There are two versions of TFP in the NBER-CES Manufacturing Database: 4-factor and 5-factor. The 5- factor version separates out energy from non-energy materials; the 4-factor uses a single materials input (which includes energy). The TFP calculation requires definitions of the cost shares, the factors, the factor changes, and the output changes. The five cost shares (`alpha_i`) vary by industry by year, defined using the variable names in the dataset:

• (`alpha_1`) Non-production workers: (pay-prodw)/vship [i.e., their pay divided by shipments]
• (`alpha_2`) Production workers: prodw/vship [i.e., their pay divided by shipments]
• (`alpha_3`) Energy: energy/vship [i.e., energy expenditure divided by shipments]
• (`alpha_4`) Materials: (matcost-energy)/vship [i.e., non-energy materials divided by shipments]
• (`alpha_5`) Capital: 1 - (sum of the above shares) [i.e., the residual]

In calculating TFP growth from one year to the next, we use the average of the two years' cost shares: `bar alpha_{it} = 0.5alpha_{it} + 0.5alpha_{it-1}` The 5 factors (`X_i`) are defined as follows, using the variable names in the dataset:

• (`X_1`) Non-production workers: (emp-prode) [i.e., the number of non-production workers]
• (`X_2`) Production workers: prodh [i.e., production worker hours, not employees]
• (`X_3`) Energy: (energy/pien) [i.e., real energy expenditures]
• (`X_4`) Materials: ((matcost/pimat) - (energy/pien)) [i.e., real non-energy materials]
• (`X_5`) Capital: cap [i.e., total capital stock, already in real terms]

The change in factor usage between one year and the next is defined as the change in natural logs, (for example): `dX_{it} = ln(X_{it}) - ln(X_{it-1})`

We also need the change in real output: (Q) Real output: vship/piship

As with factor usage, we express output change in terms of natural logs, hence: `dQ_t = ln(Q_t) - ln(Q_{t-1})`

The change in 5-factor TFP (dTFP5) between this year and last is thus defined as: `dTFP5_t = dQ_t - sum_i(alpha'_{it}dX_{it})`, `i = 1,..., 5`

Given the series of dTFP5 values, one can then "roll up" these changes to form a TFP index (TFP5), by setting the index equal to 1.0 in some initial year `t` and then growing the index forward by the following equation: `TFP5_{t+1} = exp[ ln(TFP5_t) + (dTFP5_{t+1})]`

The values of 4-factor TFP growth (dTFP4) and the corresponding TFP index (TFP4) are calculated similarly, but using total materials cost spending rather than separating it into energy and non-energy materials.