Answer:
Explanation:
It appears that the equation you provided is incomplete. Please provide the complete equations for the policy rule, IS curve, and Philips curve so that I can assist you further.
a. tₓ = 3x - 6y and tᵧ = 3y - 6x
Setting tₓ = 0 and tᵧ = 0, we have the following system of equations:
3x - 6y = 0 ...(1)
3y - 6x = 0 ...(2)
To solve this system, we can rearrange equation (1) to express x in terms of y:
3x = 6y
x = 2y ...(3)
Substituting equation (3) into equation (2), we can solve for y:
3y - 6(2y) = 0
3y - 12y = 0
-9y = 0
y = 0
Substituting y = 0 into equation (3), we find:
x = 2(0)
x = 0
Therefore, the critical point is (0, 0).
Now, let's use the second partial derivative test to determine the nature of the critical point:
The discriminant D = fₓₓ(x, y)fᵧᵧ(x, y) - (fₓᵧ(x, y))², where fₓₓ is the second partial derivative of f with respect to x, fᵧᵧ is the second partial derivative of f with respect to y, and fₓᵧ is the second partial derivative of f with respect to x and y.
Calculating the second partial derivatives of f(x, y):
fₓₓ = ∂²f/∂x² = 0
fᵧᵧ = ∂²f/∂y² = 0
fₓᵧ = ∂²f/∂x∂y = 0
Plugging these values into the discriminant:
D = 0(0) - (0)² = 0
Since the discriminant D is zero, the second partial derivative test is inconclusive.
Therefore, at the critical point (0, 0), we cannot determine whether it is a maximum, minimum, or saddle point.
b. tₓ
It seems there is a partial derivative missing in your question. Please provide the complete partial derivative equation (tₓ) for part b so that I can assist you further.