Perpetuity

Abstract: This paper provides a fairly systematic study of general economic conditions under which rational asset pricing bubbles may arise in an intertemporal competitive equilibrium framework. Our main results are concerned with nonexistence of asset pricing bubbles in those economies. These results imply that the conditions under which bubbles are possible -including some well-known examples of monetary equilibria-are relatively fragile. THIS PAPER IS CONCERNED with the conditions under which asset prices in an intertemporal competitive equilibrium are equal to the present value of the streams of future dividends to which each asset represents a claim. According to a central result of the theory of finance, this is very generally true in the case of economies with trading at only a finite sequence of dates, as long as there are no restrictions upon transactions other than that associated with possible incom- pleteness of the set of securities that are traded. (The result is sometimes called "the fundamental theorem of asset pricing.") Here we consider the extent to which such a result continues to be valid in the case of trading over an infinite horizon. It has often been observed in the econometric literature on "asset pricing bubbles" that it is possible, in principle, for the price of a perpetuity and the dividends on that security to satisfy at all times a present-value relation for one-period holding returns, while the security's price nonetheless does not equal the present value of the stream of dividends expected over the infinite future. In such a case, the price of the perpetuity is said to involve a bubble component.2 Joint stochastic processes for which this component is nonzero are sometimes argued to characterize existing assets; for a recent example, see Froot and Obstfeld (1991). But such an inference depends upon aspects of the stochastic

Abstract: The problem of demand for money and other assets has been recently studied by Douglas (1966) and Sidrauski (1965) within the framework of a rational individual faced with the choice of a consumption schedule which is optimal with respect to the individual’s time preference structure. In both papers, the intertemporal utility function upon which the consumer’s choice is based is represented by a discounted integral of the stream of instantaneous utility levels, where future utilities are discounted by a rate which is kept constant independently of time profile of the utility stream associated with each consumption schedule. Thus, if a consumer is permitted to hold his assets either in the form of real cash balances or in the form of perpetuities yielding a constant rate of interest and if his instantaneous utility function is linear and homogeneous, he will either postpone his consumption until the very last moment or will consume as much as possible, according to whether the subject rate of discount is lower than the rate of interest. The only case in which the individual would desire to possess two types of assets simultaneously is one where his subject rate of discount is precisely equal to the rate of interest. Douglas has avoided this difficulty by having the level of bond holdings as one of the components for instantaneous utility level, while Sidrauski has introduced real capital as an alternative asset for which the rate of return varies with the amount held. In this paper, we shall instead start with an analysis of an individual’s time preference structure, to derive a certain specific formulation regarding the rate by which he discounts future levels. We shall then proceed to examine the behavior of an individual consumer who decides the allocation of his income between consumption and savings and the choice of portfolio balances in such manner that the resulting consumption stream is most preferred in terms of his time 486preference structure. The analysis will be first carried out for the simple case in which the individual is permitted to hold his assets only in the form of bonds for which the expected rate of interest is constant, and then for a more general case in which he may hold his assets in the form of money and bonds and other types for which the rates of return may vary.

Abstract: If Vk is the discount factor for the kth period, then Z = Σ k⩾1V 1...Vk Ck is the discounted value of a perpetuity paying Ck at time k. In some cases Z is also the limiting distribution of St =Vt (St-1 +Ct-1 ). This paper 1. reviews the literature concerning Z and {St } 2. considers continuous-time counterparts of Z and S, at the same time deriving the distribution of ∫ exp(-γt-σWt )1(0, ∞) (t)dt when W is Brownian motion; 3. gives applications to risk theory and pension funding.

Abstract: This lecture discusses the economic losses that result from an unfunded social security retirement system and the potential gain from shifting to a funded system. The social security payroll tax distorts labor supply and the form in which compensation is paid. Although each individual's benefits are linked to that individual's previous payroll tax payments, the low equilibrium rate of return inherent in an unfunded system implies a `net' payroll tax that causes distortions. The resulting deadweight loss is 1% of each year's GDP in perpetuity, an amount equal to 20% of payroll tax revenue and a 50% increase in deadweight loss of the personal income tax. Also, there is the loss of investment income resulting from forcing employees to accept the low implicit return of an unfunded program rather than the much higher return paid on private saving or in a funded social security program. The present value of the annual losses from using an unfunded system exceeds the benefit to those who received windfall transfers when the program began and when it expanded. Shifting to a funded program cannot reverse the crowding out of capital that has already occurred. Recognizing the existing unfunded obligation only makes that piece of the national debt explicit, but shifting to a funded program limits crowding out of capital formation to the amount that already occurred. Future increases in annual saving that result from economic growth are able to earn the higher rate of return on real capital. The present value of these gains is equivalent to a perpetuity of more than 2% of GDP a year. The combi- nation of improved labor market incentives and higher real return on saving has a net present value gain of more than $15 trillion, an amount equivalent to three percent of each future year's GDP forever.