Answer to Question #216534 in Calculus for xiamiii

Question

skippy leaves on an highland where she produces two goods. According to the production possibility frontier 100=x²+y² and she consumes all goods herself. A utility function is U=xy. Find a utility maximizing X and Y as well as the value of lamda.

Answer 1

Create a new equation form the original information

$$L=xy-\lambda(x^2+y^2-100)$$

$$\dfrac{\partial L}{\partial x}=y-2\lambda x=0$$

$$\dfrac{\partial L}{\partial y}=x-2\lambda y=0$$

$$\dfrac{\partial L}{\partial \lambda}=-(x^2+y^2-100)=0$$

$$y=2\lambda x$$

$$x=4\lambda^2x$$

$$x^2+4\lambda^2x^2-100=0$$

$x>0$

$$y=2\lambda x$$

$$4\lambda^2=1$$

$$x^2+x^2-100=0$$

$x\not=0$

If $\lambda=-\dfrac{1}{2},$ then $x$ and $y$ have opposite signs. So $U=xy<0$

If $\lambda=\dfrac{1}{2},$ then

$$x=5\sqrt{2}, y=\dfrac{5\sqrt{2}}{2}$$

$x=5\sqrt{2}, y=\dfrac{5\sqrt{2}}{2}, \lambda=\dfrac{1}{2}$

$$U_{max}=25$$