Utility Maximization Problems

Cobb-Douglas utility function

Let the utility function be 

u(x,y)=x3y7.

Let the price of good xbe pand the price of good ybe qand the income be I.

Applying the method of Lagrange multiplier, find the utility maximizing consumption bundle (x∗,y∗).

Also determine the value of Lagrange multiplier λ∗.

2. Quasi-linear utility function

Let the utility function be 

u(x,y)=x+0.1ln⁡y.

Let the price of good xbe $1per unit and the price of good ybe $pper unit and the income be I.

Applying the method of Lagrange multiplier, find the utility maximizing consumption bundle (x∗,y∗).

Also determine the value of Lagrange multiplier 

Hint: It is useful to note that dln⁡ydy=1y.