A system of equations is given by:
F1(x,y,a,b) = x² + bxy + y² - a – 2 = 0
F2(x,y,a,b) = x² + y ² - b² + 2a + 3 = 0
Where x and y are endogenous variables while a and b are exogenous variables. Compute the differentials δx/δb and δy/δa at the point (x,y,a,b) = (0,1,3,4).
To compute the differentials δx/δb and δy/δa at the point (x,y,a,b) = (0,1,3,4), we first need to find the partial derivatives of the given system of equations with respect to x, y, a, and b.
Let's differentiate F1 and F2 with respect to x, y, a, and b one by one:
1. Differentiating F1 with respect to x:
∂F1/∂x = 2x + by
2. Differentiating F1 with respect to y:
∂F1/∂y = bx + 2y
3. Differentiating F1 with respect to a:
∂F1/∂a = -1
4. Differentiating F1 with respect to b:
∂F1/∂b = xy
5. Differentiating F2 with respect to x:
∂F2/∂x = 2x
6. Differentiating F2 with respect to y:
∂F2/∂y = 2y
7. Differentiating F2 with respect to a:
∂F2/∂a = 2
8. Differentiating F2 with respect to b:
∂F2/∂b = 0
Now, evaluate the partial derivatives at the point (0,1,3,4):
∂F1/∂x = 2(0) + 4(1) = 4
∂F1/∂y = 4(0) + 2(1) = 2
∂F1/∂a = -1
∂F1/∂b = 0(1) = 0
∂F2/∂x = 2(0) = 0
∂F2/∂y = 2(1) = 2
∂F2/∂a = 2
∂F2/∂b = 0
Finally, we can calculate the differentials:
δx/δb = (∂F1/∂b * δb) / (∂F1/∂x * δx + ∂F1/∂y * δy + ∂F1/∂a * δa + ∂F1/∂b * δb)
= (0 * δb) / (4 * δx + 2 * δy + (-1) * δa + 0 * δb)
= 0 / (4 * δx + 2 * δy - δa)
δy/δa = (∂F2/∂a * δa) / (∂F2/∂x * δx + ∂F2/∂y * δy + ∂F2/∂a * δa + ∂F2/∂b * δb)
= (2 * δa) / (0 * δx + 2 * δy + 2 * δa + 0 * δb)
= 2δa / (2 * δy + 2 * δa)
Therefore, the differentials δx/δb and δy/δa at the point (x, y, a, b) = (0, 1, 3, 4) are 0 / (4 * δx + 2 * δy - δa) and 2 * δa / (2 * δy + 2 * δa), respectively.