Economic growth and economic development 301

Introduction to Modern Economic Growth the future exponentially with the constant discount factor β ∈ (0, 1) He also faces a certain (nonnegative) labor income stream of {w (t)}∞ t=0 , and moreover starts life with a given amount of assets a (0) He receives a constant net rate of interest r > on his asset holdings (so that the growth rate of return is + r) Therefore, the utility maximization problem of the individual can be written as max {c(t),a(t)}∞ t=0 ∞ X u (c (t)) t=0 subject to the flow budget constraint a (t + 1) = (1 + r) [a (t) + w (t) − c (t)] , with a (0) > given In addition, we impose the assumption that the individual cannot have a negative asset holdings, so a (t) ≥ for all t.1 A couple of comments are useful at this point First the budget constraint could have been written alternatively as a (t + 1) = Ra (t) + w (t) − c (t) The difference between these two alternative budget constraints involes the timing of interest payments The first one presumes that the individual starts the period with assets a (t), then receives his labor income, w (t), and then consumes c (t) Whatever is left is saved for the next date and earns the gross interest rate (1 + r) In this formulation, a (t) refers to asset holdings at the beginning of time t The alternative formulation instead interprets a (t) as asset holdings at the end of time t The choice between these two formulations has no bearing on the results One other important comment about the flow budget constraint is that it does not capture all of the constraints that individual is subject to In particular, an individual can satisfy the flow budget constraint, but run his assets position to −∞ In general to prevent this, we need to impose an additional restriction to make sure that the asset position of the individual does not become “too negative” at infinity However, here we not need this additional restriction, since we have already imposed that a (t) ≥ for all t 1Note that in this example, the choice set is not necessarily compact But this creates no difficulty in applying the tools developed so far 287