## Household behaviour intertemporal choices

Households have to decide not only how to allocate time between work and leisure, and split income between consumption and saving today (or this year). They also have to make a plan for these same decisions tomorrow (or next year). Of course, today's decisions cannot be made independently of tomorrow's. This is most obvious in the case of consumption: if I consume less (and save more) today, I will be able to consume more tomorrow. But it also applies to leisure time: by working more today I may be able to enjoy more leisure time tomorrow.

We now analyze the intertemporal decisions of households: how they plan to spread consumption over their two-period horizon, and how to allocate their

Maths note. The two-period budget constraint states that since today's income is either consumed or saved, Y1 = C1 + S^and consumption next period is limited by the sum of period-2 income and period-1 saving plus interest, C2 = Y2 + (1 + /^S1,the present value of consumption must equal the present value of incoiri e:

This is rewritten C2 = (1 + r)Y1 + Y2 - (1 + r)C1, which defines a straight line with slope (1 + r).The concept of present values makes payments comparable that accrue at different points in time. For example, when the interest rate is 4% annually, €100,000 becomes €104,000 in a year. Then we may say that at an interest rate of 4% the present value of €104,000 next year is €100,000. Formally: present value=€100,000 = €104,000/(1 + 0.04).

work time over the two periods. These two decisions are interdependent. To obtain an unobstructed view of each decision, however, we will pretend that each can be taken in isolation, while the other has already been taken.

The consumption constraint Intertemporal consumption choices can be analyzed in a diagram that measures current consumption on the horizontal axis and future consumption on the vertical axis. Here we assume that households have already decided on how much to work today and tomorrow. If the incomes that are thus being earned today and tomorrow were being consumed immediately, in the period in which they accrue, this would generate a specific point in the intertemporal consumption diagram, which we call the endowment point. In a long-run equilibrium current and (expected) future income are identical. Then the endowment point sits on the 45° line. When the economy has been thrown off its long-run equilibrium, current income will differ from future income, and endowment points off the 45° line result.

In Figure 16.8 the household has decided to work equally in both periods. Thus the endowment point A sits right on the 45° line. Does this point also indicate the household's consumption pattern? Only in a barter economy that produces nothing but perishable goods which cannot be stored and thus need to be consumed right away. Then all income would be consumed in the period in which it was earned. Once non-perishable goods are being produced, households have the option of consuming each unit of income either today or tomorrow. All their consumption options would now be lined up

ncome

Current consumption

ncome

Figure 16.8 When credit markets exist, households may move consumption into the future or up to the present. The endowment point A in panel (a) shows the consumption pattern that results when all income is consumed during the period in which it accrues. Then period consumption equals period income. By borrowing or lending, households may enjoy any consumption pattern that lies on a straight line passing through A and having slope -(1 + r). To consume the pattern indicated by B, households must save Y1 - C1 in period 1 and spend (1 + r)(Y1 - C1) = C2 - Y2 in period 2.

along a constraint that passes through the endowment point and has slope —1. To make things even more realistic, enter credit markets, where households can lend or borrow at the real interest rate r. This improves the household's options of transforming today's income into future consumption further. After forsaking 1 euro's worth of consumption in period 1 and putting it in the bank, the household can withdraw 1 euro plus interest payment next period. Period-2 consumption thus increases by (1 + r) euros. So the constraint along which households may select their intertemporal consumption pattern is a straight line which passes through the endowment point and has slope —(1 + r). By saving in period 1 they can achieve consumption patterns such as the one characterized by point B that are related to but can be quite different from the income pattern. Or they may borrow to raise period-1 consumption above period-1 income. The intertemporal budget constraint shifts when the endowment point moves to a point such as C, and it pivots around the endowment point when the interest rate changes (see panel (b)). The latter effect will play an important role later on.

The consumption preferences Let us now return to household preferences as formalized by equation (16.11). Preferences have an intertemporal component as well. Suppose for now that leisure in periods 1 and 2 is given, and let us characterize preferences by means of indifference curves in a C-Q diagram (Figure 16.9). These have the familiar shape. The slope is negative, which means that if you take away some of tomorrow's consumption, households need to be compensated with more current consumption in order to keep them at the same level of utility. Further, indifference curves are convex to the origin. That means they bend away from the point where nothing is being consumed. This is because marginal utility decreases. Therefore, if we start at point D (where period-2 consumption is high and not valued all that highly at the margin) and reduce period-2 consumption by one unit, we need

Figure 16.9 With decreasing marginal utility of consumption, intertemporal indifference curves are convex. Households maximize utility by selecting the consumption point that is associated with the highest possible indifference curve. This is where the linear intertemporal constraint just touches an indifference curve. At such a point the marginal rate of substitution (representing the slope of the indifference curve) equals 1 plus the interest rate r. In any long-run equilibrium, in which household consumption is constant, on the 45° line, the marginal rate of substitution equals 1 plus the time discount rate b. Since this must be equal to 1 plus the interest rate, such an equilibrium is characterized by r = b (the interest rate equals the time discount rate).

The marginal rate of substitution states at what ratio households are willing to exchange small amounts of current consumption for future consumption.

Maths note. To obtain the marginal rate of substitution (MRS), let work time be constant in equation (16.11). Taking the total differential and solving this for dC2 - (1 + m0"/0C1

yields a general equation for the MRS (i.e. the slope of the indifference curve). On the 45° line we have C1 - C2and, hence, du/dC1-du/dC2. Substituting this into the general equation yields dC1 - - (1 + b) as the MRS at the point of intersection of indifference curves with the 45° line.

smaller compensation in terms of additional period-1 consumption than if we had started from point A (where period-2 consumption is relatively low and one additional unit of it raises utility a lot).

The rate at which households are willing to give up future consumption in exchange for one more unit of current consumption is called the marginal rate of substitution (MRS). Graphically, if we think in infinitesimally small units, the marginal rate of substitution equals the absolute value of the slope of the indifference curve. It is being affected by two things: first, by the marginal utilities which period-1 and period-2 consumption levels generate in a given point; and second, by the household's rate of time preference b, which states how impatient they are, how much they prefer consumption today over consumption tomorrow. As we slide along an indifference curve, the MRS changes. There is one point on each indifference curve, however, where the MRS has a very specific value and a straightforward interpretation. This is where indifference curves intersect the 45° line. At any such point, today's and tomorrow's consumption are the same. So the marginal utilities of today's and tomorrow's consumption are also the same. Therefore, if there were no discounting of the future, households would be prepared to trade current for future consumption at a rate of one to one. The indifference curve would have a slope of -1. The MRS would equal 1. But since, according to the utility function, one unit of period-2 consumption is worth only 1/(1 + b) units of period-1 consumption, households require (1 + b) units of consumption in period 2 in exchange for one unit of consumption in period 1. This means that the slope of all indifference curves equals -(1 + b) at their respective points of intersection with the 45° line. Almost trivially, utility increases as we move onto indifference curves positioned further up and to the right, as they feature higher current and/or future consumption.

The intertemporal consumption optimum The household's optimal intertemporal consumption pattern can be identified after merging indifference curves (preferences) and the budget constraint (options) onto a single diagram (Figure 16.10, panel (a)). As before, let the economy be in a long-run equilibrium. This positions the endowment point A, the intertemporal income pattern, on the 45° line. Since households maximize utility, they want to reach the highest indifference curve available. With this determining their choice from the options offered by the budget constraint, they pick the point of tangency between the budget line and an indifference curve as their preferred choice. So the slope of the budget line running through the endowment point is the key determinant of this choice.

Panel (a) reveals an interesting property of long-run equilibria (or steady states) in the real business cycle model: if the economy is in a long-run equilibrium, the budget constraint touches an indifference curve right on the 45° line. We previously found that on the 45° line the slope of any indifference curve is -(1 + b). We also know that the slope of the budget line is -(1 + r). And since in a long-run equilibrium these two slopes are equal, that is -(1 + r) = — (1 + b), the time discount rate equals the interest rate:

Long-run equilibrium condition (16.12)

This holds in any long-run equilibrium on the 45° line and should thus be noted as a long-run equilibrium condition.

Figure 16.10 Initially the economy is in a long-run optimum on the 45° line. Here the consumption point equals the endowment point A, and the interest rate equals the time discount rate. When the interest rate rises above the rate of time preference, from rA to rB, the constraint pivots around the endowment point A. Now households maximize utility at B, which implies shifting some of the current consumption into the future. The act of saving today to enjoy spending tomorrow is called intertemporal substitution of consumption. Panel (b) shows that current consumption and, hence, the magnitude of intertemporal substitution of consumption depend negatively on the interest rate.

Temporary deviations of the interest rate from its long-run equilibrium value occur in the aftermath of shocks. If r is driven up, the constraint turns steeper. This signals that saving a certain amount today buys more added consumption in period 2 than it did previously. Other than in the initial optimum at the endowment point, people now save part of current income. As a reward, period-2 consumption exceeds period-2 income by period-1 saving plus interests earned. The optimal response is to move up to the point of tan-gency between the new budget line and an indifference curve (point B).

The effect just described can be generalized: the higher the interest rate on current savings, the more households save out of current income, and the lower is current consumption. So the mechanism exemplified in Figure 16.10, panel (a), by means of two optimal points A and B suggests a negative relationship between current consumption and the interest rate. This can be displayed by a negatively sloped consumption demand line CL in an r/Y diagram (Figure 16.10, panel (b)).

Intertemporal substitution is the allocation of things like consumption spending or work-hours over time in an effort to increase utility.

The intertemporal pattern of employment The postponing of consumption to future periods when interest rates are high, or the spending of income to be earned in the future on current consumption when interest rates are low, is referred to as intertemporal substitution. This phenomenon also drives the second decision variable of households, the supply of work time, or labour. The line of argument is analogous to our discussion of intertemporal substitution in consumption. In this discussion we assumed that the work time decision with the resulting income pattern had been made, and asked how households would allocate their incomes in the form of consumption spending over time. Now we switch our perspective. We now assume that the consumption decisions for periods 1 and 2 have been made and ask how households allocate work-hours over time in order to finance their consumption plan. This question can be analyzed in a diagram with current leisure time and future leisure time on the two axes (Figure 16.11).

Indifference curves do not require any further discussion. They are convex-shaped as shown in panel (a) for familiar reasons. The rationale behind the constraint shown in panel (b) may be less straightforward than in our

Figure 16.11 With decreasing marginal utility of leisure, intertemporal indifference curves are convex (panel (a)). Utility rises as we move northeast. Panel (b) shows the intertemporal constraint. Assume that we are in a long-run optimum where household consumption does not change over time. If households work just enough to pay for planned consumption each period, work times are L1A = L2A. The consumption plan translates to point A with leisure times 1 - L1A = 1 - L2 A. The very same consumption pattern can be financed by other allocations of work and leisure across time. For instance, they may work more today and save some of this income for tomorrow, when they can afford to work less (point B). All combinations of current and future leisure that pay for the consumption plan behind A sit on a straight line passing through A. This line turns steeper when the interest rate goes up. Then one hour less leisure today buys us more added leisure tomorrow than when the interest rate is low.

Figure 16.11 With decreasing marginal utility of leisure, intertemporal indifference curves are convex (panel (a)). Utility rises as we move northeast. Panel (b) shows the intertemporal constraint. Assume that we are in a long-run optimum where household consumption does not change over time. If households work just enough to pay for planned consumption each period, work times are L1A = L2A. The consumption plan translates to point A with leisure times 1 - L1A = 1 - L2 A. The very same consumption pattern can be financed by other allocations of work and leisure across time. For instance, they may work more today and save some of this income for tomorrow, when they can afford to work less (point B). All combinations of current and future leisure that pay for the consumption plan behind A sit on a straight line passing through A. This line turns steeper when the interest rate goes up. Then one hour less leisure today buys us more added leisure tomorrow than when the interest rate is low.

discussion of intertemporal consumption decisions, however. Now the intertemporal consumption plan plays the same role the endowment point played in Figure 16.10. Suppose again that the economy is in a steady state. Then today's and tomorrow's consumption are the same. And if the income needed for each period's consumption was to be earned in the very same period, without lending or borrowing in the credit market, this would translate into a given amount of required work time. In a steady state this point that relates work-hours to intended consumption spending sits on the 45° line. We call this point the consumption plan. One way to realize the consumption plan associated with point A is by working 1 — L^a% of one's time in period 1, and 1 — L2,a% in period 2.

Credit markets offer other options as well. Households may wish to enjoy some more leisure time today in exchange for work in period 2 (point C). The options are endless, and all are lined up on a string whose slope reflects the interest rate and, possibly, anticipated changes in the wage rate. When the interest rate is high, giving up some leisure time today to work more and save the additional income provides households with more additional leisure time tomorrow than when the interest rate is low. Also, when today's wage rate exceeds tomorrow's wage rate, today's leisure time is more expensive than tomorrow's. So it is better to enjoy leisure tomorrow than it is to enjoy it today. The constraint thus turns steeper.

Figure 16.12 identifies the intertemporal allocation of leisure and work time. As in our discussion of optimal consumption patterns, utility is maximized where the constraint is tangent to one of the indifference curves. Panel (a) shows a long-run optimum at point A. Since in a long-run equilibrium neither income nor wages change, the slope of the constraint equals —(1 + r), the marginal rate of substitution is (1 + b), and, therefore, r = b. If the interest rate now rises, the constraint pivots around the consumption plan, and households reduce current leisure time more and more. The resulting increase in work time is accompanied by an increase in output. So, all other things being held constant, output increases when the interest rate increases. The mechanism behind this is the intertemporal substitution of labour.

Our analysis of household behaviour has generated two important macro-economic implications. First, the volume of current output produced by firms rises at times when the interest rate rises. Second, current consumption demand falls when the interest rate goes up. Before we can compare economy-wide supply and demand in one diagram and analyze their interaction, we need to specify the demand for the second component of economy-wide demand, investment, exercised by firms.

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### Responses

• Thomas
Why 1 plus the interest rate in the intertemporal choice?
7 years ago