Suppose that John's preferences over meat (M ) and vegetables (V ) are represented by the following utility function

U(M, V ) = ln(M) + (1 ) ln(V )

where 0 < < 1. (a) Write down the Lagrangian for John's optimization problem. (Recall that John max- imizes utility given an income, I, and prices pM and pV for the goods.)

(b) SolveforJohn'soptimalconsumptionbundle(M,V)(asafunctionofincomeand prices) using the Lagrangian method.

(c) Suppose = 15 . Suppose also that John has income I = 200 and faces prices pM = 1 and pV = 2. What is the value of John's optimal consumption bundle? What happens if John's income doubles to I = 400?