5.3 DEA in Practice : Insurance Agencies

The quantlet deahull for DEA has the following syntax :

dea = deahull (X, Y)

computes the input and output e¬ciency score for each DMU

based on DEA

with input variables: X is a n — p matrix, p dimensional inputs of n DMU™s,

and Y is n — q matrix, q dimensional outputs of n DMU™s.

The list dea contains the output variables : dea.effscore is a n dimen-

sional vector containing the input e¬ciency scores, θIN , of n DMU™s, and

dea.efflevel is n — p matrix containing the estimates of e¬cient level of

inputs, x‚ , of n DMU™s.

To illustrate how DEA works, consider an example from the empirical study

by Scheel (1999). He studied the e¬ciency of 63 agencies of a German insurance

company by the DEA method. The input (X ∈ R4 ) and output (Y ∈ R2 )

+ +

variables were as follows:

X1 : Number of clients of Type A

X2 : Number of clients of Type B

96 5 Nonparametric Productivity Analysis

Table 5.1: Summary Statistics for 63 agencies of a German insurance company

Minimum Maximum Mean Median Std.Error

X1 42 572 225.54 197 131.73

X2 55 481 184.44 141 110.28

X3 0 140 19.762 10 26.012

X4 73756 693820 258670 206170 160150

Y1 2 70 22.762 16 16.608

Y2 696 33075 7886.7 6038 7208

X3 : Number of clients of Type C

X4 : Potential new premiums in EURO

Y1 : Number of new contracts

Y2 : Sum of new premiums in EURO

The two output variables are typical for insurance agencies. Summary statistics

for this data are given in Table 5.1. The included XploRe code reads the data

agency.dat, creates the input variables for the quantlet deahull, runs the

quantlet, and lists the e¬ciency and the e¬cient level of inputs for each agency

as a result, see Table 5.2 and Table 5.3.

The input e¬cient scores effscore are useful as a benchmark in comparing the

63 DMU™s, the insurance agencies. The list of e¬cient level of inputs provides

with a ™goal™ inputs for each agency to be e¬cient.

5.4 FDH in Practice : Manufacturing Industry

The quantlet fdhull for FDH has the following syntax:

eff = fdhull (X, Y)

computes the input and output e¬ciency score for each DMU

based on FDH

5.4 FDH in Practice : Manufacturing Industry 97

Table 5.2: E¬ciency score of the 63 DMU™s insurance Agencies

E¬eciency score

1 0.38392

2 0.49063

3 0.86449

. .

. .

. .

62 0.79892

63 1

STFagency.xpl

Table 5.3: E¬ciency level of the 63 DMU™s insurance Agencies

e¬„evel

1 52.981 92.909 3.8392 108960

2 81.444 60.838 2.4531 76895

3 131.4 72.617 2.5935 96070

. . . . .

. . . . .

. . . . .

62 66.311 87.083 1.5978 111710

63 108 257 0 299910

STFagency.xpl

with input variables: X is n — p matrix, p dimensional inputs of n DMU™s, and

Y is n — q matrix, q dimensional outputs of n DMU™s.

The arbitrary name eff is used to indicate the output variable: eff which is

the n — 2 matrix containing the input and output e¬ciency scores of n DMU™s.

In order to illustrate how this quantlet works the Manufacturing Industry Pro-

ductivity Database from the National Bureau of Economic Research, USA is

considered. This database is available in the internet [http://www.nber.org]

98 5 Nonparametric Productivity Analysis

Table 5.4: Summary Statistics for Manufacturing Industry Productivity

Database (NBER, USA)

Minimum Maximum Mean Median Std.Error

X1 0.8 500.5 37.833 21 54.929

X2 18.5 145130 4313 1957.2 10771

X3 0.5 3807.8 139.96 49.7 362

X4 15.8 64590 2962.8 1234.7 6271.1

Y 34.1 56311 3820.2 1858.5 6392

with a description of the database. It contains annual industry-level data on

output, employment, payroll and other input costs, investment, capital stocks,

and various industry-speci¬c price indexes from 1958 on hundreds of manufac-

turing industries (indexed by 4 digits numbers) in the United States. We chose

the data from 1996 (458 DMU™s), for example, with 4 input variables (p = 4)

and 1 output variable (q = 1) along with the study of Park, Simar, and Weiner

(1999) :

X1 : Total employment

X2 : Total cost of materials

X3 : Cost of electric and fuels

X4 : Total real capital stock

Y : Total value added

Summary statistics for this data are given in Table 5.4. The included XploRe

code reads the MS-Excel data ¬le nber96.csv, creates the vector of the ID for

DMU™s and the input variables for the quantlet fdhull, runs the quantlet, and

lists the e¬ciency scores, θIN (·, ·) and 1/θOUT (·, ·), with ID of industries.

¿From Table 5.5, we see that the DMU indexed by 2015 is e¬cient in both input

and output oriented aspect. This means the DMU is one of the vertices of the

free disposal hull generated by the 458 observations. On the other hand, the

DMU 2298 seems to be good in input e¬ciency but poor in output e¬ciency.

With these e¬ciency scores we can estimate the e¬cient level of input (or

output) by multiplying (or dividing) the score to the corresponding observation.

5.4 FDH in Practice : Manufacturing Industry 99

Table 5.5: FDH e¬ciency scores of the 458 observations

fdh

1 2011 0.93803 0.97451

2 2013 0.64321 0.70255

3 2015 1 1