ECON2125/6012, Semester-1 2023 Assignment-4

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Assignment-4

ECON2125/6012, Semester-1 2023

Let x e R denote a vector of inputs, y e R a vector of outputs and (x, y) e T .  The input distance function is defined as

dI (x, y) = maxd {d | xd eV(y)}

where V(y) is the input requirement set.

(1) Draw a diagram with two inputs and an isoquant. Identify the distance function and provide an interpretation of this function.

(2) Given the standard assumptions on technology, show the following properties: d(x, y) > 1

d(x, y) is non-decreasing, concave and homogenous of degree 1 in x

d(x, y) is non-increasing and quasi-concave in outputs.

d(x, y) is continuous in x, y .

(3) It can be shown that the cost function can be obtained from a distance function using

( N                                                )

C(w, y) = minx〈l  wi xi  | dI (x, y) = 1J卜 .

Show that  =

(4) Consider the following (restricted) Translog function

N                            1  N    N                                           M

ln dI (x, y) = a0 + ai ln xi + 2 aij ln xi ln xj  + bk ln yk

(i)         What restrictions on the parameters need to be imposed for this function to be a proper input distance function (exclude concavity and quasiconcavity)

(ii)        Under what conditions this function is linearly homogenous in outputs?

(iii)       Using (3) derive the equation for the cost minimising input demand shares i.e.

xi(*)wi      

C (w* , y)