The economy is modelled to be in equilibrium with the system of equations f and g respectively:
$2xz + xy + z -2z^{1/2} = 11$ , $xyz = 6$,
One solution of this set of equations is (3,2,1).
If z is to be raised to 1.1 estimate the change of x and y.
My attempt at this question is as follows: I've set up two new simultaneous equations by differentiating the two initial equations w.r.t x,y,z and setting them equal to zero like so:
$df = 4 dx +3 dy + 6 dz = 0$
$dg = 2 dx + 3 dy + 6 dz = 0$
Then in attempt to solve the simultaneous equations for $dx/dz$ and $dy/dz$ I got:
$-2 = dy/dz$ and $2 dx = 0$
From here how can I 'estimate' the change of x and y, and why can it only be an estimate? Help would be much appreciated.
Putting the second equation of $y$ in terms of $x$ and $z$ and substitute it into the first equation, we have $$x = \frac{1}{2z}(11 + 2\sqrt{z} - z - \frac{6}{z})$$ $$\implies \frac{dx}{dz} = -\frac{11}{2z^2} - \frac{1}{2}z^{-\frac{3}{2}} + 6z^{-3}$$
When $z = 1$ $\frac{dx}{dz} = 0$
Similarly
$$y = \frac{12z}{11 + 2z^\frac{3}{2} - z^2 - 6}$$ $$\implies \frac{dy}{dz} = \frac{12(-z^{\frac{3}{2}} + z^2 - 6)}{(11z + 2z^{\frac{3}{2}} - z^2 - 6)^2}$$
When $z = 1 \frac{dy}{dz} = -2$
These rates of change represent an infinitesimal change from $z = 1$, say to $z = 1.00001$. If we want to find the change to $z = 1.1$, we have to find the values of $x$ and $y$ when $z = 1.1$ and find the differences for the changes.