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 need to first find the partial derivatives of the given system of equations with respect to the variables x, y, a, and b. Then, we substitute the values of (x, y, a, b) = (0, 1, 3, 4) into these partial derivatives.
Let's take the partial derivative of F1(x, y, a, b) with respect to b, denoted as ∂F1/∂b:
∂F1/∂b = (∂/∂b)(x² + bxy + y² - a – 2)
To take the derivative with respect to b, we treat x, y, and a as constants and only differentiate the term containing b:
∂F1/∂b = xy
Now, let's take the partial derivative of F2(x, y, a, b) with respect to a, denoted as ∂F2/∂a:
∂F2/∂a = (∂/∂a)(x² + y ² - b² + 2a + 3)
Again, we treat x, y, and b as constants and only differentiate the term containing a:
∂F2/∂a = 2
Now, let's substitute the values of (x, y, a, b) = (0, 1, 3, 4) into these partial derivatives:
∂F1/∂b = (0)(1) = 0
∂F2/∂a = 2
Therefore, δx/δb = ∂F1/∂b = 0, and δy/δa = ∂F2/∂a = 2.
Note that the differentials δx/δb and δy/δa represent how small changes in b affect x and how small changes in a affect y, respectively, at the point (0, 1, 3, 4).