The Ideal Inflation‐Indexed Bond and Irving Fisher's Impatience Theory of Interest with Overlapping Generations*

Irving Fisher also developed and advocated the impatience theory of interest. But in OLG economies, the rate of interest is determined by population growth, not impatience. I reconcile this contradiction by proving that in stationary OLG economies with land, the interest rate at the unique steady state does depend on impatience. Indeed, the proposition that greater impatience creates higher interest rates holds more generally in OLG with land than in Fisher's two-period model, because then income effects and substitution effects naturally work in the same direction. Irving Fisher viewed the real rate of interest as the most important price in the economy, since it gives the relative value of consumption today in terms of consumption in the future. He proposed a theory of impatience to explain what the rate of interest should be in a finite horizon economy. He lamented that many people did not directly trade off consumption today for consumption tomorrow but instead traded money today for money tomorrow, thus enabling unexpected inflation to garble the real rate of interest. In this paper I examine whether Fisher's impatience theory of interest holds in an infinite horizon world with overlapping generations, and I derive the optimal indexed bond in a two-period economy with uncertainty and quadratic utilities. Irving Fisher was profoundly interested in the correct measurement of inflation and was a leading proponent of inflation-indexed bonds. He created an inflation index and meticulously published its values. He insisted his secretary sign an inflation-indexed contract linked to his index, and his company Rand Kardex issued an inflation-indexed bond on the same day it officially opened for business under its new name. (1) He evidently regarded the proper inflation-indexed bond as an important policy question with significant welfare implications. But he did not explain precisely what the welfare benefits are of choosing the right inflation-indexed bond. The problem is that as prices evolve over time, no household will maintain exactly the same consumption or the same utility. Indeed, no household will even maintain the same ratio of consumption goods; it will substitute goods that become relatively cheaper for goods that become relatively more expensive. Neither a bond that promises the money required to purchase the same commodity bundle nor a bond that promises the money required to achieve the same utility will be sufficient by itself to provide for the needs of the holder. On what theoretical basis is there to choose one over the other? Practically speaking, it is much simpler to measure inflation by the cost of buying a fixed commodity bundle. As Fisher pointed out, then one does not have to worry about inferring what utility is or how to deal with agents with heterogeneous utilities. And indeed, in actual practice in the United States and elsewhere, inflation-indexed bonds make payments that guarantee the purchase of the same commodity bundle. But current practice should not necessarily be the last word on the subject. From a theoretical point of view, however, it is not obvious what the right indexed bond is or even what the criterion should be. (Fisher proposed 40 tests that an inflation index should satisfy.) Perhaps the best indexed bond should guarantee each agent the same marginal utility? Perhaps it should be tied to the growth of the economy? Fisher himself modified his view and suggested as an ideal inflation index the geometric average of the Laspeyre (1871) and Paasche (1874) indices of inflation. Since the Laspeyre index is greater than the Konus (1939) index and the Paasche index is smaller, their geometric average is likely to be near the Konus index. The Konus index is meant to measure the increase in cost necessary to achieve the same utility as prices change. Thus, Fisher in the end advocated a practical index that could be used to create a bond that guarantees nearly the same utility. …