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Introduction to Modern Economic Growth price of the consumption good in each date will be normalized to and the interest rates will directly give the intertemporal relative prices This is the justification for focusing on interest rates as the key relative prices in macroeconomic (economic growth) models 5.8 Optimal Growth in Discrete Time Motivated by the discussion in the previous section let us start with an economy characterized by an aggregate production function, and a representative household The optimal growth problem in discrete time with no uncertainty, no population growth and no technological progress can be written as follows: (5.18) max ∞ {c(t),k(t)}t=0 ∞ X β t u (c (t)) t=0 subject to (5.19) k (t + 1) = f (k (t)) + (1 − δ) k (t) − c (t) , k (t) ≥ and given k (0) > The objective function is familiar and represents the discounted sum of the utility of the representative household The constraint (5.19) is also straightforward to understand; total output per capita produced with capital-labor ratio k (t), f (k (t)), together with a fraction − δ of the capital that is undepreciated make up the total resources of the economy at date t Out of this resources c (t) is spent as consumption per capita c (t) and the rest becomes next period’s capital-labor ratio, k (t + 1) The optimal growth problem imposes that the social planner chooses an entire sequence of consumption levels and capital stocks, only subject to the resource constraint, (5.19) There are no additional equilibrium constraints We have also specified that the initial level of capital stock is k (0), but this gives a single initial condition We will see later that, in contrast to the basic Solow model, the solution to this problem will correspond to two, not one, differential equations We will therefore need another boundary condition, but this will not take the form of an initial condition Instead, this additional boundary condition will come from the optimality of a dynamic plan in the form of a transversality condition 245