Credit Markets and the Propagation of Monetary Policy Shocks

by the accumulated level of assets (a) and the productivity signal (z). The oc-
cupational choice is made by comparing the expected present discounted utility
obtained from being a worker with that of being an entrepreneur subject to the
financial constraint (2). To guarantee the existence of a stationary recursive equi-
librium with a positive fraction of the population in each occupation we need to
impose the next two assumptions:

Assumption 1. The upper bound for the signal ability shock, z, is such that
there exists an asset level a^s for which

I ^vW (^a,z)^ψ (d^z ) ≤ I ^vE (^α,z0⁾ Q ^(z,dz⁰⁾

for all a ≥ a^s .

Assumption 2. The lower bound for the signal ability shock, z, is such that
ʃ vW (a, z'0) ψ (dz') ≥ v vE (a, z') Q (z, dz'⁰)

for all a ∈ A.

The first assumption requires a sufficiently high productivity shock so that agents
with accumulated assets above a threshold value a^s become entrepreneurs. This
is because the expected value of creating a firm is larger than the expected value
of being a worker. The second assumption imposes the opposite condition. If
an agent receives the lowest signal, he will always choose to become a worker
independently of his accumulated asset level or prior occupation.

The properties of the value functions v(a, z), v^W (a, z), and v^E(a, z) depend
upon the assumptions on the transition functions ψ and Q and the utility function
that follow the analysis in Stokey, Lucas and Prescott [22]. All value functions,
vⁱ(a, z⁰) for i = E, W, are increasing in both arguments since the utility function
is strictly increasing and strictly concave and the agent’s constraint set is strictly
increasing in assets and the ability shock. Furthermore, the expected value for
workers is independent of the productivity z and is an increasing and continuous
function of a. On the other hand, the expected value function of entrepreneurs
is an increasing and continuous function of both a and z . This is due to the
assumption on Q being monotone and satisfying the Feller property. Finally, the
value function v(a, z) is non-decreasing in z and strictly increasing in a.⁶ The
properties of the value function together with assumptions 1 and 2 ensure that for

⁶ The value function v(a, z) is the outer envelope for the value functions at
each shock level and may not be a concave function even if the value functions
of workers and entrepreneurs are. The operator on the value function satisfies
Blackwell’s sufficient conditions for a contraction mapping.

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