The Review of Economics and Statistics | Macro Paper Warehouse

Interactions between the Multiplier Analysis and the Principle of Acceleration

This short 1939 note by Paul Samuelson, written during his time as a member of the Society of Fellows at Harvard at the suggestion of Alvin Hansen, formalizes Hansen’s combination of Keynes’s income-expenditure multiplier with the accelerator principle of induced private investment. Samuelson sets up national income in each period as the sum of a constant level of government deficit spending, consumption equal to a fixed fraction (the marginal propensity to consume, a) of the previous period’s income, and induced private investment equal to a coefficient (the “relation,” b) times the change in consumption between the previous two periods. Working through a numerical example (a = 1/2, b = 1, tabulated period by period) and then a table of alternative coefficient values, he shows that adding this accelerator term to the plain multiplier can turn an otherwise smoothly convergent income sequence into one that oscillates, and that whether the resulting path damps out, repeats indefinitely, explodes in oscillation, or instead grows or shrinks monotonically without any oscillation at all depends purely on the numerical values of a and b. He derives the exact boundaries between four such qualitatively distinct regions algebraically from the roots of the underlying difference equation’s characteristic quadratic and displays them as a chart in (a, b) space, so that the conventional multiplier (recovered as the special case b = 0) appears as one special case of a more general family in which cyclical fluctuations can arise purely from the mechanical interaction of consumption and investment lags, without any outside shock. Samuelson closes by flagging that the whole analysis is explicitly marginal – it treats a and b as constants even though they would actually shift with the level of income – and by noting that the formal structure of his model sequence parallels contemporaneous dynamic work by Lundberg and Tinbergen.

MPCs, MPEs, and Multipliers: A Trilemma for New Keynesian Models

This paper shows that New Keynesian models with frictionless labor supply cannot simultaneously match three well-established macro and micro facts: high average marginal propensities to consume (MPCs, about 0.25 quarterly), low average marginal propensities to earn (MPEs, between 0 and 0.04 annually), and fiscal multipliers that are moderate under accommodative monetary policy (0.6 to 2). Using standard consumer theory, the authors show at the individual level that the ratio of MPE to MPC is governed by a “complementarity index” (CI) between consumption and labor in preferences, together with the Frisch elasticity and the elasticity of intertemporal substitution (EIS): matching high MPCs and low MPEs simultaneously requires CI close to 1, as under Greenwood-Hercowitz-Huffman (GHH) preferences. But in a representative-agent New Keynesian model with a constant real interest rate, they derive an exact formula showing the fiscal multiplier equals 1/(1 - (1-tau)CI), where tau is the steady-state labor wedge; separable preferences (CI = 0) give Woodford’s (2011) multiplier of exactly 1, while GHH preferences (CI = 1) give a multiplier of 1/tau, typically 5 or more under standard calibrations – far outside the empirically plausible range. Solving a quantitative heterogeneous-agent New Keynesian (HANK) model with flexible “GHH-plus” preferences that span the full range of complementarity, calibrated to always match the target MPC, the authors show numerically that no value of the complementarity parameter can deliver both an acceptable MPE and an acceptable cumulative fiscal multiplier at once – the trilemma survives, and is robust to varying the EIS, the Frisch elasticity, the markup, and the progressivity of financing taxes. The authors’ proposed resolution is to introduce nominal wage stickiness and demand-determined labor, which mechanically sets every household’s MPE to zero regardless of preferences, freeing the model to use separable preferences (CI = 0) to simultaneously match high MPCs and a moderate multiplier (1.21 on impact, 1.18 cumulative in their calibration).

Technical Change and the Aggregate Production Function

Robert Solow’s 1957 paper proposes a simple method for separating shifts in the aggregate production function (“technical change,” broadly defined) from movements along it caused by capital accumulation, and applies it to U.S. private non-farm output from 1909-1949, finding that seven-eighths of the doubling in output per worker-hour is attributable to technical change and only one-eighth to increased capital per worker. Starting from an aggregate production function Q=F(K,L;t), Solow specializes to the case of neutral technical change, Q=A(t)f(K,L), where neutrality means the shift “leaves marginal rates of substitution untouched” and simply scales output at any given capital-labor ratio; under the standard (and, he argues, practically unavoidable) assumption of constant returns to scale and competitive factor markets paying marginal products, this yields a simple decomposition of the growth rate of output per worker, q-dot/q, into a technical-change term A-dot/A and capital’s income share times the growth rate of capital per worker, w_k*(k-dot/k) – requiring, to estimate it, only time series of output per worker, capital per worker, and capital’s share of income, and one new assumption (competitive factor pricing), without needing to specify the exact functional form of the production function. Applying this to U.S. private non-farm GNP per man-hour, an estimate of the capital stock (Goldsmith’s data, crudely corrected for unemployment but not for wartime multi-shift operation) and factor-share data for 1909-1949, Solow reconstructs the cumulative shift factor A(t) year by year; a scatter of the year-to-year technical-change term against the capital-labor ratio shows essentially no relationship, so he concludes technical change over the period was, on average, neutral, though the average annual rate of shift roughly doubled between the first and second halves of the sample (about 1 to 1.2 percent per year before 1929 versus roughly 2 percent per year after 1930). The paper’s headline growth-accounting result compares the near-doubling of output per man-hour ($0.623 to $1.275) against the roughly 80 percent cumulative rise in A(t): correcting the 1949 output figure for the estimated technical-change factor implies that about one-eighth of the 40-year increase in output per hour is attributable to increased capital intensity and the remaining seven-eighths to technical change broadly defined. Dividing the resulting technical-change-corrected output-per-worker series by A(t) and plotting it against capital per worker (Chart 4) reveals a scatter with a distinct, though not violent, curvature consistent with diminishing returns; several two-parameter curves (including Cobb-Douglas, semi-logarithmic, and others with upper asymptotes) fit this corrected scatter about equally well, with the linear specification performing noticeably worse, and the data show no sign of approaching capital saturation within the observed range. Solow flags a cluster of wartime and postwar observations (1943-1949) as anomalously high relative to the rest of the scatter, likely reflecting underestimated capital utilization from unmeasured multi-shift wartime operation, and, after experimentation, excludes these years from the regressions reported in the paper’s tables.

The Interest-Elasticity of Transactions Demand For Cash

This paper works out, with a simple mathematical model, whether the ordinary transactions demand for cash – the money people hold just to bridge the gap between when they receive income and when they spend it – responds to the rate of interest, challenging the then-standard view that transactions balances are essentially interest-inelastic and only asset-motive money demand responds to rates. Tobin models an individual who receives income Y at the start of a period and spends it at a constant rate until it is exhausted, and who can hold part of that balance in interest-bearing bonds rather than cash, subject to a transaction cost with a fixed component plus a component proportional to the amount transferred each time cash and bonds are exchanged. Solving in three steps – the optimal timing and size of a given number n of cash-bond transactions, the profit-maximizing number of transactions n* for a given interest rate r, and how n* (and hence average cash and bond holdings) moves as r changes – the paper shows that whether cash demand responds to the interest rate at all depends on which of four regimes the interest rate, the volume of transactions, and the transaction-cost parameters place the individual in: below a threshold interest rate, no bond transaction is worthwhile and cash demand is completely insensitive to r, while above that threshold the share of the transactions balance held in bonds rises continuously with r. That threshold, and the degree of sensitivity above it, is not the same for everyone: because the fixed component of the transaction cost does not scale with income, the interest-elastic range of r widens as the volume of transactions Y grows, so small transactors may never find it worthwhile to economize on cash while large transactors become increasingly rate-sensitive, and the ratio of cash held to Y falls as Y rises within that elastic range. An appendix both proves the results formally and situates the model against Baumol’s (1952) inventory-theoretic transactions-demand paper, noting that Tobin’s paper proves rather than assumes that optimal cash withdrawals are equal in size and equally spaced, treats the number of transactions as an integer rather than a continuous variable, and – unlike Baumol – allows for the case in which no bond transaction is worth making at all.