u(x1,x2,x3)=x1x22x3. As usual, goods 1 and 2 are perfectly divisible, meaning that x1 and x2 can take on any nonnegative real values and need not be integers. Good 3 is indivisible and unique, meaning that x3 must be either 0 or 1 (for example, good 3 could be an original painting). (a) Show that this consumer's preferences are monotone. Solution: The derivatives u/x1 and u/x2 are positive for x1,x2>0. Increasing x3 from 0 to 1 increases utility by 1/2. Therefore, if the quantity of all three goods increases, utility (strictly) increases. (b) Find this consumer's Marshallian demand. Solution: If x3=0, the optimal choice is (x1,x2,x3)=(2p1w,2p2w,0). If x3=1, the optimal choice is (x1,x2,x3)=(2p1wp3,2p2wp3,1). Note that if p3>w, we must have x3=0. Otherwise, the optimal choice of x3 is 0 if 2p1w2p2w12p1wp32p2wp321, which simplifies to p3p1p2. Therefore, x(p,w)=(2p1wp3,2p2wp3,1)(2p1w,2p2w,0)ifp3wandp3p1p2,ifp3>worp3p1p2 u(x1,x2,x3)=x1x22x3. As usual, goods 1 and 2 are perfectly divisible, meaning that x1 and x2 can take on any nonnegative real values and need not be integers. Good 3 is indivisible and unique, meaning that x3 must be either 0 or 1 (for example, good 3 could be an original painting). (a) Show that this consumer's preferences are monotone. Solution: The derivatives u/x1 and u/x2 are positive for x1,x2>0. Increasing x3 from 0 to 1 increases utility by 1/2. Therefore, if the quantity of all three goods increases, utility (strictly) increases. (b) Find this consumer's Marshallian demand. Solution: If x3=0, the optimal choice is (x1,x2,x3)=(2p1w,2p2w,0). If x3=1, the optimal choice is (x1,x2,x3)=(2p1wp3,2p2wp3,1). Note that if p3>w, we must have x3=0. Otherwise, the optimal choice of x3 is 0 if 2p1w2p2w12p1wp32p2wp321, which simplifies to p3p1p2. Therefore, x(p,w)=(2p1wp3,2p2wp3,1)(2p1w,2p2w,0)ifp3wandp3p1p2,ifp3>worp3p1p2