Malmquist Productivity Index: An Introduction

Introduction to Malmquist Productivity Index, its theoretical basis, and decomposition using frontier-based approaches in productivity analysis.

Key Takeaways

Malmquist Productivity Index does not require price data, unlike Fisher's or Törnqvist indexes.
It uses frontier-based methods to measure productivity and decompose changes into technical and scale efficiency components.
Distance functions under CRS and VRS are central to calculating technical efficiency.
The index compares productivity across different time periods to assess change.
Scale efficiency captures the effect of operating at optimal production scale.

What the video covers

The video introduces the Malmquist Productivity Index as a frontier-based approach to measure productivity changes without requiring price data.
It revisits concepts of production technology, average product, and distance functions related to efficiency measurement.
The Malmquist index decomposes productivity change into technical change, efficiency change, and scale efficiency change.
The video explains the difference between CRS (constant returns to scale) and VRS (variable returns to scale) frontiers.
Technical efficiency is measured using Shephard distance functions under CRS and VRS assumptions.
The index compares input-output bundles across two time periods to estimate productivity change.
Scale efficiency is defined as the ratio of technical efficiency under CRS to that under VRS.
Mathematical formulations and graphical illustrations are used to explain the concepts.
The video references seminal works by Färe, Diewert, and others on Malmquist productivity index decomposition.
The session concludes by highlighting the practical advantages of the Malmquist index in empirical productivity analysis.

Chapters

What is the main advantage of the Malmquist Productivity Index over Fisher's or Törnqvist indexes?

The Malmquist Productivity Index does not require price data for inputs and outputs, making it more practical for empirical productivity measurement when price information is unavailable.

How does the Malmquist Productivity Index decompose productivity change?

It decomposes productivity change into technical change, efficiency change, and scale efficiency change using frontier-based distance functions under CRS and VRS assumptions.

What role do CRS and VRS frontiers play in the Malmquist Productivity Index?

CRS (constant returns to scale) and VRS (variable returns to scale) frontiers define different production possibility sets; technical efficiency is measured against these frontiers to assess scale efficiency and productivity changes.

Full Transcript — Download SRT & Markdown

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[music] [music] Hi. Welcome back to the course of late production analysis using MATLAB. Last few sessions, we focused on Fisher's productivity index, Törnqvist productivity index, and how to estimate total factor productivity or productivity change using these approaches.

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They're very theoretically sound. There's no doubt about that. And mathematically, very simple for anyone to estimate, and there are no complications involved.

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But in reality, we lack price information. That is why we are unable to estimate allocating efficiency also in most cases.

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So, whether it is Fisher's index or Törnqvist index, it requires price data. So, having price data for both outputs and inputs is not very easy from an empirical point of view.

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As a result, we have one frontier-based approach for measuring productivity and decomposing the same.

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That is basically the Malmquist productivity index. So, in this session, I would give you a brief outline of how Malmquist productivity index works. It's basically initially proposed by Diewert et al.

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And here, we see production technology. We will revisit our concept about production technology and explain average product in this context. Technology set and distance function. Distance function is not very new to you. Basically, this is the same concept of efficiency in some

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sense that we have already done. The efficiency scores or C value that we got in the output oriented basically called as Farrell distance function or better measure of efficiency.

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We're using the same concept. And the beauty of Malmquist productivity index is it can measure productivity. Along with that, it can decompose the sources of productivity into technical change or efficiency change or in a later case, you can even

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measure scale change factor. So, production technology, the average product of a firm can be defined as f of x divided by x. Basically, f of x will be the y or the same.

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One input one output case that we have considered. And here, if you're operating at optimal scale, that is basically x star, and first difference of this f of x basically will be the marginal productivity of

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marginal productivity of the factor. And x star into the marginal productivity will give again the output that is f x star. It's not really significant in this context.

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So, here, when we are estimating productivity, we are following a frontier-based approach. We'll be using in the first instance, as I mentioned, you can think of CRS case input output.

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So, basically, this is basically say R1 or R2. We can write it as R1 and R0.

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So, here, you can see the slope of R1 and R0. R1 and R0. As compared to R0, R1 has a greater slope. That means when you see out an outcome over here and an outcome on frontier, can you see the

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average productivity of these two outcomes. You can see average productivity or productivity of this input that you're considering has changed over period.

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So, that gives us a hint that once you have the frontier, you can use the information of the frontier to see whether average productivity of the individual observations have changed over the period. Okay.

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Here we'll be using R for CRS frontier. It is basically the CRS frontier. And f of x basically for the VRS frontier.

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And as we know, so this is basically say we have one frontier. So, this one is a technically optimal production scale say.

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So, this is basically x star. Right? Here. And this observation is basically efficient under both VRS and CRS frontier. So, we can call this as a technically optimal production scale and it is basically both CRS and VRS will be higher.

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But in most cases, if we have both CRS and VRS frontier, the output under CRS or the potential output under CRS will be greater than or equal to that of the potential output under VRS for any other

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domain that we are considering because VRS is basically a subset of CRS production possibility set.

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And this wherever r of x equal to f of x as I mentioned this point, these points on this CRS frontier or this point where our VRS and CRS technical efficiencies are same, we call it the technically optimal production scale.

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Okay. Now, we'll be using a distance function framework. This is basically Shephard distance function, I would say.

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Here, we are defining d(x,y) as minimum of x,y divided by fee and such a way that the x,y by Sorry, x,y by delta should belong to T.

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In case of Farrell's, we were considering y fee, which should belong to technology set. Here, instead of that, we are minimizing delta so that y divided by delta should also belong to our technology set.

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And delta is defined as y by f of x. It is basically actual output divided by the potential output. It's basically nothing other than the technical efficiency score or one by Farrell measure of efficiency fee. Okay. And technology set

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is defined as all set of x and y such a way that the distance function that you are estimating for each data point should be less than or equal to one.

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And similarly, this was the distance function under VRS. Similarly, we can define the technology set under CRS, that set of all x and y when you consider the output y, that should be less

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than the potential output in the CRS function, or the CRS function is basically the boundary of CRS sector.

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Same instead of VRS, we are using CRS so that you basically dC(x,y) basically minimum of delta(x,y) by delta with the condition that x,y by delta belongs to technical efficiency.

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And here dC(x,y) is nothing other than the technical efficiency of input-output bundle x,y estimated against CRS boundary at the corresponding period that is the y divided by the potential output r of x.

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So what Malmquist productivity index does, it considers the individual data point that is basically x1,y1 that is for the current period and x0,y0 for the last period.

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And here you can see y1 r x1 divided by y0 x0. Okay? So that is basically the technical efficiency of observation, the change in technical efficiency I would say, right?

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y1 divided by r x1 that is basically the output divided by the potential output under CRS frontier and then output in the particular period divided by the output in the base period divided by the potential output so that will give you the

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output under the technical efficiency in the last period input-output. So this is basically nothing other than started with y1 divided by x1 divided by y0 divided by x0. So, what we did here instead we can plug in, we can make

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a reference to our CRS from here. We can write it as y1 r1 x1 or r x1 into r x1 divided by x1. So, this will cancel out and this will remain the same for our denominator.

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Okay. And by formulation r x1 x y x1 divided by r x0 x0 because since it is a line passing through origin, both the ratio will get cancelled out and it will remain this component. y1 y1 r x1 y0 r x0

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And here this y1 r x1 is basically the technical efficiency under CRS for period one divided by technical efficiency under CRS for period zero. So, we can use this ratio of distance function as a measure of technological change under CRS.

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Now, we have something beyond that. We have something called scale efficiency change. Basically, scale efficiency is something we measured as technical efficiency under CRS by technical efficiency under VRS. Or in this case, we can put it this manner. So, this is

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basically f of x divided by x. So, that is basically y divided by x divided by y. This is the potential output under VRS divided by x and the potential output under the CRS that is basically a form x0

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y star divided by x star is basically the y star is the potential output under CRS technically optimal production scale.

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Okay. So we'll be exploiting this identity also or this equation also to see the scale efficiency. So scale efficiency is defined as the x and x star we can plug in here. So scale efficiency of point x is basically f of x divided by

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the corresponding r of x. Okay. And breaking it down as I mentioned y by r.

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efficiency of input-output bundle XY estimated again CRS frontier and this is basically the technical efficiency of X and Y estimated again VRS model.

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So these are like very connected so but looks a bit uh confusing for you. So our attempt is to we having formula that is Y1 by X1 divided by Y0 by X0 will give you the productivity change. So what we are doing we are

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plugging in the frontiers into that so that we don't need a price information with reference to the potential output under CRS and VRS and exploiting that then so we can estimate the productivity.

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So basically when it comes to productivity analysis that is particularly the Malmquist productivity index there are multiple decomposition.

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The very first as I mentioned this original model was Färe et al. and Diewert and later Um but the very first decomposition we would see the estimation of productivity and the decomposition thereof. Decomposition of productivity into technical change and

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technical efficiency change and if you are assuming different returns, so we can have scale efficiency change also.

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So, what I intend to say the literature on uh Malmquist productivity index is back early days still but the Malmquist productivity index the combo the estimation and its decomposition start with Färe, Grosskopf, Lindgren and Roos. So, we call it as FGLR

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FGLR 1992. This is like the most classic paper in the context of Malmquist productivity index estimation and the decomposition thereof.

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Then we have Färe Grosskopf Norris and Zhang. This is basically FGNZ. We're not be calling all the names in the later case it will be MGNZ 1994.

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So, what is the different? Quickly this model is purely based on CRS. So, we consider only CRS technical efficiency CRS frontier for estimation of technical efficiency change and technological change.

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Moving ahead in FGNZ technical efficiency is estimated against VRS but the technological change component in the TFP change being uh estimated using CRS. The technological change remain CRS in Moving ahead, Reifschneider uh decomposition that is basically a revisiting of

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the concept covered by FGN said here. For technological change also, we consider VRS specification. And as a result, we have scale efficiency and here the moment we plug in technical efficiency change under VRS, we get FGN said scale efficiency and

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pure technical efficiency change. And then we can uh discuss about Ray and industry decomposition which has technical efficiency change, technological change, both of them are estimated against VRS frontier and then additional component basically is scale efficiency change but in the industry

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context since we are approaching that scale efficiency from a different point of view, we call it a scale change factor.

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And these concepts, whatever I covered here, are very basic or fundamental how directionally same functions are plugged into a productivity analysis being taken from the DEA textbook of Ray and Subhash Sharma. Okay.

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So, in the upcoming session, we'll see FGLR, FGN said Ray and industry decomposition and I'll try to estimate it using R and data. Thank you.

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[music] [music]

Topics:Malmquist Productivity IndexProductivity AnalysisTechnical EfficiencyScale EfficiencyDistance FunctionCRS FrontierVRS FrontierTotal Factor ProductivityProduction TechnologyEfficiency Change

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