Efficiency change over time

Transkriptio

1 Efficiency change over time Heikki Tikanmäki Optimointiopin seminaari

2 Contents Introduction (11.1) Window analysis (11.2) Example, application, analysis Malmquist index (11.3) Dealing with panel data, catch-up effect, frontier-shift effect, Malmquist index, the radial MI

3 Introduction So far, we have been considering DEA under static conditions (though, we have had examples where the DEA model was applied separately to each year) DEA model that takes into account the efficiency changes in time is introduced We define the concept of Malmquist productivity index

4 Window analysis Basic idea is that each DMU considered as a different DMU in each reporting dates Usually the model is made such that a window covers a few time steps (in the following example 4 steps) If there are N DMUs and there are T time steps in a window, the computations are made effectively over NT DMUs

5 Military Example (1/2) Efficiency analysis of recruiting units of the US army Example built on 10 inputs and 3 outputs Different kinds of DEA models applied to each quarter separately did not give satisfactory results This study has motivated the development of window analysis

6 Military Example (2/2)

7 Analysis Number of DMUs: n=10 Number of periods: k=8 Length of a window: p=4 No windows: w=k-p+1=8-4+1=5 No DMUs in a win: np/2=10*4/2=20 No of different DMU s: npw=200 Change of no DMUs: n(p-1)(k-p)=120

8 Malmquist index (MI) Malmquist productivity index is first introduced by S. Malmquist 1953 Evaluates the productivity change of a DMU between two time periods Example of comparative statistics analysis MI is a product of Catch-up (recovery) and Frontier-shift (innovation) terms

9 Dealing with panel data A set of n DMUs m inputs and q outputs: The production possibility set:

10 Catch-up effect (1/2) The catch-up term relates to the degree of a DMU improving (worsening) its efficiency The catch-up term in (input) orientation can be computed as in the following picture: Catch-up=(BD/BQ)/(AC/AP)

11 Catch-up effect (2/2)

12 Frontier-shift effect (1/2) Catch-up measures the efficience change wrt the respective frontier at each step Frontier-shift term measures the change of the frontier (e.g. the effect of new technologies etc.) In the case of the previous picture this is as follows:

13 Frontier-shift effect (2/2) The point C moves to point E, which gives the frontier shift in these reference points Analogously, F moving to D gives: The total frontier-shift is the geometric mean

14 Malmquist index The Malmquist index is the product of catchup and frontier-shift In our example: Catch-up: Frontier-shift: MI:

15 Malmquist index and DEA In non-parametric framework MI is constructed by means of DEA technologies There are many different ways for computing MI Input and output oriented radial DEA models Shortcoming: no slacks Slack-based non-radial oriented DEA models Non-radial and non-oriented DEA models

16 The radial MI (1/2) The input-oriented radial MI measures the within and intemporal scores by 4 linear programs Within score in input-orientation Intertemporal score in input-orientation Within score in output-orientation Intertemporal score in outpu-orientation

17 The radial MI (2/2)

18 Summary We consider DEA models in non-static setup Window analysis considers the DMU in different time steps (within the window) as different DMUs Malmquist productivity index catches the efficiency change of a DMU. It can be decomposed to Catch-up and Frontier-shift In non-parametric case MI is computed using DEA models

19 Exercise Why does not one allways choose the total amount of periods to be the length of a window? (4p) What are the roles of Catch-up term (2p) and Frontier-shift term (2p) in the Malmquist productivity index (2p)?