CAPACITY AND ASYMMETRIES IN MONETARY POLICY
2.1. Final Good Firms. At time t, a single final good, denoted by У, is produced
by a representative firm which sells it in a perfectly competitive market. Such
commodity can either be used for consumption or for investment. There is no fixed
input, which implies that the optimization program of these firms remain purely
static. The production activities are carried out by combining a continuum of
intermediate goods, indexed by j ∈ (0,1). The production technology is represented
by a constant return-to-scale CES function defined as follows
with e > 1 being the elasticity of substitution of inputs and where ly⅛ is the quantity
of input j used in production at date t. Here, Vj,t ≥ 0 is a productivity parameter
corresponding to input j. It is assumed to be drawn from a stochastic process i.i.d.
distributed across time and input firms,9 with a log normal distribution function
F(t>) that has unit mean and is defined over the support [y,v] with 0 < v < 1 <v.
The representative firm purchases inputs to intermediate good firms taking into
account that the supply of each input j is limited to an amount lj,t. Assuming
a uniform non-stochastic rationing scheme, the optimization program of the final
firm can be written as follows
(Pi)
max Ptyt - / Pj,tYj,tdj ,
{yt,yit} Jq
subject to
Yj,t ≤ γj,t Vj ∈ (0,1) ,
where P⅛ is the price of the final good which is taken as given by the firm. When
maximizing profits, the final firm faces no uncertainty: it knows the input prices
{Pj.t}, the supply constraints {P),t} and the productivity parameters {t⅛,t}. It is
important to notice that the inclusion of supply constraints in the problem above is
due to the particular structure of the model, where input producing firms set their
prices before the idiosyncratic shock is realized.
The solution to (P.l) determines the quantity that the final good firm is go-
ing to make for the goods produced by each intermediate firm. As the produc-
tion technology displays constant returns-to-scale, the competitive firm necessarily
makes zero profits at the prevailing prices and is willing to produce any output
level yt > 0. Moreover, under deterministic quantity constraints and a uniform
rationing scheme, effective demands are not well defined. Realized transactions can
be derived, however.10 The quantity of inputs used will be determined by the cor-
responding idiosyncratic productivity level of each intermediate firm as described
in the next result:
Lemma 1 (Realized Transactions). The optimal allocation of inputs across inter-
mediate good firms is given by the following system of equations
(2.2)
ytvj,t (⅛)
if υj,t ≤ ¾t ≤ ¾,t
otherwise
9In order to keep the model tractable, it is assumed that the idiosyncratic shock is not serially
correlated. Thus, its realization influences exclusively contemporary production and employment
decisions, but not investment decisions.
10For a detailed discussion on the theory of effective demands see Green (1980) or Svensson
(1980).