Answer to Question #120312 in Financial Math for Danelle Sammy

Question

4. A consumer has the utility function U = 4X0.75Y 0.25 and can buy good X at £12
a unit and good Y at £2 a unit. Find the combination of X and Y that they should
purchase to minimize the cost of achieving a utility level of 580 and check that
second-order conditions are met using the bordered Hessian matrix.

Answer 1

Objective: Minimize 12X + 2Y = TC

Subject to:

4X^0.75Y^0.25 = 580

x≥ 0, y≥0

L(λ, x, y) = 12X + 2Y - λ(4X^0.75Y^0.25 - 580)

∂L / ∂X = 12 – 3λ(X^-0.25Y^0.25)

∂L / ∂Y = 2 – λ(X^0.75Y^-0.75)

∂L / ∂ λ = -1(4X^0.75Y^0.25 – 580)

Ratio of FOC

(∂L / ∂X) / (∂L / ∂Y) = 12/2 – 3λ(X^-0.25Y^0.25)

X-1Y1 = 2

Y = 2X

Substituting into the utility function:

4X^0.75Y^0.25 = 580

4X^0.75(2X)^0.25 = 580

X^0.75+0.25 = 580/(4*2^0.25)

X= 121.93

Getting the value of Y

Y = 2*121.93

Y = 243.86

Solving for λ

2 – λ(X^0.75Y^-0.75) = 0

λ(121.93^0.75243.86^-0.75) = 2

λ(121.93^0.75243.86^-0.75) = 2/36.69298*0.016205)

λ = 2/0.54604 = 3.3636

Checking for second order conditions using Bordered Hessian matrix

Getting the determinant of the matrix

Det |BH| = (-1)² (0.024604) (0-[0.594604 * 0.594604]) + (-1)³ (0)( 1.189207*0.594604-0) +(-1)⁴ (1.189207) (0-[1.189207*23.42403])

Det (-1) |BH| = -0.0087 + 0 + - 33.1266

Det (-1) |BH| = - 33.1353 > 0, (this is the point of relative minimum). Thus, the condition for minimization is satisfied.

Thus, the output (x,y) that produced the minimum total cost will be given by:

X = 121.93

Y = 243.86

Minimum TC =12X + 2Y

TC= 12*121.93 + 2*243.86

TC = 1463.16 + 487.72

TC = 1950.88