Marvin has a cobb douglas utility function Uq15q25 his incom

Marvin has a cobb douglas utility function, U=q₁^.5q₂^.5

his income is Y=300 and initially he faces prices of p1=$4 and p2= $4. if p1 increases from $4 to $5, what are his compensating variaiton (CV), change in consumer surplus (CS), and equivalent variation (EV)?

Marvins compensating variation (CV) is $______

Marvins change in consumer surplus (CS) $_________

Marvins equivalent variation (EV) $________

Rounded to two decimal places and include minus sign if necessary.

Solution

When P1 = 4 and P2 = 2

MUq1 / MUq2 = q2 / q1

and at optimal combination

MUq1 / MUq2 = P1/P2

q2/q1 = 4/2

q2 = 2q1

and budget is 500 = 4q1 + 2q2

or 500 = 4q1 + 2(2q1)

q1 = 62.5

and q2 = 2(62.5) = 125 units

Now when price changes then

MUq1 / MUq2 = P1/P2

q2/q1 = 5/2

q2 = 2.5q1

and budget is 500 = 5q1 + 2q2

or 500 = 5q1 + 2(2.5q1)

q1 = 50

and q2 = 2.5(50) = 125 units.

Now initial utility is U = (62.5).5(125).5 = 88.39

Demand function for q1 is 0.5Y/P1 and for q2 is 0.5Y/P2.

Now derive expendniture function:

U = (q1.5(q2).5 from demand functions

U = (0.5Y/P1).5(0.5Y/P2).5

or U =Y (0.5/P1).5(0.5/P2).5

or Y = U * (P1/0.5).5(P2/0.5).5

The above function is expedniture function corresponding to utility function.

Now calculates how much income he needs to have same level of utility of initial.

Y = U * (P1/0.5).5(P2/0.5).5 = 88.39 * (5/0.5).5(2/0.5).5= 559.02 or approx 560

Thus the compensating variation is new income required to get initial level of utility subtract orginal income that is 560 - 500 = $60.