Asset Bubbles and Overlapping Generations

The paper investigates “whether the fundamentalist view of asset pricing,” under which “the price of an asset equals the (expected) present discounted value of its dividends, i.e., its market fundamental,” extends to overlapping-generations economies, after the author had previously shown, for an economy with a finite number of infinitely lived traders, that “any asset must indeed be valued according to its market fundamental,” a conclusion “robust to differential information and to the presence of short sales constraints” (Tirole [1982], cited in Sec. 1). The paper states plainly that in overlapping-generations settings the fundamentalist view “can readily be answered in the negative,” since Samuelson’s consumption-loan model already shows money can have positive value “in spite of the fact that it is intrinsically useless” (Sec. 1, p. 1071).

A bubble is defined as “the difference between the market price and the market fundamental” of an asset, and under perfect foresight “the bubble must bear the same yield as capital, as, by definition, it does not distribute any dividend,” and must be nonnegative (Sec. 2, p. 1075, Eqs. 12-13). The paper reports the prior “loose reasoning,” made rigorous by Scheinkman [1980], that if the long-run interest rate is nonpositive and a rent-paying asset exists, that asset’s market fundamental is infinite, while if the interest rate is positive, “the asset bubble – which must grow at the interest rate – eventually becomes so big that young generations cannot buy the asset,” appearing to rule out bubbles whenever a real rent-paying asset exists (Sec. 1, p. 1072). The paper states that “its validity however is limited to economies that do not grow,” which the paper’s own analysis of a growing economy with capital accumulation is built to show (Sec. 1, p. 1072).

Proposition 1 establishes that “if r̄ > n, there exists a unique equilibrium. This equilibrium is bubbleless and the interest rate converges to r̄” – where r̄ is the steady-state interest rate of the bubbleless, rentless Diamond economy and n is the population growth rate – while “if r̄ < n, there exists no bubbleless equilibrium [and] there exists a unique bubbly equilibrium… asymptotically bubbly [with] the interest rate converg[ing] to n” (Sec. 3, p. 1076, Prop. 1). In the intermediate case 0 < r̄ < n, the paper shows there is a continuum of equilibria indexed by the initial bubble, ranging from asymptotically bubbleless (converging to r̄) up to a maximum feasible bubble, beyond which the unique remaining equilibrium is asymptotically bubbly and the interest rate converges to n (Sec. 3, p. 1076, Prop. 1b). The intuition given is that bubbles crowd out productive savings, raising the interest rate; when the bubbleless economy is already efficient (r̄ > n), adding a bubble would eventually make it grow faster than the economy can sustain, which rules bubbles out, but this logic reverses when the bubbleless economy is itself dynamically inefficient (r̄ < n).

Proposition 2 states that “if r̄ < n, then the asymptotically bubbleless equilibria are inefficient and the asymptotically bubbly equilibrium is efficient.” The paper frames bubbles, in this inefficient case, as “the private counterpart to national debt à la Diamond,” transferring resources from the young generation to the old and thereby correcting the overaccumulation that characterizes the bubbleless Diamond equilibrium when it is dynamically inefficient (Sec. 3, p. 1079; Sec. 7, p. 1090).

With a constant total rent R per period that is capitalized from date zero, the paper’s Proposition 1 extends to show that when the long-run bubbleless-and-rentless interest rate r̄ is negative, “there exists no bubbleless equilibrium[;] there exists a unique bubbly equilibrium[,] asymptotically bubbly[,] and the interest rate converges to n,” because “if f_t does not converge to zero… this requires the interest rate to be relatively high in the long run (close to n),” while if it does converge to zero the rent’s market fundamental becomes infinite – a contradiction either way unless a bubble is present (Sec. 3, pp. 1076-1079). The paper also shows that if rents are instead created gradually over time rather than capitalized in advance – formalized by assuming each generation is born with a fixed endowment of as-yet-uncapitalized rents – then “intensive rent creation crowds out bubbles” but a bubbly steady state can still exist whenever the implied bubbleless interest rate r is below n (Prop. 3), so that “bubbles are not necessarily inconsistent with rents per period growing as fast as the economy”* (Sec. 3, pp. 1079-1082).

The paper shows that imposing a reserve requirement – consumers must hold at least a fraction δ of savings in an intrinsically useless “backed” asset (“gold”) – operates a selection among equilibria: Proposition 5 shows that “an arbitrarily small reserve requirement leads to the asymptotically bubbly path of the unbacked economy,” so that “the asymptotically bubbleless equilibria… are not robust to small reserve requirements whereas the asymptotically bubbly one is.” Proposition 6 further shows this selection is “approximately efficient”: as the reserve requirement shrinks toward zero, the resulting allocation converges uniformly to the (efficient) asymptotically bubbly allocation of the unconstrained economy (Sec. 4, pp. 1081-1087).

The paper shows that in an economy on the asymptotically bubbly path with two assets (stocks and gold) each carrying part of a deterministic aggregate bubble, “interest rates are deterministic and so is the market fundamental of stocks[;] however the bubble on stocks and therefore the price of stocks can fluctuate wildly,” so that “the variance bound [that the variance of price should not exceed that of market fundamental] is reversed,” and “the asset prices are negatively correlated” (Sec. 5, pp. 1085-1086). The paper notes this reversed-variance-bound result had already been remarked on in a partial-equilibrium setting by Blanchard and Watson [1982], and contrasts its prediction of overall negative cross-asset correlation with the Grossman-Shiller explanation of stock-price volatility via time-varying interest rates, which implies the opposite (positive correlation).

The paper derives that money’s market fundamental is “the present discounted value of transaction savings,” and that, unlike other assets, “its market fundamental, and not only its price, depends on future prices” (Sec. 6, p. 1087, Eq. 29). Proposition 7 (proved for a money-in-the-utility-function model under a stated Assumption A) states that “under Assumption A, there cannot exist a bubble on money,” provided money retains a positive transaction (or tax-payment, or reserve-requirement) value in the long run, with the same conclusion extending to Clower-constraint and money-financed-tax settings as long as the relevant shadow value of money remains bounded away from zero in the long run (Sec. 6, pp. 1088-1090, Prop. 7 and discussion).

The paper distinguishes “the ‘fundamentalist view’” – money held to finance transactions (or pay taxes, or meet a reserve requirement), which, by Proposition 7, implies “it is not held for speculative purposes as there is no bubble on money” – from “the ‘bubbly view’” in which “money is a pure store of value à la Samuelson [and] does not serve any transaction purpose at least in the long run,” implying the price of money grows at the real interest rate and money is “held entirely for speculation.” The paper states that “only the fundamentalist view can explain the rate of return dominance of money by other assets,” that “only this view seems to be consistent with money keeping some value under intensive creation,” and that “the fundamentalist view can also account for inflations while the bubbly view cannot if the real rate of return is nonnegative” (Sec. 6, pp. 1089-1090).

First, “costly bubble creation”: if a bubbly asset can be newly produced at positive cost, the paper states that “this creation… does not create any real wealth in the economy,” so the resulting equilibrium is not just inefficient but “Pareto dominated by some other perfect foresight equilibrium path in which creation of this asset is prohibited.” Second, “nonexhaustion of a market fundamental”: using the “black schmoos” example (a scarce, storable good priced above its white-schmoo substitute because of its presumed future resale value), the paper shows that “such an equilibrium is in[e]fficient” because the black schmoos are “never consumed if interest rates are positive,” which the paper identifies as “nothing but the well-known problem of potential inefficiency of private extraction of a nonrenewable resource” (Sec. 7, pp. 1090-1091).

Using the same “black schmoos”/gold-type example, the paper argues that “few people would say that the market fundamental of a black schmoo is zero” even though its financial market fundamental (the present value of its zero dividend stream) is indeed zero, because the asset retains a “real value” – its consumption value – so that “one is… led to distinguish between the financial market fundamental… and the real market fundamental,” with the paper noting that “properly speaking the real bubble is not a bubble, because it does not grow at the rate of interest[;] it rather is a measure of the overestimation of the asset relative to its real value.” The paper generalizes this to a non-dividend-paying firm (whose real market fundamental is the present value of profits net of debt) and to a backed asset under a binding reserve requirement, concluding that in each case “the real market fundamental serves as a lower bound on the market price” (Sec. 8, pp. 1091-1093).

The paper states “there are three conditions that are necessary to create a bubble: Durability, scarcity, and common beliefs,” explaining that “the common agreement requirement comes from the need for a focal point in the set of potential bubbles” and “the scarcity requirement stems from the fact that new units must have the same price as the old ones[; t]he possibility of creating too much of this asset may prevent bubbles on this asset.” The paper notes this “explains why, at first sight, bubbles often affect assets that for historical reasons cannot be reproduced,” giving rare stamps, letters of famous people, famous paintings, gold, diamonds, and land as potential examples, while cautioning that the supply of bubbles in principle is “virtually unlimited” and that common agreement is the binding constraint on which of them actually become focal (Sec. 9, pp. 1093-1094).

The paper’s conclusion states a belief “that the investigation of overlapping generations models should somewhat shift emphasis from the study of money to that of assets that are held for more speculative purposes,” while cautioning that “the models and the empirical evidence are too preliminary to settle the question of whether we should expect to observe asset bubbles in overlapping generations economies.” The paper explicitly states it has “not solved the old debate about which one of overlapping generations and infinitely lived consumers (or overlapping generations with bequests) is the ‘right’ model[; t]he difference between the two formalizations is substantial since bubbles may exist in the former but not in the latter.” The paper’s final assessment is that “in our current state of knowledge we would be best advised to believe that bubbles are not inconsistent with optimizing behavior and general equilibrium” (Sec. 10, pp. 1093-1094).