Question

6. CHOICE (10 POINTS) You consume only 2 goods: 11 and 12. Your utility function for consuming these goods is u (21,22) = 221 22 +1 21 and 22 cost Pı and P2, respectively, and your income is m. (a) (2 POINTS) Write down your utility maximization problem. (b) (6 POINTS) Solve the utility maximization problem to find the demand functions for x and 12. Remember that the demand functions express the endogenous variables as a function of the exogenous variables alone. (C) (2 POINTS) If p1 = 1, P2 = 2, and m= 20, how much of each good will you buy?

Answer 1

a) The utility maximization problem is as follows:

Maximize u(x1,x2) =2x1x2 +1

Subject to : x1p1 +x2p2 = m

b) the lagrange expression of the above maximization problem is

L= 2x1x2 +1 +$\lambda$(m - x1p1- x2p2)

the first order conditions are

$\partial L/\partial x1 =$ 2x2 - $\lambda$ p1=0 $\Rightarrow$ 2x2=$\lambda$p1 .....(1)

$\partial L/\partial x2=$2x1 - $\lambda$ p2=0 $\Rightarrow$ 2x1=$\lambda$p2 ......(2)

$\partial L/\partial \lambda =$m-x1p1 - x2p2=0 ......(3)

dividing (1) by(2) x2/x1= p1/p2 $\Rightarrow$ x2p2=x1p1......(4)

Putiing (4) in (3) m-x1p1-x1p1 = 0$\Rightarrow$ 2x1p1=m $\Rightarrow$ x1=m/2p1

x2= m/2p2

c) demand functions are x1= m/2p1 and x2=m/2p2

if p1=1 , p2=2 and m=20

x1=20/(2x1) = 10 and x2=20/(2x2)=5