1) Let u(x, y) denote a utility function where x is the quantity of good X and y is the quantity of good Y consumed. Assume the quantities can take any value greater than or equal to zero and that the utility function has strictly positive marginal utilities. Define a second utility function for the individual denoted v(x, y).

An affine transformation is an adjustment to a function that uses a slope and an intercept term (a linear transformation is a special case of an affine transformation setting the intercept term equal to zero). Thus, let v(x, y) = αu(x, y) + β where α, β > 0.

(a) Does the utility function v capture the same preference ordering that u does? Discuss and verify.

(b) Instead, define w(x, y) = u(x, y) ^γ where γ > 1. Does the utility function w capture the same preference ordering that u does? Discuss and verify.