Credit Markets and the Propagation of Monetary Policy Shocks

of government liabilities, X . Such normalized variables will be denoted by lower
case letters. So, define

_h-H._b-B. -P
X ;^b x ;^p X,

with h + b =1. At a steady state, the central bank keeps h and b fixed forever.
This means, all liabilities of the central bank, together with all nominal variables,
grow at the same rate equal to

H 0 + B⁰   _{1 ι} RB B

^μ = H + B = 1 ⁺ H + B

Notice that the central bank does not pursue policies where its liabilities grow
exogenously. The growth rate of these liabilities, μ, and therefore the inflation
rate, is linked to the interest rate and the number h. There is also a stationary
probability measure λ that determines the density of agents with each combination
of productivities z and capital a. So, this measure should satisfy

with S = {(a, z⁰): a⁰(a, z⁰) ∈ A⁰ and z⁰ ∈ Z⁰} and ∆ being the transition selector
defined above.

With these elements, we can define the steady state of this economy.

Definition 1 A stationary recursive competitive equilibrium is constant prices
(RD, RL, R^b, w, p), a constant growth rate for nominal prices μ, value functions
[v (a, z), v^W (a, z), v^E (a, z)], policy functions [k (a, z), n (a, z), c (a, z⁰), a⁰ (a, z⁰)],
a probability measure (λ) and a transition selector [∆ (z, dz⁰)] such that given the
policy of the central bank (h, b, ρ),

1. at given prices the policy functions solve the optimization problem of each
agent (a, z),

2. the probability measure λ is time invariant,

3. the aggregate asset level is constant

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