ECON2125/6012, Semester-1 2023 Final Exam Practice Test

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Final Exam Practice Test

ECON2125/6012, Semester-1 2023

Question-1: Consider the following function                                                                            (50marks)

f (x, y, z) = (x2 + 2y  )e2 −(x2 +y2 )

(i)         Find all the stationary points.

(ii)        Determine their type of stationary points.

(iii)       Is this function concave over its domain? Prove your answer.

Now suppose we have the following constrained problem:

maxx,y (x2 + 2y  )e2 −(x2 +y2 )

subject to x + y 三 0

(iv)       Find the solution for the constrained problem.

(v)        Check whether the solutions are in fact maximum.

(vi)       How much does the value of the maximum of the function changes if 2 changes to 2.1

in the objective function.

u(x1 , x2 , x3 ) =aln(x1 − a1 ) + bln(x2 − a2 ) + Y ln(x3 − a3 )

(i)         Derive the Marshalian demand function for this utility function. (ii)        Derive the indirect utility function.

(iii)       Show this indirect utility function satisfies the conditions that an indirect utility

function must satisfy.

Question-3:                                                                                                                                           (20 marks)

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

dI (x, y) = maxd {d | xd 从V(y)}

where V(y) is the input requirement set.

(i)         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

(ii)        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  =  .