# Measuring productivity with the Törnqvist index

I have a sample of companies, of which I want to calculate the productivity of each individual firm relative to a hypothetical firm in the industry.

Choosing the measurement method was not easy, but I ended up using the Törnqvist index number approach. I am aware of the assumptions of the method and the difficulties in measuring productivity. My questions therefore relate to the actual calculation of the index. Although the formula seems straight-forward, I experience some problems when calculating it.

The basic formula is given below: $$\ln A_{it}^{IN} - \overline{\ln A_{t}^{IN}} = \left( \ln Q_{it} - \overline{\ln Q_{t}} - \tilde{s}_{it} \left(\ln L_{it} - \overline{\ln L_{t}} \right) \right) - ( 1-\tilde{s}_{it}) \left(\ln K_{it} - \overline{\ln K_t} \right)$$

$$\tilde{s}_{it} = \left(s_{it}^L + \overline{s_t^L} \right) / 2$$

This is how I defined the variables:
1. Q is the output of the firm. I defined it as sales or revenue. I know its shortcomings, but given the data I cannot make extensive corrections. Based this on research of Melitz (2000).
2. L is the labour productivity, which I defined as the total hours worked. According to Ahmad et al. (2003) this is the best proxy.
3. K is the physical capital, which I defined as Plant, machinery and equipment. Excluding buildings and land (Land should not be included, while buildings should, but I don't have information separately).

Please note that the data I have is limited for each company. I have a sample of companies of which I can access global lines from the income statement and balance sheet. I therefore have to make a trade-off between the size of the sample (more and detailed data means a smaller sample) and the quality of the proxy for productivity.

## Problems

First of all, to calculate the productivity of a specific firm, I first have to calculate the productivity of the hypothetical firm (the geometric mean of the industry), but how do I need to do this? The formula says on the left I need the productivity of the hypothetical firm in order to find the firm-specific productivity. Or do I just need to calculate the geometric mean of my variables for each industry and omit the second term of the left part of the formula? I am confused because I would need to calculate the productivity of the hypothetical firm but how? The first one is always needed for the calculation of the second one and vice versa.

Second, for the calculation of the term ~s, I calculate the share of labour in the total input. But as I said before I calculate labour using total hours worked and capital using balance sheet lines (in €euro). These two are therefore measured in different units. I should multiply the hours worked with an average base wage to get the labour in monetary units. It is therefore better to just use total employment costs, right? I can find this cost in the income statements.

Third, I have a question related to determining capital. This should reflect all physical assets used in production. So should I include leasing (on the balance sheet). As of know I only include plant, machinery and equipment.

The formula you have above to solve for multifactor productivity (or total factor productivity) is correct; however, after reading your post, you are interpreting it differently.

The terms with the natural logs in the formula is asking you to obtain the rate of change (growth rates) [same as the difference in logs seen in formula] of a particular period for capital, labor , and output and the average cost shares for labor and capital. The term on the left indicates the log difference of MFP which is the same as the rate of change (growth rate) of MFP.

To answer your first question: If you are calculating firm level productivity you should be using firm level labor input (number of hours worked of all persons), capital input , output data.

For your second question: The number of hours should not be used as labor input cost share. That term would be labor compensation or total wages.

The term ~s is actually the two period average cost share of labor input [cost share of labor input is the labor cost (or labor compensation) divided by the cost of both labor and capital (which should be the same as total revenue/output/sales)].

-**So your statement is correct; you should multiply the hours worked with an average base wage to get the labor cost in monetary units. total employment costs should also be the same and work. **-

For average capital cost share, the formula provided is actually showing you a shortcut in coming up with average capital cost share. Just take the average labor cost share and subtract it by 1.

For your third question: Capital should reflect the real value all physical assets related to the firm’s final output which should include the use of:

Equipment

Structures

Inventory- items that are in the process of being prepared for sale

Intellectual property products (software, R&D, artistic originals)

Land

These fixed production assets costs should include depreciation.

To calculate MFP growth in 1987-1988 (or 1988%) Real output % = 4.3

Labor input% = 3.2

Capital input%= 3.9

Average labor cost share = ( .663 + .671) /2 = .667

Average capital cost share= (1- .667) = .333 [shortcut]

4.3 – (.667*3.2) – (.333*3.9) = .87

Actual value of MFP% is = .8

*note that doing it this way in few periods will cause rounding issues that may show a .1 (tenth decimal) difference. But in most periods you will be right on target.

• Thank you for this very clear answer! I would give you a +1, but I don't have enough reputation points. – BRCO Apr 4 at 8:33
• can you accept this as the answer? – Mike J Apr 4 at 11:07
• also on the top left of the answer post you can select the up arrow to indicate that this was a useful answer. That would be appreciated. Thanks. – Mike J Apr 4 at 11:25