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:
k (t) ≥ 0 and given k (0) = ko > 0.
The objective function is familiar and represents the discounted sum of the utility of the representative household. The constraint (5.20) is also straightforward to understand; total output per capita produced with capital-labor ratio k (t), f (k (t)), together with a fraction 1 — 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, (5.20). 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.
This maximization problem can be solved in a number of different ways, for example, by setting up an infinite dimensional Lagrangian. But the most convenient and common way of approaching it is by using dynamic programming.
It is also useful to note that even if we wished to bypass the Second Welfare Theorem and directly solve for competitive equilibria, we would have to solve a problem similar to the maximization of (5.19) subject to (5.20). In particular, to characterize the equilibrium, we would need to start with the maximizing behavior of households. Since the economy admits a
representative household, we only need to look at the maximization problem of this consumer. Assuming that the representative household has one unit of labor supplied inelastically, this problem can be written as:
subject to some given a (0) and
(5.21)
where a (t) denotes the assets of the representative household at time t, r (t) is the rate of return on assets and w (t) is the equilibrium wage rate (and thus the wage earnings of the representative household). The constraint, (5.21) is the flow budget constraint, meaning that it links tomorrow’s assets to today’s assets. Here we need an additional condition so that this flow budget constraint eventually converges (i.e., so that a (t) should not go to negative infinity). This can be ensured by imposing a lifetime budget constraint. Since a flow budget constraint in the form of (5.21) is both more intuitive and often more convenient to work with, we will not work with the lifetime budget constraint, but augment the flow budget constraint with another condition to rule out the level of wealth going to negative infinity. This condition will be introduced below.
5.10.