14.06: Section HandoutTA: Jose TessadaApril 15, 2005Topics to be covered today:• Hall’s Random Walk Hypothesis• Precautionary Savings1 Hall’s Random Walk HypothesisHall’s result on the properties of consumption marked a clear challenge to the existingview of consumption. The early models of consumption had clear implications onthe predictability of consumption. Hall’s result challenged that based mostly onthe basic intuition behind the permanent-income hypothesis, with some additionalassumptions.We shall review first a simple model of consumption with certainty and then wecan move on to analyze the case with uncertainty.1.1 The Certainty CaseConsider an agent with the following preferences:U =TXt=1u (ct) , u0(·) > 0, u00(·) < 0, (1)where ctis consumption in period t and we have assumed β = 1. The budget con-straint of the agent isTXt=1ct≤ A0+TXt=1yt(2)where we have also assumed that r = 0.1We can prove that under the assumptionthat u0(·) > 0, then the budget constraint will be binding in equilibrium.The Lagrangian for this problem is£ =TXt=1u (ct) + λÃA0+TXt=1yt−TXt=1ct!(3)1Setting both the interest rate and the subjective discount rate equal to 0 will help us simplifythe math a bit but the main result will still hold.1with FOCsu0(ct) = λ, ∀t. (4)This is the basic idea of consumption smoothing, individuals will choose a con-sumption path so as to keep a constant marginal utility of consumption. Under ourassumptions, the consumption level uniquely determines the marginal utility, thenc1= c2= . . . = cT= c. Using (2) we obtainc =1TÃA0+TXt=1yt!. (5)Equation (5) has a very intuitive interpretation. The right hand side correspondsto the permanent income, and this is the basic result from the permanent incomehypothesis, consumption is determined by the permanent level of income, not by thecurrent level; savings in this model are equal to the difference between the currentand the permanent income level. The life-cycle hypothesis relates connects the basicidea of consumption smoothing to the earnings profile, then an individual borrowswhen young, pays the debt and saves when adult (working age), and disaves whenold (particularly after retirement).1.2 The Uncertainty CaseLet us now deal with the uncertainty case. We will not use the same representationof uncertainty used in the lectures, instead we will write the individual’s objectivefunction as the expected intertemporal utility. All the assumptions made in section1.1 will be maintained here, we will introduce the additional assumption that u (ct) =ct− (a/2) c2twith a > 0. In this case we can write the objective function of theindividual as followsU = Et=1TXt=1hct−³a2´c2tiwhere Et=1is the expectations operator with the information available in time t = 1.The budget constraint is given by (2), so the consumption profile will satisfy it withequality. We can apply expectations to both sides of (2) to obtainTXt=1Et=1(ct) ≤ A0+TXt=1Et=1(yt) . (6)Using the FOCs with respect to consumption we obtain the following result1 − ac1= Et=1(1 − act) , for t = 2, 3, . . . , T. (7)Given the linearity of marginal utility we can obtain the following condition forconsumption along the optimal pathc1= Et=1(ct) , for t = 2, 3, . . . , T. (8)So the individual expects to consume the same quantity in any subsequent period.Of course, this is based on the information she has access to in t = 1, so ”news”2(innovations) in any relevant variable will induce changes in the consumption profile.Notice that if the individuals are rational, in the sense that they incorporate all theinformation available, then these changes cannot be predicted in t = 1, if they arepredictable, then the individual should reoptimize in order to smooth consumption(as it is implied by the condition on the marginal utility of consumption, equation 7).We can go a bit further and postulate an empirical model based on the basicresults we have obtained so far.2Take equation (8) for t = 2, then we can clearly seethat changes in consumption are unpredictable.Under the assumption that agents incorporate all the available information whenforming their expectations, we can writect= Et−1(ct) + εt, (9)where εtis an expectational error and Et−1(εt) = 0. Using (8) in (9)ct= ct−1+ εt(10)which is exactly Hall’s famous result: consumption follows a random walk. The mainintuition has already been explained: if the individual expects consumption to change,then she can do a better job by reallocating resources smoothing out those fluctuationsaccording to (7), which implies that we expect no changes in consumption.If we follow Romer (2001)3we can also find the actual value of εt. the ”change”in consumption. Take (8) and (6) to obtainc1=1TÃA0+TXt=1E1[yt]!,so consumption is given by the ”average” value of expected lifetime resources. Takethe same formula for t = 2c2=1T − 1ÃA1+TXt=2E2[yt]!=1T − 1ÃA0+ y1− c1+TXt=2E2[yt]!=1T − 1A0+ y1− c1+TXt=2E1[yt] +(TXt=2E2[yt] −TXt=2E1[yt])| {z }∆ in the expectation: new information2Hall’s paper presents a more general version with less assumption, but the main implicationsare the same. In particular, if utility is quadratic (so marginal utility is linear) and the discountrate equals the interest rate, then consumption follows a ”random walk”.3See Section 7.2 of Romer’s textbook.3=1T − 1A0+ y1+TXt=2E1[yt]| {z }T c1− c1+(TXt=2E2[yt] −TXt=2E1[yt])c2= c1+1T − 1ÃTXt=2E2[yt] −TXt=2E1[yt]!. (11)Equation (11) tells us that the individual will adjust current consumption accord-ing to the change in the total expected value of income during lifetime, and will do itby a fraction equal to the inverse of the periods left before T . In simpler words (hope-fully simpled by now), current consumption change in the same amount permanentincome changed because of the news that arrived.4Notice the following property, if we were to offer the individual to change hisuncertain income stream for a certain one that is equal to the expected value, shewould choose exactly the same consumption level. This is called certainty equivalence.What’s the fundamental assumption that delivers this result?2 Precautionary SavingsOur previous model assumed that utility is quadratic. There are several problem withit, starting from the fact that marginal utilities can be negative for sufficiently highlevels of consumption, and continuing from the