Economic growth and economic development 521

Introduction to Modern Economic Growth Throughout this chapter, we assume that the economy admits an infinitely-lived representative household, with household size growing at the exponential rate n The preferences of the representative household at time t = are given by # " Z ∞ c (t)1−θ − dt exp (− (ρ − n) t) (11.1) U= 1−θ Labor is supplied inelastically The flow budget constraint facing the household can be written as (11.2) a˙ (t) = (r (t) − n)a (t) + w (t) − c (t) , where a (t) denotes assets per capita at time t, r (t) is the interest rate, w (t) is the wage rate per capita, and n is the growth rate of population As usual, we also need to impose the no-Ponzi game constraint: ẵ Z t áắ [r(s) − n] ds ≥ (11.3) lim a(t) exp − t→∞ The Euler equation for the representative household is the same as before and implies the following rate of consumption growth per capita: c˙ (t) = (r (t) − ρ) c (t) θ (11.4) The other necessary condition for optimality of the consumer’s plans is the transversality condition, (11.5) ẵ Z t áắ [r(s) n] ds = lim a(t) exp − t→∞ As before, the problem of the consumer is concave, thus any solution to these necessary conditions is in fact an optimal plan The production sector is similar to before, except that Assumptions and are not satisfied More specifically, we adopt the following aggregate production function: Y (t) = AK (t) , with A > Notice that this production function does not depend on labor, thus wage earnings, w (t), in (11.2) will be equal to zero This is one of the unattractive features of the baseline AK model, but will be relaxed below (and it is also relaxed in Exercises 11.3 and 11.4) Dividing both sides of this equation by L (t), and as 507