1. Problem associated with first video: Increasing transformation Consider the f

Question

1. Problem associated with first video: Increasing transformation Consider the following utility function u(x1, x2) = x1x2.
(a) Calculate the MRS for this utility function.
(b) Compute and compare the MRS for the bundles (x1, x2) = (1, 2) and (x1, x2) = (2, 1). What does this comparison tell us about the preferences of the person with this utility function?
(c) Consider the bundles (x1, x2) = (4, 4) and (x1, x2) = (1, 9). Which bundle is the most preferred (i.e. yield the highest utility)?
(d) Consider the following function f(x) = x and then define a new utility function v(x1, x2) = f (u(x1, x2)).
i. Show that v(x1, x2) is an increasing transformation of u(x1, x2).

ii. Answer, without calculating, which of the bundles in question c) above will be the
most preferred according to v(x1, x2).

iii. Calculate the MRS for v(x1,x2). How it compares with the MRS for u(x1,x2). Why?

Explanation / Answer

Utility Function : U = x1*x2

(a) MRS = (dU/dx1) / (dU/dx2) = x2 / x1

(b) MRS for bundles (1,2) and (2,1) is calculated below:

MRS = -change in x2/change in x1 = -1/1 = -1

BY giving one unit of x2 the consumer will get exactly one additional unit of x1. Hence the consumer prefer one more unit of x1 by moving from first bundle to another.

(c) At bundle (4,4) utility = x1*x2 = 4*4 = 16

and at bundle (1.9) utility = 1*9 = 9

the first bundle gives more utility. Hence the first bundle would be more preferrable.

(d) F(x) = X1/2

v(x1,x2) = f(u(x1,x2)) = x11/2 * x21/2

(ii) the first bundle will be more preferrable which will give 4 utility and second bundle will give 3 utility.

(iii) New MRS = ( 0.5x20.5/x10.5  ) / ( 0.5x10.5/x20.5) = x2 / x1

The new MRS is same with the previous one.

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