Capacity Utilization

Transkriptio

1 Capacity Utilization Tim Schöneberg 28th November

2 Agenda Introduction Fixed and variable input ressources Technical capacity utilization Price based capacity utilization measure Long run and short run capacity utilization Summary Home Assignment

3 Introduction With help of capacity utilization measures we can determine if it is possible to increase some of a DMU s output and make it more efficient by increasing some inputs without worsening the other outputs

4 Fixed vs.variable input ressources Some inputs are variable and can be changed Employees, office hours Some are fixed and cannot be altered Invested capital, real estate

5 Fixed vs.variable input ressources We define for each Output vector Fixed input vector Variable input vector

6 Technical capacity utilization We solve the SBM-O restricted (Ch 4.7) For each given

7 Technical capacity utilization We get the optimal solution, With help of this solution we define The values define a point on the efficient frontier within the PPS defined by the constraints [ ]

8 Technical capacity utilization Now we introduce a new variable, the expansion rate This leads to because of )

9 Technical capacity utilization This tells us that is the average expansion rate of all In this case, means that the DMU can expand some outputs without worsening other outputs with the same inputs

10 Technical capacity utilization Now we solve SBM-O relaxed Constraints on variable inputs are removed New solution,

11 Technical capacity utilization For a given solution, We obtain variable inputs Which can be greater or less than the observed inputs from the restricted model ( ) This means, we can use more or less of the variable inputs than in the restricted model

12 Technical capacity utilization With the help of and we can define a capacity utilization measure means < Thus, there could be more outputs if more variable inputs would be used

13 Technical capacity utilization With the help of and vectors, we can define This is an capacity utilization measure for every single variable input We can now say which variable inputs can be increased

14 Price based capacity utilization measure We now assume to know the prices Profit for outputs Cost for variable inputs Prices are not identical for each DMU Revenue can be expressed as Outputs Inputs

15 Price based capacity utilization measure We have a price-based PPS with Where

16 Price based capacity utilization measure Now we solve three LP s for each DMU to determine Loss due to technical inefficiency Loss due to allocative inefficiency Loss due to capacity utilization

17 Price based capacity utilization measure Loss due to Technical inefficiency

18 Price based capacity utilization measure With the help of the optimal solution We define the efficient as This means, profits are always greater or equal and cost are always less or equal.

19 Price based capacity utilization measure In mathematical terms: and This leads to Thus, we can say that is the loss due to technical inefficiency

20 Price based capacity utilization measure Loss due to Allocative inefficiency Profit o Restricted Model (Ch 8.3.4)

21 Price based capacity utilization measure Loss due to Allocative inefficiency describes loss due to the price based input and output mix is called the loss due to allocative inefficiency

22 Price based capacity utilization measure Until now, we behaved like learned in chapter 8. But this time, we go further. We analyze how efficient the DMU could be if the variable inputs would not be restricted We relax to

23 Price based capacity utilization measure We get new relaxed profit model

24 Price based capacity utilization measure And now we can define the loss due to capacity utilization

25 Price based capacity utilization measure This leads us to the maximum Profit for this DMU and it s decomposition

26 Price based capacity utilization measure example

27 Long run and short run capacity utilization In this analysis, we have considered fixed variables as fixed This is not true in long run, because new investments can be made or new real estate can be bought To deal with this issue, we can also relax the fixed variables for long run capacity utilization

28 Long run and short run capacity utilization We get a new model

29 Long run and short run capacity utilization Now we can define the loss of profit due to the gap in long run and short run This gives us a new definition of maximum profit (Home Assignment)

30 Summary We learned the difference between fixed and variable inputs We learned how to measure capacity utilization When prices are unknown When prices are known Consider long and short run

31 Home Assignment There is one important assumption concerning returns to scale which makes this kind of analysis possible. What is it, and why is it necessary? (4P for a clear answer) Give a reasonable example for a fixed cost which can be relaxed in long run analysis. (3P for a reasonable answer) Make a decomposition of like shown on the slides for.(3p for decomposition)

32 Thank you for Listening!