Monopolistic Pricing in the Banking Industry: a Dynamic Model

The interest rate on loans

The interest rate on loans can easily be obtained substituting the solution (44) for the quantity of
loans in the demand condition:

^Lt+j = a - brL + drB₊ j + ^ηt+j^,

(88)

For simplicity we use the solution of (26), but the result is general.

1 1 [(1 - i)⁽γγ^β - ^h^{) + (1} - q)^κH∖^rB+ j+■⁺^κ(^rR+ jq - ^rD+ j)

^L^t+j+1 = ^τγ^βL^t+j⁺ γγβ ^nw γγβα

⁽^βγY - ¹) [^z - a ] - ^κuγγ βα

¹ - b )⁽ γγ^β -¹)^{+ (1} - q )^κ

-■------^j------------------------COV

γγ βα

+ ' COV ( L^d ,¹ ),(89)

βγγ α α^β

^rL+ j = â{ ^a⁺^drB+ j ⁺^η^t+j--Lt++j--N^W⁺

⁺^j b I ⁺ j γγ β γγ β

[(¹ - d )⁽γγ^β - ^h)^{+ (1} - q) ^κH] ^rB+ j+ι⁺^κ (^rR+ j q- ^rD+ j ) ⁽^βγγ -¹) [^z - a] - ^κu

γγβα γγβα

(¹ - τ)⁽γγ^β-¹)^{+ (1} -q)^κ

^b----^j------------------COV

γγ βα

(r^B,¹ ) - ^γ^xβ~¹ COV(L^d,¹ )};
∖ α/ βγγ α α^β J

(90)

focusing just on the relationship with the rate on bonds, we obtain:

L -ɪ^{_d,в ₊ K¹ -^β⁽γγ^β^-^h⁾⁺⁽¹ -’⁾^κβrB+ j+■ ₊_ct⁺j = b∖^dr^t+j⁺ γ-βα ⁺^g

(91)

where

^g = a + ^η^t+^j - Lt+^j - NW +

^κ(^rR+ j q - ^rD+ j)
γγ βα

⁽^βγγ - ¹) [^z - a ] - ^κuγγ βα

(¹ - τ)⁽ γγ^β - ¹)^{+ (1} - q )^κ

^b----^j------------------C^V

γγ βα

(r^B,¹ ) - ^γ^γβ-¹ COV(L^d,¹ )};
∖ α/ βγγ α α/ J

(92)

It is useful to study in first instance the result when the bank has no market power and the rate on
loans is taken as exogenous. Considering the bank as representative of the sector and aggregating
we would obtain:

	_l [-⁽ γγ ^β - ^h ) ^rL+ j+■^{+ (1} - q ) ^κH∖^rB+ j+■ y, ^r^t+j = γγφ_v ⁺^g Γ	(93)
and	_rL ₊ γγ^β - H _rL = ⁽¹ - ^q) ^κrB+ j ₊ G ^t+j γγ βv ^{t + j}^{+ 1} γγ βv	(94)
Finally,	rL = ¹ - ^γγ^βv rL . ⁽¹ - ^q)^κ rB + vG r^{t+j +1} = ₇γ β r^t+j⁺ ₇γ β r^t+j⁺ ,	(95)
where	, 1 1 κ(rR₊_jq - rD₊_j - u) ^g = - γγβ^L^t+j - γγβ ^nw⁺ γγβv ⁺ + (¹-q)^κCOV(r^B,¹ ) - ^γ^γβ^-¹ COV(L^d,¹ )}; γγ βv V V/ βγγ V V^β>	(96)

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