Question #58772

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

Constraint is X + 2Y = 100. What are the optimal amounts of X and Y

Expert's answer

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

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

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