An indifference curve that is convex to the origin has diminishing marginal rate of substitution:

a. Find the marginal rate of substitution between x and y for the following function:

i. Cobb-Douglas: U=(Ax^a)(y^b)

ii. Constant elasticity of substitution: U=A(ax^p +(1-a)y^p)^1/p

iii. Linear: U= ax+by

b. Demonstrate mathematically whether each utility function above exhibits a diminishing marginal rate of substitution. Include in your answer an explanation of diminishing marginal rate of substitution.

c. Explain whether diminishing marginal utility is necessary for a diminishing marginal rate of substitution.