67 James Ballard James Bu//ard is an economist at the Federal Reserve Bank of St. Louis. Lynn Dietrich provied research assistance. The author would like to thank Chris Sims for a valuable comment on an earlier draft of this paper. Samuelson’s Model of Money with n-Period Lifetimes AUL SAMUELSON’S OVERLAPPING genera- and Solow (1956). Yet, while these rivals in the tions model is a classic in modern economic 1980s have begun confronting the data directly, literature. It has enjoyed a renaissance in the the overlapping generations approach for the last decade or so as a framework for analyzing most part remains the province of theorists.’ fundamental issues in many areas of economics, This is so primarily because a “time period,” including pure theory, public finance and, of instead of being interpretable as a month ot’ a special concern for this paper, monetary theory.’ quarter, has a biological basis as a fraction of Samuelson’s (1958) model continues to attract an adult human lifetime; in standard two-period interest in the latter field because it has the formulations, it would be interpreted literally potential to offer a convincing explanation of as something on the order of 25 or 30 years. why unbacked paper currency has value without Conventional data sets preclude most empirical resort to special assumptions.’ This paper will analysis on such a time scale. This fact forms focus on the value of paper currency in a the foundation for a great deal of criticism of generalized version of Samuelson’s otiginal the overlapping generations approach. approach. The purpose of this paper is to argue that Samuelson’s essential insight was to introduce some of the key results from conventional over- demographic structure. The economic actors in lapping generations models in which agents live the model actually die, so that people have finite for two periods extend surprisingly well to the planning horizons even though the economy case where agents live for many periods—at itself continues without end. This is in stark least for the example studied here. Consequently, contrast to the immortal people that occupy the some of the typical criticisms of Samuelson’s chief rival models in use in macroeconomics model of money should exert less force on today, most of which are sophisticated versions economists than they commonly do- In addition, of growth models pioneered by Ramsey (1928) the n-period approach opens the possibility, I See, for instance, Wallace (1980). ‘See Kydland and Prescott (1962) for an example of com- 2 Special assumptions that have been used include placing paring the predictions of a Ramsey-Solow model with data. money in the utility function, or imposing a cash-in-advance constraint on the purchase of some goods. For discus- sions of these alternative approaches, see Sargent (1987). 68 already pursued by some researchers, of con- The results described above, it should be em- fronting overlapping generations models with phasized, are based entirely on an example in available data front macroeconomic time series. which the preferences of the people in the model are described by particularly simple func- 8 Recent general theoretical results on n-period tions. This allows key results to be derived overlapping generations models are developed algebraically. The model will be described in the in Kehoe, el aL (1991) and Aiyagari (1988, 1939).4 following section. The results concerning the This paper illustrates sonic key points developed existence of stationary equilibria and the condi- by these authors. Some results are new, how- tions for fiat currency to be valued are described ever, especially those concerning the conditions subsequently, and will be contrasted and com- for fiat currency to have value in equilibrium. pared to the conventional two-period case. In particular, previous studies have suggested that, if the people in the model discount the ~EE1(JN’STt iRUL future, letting the number of periods in the model become arbitrarily large implies that fiat money cannot be valued in equilibrium.’ Since most economists believe that people in real Given a disturbing disadvantage such as an in- economies do discount the future, this result appropriately long time period, one might won- seemed to sink hopes that the overlapping der if retaining the overlapping generations generations approach could convincingly explain framework is worthwhile. But Samuelson’s why unbacked paper currency has value. The approach has important advantages that have results presented here suggest, in contrast, that induced continuing interest in the model, time discounting the future is actually less important period problems notwithstanding. A few of than the previous research seemed to suggest. these positive aspects will be reviewed here. The condition for fiat currency to have value in In Samuelson’s model, a new generation is equilibrium in the n-period model is instead born in every period, at the same time that the found to be analogous to the condition in the oldest generation dies. This structure implies a two-period model. In fact, there is a sense in certain heterogeneity among individuals, where which adding periods to a model with discount- younger people have a relatively long horizon in ing makes it easier, instead of more difficult, to which to wot* and save, and older people have satisfy the condition. a relatively short horizon. One can infer that this will affect the way these people behave. Whether fiat currency has value in equilibrium Although heterogeneity of this type is a feature also depends on the lifetime productivity profiles of observed economies, it is absent from most of the economic actors in the model.~A stand- competing models.° ard result from the two-period model is that this profile would have to be declining over a As has already been emphasized, fiat money-- person’s lifetime in order for fiat currency to be intrinsically worthless pieces of paper issued by valued in equilibrium! In actual economies, the government—can have value in equilibrium however, productivity tends to rise with age, in Samuelson’s model without resort to special dt-opping off quickly only near retirement. assumptions. This is the primary reason mone- A key result of the present paper is that in tary theorists have paid close attention to the the n-period model fiat money can still be model. In contrast, the Ramsey-Solow model valued when the lifetime productivity profile generally does not admit equilibria with valued is plausibly hump-shaped. fiat money unless special assumptions are invoked. 7 ~Strictlyspeaking, the Kehoe, et at (1991) results apply to This is also an oversimplification, the meaning of which will “large square economies,” that is, those with many goods be clarified in the discussion of the model. and many participants, but where consumers live for only People will be endowed with logarithmic, time-separable two periods. However, they argue that, analytically speak- utility functions. ing, these models are equivalent to those with, say, a 9 single good and n-period lifetimes. There are some models in which all people have infinite lives but heterogeneity of a similar type plays a role. ~Thisis an oversimplification; more exact statements will be See, for instance, Becker and Foias (1987). made in the next section. °Theseare represented by the endowment patterns in the subsequent analysis. There ar-c two stationary equilibria in conven- Tobin’s reasons is that identifting money as an tional versions of Samuelson’s two-period asset that would be held for 25 years is “slightly model.’’ One is the monetary steady state, ridiculous,” in part because “the average holding where fiat currency has value and the price period of a dollar of demand deposits is about level is constant (pt-ovided the currency stock is two days.” He also suggests that the real world constant). ‘I’he other is the autarkic (no trade) analog of the asset in the model might he better steady state, where fiat currency has no value viewed as land. Social security schemes, in (currency is not held) and the price level grows ‘robin’s view, would he better- mechanisms for without bound. One concern ahout n-period accomplishing intergenerational transfers be- versions of the overlapping generations model tween the old and the young. In short, the has been that the number of stationary “money” in the overlapping generations model, equilibria might multiply uncontrollably as n according to Tohin, “is not the money of com- increased to a value that would allow researchers mon parlance.” Since all of these criticisms are to interpret a time period as, say, a quarter. tied to the notion that the time period in the Presumably, if one thinks of adult lifetimes as model is very long, an n-period model in which 55 or 60 years, n would have to he 220 or 240 the pet’iod could he much shorter, hut could for such an inierpretation to be valid. It is share conclusions similar to the two-period therefore somewhat surprising that the version model, presumably would allay some of these of the n-period model examined heie has only concerns. two stationary equilibria, and that these are the analogs of the two steady states that exist when Another aspect of the time period prohletn n =2.” and its treatment in the literature deserves mention. Some authors have argued that many The fact that two steady states can exist is of the central insights would carr’y over from important, because the conventional overlapping the two-period case to the n-period case and genet-ations model also serves as a classic example that, for clarity’s sake, the two-period model of a ft-amework that may produce inefficient should be the version of choice.