Another look at yield spreads: the role of liquidity

Southern Economic Journal, April, 2008 by Dong Heon Kim

The bank's problem is to maximize the value function [V.sub.j,t](.) subject to the CIA constraint (Eqn. 3) at the first session of period t and the value function [U.sub.j,t](.) subject to the balance-sheet constraint (Eqn. 1) and required reserve constraint (Eqn. 2) at the second session of period t. In other words, bank j chooses federal funds to maximize the value function [U.sub.j,t](.) subject to the bank's balance-sheet constraint and the required reserve constraint at the end of period t. The first-order conditions are as follows:

[r.sup.F.sub.t] [beta][E.sub.t][partial derivative][V.sub.j,t 1]/[partial derivative][F.sub.j,t] = [beta][E.sub.t][partial derivative][V.sub.j,t 1]/[partial derivative][R.sub.j,t] [[lambda].sub.j,t], (4)

[R.sub.j,t][greater than or equal to] [theta] [[bar.D].sub.j,t], with equality if [[lambda].sub.j,t] > 0, (5)

where [[lambda].sub.j,t] is the Lagrange multiplier of the required reserve constraint and represents the implicit price of this constraint. Note that the balance-sheet constraint can be substituted for [R.sub.j,t] in the value function and the required reserve constraint, so we do not have a Lagrange multiplier for the balance-sheet constraint.

At the beginning of period t, bank j starts with reserve balances given by [R.sub.j,t-1] [L.sub.j,t-n]. Given the loan rate and the federal funds rate, bank j must choose its loan supply, [L.sub.j,t], before knowing the deposits, [[bar.D].sub.j,t]. The state variables for the decision at the beginning of period t are [L.sub.j,t- 1], [L.sub.j,t-2], ..., [L.sub.j,t-n], [F.sub.j,t-1], [R.sub.j,t-1], and [[bar.D].sub.j,t-1]. The value function of the beginning of period at time t is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

subject to

[L.sub.j,t] [less than or equal to] [R.sub.j,t-1] [L.sub.j,t-n].

Maximizing the value function [V.sub.j,t](.) subject to the CIA constraint at the beginning of period t, I have the following first-order conditions:

[r.sup.L.sub.n,t] [partial derivative][U.sub.j,t]/[partial derivative][L.sub.j,t] - ([[delta].sub.0] [delta][L.sub.j,t]) = [[eta].sub.j,t], (6)

[L.sub.j,t-n] [R.sub.j,t-1][greater than or equal to][L.sub.j,t], with equality if [[eta].sub.j,t] > 0, (7)

where [[eta].sub.j,t] is the Lagrange multiplier of the CIA constraint and represents the shadow price of this constraint. Using the envelope conditions from the value functions at the beginning of period t and at the end of period t, I have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

since [beta][E.sub.t][[eta].sub.j,t 1] [[lambda].sub.j,t] = [r.sup.F.sub.t], ..., [[beta].sup.n-1][E.sub.t][[eta].sub.j,t n-1] [[beta].sup.n-2][E.sub.t] [[lambda].sub.j,t n-2] = [[beta].sup.n-2][E.sub.t][r.sup.F.sub.t n-2] and [[beta].sup.n-1][E.sub.t][[lambda].sub.j,t n-1] = [[beta].sup.n-1][E.sub.t] [r.sup.F.sub.t n-1] - [[beta].sup.n][E.sub.t][[eta].sub.j,t n]. Using the above equations and the first-order conditions, it is straightforward to characterize the equilibrium as satisfying the following equations: (12)

[r.sup.F.sub.t] = [beta][E.sub.t][[eta].sub.j,t 1] [[lambda].sub.j,t], (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Equation 9 can be rewritten as follows: