Technical Change and the Aggregate Production Function

Solow describes his contribution as “an elementary way of segregating variations in output per head due to technical change from those due to changes in the availability of capital per head” (Theoretical Basis, p. 312). He is explicit about the cost of this extra information: “the price consists of one new required time series, the share of labor or property in total income, and one new assumption, that factors are paid their marginal products” (p. 312) – and adds that he is not attempting to justify the exercise “by calling on fancy theorems on aggregation and index numbers,” acknowledging the difficulties Joan Robinson and others (including Solow himself, in earlier work) have raised about the concept of an aggregate capital stock (p. 312, fn. 1).

Solow is deliberately expansive: “I am using the phrase ’technical change’ as a shorthand expression for any kind of shift in the production function. Thus slowdowns, speed-ups, improvements in the education of the labor force, and all sorts of things will appear as ’technical change’” (p. 312). Formally, starting from Q=F(K,L;t) and specializing to the neutral case Q=A(t)f(K,L) – where neutral shifts “leave marginal rates of substitution untouched but simply increase or decrease the output attainable from given inputs” – total differentiation with respect to time, division by output, and substitution of the marginal-productivity relations for capital’s and labor’s income shares (w_k and w_L, which sum to one under constant returns and competitive factor payment) yields equation (2a): q-dot/q = A-dot/A + w_k(k-dot/k)*, where q=Q/L and k=K/L (p. 312-313). This equation can be estimated from time series on output per worker, capital per worker, and capital’s income share alone, without needing to know the underlying function f(K,L) (p. 313).

Solow restricts the exercise to U.S. private non-farm economic activity (to skirt the problem of measuring government output and to move toward homogeneity), using Kendrick’s real private non-farm GNP per man-hour as his output series (Application, p. 313-314). He flags the capital series as the part that “will really drive a purist mad”: it uses Goldsmith’s capital stock estimates (land, mineral deposits, etc. included; government, agricultural, and consumer durables excluded), reduced by the fraction of the labor force unemployed each year as a crude correction for capacity utilization (p. 314). Solow is candid that this correction is “undoubtedly wrong,” and separately flags an uncorrected problem of wartime multi-shift operation understating true capital input, plus a hodgepodge share-of-capital series pieced together from various sources and ad hoc assumptions (p. 314-315).

Cumulating the year-to-year A-dot/A estimates from equation (2a), fixing A(1909)=1, Solow reconstructs the full A(t) time series (Chart 3), finding “an average upward shift of about 1.5 per cent per year” over the whole 40-year period (p. 316) – a somewhat higher figure than Valavanis-Vail’s roughly 0.75 percent per year found by a different method over 1869-1948. The series shows sharp dips after each World War (which “can easily be rationalized”), a distinct levelling-off in the second half of the 1920s, and a sustained rise beginning again in 1930; more systematically, Solow finds the average rate of progress in 1909-1929 (about 1.2 percent per year) was smaller than in 1930-1949 (about 1.9 percent per year), while cautioning that “such post hoc splitting-up of a period is always dangerous” (p. 316).

A scatter of the estimated shift term (Delta-F/F) against the capital-labor ratio “indicates no trace of a relationship,” which Solow states as “a formal conclusion that over the period 1909-49, shifts in the aggregate production function netted out to be approximately neutral” (The Outlines of Technical Change, p. 316). He is careful to specify exactly what “neutral” means in this context: “the shifts were pure scale changes, leaving marginal rates of substitution unchanged at given capital/labor ratios” – and notes, in addition, that the shift rate itself looks close to constant over time apart from ordinary fluctuations, “though not quite,” given the 1929-1930 break already discussed (p. 316).

Over the 40-year period, real private non-farm GNP per man-hour rose from $0.623 to $1.275, while the cumulative shift factor A(t) rose about 80 percent. Solow’s calculation: dividing the 1949 output figure ($1.275) by the estimated A(1949) value of 1.809 (the full 40-year shift factor) yields a “corrected” GNP per man-hour, net of technical change, of $0.705 – so of the 65-cent total increase, about 8 cents is attributable to increased capital intensity and the remaining approximately 57 cents (roughly seven-eighths of the total) to technical change (Outlines of Technical Change, p. 316-317). Solow separately notes that a similar computation for just the first half of the period (1909-1929) attributes about one-third of the observed increase in output per man-hour to increased capital intensity, implying the relative importance of capital deepening was larger in the earlier sub-period (p. 317, fn. 6).

Solow compares his estimate to Solomon Fabricant’s finding that about 90 percent of the increase in output per capita over 1871-1951 is attributable to technical progress, and to Jacob Schmookler’s output-per-unit-of-input computations showing about a 36 percent increase in output per unit of input between 1904-13 and 1929-38 – noting his own A(t) rises 36.5 percent between 1909 and 1934, “but these are not really comparable estimates, since Schmookler’s figures include agriculture” (p. 316). He also directly compares his approach to the standard output-per-unit-of-input method, arguing that despite an initial appearance of being more assumption-free, “the implicit load of assumptions is quite heavy, and if anything the method proposed above is considerably more general,” since the usual method tacitly assumes both neutral technical change and a strictly linear (constant-returns, no diminishing-returns) aggregate production function (The Aggregate Production Function, p. 317).

Plotting q(t)/A(t) against k(t) collapses all the observed points onto a single member of the family of production curves; Chart 4 shows “an inescapable impression of curvature, of persistent but not violent diminishing returns” (p. 318). Solow fits five two-parameter functional forms (linear, semi-logarithmic, hyperbolic with an asymptote, Cobb-Douglas, and a logarithmic form with an asymptote), finding correlation coefficients “uniformly so high that one hesitates to say any more than that all five functions, even the linear one, are about equally good at representing the general shape,” though a runs test on the regression residuals shows the linear form is a “systematically poor fit” while the Cobb-Douglas and semi-logarithmic forms perform best (p. 318-319). On saturation: “it has already been mentioned that the aggregate production function shows no signs of levelling off into a stage of capital-saturation” within the observed range of the data, though Solow offers a “tongue in cheek” guess (using an assumed saturation output level of 0.95 and the linear fit) that saturation might occur at a capital-labor ratio “more than twice its present value” – a extrapolation he flags as “necessarily treacherous” (p. 319).

A cluster of points for 1943-1949 lies systematically above the main scatter in Chart 4, almost parallel to it, tempting the conclusion that “in 1943 the aggregate production function simply shifted” – but Solow rejects this reading since the whole point of the procedure was to purify the scatter of exactly such shifts (p. 317-318). His preferred explanation is “some systematic incomparability of the capital-in-use series,” specifically that wartime multi-shift operation meant capital services were more intensively used than the capital stock figures (even after the crude unemployment correction) would show, leading to an underestimate of capital input and hence an overestimate of the productivity increase for those years (p. 318). After experimenting with including these years and finding they produced “noticeable distortion,” Solow omits the 1943-1949 observations from the regressions reported in Table 2, calling the outcome unsatisfying: “it would be better if they could be otherwise explained away” (p. 318).

Solow’s own four-point summary states: (1) technical change during 1909-1949 was neutral on average; (2) the upward shift in the production function proceeded at roughly 1 percent per year in the first half of the period and roughly 2 percent per year in the second half; (3) gross output per man-hour doubled over the interval, with seven-eighths of the increase attributable to technical change and one-eighth to increased capital; and (4) the technical-change-corrected aggregate production function shows a distinct, though not violent, impression of diminishing returns (Summary, p. 320). He is careful to add a caveat about the counterfactual interpretation of these results: “this is not meant to suggest that the observed rate of technical progress would have persisted even if the rate of investment had been much smaller or had fallen to zero. Obviously much, perhaps nearly all, innovation must be embodied in new plant and equipment to be realized at all” (p. 316-317) – flagging that his neutral, additively-separable accounting does not settle the separate question of whether technical progress is itself dependent on the pace of investment.