Lumpy Investment, Sectoral Propagation, and Business Cycles

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)(λj^κt - 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 k_j∙,_t(1 — δ_j∙)/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 w_t 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∣,t^zI,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:

∞

∑ β^tU(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

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