This document presents a model of the over-the-counter federal funds market. It uses a search-and-bargaining framework to model how banks with reserve shortages and surpluses are matched to reallocate balances. The model shows the fed funds rate is determined by banks' discount rates and the distribution of reserve balances held by banks. It also shows under certain conditions, the market structure achieves efficient reallocation of funds among banks.

1. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
The OTC Theory of the Fed Funds Market:
A Primer
Gara Afonso Ricardo Lagos
FRB of New York New York University

2. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
The market for federal funds
A market for loans of reserve balances at the Fed.

3. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
The market for federal funds
What’s traded?
Unsecured loans (mostly overnight)
How are they traded?
Over the counter
Who trades?
Commercial banks, securities dealers, agencies and branches of
foreign banks in the U.S., thrift institutions, federal agencies

4. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
The market for federal funds

5. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Why is the fed funds market interesting?
It is an interesting example of an OTC market
(Unusually good data is available)
Reallocates reserves among banks
(Banks use it to o¤set liquidity shocks and manage reserves)
Determines the interest rate on the shortest maturity
instrument in the term structure
Is the “epicenter” of monetary policy implementation

6. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Why is the fed funds market interesting?
It is an interesting example of an OTC market
(Unusually good data is available)
Reallocates reserves among banks
(Banks use it to o¤set liquidity shocks and manage reserves)
Determines the interest rate on the shortest maturity
instrument in the term structure
Is the “epicenter” of monetary policy implementation

7. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Why is the fed funds market interesting?
It is an interesting example of an OTC market
(Unusually good data is available)
Reallocates reserves among banks
(Banks use it to o¤set liquidity shocks and manage reserves)
Determines the interest rate on the shortest maturity
instrument in the term structure
Is the “epicenter” of monetary policy implementation

8. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Why is the fed funds market interesting?
It is an interesting example of an OTC market
(Unusually good data is available)
Reallocates reserves among banks
(Banks use it to o¤set liquidity shocks and manage reserves)
Determines the interest rate on the shortest maturity
instrument in the term structure
Is the “epicenter” of monetary policy implementation

9. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
In this paper we ...
(1) Propose an OTC model of trade in the fed funds market
(2) Use the theory to address some elementary questions, e.g.,
Positive: How is the fed funds rate determined?
Normative: Is the OTC market structure able to achieve an
e¢ cient reallocation of funds?

10. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
The model
A trading session in continuous time, t 2 [0, T], τ T t
Unit measure of banks hold reserve balances k (τ) 2 f0, 1, 2g
fnk (τ)g : distribution of balances at time T τ
Linear payo¤s from balances, discount at rate r
Uk : payo¤ from holding balance k at the end of the session
Trade opportunities are bilateral and random (Poisson rate α)
Loan and repayment amounts determined by Nash bargaining
Assume all loans repaid at time T + ∆, where ∆ 2 R+

11. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Institutional features of the fed funds market
Model Fed funds market
Search and bargaining

12. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Institutional features of the fed funds market
Model Fed funds market
Search and bargaining Over-the-counter market

13. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Institutional features of the fed funds market
Model Fed funds market
Search and bargaining Over-the-counter market
[0, T]

14. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Institutional features of the fed funds market
Model Fed funds market
Search and bargaining Over-the-counter market
[0, T] 4:00pm-6:30pm

15. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Institutional features of the fed funds market
Model Fed funds market
Search and bargaining Over-the-counter market
[0, T] 4:00pm-6:30pm
fnk (T)g

16. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Institutional features of the fed funds market
Model Fed funds market
Search and bargaining Over-the-counter market
[0, T] 4:00pm-6:30pm
fnk (T)g
Distribution of reserve
balances at 4:00pm

17. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Institutional features of the fed funds market
Model Fed funds market
Search and bargaining Over-the-counter market
[0, T] 4:00pm-6:30pm
fnk (T)g
Distribution of reserve
balances at 4:00pm
fUk g

18. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Institutional features of the fed funds market
Model Fed funds market
Search and bargaining Over-the-counter market
[0, T] 4:00pm-6:30pm
fnk (T)g
Distribution of reserve
balances at 4:00pm
fUk g Reserve requirements,
interest on reserves...

19. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Bellman equations
rV1 (τ) + ˙V1 (τ) = 0
rV0 (τ) + ˙V0 (τ) = αn2 (τ) max fV1 (τ) V0 (τ) ¯R (τ) , 0g
rV2 (τ) + ˙V2 (τ) = αn0 (τ) max fV1 (τ) V2 (τ) + ¯R (τ) , 0g
Vk (0) = Uk

20. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Bellman equations
rV1 (τ) + ˙V1 (τ) = 0
rV0 (τ) + ˙V0 (τ) = αn2 (τ) max fV1 (τ) V0 (τ) ¯R (τ) , 0g
rV2 (τ) + ˙V2 (τ) = αn0 (τ) max fV1 (τ) V2 (τ) + ¯R (τ) , 0g
Vk (0) = Uk
¯R (τ) = arg max
R
[V1 (τ) R V0 (τ)]θ
[V1 (τ) + R V2 (τ)]1 θ
= θ [V2 (τ) V1 (τ)] + (1 θ) [V1 (τ) V0 (τ)]
¯R (τ) e r(τ+∆)
R (τ) (PDV of repayment)

21. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Bellman equations
rV1 (τ) + ˙V1 (τ) = 0
rV0 (τ) + ˙V0 (τ) = αn2 (τ) φ (τ) θS (τ)
rV2 (τ) + ˙V2 (τ) = αn0 (τ) φ (τ) (1 θ) S (τ)
where:
S (τ) 2V1 (τ) V0 (τ) V2 (τ)
φ (τ) =
1 if 0 < S (τ)
0 if S (τ) 0

22. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Time-path for the distribution of balances
˙n0 (τ) = αφ (τ) n2 (τ) n0 (τ)
˙n1 (τ) = 2αφ (τ) n2 (τ) n0 (τ)
˙n2 (τ) = αφ (τ) n2 (τ) n0 (τ)

23. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
De…nition
An equilibrium is a value function, V, a path for the distribution of
reserve balances, n (τ), and a path for the distribution of trading
probabilities, φ (τ), such that:
(a) given the value function and the distribution of trading
probabilities, the distribution of balances evolves according to the
law of motion; and
(b) given the path for the distribution of balances, the value
function and the distribution of trading probabilities satisfy
individual optimization given the bargaining protocol.

24. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Analysis
˙S (τ) = δ (τ) S (τ)
δ (τ) fr + αφ (τ) [θn2 (τ) + (1 θ) n0 (τ)]g
)
S (τ) = e
¯δ(τ)
S (0)
¯δ (τ)
Z τ
0
δ (x) dx
S (τ) 2V1 (τ) V0 (τ) V2 (τ)

25. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Analysis
˙S (τ) = δ (τ) S (τ)
δ (τ) fr + αφ (τ) [θn2 (τ) + (1 θ) n0 (τ)]g
)
S (τ) = e
¯δ(τ)
S (0)
¯δ (τ)
Z τ
0
δ (x) dx
S (τ) 2V1 (τ) V0 (τ) V2 (τ)

26. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Assumption: S (0) 2U1 U2 U0 > 0
)
S (τ) = e
¯δ(τ)
S (0) > 0 for all τ
)
φ (τ) = 1 for all τ

27. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Assumption: S (0) 2U1 U2 U0 > 0
)
S (τ) = e
¯δ(τ)
S (0) > 0 for all τ
)
φ (τ) = 1 for all τ

28. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Assumption: S (0) 2U1 U2 U0 > 0
)
S (τ) = e
¯δ(τ)
S (0) > 0 for all τ
)
φ (τ) = 1 for all τ

29. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Proposition
Suppose 2U1 U2 U0 > 0. Then the unique equilibrium is:
n2 (τ) = n0 (τ) + n2 (T) n0 (T)
n1 (τ) = 1 2n0 (τ) + n0 (T) n2 (T)
n0 (τ) = [n2(T ) n0(T )]n0(T )
eα[n2(T ) n0(T )](T τ)n2(T ) n0(T )
S (τ) =
[n2(T ) e α[n2(T ) n0(T )](T τ)n0(T )]e fr+αθ[n2(T ) n0(T )]gτ
n2(T ) e α[n2(T ) n0(T )]T n0(T )
S (0)
R (τ) = er∆
fβ (τ) (U2 U1) + [1 β (τ)] (U1 U0)g
β (τ)
θ[n2(T ) e α[n2(T ) n0(T )](T τ)n0(T )]e αθ[n2(T ) n0(T )]τ
n2(T ) e α[n2(T ) n0(T )]T n0(T )
+
[1 e αθ[n2(T ) n0(T )]τ
]n2(T )
n2(T ) e α[n2(T ) n0(T )]T n0(T )

30. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Proposition
Suppose 2U1 U2 U0 > 0. Then the unique equilibrium is:
n2 (τ) = n0 (τ) + n2 (T) n0 (T)
n1 (τ) = 1 2n0 (τ) + n0 (T) n2 (T)
n0 (τ) = [n2(T ) n0(T )]n0(T )
eα[n2(T ) n0(T )](T τ)n2(T ) n0(T )
S (τ) =
[n2(T ) e α[n2(T ) n0(T )](T τ)n0(T )]e fr+αθ[n2(T ) n0(T )]gτ
n2(T ) e α[n2(T ) n0(T )]T n0(T )
S (0)
R (τ) = er∆
fβ (τ) (U2 U1) + [1 β (τ)] (U1 U0)g
β (τ)
θ[n2(T ) e α[n2(T ) n0(T )](T τ)n0(T )]e αθ[n2(T ) n0(T )]τ
n2(T ) e α[n2(T ) n0(T )]T n0(T )
+
[1 e αθ[n2(T ) n0(T )]τ
]n2(T )
n2(T ) e α[n2(T ) n0(T )]T n0(T )

31. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Proposition
Suppose 2U1 U2 U0 > 0. Then the unique equilibrium is:
n2 (τ) = n0 (τ) + n2 (T) n0 (T)
n1 (τ) = 1 2n0 (τ) + n0 (T) n2 (T)
n0 (τ) = [n2(T ) n0(T )]n0(T )
eα[n2(T ) n0(T )](T τ)n2(T ) n0(T )
S (τ) =
[n2(T ) e α[n2(T ) n0(T )](T τ)n0(T )]e fr+αθ[n2(T ) n0(T )]gτ
n2(T ) e α[n2(T ) n0(T )]T n0(T )
S (0)
R (τ) = er∆
fβ (τ) (U2 U1) + [1 β (τ)] (U1 U0)g
β (τ)
θ[n2(T ) e α[n2(T ) n0(T )](T τ)n0(T )]e αθ[n2(T ) n0(T )]τ
n2(T ) e α[n2(T ) n0(T )]T n0(T )
+
[1 e αθ[n2(T ) n0(T )]τ
]n2(T )
n2(T ) e α[n2(T ) n0(T )]T n0(T )

32. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Proposition
Suppose 2U1 U2 U0 > 0. Then the unique equilibrium is:
n2 (τ) = n0 (τ) + n2 (T) n0 (T)
n1 (τ) = 1 2n0 (τ) + n0 (T) n2 (T)
n0 (τ) = [n2(T ) n0(T )]n0(T )
eα[n2(T ) n0(T )](T τ)n2(T ) n0(T )
S (τ) =
[n2(T ) e α[n2(T ) n0(T )](T τ)n0(T )]e fr+αθ[n2(T ) n0(T )]gτ
n2(T ) e α[n2(T ) n0(T )]T n0(T )
S (0)
R (τ) = er∆
fβ (τ) (U2 U1) + [1 β (τ)] (U1 U0)g
β (τ)
θ[n2(T ) e α[n2(T ) n0(T )](T τ)n0(T )]e αθ[n2(T ) n0(T )]τ
n2(T ) e α[n2(T ) n0(T )]T n0(T )
+
[1 e αθ[n2(T ) n0(T )]τ
]n2(T )
n2(T ) e α[n2(T ) n0(T )]T n0(T )

33. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Proposition
Suppose 2U1 U2 U0 > 0. Then, the equilibrium supports an
e¢ cient allocation of reserve balances.

34. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intuition for e¢ ciency result
rV0 (τ) + ˙V0 (τ) = αn2 (τ) θS (τ)
rλ0 (τ) + ˙λ0 (τ) = αn2 (τ) S (τ)
rV1 (τ) + ˙V1 (τ) = 0
rλ1 (τ) + ˙λ1 (τ) = 0
rV2 (τ) + ˙V2 (τ) = αn0 (τ) (1 θ) S (τ)
rλ2 (τ) + ˙λ2 (τ) = αn0 (τ) S (τ)
S (τ) = e
¯δ(τ)
S (0)
S (τ) = e
¯δ (τ)
S (0)

35. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intuition for e¢ ciency result
rV0 (τ) + ˙V0 (τ) = αn2 (τ) θS (τ)
rλ0 (τ) + ˙λ0 (τ) = αn2 (τ) S (τ)
rV1 (τ) + ˙V1 (τ) = 0
rλ1 (τ) + ˙λ1 (τ) = 0
rV2 (τ) + ˙V2 (τ) = αn0 (τ) (1 θ) S (τ)
rλ2 (τ) + ˙λ2 (τ) = αn0 (τ) S (τ)
S (τ) = e
¯δ(τ)
S (0)
S (τ) = e
¯δ (τ)
S (0)
¯δ (τ) ¯δ (τ) = α
Z τ
0
[(1 θ) n2 (z) + θn0 (z)] dz 0

36. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intuition for e¢ ciency result
rV0 (τ) + ˙V0 (τ) = αn2 (τ) θS (τ)
rλ0 (τ) + ˙λ0 (τ) = αn2 (τ) S (τ)
rV1 (τ) + ˙V1 (τ) = 0
rλ1 (τ) + ˙λ1 (τ) = 0
rV2 (τ) + ˙V2 (τ) = αn0 (τ) (1 θ) S (τ)
rλ2 (τ) + ˙λ2 (τ) = αn0 (τ) S (τ)
S (τ) = e
¯δ(τ)
S (0)
S (τ) = e
¯δ (τ)
S (0)
¯δ (τ) ¯δ (τ) = α
Z τ
0
[(1 θ) n2 (z) + θn0 (z)] dz 0

37. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intuition for e¢ ciency result
rV0 (τ) + ˙V0 (τ) = αn2 (τ) θS (τ)
rλ0 (τ) + ˙λ0 (τ) = αn2 (τ) S (τ)
rV1 (τ) + ˙V1 (τ) = 0
rλ1 (τ) + ˙λ1 (τ) = 0
rV2 (τ) + ˙V2 (τ) = αn0 (τ) (1 θ) S (τ)
rλ2 (τ) + ˙λ2 (τ) = αn0 (τ) S (τ)
S (τ) = e
¯δ(τ)
S (0)
S (τ) = e
¯δ (τ)
S (0)
¯δ (τ) ¯δ (τ) = α
Z τ
0
[(1 θ) n2 (z) + θn0 (z)] dz 0

38. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intuition for e¢ ciency result
Equilibrium:
Gain from trade as perceived by borrower: θS (τ)
Gain from trade as perceived by lender: (1 θ) S (τ)
Planner:
Each of their marginal contributions equals S (τ)
δ (τ) δ (τ) for all τ 2 [0, T], with “=” only for τ = 0
) The planner “discounts”more heavily than the equilibrium
) S (τ) < S (τ) for all τ 2 (0, 1]
) Social value of loan < joint private value of loan

39. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intuition for e¢ ciency result
Equilibrium:
Gain from trade as perceived by borrower: θS (τ)
Gain from trade as perceived by lender: (1 θ) S (τ)
Planner:
Each of their marginal contributions equals S (τ)
δ (τ) δ (τ) for all τ 2 [0, T], with “=” only for τ = 0
) The planner “discounts”more heavily than the equilibrium
) S (τ) < S (τ) for all τ 2 (0, 1]
) Social value of loan < joint private value of loan

40. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intuition for e¢ ciency result
Equilibrium:
Gain from trade as perceived by borrower: θS (τ)
Gain from trade as perceived by lender: (1 θ) S (τ)
Planner:
Each of their marginal contributions equals S (τ)
δ (τ) δ (τ) for all τ 2 [0, T], with “=” only for τ = 0
) The planner “discounts”more heavily than the equilibrium
) S (τ) < S (τ) for all τ 2 (0, 1]
) Social value of loan < joint private value of loan

41. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intuition for e¢ ciency result
Equilibrium:
Gain from trade as perceived by borrower: θS (τ)
Gain from trade as perceived by lender: (1 θ) S (τ)
Planner:
Each of their marginal contributions equals S (τ)
δ (τ) δ (τ) for all τ 2 [0, T], with “=” only for τ = 0
) The planner “discounts”more heavily than the equilibrium
) S (τ) < S (τ) for all τ 2 (0, 1]
) Social value of loan < joint private value of loan

42. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intuition for e¢ ciency result
Equilibrium:
Gain from trade as perceived by borrower: θS (τ)
Gain from trade as perceived by lender: (1 θ) S (τ)
Planner:
Each of their marginal contributions equals S (τ)
δ (τ) δ (τ) for all τ 2 [0, T], with “=” only for τ = 0
) The planner “discounts”more heavily than the equilibrium
) S (τ) < S (τ) for all τ 2 (0, 1]
) Social value of loan < joint private value of loan

43. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intuition for e¢ ciency result
Equilibrium:
Gain from trade as perceived by borrower: θS (τ)
Gain from trade as perceived by lender: (1 θ) S (τ)
Planner:
Each of their marginal contributions equals S (τ)
δ (τ) δ (τ) for all τ 2 [0, T], with “=” only for τ = 0
) The planner “discounts”more heavily than the equilibrium
) S (τ) < S (τ) for all τ 2 (0, 1]
) Social value of loan < joint private value of loan

44. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intuition for e¢ ciency result
Planner internalizes that searching borrowers and lenders
make it easier for other lenders and borrowers to …nd partners
These “liquidity provision services” to others receive no
compensation in the equilibrium, so individual agents ignore
them when calculating their equilibrium payo¤s
The equilibrium payo¤ to lenders may be too high or too low
relative to their shadow price in the planner’s problem:
E.g., too high if (1 θ) S (τ) > S (τ)

45. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intuition for e¢ ciency result
Planner internalizes that searching borrowers and lenders
make it easier for other lenders and borrowers to …nd partners
These “liquidity provision services” to others receive no
compensation in the equilibrium, so individual agents ignore
them when calculating their equilibrium payo¤s
The equilibrium payo¤ to lenders may be too high or too low
relative to their shadow price in the planner’s problem:
E.g., too high if (1 θ) S (τ) > S (τ)

46. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intuition for e¢ ciency result
Planner internalizes that searching borrowers and lenders
make it easier for other lenders and borrowers to …nd partners
These “liquidity provision services” to others receive no
compensation in the equilibrium, so individual agents ignore
them when calculating their equilibrium payo¤s
The equilibrium payo¤ to lenders may be too high or too low
relative to their shadow price in the planner’s problem:
E.g., too high if (1 θ) S (τ) > S (τ)

47. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Frictionless limit
Proposition
Let Q ∑2
k=0 knk (T) = 1 + n2 (T) n0 (T).
For τ 2 [0, T], as α ! ∞, ρ (τ) ! ρ∞, where
1 + ρ∞
=
8
<
:
er∆
(U1 U0) if Q < 1
er∆
[θ (U2 U1) + (1 θ) (U1 U0)] if Q = 1
er∆
(U2 U1) if 1 < Q.

48. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Positive implications
The theory delivers:
(1) Time-varying trade volume
(2) Time-varying fed fund rate
(3) Intraday convergence of distribution of reserve balances

49. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Trade volume
Flow volume of trade at time T τ:
υ (τ) = αn0 (τ) n2 (τ)
Total volume traded during the trading session:
¯υ =
Z T
0
υ (τ) dτ

50. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Fed funds rate
The gross rate on a loan at time T τ is:
1 + ρ (τ) = er∆
fβ (τ) (U2 U1) + [1 β (τ)] (U1 U0)g
with
β (τ) 2 [0, 1]
The average daily rate is:
¯ρ =
1
T
Z T
0
ρ (τ) dτ

51. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Cross-sectional distribution of reserve balances
Let µ (τ) and σ2
(τ) denote the mean and variance of the
cross-sectional distribution of reserve balances at t = T τ
µ (τ) = 1 + n2 (T) n0 (T) Q
σ2
(τ) = σ2
(T) 2 [2 + n2 (T) n0 (T)] [n0 (T) n0 (τ)]

52. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
U1 = e r∆
(1 + ir
f )
U2 = e r∆
(2 + ir
f + ie
f )
U0 = e r∆
(iw
f ir
f + Pw
)

53. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
U1 = e r∆
(1 + ir
f )
U2 = e r∆
(2 + ir
f + ie
f )
U0 = e r∆
(iw
f ir
f + Pw
)
Proposition
ρ (τ) = β (τ) ie
f + [1 β (τ)] (iw
f + Pw
) where
1 If n2 (T) = n0 (T), β (τ) = θ
2 If n2 (T) < n0 (T), β (τ) 2 [0, θ], β (0) = θ and β0
(τ) < 0
3 If n0 (T) < n2 (T), β (τ) 2 [θ, 1], β (0) = θ and β0
(τ) > 0.

54. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intraday interest rate in a balanced market

55. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intraday interest rate in a market with shortage of reserves

56. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intraday interest rate in a market with excess reserves

57. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Interest rate and the quantity of reserves

58. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
¯k = 1
Two scenarios
nH
0 (T) , nL
2 (T) nL
0 (T) , nH
2 (T)
f0.6, 0.3g f0.3, 0.6g
Experiments
Bargaining Power (θ) Discount Rate (iw
f ) Contact Rate (α)
0.1 0.5 0.9 .0050
360
.0075
360
.0100
360 25 50 100

59. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Bargaining power
16:00 16:30 17:00 17:30 18:00 18:30
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x 10
- 5
Surplus
Eastern Time
θ=0.1
θ=0.5
θ=0.9
16:00 16:30 17:00 17:30 18:00 18:30
1.000006
1.000008
1.00001
1.000012
1.000014
1.000016
1.000018
1.00002
V
2
-V
1
Eastern Time
θ=0.1
θ=0.5
θ=0.9
16:00 16:30 17:00 17:30 18:00 18:30
0.3
0.4
0.5
0.6
0.7
0.8
ρ(%)
Eastern Time
θ=0.1
θ=0.5
θ=0.9
16:00 16:30 17:00 17:30 18:00 18:30
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x 10
- 5
Surplus
Eastern Time
θ=0.1
θ=0.5
θ=0.9
16:00 16:30 17:00 17:30 18:00 18:30
1.000007
1.0000075
1.000008
1.0000085
1.000009
V
2
-V
1
Eastern Time
θ=0.1
θ=0.5
θ=0.9
16:00 16:30 17:00 17:30 18:00 18:30
0.3
0.4
0.5
0.6
0.7
0.8
ρ(%)
Eastern Time
θ=0.1
θ=0.5
θ=0.9

60. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Contact rate
16:00 16:30 17:00 17:30 18:00 18:30
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x 10
- 5
Surplus
Eastern Time
α=25
α=50
α=100
16:00 16:30 17:00 17:30 18:00 18:30
1.000006
1.000008
1.00001
1.000012
1.000014
1.000016
1.000018
1.00002
V
2
-V
1
Eastern Time
α=25
α=50
α=100
16:00 16:30 17:00 17:30 18:00 18:30
0.54
0.56
0.58
0.6
0.62
0.64
0.66
0.68
0.7
ρ(%)
Eastern Time
α=25
α=50
α=100
16:00 16:30 17:00 17:30 18:00 18:30
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x 10
- 5
Surplus
Eastern Time
α=25
α=50
α=100
16:00 16:30 17:00 17:30 18:00 18:30
1.000007
1.0000075
1.000008
1.0000085
1.000009
V
2
-V
1
Eastern Time
α=25
α=50
α=100
16:00 16:30 17:00 17:30 18:00 18:30
0.35
0.4
0.45
0.5
0.55
ρ(%)
Eastern Time
α=25
α=50
α=100

61. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Discount-Window lending rate
16:00 16:30 17:00 17:30 18:00 18:30
0
0.5
1
1.5
2
2.5
x 10
- 5
Surplus
Eastern Time
i
f
w
=0.5%
i
f
w
=0.75%
i
f
w
=1%
16:00 16:30 17:00 17:30 18:00 18:30
1.000006
1.000008
1.00001
1.000012
1.000014
1.000016
1.000018
1.00002
V
2
-V
1
Eastern Time
i
f
w
=0.5%
i
f
w
=0.75%
i
f
w
=1%
16:00 16:30 17:00 17:30 18:00 18:30
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
ρ(%)
Eastern Time
i
f
w
=0.5%
i
f
w
=0.75%
i
f
w
=1%
16:00 16:30 17:00 17:30 18:00 18:30
0
0.5
1
1.5
2
2.5
x 10
- 5
Surplus
Eastern Time
i
f
w
=0.5%
i
f
w
=0.75%
i
f
w
=1%
16:00 16:30 17:00 17:30 18:00 18:30
1.000007
1.0000075
1.000008
1.0000085
1.000009
1.0000095
1.00001
1.0000105
V
2
-V
1
Eastern Time
i
f
w
=0.5%
i
f
w
=0.75%
i
f
w
=1%
16:00 16:30 17:00 17:30 18:00 18:30
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
ρ(%)
Eastern Time
i
f
w
=0.5%
i
f
w
=0.75%
i
f
w
=1%

62. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
More to be done...
Fed funds brokers
Banks’portfolio decisions
Random “payment shocks”
Sequence of trading sessions
Ex-ante heterogeneity (αi , θi
, Ui
k )
Generalized inventories
Quantitative work

63. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
The views expressed here are not necessarily re‡ective of
views at the Federal Reserve Bank of New York
or the Federal Reserve System.

64. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Evidence of OTC frictions in the fed funds market
Price dispersion
Intermediation
Intraday evolution of the distribution of reserve balances
There are banks that are “very long” and buy
There are banks that are “very short” and sell

65. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Price dispersion
-.15
-.1
-.05
0
.05
.1
Percent
16:00 16:30 17:00 17:30 18:00 18:30
10th/90th Percentiles 25th/75th Percentiles
Median Mean
Intraday Distribution of Fed Funds Spreads, 2005

66. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intermediation: excess funds reallocation
0
50
100
150
BillionsofDollars
2005 2006 2007 2008 2009 2010
Excess Funds Reallocation

67. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intermediation: proportion of intermediated funds
0
.1
.2
.3
.4
.5
.6
2005 2006 2007 2008 2009 2010
Proportion of Intermediated Funds

68. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Intraday evolution of the distribution of reserve balances
-.5
0
.5
1
16:00 16:30 17:00 17:30 18:00 18:30
10th/90th Percentiles 25th/75th Percentiles
Median Mean
Normalized Balances, 2007

69. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Banks that are “long”...and buy...
0
1,000
2,000
3,000
MillionsofDollars
16:00 16:30 17:00 17:30 18:00 18:30
10th/90th Percentiles 25th/75th Percentiles
Median Mean
Purchases by Banks with Nonnegative Adjusted Balances, 2007

70. Intro Model Equilibrium E¢ ciency Frictionless Implications Policy Examples Conclusion Appx.
Banks that are “short”...and sell...
0
100
200
300
400
500
MillionsofDollars
16:00 16:30 17:00 17:30 18:00 18:30
10th/90th Percentiles 25th/75th Percentiles
Median Mean
Sales by Banks with Negative Adjusted Balances, 2007