# Production economics and efficiency analysis

INSTRUCTIONS: Do the following exercises and submit your results as
a single PDF file using the link provided in the LMS. Provide complete and
clear answers. Hand written answers are acceptable but need to be neat and
well formatted. The weight marks for exercises 1, 2 and 3 below are 30, 50
on the discussion board or ask during the last session so that any clarificaitons
can be provided in class.
1) Given the single-input production function y = 3x
1
questions.
a) Show that the input distance function is ID(x, y) = 9x
y
2
.
b) Derive the output distance function OD(x, y).
c) Derive the directional distance function DD(x, y). For this exercise
assume that the direction vector used for the translation of the inputoutput vector (x, y) is g = (0, gy), i.e. this is an output directional
distance function.
d) Complete the following table by inserting for each observed input output combination the input-oriented technical efficiency (T Ex), outputoriented technical efficiency (T Ey) and the input/output/directional
distance function values. Assume gy = 1 in the calculation of the directional distance function values.
1
Table 1: Efficiency and distance function values by observation
Producer Input Output TEx TEy ID OD DD
1 1 3 1.0
2 1 1.5 0.25
3 4 5
4 4 4
5 9 7
6 16 8
2
2) Consider the the following two-input production function y = AertL
αKβ
where: A, t, y, L and K are, respectively, a constant, a time trend, output
level, labour and capital input. The prices for inputs are w1 for labour
and w2 for capital while the output price is p.
a) Define technological progress in terms of output growth (T Cy) and show
that the rate is r.
b) Derive the cost function, C(w1, w2, y), for this technology. (Note: You
do not have to start from the scratch; you could adapt results from
previous exercises (assignment 1) to specify the cost equation.)
c) What is the rate of technological progress expressed in terms of cost
changes (percentage reduction in cost over time)?
d) Show that the output distance function corresponding to the the production function above is: ODt
(y, K, L) = y
AertLαKβ
e) Describe how you would estimate the above output distance function
as stochastic frontier model, including writing out the composed error
SFA function you would estimate.
f) Derive the input distance function, IDt
(K, L, y), corresponding to the
the production function above.
3) The output-oriented Malmquist productivity index (of period 1 relative to
0) is defined as:
M =

D0
(K1
,L1
,y1
)
D0(K0,L0,y0)
.
D1
(K1
,L1
,y1
)
D1(K0,L0,y0)
#
1
2
And this index can be decomposed into an efficiency change (EC, first
ratio on RHS below) and technical change (TC), with the latter evaluated
as the geometric average of the data points for periods 1 (T C1
) and 0
(T C0
):
M =
D1
(K1
,L1
,y1
)
D0(K0,L0,y0)
.

D0
(K1
,L1
,y1
)
D1(K1,L1,y1)
.
D0
(K0
,L0
,y0
)
D1(K0,L0,y0)
#
1
2
Suppose the production technology is represented by the function in 2(d)
above. Calculate the productivity growth (PR), efficiency change (EC)
3
and technological progress (TC, the remainder of the RHS) for a producer
who has been observed to use the following input output combinations
over 3 production years Years 0, 1 and 2). Assume A, α and β in the
production function in exercise 2 above are, respectively, 1.0, 0.5 and 0.3.
And r is 0.03 or 3%. Use the above decomposition formula complete the
table below.
Table 2: Productivity decomposition for actual input and output use
t Labour Capital Output TEy EC TC PR
0 2 3 1.5
1 6 8 4.0
2 10 12 6.5

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