where δj is an industry specific depreciation rate. Investment ij,t is a composite good
produced by combining all the goods symmetrically as:
N
ij,t = N 1'<1-ξ>(∑⅛1)(1-ξvξ )' (1 ' (3)
l=1
where ξ > 1 is the elasticity of substitution between inputs in the production of invest-
ment good.
We assume that the investment rate is chosen from a discrete set. Specifically, we
assume that:
ij,t/kj,t ∈ {(1 - δj)(λjκt - 1)}κt=0,±1,±2,... (4)
where λj > 1. Note that the choice space for kj,t is indepent of the path: kj,t ∈
{((1 — δ)∕g)tkj,0λKt}κt=0,±1,±2,...∙ The assumption implies that the next period capital
kj,t+1 has to be either the naturally depreciated level kj∙,t(1 — δj∙)/g or its multiplication
or division of λj . By this assumption, the producer is forced to invest in a lumpy
manner. Thus this constraint is a shortcut for the lumpy behavior which typically
occurs when a fixed cost incurs in investment. This is the only modification from
the usual model of monopolistic economies. The main objective of this paper is to
examine the aggregate consequence of a non-linear behavior of producers induced by
the discreteness constraint.
Let pj,t denote the price of good j at t. Define a price index pt ≡
(∑jN=1 pj1-ξ/N)1^1-ξ> and normalize it to one. Let wt denote a real wage for an ef-
ficiency unit of labor. Then the monopolist’s profit (normalized by At) at t is written
as:
N
πj,t ≡ Pj,tyj,t - wthj,t - ∑P∣,tzI,j,t (5)
l=1
The demand function for good j is derived by usual procedure as in Dixit and
Stiglitz (1977). Let us suppose that the representative household has a preference over
the sequence of consumption and labor:
∞
∑ βtU(Ct, ht) (6)
t=0
where Ct = Atct is a composite consumption good produced identically as the invest-
ment good:
N
ct = N W-ξ>(∑(zg)<1-ξ>'ξ )ξ∕<1-ξ>. (7)
l=1