2 A model with contemporaneous feedback
The model is in discrete time, and has the following time structure. At the beginning of every
period, households and firms dispose of a certain amount of funds that are the bequest of previous
periods. Households take decisions regarding their portfolio allocation and their consumption
plans for the period. Firms plan their investments for the period and evaluate their finance needs.
Deposits are necessary for households in order to carry out transactions, since there is no currency.
At the end of every period households and firms dispose of an amount of funds that reflects the
evolution of the value of their assets, the income of the period and their consumption choices.
Firms obtain the liquidity that is necessary to carry out their transactions by means of loans.
The feedback process of loans on deposits that we describe can be understood as the result of the
liquidity creation at the firm level: at the end of the period part of the liquidity generated by
means of loans is distributed to households. This assumption fits well with Ramey’s [20] findings
of cointegration between M1 and business M1.
The bank can invest its deposits in loans or other assets, that we implicitly assume to be bonds.
Besides it is compelled to hold a fraction of its deposits as reserves that can eventually provide a
return.3 Deposits are immediately invested in loans or assets by the bank, except the share that
is kept as reserves. At the end of the period loans are paid back and depositors are reimbursed.
With the beginning of the new period there is a new inflow of deposits and so on. Loans feedback
in deposits because firms invest the sums received, creating deposits in the system proportionally
to the amount of the loans. The circulation of money allows banks to provide payment services
with the same funds they have loaned, just keeping a fraction of deposits as reserves.
2.0.1 The budget constraint
The budget constraint is the following:
Lt + Ft + Rt = Dt + NWt. (1)
The bank can buy securities or lend as loans only the part of deposits that it does not keep as
reserve. Defining with q the legal reserve coefficient, so that Rt = qDt , the equation becomes:
Lt + Ft =(1- q)Dt + NWt. (2)
The value of Ft represents the amount of assets that are invested on assets, such as bonds. The
value of Lt represents the amount of loans issued by the bank. NWt is the net worth of the bank,
and we assume that it remains constant over time: N Wt+1 = NWt = NW. The bank cannot get
access to the capital markets to increase its capital. Since we assume the existence of a monopolistic
framework, profits are not pushed down to the normal rate by competition, but we assume that
there is a one hundred per cent dividend payout, so that all profits are distributed to shareholders
in every period.
2.0.2 Cost functions
The analysis of the problem of the banking firm in its most general form is impossible without
specifying a simplified cost structure that allows an analytical treatment. A solution often adopted
is to specify the cost structure in terms of the different components of the portfolio, such as bonds
loans and deposits. And a further simplification used is to assume a cost function that is separable
in the arguments. Formally:
C(K, L) = C(D(K, L)) + C(L(K, L)) + C(B(K, L)).
Using this formulation the existence of a separate production function for each class of assets and
for deposits is implicitly assumed. The last simplification is not a big problem as long as the
eventual economies of scope between assets and liabilities or among assets are not crucial for the
problem studied. The empirical evidence regarding the relevance of the economies of scope among
3 The model could be structured in order to allow the possibility for the bank to issue other liabilities, and the
results would not change radically.