Introduction to Modern Economic Growth thus ν a (t) 1−ν The other important element of the model is the evolution of the demographics (9.33) z (a (t)) = Since each agent faces a probability of death equal to ν at every date, there is a natural force towards decreasing population We assume, however, that there are also new agents who are born at every date Differently from the basic neoclassical growth model, we assume that these new agents are not born into a dynasty; instead, they become separate households themselves We assume that when the population at time t is L (t), there are nL (t) new households born Consequently, the evolution of total population is given by (9.34) L (t + 1) = (1 + n − ν) L (t) , with the boundary condition L (0) = 1, where we assume that n > ν, so that there is positive population growth Throughout this section, we ignore technological progress Perpetual youth, together with exponential population growth, leads to a simple pattern of demographics in this economy In particular, it is easy to verify that at some point in time t > 0, there will be n (1 + n − ν)t−1 one-year-olds, n (1 + n − ν)t−2 (1 − ν) two-year-olds, n (1 + n − ν)t−3 (1 − ν)2 three-year-olds, etc (See Exercise 9.21) The production side of the economy is standard and it is represented by an aggregate production function satisfying Assumptions and 2, F (K (t) , L (t)) Suppose that capital depreciates at the rate δ Factor markets are competitive and factor prices are determined in the usual fashion The rental return of capital at time t is again given by R (t) = f (k (t)), so that the net return on saving is r (t + 1) = f (k (t)) − δ, and the wage rate is w (t) = f (k (t)) − k (t) f (k (t)) An allocation in this economy is similar to an allocation in the neoclassical growth model and involves time paths for the aggregate capital stock, wage rates and rental rates of capital, {K (t) , w (t) , R (t)}∞ t=0 However, it is no longer sufficient to specify aggregate consumption, since the level of consumption is not the same for 443