,
\[\varepsilon_{\varsigma}^{i_{B}} = \frac{RF}{T}\frac{\varepsilon_{\varsigma}^{RF}}{{(\varepsilon}_{i_{B}}^{L_{B}} - \varepsilon_{i_{B}}^{\mu})}\]
where and are respectively, the central bank interest rate elasticity of the refinancing, the lending interest rate elasticity of NFAs demand for bank credit and the lending interest rate elasticity of loan share in the bank’s portfolio.
End of proof.
The sensitivity of the lending interest rate to the monetary policy instrument therefore depends on the three elasticities. Consequently, ineffectiveness can occur if and only if, or or As the last two cases are less plausible, the sensitivity of refinancing to the monetary policy instrument appears significant given the financial architecture in LDCs. Otherwise, if that is to say, the bank does not resort to central bank for refinancing, the pass-through mechanism is not working and monetary policy can be ineffective. In opposite, even if both demand and supply of bank credit are inelastic to the lending interest rate, the sensitivity of the refinancing to the monetary policy instrument is sufficient at this stage to guarantee a greater effectiveness of the monetary policy.
Result 4
When the financial system shows dualism, the monetary transmission mechanism works not only through the channel of interest rate on bank credit to NFAs but also through that of interest rate on micro-refinancing to MFIs. More precisely, if the bank’s financial intermediation is such that the elasticity of the credit interest rate to NFAs to the micro-refinancing interest rate to MFIs, is less than unity, refinancing the MFIs is the most effective channel for transmitting monetary policy.
Proof
From the derivative it is possible to know the effect of the monetary policy instrument on the micro-refinancing interest rate. That is,
\[\frac{\partial i_{B}}{\partial\varsigma} = \frac{\partial i_{B}}{\partial i_{BL}}\frac{\partial i_{BL}}{\partial\varsigma}\]
Given the previous result we have and implies Consequently, the analytical framework helps to confirm the first mechanism of monetary policy transmission, not only through the channel of loan interest rate to NFAs but also through the channel of the micro-refinancing of MFIs by banks. The changes in the monetary policy instrument are reflected in the same direction on the two bank lending interest rates: Multiplying the two members by the monetary policy instrument elasticity of the micro-refinancing interest rate can be obtained as,
\[\varepsilon_{\varsigma}^{i_{BL}} = \frac{\partial i_{BL}}{\partial\varsigma}\frac{\varsigma}{i_{BL}} = \left( \frac{\partial i_{B}}{\partial\varsigma}\varsigma \right)\bigg/\left( \frac{\partial i_{B}}{\partial i_{BL}}i_{BL} \right)\]
Dividing both denominator and numerator by we get
\[\varepsilon_{\varsigma}^{i_{BL}} = \left( \frac{\partial i_{B}}{\partial\varsigma}\frac{\varsigma}{i_{B}} \right)\bigg/\left( \frac{\partial i_{B}}{\partial i_{BL}}\frac{i_{BL}}{i_{B}} \right) = \varepsilon_{\varsigma}^{i_{B}}/\varepsilon_{i_{BL}}^{i_{B}}\]
where, and represent respectively, the monetary policy instrument elasticity of the credit interest rate for NFAs (result 1) and the micro-refinancing interest rate elasticity to the same interest rate. The usefulness of this result is that it allows assessing the sensitivity of the two bank interest rates to the monetary policy instrument. The relationship can indeed be rewritten as well,
\[\varepsilon_{i_{BL}}^{i_{B}} = \varepsilon_{\varsigma}^{i_{B}}/\varepsilon_{\varsigma}^{i_{BL}}\]
End of proof.
Consequently, when the bank financial intermediation is such that the elasticity of interest rate to the interest rate is greater than unity that is to say the changes in interest rate to monetary policy instrument changes are of greater magnitude than those of the interest rate to the same monetary policy instrument variations. Then, the channel of credit for NFAs appears to be more effective than that of the micro-refinancing. In opposite, if the financial intermediation is such that refinancing the MFIs appears to be the most effective channel for transmitting monetary policy impulses.
4. The MFI credit market: equilibrium and its implications
Like the banking firm, the representative MFI chooses the levels of decentralized loans and deposits that maximize its profit in each period under the resources constraints and the cost of financial intermediation. The MFI is assumed to be refinanced by bank at the rate and to constitute the macro-deposits remunerated at the rate It profit should be specified as follows,
\[ \begin{align} \pi_{M}\left( . \right) = &i_{M}L_{M} - i_{BL}L_{B}^{MFI} - r_{M}D_{M} \\ &+ r_{BD}D_{B}^{MFI} - C_{M}(.) \end{align} \]
with and and measure respectively, the proportion of micro-refinancing relative to decentralized loans and the transformation ratio of MFI. Then, the profit can be rewritten,
\[ \begin{align} \hspace{2mm} \pi_{M}\left( . \right) = &{(i}_{M} - i_{BL}{w)L}_{M} \\ &+ \left\lbrack \left( 1 - \tau \right)r_{BD} - r_{M} \right\rbrack D_{M} - C_{M}(.) \hspace{4mm} [14] \end{align} \]
Likewise, under these conditions, the budget constraint of the MFI defined in the simplified balance sheet is reduced to,
\[ \begin{align} \hspace{12mm} \tau D_{M} &= (1 - w)L_{M} \text{ with, } L_{M} \\ &= \tau{{(1 - w)}^{- 1}D}_{M} > \tau D_{M} \hspace{13mm} [15] \end{align} \]
More specifically for the relation [15] is equivalent to the case where the MFI performs the financial transformation without the micro-refinancing.
Result 5
The MFI’s loan supply adjusts to the decentralized deposit from non-financial agents, confirming the absence of money creation ex-nihilo; however, when the micro-refinancing mechanism exists between banks and MFIs, this loan supply is greater than what prevails in an integrated financial system.
This new MFIs constraint represented by [15] allows to deduce a broad financial transformation ratio, considering the micro-refinancing. That is,
\[\hspace{10mm}\frac{L_{M}}{D_{M}} = \frac{\tau}{1 - w} \hspace{10mm}\]
As assumed, two regimes may prevail: depending on the net flow of deposits and loans between the two types of institution, the MFI is “out of the bank” or it is “in the bank”.
In the first “out of the bank” regime, the macro-deposits are greater than the micro-refinancing, that is to say Equivalently or By substituting the broad transformation ratio, it comes That is after transformation In contrast, in the second regime where micro-refinancing is greater than macro-deposits with “in the bank” regime, At the equilibrium which also corresponds to a classical regime without interactions between the two institutions, macro-deposits and micro-refinancing are equal and therefore,
Given the profit function defined by [14] and the constraint in [15], the objective of profit maximization is written,
\[ \begin{align} \hspace{0mm} \underset{\left\{ L_{M} \right\}}{Max} \; \pi_{M}\left( . \right) = &- C_{M}(.) \\ &+ \biggl\lbrack i_{M} + \left( 1 - \tau \right)(1 - w)\tau^{- 1}r_{BD} \\ &- i_{BL}w - r_{M}(1 - w)\tau^{- 1} \biggr\rbrack L_{M} \hspace{3mm} [16] \end{align} \]
Unlike the bank, the MFI’s profit is therefore the intermediation margin on loans, net of the intermediation costs represented by The specification of this function is also critical in the characterization of the decentralized credit supply function.
More specifically, when the transformation ratio is equal to 1 without a micro-refinancing mechanism, that is to say the maximization objective represented by [16] is reduced to a maximization program in a dualistic system. Then this program becomes a generalization of the standard analysis framework of the MFI maximization behavior in Van Wijnbergen (1983), Eboué (1990) and Ary Tanimoune (2007). The first order condition is:
\[ \begin{align} \hspace{2mm} \frac{\partial\pi_{M}}{\partial L_{M}} = &i_{M} - i_{BL}w + \left( 1 - \tau \right)(1 - w)\tau^{- 1}r_{BD} \\ &- r_{M}(1 - w)\tau^{- 1} - \frac{\partial C_{M}(.)}{\partial L_{M}} = 0 \hspace{10mm} [17] \end{align} \]
As in dualistic system, at equilibrium, the credit supply and deposit demand functions are such that the intermediation margin equalizes the corresponding marginal costs. In addition, the second-order condition allows to establish that the intermediation cost is non-decreasing at the optimum, given,
\[\partial^2\pi_{M}/\partial L^2_{M} = - \frac{\partial^{2}C_{M}(.)}{\partial L_{M}^{2}} \leq 0\]
That is In the rest of this section, it is assumed that the cost function depends solely on the input and that the profit function is differentiable in and Hotelling’s lemma allows us to deduce the supply and demand functions. For the loan supply, it follows
\[\partial\pi_{M}/\partial i_{M} = L_{M}(i_{M},r_{M})\]
The relation [15] gives the corresponding demand for decentralized deposits as,
\[D_{M}\left( i_{M},r_{M} \right) = {(1 - w)\tau}^{- 1}L_{M}(i_{M},r_{M})\]
How do these two functions behave when interest rates change?
Result 6:
The supply of decentralized loans and the demand for decentralized deposits depend positively on the lending interest rate of MFIs; in absolute terms, the MFIs financial intermediation shows economies of scale.
If additionally, the MFI works within “out of the bank” regime, the demand for decentralized deposit is more sensitive to changes in the lending interest rate than the supply of decentralized credit; that is .
Proof
The total differential of the equation [17] with respect to the rate gives,
\[\hspace{10mm}\frac{\partial^{2}C_{M}(L_{M})}{\partial L_{M}^{2}}\frac{dL_{M}}{di_{M}} = 1\hspace{10mm}\]
This follows,
\[\hspace{10mm} \frac{dL_{M}}{di_{M}} = \left\lbrack \frac{\partial^{2}C_{M}(L_{M})}{\partial L_{M}^{2}} \right\rbrack^{- 1}\hspace{10mm}\]
with, The demand for decentralized deposits of the MFI is deduced from the constraint [15] with When the MFI is in a “out of the bank” regime, the parameters and are such that or equivalently, It comes out that, Hence the result 6 is obtained.
End of proof.
The direct effect of the interest rate on the loan supply is an evidence. Unlike the case of the banking system, the effect of a change in on the demand for deposits is determined and positive. This behavior characterizes the banking system when the increase in credits induces a reduction of the marginal costs of deposits. By analogy, there is an economy of scale due to the MFI financial intermediation. This reduces the marginal costs of decentralized deposits and allows demand increase when the supply of decentralized credit and the corresponding interest rate raise. More specifically, if the MFI is in the “out of the banking” regime, the increase in the supply of decentralized loans after lending interest rate increase, induces more than proportional increase in demand for decentralized deposits. In absolute terms, any loan interest rate policy will have a larger positive effect on the demand for decentralized deposits than on the credit supply of MFIs. When the “in-bank” regime prevails, any lending interest rate policy will have a greater positive effect on decentralized credit supply than on demand for decentralized deposits.
Result 7: et
The supply of decentralized loans and the demand for decentralized deposits negatively depend on the deposit interest rate of MFI; in absolute terms, the financial intermediation of the MFI shows economies of scale.
If the MFI is working in “out of the bank” regime, the demand for decentralized deposit is more sensitive to changes in the deposit interest rate than in the MFI credit supply .
Proof
By completely differentiating the equation [17] with respect to the deposit interest rate it comes,
\[\hspace{10mm} \frac{\partial^{2}C_{M}(L_{M})}{\partial L_{M}^{2}}\frac{dL_{M}}{dr_{M}} = {- (1 - w)\tau}^{- 1} \hspace{10mm}\]
That is,
\[\hspace{10mm} \frac{dL_{M}}{dr_{M}} = {- (1 - w)\left\lbrack \tau\frac{\partial^{2}C_{M}(L_{M})}{\partial L_{M}^{2}} \right\rbrack}^{- 1} \hspace{10mm}\]
with and The demand for decentralized deposits of the MFI is deduced from [15] with Under the “out of the bank” regime, the constraint [15] implies Consequently, it results that Hence the result 7.
End of proof.
As previously, the negative effect of the deposit interest rate on the demand for decentralized deposits is obvious. Also, there is no indeterminacy about the negative effect of the rate on the decentralized credit supply like in the banking system. This is characterizing the banking system when deposits increase induces a reduction in the marginal cost of loans. Thus, by analogy, there is an economy of scale due to the financial micro intermediation; it reduces the marginal costs of decentralized loans and allows the supply increase as a result of fall in the rate More specifically, when the MFI is working with “out of the banking” regime, the increase in the demand for decentralized deposits induces a less than proportional increase in decentralized credit supply. In absolute terms, any deposit interest rate policy will have higher negative effect on the demand for deposits than on the MFI credit supply. In opposition, when “in-bank” regime prevails, any deposit interest rate policy will have a greater negative effect on credit supply than on demand for decentralized deposits.
Globally, unlike the situation prevailing in the banking system,[10] the cross-effects of a change in the interest rates and on demand for deposits and credit supply respectively, are determined and, depending on the regime, they can be higher or lower than the price-direct effects. Within the “out of the bank” regime, the MFI constitutes more macro-deposits than it benefits from micro-refinancing: then The decrease of intermediation margin will have the effect of contracting the demand for decentralized deposits more than the supply of decentralized credit. With the “in bank” regime, the MFI benefits from more micro-refinancing than it constitutes the macro-deposits: then Any deposit interest rate policy will have a greater effect on the loan supply than on the demand for deposits; the same result holds about the reduction in the intermediation margin with the effect of contracting the loan supply more than the demand for deposits. Finally, if neutral regime prevails, the macro-deposits compensate the micro-refinancing with, and In absolute term, any interest rate policy will have a similar effect on both loan supply and demand for deposits; this neutral regime is specific case on which previous studies have been based, notably in Eboué (1990) and Ary Tanimoune (2007).
Corollary
The regime of complementarity between banking and MFI sectors is a major determinant of the effect of liquidity regulation policies on the latter. Since for the MFI, the credit supply adjusts to the demand for deposits, one determines the other and vice versa.
To guarantee the effectiveness of the monetary policy instrument, the loan supply must be targeted when “in the bank” regime is dominant, whereas the demand for deposits must be targeted if the “out of the bank” scheme prevails; in each case, the second variable adjusts.
In order to find the decentralized credit supply function, the MFI constraint given by [15] is written,
\[{L_{M} + D_{B}^{MFI} = D}_{M} + L_{B}^{MFI}\]
In the same logic as above, it is assumed that the MFI composes its asset portfolio for a proportion of decentralized loans to NFAs. Under this hypothesis, the constraint allows writing the expression of MFI loan supply function for non-financial agents as,
\[\hspace{10mm} L_{M} = \eta\left( i_{M},\ r_{BD} \right)\left\lbrack L_{B}^{MFI} + D_{M} \right\rbrack \hspace{10mm} [18]\]
with, and The ratio depends therefore on the different returns of the two components of the MFI portfolio, the lending interest rate applied to NFAs and the interest rate of the macro-deposits
At equilibrium, demand for decentralized deposits adjusts to supply of non-financial agents. This supply is assumed to be that of Babatoundé (2015). Indeed, including the decentralized deposits and their preference compared to bank deposits in the analysis framework of Sidrauski (1967) and Agénor and Alper (2009), and in optimizing an intertemporal utility function, we derive the aggregate solution represented by,
\[ \begin{align} \hspace{4mm} D_{M} = &{C\left( Ki^{uρ} \right)}^{- \sigma}\biggl\lbrack \left( \frac{u}{1 - u} \right)\bigl( \left( i - r_{B} \right)q^{- 1} \\ &+ \left( i - r_{M} \right) \bigr) \biggr\rbrack^{\sigma(u\rho - 1)} \hspace{28mm} [19] \end{align} \]
The supply of decentralized deposits is written as a function of the consumption and the behavior parameters with, and or and represent respectively, the money market interest rate, the bank deposit interest rate and the MFI deposit interest rate. Besides the parameter which accounts for the agent’s behavior, there is the parameter indicating the share of the real cash balances in the monetary aggregate composite. Finally, is a substitution parameter defined by given the intra-temporal elasticity of substitution between consumption and monetary aggregate composite noted is a parameter. Thus, for constant level of consumption, equation [19] shows that the supply of decentralized deposits is determined by the two lending interest rates prevailing in the two sectors, but also by the nominal loan interest rate and the preference for decentralized deposits. It also depends on the shares of the components of monetary aggregate composite namely, the real cash balance and the two types of deposits, but also the intra-temporal elasticity of substitution
Substituting this expression in [18], the MFI credit supply is therefore written as,
\[\hspace{11mm} L_{M}^{s} = \left\lbrack L_{B}^{MFI} + λC \right\rbrack\eta\left( i_{M},\ r_{BD} \right) \hspace{11mm} [20]\]
The function is defined with reference to Asako (1983) and Rajhi and Villieu (1998) by,
\[ \begin{align} \lambda(.) = D_{M}/C = &\left( Ki^{uρ} \right)^{- \sigma}\biggl\lbrack \left( \frac{u}{1 - u} \right)\bigl( \left( i - r_{B} \right)q^{- 1} \\ &+ \left( i - r_{M} \right) \bigr) \biggr\rbrack^{\sigma(u\rho - 1)} \end{align} \]
Before determining the equilibrium of the MFI credit market, it is useful to analyze the credit supply determinants. The partial derivatives allow to appreciate the different interactions and in particular, the effect of the monetary policy variables on the supply function. Differentiating [20] with respect to the interest rates and it comes,
\[\hspace{5mm} \frac{\partial L_{M}^{s}}{\partial i_{M}} = \left( L_{B}^{MFI} + λC \right)\frac{\partial\eta}{\partial i_{M}} > 0 \hspace{5mm}\]
\[\hspace{5mm} \frac{\partial L_{M}^{s}}{\partial r_{BD}} =\left( L_{B}^{MFI} + λC \right)\frac{\partial\eta}{\partial r_{BD}} < 0 \hspace{5mm}\]
By definition, the decentralized credit supply is an increasing function of the lending interest rate and a decreasing function of the interest rate paid for macro-deposits. An increase in the rate induces a rise in the demand for macro-deposits to the detriment of the loan supply for NFAs.
Differentiating the supply function with respect to quantities and it comes
\[\frac{\partial L_{M}^{s}}{\partial C} = λη\left( i_{M},\ r_{BD} \right) > 0\]
\[\frac{\partial L_{M}^{s}}{\partial\lambda} = Cη\left( i_{M},\ r_{BD} \right) > 0\]
\[\frac{\partial L_{M}^{s}}{\partial D_{M}} = \eta\left( i_{M},\ r_{BD} \right) > 0\]
\[\frac{\partial L_{M}^{s}}{\partial L_{B}^{MFI}} = \eta\left( i_{M},\ r_{BD} \right) > 0\]
Those partial derivatives allow us to conclude that the variables of consumption, supply of decentralized deposits of NFAs and micro-refinancing of the MFI have a positive effect on the supply of decentralized loan.
Analysing the effects of the MFI credit interest rates on the supply of decentralized deposits, Babatoundé (2015) indeed shows that the nature of the sign being a function of the interest rate differential between the two sub-sectors of the financial system. With reference to this comparative static, it is possible to derive the effect of the two borrowing rates on the MFI’s credit supply. That is,
\[\hspace{10mm} Sign\left( \frac{\partial L_{M}^{s}}{\partial r_{M}} \right) = Sign\left( \frac{\partial L_{M}^{s}}{\partial r_{B}} \right) \hspace{10mm}\]
with,
\[\hspace{7mm}\biggl\{ \begin{matrix} \left( i - r_{B} \right)p^{- 1} > \mathrm{\Delta}r \Longrightarrow \frac{\partial L_{M}^{s}}{\partial r_{M}} > 0\ et\ \frac{\partial L_{M}^{s}}{\partial r_{B}} > 0 \\ \left( i - r_{B} \right)p^{- 1} < \mathrm{\Delta}r \Longrightarrow \frac{\partial L_{M}^{s}}{\partial r_{M}} < 0\ et\ \frac{\partial L_{M}^{s}}{\partial r_{B}} < 0 \\ \end{matrix} \biggr.\ \hspace{7mm}\]
Therefore, counter-intuitively, the two interest rates applied by banks and MFIs have the same effect on the decentralized credit supply, which can be positive or negative depending on the MFIs market share parameters the bank margin and the differential interest rates More specifically, for low differential levels, and any increase in the two deposit interest rates implies increasing supply of decentralized credit. In contrary, for high differentials, a negative effect of the interest rates on the supply will prevail. If, given the MFI deposit interest rate, the result is slight, it is not the same for the bank deposit interest rate. Here, the effect is justified notably by the relative insensitivity of demand for bank deposits to variations of interest rate when the differential is low: the social logic outweighing the logic of performance. The increase in demand for MFI deposits is helping to strengthen the MFI credit supply capacity. Generally, if the structure of deposit interest rates is such that the differential is relatively small, any bank interest rate policy will have positive effect on demand for decentralized deposits but also on supply of decentralized credit in the economy. This result reinforces that of Babatoundé (2017) justification for the persistence of financial dualism and the failure of financial liberalization policies in the WAEMU[11] area given the differentials that characterize the financial system in these countries.
Finally, given it is also possible to appreciate the effect of the money market interest rate on the credit supply of MFIs. That is,
\[\frac{\partial L_{M}^{s}}{\partial i} = Cη\left( i_{M},\ r_{BD} \right)\frac{\partial\lambda}{\partial i} < 0\]
There is therefore an inverse relationship between the two variables; the rise in the interest rate induces a decline in the supply of decentralized deposits through different mechanisms; consequently, there is a decline in the supply of decentralized credit. In another sense, the attractiveness of the interbank market has a reducing effect on the micro-refinancing provided to MFI; finally, this lowers the supply of decentralized credit.
After this comparative static of the MFI’s credit supply function, market equilibrium implies to define the demand function. Since agents choose to demand one of the two types of loan based on the respective lending interest rates, the demand function of decentralized credit is given by,
\[\hspace{19mm} L_{M}^{d} = L_{M}\left( y,\ i_{B},i_{M} \right) \hspace{19mm} [21]\]
with and The conditions and are trivial and arises from the transaction motive, put there for the income.
The condition of equilibrium in the decentralized credit market is obtained by equalizing supply and demand, that is
\[\hspace{2mm} L_{M}\left( y,\ i_{B},i_{M} \right) = \left\lbrack L_{B}^{MFI} + λC \right\rbrack\eta\left( i_{M},\ r_{BD} \right) \hspace{3mm} [22]\]
Resolving equation [22], it is possible to derive the lending interest rate of MFI with respect to the variables and That is,
\[\hspace{5mm} i_{M} = \Pi(i_{B},\ i,r_{BD},r_{B},r_{M},y,L_{B}^{MFI},\lambda,\ C) \hspace{5mm} [23]\]
with
\[\Pi_{1} = \frac{\partial i_{M}}{\partial i_{B}} = - \nabla^{- 1}\partial L_{M}/\partial i_{B} > 0\]
\[\Pi_{2} = \frac{\partial i_{M}}{\partial i} = \nabla^{- 1}Cη\left( i_{M},\ r_{BD} \right)\partial\lambda/\partial i > 0\]
\[\Pi_{3} = \frac{\partial i_{M}}{\partial r_{BD}} = \nabla^{- 1}\left( L_{B}^{MFI} + λC \right)\partial\eta/\partial r_{BD} > 0\]
\[\Pi_{4} = \frac{\partial i_{M}}{\partial r_{B}} = \nabla^{- 1}Cη\left( i_{M},\ r_{BD} \right)\partial\lambda/\partial r_{B} \gtrless 0\]
\[\Pi_{5} = \frac{\partial i_{M}}{\partial r_{M}} = \nabla^{- 1}Cη\left( i_{M},\ r_{BD} \right)\partial\lambda/\partial r_{M} \gtrless 0\]
\[\Pi_{6} = \frac{\partial i_{M}}{\partial y} = {- \nabla}^{- 1}\partial L_{M}/\partial y > 0\]
\[\Pi_{7} = \frac{\partial i_{M}}{\partial L_{B}^{MFI}} = \nabla^{- 1}\eta\left( i_{M},\ r_{BD} \right) < 0\]
\[\Pi_{8} = \frac{\partial i_{M}}{\partial\lambda} = \nabla^{- 1}Cη\left( i_{M},\ r_{BD} \right) < 0\]
\[\Pi_{9} = \frac{\partial i_{M}}{\partial C} = \nabla^{- 1}λη\left( i_{M},\ r_{BD} \right) < 0\]
where,
\[\nabla = \partial L_{M}/\partial i_{M} - \left( L_{B}^{MFI} + λC \right)\partial\eta/\partial i_{M} < 0\]
Equation [23] represents the curve in Carpenter (1999) which determines the MFI interest rate from the banking sector variables including reserves and refinancing; instead of volumes, the model considers the interest rate and the micro-refinancing since there is no formal relationship between MFI and central bank. This has the advantage to lift the indeterminacy in terms of the effect of the variables related to bank on the MFI interest rate.
Following a rise in interest rates and increase in the cost of bank credit lowers its demand; with the substitution effect, everything being equal, the excess demand in the MFI credit market leads to a rise in the interest rate It is the same for an increase in the income with the income effect, it induces a demand excess and therefore an increase in the MFI lending interest rate. In contrast, any positive change in the micro-refinancing and the demand for decentralized deposits through lowers the MFI lending interest rate, given the falling intermediation cost. Finally, the effect of deposit interest rates is ambiguous; as indicated above, it depends on the differential levels between the two sub-sectors of the financial system. If the differential is small, and affect negatively the MFI lending interest rate since and A positive effect of the first one on the second occurs when the differential is quite high. This result of the ambiguous effect of the bank credit interest rate is particularly equivalent to the result of Montiel (1991) who beyond financial dualism, also considers a parallel exchange market.
Past the traditional transmission channels, the hypothesis of complementarity between the two sub-sectors of the financial system allows to surmise on alternative mechanisms including the financial micro intermediation and transformation of MFIs. Given the micro-refinancing of MFI by bank, allows verifying the effectiveness of an amplifying mechanism of monetary transmission through a direct channel of bank interest rates and an indirect channel of MFI interest rates. In this case, according to the direct channel, changes in the central bank’s interest rate policy affect the lending interest rates of commercial banks. In the sense of the indirect channel, complementarity relationships allow commercial banks transmitting changes in monetary policy instrument to the MFI lending rates and then to the real variables.
Result 8
The sensitivity of the MFI loan interest rate to the micro-refinancing interest rate depends on three elasticities: that of micro-refinancing to the bank interest rate and those of both MFI credit demand and supply to the loan interest rate.
If the MFI’s financial micro intermediation is such that the demand for micro-refinancing is sensitive to the rate applied by the bank, the MFI credit represents a transmission channel for the central bank’s monetary shocks like the bank credit.
Indeed,
\[\frac{\partial i_{M}}{\partial i_{BL}} = \frac{\partial i_{M}}{\partial L_{B}^{MFI}}\frac{\partial L_{B}^{MFI}}{\partial i_{BL}}\]
By definition is the rate at which bank provides credit to MFI as micro-refinancing; consequently, The derivative implies therefore Then, like the traditional theories of channels, the analytical framework enables to verify a mechanism of monetary transmission from the banks to the MFIs. All variations in the micro-refinancing interest rate passed on the MFI’s lending interest rates in the same sense. At what extent does this take place? From it is possible to write explicitly with,
\[\frac{\partial i_{M}}{\partial i_{BL}} = \frac{\frac{\partial L_{B}^{MFI}}{\partial i_{BL}}}{\frac{\partial L_{M}}{\partial i_{M}} - \left( L_{B}^{MFI} + λC \right)\frac{\partial\eta}{\partial i_{M}}}\eta\left( i_{M},\ r_{BD} \right)\]
Multiplying the two members of this equation by the elasticity of the MFI lending interest rate to the micro-refinancing interest rate is obtained, as
\[ \begin{align} \varepsilon_{i_{BL}}^{i_{M}} &= \frac{\partial i_{M}}{\partial i_{BL}}\frac{i_{BL}}{i_{M}} \\ &= \frac{\frac{\partial L_{B}^{MFI}}{\partial i_{BL}}}{\frac{\partial L_{M}}{\partial i_{M}} - \left( L_{B}^{MFI} + λC \right)\frac{\partial\eta}{\partial i_{M}}}\frac{i_{BL}}{i_{M}}\eta\left( i_{M},\ r_{BD} \right) \end{align} \]
\[ \begin{align} \varepsilon_{i_{BL}}^{i_{M}} = &\frac{L_{B}^{MFI}\left(\frac{\partial L_{B}^{MFI}}{\partial i_{BL}}\right)\left(\frac{i_{BL}}{L_{B}^{MFI}}\right)}{L_{M}\left(\frac{\partial L_{M}}{\partial i_{M}}\right)\left(\frac{i_{M}}{L_{M}}\right) - \eta\left( L_{B}^{MFI} + λC \right)\left(\frac{\partial\eta}{\partial i_{M}}\right)\left(\frac{i_{M}}{\eta}\right)} \\ &\times \eta\left( i_{M},\ r_{BD} \right) \end{align} \]
At the equilibrium of the decentralized loan market, By substituting this equilibrium condition in the elasticity expression above, it comes
\[\varepsilon_{i_{BL}}^{i_{M}} = \frac{L_{B}^{MFI}}{\left( L_{B}^{MFI} + λC \right)}\frac{\varepsilon_{i_{BL}}^{L_{B}^{MFI}}}{\left( \varepsilon_{i_{M}}^{L_{M}} - \varepsilon_{i_{M}}^{\eta} \right)}\]
and are respectively, the bank lending interest rate elasticity of the micro-refinancing, the MFI lending interest rate elasticity of the demand for decentralized credit and the deposit interest rate elasticity of the decentralized credit supply. Then, the sensitivity of the MFI interest rate to the micro-refinancing interest rate depends on these elasticities; this new channel may become defective only in each of the following three cases:
\[\varepsilon_{i_{BL}}^{L_{B}^{MFI}} \rightarrow 0,\ \varepsilon_{i_{M}}^{L_{M}} \rightarrow - \infty,\ \varepsilon_{i_{M}}^{\eta} \rightarrow + \infty\]
Otherwise, the last two cases are less plausible. Then as long as the micro-refinancing will be sensitive to the interest rate applied by banks, the MFIs loan will be able to serve as channel of monetary transmission. Despite the inelasticity of supply and demand for decentralized credit, this condition is sufficient to guarantee the effectiveness of this amplifying mechanism.
5. Conclusion
This paper reviews the channels of monetary transmission by analyzing the credit market when the financial system is dualistic, with banks and Microfinance Institutions (MFI). Two mechanisms of the articulation process between the two sub-sectors of the financial system are highlighted: the constitution of deposits by MFI with banks on one hand and the micro-refinancing of MFI by the bank, on the other hand. Agents choose to hold their assets either in bank deposits or in decentralized deposits, while bank credit and decentralized credit are the two sources of financing for consumption and investment needs. To account for optimization behaviors in each credit market, the determinants of supply and demand functions were derived, taking into account the profit maximization objective.
Within this specific framework analysis, the examination of the bank credit market leads to different results. In particular, the bank’s financial intermediation margins are superior to those prevailing in an integrated system; the credit supply is a negative function of the lending interest rate applied by MFI. Theoretically, the bank interest rate should reflect changes in the central bank monetary policy interest rate. The model allows to show that this sensitivity depends on three elasticities: that of the refinancing to the instrument and those of both credit demand and credit supply to the loan interest rate. This leads to the conclusion that when the financial system is dualistic, the transmission mechanism of the monetary policy takes place not only through the channel of bank loan interest rate but also through the channel of micro-refinancing. Besides, there is a condition that makes the MFI’s refunding by the bank, the most effective channel for transmitting monetary shocks.
Symmetrically, the analysis of the decentralized credit market allows to confirm these various results, notwithstanding the absence of monetary creation ex-nihilo since the MFI decentralized credit supply adjusts to the decentralized deposit of non-financial agents. However, the model shows that, when the micro-refinancing relationship exists between banks and MFI, the MFI credit supply is higher than what prevails in an integrated financial system. The sensitivity of the MFI lending interest rate to the micro-refinancing interest rate depends on three elasticities: that of micro-refinancing to the bank interest rate and those of both MFI credit demand and supply to the loan interest rate. Therefore, we show that the MFI loan represents a channel for transmitting monetary shocks of the central bank when the demand for micro-refinancing is sensitive to the interest rate applied by the bank.
At the end of this analysis of the transmission channels of the central bank monetary policy in a segmented financial system, two plausible mechanisms must be indicated, one direct and the other indirect. On the one hand, represents the traditional transmission mechanism by which the bank disseminates changes in monetary policy instrument to real variables through its own lending interest rates. On the other hand, and represent the alternative and indirect mechanism, given the financial intermediation of MFI between the bank and NFAs. When the MFI micro-refinancing mechanism is working, banks pass on the changes in the monetary policy interest rate through its own credit interest rates applied not only to the NFAs but also to the MFIs; in return, the MFIs pass on these changes to the real variables through their own decentralized credit interest rates.
Given the complementarity of the two markets and the specificities of the two products, the hypothesis of perfect substitutability is not plausible. Traditionally, the variable of interest on which the substitution effect depends is the price; beyond price, the substitutability invoked here can be justified by the social factors, the proximity and the collateral which govern the loan contracts.
Indeed, if the rate applied by the bank is lower than that of IMF, there is an arbitration of the rational agent: borrow in the bank and place with IMF. The result is an increase in the bank interest rate and a lower interest rate in the IMF, until the two rates become equal. Thus, over time, the interest rate differential will dissipate with an integrated financial system. This difficult mechanism to achieve by NFAs, can also be difficult with financial agents and then, the link between the two institutions can lead to financial unification, as Lelart (2000) suggests.
There are various reasons for this relationship: there is the reason for security, for deposit insurance, for liquidity risk, for profitability, for interest rate risk, for collateral, etc.
The simplest case for analyzing the effect of monetary policies is certainly the one where the cost function is assumed to be linear, with constant returns (Grimaud and Rochet 1994).
This is also the hypothesis in Montiel (1991), who also assume, instead of MFI deposits, foreign exchange reserves in a perspective of dualism exchange rate analysis.
When the interest elasticity of the multiplier is low, any variation in the interbank interest rate will have a negative effect on the bank credit supply.
The same result can be obtained from the profit maximization of the bank; the effect of the credit interest rate on the supply of credits depends on the sign of the second derivative. If the cost function exhibits an economy of scale, the supply of bank credits is a decreasing function with respect to the credit interest rate; otherwise, it is an increasing function where,
Here, the signs we derived are identical to the results of Bernanke and Blinder (1988) for the variables and (through which are common in both specifications.
When the interest rate elasticity of the multiplier is low, any variation in the interbank rate will have a positive effect on the bank loan interest rate.
The analysis of bank credit supply and demand for bank deposits shows that the cross-effects of credit and deposit interest rates depend fundamentally on the separability of the intermediation cost function in relation to economy of scale (see Freixas and Rochet 2008).
West African Economic Monetary Union, including Benin, Burkina Faso, Cote d’Ivoire, Guinea Bissau, Mali, Niger, Senegal and Togo.