Introduction to Modern Economic Growth for renting one unit of capital at time t in terms of date t + goods Notice that the gross rate of return on assets is defined as + r, since r often refers to the net interest rate In fact in the continuous time model, this is exactly what the term r will correspond to This notation should therefore minimize confusion In addition, to capital income, households in this economy will receive wage income for supplying their labor at the market wage of w (t) = f (k (t))−k (t) f (k (t)) Now consider the maximization problem of the representative household: max {c(t),a(t)}∞ t=0 ∞ X β t u (c (t)) t=0 subject to the flow budget constraint (6.40) a (t + 1) = (1 + r (t + 1)) a (t) − c (t) + w (t) , where a (t) denotes asset holdings at time t and as before, w (t) is the wage income of the individual (since labor supply is normalized to 1) The timing underlying the flow budget constraint (6.40) is that the individual rents his capital or asset holdings, a (t), to firms to be used as capital at time t + Out of the proceeds, he consumes and whatever is left, together with his wage earnings, w (t), make up his asset holdings at the next date, a (t + 1) In addition to this flow budget constraint, we have to impose a no Ponzi game constraint to ensure that the individual asset holdings not to go to minus infinity Since this constraint will be discussed in detail in Chapter 8, we not introduce it here (though note that without this constraint, there are other, superfluous solutions to the consumer maximization problem) For now, it suffices to look at the Euler equation for the consumer maximization problem: (6.41) u0 (c (t)) = (1 + r (t + 1)) βu0 (c (t + 1)) Imposing steady state implies that c (t) = c (t + 1) Therefore, in steady state we must have (1 + r (t + 1)) β = 298