Question

A utility function is given by U = 2x² + y² .

(a) State the equation of the indif erence curve which passes through (4, 2).

(b) Calculate the marginal utilities at (4, 2) and hence work out the gradient of the curve at this point.

Answer 1

a)

U = 2X² + Y²

Passing point (4,2)

U = 2(4)² + (2)²

U = 32 + 4  

U = 36

Equation of the indifference curve is

36 = 2X² + Y²  

Y² = 36 - 2X²  

Y = (36 - 2X² )^1/2   

b)

U = 2X² + Y²

U/X = MU_X = 4X  

MUx = 4X  

At point (4,2)  

MUx = 4X = 4(4) = 16  

U/Y = MU_Y = 2Y  

MU_Y = 2Y  

At (4,2)  

MU_Y = 2Y = 2(2) = 4  

Slope or gradient of the curve or the indifference curve is

MRS = MUx/MU_Y = 4X/2Y = 2X/Y  

At (4,2)

MRS = 2X/Y = 2(4)/2 = 4