Fisher Productivity Index

Learn how to estimate productivity change using the Fisher productivity index with MATLAB, focusing on output and input quantity indexes.

Key Takeaways

Fisher productivity index uses geometric mean of Laspeyres and Paasche quantity indexes to measure productivity change.
Productivity improvement is indicated when output quantity index exceeds input quantity index.
The index compares productivity relative to a base period, typically the previous year.
Fisher index methodology parallels price index calculation but focuses on quantities weighted by prices.
Despite its theoretical appeal, Fisher's index is less commonly applied in empirical productivity studies.

What the video covers

Introduction to Fisher productivity index as an approach to estimate productivity change using price data of outputs and inputs.
Explanation of the difference between price index and quantity index, with Fisher index focusing on quantity indexes weighted by prices.
Step-by-step method to estimate output and input quantity indexes using simple one input-one output cases.
Definition and calculation of Laspeyres and Paasche quantity indexes for both output and input.
Use of geometric mean of Laspeyres and Paasche indexes to compute Fisher's output and input quantity indexes.
Explanation of productivity index as the ratio of output quantity index to input quantity index.
Mathematical formulation of productivity indexes for multiple inputs and outputs.
Interpretation of Fisher's productivity index values, where values greater than one indicate productivity improvement.
Comparison of Fisher index with other productivity measurement frameworks like Solow's.
Mention of limited empirical application of Fisher's index in productivity change literature.

Chapters

What is the Fisher productivity index?

The Fisher productivity index is a measure of productivity change that uses the geometric mean of Laspeyres and Paasche quantity indexes for outputs and inputs, weighted by prices.

How do you calculate the Fisher output quantity index?

You first calculate the Laspeyres output quantity index using base period prices and the Paasche output quantity index using current period prices, then take the square root of their product.

What does it mean if the Fisher productivity index is greater than one?

A Fisher productivity index greater than one indicates that productivity has improved in the current period compared to the base period.

Full Transcript — Download SRT & Markdown

Speaker A

Hi. Welcome back to the course of total factor productivity analysis using MATLAB. In today's session, we'll be covering one of the approaches for estimating productivity change, especially when you're having price data, particularly that of outputs and inputs. That is basically the Fisher productivity index. So, the Fisher index is very familiar to most of you, especially in the context of price index.

Speaker A

the Fisher productivity index. So, Fisher index is very familiar to most of you, especially in the context of price index.

Speaker A

We estimate Fisher price index. We'll be following the same framework, more or less.

Speaker A

But here, instead of price index, we'll be focusing on the quantity index. So, in place of quantity, price plays the uh weightage.

Speaker A

But here, instead of price index, we'll be focusing on the quantity index. So, in place of quantity, price plays the weightage.

Speaker A

And for estimating Fisher's productivity index, first we need to estimate the Laspeyres quantity index.

Speaker A

As against the price index, well, quantity plays a weightage role. So, in this session, I will take you through how to estimate output and input quantity index by considering a very simple one input one output case.

Speaker A

These two. And for estimating Fisher's input quantity index, we need both Paasche and Laspeyres input quantity index indexes and we'll be using geometric mean of that.

Speaker A

And for estimating Fisher's productivity index, first we need to estimate the Laspeyres quantity index.

Speaker A

So, in the first instance we can consider a productivity index. So, basically when you talk of an index, it is basically mentioning about a change with a base period.

Speaker A

Or for estimating Fisher's output quantity index, we need Laspeyres and Paasche's output quantity index, and we'll be taking the geometric mean of them.

Speaker A

So, we'll be following the same framework. We need output how much out output has been changed and how much input has been changed proportionately and then the ratio of that will give us an idea whether output has changed more than

Speaker A

These two. And for estimating Fisher's input quantity index, we need both Paasche and Laspeyres input quantity indexes and we'll be using the geometric mean of that.

Speaker A

So, here we are defining pi i is basically the productivity of observation in the period one that is basically y1 divided by x1 in a simple one input one output case.

Speaker A

So, before directly estimating or directly jumping into estimation of Fisher's index, we need to estimate Laspeyres and Paasche and then using a formula we'll be estimating the Fisher's index.

Speaker A

It is productivity in the period zero, and we call it as the base period.

Speaker A

So, in the first instance we can consider a productivity index. So, basically when you talk of an index, it is basically mentioning about a change with a base period.

Speaker A

Basically nothing other than Y1 by X1 divided by Y0 by X0. So, alternatively we can write it as Y1 by Y0 divided by X1 by X0.

Speaker A

So, when I say the productivity of period one has changed certain percentage that is basically as against the last year the productivity has improved.

Speaker A

input quantity index. Whenever you are output quantity index is greater than that of the input quantity index, that means productivity has improved, or there is a portion of change in output which is not being explained by the uh change in input.

Speaker A

So, we'll be following the same framework. We need output, how much output has been changed and how much input has been changed proportionately and then the ratio of that will give us an idea whether output has changed more than

Speaker A

For that, it is going to be the geometric mean of Laspeyres index and Paasche's index, but not in a very direct sense.

Speaker A

proportionately than that of the input change which will tell us whether the productivity has changed. It is similar to the Solow framework.

Speaker A

step that we follow. It is not like first you estimate Laspeyres productivity index, and then you estimate Paasche's productivity index, and then take the geometric mean of that. Logic doesn't work over here.

Speaker A

So, here we are defining pi. Pi is basically the productivity of observation in the period one that is basically y1 divided by x1 in a simple one input one output case.

Speaker A

Then, taking the ratio of that will give you a Fisher's productivity index. So, here we define our output vector.

Speaker A

Alternatively, we can have a productivity in the period zero that is going to be Y0 divided by X0.

Speaker A

So, we can define our output bundle for period zero as basically Y10 {comma} Y20 up to YM0.

Speaker A

It is productivity in the period zero, and we call it as the base period.

Speaker A

Okay? Correspondingly, we will be having the price index as well or the price information.

Speaker A

And now, small pi one, that is basically the productivity index I'm referring to, which is basically with the base of last year. That is defined as pi one by pi zero.

Speaker A

And then the input prices are defined as W0, which is basically W10 W20 up to W N0 where you have N number of inputs.

Speaker A

Basically nothing other than Y1 by X1 divided by Y0 by X0. So, alternatively we can write it as Y1 by Y0 divided by X1 by X0.

Speaker A

P21 up to PM1 and correspondingly we can have a input vector for period one and input vector for or input price vector for period one.

Speaker A

Here, this component I can call as output quantity index, and this component we can call as input quantity index, and the productivity index that we are going to estimate is basically estimated as a ratio of output quantity index and

Speaker A

So here we define our Laspeyres output quantity index as L Q Y. It is going to be a quantity index that uses last year price information as the weightage.

Speaker A

input quantity index. Whenever your output quantity index is greater than that of the input quantity index, that means productivity has improved, or there is a portion of change in output which is not being explained by the change in input.

Speaker A

Now we define Laspeyres output quantity index as sum of Basically it is going to be P J zero J one up to M we have output prices as P vector and Y J one divided by sum of J one up to

Speaker A

That is basically considered as a productivity change. Moving ahead, now we consider the Fisher productivity index.

Speaker A

Similarly we can define Paasche's output quantity index. So same formula but instead of last year weightage, we use present year weightage, so it become sum of J1 of two M PJ1 YJ1 divided by sum of J1 of two

Speaker A

For that, it is going to be the geometric mean of Laspeyres index and Paasche's index, but not in a very direct sense.

Speaker A

And once you have this Laspeyres and Paasche's output quantity index, we can define Fisher's output quantity index as square root of Laspeyres output quantity index into Paasche's output quantity index.

Speaker A

First, we need to estimate Fisher's output quantity index. For that, we need to estimate Laspeyres output quantity index and Paasche's output quantity index, and square root of that will give you the Fisher's output quantity index. So, that is the

Speaker A

R1 of two N We have WR zero X R one divided by sum of R1 of two N We have N inputs.

Speaker A

step that we follow. It is not like first you estimate Laspeyres productivity index, and then you estimate Paasche's productivity index, and then take the geometric mean of that. Logic doesn't work over here.

Speaker A

Similarly, Paasche's input quantity index we can define as sum of R1 up to M WR1 XR1 divided by sum of R ending from 1 to M WR1 XR zero.

Speaker A

First, you need to estimate the Fisher's output and input quantities. That is output and input quantity indexes.

Speaker A

Okay? It's very simple. Just the logic of our uh price index where price index we were using output as the weightage.

Speaker A

Then, taking the ratio of that will give you a Fisher's productivity index. So, here we define our output vector.

Speaker A

So, now once we have Fisher's uh quantity output quantity and input quantity index we can define Fisher's productivity index.

Speaker A

So, say here we have multiple inputs and multiple outputs. So, here we have M outputs, and say N inputs.

Speaker A

And if it is greater than one if this output con- Fisher's index gives you a value greater than one, that means productivity has improved in the period one as compared to the period zero, all the way here.

Speaker A

So, we can define our output bundle for period zero as basically Y10, Y20 up to YM0.

Speaker A

So, just to summarize, uh here in this case, we were discussing Fisher's productivity index as a uh simple measure for measuring productivity change.

Speaker A

And you can define input quantity index, basically it's going to be for the period zero, X10, X20 up to XN0.

Speaker A

change. Using the same logic, we use Fisher's output quantity index and Fisher's input quantity index for getting the uh Fisher's productivity index.

Speaker A

Okay? Correspondingly, we will be having the price index as well or the price information.

Speaker A

And for Laspeyres output quantity index, we were using past year uh price of in- outputs, and for Pas- Laspeyres input quantity index, we were using last year price of inputs.

Speaker A

So, we're defining P0 as the price vector for period zero for each output. So, it is basically P10, P20 up to PM0.

Speaker A

And once you get this uh output quantity index of Laspeyres and Paasche's, we take the geometric mean of Laspeyres and Paasche's to get the Fisher's index. Similarly, once you get the input quantity index under Paasche's and Laspeyres, we take the geometric mean of

Speaker A

And then the input prices are defined as W0, which is basically W10, W20 up to WN0 where you have N number of inputs.

Speaker A

And finally, we took the ratio of output quantity index to input quantity index, which tells you a story whether productivity has improved over the period.

Speaker A

Using the same logic, we can have Y1 that is basically the output bundle for period one that is basically Y11, Y21 up to YM1 and the corresponding price vector you can consider as P1, basically P11,

Speaker A

However, in empirical literature, you see very less application of Fisher's index as a measure of productivity change.

Speaker A

P21 up to PM1 and correspondingly we can have an input vector for period one and input price vector for period one.

Speaker A

And for Fisher's index, we need to get it for both the years. Right? As a result, in empirical literature, you will not be seeing much of the application of Fisher's productivity index.

Speaker A

Moving ahead, using this information we need to create our Laspeyres and Paasche's output quantity index in the first instance and then we can create using the same information we can create the Laspeyres and Paasche's input quantity indexes.

Speaker A

Here, there is no frontier estimation involved. Uh there is no technicalities involved in the uh assumptions about the production function or assumptions about the distributional uh assumptions so on and so forth.

Speaker A

So here we define our Laspeyres output quantity index as LQY. It is going to be a quantity index that uses last year price information as the weightage.

Speaker A

Thank you. Woo!

Topics:Fisher productivity indextotal factor productivityLaspeyres quantity indexPaasche quantity indexproductivity measurementinput quantity indexoutput quantity indexproductivity changeMATLAB productivity analysisNPTEL IITM

Get More with the SozAI App

Transcribe recordings, audio files, and YouTube videos — with AI summaries and speaker detection. 30 minutes free.

App Store Google Play

Or transcribe another YouTube video here →

Free tools: TXT to SRT · SRT Validator · Merge SRT · Subtitle to Text · All tools