Lumpy Investment, Sectoral Propagation, and Business Cycles

The representative household maximizes the utility function subject to the sequence
of budget constraints:

NN

∑ PjtzCt = wtht + ∑((πj,t + qj∙,t)vj,t^- qj,tvj,t+ι).                 (8)

j=1                   j=1

where vj,t is the stock holding for firm j and qj,t is its price.

The cost minimization of the consumer given the level of consumption C_t implies
zC_t = (Pj,t∕Pt)^-ξct/N. Similarly, the derived demand for good j by the monopolist
l given the level of investment i_l,_t is obtained as z∣,_l,t = (Pj,_t∕p_t)^-ξil,_t∕N. With the
equilibrium condition for good j, y_j∙,_t = zC_t + ∑N=₁ z∣,ι,_t, these yield the demand function
for good j as: y_j∙,_t = (Pj,_t∕p_t)^-ξ(c_t + it)/N where it ≡ ∑jN=₁ ij,t. Define a production
index yt ≡ N ¹^^1-ξ)(∑j=₁ yj^ξ_t^-1)^^ξ∑(^ξ-1). Then we have relations ∑j=₁ Pj,tyj,t = Ptyt,
∑j=1 Pj,tzCt = Ptct, and ∑NN=ιPj,tzI,_lt = Ptil,t. Combining with the consumer’s budget
constraint (8) and the equilibrium condition for labor, ht = ∑_j∙ hj,t, we obtain the
demand function:

yj,t = (Pj,√pt)^-ξ yt^/N                                 (9)

The monopolist maximizes its discounted future profits as instructed by the rep-
resentative household. The discount rate, r_t^-1 , is the intertemporal ratio of marginal
utility of consumption. Then the monopolist’s problem is defined as follows.

∞∞   N

^max I   ∑(rι ∙ ∙ ∙ rt)^-1 At^πj,t = Ao ∑(rι ∙ ∙ ∙ ^rt)^-1gt(Pj,tyj,t^-wt^hj,t^-∑Pι,tzIj,t)

{yj,t,kj,t+1,hj,t,ij,t,z_l,j,t} t=0                                        t=0                                                   l=1

(10)
subject to the production function (1,3), the capital accumulation (2), the discreteness
of investment rate (4), and the demand function (9).

Let us define the aggregate capital index k_t as follows.

By using the optimality condition for hj_,t , the profit at t is reduced to a function of
(kj,t, kj,t+1) as:

πj,t = Dow(^{α- 1}V^αk1∕(^ξα+1-α)kα(^ξ- ≡^α+1^-α) - gkj,t+ι + (1 - δj)kj,t      (12)

where D₀ ≡ (1 — (1 — 1∕ξ)(1 — α))((1 — 1∕ξ)(1 — a))(^1-a)/a. The discounted sum of
the profit sequence is concave in kj_,t . Thus the optimal policy is characterized by an

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