The Intertemporal Budget Constraint PDF

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The Intertemporal Budget Constraint Combining the period budget constraints (3.1) and (3.2) with the transversality condition (3.3) to eliminate B_1 and B_2, gives rise to the intertemporal budget constraint of the household C1 + C2 1 + r1 = (1 + r0)B_0 + Q1 + Q2 1 + r1 . (3.4) The intertemporal budget constraint requires that the present discounted value of consumption (the left-hand side) be equal to the initial stock of wealth plus the present discounted value of the endowment stream (the right-hand side). The household chooses consumption in periods 1 and 2, C1 and C2, taking as given all other variables appearing in (3.4), namely, r0, r1, B_0 , Q1, and Q2. Figure 3.1 displays the pairs (C1, C2) that satisfy the household’s intertemporal budget constraint (3.4). For simplicity, we assume for the remainder of this section that the household’s initial asset position is zero, that is, we assume that B_0 = 0. Then, clearly, the basket C1 = Q1 and C2 = Q2 (point A in the figure) is feasible in the sense that it satisfies the intertemporal budget constraint (3.4). In words, it is feasible for the household to consume its endowment in each period. But the household’s choices are not limited to this particular basket. In period 1 the household can consume more or less than its endowment Q1 by borrowing or saving the amount C1 −Q1. If the household wants to increase consumption in one period, it must sacrifice some consumption in the other period. In particular, for each additional unit of consumption in period 1, the household has to give up 1 + r1 units of consumption in period 2. This means that the is downward sloping, with a slope of −(1 + r1). Note that points on the budget constraint located southeast of point A correspond to borrowing (or dissaving) in period 1. Letting S1 denote savings in period 1, we have that S1 = r0B_0 +Q1−C1 = Q1−C1 (recall that we are assuming that B_0 = 0). So, all points on the intertemporal budget constraint located southeast of point A imply that S1 < 0. In turn, the fact that S1 < 0 implies, by the relation S1 = B_1−B_0 , that the household’s asset position at the end of period 1, B_1 , is negative. This implies that a point on the budget constraint located southeast of the endowment point A is also associated with positive saving in period 2 because S2 = B_2−B_1 = −B_1 > 0 (recall that B_2 = 0 by the transversality condition). Similarly, points on the budget constraint located northwest of A are associated with positive saving in period 1 and dissaving in period 2. If the household chooses to allocate its entire lifetime income to consumption in period 1, then C1 would equal Q1 + Q2/(1+ r1) and C2 would be nil. In figure 3.1, this basket corresponds to the intersection of the intertemporal budget constraint with the horizontal axis. At the opposite extreme, if the household chooses to allocate all its lifetime income to consumption in period 2, then C2 would equal (1+r1)Q1+Q2 and C1 would be nil. This basket is located at the intersection of the intertemporal budget constraint with the vertical axis....