Abstract

This note describes methods for solving deterministic and stochas-tic versions of the discrete-time Ramsey model of economic growth. We derive an iterative procedure for solving the Euler equation and apply it to an example adapted from Pan (2007). The deterministic Ramsey model Consider the following discrete-time intertemporal optimization problem max {kt} t=0 βtu(ct) subject to ct = f(kt) − kt+1 k0 = given, where 0 < β < 1. The problem is to find a sequence of accumulated stock {kt} that will result in a consumption flow {ct = f(kt)−kt+1} that maximizes the discounted utility sum. We will not discuss here conditions under which this particular problem has a finite solution. Instead we will simply assume that a unique solution exists for every non-negative value of initial capital k0. The above problem is an example of the deterministic Ramsey model. It is stationary and it is not difficult to show that solutions satisfy kt+1 = ϕ(f(kt)) for some ‘investment function ’ ϕ(w), where ϕ(w) does not depend on k0. Hence the problem may be rephrased as max ϕ(.) t=0 βtu(ct) subject to ct = f(kt) − kt+1 kt+1 = ϕ(f(kt)) k0 = given For ease of notation we will write w = f(k).1 It is well-known2 that the optimal investment path {kt+1} satisfies the following intertemporal condi-tion for t = 0, 1, 2,... (the Euler equation): u′(wt − kt+1) = βu′(wt+1 − kt+2)f ′(kt+1) (1) 1In a typical application w would be the sum of current income (y, say) and the remaining value of the capital stock ((1 − δ)k): w = y + (1 − δ)k. 2E.g., see Stokey and Lucas (1989).