Introduction to Modern Economic Growth all individuals Instead, individuals born at different times will have accumulated different amounts of assets and will consume different amounts Let us denote the consumption at date t of a household born at date τ ≤ t by c (t | τ ) An allocation must now specify the entire sequence {c (t | τ )}∞ t=0,τ ≤t Using this notation and the life insurance contracts introduced by (9.33), the flow budget constraint of an individual of generation τ can be written as: ¶ µ ν a (t | τ ) − c (t | τ ) + w (t) (9.35) a (t + | τ ) = + r (t + 1) + 1−ν A competitive equilibrium in this economy can then be defined as follows: Definition 9.3 A competitive equilibrium consists of paths of capital stock, wage rates and rental rates of capital, {K (t) , w (t) , R (t)}∞ t=0 , and paths of consumption for each generation, {c (t | τ )}∞ t=0,τ ≤t , such that each individual maximizes utility and the time path of factor prices, {w (t) , R (t)}∞ t=0 , is such that given the time path of capital stock and labor {K (t) , L (t)}∞ t=0 , all markets clear In addition to the competitive factor prices, the key equation is the consumer Euler equation for an individual of generation τ at time t Taking into account that the gross rate of return on savings is + r (t + 1) + ν/ (1 − ν) and that the effective discount factor of the individual is β (1 − ν), this Euler equation can be written as (9.36) u0 (c (t | τ )) = β [(1 + r (t + 1)) (1 − ν) + ν] u0 (c (t + | τ )) This equation looks similar to be standard consumption Euler equation, for example as in Chapter It only differs from the equation there because it applies separately to each generation τ and because the term ν, the probability of death facing each individual, features in this equation Note, however, that when both r and ν are small (1 + r) (1 − ν) + ν ≈ + r, and the terms involving ν disappear In fact, the reason why these terms are present is because of the discrete time nature of the current model In the next section, we will analyze the continuous time version of the perpetual youth model, where the approximation in the previous equation is exact Moreover, the continuous time model will allow us to obtain closed-form solutions for aggregate consumption and 444