Summarizing, each period agents first make decisions based on their individual
state (a, z) and the prices of the economy. These decisions are whether to become
a worker or an entrepreneur and the amount of labor, n (a, z), and capital, k (a, z),
to hire. Then, after productivity for the period (z0) is revealed, they decide how
much to consume, c (a, z0), and how many assets to transfer to the next period,
a0 (a, z0). At the aggregate level, the equilibrium outcome of these decisions is a
probability measure λ that determines the density of agents with each combination
of productivities z and capital a. This measure evolves as
where S = {(a, z0): a0(a, z0) ∈ A0 and z0 ∈ Z0} and ∆ is a transition selector
λ0
(A0,Z0)=
S
∆ (z, dz0) λ(da × dz),
∆ (z, dz0) ≡ Q(z,dz‰ + ψ(dz0)∣w
that determines the end of period productivities from the beginning of period
productivities. This transition selector is endogenous because how productivities
change within the period is different between workers and entrepreneurs, and this
occupational choice is a decision of the agents. So, from the law of motion of λ
the measure of agents with next period’s state in the set (A0, Z0) are those whose
skills evolve to the set Z0 and whose optimal decision is to accumulate assets that
belong to the set A0 .
The problem of the banking sector is to maintain the nominal portfolio of the
economy. The balance sheet of a representative bank is
D = L + Hd + Bd
where D are deposits (sources of funds for the bank), L denotes the loans to
entrepreneurs, Hd is the demand for reserves at the central bank, and Bd represents
certificates of deposits and other forms of lending to the public sector. Reserves
earn no interest. Banks have to hold them because there is a requirement that
links the amount of deposits they can provide with the amount of reserves they
5
own5
Hd = ρD.
5 The reserve requirement can be interpreted in literal terms for countries where
this regulation is in place or as a reserve demand for settlement accounts for
payment systems in countries without such a rule.
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