Speaker A
[music] [music] Welcome back to the course Applied Production Analysis Using MATLAB. Last session, we were seeing aspects related to technology.
This lecture covers theoretical and empirical aspects of production functions, focusing on technology sets, returns to scale, and efficiency using MATLAB.
Key Takeaways
- Technology sets satisfy convexity and free disposability, forming the basis for empirical production function estimation.
- CRS assumes proportional scaling of inputs and outputs, while VRS allows for variable scaling and more flexibility.
- TOPS marks the point of maximum average productivity and divides regions of increasing and diminishing returns to scale.
- NIRS is a special case where returns to scale do not increase, transitioning directly from CRS to diminishing returns.
- Understanding these concepts is crucial for analyzing technical efficiency and production performance in empirical studies.
What the video covers
- Introduction to the relationship between production functions and technology sets, including properties like free disposability and convexity.
- Empirical construction of technology sets using observed input-output data points and the concept of convex hull.
- Explanation of free disposability of inputs and outputs and its impact on feasible production regions.
- Discussion on the production possibility frontier as the boundary of the technology set.
- Comparison between variable returns to scale (VRS) and constant returns to scale (CRS) production functions.
- Mathematical characterization of CRS where output scales proportionally with input.
- Definition and identification of the technically optimal production scale (TOPS) where average productivity is maximized.
- Analysis of regions of increasing and diminishing returns to scale relative to TOPS.
- Introduction to the non-increasing returns to scale (NIRS) case as a special but rare production function type.
- Implications of these production function concepts for estimating technical efficiency in empirical applications.
Chapters
- 00:00Introduction and Recap of Technology Set Properties
- 00:57Types of Returns to Scale: VRS, CRS, and NIRS
- 02:25Conceptualizing the Technology Set with Data Points
- 03:27Convex Combinations and Convex Hull
- 04:30Free Disposability of Output
- 05:53Free Disposability of Input
- 07:16Smoothing the Technology Set with More Observations
- 08:14Constant Returns to Scale Explained
- 08:58Average Productivity and Its Variation
- 09:53CRS Frontier and Radial Expansions
Full Transcript — Download SRT & Markdown
Speaker A
How production function and technology set are correlated or related. We also saw two important properties of technology set: free disposability and convexity.
Speaker A
Within free disposability, we saw free disposability of inputs and free disposability of output. In this session, we'll be empirically constructing a technology set and the production function as its boundary.
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We also see mainly two variations of production function. One is the variable return to scale and the other one is constant returns to scale. Also, we see a very rare case that is the non-increasing return to scale and its aspects.
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Moving ahead, suppose we have six data points and we are considering the case of one input and one output.
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Say we have six data points. Okay. Okay. And these are actually observed input-output bundles, meaning technologically feasible input-output bundles coming from your data set being reported by a firm or decision-making unit. So now
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we'll try to conceptualize the technology set in this context which satisfies the properties that we discussed already.
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So the very first property that we had was convexity. So by convexity, if this point is feasible, this point is feasible, any point connecting these two lines, that is the convex combination of these two points, should also be feasible.
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Same for the case of these two points. These two points. These two points. These two points. Sometimes convexity goes beyond two points. You can conceptualize three points and their convex combination such that the weights are summing to one, which can also be conceptualized
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here. So here, if I take an arbitrary point here, this can be conceptualized as a convex combination of these two points or these two points or these three points in some sense.
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So hereafter, we call this region, which is basically the inner region that can be conceptualized as the convex combination of these six data points, that you consider as the convex hull.
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Now we apply our understanding of free disposability of output. So here, say consider this point where a firm is using x1 unit of input and producing y1 unit of output. So by free disposability of output, any point below
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this y1 level of output should also be feasible under the technology. That means by using x1 unit of input, we should be able to produce any output which is less than y1 unit. That is the case of free
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disposability of output. So with free disposability of output, especially when you extend this same into the convex hull, any point below this convex hull also will become feasible. That is free disposability from an output point of view. So now
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we extend our understanding of free disposability of input. Disposal of input here. Say consider the case where this firm is producing at this point by using x2 amount of input and say producing y2 amount of output.
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Given the technology, this region should also become feasible. That means the firm should be able to produce y2 amount of output even by using an input which is beyond the point x2.
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So now with that, our production possibility set satisfying convexity and free disposability will become the technology set and the boundary of the technology set, as I have highlighted over here, is the production possibility frontier that you can conceptualize in an empirical
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setting. So here, we have started with a very small number of observations, that is six observations. Because of that, we see a production possibility set. It's like a piecewise linear but not that smooth. The moment we have more and more number of
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observations, we get an even smoother line as the border of the technology set, which is going to be the case in most of the empirical applications.
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So now whatever we conceptualized over here are basically the cases where the returns to scale are changing. By returns to scale, we have two cases: constant returns to scale and variable returns to scale.
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From a mathematical point of view, if f of x is able to produce y units of output given the technology, we should be able to double the input.
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Say use k times that of the input being used. Suppose it is resulting in an output which is basically k times the same proportion as what we did for our input. This is the case of constant returns to scale.
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For constant returns to scale to be present in the production function, the average productivity should remain constant throughout.
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But in this case, average productivity, that is the output that we are able to produce per unit of input, keeps on changing.
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Suppose you take an arbitrary point here. If I take this point, we can conceptualize the average productivity of this point as the slope of this line. Similarly, I can take a point over here where the average productivity is the same from that. So here you can
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see the first line had a slope less than that of the second line. That means average productivity at this point is higher than the average productivity at this point.
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So now coming to the CRS frontier, constant returns to scale frontier, we can conceptualize it as an extension of the VRS that we are having over here. By CRS, what we are conceptualizing is we are imposing an assumption that radial
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expansions and radial contractions are also feasible. That means if this point is feasible, if you multiply the input by k, you should be able to get the output k times more than what we are originally producing here. So with that, it becomes
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a straight line and we conceptualize it as R of X, which is basically the CRS production function, and the VRS production function we generally represent in terms of f of X.
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Talking of CRS production function, the CRS production function passes through the point at which the VRS production function has the maximum average productivity. So this is basically the point on the VRS production function which has the maximum average
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productivity and we can call it the technically optimal production scale or TOPS, technically optimal production scale.
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That means that will be the point at which the average productivity is maximum. And you can see, as I mentioned, any point toward the right side of the CRS point has a lower average productivity as compared to the point
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CRS or toward the right side. So this is basically the region of increasing returns to scale. That means by increasing the input for one unit, you get more than proportionate increase in your output.
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And in case of points toward the right side of the CRS point, that is technically optimal production scale, that region in the VRS frontier,
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a proportionate increase in input may result in less than proportionate increase in the output. That is basically called the region of diminishing returns to scale.
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And as you can see, the region below the CRS frontier is basically the CRS technology set. And here you can see that variable returns to scale technology set is a subset of the CRS technology set and the CRS production function that we
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already conceptualized is basically the boundary of the CRS production possibility set. It has a lot of implications when you estimate technical efficiency that we are going to discuss in the next class.
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Now I'll quickly take you through a very special case of production function which is not very common in the empirical literature but still you can conceptualize something of that. So that is basically the non-increasing return to scale or we can call it as NIRS.
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So non-increasing returns to scale is a special case where there is no increase in returns to scale. We start from the origin.
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We start from the origin. That is up to CRS it is the CRS frontier and after that we have the VRS frontier.
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So that I repeat, there is no increase in returns to scale context. We have constant returns to scale, and indirectly the production function enters into the region of diminishing returns scale from an industry manner.
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Sometimes we can conceptualize non-increasing returns to scale as
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One unit improvement in or one unit increase in funding utilization may keep equal proportionate improvement in the outcome or output. But after a point the moment the funding becomes more and more. There might be lot of technicalities and bureaucracy involved
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which may result in less than proportionate uh outcome or output in a quantified sense.
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So in that case actually in the initial stage equal proportionate improvement or equal proportionate uh increase in the output we can conceptualize as the case of CRS region and beyond that we can have a case where more amount of output
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is resulting in a less than proportionate improvement in the output or outcome. That is the case of diminish tens of scale and we can conceptualize such production functions also in our empirical literature.
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To summarize uh in today's session what we did we took the concepts or the fundamental properties of technology set that is basically convexity and predisposivity of input and predisposivity of output. And using a very simple example of six data points,
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we constructed the technology set which satisfy the convexity and predisposability properties. And then the boundary of this technology set which was conceptualized as the production function. The initial example what we had it was the variable return to scale and by allowing for radial
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expansion and radial contraction we came across the case of constant retentive scale technology or boundary of that we can conceptualize as the constant retention sale production function and we saw the relationship between CRS and VRS where CRS will subsume the VRS
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frontier or VRS frontier or VRS production technology set as subset of CRS and we saw the very special case of production function there is no increase in retention scale only constant retentive scale and directly you are entering into the
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region of diminish retentive scale that is basically the case of non-inreasing retentive scale so throughout this course we'll be using mainly these two fundamental cases that is CRS and VRS for measuring technical efficiency and productivity in some sense
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Hey. [music] [music]
Topics:production functiontechnology setreturns to scaleconstant returns to scalevariable returns to scalenon-increasing returns to scaleconvexityfree disposabilityproduction possibility frontiertechnical efficiency
