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 Ct implies
zCt = (Pj,t∕Pt)-ξct/N. Similarly, the derived demand for good j by the monopolist
l given the level of investment il,t is obtained as z∣,l,t = (Pj,t∕pt)-ξil,t∕N. With the
equilibrium condition for good j, yj∙,t = zCt + ∑N=1 z∣,ι,t, these yield the demand function
for good j as: yj∙,t = (Pj,t∕pt)-ξ(ct + it)/N where it ≡ ∑jN=1 ij,t. Define a production
index yt ≡ N 1^1-ξ)(∑j=1 yjξt-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, rt-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 Atπj,t = Ao ∑(rι ∙ ∙ ∙ rt)-1gt(Pj,tyj,t-wthj,t-∑Pι,tzIj,t)
{yj,t,kj,t+1,hj,t,ij,t,zl,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 kt as follows.
kt ≡ ( k
j =1
α(ξ- 1)Λξα+1-α)
j,t
/N ) (ξα+1-a)/(a(€—1))
(11)
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- 1Vαk1∕(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