**An introduction to Parametric approaches of measuring productivity — Transcript & Summary | SozAI**
Source: https://sozai.app/transcript/parametric-approaches-measuring-productivity/

Introduction to parametric approaches for measuring and decomposing total factor productivity using production functions and panel data.

## Key Takeaways

- Parametric approaches assume a functional form to estimate and decompose productivity.
- TFP change can be decomposed into technological change, technical efficiency change, returns to scale, and allocative efficiency components.
- Single input-single output models use the difference between output and input growth rates to measure productivity change.
- Multiple inputs require weighting input changes by their cost shares to calculate TFP change.
- Parametric methods face criticisms and are less common than nonparametric or semiparametric methods in productivity analysis.

## What the video covers

- The video introduces parametric approaches to measuring total factor productivity (TFP) and decomposing it into components.
- It contrasts parametric methods with nonparametric and semiparametric approaches, noting criticisms of parametric methods.
- The single input-single output case is explained using the Solow residual concept where productivity change is output growth minus input growth.
- The production function approach is presented, including the use of panel data and functional forms like Cobb-Douglas or translog.
- The inefficiency term (u) and its role in stochastic frontier analysis is discussed.
- Extension to multiple inputs with one output is covered, including the use of cost shares to weight input changes.
- Decomposition of TFP change into technological change, technical efficiency change, returns to scale, and allocative efficiency is explained.
- Mathematical techniques like differential calculus are used to derive components of TFP change.
- The video highlights practical considerations and controversies around interpreting productivity changes in firm performance.
- The session aims to strengthen theoretical understanding of TFP decomposition using parametric approaches, especially in the Indian context.

## Chapters

1. 00:00 Introduction to Parametric Approaches and Course Overview
2. 02:02 Context and Relevance of Parametric Methods in India
3. 03:44 Single Input-Single Output Productivity Change Concept
4. 05:28 Production Function Approach and Inefficiency Term
5. 07:13 Extension to Multiple Inputs and Cost Shares
6. 08:59 Decomposition of TFP Change Using Differential Calculus
7. 10:45 Components of TFP: Technological Change and Efficiency
8. 14:25 Allocative Efficiency and Input Misallocation
9. 15:48 Summary and Further Explanation on Parametric Productivity Measurement

Answers

## Questions about this video

What is the main difference between parametric and nonparametric approaches to measuring productivity?

Parametric approaches assume a specific functional form for the production function, while nonparametric approaches do not require such assumptions and are more flexible but less structured.

How is total factor productivity change calculated in a single input-single output model?

It is calculated as the difference between the rate of change in output and the rate of change in input, representing productivity improvements beyond input growth.

Why are parametric approaches less common in productivity analysis compared to nonparametric or semiparametric methods?

Parametric approaches face criticisms related to functional form assumptions and flexibility, leading many researchers to prefer semiparametric or nonparametric methods that avoid some of these limitations.

## Full Transcript — Download SRT & Markdown

00:05

Speaker A

[music] [music] Hi, welcome back to the course Applied Production Analysis Using MATLAB. We have already covered the nonparametric approaches for estimating total factor productivity and decomposing the same. From this session onwards, we'll take a deviation as we had in the case of technical efficiency estimation. We can a priori assume a functional form and do the same procedure or more or less similar procedure what we did for technical efficiency estimation, which is called the parametric approach. So in this session, I'll be giving you a brief introduction to parametric approaches of measuring total factor productivity and decomposing the same to the components as similar to the nonparametric approach.

00:33

Speaker A

Quick disclaimer: as a researcher who is involved in parametric or productivity analysis, I never come across so many works in the context of parametric approach. I can map it very similar to the CS or CM what we had because parametric approach which uses a functional form we can use as a starting point for the approaches based on a functional form but it is open to several criticisms. So based on that, there are semiparametric approaches coming which are free from few of the criticisms that are posed against the parametric approach. So when you explore further, I think most of the studies you will see either semiparametric or nonparametric and very few studies are there that use parametric approaches for estimating TFP and decomposing the same, particularly in the Indian context. Still, we cover this thing so that your theoretical understanding of TFP decomposition using parametric approach or even nonparametric approach becomes a bit more strong.

00:50

Speaker A

So in this session, we start with the single input, single output case which we have already covered in terms of Solow residual and we see the production function approach which is going to be the parametric approach in this context and we also see the total factor productivity change and the components of the same in the context of parametric approach. So in a simple single input single output case as we have already designed in the case of Solow residual, productivity change is simply the difference between the rate of change in output and input change. So in econometric or even in the growth framework, we can use dot as a change in a variable. So here as you can see, productivity change is defined as difference between change in output and change in input. Y dot, define Y dot is basically the rate of change in output and X dot is basically the rate of change in input and since we are in a single input single output case, the difference between change in output and change in input will give you the productivity change and a positive productivity change means the output that you are referring to has grown or changed positively faster than the input change. That means there's a portion of output change which is not being explained by change in input and which we attribute as a change in productivity and going beyond you can decompose the same productivity into several components as we saw in the context of nonparametric approach and one interesting question comes whether having a higher productivity or having a positive productivity change is at all the final case that a decision-making unit is performing better. That is a very controversial question to have in this context. Not necessarily because firms are not maximizing their productivity. Sometimes they have to maximize their profitability or in some cases maximize their sales. There are different behavioral models that come into the picture. So having higher productivity or having a positive productivity change, you cannot necessarily claim the fact that it has improved the overall performance of the firm or performance of a firm from its objective point of view. But still, we can consider this is one factor that results in an improvement in the performance of firm and also suppose you can consider the case of agricultural production. Say in some periods we face a very improved productivity in context of some crops but sometimes that improved productivity results in higher supply of the commodity and it may result in lower price and sometimes it may result even in a loss to the farmers and you can see that in the case of potato, tomato.

01:02

Speaker A

So we are not jumping into the controversy. So this is like a question that you can pose in between. So the production function approach in a single input single output, we again conceptualize a production function, not a very explicit functional form. We can say yit as a function of xit, t and exponential minus u. This is very similar. It's very familiar to you already which we have already considered in the case of stochastic frontier analysis. So here yt as you know it is the output, we have a t subscript that is for the time period t, i stands for the i firm. So yit is basically the i firm's output for the period t. For the productivity change, you'll be using panel data. Similarly, you will have XIT as the vector of J inputs and F of XIT is the production technology that you are using in the context. So sometimes you'll be using it as a Cobb-Douglas production function or translog production function but for simplicity we are keeping it as a f of xit, t and t is the time trend variable which will be used for seeing particularly the technological change happening into the production function and u is basically the one-sided inefficiency component which always takes an individual observation below the production frontier. So it takes always since we are putting a negative here it will take u greater than or equal to zero always and y star can be conceptualized as a potential output which is like the free from inefficiency in this context talking of total factor productivity. So in the context of multiple inputs and even multiple output, here multiple output you will not be able to incorporate in a parametric approach. We'll be considering the multiple input. You can do but using some composite output factor which will be very complicated. But here we'll be considering one output and multiple input. In that case actually you have to go beyond X dot, you will be having J number of inputs. So TFP will be Y dot that is change in rate of change in output minus sum of SJA into XJ dot where XJ dot stands for the rate of change in input J. And this SJA is basically the ratio of particular input's share in the total cost of the firm. So this is very similar to what we saw in the Tornqvist index. So we can define sja as the wj xj divided by ca and here ca is the total cost that is going to be the sum of wj xjs over the entire input that you are considering. And here wj as you can see it is the input price, price of j input and xj is basically the quantity of input j and sja as I mentioned it is basically the cost share of each input j in the total cost of the firm that you are referring or decision-making unit that you are referring to.

01:22

Speaker A

So for decomposing this thing, what we can do, we can use the techniques of differential calculus and by differentiating the function that we are having with respect to t, you'll be getting a technological change and also by differentiating with inputs and all you will be getting the different components of TFP. So here this function is being differentiated. So you get a TFP change as a component TG TC that is technological change, do U by do T that is basically the change in U term which captures the inefficiency with respect to T, that is basically the technical efficiency change and you have an additional component which can be decomposed into return to scale component also. Alternatively, TFP change is basically RTS minus one into sum of lambda J XJ dot and here RTS stands for return to scale and this lambda J we'll be discussing further in detail plus TC. TC basically the technological change, TC is the technical efficiency change and this sum of lambda j minus s j a

01:44

Speaker A

criticism that is posed again the parametric approach. So when you explore further I think most of the studies you will see either semiparametric or nonparametric and very few studies are there that uses parametric approaches for estimating TFP and decomposing the

02:02

Speaker A

same particularly in the Indian context. Still we cover this thing so that your theoretical understanding of TFP decomposition using parametric approach or even nonparametric approach become bit more strong.

02:17

Speaker A

So in this session we start with the single input uh single output case which we have already covered in terms of solo residual and we see the production function approach which is going to be the parametric approach in this context and

02:35

Speaker A

we also see the total factor productivity change and the components of the same in the context of parametric approach. So in a simple single input single output case as we have already designed in the case of solo residual

02:49

Speaker A

productivity productivity change is simply the difference between the rate of change in output and input change. So in econometric or or even in the growth framework we can use dot as a change in a variable. So here as you can see

03:08

Speaker A

productivity change is defined as difference between change in output and change in input. Y dot define Y dot is basically the rate of change in output and X dot is basically the rate of change in input and since we are in a

03:25

Speaker A

single input single output case the difference between change in output and change in input will give you the productivity change and a positive productivity change means uh the output that you are referring to has grown or changed positively faster than the input

03:44

Speaker A

chain. That mean there's a portion of output change which is not being explained by change in input and which we attribute as a change in productivity and going beyond you can decompose the same productivity into several components as we saw in the context of

04:00

Speaker A

nonparametric approach and uh one interesting question comes whether having a higher productivity or having a positive productivity change is at all the final case that a decision-m unit is performing better. That is a very controversial question to have in this context. Not

04:22

Speaker A

necessarily because firms are not maximizing their productivity. Sometimes they have to maximize their profitability or in some cases maximize their sales. There are different behavioral model that comes into the picture. So having higher productivity or having a positive productivity change

04:41

Speaker A

you cannot necessarily claim the fact that it has improved the overall performance of the firm or performance of a firm from its uh objective point of view. But still we can consider this is one factor that result in a improvement

04:57

Speaker A

in the performance of firm and also suppose you can consider the case of agricultural production. Say in some periods we face a very improved uh productivity in context of some crops but sometime that improved productivity result in higher

05:17

Speaker A

supply of the commodity and it may result in lower price and sometime it may result even in a loss to the farmers and you can see that in the case of potato tomato.

05:28

Speaker A

So we are not jumping into uh the controversy. So this is like a question that you can pose in between. So the production function approach in a single input single output we again conceptualize a production function not a very uh explicit functional form. We

05:46

Speaker A

can say yit as a function of xit t and exponential minus u. This is very similar. It's very familiar to you already which we have already considered in the case of stoastic frontier analysis. So here yt as you know it is

06:04

Speaker A

the output we have a t subscript that is for the time period t i stands for the i form. So yit is basically the i form's output for the period t for the productivity change you'll be using panel data. Similarly, you will have XIT

06:20

Speaker A

as the vector of J inputs and F of XIT is the production technology that you are using in the context. So sometime you'll be using it as a coalas production function or trans production function but for the simplicity we are keeping it as a f of

06:38

Speaker A

xit t and t is the time trend variable which will be used for seeing particularly the technological change happening into the production function and u is basically the one-sided inefficiency component which is always uh takes an individual observation below

06:58

Speaker A

the production frontier. So it takes always since we are putting a negative here it will take u greater than or equal to zero always and y star can be conceptualized as a potential output which is like the free from inefficiency

07:13

Speaker A

in this context talking of total factor productivity. So in the context of multiple inputs and even multiple output here multiple output you will not be able to incorporate in a parametric approach. We'll be considering the multiple input. You can do but using uh

07:32

Speaker A

some composite uh output factor which will be very complicated. But here we'll be considering one output and multiple input. In that case actually you have to go beyond X dot you will be having J number of inputs. So TFP will be Y dot

07:53

Speaker A

that is change in rate of change in output minus sum of SJA into XJ dot where XJ dot stands for the rate of change in input J. And this SJ A is basically the ratio of particular inputs share in the total cost of the

08:17

Speaker A

firm. So this is very similar to what we saw in the tonquist index. So we can define sja as the wj xj divided by ca and here ca is the total cost that is going to be the sum

08:31

Speaker A

of wj xjs over the entire input that you are considering. And here wj as you can see it is the input price price of j input and xj is basically the quantity of input j and sja as I mentioned it is

08:48

Speaker A

basically the cost of cost share of each input j in the total cost of the firm that you are referring or decision making unit that you are referring to.

08:59

Speaker A

So for decomposing this thing we can what we can do we can use the uh techniques of differential calculus and by differentiating the function that we are having with respect to t you'll be getting a uh technological change and

09:14

Speaker A

also by differentiating with inputs and all you will be getting the different components of TFP. So here this function is being differentiated. So you get a TFP change as a uh component TG TC that is technological change do U by do T

09:33

Speaker A

that is basically the change in U term which capture the inefficiency of with respect to T that is basically the technical efficiency change and you have an additional component which can be decomposed into uh return to scale component also. Alternatively, TFP

09:50

Speaker A

change is basically RTS minus one into sum of lambda J XJ dot and here RTS stands for return to scale and this lambda J we'll be discussing further in detail plus TC TC basically the technological chain tc is the technical

10:08

Speaker A

efficiency chain and this sum of lambda j minus s j a into x dot j is basically component that we'll be considering as the scale component in this framework.

10:25

Speaker A

So to repeat TC is basically the technical change that is basically the shift in the frontier with respect to uh time. So this is basically from f of dot for the period one and f for dot for the period 2. So this is basically um

10:45

Speaker A

shift in the frontier that so that will give you the technical change or technological change and technical efficiency change is basically as I mentioned it is basically the change in inefficiency over the period of time that is basically d u by dot t with a

10:58

Speaker A

negative sign and return to scale component is basically the sum of do log y and do log x. So here we have multiple inputs. So you have to take the sum over the entire J1 up to number of input that

11:14

Speaker A

you are having. Unpacking the scale component. RTS minus one sum of lambda J exj dot is the RTS component that you can have in this TFP film TFP formulation and uh that RTS capture the effect of changing in the

11:33

Speaker A

scale of production very similar to what we had it in the context of scale chain component in a nonparametric approach and here lambda J is the output elasticity of input J and as we already seen that X do. J is the growth in the

11:50

Speaker A

uh input J and under CRS RTS is going to be one. Therefore, RTS -1 lambda JX X J dot will become zero. So, we cannot expect if you impose a CRS assumption to the parametric approach, you'll not be

12:09

Speaker A

having any return to scale change uh that you see in the decomposition. To revisit each components of that TC is basically the do log F XT do T which measures the shift in the production front as I have already shown over the

12:25

Speaker A

period of time and it changes it capture both technological progress as well as technological regress. If TC is greater than zero that means there is a improvement in technology which is the case in most of the cases and if come

12:38

Speaker A

some cases you may see a deterioration in the technology as well and technical efficiency component it is bit technical in this case basically for technical efficiency measurement you may have to do a uh stoifive frontier analysis in the back end assuming a functional form

12:55

Speaker A

assuming a distributional assumption for the u term and so and so and uh in a straightforward manner. It is basically minus d u dot t as I already mentioned and which measures uh the movement of an individual observation from its

13:10

Speaker A

technically efficient point over the time period and uh and in a reduction in technically technical inefficiency means a positive technical efficiency change and sometime we can impose the technical inefficiency to be time invariant that is basically u we can Say the case that

13:31

Speaker A

it is persistent over the period here DMU remain same inefficient over the period. So in that case actually dot t become zero and in that framework you will not be having a technical inefficiency component. So we are going

13:47

Speaker A

to have a matlab practice for the same uh framework where we'll not be going to the framework which is having a technical efficiency change or will be considered in the case that technical efficiency change is assumed to be zero

14:02

Speaker A

in that case and allocative component which is which looks a bit complicated but here it can be written as sum of lambda j minus sj into xj dot which captures the effect of input misallocation.

14:16

Speaker A

So and as I mentioned lambda J is the elasticity output elasticity of input J and SJ is already defined as the cost of share in cost of J input in the total share of the firm total cost of the firm

14:30

Speaker A

and uh allocative inefficiency arises when firms are not u using the like WJ that when they are PFJ is not equal to where WJ that is the case of misallocation of resources as I was mentioning and it can be defined as

14:54

Speaker A

in a two input case FJ FK not equal to WJ WK can be considered as a allocative component or misallocation in that case where uh this formula capture the same theory that we have in context of micro foundation that is basically the margin

15:14

Speaker A

rate to technical substitution solution is deviating from the or it's not equal to the rate of ratio of input prices in this multiple input context and uh finally the techn total factor productivity change you can assume even the context of return to

15:33

Speaker A

scale equal to one that is constant return to scale in that case RTS will become one there is no technical uh return to scale component and also you can impose uh as we did for the example you can impose there is no

15:48

Speaker A

change in the technical efficiency that is basically do u dot t equal to0 in that case actually our end date will have only two component that is basically the techn technological change and the uh misallocation or allocative efficiency component that we are

16:04

Speaker A

discussing. So uh just to summarize basically uh you can of course estimate the productivity following a parametric approach where you will have to priorly assume a functional form like cob loss or transfer production function and using the tools of calculus by differentiating

16:24

Speaker A

with respect to inputs or time you will be able to get uh the components of TFP very similar to what we had it in the context of parameter. approach including technical efficiency change, scale change factor and the technological

16:42

Speaker A

change component. But problem is that uh these models are criticized mainly because uh there can be productivity shocks based on which firms may change their input output component. So this productivity shocks if not being included in the model it

16:59

Speaker A

can result in endogenity due to omitted variant. So now the upcoming model that we are going to discuss that is basically the semiparametric approaches are building upon the the weakness of parametric couple that is basically the uh bias

17:16

Speaker A

happening due to not including or the endogenity happening due to not including productivity shocks which is a significant variable into the model. So this content I have taken from Kumbakar a practitioner guide to stoop analysis using STA. Of course you can refer to

17:35

Speaker A

that and see the even detailed uh explanation on the measurement of productivity following a parametric approach and the decomposition of the same. Thank you.

17:50

Speaker A

[music]

Topics: parametric approach total factor productivity TFP decomposition production function stochastic frontier analysis technical efficiency technological change returns to scale allocative efficiency Applied Production Analysis

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