A consumer consuming two commodities X and Y has the following utility function U= XY+2X. if the price of the two commodities are 4 and 2 respectively and his/her budget is birr 60
A. find the quantities of good X and Y which will maximize utility
B. Find MRSx,y at optimum
A. To maximize utility, we first need to set up the Lagrangian:
L = XY + 2X + λ(60 - 4X - 2Y)
Taking the partial derivative of L with respect to X, Y, and λ and setting them equal to zero, we get:
∂L/∂X = Y + 2 - 4λ = 0
∂L/∂Y = X - 2λ = 0
∂L/∂λ = 60 - 4X - 2Y = 0
From the first two equations, we can solve for Y and X:
Y = 4λ - 2
X = 2λ
Substitute these into the budget constraint:
60 - 4(2λ) - 2(4λ - 2) = 0
60 - 8λ - 8λ + 4 = 0
16λ = 64
λ = 4
Now, substitute λ back into Y and X to get:
Y = 4(4) - 2 = 14
X = 2(4) = 8
Therefore, the quantities of good X and Y that will maximize utility are X = 8 and Y = 14.
B. To find the Marginal Rate of Substitution (MRS) at the optimum, we need to find the ratio of the marginal utilities of X and Y:
MRSx,y = ∂U/∂X / ∂U/∂Y
First, find the partial derivatives of utility with respect to X and Y:
∂U/∂X = Y + 2
∂U/∂Y = X
Now, substitute the values of X and Y at the optimum:
∂U/∂X = 14 + 2 = 16
∂U/∂Y = 8
MRSx,y = 16 / 8 = 2
Therefore, the Marginal Rate of Substitution of X for Y at the optimum is 2.