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1. Consider the following utility functions:

(i)

(, ) =

(ii)

(, ) = +

(iii) (, ) = +

(iv) (, ) = min{, } (that is, (4,3) = 3, (1,1) = 1, (2,3) = 2, ..)

For each case:

(a) Sketch an indifference curve for (, ) = (1,1) and another one for (4,2).

(b) Derive a formula for the MRS for each of the first 3 examples and use it to

calculate the MRS at (1,1). (In the last example (iv), calculus cannot be used.

However, think about the concept of MRS as how much y you are willing to give

up for a bit more of x as represented by the steepness of the indifference curve at a

point. Can you find some value for MRS in this case?)

(c) Calculate the MRS at the points (1,1) and (4,2).

(d) Which of the examples exhibit diminishing MRS?