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Intermediate Macroeconomics:
Consumption
Eric Sims
University of Notre Dame
Fall 2012
1 Introduction
Consumption is the largest expenditure component in the US economy, accounting for between 60-70 percent of total GDP. In this set of notes we study consumption decisions. In micro you probably studied how people choose consumption among different goods in the cross-section: for example, how many apples and oranges to consume. In macro we study consumption in the time series dimension: how much total consumption does one do today versus in the future. So as to study the behavior of consumption as a whole in the time series dimension, we engage in the fiction that households only consume one good. I will consistently refer to this one good throughout the course as “fruit,” though in reality it is more like a composite good or a basket of goods. Think about it this way – a household has some income to spend each period, and it must decide how much of that income to spend on consumption goods. We are going to study that decision. How that expenditure is split among different types of goods (e.g. apples and oranges) is the purview of microeconomists.
2 The Basic Two Period Model
Assume that a household lives for two periods: the present (t) and the future (t + 1). This is a useful abstraction to a multi-period horizon. The household has an exogenous stream of income in the two periods: Yt and Yt+1. We abstract from any uncertainty, so that Yt+1 is known at time t. The household begins life (period t) with no existing assets, though it would be straightforward to modify the environment to allow for that. The household can consume each period, Ct and Ct+1. It can also save or borrow in the first period, St = Yt − Ct (borrowing is negative saving). It earns/pays interest rt on saving/borrowing, so that St today yields (1 + rt)St in income tomorrow.^1 Everything here is in real terms, which means that everything (including the real interest rate, rt) is denominated in physical units of goods. It is helpful to think about income and consumption as being in the same units, and I like to use the fruit analogy. A household has an exogenous stream (^1) Since the household effectively dies after period t + 1, it will not choose to do any saving in t + 1.
of fruit available to it each period; this is its income. Ct and Ct+1 is how much fruit it actually eats each period. If it chooses to not consume some of its fruit in period t, so that St > 0, it can enter into a financial contract in which it gives up its fruit today in return for (1 + rt)St units of fruit tomorrow. In contrast, if it wants to consume more fruit today than it has, it can borrow some extra fruit, with St < 0, and will have to pay back (1 + rt)St units of fruit to the lender in period t + 1. The fruit is not storable on its own – if the household wants to save some of its fruit to eat tomorrow, it has to “put it in the bank” and earn rt. Finally, the household is a price-taker: it takes rt as given, and does not behave in any strategic way to try to influence rt. Thus, from the household’s perspective rt is exogenous, though from an economy-wide perspective (as we will see), it is endogenous. The household thus faces two budget constraints: one in period t, and one in period t + 1, which I assume hold with equality:
Ct + St = Yt Ct+1 = Yt+1 + (1 + rt)St
These two budget constraints can be combined into one: you can solve for St from either the first or the second period constraint, and then plug into the other one. Doing so, I obtain what is called the “intertemporal budget constraint”:
Ct + Ct+ 1 + rt = Yt + Yt+ 1 + rt
In words, the intertemporal budget constraint (“intertemporal” = “across time”) says that the present discounted value of consumption expenditures must equal the present discounted value of income. C 1+t+1rt is the (real) present value of Ct+1. Why is that? The present value is the equivalent amount of consumption I would need today to achieve a given level of consumption in the future. Since saving pays a return of 1 + rt, the present value of future consumption would have to satisfy: (1 + rt)P Vt = Ct+1 ⇒ P Vt = C 1+t+1rt. Households get utility from consumption. Loosely speaking, you can think about utility as happiness or overall satisfaction. We assume that overall lifetime utility, U , is equal to a weighted sum of utility from consumption in the present and in the future periods:
U = u(Ct) + βu(Ct+1), 0 ≤ β < 1
β is what we call the discount factor, and it is constrained to lie within 0 and 1. It is a measure of how the household values current utility relative to future utility. We assume that β must be less than 1, so that the household puts less weight on future utility than the present. This does not seem to be a particularly controversial assumption when looking at how people actually behave in the real world. The bigger is β, the more patient the household is, in the sense that it places a large value on future utility relative to current.
function converges to the log utility function plus a constant.^2 The problem of the household at time t is to choose current and future consumption to maximize lifetime utility, subject to its unified budget constraint. What it is really doing is choosing current consumption and current saving, with saving effectively determining how much consumption it can do in t + 1. But because of the way I’ve written the unified budget constraint, we’ve eliminated St from the analysis. Formally, the problem is:
Cmax t,Ct+ U = u(Ct) + βu(Ct+1)
s.t.
Ct + (^) 1 +Ct+1 r t
= Yt + (^) 1 +Yt+1 r t As written, this is a constrained, multivariate optimization problem. We will reduce it to an unconstrained, univariate optimization problem by eliminating the constraint. In particular, solve for Ct+1 from the constraint:
Ct+1 = (1 + rt)(Yt − Ct) + Yt+
Plug this back into the lifetime utility function, re-writing the maximization problem as just being over Ct:
max Ct U = u(Ct) + βu ((1 + rt)(Yt − Ct) + Yt+1)
To find the optimum, take the derivative with respect to the choice variable, Ct, making use of the chain rule:
dU dCt^ =^ u
′(Ct) − βu′ (^) ((1 + rt)(Yt − Ct) + Yt+1) (1 + rt)
Now set this equal to zero and simplify, taking note of the fact that (1 + rt)(Yt − Ct) + Yt+1 = Ct+ in writing out the first order condition:
u′(Ct) = β(1 + rt)u′(Ct+1) This first order condition has the following interpretation: at an optimum, the marginal utility from consuming a little extra today, u′(Ct), must be equal to the marginal utility of saving a little extra today. If you save a little extra today this will leave you with 1 + rt extra units of fruit tomorrow, which will yield extra utility of u′(Ct+1)(1 + rt). The multiplication by β factors in that you discount the future utility payoff. So, in other words, this condition simply says that
(^2) To see this, re-write the isoelastic utility function as u(ct) = c^1 t 1 −−σσ− 1 = c 11 t−−σσ − (^1) − (^1) σ. In other words, I’m just subtracting a constant, (^1) −^1 σ , from what I showed in the main text. Utility is an ordinal concept, and so we are free to add and subtract constants to them without altering any of the implications of the actual functional form. As σ → 1, we have that u(ct) → 00 , so you can use L’Hopital’s rule to find the limit, which works out to the natural log.
the household must be indifferent between consuming some more and saving some more at an optimum. If this were not true, the household could increase utility by either consuming or saving some more. We sometimes also refer to this optimality condition as an Euler equation: an Euler equation is a dynamic optimality condition, and this is a dynamic (across time) optimality decision for consumption in the present and in the future. If you’ve taken intermediate micro, you might recognize this condition as something like a MRS = price ratio condition, where MRS stands for “marginal rate of substitution.” The first order condition can be re-written:
u′(Ct) βu′(Ct+1) = 1 + rt
Think about Ct+1 and Ct as just two different goods. They are different in the time dimension, just as apples and oranges are different in another dimension. 1 + rt is the relative price between first and second period consumption. Consuming an extra fruit today means saving one fewer fruit, which means giving up 1 + rt units of consumption tomorrow. Hence, the price of current consumption relative to the future is 1 + rt. Note that this optimality condition is not a “consumption function.” A consumption function would express current consumption as a function of income, the interest rate, etc. This condi- tion just relates current to future consumption. It could hold for two low values of current and future consumption, or two higher values of current and future consumption. It is a condition characterizing an optimal consumption allocation. It does not determine current consumption on its own. We can analyze the consumption problem graphically using an indifference curve - budget line diagram. An indifference curve shows combinations of two goods for which the household achieves the same overall level of utility. Again, just think about consumption of fruit at different points in time as different goods. Suppose we start with an initial consumption bundle, (C t^0 , C t^0 +1). This would yield overall utility:
U 0 = u(C t^0 ) + βu(C t^0 +1) Now we use the concept of the total derivative, which was introduced in the math review notes. The total derivative says that the change in a function is approximately equal to the sum of its partial derivatives evaluated at a point times the change in each of the right hand side variables at a point. Applying that here, we have:
dUt = u′(C t^0 )dCt + βu′(C t^0 +1)dCt+
Where dUt = Ut − U (^) t^0 , dCt = Ct − C t^0 , dCt+1 = Ct+1 − C t^0 +1. Now for utility to stay constant – which is what defines an indifference curve, it must be that dUt = 0. Imposing that and solving, we get:
There are some other properties of indifference curves that we will not mention in any depth here. One obvious one is that indifference curves cannot cross. Crossing indifference curves is a logical impossibility when one recognizes that each indifference curve is associated with a particular level of overall utility. If indifference curves crossed, this would mean that one gets different levels of utility at the same consumption bundle, which is impossible. Wikipedia has a pretty good page on indifference curves: http://en.wikipedia.org/wiki/Indifference_curve. The budget line is a graphical depiction of the intertemporal budget constraint derived above. Solve for Ct+1 in terms of Ct, the real interest rate, and income levels:
Ct+1 = (1 + rt)Yt + Yt+1 − (1 + rt)Ct
The budget line intercepts the vertical axis at Ct+1 = (1 + rt)Yt + Yt+1 and intersects the horizontal axis at Ct = Yt + Y 1+t+1rt. It has slope dC dCt+1t = −(1 + rt). Note also that it must cross through the “endowment point” at which Ct = Yt and Ct+1 = Yt+1. It is always possible to just consume the entirety of income each period, and doing this exhausts resources. It is sometimes helpful to label the endowment point when drawing the budget line.
Points inside the budget line are feasible but leave resources unused. Points outside (e.g. to the northeast) of the budget line are not feasible. Graphically, we can think about the household’s problem as to pick a point on the highest possible indifference curve that is consistent with the budget constraint. This will occur when the indifference curve just “kisses” the budget constraint. Mathematically, this occurs where the curve is tangent to the budget line, so this optimality con- dition is sometimes called a “tangency condition.” This tangency/optimality condition is shown in the plot below. Tangency means that the slopes are equal. As we can see, the slopes being equal is exactly the mathematical condition characterizing the optimum we derived above.
Now that we’ve graphically characterized the optimal consumption bundle, we can do a couple of exercises. First, suppose that current period income increases, that is that Yt becomes bigger. We can see in the plot below that this will shift the budget line outward – both the horizontal and vertical axis intercepts increase, with the horizontal intercept increasing by the change in Yt and
Finally, let’s look at what happens when rt increases. This is going to have the effect of pivoting the budget line through the endowment point, with the budget line becoming steeper. We can see that, holding income in each period fixed, the horizontal axis intercept must shift in. In contrast, the vertical axis intercept must shift up. The budget line must pivot through the endowment point because it is always possible to consume the endowment each period regardless of what rt is. Below I have shown in the indifference curve-budget line diagram what might happen when rt increases.
In the paragraph above I said what “might” happen when rt increases. It turns out that it is theoretically ambiguous what effect an increase in the real interest rate will have on current consumption. The reason why is that, unlike the cases where income changed, there are both income and substitution effects at work here. The substitution effect is always to reduce current consumption and increase future consumption – 1 + rt is the relative price of current consumption, so an increase in rt makes people want to shift away from current consumption and into future
consumption by saving more. The sign of the income effect depends on whether the household is initially a saver (consumption less than current income) or a borrow (consumption greater than current income). I drew the figure above where the household is initially a saver, with C t^0 < Yt. If the household is initially a saver, then the income effect is positive – the household feels richer, and will want to consume more in both periods. Intuitively this is clear – the household was already saving, so an increase in the real interest rate means it will get a bigger return on that saving and hence more income in the next period. You can also see this by simply looking at the plot – if the household is initially a saver, then it is guaranteed to be able to get to a higher indifference curve when rt increases. We can see that by noting that the initial optimality point is strictly inside the new budget line. The higher income the household receives in the second period from its existing saving makes it want to consume more in both periods. Hence, the income effect goes in the opposite direction of the substitution effect for first period consumption – the substitution effect says to consume less, whereas the income effect is to consume more. Which one dominates is unclear – it depends on the nature of preferences and how much the household was initially saving. Now if the household were a borrower, with C t^0 > Yt, then the income effect would be negative. Intuitively the household would have to pay back more on its borrowing, reducing its future income. This effect would make the household feel poorer, which would lead it to want to reduce both future and current consumption. Hence, if the household is a borrower, the income and substitution effects go in the same direction, leading the household to definitely reduce first period consumption. I drew the figure above where the household is initially a saver. This means that it is ambiguous as to whether the income or the substitution effect dominates, and hence ambiguous as to whether current period consumption will increase or decrease. I drew the diagram where current period consumption nevertheless decreases – that is, I assumed that the substitution effect dominates. Unless otherwise noted, we are going to in this class assume that the substitution effect dominates, so that current consumption is decreasing in the real interest rate. This seems to be the empirically plausible case.
2.1 The Consumption Function
As emphasized above, the tangency condition or Euler equation is not a consumption function. The Euler equation is a condition between current and future consumption (and the relative price between the two, the real interest rate). A consumption function expresses current consumption as a function of things other than future consumption: income, the interest rate, and parameters. From the indifference curve exercises above, we see that consumption evidently depends on current income, future income, and the real interest rate. Generally:
Ct = C(Yt, Yt+1, rt)
Here C(·) is a function mapping income and the interest rate into consumption. We know something about its partial derivatives. In particular, ∂C ∂Ytt > 0 and (^) ∂Y∂Ct+1t > 0. That is, consumption is
income, or what is sometimes called the “marginal propensity to consume,” is positive, and bound between 12 (β → 1) and 1 (β → 0). In other words, the bigger is β, so the more patient is the household, the lower is the marginal propensity to consume. Second, the partial with respect to future income is also positive. This being positive relies on rt > −1. Note that the real interest rate can be negative. Remember, all the real interest rate says is how many goods you get back tomorrow for giving up a good today. You may be willing to take a deal in which this return is negative, so that you get back fewer goods tomorrow than you give up today. This is driven by the assumption (noted above), that fruit is not directly storable – if you could store your fruit, one fruit today left in the pantry would be one fruit tomorrow still in the pantry, and no household would accept a negative real interest rate, because the best outside option would be holding on to it and earning zero return. But if storability is not an option, the household may be better off taking a negative real interest rate than not saving at all. Nevertheless, rt must be > −1. If the real interest rate were −1, this would mean that you give up a fruit today for nothing in return. You would never do that – you’d be better off eating the fruit today and having nothing tomorrow. Since rt > −1, the partial with respect to future income must be positive. Whether the marginal propensity to consume out of future income is bigger or smaller than current income depends on the sign of rt. If rt > 0, the partial with respect to future income is smaller than with respect to current income. If rt < 0, the reverse is true – a one fruit increase in future income would have a bigger effect on current consumption than would a one fruit increase in current income. This is perhaps easiest to see by nothing that (^) 1+Yt+1rt is the present value of future income. Looking at the derivatives, (^) 1+^1 β is the partial with respect to the present value of income in either period (the present value of current income is just current income). If rt > 0, the present value of future income is smaller than current income, and so the overall MPC is smaller. If rt < 0, then the reverse is true. Finally, let’s look at the derivative with respect to the real interest rate. Here we see that, as long as Yt+1 > 0, then the derivative must be negative. That is, consumption is decreasing in the real interest rate, unless the household was going to have no income in the future. The case of no income in the future would correspond to the case of the income and substitution effects perfectly canceling out that we discussed above in analyzing indifference curves. This seems reasonable to assume that people have at least some income in the future, and hence, with log utility, we see that current consumption is in fact decreasing in the real interest rate.
3 Some Extensions of the Two Period Model
In this section we are going to look at three different implications of the two period model. These are (i) “permanent” versus “transitory” changes in income, with an application to tax cuts; (ii) the role of wealth; (iii) and the role of uncertainty.
3.1 “Permanent” vs. “Transitory” Income
For a generic specification of utility, the consumption function can be characterized as:
Ct = C(Yt, Yt+1, rt) As shown above, the partial derivatives of the consumption function with respect to both current and future income are positive. Given this, it should come as no surprise that consumption ought to react more to changes in income the more persistent the change in income is. Suppose that we have a simultaneous increase in both current and future income. Using the total derivative, the approximate change in consumption would be the sum of the partial derivatives (evaluated at a point) times the changes in income:
dCt ≈
∂C
∂Yt^ dYt^ +^
∂C
∂Yt+1^ dYt+ To see this most clearly, suppose that there is a “permanent” change in income, so that current and future income increase by exactly the same amount. Call this common amount d Y¯. Simplifying, we see that the approximate change in consumption for a one unit “permanent” increase in income is equal to the sum of the partials:
dCt d Y¯
≈ ∂C
∂Yt
+ ∂C
∂Yt+ As we discussed in the previous section, both of these partial derivatives should be positive. This means that a change in income that persists into both periods ought to have a much larger effect on consumption than a change that only lasts one period. To use a concrete example, we derived a consumption function for the case of log utility in the last section. If we approximate rt ≈ 0 and β ≈ 1, then the partial derivative of the consumption function with respect to income in either period is 12. Continuing with that approximation, we can deduce that a “transitory” change in income in just the first period ought to be half consumed and half saved. In contrast, a “permanent” change in income should result in a roughly one-for-one reaction of consumption. Another way to phrase this is as follows: the marginal propensity to consume out of income will depend on how persistent the change in income is. The more persistent the change in income, the more consumption ought to react. This insight has potentially important implications for policy. During the last several recessions there have been “stimulus” packages which, among other things, cut taxes for most households. We can fairly easily modify the household problem to include taxes. Let Tt and Tt+1 be tax payments in periods t and t + 1, respectively. Since income is exogenously given in this exercise, just think of taxes as the government taking away some of the household’s endowment of fruit. The period-by-period budget constraints look like:
Ct + Ct+ 1 + rt = B 0 + Yt + Yt+ 1 + rt This looks very much like the original intertemporal budget constraint, and as I alluded to above, we can really just think about B 0 as being another source of first period income. Changes in B 0 will function just like changes in first period income – they will shift the budget constraint in or out, leading to changes in consumption and saving:
Although it may seem somewhat trivial since you can real think about wealth as first period income here, I think it is worth pointing the wealth channel out directly. In the last fifteen years there have been episodes where “wealth” seems to have played a role in actual consumption patterns. The two most important sources of wealth for most households are (i) housing and (ii) stock market holdings. In the mid-1990s, stock prices rose. This had the effect of raising wealth for households, which made them feel richer and led them to spend more, helping to fuel a boom. In the mid-2000s home prices soared, leading to increasing consumption for similar reasons. In late 2006 and early 2007, however, home prices came crashing down. This had a negative wealth effect that made households want to reduce their consumption, and was an important contributor in the recent recession.
3.3 Uncertainty
Up until this point we have assumed that future income is known in the present – that is, at time t households know Yt+1 with certainty. In this subsection we consider the implications of how uncertainty impacts consumption decisions. Suppose that future income can take on two values: Y (^) th+1 > Y (^) tl+1. Let the probability of the “high state” occurring be p, with the probability of the “low state” of 1 − p. Then the expected value of Yt+1 is equal to:
E(Yt+1) = pY (^) th+1 + (1 − p)Y (^) tl+ The optimization problem of the household has to be re-cast slightly. In particular, the house- hold will want to maximize expected utility subject to the intertemporal budget constraint. It is expected utility because, if future income is not known, then future consumption cannot be known with certainty. If income ends up high, consumption in the future will be relatively high. But if income ends up low, then consumption will be low. Not knowing future consumption with certainty means that one cannot know future utility with certainty. Hence, when making consumption/saving decisions in the present, one seeks to maximize expected utility. The first order condition ends up looking the same as in the case with certainty, but with an expectation operator. The optimality condition, or Euler equation, is:
u′(Ct) = β(1 + rt)E(u′(Ct+1)) Note that what shows up on the right hand side is the expected marginal utility of consumption, which is not the same thing as marginal utility of expected consumption. With two possible realizations of future income, there are two possible realizations of future consumption, given current consumption, current income, and the real interest rate:
Cth+1 = Y (^) th+1 + (1 + rt)(Yt − Ct) Ctl+1 = Y (^) tl+1 + (1 + rt)(Yt − Ct)
The expected value of consumption in the second period is E(Ct+1) = pCht+1 + (1 − p)Ctl+1. The key insight to understanding how uncertainty impacts consumption is that expected marginal utility is not, in general, the same thing as marginal utility evaluated at the expected value of future consumption. Only in the special case in which marginal utility is linear, which occurs with the quadratic utility function shown above, will expected marginal utility coincide with the marginal utility of expected consumption. Let’s suppose that the third derivative of the consumption function is positive: u′′′(Ct) > 0. The third derivative of the utility function is the second derivative of marginal utility. Hence, the third derivative is a measure of the curvature of marginal utility. If the third derivative is positive, then it means that marginal utility is decreasing, but it flattens out as Ct gets big. In words, that means that marginal utility is decreasing in Ct, but it flattens out as Ct gets big (i.e. the slope of marginal utility becomes less negative). The log utility function has this property, for example. When u(Ct) = ln Ct, then u′(Ct) = C t− 1 > 0, u′′(Ct) = −C t− 2 < 0, and u′′′(Ct) = 2C t− 3 > 0. Below is a plot of marginal utility of future consumption, u′(Ct+1), as a function of Ct+1. Graphically, we can see the third derivative being positive means that marginal utility is “bowed in” (the plot of marginal utility actually looks like an indifference curve, or the lower southwest portion of a circle). Expected consumption is equal to pCth+1 + (1 − p)Ctl+1. Expected marginal
x − u′(Ctl+1) E(Ct+1) − CtL+
u′(Cht+1) − u′(Clt+1) Cth+1 − Ctl+
First, plug in for the expected value of consumption, which can be written E(Ct+1) = p(Cth+1 − Ctl+1) + Clt and then start simplifying:
x − u′(Ctl+1) p(Cth+1 − Ctl+1)
u′(Cth+1) − u′(Ctl+1) Cth+1 − Ctl+
x − u′(Ctl+1) = (p(Cth+1 − Clt+1))(u′(Cth+1) − u′(Ctl+1)) Cth+1 − Ctl+
x = (p(Cth+1 − Clt+1))(u′(Cth+1) − u′(Ctl+1)) Cth+1 − Ctl+
x = (p(Cth+1 − Ctl+1))(u′(Cht+1) − u′(Clt+1)) + (Cth+1 − Ctl+1)u′(Ctl+1) Cth+1 − Ctl+
x = p(Cth+1 − Ctl+1)u′(Cth+1) + (1 − p)(Cth+1 − Ctl+1)u′(Ctl+1) Cth+1 − Clt+ x = pu′(Cht+1) + (1 − p)u′(Ctl+1) = Eu′(Ct+1)
This proves that the line connecting marginal utilities at the two possible consumption values evaluated at the mean consumption value is equal to expected marginal utility. And since the the line lies above the curve, expected marginal utility is greater than marginal utility of expected consumption. If there is uncertainty over future income, and the third derivative of utility is positive, then marginal utility of future consumption will be higher than if there were no uncertainty. Re-written slightly, the Euler equation says:
u′(Ct) β(1 + rt) =^ E(u
′(Ct+1))
Looking at this first order condition, we can qualitatively think about what happens when there is an increase in uncertainty. An increase in uncertainty that leaves the expected value (mean) of future income unaffected must raise expected marginal utility. We can see this depicted graphically below. The right hand side in the above first order condition thus increases. To make the optimality condition hold, then the left hand side must also get bigger. For a given interest rate, this means that the current marginal utility of consumption must increase. Marginal utility of current consumption getting bigger requires current consumption to fall. Put differently, households will react to higher uncertainty by trying to reduce current current consumption, or equivalently by trying to increase current saving. We call this precautionary saving.
The intuition for precautionary saving can be seen from the graph above. If marginal utility is convex (i.e. if the third derivative of the utility function is positive), and uncertainty increases then the “pain” from the bad state being realized is worse than the “gain” from the good state being realized. Households will react to this by trying to “save for the rainy day” – by building up a larger stock of savings, the household will be better able to minimize its utility losses from the bad state occurring. The end result is the following: an increase in uncertainty over future income, holding all other factors constant, will lead the household to try to reduce its current consumption.
4 Multi-Period Generalization and the Life Cycle
All of the basic insights of the two period model carry over to a multi-period extension of the model. The distinction between “permanent” and “transitory” income becomes even stronger in a multi-period setting, however, and we can also think about how consumption and income ought to vary over the “life cycle” when there are multiple periods. For this section we revert to assuming that future income is known with certainty. Instead of assuming that households just live for two periods (t and t + 1), let’s assume that they live for many periods: t, t + 1, t + 2,... , t + T (T + 1 total periods). Lifetime utility is now:
U = u(Ct) + βu(Ct+1) + β^2 u(Ct+2) + β^3 u(Ct+3) + · · · + βT^ u(Ct+T ) Here each period into the future gets multiplied again by β. β is still the relative weight placed on future consumption between any two adjacent periods. Between today and several periods into the future, however, the weight on future utility is β raised to the number of periods in between. For example, if T = 50 and β = 0.95, β^50 = 0.08, so that the household places relatively little weight (0.08) on utility flows 50 periods from now relative to the present. We can equivalently write lifetime utility using the summation operator:
