Decomposition of Malmquist Productivity Index - Part I

Detailed explanation and decomposition of the Malmquist Productivity Index using CRS and VRS technologies in production analysis.

Key Takeaways

Malmquist Productivity Index does not require price data, unlike other productivity indices.
Decomposition varies depending on whether CRS or VRS assumptions are used.
FGLR decomposition focuses on CRS-based components, while Ray and Desi incorporate VRS and scale change factors.
Technical efficiency change and technological change are core components of productivity change.
Scale efficiency/change is an additional component when VRS is considered.

What the video covers

Introduction to Malmquist Productivity Index and its advantage of not requiring price data compared to Fisher and Törnqvist indices.
Explanation of productivity measurement using directional distance functions and DEA framework.
Discussion on decomposition of Malmquist Productivity Index under Constant Returns to Scale (CRS) and Variable Returns to Scale (VRS).
Presentation of three main decompositions: FGLR (1992), FGN set, and Ray and Desi (1997) revisited decomposition.
FGLR decomposition uses CRS technology for both technological change and technical efficiency change components.
Ray and Desi decomposition applies VRS technology and introduces a scale change factor instead of scale efficiency change.
Mathematical formulation of productivity change using output and input bundles with directional distance functions.
Explanation of technical efficiency change, technological change, and scale efficiency/change components in the decompositions.
Use of geometric mean of productivity components from two periods for final Malmquist productivity index calculation.
Summary of how scale efficiency is defined as the ratio of technical efficiencies under CRS and VRS technologies.

Chapters

What is the main advantage of the Malmquist Productivity Index over other productivity measures?

The Malmquist Productivity Index does not require any price data for inputs or outputs, unlike Fisher or Törnqvist productivity indices, making it useful when price data is unavailable.

How does the assumption of CRS versus VRS affect the decomposition of the Malmquist Productivity Index?

Under CRS, the decomposition includes only technological change and technical efficiency change, while under VRS, an additional component called scale efficiency change appears.

What are the key components in the FGLR and Ray and Desi decompositions?

FGLR decomposition uses CRS technology focusing on pure technical efficiency and technological change, whereas Ray and Desi use VRS technology and replace scale efficiency change with a scale change factor.

Full Transcript — Download SRT & Markdown

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[music] [music] Hi, welcome back to the course Applied Production Analysis using MATLAB. In the last session, we covered very basic foundations of Malmquist productivity index. Just to summarize, as against other measures of productivity that we familiarize with,

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that is Fisher productivity index and Törnqvist productivity index, Malmquist productivity index has an advantage that it does not require any price data. It is basically a directional distance function-based approach of measuring productivity, particularly DEA. So in that case, actually productivity is

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measured with reference to distance functions corresponding to years and particular observation that you are considering. As a result, we don't require any price data. When I say price data for Fisher's or Törnqvist, we need both input prices and output prices, but in case of

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Malmquist productivity index, we don't need any price data. And we already familiarized ourselves with the very foundational formula for measuring productivity and how to plug in the distance functions into that. Then in the last session, we already discussed what are the

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components or potential components of productivity. This is basically the measurement of productivity that we are referring to in this session. We will decompose the Malmquist productivity into multiple components, and based on the fundamental assumption that you are keeping, whether returns to scale VRS or

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CRS, you will get different components. So in the first instance, we keep CRS throughout. Then both technological change and technical efficiency change will be estimated using CRS. So in that case, actually when you are keeping CRS throughout, you will be having only

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technological change and technical efficiency change. As I mentioned, the moment you plug in VRS into the picture, particularly for measuring technical efficiency, then the change in productivity or its components will have one more component, that is basically the

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scale efficiency change that we have already seen. Okay. So here in this session, I'll be familiarizing you with particularly three decompositions.

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The very first decomposition that I'm mentioning is basically the FGLR Farrell 1992 decomposition. This is basically a purely CRS-based approach.

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Both technological change and technical efficiency change are projected against a CRS frontier. As against that, we have FGN set. Just to summarize what we'll be doing, FGN set we'll be using VRS for technical efficiency change. As a result,

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you'll be having scale efficiency change and technical efficiency change. So here, this is going to be pure technical efficiency change and then the technological change component of our FGL. Moving ahead, in 1997 Ray and Desi revisited the decomposition

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of Farrell 1994, and there are slight modifications for the TE component. They follow the, or TEC component, they follow the VRS technical efficiency. Even for the technological change component, we'll be using VRS technology. Then for the

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decomposition, instead of scale efficiency change, we will be having scale change factor, which is slightly different from our FGLR model. That is a summary of what we are going to discuss in this session. So productivity we can define as y1 / x1 divided by y0 / x0. This is

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basically the productivity in period 1, and this is basically the productivity in period 2. The ratio of productivity in period one with the base period, that is period zero, will give you the productivity change. Now we are slowly infusing our

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directional distance functions into this model. So here what we can do, we can plug in the potential output, potential output under VRS basically. So we had R0, that is basically with reference to CRS technology for period zero, and if you see f0, this is

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basically VRS potential output for the period zero. So having said that, we can change this ratio into y1 / f_sub_0 x1 times f_sub_0 x1 / x1. The formula remains the same divided by y0 / f_sub_0 x0 times f_sub_0 x0 / x0. That is basically the

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potential output divided by the actual input that we are using. So alternatively, we can have the same decomposition with reference to CRS technology, and here you can see this component is going to be one because

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here this entire component is basically a distance that you are considering or the efficiency that you are considering point for a line passing through origin, and so this will be one in that case. And finally, we can measure

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pi 0 again as the components, two components: the technical efficiency component and the ratio. This is basically the technical efficiency change component; this is basically the scale efficiency component. Okay, so now we'll see the Randel decomposition, how

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it works. Frankly speaking, I'm not going to the each and every step of the decomposition. When I say decomposition, I'm referring to the final components of productivity under the Malmquist productivity framework and the components thereof under the framework

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that I'm referring to. So here the same formula we had in the earlier case, and in that case, actually this one was basically with reference to zero period technology, and the second one was basically pi 1 for the first period.

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And the technological, the productivity change, we can take it as a geometric mean of this pi 1 and pi 0, which will give you this component, and it can be decomposed into three components. So here you can see this is

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like the crux of or the final outcome of Randel decomposition. MPI is basically the Malmquist productivity index, and here f_sub_1 x1 divided by f_sub_0 x1 is basically potential output in the context of

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first period technology, that is with reference to VRS technology, divided by the potential output in period zero under VRS technology for the input bundle x1. So this is basically the ratio of, say, you have x0 and x1. So what we are searching here? So

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this is basically f_sub_1 f_sub_1 x0. Okay. And here we can say this one is basically x0 x0. Okay. This is basically f_sub_1 x1 superscript. This is basically f_sub_0 x1. So we are referring to these two points when I am typing.

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So let's see what is given in this component. This is basically the ratio of potential output in period one VRS technology for input bundle one divided by the same input bundle but

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for period zero. So basically this will capture this component, this shift component. Okay. And alternatively, if you consider x0 as the input bundle,

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so when you consider x0 as the input bundle, we are referring to this shift. So that is basically f_sub_1 x0 divided by f_sub_0 x0. So this whole component that we are having over here, this part, this is basically the

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technological change component. Moving ahead, we have y1 / f_sub_1 x1 divided by y0 / f_sub_0 x0. That is basically we will be having a data point over here.

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Say this was the data point, and say this was the data. So this was the data point for x1. So this is basically y1 x0 y0, right? Or we can say this point is basically the f of x0, and this

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is basically the actual point. So y1, y1 is given here, y1 x1, y1.

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So what we are doing, this is y1 divided by f_sub_1 x1. You see it's already here x1, x1. That is basically the technical efficiency in period t, output oriented in period 1, and this denominator is basically t output

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oriented in period zero. The ratio of TE output oriented, that is with reference to VRS technology for period 1 to period zero, will give you the technical efficiency change. This is with that actually we get this component as

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technical efficiency change. The last component of Randel given over here is a bit tricky. So here you can see scale efficiency is basically defined as technical efficiency under CRS divided by technical efficiency under VRS. And here f0 and R0 represent VRS

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potential output and CRS potential output respectively. So with that, actually ratio of f_sub_0 x1 divided by r0 x1 is basically the scale efficiency that you are estimating again.

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particular bundle that we are considering that is particular bundle of X0 and that ratio of Right. Okay.

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X1 X F0 X1. So this is basically the technical efficiency of the ratio of technical efficiency.

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This is the CRS VRS. So, so consider this as R0, this as F0 and we are referring to X1 period technology. This is like a bit of a pseudo case. Actually in period 0, R0, F0 the technology set CRS and VRS but X1

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was not observed. Here the randomly decomposition consider bit of a pseudo case also that gives us an idea for each bundle how much uh scale efficiency change has happened in period 0 and one.

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So what I'm referring here in this case actually say this was the point x1 y1 y1 and this is basically f_sub_1 f0 x1 and this is basically our r0 x1. Okay. So now our preliminary understanding of u scale efficiency. So ideally scale

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efficiency in this context should have been u y1 divided by r0 x1 right divided by y1 all / x1. So this get cancel it will become f0 x1 / r0 x1. So that's what we are seeing over here. So that is basically

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the entire component is reference to zero period technology. Then we are seeing input output bundle for period 1 and period zero.

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And then randomously has another component f_sub_1 x1 / r1 x1 or divide by f_sub_1 x0 / r1 x0. So that will be the uh here in place of R1 it become in place of R0 f_sub_0 it becomes R1

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F_sub_1 and with reference to that X1 and X0 you get a uh scale efficiency ratios and then the geometric mean you know these two component that we consider here will give you the scale change factor. So this is basically not very

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straightforward manner that we have it in the context of earlier decomposition. So here you can see earlier de composition scale change factor basically it will be only uh f_sub_1 x1 r1 x1 divid by f_sub_0 x0 r0 x0 but uh in this context for randly

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we are considering both the periods technologies and and then we are taking geometric mean of both the cases period zero input bundle and periods one input one. Okay.

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And here now we can define the same thing from a distance function point of view. Here distance function we have already defined in the last session.

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Whatever given over here is basically the shuffer distance function. So this is basically f_sub_1 x1 you can define as d0 d1 x1 right the technical efficiency. Now we need to take the potential output of that particular data point. So it will come the reciprocal.

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So f_sub_1 x1 / f_sub_0 x1 will become d 0 x1 y1 divide by d1 x1 y1 and f_sub_1 x0 / f_sub_0 x0 will become d 0 x0 y by d1 x0 y and the geometric mean will give you the technological change.

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Similarly the technical efficiency change also you can express in terms of distance function or the t score. So this will become the reciprocal of uh the potential output or the not reciprocal it is basically the distance function d1 x1 divid by d0 x1 d 0 x0 y.

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So this will give you the technical efficiency. So here you can not uh notice that none of them have a C subscript that means all of them are estimated against the VRS technology or the distance function that you are

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mentioning over here at the VRS technology. Now it comes the scale change factor componently.

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So here this thing is something we have already arrived at in the last uh final decomposition value. And now expressing these things in a distance function form you get the same components from a uh distance function component point of

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view. But here along with the VR distance function that is dx x x t or yt here we'll be having d0 c and d1 cb basically the d 0 c or d1 c or the crs distance function that we are

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getting. This is very straightforward over here. Here this is basically the technical efficiency score. is very consistent with the technical efficiency score that we estimate in our conventional framework.

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In the full decomposition we can say that the productivity in period one with the last period as the base under randomly decomposition is basically TC into technical efficiency change into scale change factor. So they are using scale change factor because it's not

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very straightforward as the scale efficiency chain that we have seen in the earlier case. Now uh this is this one is the latest decomposition. Once you understand the reindustry decomposition it's very easy for you to extrapolate or like extend or get the

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FGL FGLR or FGNZ decomposition. For FGR decomposition it follows CRS throughout. So in that case actually Yi / R0 X1 / Y / R0 X0 and this will be the technological change for period 0. Okay. Then uh we have y1 / r1

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x1 / y0ide by r1 x1 x0. So this is basically our r1 r0 and this shifts we are taking the geometric mean of this shift x0 x1. So what we are taking basically the shift in R0 that is basically the CRS from here

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both for period uh in period one I say again period zero both for uh input bundle x0 and x1. Okay. Then we can decompose this part into basically this one is our R0 R1 X U R1 X0 R0 X0. This

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is basically the uh technological change with reference to zero period bundle that I was referring to. And this is basically the technological change again but with reference to first period bundle. And the final part is basically Y1 divided

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by R1 X1 that gives you actual output divided by the potential output under CRS technology divided by Y R0 X0 that is basically actual output in zero period divided by the potential output in zero period with reference to the CRS

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technology. So this is basically very straightforward technical efficiency change. So FGLR you don't see any F0 over here. That means FGLR decomposition is purely based on CRS technology. So it has technological change with reference to R0 R1. The scale efficiency change

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with reference to R0 R1. And there is no cross bundling happening here. Y1 divided by R1 is basically the technical efficiency. Since we are referring only CRS technology there is no scale efficiency that it going to get in the

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context of FGLR decomposition final outcome. Moving ahead FGNZ decomposition it follows CRS for technological change but it consider VRS for technical efficiency change. The moment we consider VRS for technical efficiency chain, we'll be having pure technical efficiency chain as well as the scale

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efficiency chain. So here the decomposition goes this manner. The technological change under FGNZ is the same what we have it in the context of technological change under FGLR. But scale efficiency change under FG and Z become f of f_sub_1 x1 / r1 x1 / f0 x0 /

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r0 x0. Okay. And the technical efficiency change that we are having in the context of fgz is basically the same technical efficiency change that we are having in the context of randity decomposition that will become y1 with reference to the

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y1 period. uh VRS technology and Y0 with reference to our Y y per grid VRS technology.

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So these are the component uh we are taken mostly from uh the final reference that is basically the data envelopment analysis by sub 2004 book and the earlier uh the technicalities or the logic behind decomposition and so and so you can get in the context of

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FGLR uh 1992 paper FGNZ 1994 paper and the 1997 factor. I know it might have come a bit abstract for most of you. So in the next session we'll be trying to approach the same problem from a uh diagrammatic

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point of view which will give you bit more cleared or clarity uh about the final outcome of these 3D composition that we have considered. So just to summarize in this session we saw the final outcome of three decomposition

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FGLR being the very basic decomposition. It considers CRS throughout. So it has total factor productivity decomposed into technical efficiency change and technological change. Both of them are estimated again CRS technology. Moving ahead FG NZ consider CRS for technological change. For technical

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efficiency change we consider VRS technology. So we will be having technical efficiency change, scale efficiency chain and the technological change.

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The ray industry decomposition is bit more u advanced in the sense that it consider VRs for technological change also. So as a result we'll be having techn technological change estimated the shift of VRS from India for both zero period

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and first period input bundle. Then we have technical efficiency change which is the same what we have it in the context of FGL NZ. Then scale instead of scale efficiency change we'll be having scale change factor because both the

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periods techn uh scale uh efficiency change we are estimating again both the um input output bundle for period 0 and one then we'll be taking the geometric mean of that and we'll get the scale change factor which is slightly

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different from what we have it in the context of fg and zed thank Okay.

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Topics:Malmquist Productivity IndexProductivity DecompositionDirectional Distance FunctionData Envelopment AnalysisConstant Returns to ScaleVariable Returns to ScaleTechnical EfficiencyTechnological ChangeScale EfficiencyApplied Production Analysis

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