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, we'll need to first find the partial derivatives of the given equations with respect to x, y, a, and b.
Taking the partial derivative of F1 with respect to x:
∂F1/∂x = 2x + by
Taking the partial derivative of F1 with respect to y:
∂F1/∂y = bx + 2y
Taking the partial derivative of F2 with respect to x:
∂F2/∂x = 2x
Taking the partial derivative of F2 with respect to y:
∂F2/∂y = 2y
Taking the partial derivative of F2 with respect to a:
∂F2/∂a = 2
Now we can compute the differentials δx/δb and δy/δa at the point (x,y,a,b) = (0,1,3,4) by substituting the values into the partial derivatives we found.
δx/δb = ∂F1/∂x = 2(0) + (4)(1) = 4
δy/δa = ∂F2/∂y = 2(1) = 2
Therefore, at the point (x,y,a,b) = (0,1,3,4), the differentials are:
δx/δb = 4
δy/δa = 2.