The Lifetime Utility Function PDF

Preview

Summary

Macroeconomics...

Description

The Lifetime Utility Function Which consumption bundle on the intertemporal budget constraint the household will choose depends on its preferences. We will assume that households like to consume in periods 1 and 2, that is, they enjoy both C1 and C2. Further, we will assume that their preferences for current and future consumption can be described by the intertemporal utility function U(C1, C2), (3.5) where the function U is strictly increasing in both arguments and concave. An example of a utility function that satisfies these assumptions is U(C1, C2) = lnC1 + lnC2, Figure 3.2 displays the household’s indifference curves. All consumption baskets on a given indifference curve provide the same level of utility. Because consumption in both periods are goods, that is, items for which more is preferred to less, as one moves northeast in figure 3.2, utility increases. Note that the indifference curves drawn in figure 3.2 are convex toward the origin, so that at low levels of C1 relative to C2 the indifference curves are steeper than at relatively high levels of C1. Intuitively, the convexity of the indifference curves means that at low levels of consumption in period 1 relative to consumption in period 2, the household is willing to give up relatively many units of period-2 consumption for an additional unit of period-1 consumption. On the other hand, if period-1 consumption is high relative to period-2 consumption, then the household will not be willing to sacrifice much period-2 consumption for an additional unit of period-1 consumption. For example, for the utility function U(C1, C2) = lnC1 + lnC2, the indifference curve corresponding to a utility of 3 is implicitly given by lnC1+ lnC2 = 3. Solving for C2, we obtain C2 = 20.08 C1

. This expression is in line with the properties of indifference curves given above, because it states that C2 is a decreasing and convex function of C1. The negative of the slope of an indifference curve is known as the marginal rate of substitution of C2 for C1. Therefore, the assumption of convexity means that along a given indifference curve, the marginal rate of substitution decreases with C1....