So what if the slacks are zero but the TE is still not equal to one?

When the slacks are zero PTE will be equal to one.

We get PTE equal to 1 because the firm is benchmarked with it’s fellow IRS or DRS.

Let’s remember the main difference between CCR and BCC is the constraint on λ, where by in the CCR model it is just non zero value and that expands the Efficiency Reference Set (ERS). However with BCC the constrain is summation of λ is equal to 1, which limits the ERS to only those firms in the same scale. Therefore if it is benchmarked with firms at same productive scale they are efficient but they are still not performing similar to those at CRS. This is equivalent to say that there is no excessive usage of inputs but the output to input ratio is still not the same as the most efficient firm, the firm is still using more inputs to produce one output.

Where did the concept of scale come from in BCC?

Lets consider this example, if we have Car A and car B. Car A is designed to travel 20miles per Gallon of gas and car B is designed to travel 30 miles per Gallon of gas. It will be unfair to say that car A is inefficient because it only travels 20 miles per gallon ( it will only be 66.7% efficient) if it is benchmarked with the car B. But this information does not consider the weight capacity of the car and other qualitative information. So with BCC, car A will be benchmarked with its peers in similar scale by assigning different weights on the inputs and outputs. Therefore BCC will find that if the car is actually traveling at 20 miles per gallon, then it is efficient interms of utilizing the inputs to produce outputs. However it will still declare that the car A is not scale efficient, and will compute the distance that car A needs to move to achieve an efficient scale (1-SE).

Before going further into theoretical analysis, I would suggest that the DMU that has PTE =1 but scale inefficient can enhance its output interms of quality or accompanied service. And if we are input oriented, outsourcing can also be a good idea, like opening a new department with in and then outsource service or some inputs from there.

What determines if DRS or IRS?

It is the summation of λ. λ is the weight of inputs assigned by the DEA, and the results of this summation can be computed by the Joe Zhu software.

Under CCR model an efficient firm will have λ =1, and in BCC Summation of λ will be equal to one. If less than one than the firm is operating at IRS and if greater than one than it is operating at DRS.

If we remember the Cob-Douglas production function the implications of return to scale are the same.

q=Akalb (a and b are exponents to k and l respectively)

Where q represents the firm’s services during a period, k represents the capital usage during the period, l represents hours of labor input and A is the technology used in the production of service.

when summation of input elasticity (a+b) equal to one then the firm is operating at CRS, less than one is IRS and more than one is DRS.

Joe Zhu suggest that in certain situations the λ can be used/ translated as Marginal Rate of Technical Subsititution (MRTS), how much of input X1 can be substituted with input X2 to maintain the same level of outputs. As we know that MRTS is actually related to the Marginal productivity of both inputs at that particular combination.

MRTS x1 for x2 = MPx1/MPx2

Normally we need to understand the cost of inputs, and the best combination is found at the point of tangent between Iso-cost and Isoquant. So how do we know the best combination before including the cost? And what is the relationship between MRTS and RTS?

So how do we know the best combination before including the cost?

When the summation of λ equals to 1 then that means MRTS is equal to one which implies that we can substitute one unit of input X1 with one unit of input X2 and still maintain the same level of inputs.

Relationship between RTS (Returns To Scale) and MRTS.

They are both related to Marginal productivity. With CRS the MRTS is equal to one which will mean that the marginal productivities of both inputs are similar. The Cob-Douglas production function is homothetic that at any isoquant (any increase in inputs by scalar t) should yield another isoquant with the same RTS, same slope. When the next slope is is greater than 1 (DRS) Marginal productivity of input 1(MPx1) will be greater than MPx2 and vice versa for IRS.

And yes, it is wise to use more of the input that still has high Marginal productivity than the one that has less. I think this is how a firm can move towards CRS. But how you determine the decomposition in a situation with more than 2 inputs? (Beyond the time that I have for now).

I hope this information is somehow useful for the interpretation of BCC DEA results. And ofcourse further analysis and discussion is needed.

References:

1. Nicholson, W., & Snyder, C. (2008). Microeconomic theory: basic principles and extensions. Cengage Learning

2. Sherman, H. D., & Zhu, J. (2006). Data Envelopment Analysis Explained. Service Productivity Management: Improving Service Performance using Data Envelopment Analysis (DEA), 49-89.