Problem associated with second video: Demand with Cobb-Douglas utility function

Question

Problem associated with second video: Demand with Cobb-Douglas utility function
All the workers in a factory have the same utility function u(x1, x2) = x1x2, earn the income of 12 and face the same prices p1 = 1 and p2 = 1.
(a) What is the optimal consumption bundle for the workers? What utility do they obtain?.
(b) Graph the indifference curve that goes through the optimal consumption bundle. In the same graph draw the budget set.
(c) Unfortunately for the workers there is an increase in p1 of 100%. What is the new p1? What is the new optimal consumption bundle and what utility do they obtain?
(d) In the same graph as in point b, draw the indifference curve that goes through the new optimal consumption bundle and also draw the new budget set.

Explanation / Answer

a) max u=x1x2 subject to x1+x2 = 12

u = x1 (12-x1)

Differentiating w.r.t x1 and equating to 0 :

12 - 2x1 = 0

x1 = 6

x2 = 6

max utility = 36

c) new p1 = 1 + 1 = 2

Max u subject to 2x1 + x2 = 12

u = x1 (12 - 2x1)

differentiating w.r.t x1

12 - 4x1 = 0

x1 = 3

x2 = 6

utility = 6*3 = 18

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