Consider the following utility function (referred as a quasi-linear utility function as it is linear in the second element):

u(x,y) = lnx + y

with prices and income given by: px = 1, py E R+ and I E R+.

(a) For the special case of py = 2 and I = 1, show analytically and graphically that there is no positive consumption choice (x^*,y^*) greater than or equal to 0 which satisfies the tangency condition:

ux(x,y)/ uy(x,y) = Px/Py

Briefly describe where the failure of the tangency condition stems from and what does it mean. Hint: Setup the tangency condition and solve it. What do you get? Is it affordable?

(b) If we insist that (x^*, y^*) greater than or equal to 0 ; then describe, analytically and graphically, the optimal con- sumption choice for py = 2, I = 1. Hint: Think about corner solutions.

(c) Now suppose that py E R+ and I E R+ are unspecified. Suppose that we don’t insist on positive values ofeither x or y. For which values of py and I do you get a nonsensical solution in the sense that y^* < 0 ?