We develop a simple theoretical framework linking mobile money to private sector credit in a country with limited access to financial services. We develop our framework within financial development theory, which suggests that financial innovations influence intermediation through increased access to financial services (Levine 2005). We assume a household can hold financial assets (\( {H}_t \)) as cash, deposits in financial institutions, and mobile money balances (Eq. 1).

$$ {H}_t={c}_t+{d}_t+{m}_t, $$

(1)

\( {H}_t \) is the household’s total financial assets, \( {c}_t \) denotes real cash balances, \( {d}_t \) deposits, and \( {m}_t \) real mobile money balances \( {d}_t=0 \) for households without bank accounts. We next define \( {B}_t \), as a financial institution where households \( {H}_t \) deposit real money balances. \( {B}_t \) holds these deposits in an escrow account. The commercial bank’s total deposits \( {D}_t \) from household \( {H}_t \) are the sum of the household’s bank and mobile money balances as detailed in Eq. 2.

$$ {D}_t={d}_t+{m}_t, $$

(2)

\( {d}_t \) denotes a household’s deposits and \( {m}_{t,} \) its real mobile money balances at any time t.

Evidence suggests that mobile money is easy and less risky to use compared to holding cash (Jack et al. 2013). Evidence further suggests that mobile money increases demand for and access to banking products (Weil et al. 2014). Thus, households lacking bank accounts can cash balances as mobile money. Mobile money balances including subscribers’ and agents’ balances, are in escrow at a commercial bank. This implies that commercial bank deposits, \( \left({D}_t\right) \) rise as households replace cash balances with mobile money. As a result, for all (\( \forall \Big) \) increases in mobile money balances cash in circulation declines and bank deposits increase:

$$ \begin{array}{l}\varDelta {D}_t=\varDelta \left({d}_t+{m}_t\right),\hfill \\ {}\forall\ \frac{\partial {D}_t}{\partial {m}_t}>0,\ \mathrm{a}\mathrm{s},\ {c}_t\to\ 0\hfill \end{array} $$

(3)

Commercial banks facilitate intermediation. They mobilize as many deposits from as many economic agents as they can and reallocate them as credit. Because banks’ balance sheets record deposits as liabilities and loans as assets, increased deposits enable them to create credit through balance sheet expansion. Therefore, the total deposits (\( {D}_t\Big) \) of commercial bank (\( {B}_t\Big) \) are an increasing function of aggregate deposits from other sources and mobile money balances from households, such that

$$ {D}_t=f{\displaystyle {\sum}_1^t}\left({d}_t+{m}_t\right)>0, $$

(4)

We assume banks’ accumulated deposits are demand deposits, time deposits and required or precautionary reserves. Demand deposits (\( {\mathrm{D}}_1{\mathrm{D}}_{\mathrm{t}} \)) are short-term liabilities on the bank balance sheet that depositors can claim at any time. Thus, demand deposits constitute a proportion of total deposits that are not loaned. We assume mobile money may boost long-term time deposits/loanable funds through overnight/short-term interbank deals. Time deposits (\( {\mathrm{D}}_2{\mathrm{D}}_{\mathrm{t}} \)) are longer-term deposits a proportion of which can be loaned or invested in government securities. Thus, loan supply is a function of time deposits available for credit:

$$ {L}_t^s=\gamma\ {D}_2{D}_t, $$

(5)

\( \gamma \) is the proportion of time deposits available for credit or government securities. We define the balance sheet of a profit-maximizing commercial bank, as

$$ {L}_t+{S}_t={D}_t+{C}_t, $$

(6)

\( {L}_t, \) is the loan volume, \( {S}_t \) government securities, \( {D}_t \) deposit volume, and \( {C}_t \) bank capital.

We assume that banks would rather make loans than buy government securities. However, extending credit presents the possibility of defaults and the bank credit market is confronted with frictions of asymmetric information and contract enforcement that make lending costly. Hence, we regard intermediation costs as an increasing and convex function of the volume of intermediated loans, such that

$$ C=C(L),\kern0.5em \mathrm{where}\ {c}^{\hbox{'}}>0\ \mathrm{and}\kern0.5em {c}^{\hbox{'}\hbox{'}}>0, $$

(7)

Consequently, banks charge a higher interest rate on loans to cover the costs and assure profits. From Eq. 5, their loan supply function becomes

$$ {L}_t^s=L\left(\mathrm{i}\mathrm{L}\gamma\ {D}_2{D}_t\right),\ {\mathrm{L}}^{\hbox{'}}>0, $$

(8)

where \( \mathrm{i}\mathrm{L} \) is the loan rate.

In addition, a less favourable domestic macro-economy constrains the volume of intermediated funds, such that:

$$ {L}_t^s=\frac{1}{\tau}\mathrm{L}\left(iL\gamma\ {D}_2{D}_t\right)\ {L}^{\hbox{'}}>0, $$

(9)

where: \( \tau \) captures the effect of the macro-economy on loans supply.

Recall from Eq. 4 that

$$ {D}_t=f{\displaystyle {\sum}_1^i}{d}_t+{m}_t>0, $$

Substituting Eq. 4 into Eq. 9, the bank’s loan supply function becomes an increasing function of total deposits of which mobile money balances are a significant portion. This is summarised in Eq. 10.

$$ {L}_t^s=\frac{1}{\tau}\mathrm{L}\left(iL\gamma\ {D}_2f{\displaystyle {\sum}_1^i}{d}_t+{m}_t>{0}_t\right), $$

(10)

Such that;

$$ \frac{\ \partial {L}_t^s}{\partial {m}_{\mathrm{t}}}>0\ \mathrm{and}\ {L}^{\hbox{'}}>0 $$

From Eq. 10, we conclude the following. First, loan supply is related to the macro- economy (\( \tau \)), such that the more unfavourable the macro-economy, the lower the loan supply. We proxy the macro-economy using the inflation rate and the exchange rate. Second, loan supply relates positively to interest rates. Third, loan supply relates positively to mobile money balances. Deposits rise with mobile money balances, increasing loanable funds. This is summarized in the expression \( \tau <0;\;iL>0;m>0 \).