Answer to Question #58772 in Microeconomics for pradeep

Question

An agent’s utility function is written as U = X2 Y, and his budget
Constraint is X + 2Y = 100. What are the optimal amounts of X and Y

Answer 1

Marginal Rate of Substitution (MRS) is a slope of indifference curve.
MRS = -dy/dx = 2y/x (derivative of Cobb-Douglas function)
The slope of budget line is: 1/2
The utility function reaches its maximum when the indifferent curve and constraint line are tangent:
2y/x = 1/2
from that:
y = x/4 (*)
Taking into account the budget constraint:
x + 2y = 100 (**)
From (*) and (**) optimal x equals:
x + 2 (x/4) = 100
x = 66,66
and optimal y equals:
y = 66,66 / 4
y = 16,66
Answer: optimal amount of X is 66,66; optimal amount of Y is 16,66