Total Factor Productivity (TFP): Formula and Solow Residual

Total factor productivity (TFP) is the part of output growth that cannot be explained by growth in measured inputs such as labor and capital. Researchers use it to ask how much of an economy’s, industry’s or firm’s growth came from using inputs better rather than from using more of them. After reading this page you should be able to compute a TFP growth rate from four numbers, state the assumptions behind it, and report it correctly.

What is total factor productivity (TFP)?

TFP relates real output to the combined use of inputs. The US Bureau of Labor Statistics (BLS) describes multifactor productivity (MFP) as “also called total factor productivity or the ‘residual'”, and its Productivity program uses “total factor productivity (TFP), also known as multifactor productivity (MFP)” for the same idea: output growth compared with the growth of combined inputs (labor, capital and, in industry measures, energy, materials and purchased services). It is therefore a residual: whatever output growth is left after the measured inputs have been accounted for.

The residual captures more than “technology”. BLS notes that it reflects technological change, efficiency, returns to scale, reallocation of resources, worker skills and management practice, and that it does not isolate the contribution of any single input.

TFP formula: the Solow residual

The standard growth-accounting form, introduced by Robert Solow (1957), writes output as Y = A × K^α × L^1-α and expresses TFP growth as the residual:

g_A = g_Y – α × g_K – (1 – α) × g_L

• g_A: growth rate of TFP (the Solow residual), per period.
• g_Y: growth rate of real output (for example real GDP or real value added).
• g_K: growth rate of the capital input.
• g_L: growth rate of the labor input (hours worked is better than headcount).
• α: capital’s share of income or cost, between 0 and 1. Labor’s share is 1 – α.

BLS builds its measure the same way: output growth minus the cost-share-weighted growth of inputs, with each weight set to the average of the two periods’ shares of total costs. The level version is A = Y / (K^α × L^1-α).

Worked example with real arithmetic

The numbers below are hypothetical, chosen to show the arithmetic; they are not data for any real economy.

Suppose real output grows 4.0% in a year, capital input grows 5.0%, labor input grows 2.0%, and capital’s cost share is α = 0.35 (so labor’s share is 0.65).

1. Capital contribution: 0.35 × 5.0 = 1.75 percentage points.
2. Labor contribution: 0.65 × 2.0 = 1.30 percentage points.
3. Combined input contribution: 1.75 + 1.30 = 3.05 points.
4. TFP growth: 4.0 – 3.05 = 0.95% per year.

Of the 4.0% output growth, 3.05 points come from more inputs and 0.95 points (about 24%) are unexplained by inputs, which is the TFP residual.

How to interpret a TFP growth rate

Result	Reading
gA > 0	Output grew faster than the weighted growth of inputs: more output per unit of combined input.
gA = 0	Output growth is fully accounted for by input growth.
gA < 0	Output grew more slowly than inputs. BLS reported, for example, that TFP fell in 70 of 86 four-digit NAICS manufacturing industries in 2023 (release of 24 June 2026).

A positive residual is not proof of faster invention. Because it absorbs everything the inputs do not explain, it also moves with measurement error, changes in how intensively capital and labor are used, and reallocation across firms.

TFP vs multifactor productivity vs labor productivity

• TFP and MFP are used interchangeably by BLS; the BLS Handbook of Methods lists both names for one measure.
• Labor productivity compares output with labor alone, so it rises when workers have more capital, not only when production becomes more efficient. TFP nets out capital as well.
• Utilization-adjusted TFP removes the effect of varying capacity use over the business cycle. The Federal Reserve Bank of San Francisco publishes a quarterly US business-sector series on this basis (method of Basu, Fernald and Kimball, 2006) alongside the unadjusted series, labor productivity and a utilization estimate, so the two TFP versions can be compared directly.

In cross-country datasets TFP appears as a precomputed variable; the guide to the Penn World Table explains the ctfp and related variables.

Assumptions and common mistakes

Assumption (BLS MFP method)	What it means for you
Hicks-neutral technical change	Technology scales output without changing the balance between capital and labor.
Constant returns to scale	Doubling all inputs doubles output.
Input elasticities equal cost shares	Holds under perfect competition in product and input markets; marginal products cannot be observed, so shares stand in for them.
Well-defined, independently measured output and inputs	Errors in measuring capital or hours flow directly into the residual.

• Treating the residual as pure technology; it also captures efficiency, skills, management and reallocation.
• Mixing growth rates and levels, or using headcount where hours are available.
• Using a fixed α without checking that it matches the sector and period; BLS averages shares across the two periods compared.
• Comparing TFP growth across studies that use different input definitions (labor and capital only, or also energy, materials and services).

How to report TFP (APA example)

State the output and input definitions, the share α and its source, the period and the method. Example in APA 7th style:

“Total factor productivity growth, computed as the Solow residual with a capital share of .35, was 0.95% per year (output growth 4.0%, capital growth 5.0%, labor growth 2.0%).”

Cite the method (Solow, 1957) and the data source in the reference list, and report TFP growth in percent per year.

How to calculate TFP growth in Excel, R or Python

Excel (growth rates in B1 to B3, share in B4): =B1-B4*B2-(1-B4)*B3

R

gY <- 4.0; gK <- 5.0; gL <- 2.0; alpha <- 0.35
gA <- gY - alpha*gK - (1 - alpha)*gL
gA  # 0.95

Python

gY, gK, gL, alpha = 4.0, 5.0, 2.0, 0.35
gA = gY - alpha*gK - (1 - alpha)*gL
print(round(gA, 2))  # 0.95

For a panel of series, apply the same line column-wise. If your data are index levels, take the difference of natural logs first (for example ln(Y_t) – ln(Y_t-1)) so the growth rates are consistent. For related methods see how to interpret R-squared and causal analysis; for how research output itself is measured, see scientometrics.

Frequently asked questions

What does a TFP of 1% mean?

It means output grew 1% more than the weighted growth of inputs over the period; it is a growth rate of efficiency, not a level.

Is TFP the same as the Solow residual?

Yes, in the growth-accounting sense: TFP growth is the part of output growth left after subtracting share-weighted input growth.

Is TFP the same as technology?

No. BLS states that the measure also reflects efficiency, returns to scale, reallocation, skills and management.

Why is capital share usually between 0 and 1?

Because α and 1 – α are shares of total income or cost and must sum to one under constant returns to scale.

Sources

• US Bureau of Labor Statistics, Handbook of Methods, “Multifactor productivity: concepts” (bls.gov/opub/hom/msp/concepts.htm), read 2026-11-11.
• US Bureau of Labor Statistics, Productivity program (bls.gov/productivity), read 2026-11-11.
• Federal Reserve Bank of San Francisco, Total Factor Productivity (frbsf.org), read 2026-11-11.
• Solow, R. M. (1957). Technical change and the aggregate production function. Review of Economics and Statistics, 39(3), 312-320.
• Basu, S., Fernald, J. G., & Kimball, M. S. (2006). Are technology improvements contractionary? American Economic Review, 96(5), 1418-1448.