Assessing Dynamic Efficiency: Theory and Evidence

The paper opens by contrasting two established ways of diagnosing dynamic inefficiency: comparing the measured marginal productivity of capital (inferred from accounting profit rates, which “suggest that the marginal productivity of capital is about 10 percent, far above the economy’s growth rate”) against comparing the safe real interest rate, such as the Treasury bill rate, to the growth rate (Introduction, NBER WP pp. 1-2). The puzzle is that these two approaches disagree: the accounting-profit approach “indicate[s] the economy is dynamically efficient,” while “numerous…authors in a variety of contexts…note that real interest rates on safe assets are typically less than economic growth rates” — Ibbotson (1984) reports the mean real Treasury bill return over 1926-1983 was “roughly zero percent, well under the economy’s average growth rate” — which “seems to suggest that mature capitalist economies have overaccumulated capital” (Introduction, pp. 3-4).

The authors study an overlapping-generations economy in the tradition of Diamond (1965), but allow both population growth and the production technology to be stochastic, and allow a very general production technology that nests costs of adjustment, time-to-build, vintage capital, and embodied or disembodied technological progress (Section I, p. 5). Their central result, Proposition 1, states: “If D_t/V_t ≥ ε > 0 in all periods and all states of nature, then the equilibrium is dynamically efficient. If D_t/V_t ≤ -ε < 0 in all periods and all states of nature, then the equilibrium is dynamically inefficient,” where D_t is the “dividend” (defined below) and V_t is the market value of the capital stock (Section I.B, p. 10, Proposition 1). The authors note this is “a generalization of the Golden Rule result of Phelps (1961): an economy that invests more than its total profit in steady state is dynamically inefficient” (p. 10).

The dividend is defined by national-accounting identity as D_t ≡ Π_t − I_t, profit net of investment, which the paper shows is equivalently aggregate consumption less labor income: “the dividend as we define it equals profit less investment. (Equivalently, the dividend equals consumption less labor income.)” (Section I.A, eq. 1.6-1.7, p. 13). The authors emphasize that, “unlike the criteria applied in many previous applications of optimal growth models,” this criterion “does not depend critically on accounting judgments about depreciation or the measurement of profits in an inflationary environment. It requires only the comparison of the cash flows going into and coming out of an economy’s production sector” (Section I.D, p. 13). The applicability of the criterion does, however, rest on the assumption “that capital receives its marginal product,” which “excludes the possibility that capital income includes substantial monopoly profit” (p. 13).

Corollary 1 restates Proposition 1 in rate-of-return terms: “If R_t/G_t ≥ 1+ε > 1 in all periods and all states of nature, then the equilibrium is dynamically efficient. If R_t/G_t ≤ 1−ε < 1…the equilibrium is dynamically inefficient,” where R is the return on the market portfolio and G is the growth rate of the market value of that portfolio (Section II, eq. 2.1, p. 17, Corollary 1). Proposition 2 generalizes this further to the return on any single asset, including the safe rate, compared against the growth rate of the market value of the capital stock — but the authors immediately caution that “neither proposition is very helpful in judging the dynamic efficiency of actual economies, where capital gains and losses cause the growth of the market value of the capital stock sometimes to exceed and other times to fall short of the safe interest rate,” so that comparisons of the safe rate against the average growth rate of output or the capital stock, as is common in the literature, “are in general not sufficient to resolve the issue of dynamic efficiency” (Section II, p. 18).

Using NIPA data, Table 1 reports gross capital income (profit, rental, and interest income) and gross investment as shares of GNP for 1929-1985, and the authors report that “the difference between capital income and investment has exceeded 15 percent of GNP in every year since 1929,” with the dividend-to-wealth ratio D/V (available from 1952) having “consistently exceeded 4 percent of total wealth” (Section I.D, p. 15). The U.S. nonfinancial corporate sector shows an even larger gap, “not surprising given the heavier tax burdens placed on corporate capital” (p. 15), and cross-country data for England, France, Germany, Italy, Canada, and Japan over 1960-1984 show the criterion “comfortably satisfied for all countries in all years” — even Japan, “with its high rate of accumulation and tradition of low real interest rates,” which in 1984 “invested 16 percent of GNP and had gross capital cash flow of 28 percent of GNP” for the U.S. versus “28 percent and 38 percent” for Japan (Section I.D, pp. 15-16). The authors caveat, following Phelps, that “a path cannot be judged as dynamically efficient or dynamically inefficient prior to eternity,” so the finding is conditional on the future resembling the past (p. 16).

The authors construct an infinite-horizon representative-consumer economy, following Lucas (1978), with a single productive asset (a “fruit tree”) whose output evolves as a geometric random walk with positive drift, y_{t+1} = (1+g+ν_{t+1})y_t, consumed entirely each period; since the tree cannot be produced or felled, investment is always zero, so the criterion of Section I is trivially satisfied and the economy is dynamically efficient by construction (Section III, eq. 3.1-3.2, pp. 19-20). With logarithmic utility, they show the riskless gross return is R^F = [(1+g)/β]·exp(−σ²), so that “if the variance of ν is sufficiently large, it is clear R^F can be less than one, so that the net interest rate can be negative despite the upward drift in output,” and because the safe rate is constant, “safe interest rates over any horizon are also negative as long as output is sufficiently volatile” (Section III, eq. 3.5, p. 20). The mean return on the risky tree itself, by contrast, is (1+g)/β “regardless of the amount of uncertainty,” so “increases in uncertainty raise the risk premium solely by lowering the risk-free rate” — demonstrating that a low, or even permanently negative, safe rate is compatible with an economy that is unambiguously dynamically efficient (pp. 20-21).

The authors use the example to push back on arguments like Feldstein’s (1976), which infer overaccumulation (and the feasibility of Pareto-improving pay-as-you-go social security) directly from a safe rate below the growth rate: “as long as output can conceivably fall in each period, the feasibility of such a scheme is not guaranteed,” even though output has “an arbitrarily high probability of rising as the horizon is lengthened” — a fallacy the authors liken to the one “discussed monosyllabically by Samuelson (1979)” about investing only in high-yielding assets over long horizons: “‘almost surely’ is not certain enough” (Section III, pp. 21-22). The authors distinguish their question from the separate equity-premium puzzle of Mehra and Prescott (1985), noting that capital price risk “explain[s] the low safe rate” in principle even though “we have made no attempt to measure this risk” quantitatively, and that market imperfections such as liquidity constraints may still be needed to fully explain the magnitude of the observed gap (Section III-IV, pp. 19, 22-23).

The authors note two corollaries of their finding that the U.S. is dynamically efficient: first, “Tirole (1985) shows that rational speculative bubbles are ruled out in dynamically efficient economies,” and the paper’s “finding of a strictly positive dividend indicates that the capital stock is strictly below the Golden Rule, in which case bubbles cannot exist” (though the authors only “suspect, but have not proven” that Tirole’s deterministic result extends to their stochastic setting) (Conclusion, p. 23, and footnote 6). Second, Weil (1986) shows that an interior bequest motive of the kind Barro (1974) requires for Ricardian equivalence “can be ruled out in dynamically inefficient economies” — so the authors’ finding of dynamic efficiency means such a bequest motive “cannot be ruled out,” directly contradicting “the conclusion of Feldstein (1976)” that the low safe rate of interest precludes Ricardian equivalence (Conclusion, pp. 23-24).

The authors are explicit that their criterion is “the dynamic analogue of the standard Pareto criterion” and is therefore “relatively weak — passing it does not imply that a path is desirable for any very attractive social welfare function”; they call for future work using “stronger criteria than the dynamic efficiency criterion” to rank among efficient paths (Conclusion, p. 24). They also flag that their social welfare function (eq. 1.12) deliberately allows “separate social planning weights” for the same individual born under different states of nature, precisely so that the analysis speaks only to overaccumulation and not to intergenerational risk-sharing — “risk-sharing considerations, which we have systematically excluded from our analysis, provide yet another reason for introducing a social welfare function” of the kind social security programs embody (Conclusion, p. 25). They further note that dynamic (in)efficiency “cannot in principle be judged by observing only a particular segment of time” (Section I.D, p. 16).

The authors list, as explicit open questions: (1) characterizing the social welfare functions that would justify or rule out particular accumulation paths beyond the weak Pareto/dynamic-efficiency test; (2) incorporating risk-sharing considerations (e.g., via social security or tax policy) that they “systematically excluded” by construction; and (3) extending the analysis to economies with market imperfections such as liquidity constraints, since “considering dynamic efficiency in ‘distorted’ economies is a worthwhile subject for future research” (Section III-IV, pp. 22, 24-25). They also note they “suspect, but have not proven” that the Tirole and Weil results they invoke as corollaries generalize from the deterministic settings in which those authors proved them to the stochastic setting of this paper (Conclusion, p. 23-24).