# 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.