Credit Markets and the Propagation of Monetary Policy Shocks

(a, z ) can be summarized by the value function

v^W (a, z⁰) ψ (dz⁰) , max v^E (a, z⁰) Q (z, dz⁰) ,

where

vⁱ (a, z⁰)=max[u (c)+βv (a⁰, z⁰)]

c,a⁰

with i = W, E indexing the type of agent, either worker (W) or entrepreneur (E).

The worker’s budget constraint is
or

c + a⁰ ≤ (1 - δ) a + R^Da + wz⁰,

where w is the real wage. For the entrepreneur, the budget constraint is

c + a⁰ ≤ (1 - δ) a + π^E,

with real profits equal to

R^D max [0,Pa- Pk]  R^L max [0,Pk- Pa]

πE = z⁰f (k, n) - wn +--------P--------J--LP--------J

or

π^E = z⁰f (k, n) - wn + R^D max [0,a- k] - R^L max [0,k- a] .

Notice agents make their occupational and production decisions after observing the
market interest rates R^D and R^L but before their productivities (z⁰) are realized
for the period. When deciding on the size of the firm, the entrepreneur has to
satisfy the financial constraint

wn + RL max [0, k — a] < (1 — δ) a + zf (k, n) + RD max [0, a — k]     (2)

by which it is ensured that the production costs are covered even for the lowest
productivity level. Thus, an agent with an initially low level of assets (a) but with
the prospects of having good skills (a high expected z⁰) may not be able to reach
the optimal size of the firm because of the possibility of drawing a low realization
of his productivity and realizing negative profits he will not be able to cover with
his accumulated assets.

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