**DEA: CCR and BCC Models โ€” Transcript & Summary | SozAI** Source: https://sozai.app/transcript/dea-ccr-bcc-models/ An in-depth exploration of DEA, focusing on CCR and BCC models, assumptions, and technical efficiency estimation using MATLAB. ## Key Takeaways - CCR model benchmarks DMUs against the best performing unit using shadow prices and constant returns to scale. - DEA does not assume a specific functional form but constructs the production frontier empirically based on data and assumptions. - Convexity and free disposability are critical assumptions ensuring the feasibility of interpolated and extrapolated production points. - VRS model relaxes the CRS assumption by enforcing the sum of lambda weights to equal one, allowing for variable returns to scale. - Efficiency measurement can be input-oriented or output-oriented, and peers identified through lambda weights help in benchmarking. ## What the video covers - Introduction to CCR model as the foundational DEA model for estimating technical efficiency with one input and one output. - Explanation of average productivity standardization to derive technical efficiency scores between 0 and 1. - Discussion on shadow prices from linear programming used in CCR to benchmark DMUs against best practice. - Overview of DEA as a nonparametric approach constructing production frontiers without assuming functional forms. - Detailed assumptions of DEA: availability of input-output data, convexity, free disposability of inputs and outputs. - Construction of technology sets under VRS (Variable Returns to Scale) and CRS (Constant Returns to Scale) assumptions. - Use of lambda weights in linear programming to define feasible production frontiers and identify peers. - Graphical and conceptual explanation of convexity, disposability, and scale efficiency in DEA. - Comparison of CRS and VRS models and their implications for efficiency measurement and returns to scale. - Application of DEA framework to multiple inputs and outputs with MATLAB for practical efficiency analysis. ## Chapters 1. 00:00 Introduction and CCR Model Overview 2. 02:25 DEA Assumptions and Nonparametric Approach 3. 04:33 Free Disposability of Inputs and Outputs 4. 07:26 VRS Technology Frontier and Relaxation of CRS 5. 10:34 Constructing the Technology Set in DEA 6. 12:36 Convexity, Disposability, and Feasibility in DEA 7. 15:12 Efficiency Measurement and Orientation 8. 19:33 Multiple Inputs and Outputs in DEA Framework Answers ## Questions about this video What is the CCR model in DEA? The CCR model is the foundational DEA model that estimates technical efficiency assuming constant returns to scale. It benchmarks decision-making units against the best performing unit using shadow prices derived from linear programming. What are the key assumptions behind DEA? DEA assumes availability of accurate input-output data, convexity of the production possibility set, and free disposability of inputs and outputs. These assumptions ensure that observed production points are feasible and that interpolated or extrapolated points remain within the technology set. How does the VRS model differ from the CCR model? The VRS model introduces variable returns to scale by enforcing that the sum of lambda weights equals one, allowing for more realistic modeling of production technologies compared to the CCR model's constant returns to scale assumption. ## Full Transcript โ€” Download SRT & Markdown 00:05 Speaker A [music] [music] Hi, welcome back to the course Applied Production Analysis using MATLAB. Last session, using a very simple one input one output case, we were estimating the efficiency, and at the end, we saw how the CCR model, as the foundational model for estimating technical efficiency using DEA, was arrived at. So we started with average productivity, and then we needed to standardize it or get a measure that can be used in any context. 00:37 Speaker A For that purpose, we took the maximum value of average productivity and adjusted the other DMUs accordingly, and we got a value which lies between 0 and 1, and that can be considered as technical efficiency. 00:56 Speaker A And here the fundamental principle was we were picking the observation with maximum average productivity as the benchmark. I use the term benchmark or against which we are comparing. With that, we can get a hint that the model that we had was something that compares individual observations against the best practice or technically optimal production scale. That gives us a hint that the model was something based on constant returns to scale. 01:13 Speaker A So the model, after getting the estimates, and how we did, we started with average productivity, and then after getting estimates, we named it as a model that is basically the CCR model. 01:32 Speaker A So that is a framework where ideally we should have used the input price and output price, but in reality, you don't get input prices and output prices. So you use shadow prices, which come from the context of operations research or linear programming. And here, in that context, the shadow prices were determined such a way that for each observation, when you are calculating the average productivity, we get a set of or a vector of shadow prices for input and shadow prices for output such that once you plug in these shadow prices for other observations, none of them should get an average productivity greater than one, including for the firm under consideration. 01:50 Speaker A So that is like a just summary for our CCR model. Now, in today's session, we'll delve into detail on the technical aspect or the assumptions and the same variation we had, that CCR model or CRS-based model, and then an advanced model that is basically the VRS-based or more realistic VCC model. So today's session will cover the overall picture of data envelopment analysis, how it works, especially in the context of more than one input or more than one output, and we see what is the foundation. 02:07 Speaker A So while talking about the parametric approach, we mentioned that the parametric approach's starting point is assuming a functional form, Cobb-Douglas or translog. In the empirical literature, when it comes to the nonparametric approach, we don't a priori assume any functional form. What we do is we empirically construct the production frontier, which is not necessarily a translog or Cobb-Douglas production function, but for constructing that production frontier or technology set in the boundary of the chain, we need a set of properties of the production set that need to be plugged into our model to the data and then construct the technology set, and the boundary of that is called the production frontier in this case. So we'll 02:25 Speaker A [clears throat] familiarize ourselves with the set of assumptions that need to be satisfied or need to be imposed while estimating productivity from a nonparametric approach. So DEA, that is data envelopment analysis, it is used, it comes under the nonparametric. Here, 02:40 Speaker A we construct the technology set or production frontier by using a set of assumptions that we have already mentioned at some point. 02:55 Speaker A Assumption one, all actually input-output data are available. That means we are using a set of input-output data X, Y where i is one up to n. This is the data point that we have in our hand for our efficiency analysis in observation. So once you get this data point, we need to start with the assumption that whatever is being observed is technically feasible or all actually observed input-output bundles are feasible. So if one firm is producing 50 units of output by using 50 units of input, that means other firms also could have done the same thing for the technology that we are having. 03:16 Speaker A For that, we can say that all input-output bundles belong to the technology. Say there are no anomalies or there is no unrealistic picture that is coming in the data that you have collected for efficiency and productivity analysis. 03:32 Speaker A The second assumption is convexity. So for convexity, it's a property of the production possibility set that we are having. Say we have X\_A as one input that is being used by firm A, and they are producing Y\_A amount of output. 03:54 Speaker A That means it is technologically possible. And say another firm is working under X\_B, Y\_B, which is also technologically feasible. Given that lambda X\_A + (1 - lambda) X\_B and the same weightage lambda Y\_A + (1 - lambda) Y\_B, this becomes a new convex combination of firm A and B. If A and B are feasible, this should also belong to the technology set and it should also be feasible. 04:13 Speaker A Okay. So that is the convexity assumption. Moving ahead, we have assumption three, free disposability of inputs. 04:33 Speaker A If disposability of input, we can say that if X, Y is feasible, that implies any X\_0 which is greater than X and X\_0, Y should also be feasible. 05:04 Speaker A Right? Similarly, we have free disposability of output. That means if X and Y are feasible, it implies any Y\_0 which is less than Y, where we should be able to produce the same amount of like Y\_0 by using the same amount of input. So these are the properties we have already covered at some point. Now we'll formalize it for the DEA framework. 05:16 Speaker A Additionally, if you're assuming CRS, if X, Y is feasible for any value K greater than or equal to zero, 06:08 Speaker A also, sometimes the point zero is also feasible. KX, KY will also become feasible under CRS, including the origin. 06:28 Speaker A Okay. So these are the set of assumptions that we are having in the context of data envelopment analysis. 06:44 Speaker A Now, given these assumptions, we can conceptualize or we can construct our technology set. Here, the technology set basically is the set of all X and Y such that the X should be greater than or equal to the sum of lambda\_j X\_j, j=1 up to n, and the Y that you are referring to should be less than or equal to the sum of lambda\_j Y\_j. 07:01 Speaker A Here, we are conceptualizing sum of lambda\_j to be equal to 1, and lambda\_j should be found positive value zero. 07:26 Speaker A This is the case of VRS technology frontier. The moment we relax this sum of lambda equal to one case, that will give us the technology set under our CRS case. The others remain the same. So this technology set consists of all sets of or all combinations of X and Y such that that X that you are referring to is greater than that of the convex combination that we are referring to. 08:04 Speaker A This is basically here this part will take into care of the convexity part, and this greater than or equal to will take into account disposability of input, and here this Y less than or equal to this convex combination takes into account the disposability of output, and then sum of lambda\_j greater than or equal to is the case that you can give weightage to firms including zero, but there is no negative that we can assign for decision making under con. So 08:28 Speaker A now we'll see how DEA works by using this set of assumptions, how they construct the technology from DMU. For that, see we have DMUs like here. What we can do? So these are the DMUs that we are having. By convexity, every point between this will become feasible. This is basically the convex. By disposability of input, this point will become feasible, and by free disposability of output, this region will become feasible, and free disposability of output and input, this region we can extend towards infinity parallel to input axis. So now our objective is to get the efficiency of this point, say firm K, whether the firm K is technically efficient or not. That you can approach from input-oriented or output-oriented way. In case of output-oriented, we'll try to identify the maximum possible output t 08:50 Speaker A If dispossibility of input, we can say that if XY is feasible, that implies any X0 which is greater than X and x0 y should also be visible. 09:19 Speaker A Right? Similarly, we have three predisposivity of output. That means if X and Y are feasible implies any Y zero which is less than Y where we should be able to produce same amount of like y0 by using same amount 09:58 Speaker A of input. So these are the properties we have already covered at some point. Now we'll formalize it for the uh DEA framework. 10:17 Speaker A Additionally, if you're assuming CRS, if X Y is feasible for any value K greater than or equal to Z, it can be greater than equal to zero. 10:34 Speaker A Also, sometimes the point Z is also visible. kx k y will also become feasible under crs including the origin. 10:52 Speaker A Okay. So these are the set of assumption that we are having in the context of data envelopment analysis. 11:01 Speaker A Now given these assumptions we can conceptualize or we can construct our technology set. Here technology set basically set of all x and y such a way that the x should be greater than or equal to sum of lambda j 11:28 Speaker A xj j1 up to n and the y that you are referring to should be less than or equal to sum of lambda j y. 11:45 Speaker A Here we are conceptualizing sum of lambda j to be equal to 1 and lambda j should be found positive value zero. 12:00 Speaker A This is the case of VRS technology frontier. In the moment we relax this sum of lambda is equal to one case that will give us the technology set under our CRS case the others remain the same. So this 12:23 Speaker A technology set consist of all set of or all combination of X and Y such a way that that X that you are referring is is like greater than that of the convex combination that we are referring to. 12:36 Speaker A This is basically here this part will take into care of the convexity part and this greater than or equal to will take into account predisposibility of input and here this y less than or equal to this convex combination take into account the 12:55 Speaker A predisposability of output and then sum of lambda j greater than or equal to is the case that you can give weightage to uh firm including zero but there is no negative that we can assign for decision making under con so 13:16 Speaker A now we'll see how DEA works by using this set of assumptions how they construct the technology from DM for that see we have DM like is here what we can do. So these are the DMU that we are having by convexity 13:45 Speaker A every point between this will become feasible. This is basically the convex by predisposability of input. This point will become visible and by free dispos by free disposity of output this region will become visible and free disposity output in input this 14:11 Speaker A region we can extend towards the infinity parall to input axis. So now our objective is to get the efficiency of this point say firm K whether the firm K is technically efficient or not that you can approach from input oriented or 14:38 Speaker A output oriented way. In case of output oriented, we'll try to identify the maximum possible output the firm K could have produced if it was technically efficient given the level of input and technology. 14:58 Speaker A In case of input oriented, what we try? We try to minimize the input and try to identify what is the level of input the firm could have used if it was technically efficient. 15:12 Speaker A In case of firm K, let us conceptualize it as XK and it is producing YK amount of output. 15:27 Speaker A But our production possibility frontier says that the firm could have produced YK star as the potential output. So this is basically the YK star. 15:56 Speaker A So what is technical efficiency of firm K under output oriented over here yk divided by yk star which is going to be less than one here and says that the firm k is technically inefficient and this ratio will say how inefficient the firm is. 16:19 Speaker A So how to approach this problem from a DEA point of view? We'll conceptualize it here. 16:29 Speaker A We want to find proportionate increase in YK that the firm could have done to achieve maximum possible output but remaining in the same feasible region or the technologically feasible point or within the technology set that the way we defined earlier. So in output 16:53 Speaker A oriented case we are using one into one case one input one output. Our objective is to get maximum value of fee such a way that we take the convex combinations of individual observation that is in the data sum of lambda j yj. 17:24 Speaker A It is greater than or equal to yk. We are trying to identify a point which is greater than yk or equal to yk. But remaining in the technology set. So we add this product here. So what here we are going to get 17:43 Speaker A yk into fe is basically yk star. So from if it was technically efficient it could have produced from this point to this point. That is what we are searching. 18:03 Speaker A At the same time we need to confine ourself to the same level of input or we need to make sure that the new combination convex combination that you are considering should be using input less than or equal to that of the firm that we are 18:21 Speaker A discussing or talking about for keeping that sum of lambda j1 up to n it should be less than or equal to the firm X K or the input value X K the form K is using and since it is a convex combination we 18:44 Speaker A are referring to we'll have sum of lambda J should be equal to 1 and lambda J will take only values greater than or equal to zero. 18:59 Speaker A So this is basically a linear programming problem. What we are doing? We are changing the values of lambda and fe. We are trying to maximize the yk such a way that we still remain in the technology set or the convex 19:16 Speaker A and predisposability hole that we have constructed and we try to identify okay how much proportionate increase the firm could have done over here. 19:30 Speaker A So here for example here say firm was producing up to this point and say 1.25 is the value 1 by 1.25 will be the technical efficiency of the firm which is less than one. That means the potential output of this firm is 19:53 Speaker A basically 0.25 0.25 greater than what it is already being produced or conceptualized. So here this data what we discussed it is basically the case of or the diagram that we discussed here it is basically the VRS case and in case of CRS 20:17 Speaker A radial expansions and contractions will be allowed. So here say this observation is observation with maximum average productivity. 20:35 Speaker A So if this point is visible any expansion should also be feasible and contraction should also be feasible. 20:44 Speaker A In that case for firm K in VRS we were getting YK star as the output and in case of CRS we get YK greater than the potential output that we were getting in the conductor VRS technical efficiency. So here we can 21:07 Speaker A extend the same thing for multiple input multiple output. The simple way we can do say you have two output. So you will have uh one constraint for output one and another constraint for output two. Say here we had uh we had output one and 21:26 Speaker A two. So we can make it as J1 K1. And if you had one more output say sum of lambda j y z j2 greater than or equal to f of yk2 that will be the constraint that you can 21:46 Speaker A have it for the second output. So we can extend it or else you can just change the subscript. 21:54 Speaker A uh say we can have a R over here and R is basically 1 to M which is the case where you are having M output. Similarly, you can have J say um V XJ V and X K V and say V 22:23 Speaker A = V = 1 up to N. So this will become M output N input case. It can be easily generalized here in place of 1 into we can write it as m into case we'll approve the same problem from 22:58 Speaker A input oriented manner new diagram. So these are the data point same. So this is going to be the say this is going to be the VRS from here. We have one observation say here we need to try We need to estimate x k 23:42 Speaker A yk x k yk and we need to identify x k star which is basically reduction in input that we are considering over here such a way that keeping the same level of output what could have been the input the firm K should have used if it 24:13 Speaker A was technically efficient. So it is basically a contraction in input. So this is basically we consider it as theta XK and here our LP will become minimization problem. 24:32 Speaker A Minimize theta by changing the values of lambda and theta. Lambda j should be greater than or equal to yk. 24:53 Speaker A Sum of lambda j xj should be less than or equal to x k. We plug in this theta over here and we see how less uh the firm theta could have been but remaining in the same region. 25:11 Speaker A This can also be generalized for r = 1 up to m = 1 up to small n. So we have n input and m output and sum of lambda j equal to 1 for vrs and lambda j should be greater than or 25:40 Speaker A equal to zero for all values of j 1 up to n. So this is a case of input oriented technical efficiency we'll be getting a value of theta. Say here for this example we can see that X could have 25:55 Speaker A produced the firm K should have produced YK amount of output by using somewhat half of the uh input XK. So say value of theta will become 0.5 which gives us an idea that the firm uh K is half 26:13 Speaker A efficient and same with the CRS front. Now we will see what are the implications of the lambda weights that we are getting. 26:24 Speaker A We call it as the peer weights down weights can be considered as peer weights. 26:36 Speaker A Let's take a case that firm A is here, FM form B is here, firm C is here. Under VRS for FM C projecting here, we are getting a point over here. So you'll get lambda A and lambda will take a volume 26:58 Speaker A non zero. So here for firm C for getting the benchmark point on the frontier or for getting the convex combination of other two performing forms we are getting we are taking the case of A and B. So we call a 27:26 Speaker A and b as the peers of c. And once you solve this linear programming problem, whichever firms for which you are getting lambda weight greater than zero or a positive lambda weight by formulation, we call them as the peers. 27:44 Speaker A And those are the firms who are performing as compared to the firm the the closely comparable firms against the firm for which we are estimating the technical efficiency. 28:00 Speaker A So what is the implication of this lambda weights or peer weights from a managerial point of view? We can say that for C the comparable combination is coming from firm A and B. 28:15 Speaker A So we can say that for firm C to become technically efficient, it should follow the managerial practices of firm A and B. That is basically the interpretation of peer weight. Once we have this peer weight, will the peer weight remain the 28:32 Speaker A same under VRS and CRS? Not necessarily because VRS has an additional additional constraint sum of lambda G equal to 1. 28:44 Speaker A With that the sum of lambda that we are getting the sum of P weight we are getting it will be always equal to one. 28:53 Speaker A But in case of CS the peer weights can go greater than one or it can become less than one also. 29:04 Speaker A So for example say here from A is efficient under CRS also in this case see I'm just slight deviation is there here B is lying below the CRS frontier under VRS frontier we were getting A and B as the convex 29:30 Speaker A combination but under CRS respond here since A is slightly productive than firm B. Uh so firm C will be compared against firm A and this point it is basically firm A and the lambda weights greater than one. For getting this point we need to 29:55 Speaker A give an assign a lambda A for firm C's case greater than one. So that means under VRS the sum of lambda weights are going to be the same like sum of lambda weights are going to be within the 30:12 Speaker A constraint that is equal to one. But under CRS the sum of lambda weight can go greater than or less than that one which has an implication by using or by interpreting that we can say whether the firm is under increase in return to 30:27 Speaker A scale or diminish return to scale case. For example, consider a case here. So we had a observation here and for this observation the CRS point is somewhere here and V sorry VRS point is somewhere here and CRS point is somewhere here. 30:54 Speaker A So we are referring to the CRS frontier or the CCR model. Under CCR model for getting this point we have to take a lambda a which is less than one that will give you a convex combination there's no convex 31:11 Speaker A combination involved over here basically it is a yeah it's a sub case of convex combination where we are taking only one form there's no combination one form itself is scaling down to see the point so here actually the for getting this point we 31:28 Speaker A have to multiply y a with a value less than one that is lambda e weight. So looking at here we can say basically this region is basically the increase in attentive scale region and this region basically the which is very parallel to 31:51 Speaker A that but uh if I draw it bit more differently it would have been slightly bit more gap. So consider this as the point on VRS which is basically increase in return scale and this is basically the point on VRS from here with 32:14 Speaker A diminishing return scale. What is the interpretation of that? Under CS we are conceptualizing firm A as the optimal one. That is the one with which has the maximum average productivity or technically optimal production scale. 32:35 Speaker A And under that framework any firm operating toward the left side of the that that point or on this region this region that we are referring to they have to scale up themsel to become technically optimal. 32:56 Speaker A And opposite the case for a firm operating toward the right side of this point. That means this point is inefficient from the point that they are operating beyond the technically optimal production scale. That means for them to become 33:12 Speaker A technically efficient they need to reduce the scale of operation or reduce the input. That means this region is basically the region of diminish return and the point A is the one which we have we are having the CRS region. So that CCR model that 33:31 Speaker A is basically the CRS front here and estimate the CCR model and get the sum of lambda weights. And if the sum of lambda weight is greater than one that means the firms are operating toward the right side of the optimal technically 33:49 Speaker A optimal production scale that means the firms are facing the diminish attention scale and if it is toward the left side of the region that mean the firms are operating with a scope for improving their efficiency by scaling up or there 34:04 Speaker A is a increase in return to scale here from here to here this point to this point increasing input will increase the average productivity and from here to here the average productivity declines the moment you increase the scale of operation or amount of 34:28 Speaker A input. So that is the implication of CCR model. By looking at the lambda weights of CCR model, we get an inference whether the firms are operating under CRS or increase in attention scale or diminishing return to scale and that gives us an idea whether the 34:49 Speaker A firms has to increase their scale of operation to become scale efficient or technically efficient. 34:57 Speaker A So that's all. So just to summarize so in this uh session we formalized the DEA in a bit more comprehensive manner. So we had five assumption the fifth assumption is basically for the CRS frontier and the four assumptions of VRS 35:18 Speaker A [snorts] and uh building upon this four plus one assumption we can construct the technology frontier. If it is VRS for first four assumption and if it is CRS we extend the convexity assumption for um radial expansion and radial 35:36 Speaker A contraction. Then uh we saw two cases. One is input oriented and another one is output oriented. How the linear programming problem being formalized in input oriented and output oriented case. And the same constraint with a uh subscript for input and output we can extend to m 35:58 Speaker A output and n small n input case. And then uh we will see in the case of return to scale and how based on the lambda weights that we are getting from CCR model you can classify the firms as 36:19 Speaker A working under the IRS and DRS or at the point of constant return state and also the policy implications of peer weights. It says based on the peer wage that you are getting we can give prescriptive uh odices for the firms telling that for 36:40 Speaker A becoming efficient what they need to do from a peers perspective. So in the upcoming session we'll try to estimate the model and see how the models are being estimated and how to interpret this CRS VRS technical efficiency scores and the peer weights 36:58 Speaker A and all. Okay. Thank you. Hey, [music] Topics: Data Envelopment Analysis DEA CCR model BCC model Constant Returns to Scale Variable Returns to Scale Technical Efficiency Production Frontier Shadow Prices Linear Programming Study this video - [Flashcards from this video](https://sozai.app/tools/ai-flashcard-generator/?from=transcript&id=75859) - [Quiz on this video](https://sozai.app/tools/quiz-generator/?from=transcript&id=75859) Free, made by AI from this transcript. No signup. --- This is the markdown twin of https://sozai.app/transcript/dea-ccr-bcc-models/ โ€” the same content, without the markup. Published by SozAI (https://sozai.app). Reuse and quotation are allowed with attribution and a link back. Machine-readable index: https://sozai.app/llms.txt ยท data API: https://sozai.app/api/