Consider an economy populated by a set of agents distributed uniformly along the unit interval with a total population mass of 1. That is, for $i ∈ [0, 1]$ the probability distribution function is $f(i) = 1$ and $f(i) = 0$ elsewere; the CDF on the $[0, 1]$ interval is therefore $F (i) = i$ , implying an aggregate population mass of $F (1) = 1$ .

Agent $i$ ’s value of variable $∙$ at date $t$ is $∙t,i$ . Thus aggregate consumption is

Since the aggregate population is normalized to 1, capital letters refer not only to aggregate variables but also to per capita variables, since per-capita consumption is aggregate consumption divided by aggregate population:

Each individual agent is inﬁnitesimally small, and can therefore neglect the eﬀects of its own actions on aggregates.

1 Blanchard Lives

For many purposes the assumption that economic agents live forever is useful; but for other purposes it is necessary to be able to analyze agents with ﬁnite horizons. Blanchard (1985) introduced a tractable framework that permits analysis of many of the key issues posed by ﬁnite lifetimes.

The key assumption is that the probability of death is independent of the agent’s age. (This is similar to the Calvo (1983) assumption that the probability that a ﬁrm will change its prices is independent of the time elapsed since the last price change).

The most convenient formulation of the model is one in which the number of dying individuals is always equal to the number of newborn individuals, so that the population remains constant.

1.1 Discrete Time

As above, suppose that the population alive at time $t$ is arranged on the unit interval. The probability of death is $d$ (and the probability of not dying is $//D = 1 − d$ ). Then for a person living at any location $i ∈ [0, 1]$ , expected remaining lifetime including the current period will be

If a new cohort of size $D$ has been born each period since the beginning of time, the total population will be given by the size of a new cohort $D$ multiplied by the expected lifetime $− 1 D$ :

1.2 Continuous Time

Blanchard’s original treatment was in continuous time, with a constant rate of death $d$ , so that the probability of remaining alive (not dead) after $t$ periods for a consumer born in period 0 is¹

and if the ﬂow arrival rate of new population is $d$ (that is, at each instant a ﬂow of new population arrives at rate $d$ ) then again the population mass is constant at

1.3 Population Growth

Now suppose that the population in the discrete-time model is growing by a factor $Ξ = (1 + ξ)$ from period to period; if the number of newborns in period 0 was 1, then the number of newborns in period $t$ is given by

In this framework we want to keep track of the relative population of each cohort compared to the size of the newborn cohort. At age $z$ , the cohort that was born in period 0 will be of relative size

so that if in period 0 the population was of size $(1 − //D ∕Ξ )$ then the sizes of the relative populations will add up to one even as the absolute population grows by the factor $Ξ$ .

References

Blanchard, Olivier J. (1985): “Debt, Deﬁcits, and Finite Horizons,” Journal of Political Economy, 93(2), 223–247.

Calvo, Guillermo A. (1983): “Staggered Contracts in a Utility-Maximizing Framework,” Journal of Monetary Economics, 12(3), 383–98.