Decomposition of Malmquist Productivity Index - Part II

Detailed explanation of Malmquist Productivity Index decomposition focusing on Ray and FGNZ models with MATLAB examples.

Key Takeaways

Malmquist Productivity Index can be decomposed using different models with overlapping but distinct components.
Technical efficiency change is consistently estimated against the VRS frontier in Randly and FG decompositions.
Technological change involves shifts in technology frontiers for different input bundles across periods.
Scale change factor differs from scale efficiency ratio and is critical in understanding productivity changes.
Imposing CRS on VRS data can lead to infeasible solutions, highlighting the importance of correct model assumptions.

What the video covers

Introduction to Malmquist Productivity Index and its various decompositions including FG, Z, FGLR, and Ray industry models.
Discussion on overlapping components of FG and Z models and their relation to technological change and technical efficiency.
Explanation of CRS (Constant Returns to Scale) and VRS (Variable Returns to Scale) technologies and their frontiers for two periods.
Diagrammatic representation of productivity change using data points A, B, C, D for two periods.
Detailed walkthrough of technical efficiency change calculation using VRS frontier in Randly decomposition.
Breakdown of technological change into two parts based on input bundles X0 and X1 and their shifts in technology.
Introduction and estimation of scale change factor and distinction from scale efficiency ratio.
Discussion on challenges and potential infeasibility issues in decomposition when imposing CRS on VRS data.
Clarification on how output points affect feasibility and decomposition results.
Summary of decomposition approach from a diagrammatic perspective for better conceptual understanding.

Chapters

What is the main focus of the Malmquist Productivity Index decomposition in this video?

The video focuses on explaining the decomposition of the Malmquist Productivity Index, particularly using Ray and FGNZ models, highlighting technical efficiency, technological change, and scale change factors.

How is technical efficiency change estimated in the Randly decomposition?

Technical efficiency change is estimated by comparing the ratio of actual output to potential output against the VRS frontier for two periods, calculating the change in efficiency for a given input bundle.

What challenges arise when imposing CRS technology on VRS data in productivity analysis?

Imposing CRS technology on VRS data can lead to infeasible solutions and inaccurate estimates, as it may not reflect the true nature of the production technology, causing deviations from the actual productivity measures.

Full Transcript — Download SRT & Markdown

Speaker A

Hi, welcome back to the course Applied Production Analysis using MATLAB. In the last session, we saw the Marcus productivity index and various decompositions there, particularly the FG and Z. Before that, we had FGLR and the Ray industry decomposition. As I mentioned, we didn't go through the entire decomposition process. What we saw was the final outcome of decomposition, and we had overlapping components of models FG and Z. It has the same technological change that we are having in the context of FGLR.

Speaker A

mentioned we didn't go through the entire decomposition process what we saw was the final outcome of decomposition and we had overlapping components of models FG and Zed It has same technological change that we are having in the context of FGLR.

Speaker A

Then the technical efficiency change of FG NZ is being replicated or considered in the context of Rendously. But when it comes to technological change, then Randly becomes very different from what FGNZ and FGLR had.

Speaker A

But when it comes to technological change, then Randly becomes very uh different from what FGNZ and FGLR had.

Speaker A

Against CRS technology, Rendously considers VRS technology. The shift in this that is being considered as the technological change, and we had scale efficiency change, which was in the case of FG NZ. We have scale efficiency change. When it comes to Randely, it becomes scale change factor. So in this session, we'll be revisiting the entire procedure, not the entire concept, but particularly the components of TFP and the decomposition part. I'll be showing you the Ray and display component, and once you understand the Ray industryly and once you have an understanding of what FGNZ and FR were using, you can get the same outcome.

Speaker A

scale change factor. So in this session we'll be revisiting the entire uh procedure not entire concept but uh particularly the components of TFP and the decomposition part I'll be showing you the ray and display component and once you understand the ray industryly

Speaker A

So this one is basically a diagrammatic representation of productivity change that you can consider here. Particularly, I'll be considering the Ray and 9097 decomposition. Okay, so here we have four data points A, we have A, B, C, D, that two for period 1 and period 2. So we'll be having A0, B0, C0, D0 for period 0 and A1, V1, C1, D14 period 1, period, period 1.

Speaker A

So this one is basically a diagrammatic representation of productivity chain that you can consider here particularly I'll be considering the ray and 9097 decomposition okay so here we have four data points A we have A B C D that two for period 1

Speaker A

Now we have two returns to scale that you can conceptualize here. R stands for CRS technology and S stands for the VRS technology. So here what you can see using this A0, B0, C0, and D0, we can construct the technology set. This is going to be the technology set in the boundary of this technology set that is basically the S0, particularly the VRS frontier.

Speaker A

Now we have two returns to scale that you can conceptualize here. R stands for CRS technology and S stands for the VRS technology. So here what you can see using this a 0, b 0, c 0 and d 0 we can

Speaker A

Okay. So here in period zero, you can have VRS and CRS technology frontier. So this is our VRS frontier, and this is basically our CRS frontier for period zero. Okay.

Speaker A

Okay. So here in period zero you can have VRS and CRS technology frontier. So this is our VRS frontier and this is basically our CRS frontier for period zero. Okay.

Speaker A

Similarly, for period one, we can have VRS and CRS from here. And here, without thinking much about the estimation of market productivity index or decomposition, we can see that both CRS and VRS frontier has shifted. As against period 1, Z period zero technology, period 1 is coming above. That gives us an idea that there is a technological change the way Sodo has conceptualized in his classic.

Speaker A

P period 1 Z period period zero technology period 1 is coming above that gives us an idea that there is a technological change the way Sodo has conceptualized in his classic.

Speaker A

So here we will be particularly considering the Ray decomposition. So here, let's consider the most easy component.

Speaker A

I would write it as a technical efficiency change. So here when I say randly decomposition technical efficiency is estimated against VRS frontier.

Speaker A

I would write it as a technical efficiency change. So here, when I say Randly decomposition, technical efficiency is estimated against VRS frontier.

Speaker A

So here we'll consider bundle x1 that is basically the bundle in the period uh one and here potential output is basically a1 sorry actual output is a1 and the potential output is u1 right. So here the technical efficiency of bundle x1 is basically

Speaker A

So for that, we have both period zero and one. So period one technical efficiency and its ratio to the period zero technical efficiency will give you the technical efficiency change.

Speaker A

a 0 x0 divided by t 0 x0. This is basically the change in technical efficiency that has happened for observation uh a for period in period 1 as against period zero.

Speaker A

So here we'll consider bundle x1, that is basically the bundle in the period one, and here potential output is basically a1. Sorry, actual output is a1, and the potential output is u1, right? So here the technical efficiency of bundle x1 is basically u1 x1. So actual output is a1 x1 divided by u1 x1. So that is basically the technical efficiency in period 1, all divided by we now we'll consider what was the technical efficiency of A in period 0. So that is basically a0 x0 divided by t0 x0. This is basically the change in technical efficiency that has happened for observation A for period 1 as against period zero.

Speaker A

actual output and here s0 with that we have t 0 x0 as the potential output under vrs. So that gives us the technical efficiency change. So this technical efficiency change is basically technical efficiency change under randly as well as FG

Speaker A

We'll just cross-check it. So this a1 x1, a1 x1 is basically the actual output, and its potential output is basically u1 x1 with reference to S1 technology. When it comes to period zero, a z x0 is the actual output, and here S0 with that we have t0 x0 as the potential output under VRS. So that gives us the technical efficiency change. So this technical efficiency change is basically technical efficiency change under Randly as well as FG and Z also. You get the same technical efficiency thing. Moving ahead, we consider the technological change component.

Speaker A

So here technological change has two parts basically. If you if I consider uh X1 technology has shifted from here to here. If I consider X0 input bundle technology has shifted from here to here. Okay. So we'll consider both

Speaker A

So here technological change has two parts basically. If you consider X1, technology has shifted from here to here. If I consider X0 input bundle, technology has shifted from here to here. Okay. So we'll consider both this information while measuring a technological change component under Rand.

Speaker A

U1 it just shifted to U1 X1 and earlier case it was T1 X1. This is the first component considering X0 X1 as the bundle.

Speaker A

Okay. For that, first let's consider X1 case. For X1, basically technology has shifted to U1.

Speaker A

u x0 divided Okay, t 0 x0 gives you the complete picture and you get square root of that or the geometric mean of u1 x1 / t1 x1 into u0 x0 / t0 x0 will give you the technological change

Speaker A

U1, it just shifted to U1 x1, and earlier case it was T1 x1. This is the first component considering X0, X1 as the bundle.

Speaker A

of FGNZ and FGL technological change. So this is basically purely the uh technological change component that you are referring to in the context of ray industry decomposition.

Speaker A

Similarly, for x0, if you consider in period 0, it was t0 x0, the technology point, and for period 1, it has become u0 x0. So we'll consider u0 x0. This is basically how much the technology has shifted.

Speaker A

So the scale change factor is estimated in the first instant. We estimate the scale change that has happened with reference to zero period CRS and VRS technologies.

Speaker A

u x0 divided. Okay, t0 x0 gives you the complete picture, and you get the square root of that or the geometric mean of u1 x1 / t1 x1 into u0 x0 / t0 x0 will give you the technological change that has happened with reference to VRS technology that I'm mentioning. If you do a CRS, we'll be referring to Q1 or Q0 or P0 or P1 for period 0 and Q0, Q1 for period 1, which is basically the case of FGNZ and FGL technological change. So this is basically purely the technological change component that you are referring to in the context of Ray industry decomposition.

Speaker A

So here I can say that now we are referring to u x1 bundle. Okay. And now we are seeing uh periods zero uh first period technologies basically. So here you can see T1 X1 that is basically the potential

Speaker A

Moving ahead, we have the most complicated component of brain dis decomposition that is basically scale change factor.

Speaker A

t1 x1 divided by t1 x1. This is basically uh the scale efficiency that we are having for zero period technology but for input output bundle one. Now the same uh period technology we are referring to this highlighted component that is basically

Speaker A

So the scale change factor is estimated in the first instant. We estimate the scale change that has happened with reference to zero period CRS and VRS technologies.

Speaker A

input bundle one. Now we can estimate the same for input bundle uh x0 that is basically t 0 x0 divided by the CRS counterpart that is going to be p 0 x0.

Speaker A

Then we do the same for period one technologies for both the input-output bundle. So here the first component we can see here is basically going to be t1 x1 / p1 x1.

Speaker A

then how scale efficiency has changed for period one that is x0 value as again x period 1 that is x1 one as again the x0.

Speaker A

So here I can say that now we are referring to u x1 bundle. Okay. And now we are seeing periods zero, first period technologies basically. So here you can see T1 X1 that is basically the potential output under VRS. So scale efficiency is basically TE CRS by TE VRS. Alternatively, we can write it as the potential output Y star under VRS to Y star under CRS, or we were mentioning it as F of X / R of X, right? So here t1 x1 divided by t1 x1. This is basically the scale efficiency that we are having for zero period technology but for input-output bundle one. Now the same period technology we are referring to this highlighted component that is basically CRS technology in the period zero and the VRS technology in the period zero. So you can ignore the VRS and CRS in the context of our period 1 technology. Now we have scale efficiency estimated against period zero technology but for period input bundle one. Now we can estimate the same for input bundle x0 that is basically t0 x0 divided by the CRS counterpart that is going to be p0 x0.

Speaker A

from u1 x1 to u1 x1 / q1 x1. Okay. divide by for this thing it will become u 0 x0 divided by q0 x0.

Speaker A

I repeat here scale efficiency we are not calculating scale efficiency as the ratio of scale efficiency in period 1 to that in period zero. What we are considering, we consider the particular year's technology that is CRS and VRS technologies, then how scale efficiency has changed for period one that is x0 value as again x period 1 that is x1 one as again the x0.

Speaker A

period 1 technologies but for both input output wonder. So here from a more realistic sense only q1 u u1 x1 divid by q1 x1 is basically in a true sense that has happened that is basically the scale efficiency the firm

Speaker A

Similarly, now we can refer only to the highlighted over here. This is basically the CRS and VRS in period one. Now we'll see how change, how the scale efficiency has changed. For that, we'll see for x1. For x1, it has changed from u1 x1 to u1 x1 / q1 x1. Okay. Divide by for this thing it will become u0 x0 divided by q0 x0.

Speaker A

So this is basically scale efficiency. This diagonal component can be considered as the scale efficiency chain. The moment we have the endear component that we discussed over here become it becomes the scale change component that's or scale change factor

Speaker A

So here we can write this as scale efficiency change one and scale efficiency change zero. And here SC0 is completely estimated for X1 and X0, particularly for zero period technologies with reference to that, and X SC1 is basically estimated particularly for period 1 techn...

Speaker A

So as compared to earlier decomposition of course rand decomposition is bit more realistic particularly from the fact that it consider VRS technology for shifting the frontier. So imposing CRS technology it will be bit complicated when when you when your true technology

Speaker A

follows VRS and you are imposing CRS it can create some uh unfair estimates or it can it can make your estimate bit away from the true estimate. Okay. So with that uh there are how to decide. So these days in the literature you can see

Speaker A

non-parametic test for return to scale. Okay. So using this nonparametric test for returns to scale what we do we estimate efficiency under VRS and efficiency under CRS and see whether they are statistically significantly different and if they are statistically

Speaker A

significantly different we follow VRS. Uh if not we we can follow the CRS. So one can approach this problem from an empirical perspective. We can test for the return to scale and accordingly we can go for the technological change or technical

Speaker A

efficiency change or the scaling factor component in the TFP decomposition. Moving ahead though decomposition is an advancement over the earlier decomposition of FGN zed it has a problem. I'll show you here.

Speaker A

If A1 here we had A1 say suppose A1 was coming somewhere here. Say A1 was here slightly below what A1 is there. My objective was to see one case where A1 is given here. So A1 A1 being here it

Speaker A

will not affect our VRS from here because it's still become feasible and it is still an inefficient point over here.

Speaker A

But A1 here I'm writing it as A1 dash. Okay. So if our A1 was here at some point when you are estimating the distance function of X1. So A1 X1 will become at this point say X1 dash.

Speaker A

The moment you are trying to estimate the distance function of X1 dash against zero period technology that to VRS technology I'm referring to what will happen the algorithm will search for potential the actual output divide by the potential output either

Speaker A

greater or lesser than that of potential output with reference to zero period technology S0 I'm referring to it will not be finding any point that comes within the technology set of s0 that is VRS technology set under period zero. So

Speaker A

this will result in a case that if we had a1 dash with a x1 y1 dash and say x1 dash and y1 the moment we plug in this as the point and the reference set is basically the zero period technology. This will not

Speaker A

belongs to uh this not necessarily it belongs to s0 that is fine. It can lie above the s0 like the way we had it in the context of a1. Still we can have a feasible solution.

Speaker A

But here what has happened the x1 dash is less than the minimum of ender uh x0 js that is basically the end observation that we are having in the context of zero period technology particularly the input values that is basically

Speaker A

coming greater than the x1 dash that I have conceptualized over here. So now you know where I'm taking you.

Speaker A

The moment you are trying to estimate efficiency of one input output bundle in period one again period zero's technology that is a framework that period 1 input output bundle is not coming into the reference set of period zero technology that means

Speaker A

the moment when you are formulating the LP program problem these values are not coming in the uh LP formulation only in the right hand side of the LP you'll be getting this value. As a result the framework that we see in the

Speaker A

context of molecules productivity index estimation or its decomposition we see it follows very similar to what we had it in the context of super efficiency.

Speaker A

And in case of super efficiency we have seen the moment you have value for which you are estimating the technical efficiency the x1 dash here and if it is less than the values being included in the reference set of uh the

Speaker A

technology that you are creating that is xj0 x that is this is basically j one up to n here it will be j one up to a b cde e that we are having. So in that case it

Speaker A

will give you a this condition will give you an infeasible solution. So the moment we have an infeasible solution uh you'll not be able to get the complete picture or complete decomposition of our rand decomposition. So if you are

Speaker A

running a randomly decomposition and if somewhere you see the technological change or particularly the scale change factor you are getting I enough that means uh the observation that you are referring to it had an input value less than the minimum of uh the input values

Speaker A

that we had in the period zero or sometime it can happen even for the period one.

Speaker A

So how to solve that uh the invisible solution when your research objective is to get an average of entire units that you are considering it might not create a very huge issue because you can leave them out and then you can estimate a uh

Speaker A

tremendously average values scale change factor or scale efficiency change or technological change. But if you had a very limited number of observation and you are reporting the observations and their average values or their particular values in feasibility become very

Speaker A

difficult point for us to explain what was the uh condition of these observation with infasibility in the context of uh components of rainyly molecules productivity index decomposition.

Speaker A

So that is one case here. Uh output will not create a infusibility. Output will not result in an infusibility. Whatever even if the output of form a is less less than or greater than the greater than the maximum of that still you will

Speaker A

have a feasible solution. Even if it is less than the minimum that what will happen? It get adjusted in the far distance function or the fee that you are estimating estimating and it will not create you or it will not give you

Speaker A

any infeasible solution. In the context of output oriented approach the inputs values if it satisfy the condition that I was mentioning if it is less than the minimum of inputs in the reference period it will gives you a invisible

Speaker A

solution. And if you are following input oriented approach, if you are getting your particular output of a firm in a particular year is greater than the maximum of output in the reference period or the reference technology that you are constructing,

Speaker A

you'll be getting an invisible solution. The same way what in the context of u this randly decomposition output oriented approach. So just to summarize the entire component is basically taken from uh DEA by Subash and the roots are

Speaker A

basically uh 1997 paper by R industryly productivity growth technical progress and efficiency change in industrial countries combined. Basically this one is a uh revisiting the paper by FGNZ with a different perspective or considering technological change from a VRS point of view. So just to summarize

Speaker A

we was approaching the end decomposition from diagrammatic point of view. I'm very sure this might be bit more contri concrete for most of you as again the abstract formula that we followed what we focused was particularly the randity

Speaker A

de composition and somewhere fgnz technical efficiency was also the and taking the diagonal component we have already seen that is going to be the scale efficiency change and fgnz understanding that you can easily extrapolate the same concepts into the

Speaker A

context of fgls to summarize we were seeing the particularly the rain dusty decomposition from a diagrammatic point of view. So using this diagram where we have both period one and period zero observation and corresponding period zero and period 1

Speaker A

technologies both VRS and CRS we decomposed our uh randly values or rain the malus productivity index into technological change ratio of techn technical efficiency change ratio of technical efficiency in periods one and to that that in period zero

Speaker A

technological change with reference to particularly the VRS frontier and the geometric mean of scale efficiency with reference to zero period and first period technologies for both in both the input bundle is basically the geometric mean and we consider that as the scale

Speaker A

change factor it is slightly different from our scale efficiency change that we having in the context of fg Right.

Speaker A

And uh also we saw how randly decomposition result in inible solution particularly when you are having uh the input bundle which is less than the minimum of the input bundle that you are considering for the reference. So if

Speaker A

input bundle in period one that you are considering is less than the minimum of input bundle in period zero when you are estimating technical efficiency of the particular input bundle against the uh input bund the technology from DA that

Speaker A

you constructed for zero period technology it gives you an infasible solution and it becomes complicated. So whatever we discussed so far is basically particularly the randly decomposition.

Speaker A

uh but our technological change technical efficiency change remains the same under FG NZ and by taking the diagonal component of the two components of scale change factor geometric min which we took that we can get the uh technical efficiency change under or

Speaker A

scale change scale efficiency change under FGN zed. Thank you. Hey,

Topics:Malmquist Productivity Indexproductivity decompositiontechnical efficiencytechnological changescale efficiencyCRS technologyVRS technologyRay decompositionFGNZ modelApplied Production Analysis

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