# Growth Accounting

# Growth Accounting

Growth accounting describes the growth in the total product (output) of an economy or industry by decomposing it into its underlying determinants. This idea is very closely related to productivity growth (or technical change), typically defined as the amount of output growth not accounted for by its identifiable determinants, a residual often called a “measure of our ignorance” (Abramovitz 1962).

The economic building block underlying growth accounting is the production function Y = f(X, t), where Y is output (total production of goods and services), X is a vector (group) of inputs used for production, and t is a time counter. For example, the production function written as Y = f(K, L, M, t) says that the amount of output produced during a particular time period (say, a year) depends on the amount of physical capital (K, such as equipment and buildings) used, labor (L) employed, and primary and intermediate materials (M) purchased.

Based on this production function, and assuming that it represents all factors contributing to production, observed output growth over time can be attributed to (accounted for by) four causes—changes in each of the three inputs (j = K,L,M) and the passage of time (t, proxying technical change). Formally, this is written as the total derivative dY/dt = Σ_{j} ∂Y/∂X_{j} dX_{j}/dt + ∂Y/∂t. Because, mathematically, d refers to the full observed change in a variable and ∂ to a change “holding all else fixed,” this equation says that the full output change observed in the data is accounted for by the contribution of each input to output (marginal product of input j, ∂Y/∂X_{j}) multiplied by the input change observed in the data, plus the remaining time trend (residual).

Usually this expression is written in proportional terms by taking the derivative in logarithmic form, as dlnY/dt = Σ_{j} ∂lnY/∂lnX_{j} · dlnX_{j}/dt + ∂lnY/∂t, so each term becomes a percentage change. To then measure productivity growth or technical change (the percentage of output growth not accounted for by changes in input use), this expression is inverted to read ∂lnY/∂t = dlnY/dt-Σ_{j} ∂lnY/∂lnX_{j} · dlnX_{j}/dt, so the left-hand side of the expression, productivity growth, captures growth in output not accounted for by the measured inputs.

Two primary implementation issues immediately arise for measuring these expressions. First, dlnY/dt and dlnX_{j}/dt are simply data; assuming one has valid measures of real (not nominal) output and inputs, these are just the percentage changes in these measures between two time periods. However, ∂lnY/∂lnX_{j} is defined as the change in output from *only* changing X_{j}, which is not evident from actual data. Second, many other potentially measurable factors that contribute to output growth should not just be lumped into a residual, because that seriously limits its interpretability. For example, research and development (R&D) may be a specific cause of greater production from given input levels, which if distinguished separately would facilitate interpreting or explaining the “drivers” accounting for output growth. Further, even for standard factors of production, their measured changes may not reflect quality variations (say, education of laborers), which one would want to recognize to explain output growth patterns.

The first issue is typically dealt with by the theoretical assumption that firms maximize profits, which implies that inputs are paid the value of their marginal products (for example, a manager is willing to pay a worker up to the amount of money his labor generates). More formally, this means the input revenue share S_{j} = p_{j} X_{j}/p_{y} Y (where p_{j} and p_{y} are the prices of X_{j} and Y, so p_{j} X_{j} is the total amount of money paid to input j and p_{y} Y is the total value of output) can be substituted for ∂lnY/∂lnX_{j}. The output growth expression then becomes dlnY/dt = Σ_{j} S_{j} dlnX_{j}/dt + ∂lnY/∂t. Although this substitution is based on restrictive assumptions such as full equilibrium and perfectly competitive markets, it means all the components of the growth accounting expression are measurable directly from data on output and input levels and prices (in contrast to econometric estimation).

The second issue is somewhat more difficult to deal with, because even if factors such as R&D are included in the production function, there is no measurable “value” or “weight” for their productive contribution, such as the share paid to an input (S_{j}). That is, if the production function is written as Y = f(X, R, t) (where R = R&D), the growth accounting expression becomes dlnY/dt = Σ_{j} S_{j} dlnX_{j}/dt + ∂lnY/∂lnR . dlnR/dt + ∂lnY/∂t. This equation states that the percentage total output growth observed in the data is explained by the cost share of each of the j inputs multiplied by its percentage change, the marginal contribution to production of R&D multiplied by its percentage change, and the passage of time (trend). However, there is no measurable weight to substitute for ∂lnY/∂lnR without econometrically estimating the production function. Similarly, there is no clear way to separately identify changes in the quality or characteristics of the j inputs.

Early research in the growth accounting literature initially adjusted measures of capital and labor inputs by measures of quality or composition changes to put them in “effective” units. For example, Dale Jorgenson and Zvi Griliches (1967) “explained” the large productivity growth residual found by Robert Solow (1957) by direct adjustments of capital measures. The typically ad hoc nature of such adjustments, however, caused disputes among researchers about appropriate adjustment methodologies.

Additional issues arose from the more detailed growth accounting models of researchers such as Edward Denison (1979, 1985) and John Kendrick (1979), who included many additional potential growth drivers such as R&D, scale economies, capital obsolescence, and allocation, environmental, and “irregular” factors (such as weather). The multitude of factors that may affect output growth are, however, virtually impossible to measure and weight reliably and consistently for the growth accounting measure. Methodological problems emerging from the limited theoretical and empirical bases of many of the adjustments thus raised serious questions about their validity.

Due to these measurement and consistency issues, detailed growth accounting models have rarely been implemented recently, although this literature has provided the methodological foundation for growth accounting–based productivity or technical change measures. The focus of such studies has been identifying as carefully as possible the various inputs one can measure and find cost share weights for, so the components of the measure can be calculated without econometric estimation.

**SEE ALSO** *Change, Technological; Solow Residual, The; Solow, Robert M.*

## BIBLIOGRAPHY

Abramovitz, Moses. 1962. Economic Growth in the United States. *American Economics Review* 52: 762–782.

Denison, Edward F. 1979. *Accounting for Slower Economic Growth*. Washington, DC: The Brookings Institution.

Denison, Edward F. 1985. *Trends in American Economic Growth, 1929–1982*. Washington, DC: The Brookings Institution.

Jorgenson, Dale W., and Zvi Griliches. 1967. The Explanation of Productivity Change. *Review of Economic Studies* 34 (3): 249–282.

Kendrick, John W. 1979. Productivity Trends and the Recent Slowdown. In *Contemporary Economic Problems*, ed. William Fellner, 17–69. Washington, DC: American Enterprise Institute.

Solow, Robert M. 1957. Technical Change and the Aggregate Production Function. *Review of Economics and Statistics* 39 (5): 312–320.

*Catherine J. Morrison Paul*

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