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Introduction to Modern Economic Growth thus it is optimal for each household i > to consume one unit of the good cii+1 , which is within its budget set and gives as high utility as any other allocation within his budget set, establishing that x ˜ is a competitive equilibrium Ô 9.2 The Baseline Overlapping Generations Model We now discuss the baseline two-period overlapping generation economy 9.2.1 Demographics, Preferences and Technology In this economy, time is discrete and runs to infinity Each individual lives for two periods For example, all individuals born at time t live for dates t and t + For now let us assume a general (separable) utility function for individuals born at date t, of the form (9.1) U (t) = u (c1 (t)) + βu (c2 (t + 1)) , where u : R+ → R satisfies the conditions in Assumption 3, c1 (t) denotes the consumption of the individual born at time t when young (which is at date t), and c2 (t + 1) is the individual’s consumption when old (at date t + 1) Also β ∈ (0, 1) is the discount factor Factor markets are competitive Individuals can only work in the first period of their lives and supply one unit of labor inelastically, earning the equilibrium wage rate w (t) Let us also assume that there is exponential population growth, so that total population is (9.2) L (t) = (1 + n)t L (0) The production side of the economy is the same as before, characterized by a set of competitive firms, and is represented by a standard constant returns to scale aggregate production function, satisfying Assumptions and 2; Y (t) = F (K (t) , L (t)) To simplify the analysis let us assume that δ = 1, so that capital fully depreciates after use (see Exercise 9.3) Thus, again defining k ≡ K/L, the (gross) rate of return to saving, which equals the rental rate of capital, is given by (9.3) + r(t) = R (t) = f (k (t)) , 421