firms. The expected completed duration T for a contract that has survived
for s periods is E [T ∣ s] = s + ω~1 for all s > 0 (the since the hazard rate ω
is constant).
The steady-state cross-section of contract ages can be described by the
proportions aS of firms surviving at least s periods:
asi = ω (1 — ω)i 1 : i = 1..∞
(24)
with mean s = ^1. In demographic terms, s is the age of the contract:
aS is the proportion of the population of age s; s is the average age of the
population.
The corresponding distribution of completed contract lengths i is given
by11:
ai = ωii (1 — ω)i 1
(25)
with mean T = ^fij-. In demographic terms, T gives the distribution of ages
at death (for example as reported by the registrar of deaths): ai being the
proportion of the steady state population who will live to die at age T.
Assuming that we are in steady state (which is implicit in the use of the
Calvo model), we can assume that there are in fact ex ante fixed contract
lengths. We can classify firms∕unions by the duration of their "contract".
The fact that the contract length of a firm∕union is fixed is perfectly compat-
ible with the notion of a reset probability if we assume that the firm does not
know the contract length. We can think of the firm-union having a proba-
bility distribution over contract lengths given by aS in (24): Nature chooses
the contract length, but the firm-unions do not know when they have to set
the price (when the contract begins).
Having redefined the Calvo economy in terms of completed contract
lengths, we can now describe it as a GTE. There are an infinite number
of sectors, each strictly positive integer corresponding to a contract length:
N = ∞ Ti = i,i = 1...∞
The proportions of each completed contract length are given by (25) .The
wage setting process in each sector is uniform: there are Tfi1 cohorts of equal
size.
Ni = T71∙V∙ 1
t
11See Dixon and Kara (2004), Proposition 1.
19