Tornqvist Productivity Index

Introduction to Törnqvist productivity index, its calculation, comparison with Fisher's index, and application in production analysis using MATLAB.

Key Takeaways

Törnqvist productivity index is a weighted geometric mean-based measure of productivity change.
Weights in Törnqvist indexes reflect the proportional revenue or cost shares of outputs and inputs.
Törnqvist and Fisher productivity indexes are closely related but differ in their weighting schemes.
Reliable price data is crucial but often difficult to obtain for these productivity measures.
The Törnqvist index provides a theoretically sound and practical approach for productivity analysis in economics and production.

What the video covers

The video introduces the Törnqvist productivity index as a measure of productivity using output and input quantity indexes.
It explains the calculation of Törnqvist output quantity index as a weighted geometric mean of output changes with weights based on revenue shares.
Similarly, the Törnqvist input quantity index is calculated using input quantities weighted by their cost shares.
The productivity index is the ratio of Törnqvist output quantity index to input quantity index, indicating productivity change over time.
The video compares Törnqvist index with Fisher's productivity index, highlighting that Fisher's uses weighted arithmetic means while Törnqvist uses weighted geometric means.
It discusses the importance of price information for calculating these indexes and the challenges in obtaining reliable price data.
The relationship between Laspeyres, Paasche, Fisher, and Törnqvist indexes is elaborated with formulas and weight definitions.
The video emphasizes that Törnqvist index often yields values close to Fisher's index but tends to be slightly lower.
It summarizes the methodology and theoretical foundation for applying Törnqvist productivity index in production analysis.
The session is part of a course on Applied Production Analysis Using MATLAB, building on previous discussions about Fisher's productivity index.

Chapters

What is the Törnqvist productivity index?

The Törnqvist productivity index measures productivity change as the ratio of weighted geometric means of output quantities to input quantities, with weights based on revenue and cost shares.

How does the Törnqvist index differ from Fisher's productivity index?

While Fisher's index uses weighted arithmetic means of output and input quantities, the Törnqvist index uses weighted geometric means, often resulting in slightly lower values but closely related measures.

Why is price information important for calculating these productivity indexes?

Price data is needed to determine the weights reflecting the proportional revenue or cost shares of outputs and inputs, which are essential for accurate calculation of both Törnqvist and Fisher productivity indexes.

Full Transcript — Download SRT & Markdown

Speaker A

[music] [music] Hi. Welcome back to the course Applied Production Analysis Using MATLAB. In the last session, we discussed Fisher's productivity index, which was estimated as a ratio of Fisher's quantity index to Fisher's input quantity index. Fisher's output quantity index was created as a geometric mean of Laspeyres output quantity and Paasche's output quantity indexes, and similarly for the input quantity index.

Speaker A

So, in this session, we'll introduce a new measure of productivity that you can think of, given that you have prices that are very similar to the conditions of our Fisher's productivity index.

Speaker A

For that, first we estimate Törnqvist output quantity index. Then we see how to estimate Törnqvist input quantity index.

Speaker A

And then, using this output quantity index and input quantity index, how to estimate the Törnqvist productivity index.

Speaker A

Also, at a later stage, we compare or see how this Fisher's quantity index that we discussed earlier is connected or comparable with the Törnqvist index that I'm going to introduce.

Speaker A

So, we follow the same approach. Here we have two time periods, 1 and 0.

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Similarly, the way we have it for Fisher's index, we can conceptualize our output bundle for period one as Y1. It is going to be Y1 1, Y2 1, up to YM 1, basically M outputs that we are considering.

Speaker A

And similarly, we will have P1, that is basically the price vector for period one, that is the output price vector.

Speaker A

We can conceptualize it as P1 1, P2 1, up to PM 1. By changing the superscript, we can get the output and output price vectors for period zero as well.

Speaker A

In the same manner, we have input vectors, so let me define X0 as X1 0, X2 0, up to XN 0. We have N inputs. And W0 is basically the input price vector.

Speaker A

We can define it as W1 0, W2 0, to WN 0. Okay. The other two counterparts of the same you can conceptualize by changing the superscript.

Speaker A

Okay. So now we are going for the estimation of Törnqvist productivity index. So, in the first instance, we estimate Törnqvist output quantity index. That is basically we have Y1 for period one divided by Y1 for period zero.

Speaker A

Not complete. I'll come back to the weightage. Y2 for period one divided by Y2 for period zero.

Speaker A

Up to YM for period one to YM for period zero. So, this gives you an idea of how outputs one to M have changed over the period as compared to period zero for period one. But,

Speaker A

this formula is not complete because it gives equal weightage for all output changes.

Speaker A

But, in reality, we may have to give more weightage for outputs with higher price and less weightage for outputs with less price.

Speaker A

So, for that, we define V1, V2, up to VM as the weightage, and the moment you are taking these weightages and the product of these ratios, it becomes a geometric mean, a weighted geometric mean, I would say.

Speaker A

Now, how to define V1? So, or VJ has to be defined logically from the fact that the ratio or the weightage should capture what is the contribution of the Jth output in the overall revenue of the firm. Okay?

Speaker A

For that, we define it as PJ YJ divided by sum of K 1 to M PK VK. Here, the numerator will capture the price times output of the Jth output.

Speaker A

And the denominator is basically the total revenue, somewhat similar to total revenue. So, it says how much is the proportionate share of the Jth output in the total revenue of the firm that we are considering.

Speaker A

But, something is missing in this VJ. This PJ and YJ, you can conceptualize both from period zero and one.

Speaker A

We need to incorporate that. So, in reality, what we do, VJ is basically defined as VJ0 plus VJ1 divided by two.

Speaker A

Where VJ1 is defined as PJ1 YJ1 divided by sum of K 1 to M PK1 YK1.

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And we plug in our PK with a superscript one and YK one.

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And by changing the superscript, you can get a VJ0, and the arithmetic mean of our VJ0 and VJ1 will give you the VJ as the weightage. One condition is there, which is something very important. Sum of

Speaker A

VJ, J from one up to M, should be equal to one.

Speaker A

That is the condition that we impose, with the condition that makes the output quantity index under Törnqvist a weighted geometric mean framework.

Speaker A

Moving ahead, we can conceptualize Törnqvist input quantity index, that is basically going to be how these inputs have changed.

Speaker A

Ratio of X1 1 by X1 0. And here, in place of V1 that we had for our output case, we define it as S1 into X2 1 by X2 0.

Speaker A

S2 up to XN 1 by XN 0 with the weightage of SN, and the condition is that SR, R from one up to N in this context, should be equal to one.

Speaker A

So, this is the same as our output quantity index or Törnqvist output quantity index. Here, we are assigning a separate weight, that is basically SR. We can define it basically as WR XR divided by sum of R 1 to N WR XR.

Speaker A

Sorry, sum of can define it as for input J, WJ XJ, and R 1 to N WR XR. This is how we are going to estimate each SJ running from 1 to N in this context. And once we have Törnqvist output quantity and Törnqvist input quantity index, the ratio of that will give you productivity,

Speaker A

or Törnqvist productivity index, that is basically the ratio of output quantity index to input quantity index.

Speaker A

Similarly, in the case of Fisher's productivity index, if Törnqvist output quantity index is greater than Törnqvist input quantity index, we can say the productivity has improved over time.

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And then we can make an inference about the productivity change. So here, moving ahead, we can see a relationship between our Fisher's and Törnqvist productivity index.

Speaker A

And basically, Fisher's index and Törnqvist index, or I can use the term versus. Here we had a formula for our, say, Laspeyres quantity index, output quantity index I'm referring to, which was basically sum of J 1 up to

Speaker A

M PJ0 YJ1 divided by sum of J 1 up to M PJ0 YJ0. So, for keeping the rules of summation and doing slight modification, this can be written as

Speaker A

sum of lambda J into YJ1 by YJ0, J 1 up to M in this context. This is like has been done by opening the summation and following the summation rule. And now we are taking lambda J as a new variable. Basically, lambda J can be defined as

Speaker A

PJ0 YJ0 divided by sum of K 1 up to M PK0 YK0 in the context of Laspeyres price index. And similarly, you can define a mu J. So, here we can define lambda J0.

Speaker A

And similarly, you can define mu J1 following the same logic in the context of Paasche's index.

Speaker A

So, what is coming here? This is very similar to our VJ that we defined in the context of Törnqvist index. It is basically the same formula.

Speaker A

It says in period zero, what is the price and output of YJ and what is the share of that particular product in the overall revenue, that is basically defined as PK0 YK0.

Speaker A

So here, that gives us a hint where Laspeyres or Paasche's or the derived index, that is Fisher's, uses weighted arithmetic mean in the ratio of their relative output or input quantities.

Speaker A

At the same time, Törnqvist is going to be a weighted geometric mean of the relative output or input quantities.

Speaker A

So, that is the relationship. So here, in most cases, you'll get a very close value, but here we can see that we can simply check which Törnqvist is going to be slightly lower than that of the

Speaker A

Fisher's index that we are going to get. Just to summarize, following our discussion on Fisher's quantity index, in this session we were discussing Törnqvist productivity index.

Speaker A

So, for that, we estimated Törnqvist output quantity index by using a formula with an additional weightage that gives an idea of how much is the proportional share of the particular output in the total revenue of our firm under consideration. Similarly, by taking a weighted geometric mean of the relative inputs for period zero and one, we get Törnqvist output input quan

Speaker A

So, for that we estimated Tornqvist output quantity index by using a formula with an additional weightage that gives an idea that how much is the proportional share of the particular output in the total revenue of our con- the

Speaker A

firm under consideration. Similarly, by taking a geo- weighted geometric mean of the input relative inputs for period zero and one, we get Tornqvist output input quantity index where this weightage was decided such that uh it capture the relative share of each

Speaker A

input factors in the total cost of the firm. Here also, similarly, the way we have VJ, we can have SJ SJ0 plus SJ1 divided by two. by two.

Speaker A

It's basically the ratio that referring to. So, in the next session we'll try to estimate it using MATLAB.

Speaker A

So, as I mentioned, uh any index that we are estimating using Fischer's or Tornqvist index need a price information which will be very difficult for us to have. And even if you have the price information sometime uh it might not be very reliable. This

Speaker A

price may change because of the factors external to the firm, say inflation, so and so.

Speaker A

And in sectors like banking or health or education, getting a price data is going to be even difficult, especially the uh services being provided to the public, general public. From a welfare point of view, quantifying the price of

Speaker A

these services or inputs or output is going to be very complicated. Having said that, you won't get much of the empirical literature that talks about Tornqvist or Fischer's index that we discussed earlier.

Speaker A

But still, these two indexers gives you a theoretical foundation how productivity being estimated. [music] Thank you.

Speaker A

[music]

Topics:Törnqvist productivity indexFisher productivity indexApplied Production AnalysisMATLABoutput quantity indexinput quantity indexweighted geometric meanLaspeyres indexPaasche indexproductivity measurement

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