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.

Question

a) Find the quantities of good X and Y which will maximize utility.
b) Find the MRSxy at optimum.

Answer 1

a) The consumer's budget constraint can be written as PₓX + PᵧY = M, where Pₓ = 4, Pᵧ = 2, and M = 60. Solving for Y, we get:

Y = (M - PₓX)/Pᵧ = (60 - 4X)/2 = 30 - 2X

Substituting this into the utility function and differentiating with respect to X, we get:

MUₓ = Y + 2 = 32 - 2X

Setting MUₓ equal to the price of X (i.e. 4), we get:

32 - 2X = 4
2X = 28
X = 14

Substituting X = 14 into the budget constraint, we get:

PᵧY = M - PₓX
2Y = 60 - 4(14)
2Y = 12
Y = 6

Therefore, the quantities of X and Y that will maximize utility are 14 and 6, respectively.

b) The marginal rate of substitution (MRS) between X and Y is the absolute value of the slope of the indifference curve, which can be found by taking the partial derivative of U with respect to X and dividing it by the partial derivative of U with respect to Y:

MRSₓᵧ = |MUₓ/MUᵧ| = |(32 - 2X)/X|

At the optimum, X = 14, so the MRSₓᵧ is:

MRSₓᵧ = |(32 - 2(14))/14| = 1.14

Therefore, the MRSxy at optimum is 1.14.