monopolistically the price of deposits, while the demand for deposits is negatively affected by the
market interest rate on bonds that intermediaries pay when providing payment services.
In order to obtain a demand schedule for deposits services, we study separately the demand
of households and firms. The demand of both classes of agents is assumed to depend on two
different interest rates, the own rate on deposits and the rate on bonds, that is an opportunity
cost. For simplicity, we assume that transaction fees do not affect the demand for deposits. This
assumption can be justified, following Fama [10], considering that the market for payment services
is competitive, marginal costs are constant and the supply of these services is normally infinitely
elastic.
Interest rates on bonds are assumed to follow a pure random walk process:
B = rB + eBi with E[eB]=0 E[eBiBi, ∙] = σ2 i = j, E[eBiB, ■] = 0 i = j. (8)
t+1 t t+1 t t+i t+j B , t+i t+j .
Interest rates on deposits are the result of the equilibrium condition of the market, and are set
by the banking system in function of the rate on bonds. So they are assumed to follow another
random walk process, correlated with the process of bonds:
rD+1 = rD + et+1 with E [ eD ]=0 E [ et+ieD+'∣]= σD i = j, E [ eD+ ieD+]=0 i = j and
t+1 t t+1 t t+i t+j D t+i t+j
e [ eD+ ieB+ j ] = Cov ( DB ) i = j, e [ eD+ ieB j ]=0 i = j. (9)
Household’s deposits are assumed to depend (positively) on nominal income,7 the own interest
rate on deposits and (negatively) on the interest rate on bonds. The coefficients are assumed to
be constant over time.
Dth =f1Yt+f2rtD -f3rtB. (10)
The first term can be considered to capture mainly the behaviour of demand deposits while the
second two of time deposits. It has in fact been shown that both demand deposits and M 1are
not very sensitive to interest rates.8 We assume that nominal income is an AR(1) process, whose
trend and error coefficients depend on the growth of both real income and prices:
Yt +1 = γY Yt + eY+1 = γYr γP Yt + eY+1 + eP+1. (11)
The expected value of deposits of the following period is:
E[Dth+1]=f1γYYt+f2E[rtD+1]-f3E[rtB+1]. (12)
this can be rewritten as:
E[Dth+1]=γYDth - (γY - 1)f2E[rtD+1] + (γY - 1)f3E[rtB+1]. (13)
Firm’s deposits are assumed to depend on both rates as before, and on the quantity of loans issued
by the bank. This dependence is due to the liquidity creation that loans allow because of the
convertibility on demand of deposits.9 Besides, banks compel firms to deposit a fraction of the
loans they issue. In this way they manage the payments of the borrower, earning fees, as was
suggested by Sprenkle,10 and they can monitor his liquidity. We assume that deposits depend on
the amount of loans of the current period.11
Dtf = κLt + f4rtD - f5rtB. (14)
The coefficient κ synthesizes the effect of the feedback of loans on deposits. Loans have a direct
impact on deposits through the effect of firms’ deposits, indirectly through firms’ expenditure.
The effect of the increase of loans is assumed to be analogous among different banks of the system,
7Since their transaction demand is assumed to be a function of income.
8See Hess [14] and [15] and Moosa [17].
9 See Diamond and Rajan [7].
10See Sprenkle [26] and [27].
11Alternatively the dependence can be assumed to be lagged, and deposits of the current period depend on loans
of the previous one. It can be shown that the results do not change in a relevant way.