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

and 20, respectively. If you have questions about this assignment, post them

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

2 , answer the following

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 inputoutput 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 output combination the input-oriented technical efficiency (T Ex), outputoriented technical efficiency (T Ey) and the input/output/directional

distance function values. Assume gy = 1 in the calculation of the directional 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 production 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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