Default costs shrink the size of the forward looking part of the equation, reducing the portfolio
of bonds, and the same effect is a function of the elasticity of the demand for loans. The lower
the elasticity the smaller the portfolio. This result is due to the fact that both default costs and
inelastic demand for loans reduce the optimal quantity of loans and, as a consequence, the size of
the whole portfolio.
2.1.3 Deposits
The solution for deposits is given by:
∞ ∞ -Ixi Z
Dt+ j +1 = YYβ Dt+j + TY- Σ ( YY ) Et+' [ α>' ] - βγγ α (1 - Y ) C +
-Σ (YY)'Et+'[Xt + j+'+1], (42)
Under the assumption that interest rates follow a random walk process, we can for simplicity
assume that deterministic component of the rates remain constant and treat the values of the rates
at time t and t + 1 that enter in the solution as constants. In this case the result can be simplified
as:
1 κ[(1 - d)(Yyβ- H) + (1 - q)κH]+ (Yyβ-1)αg4H B
Dt+ j+1 = YYβ Dt+j + βγγ α (γγ - 1) rt + j+1 +
+ κ R _ κ2 + α(YYβ - 1)g3 D + κ{(βγY - 1) z - a] - κ^ +
βγγ α ( yy - 1 f'^ t++j q βγγ α ( γγ - 1) t++j βγγ α ( γγ - 1)
+ Л—K—n [(1 - d)(YYβ -1) + (1 - q)κ] COV (rB,1 ) - jYYβ -1) κ COV (LD,1 ). (43)
βYY α(YY - 1) b α βYY (YY - 1) α
It can be observed that the intertemporal equilibrium value of deposits depends negatively on the
costs of deposits as it should be the case. The variation of the quantity of deposits between two
periods is a negative function of the industrial cost of deposits. It is much more surprising that
interest rate on deposits has a negative effect. Solving the equation for the rate on deposits, it can
easily be realised that interest rates on deposits at time t increase with the level of deposits at time
t and vice-versa, as we would expect. The negative relationship is with the level of deposits at time
t + 1. This result can be understood reading the equation in the opposite way. The current level
of deposits has a direct relationship with both the rate on deposits and the level of next period.
As a consequence when the rate of interest is high because the demand for deposits is elastic, the
higher rates imply a lower level for next period. The other negative component is small since it is
of second order and reflects the reduction in profits due to the interest cost. The results regarding
the dependence on the rate on deposits, anyway, are not very robust, since they depend on the very
strong assumption that their deterministic component is expected by the banker not to change in
the future.
The quantity of deposits grows with the rate on bonds. This result is counterintuitive, but it
can easily be explained, as the effect of the feedback. It is useful to underline what would the
result be if the market for loans were to be competitive, rather than monopolistic. The structure
of the model would be the same, the only difference would be that in the final result we would have
-rL+1 instead of (1 - d)rB+ι, and of course the intercept term a would disappear and the term in
default cost would be v rather than α. The final result would show a negative relationship with the
rate on loans, the impact of which depends on the value of the feedback coefficient. In fact lower
rates would allow a greater issuance of loans generating more liquidity and vice-versa. Since higher
rates on bonds imply lower rates on loans in the monopolistic model, the first part of the result is
explained. The second component, κ2 (1 - q)rtB is of a smaller order, and is due to the fact that
higher rates imply higher returns of the part of the portfolio invested in bonds. This component
affects the level of deposits with a one period lag, and is small. The final term αg4rtB is due to the
standard demand effect, and measures the reduction in the demand of deposits services due to the
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