Utility Maximizer

Please see the attached file.

The average online user's objective function is:
Max U = Uc + Ub = Bc*Tc3/4 + Bb*Tb3/4
where subscripts "c" for communication and "b" for browsing.
The constraint is Tc + Tb = 2.5
Then the lagrangian function is:
L= Bc*Tc3/4 + Bb*Tb3/4 + λ(2.5 - Tc - Tb)
The first order condition is:
(1) dL / dTc = 3/4 Bc Tc-1/4 - λ = 0 or 3/4 Bc Tc-1/4 = λ
(2) dL / dTb = 3/4 Bb Tc-1/4 - λ = 0 or 3/4 Bb Tb-1/4 = λ
(1) / (2):
(3/4 Bc Tc-1/4 ) / (3/4 Bb Tb-1/4 )= λ /λ
(Bc / Bb) (Tc/Tb) -1/4 = 1
Bc / Bb = (Tc/Tb) 1/4
Also, in the reading we know that Tc / Tb = 57% / 43% = 1.326
Thus, Bcommunication / Bbrowsing = (1.326) 1/4= 1.07

b. Uc / Ub = (Bc*Tc3/4) / ( Bb*Tb3/4)
= (Bc/ Bb) *(Tc/Tb) 3/4
= 1.07*(57% / 43%) 3/4
= 1.32
Note, Uc / Ub = Tc / Tb
This means that people in surveys allocate their time in online activities in accordance with the utility obtained from each activity.

c. If an activity, say online gambling, is socially less acceptable than others, people being surveyed will avoid reporting such activities and attribute time to other communication activities. Then Tc will be overestimated and Tb underestimated. In this way, the estimated utility of Uc and Ub will be biased.
In part b we find Uc / Ub = Tc / Tb. When Tc is incorrectly over-reported (greater than the true Tc) and Tb underreported (less than the true Tb), the calculated Uc / Ub ratio will be positively biased, or higher than the true ratio.