(a) What is Luke's marginal rate of substitution (MRS)?

(b) Does Luke have a diminishing marginal rate of substitution?

Luke's utility is U(x1,x2)=2x1+lnx2.U(x1,x2)=2x1+ln⁡x2.

Tom, Jim , and Harry, the only three people on a desert island with two consumption goods, apples and bananas, have utility functions UT, UD, and UH, respectively, where:

UT(xa,xb) = 6xa+9xb

UD(xa,xb) = 8xa+12xb

UH(xa,xb) = xaxb2

(a) Relative to Tom, does Jim require more bananas, less bananas, or the same number of bananas to give away an apple? Why?

(b) Suppose Harry has 3 apples and 2 bananas. Would he be willing to trade an apple for a banana? Show why or why not.