Calculus for Economics, Exercises (Nov. 06, 2017）

I Find 2nd order partial derivatives $z_{xx}$, $z_{xy}$, $z_{yy}$

(1) $z=x^2+xy+y^2-4x-8y$

(2) $z=x^3+y^3-9xy+27$

(3) $z=x^2+xy-y^2-4x-2y$

(4) $z=x^2+4xy+2y^2-6x-8y$

(5) $z=x^3-xy-y^2$

(6) $z=e^{-x^2-y^2}(2x^2+y^2)$

(7) $z=(x^2+y^2)^2-2(x^2-y^2)$

(8) $z=x^3+y^3+6xy$

II

Simplify the expression of the quadratic curve $$ x^2-xy+y^2-x+2y=0 $$ by the rotational change of coordinates $$ \begin{pmatrix} x\\y \end{pmatrix} = \frac 1{\sqrt 2} \begin{pmatrix} 1&-1\\1&1 \end{pmatrix} \begin{pmatrix} X\\Y \end{pmatrix} $$

III

Simplify the expression of the quadratic curve $$ x^2+3xy+y^2-1=0 $$ by the rotational change of coordinates $$ \begin{pmatrix} x\\y \end{pmatrix} = \frac 1{\sqrt 2} \begin{pmatrix} 1&-1\\1&1 \end{pmatrix} \begin{pmatrix} X\\Y \end{pmatrix} $$

IV

We consider the CES function defined by \begin{equation} Y=F(K,L)= \left( \alpha K^\rho+\beta L^\rho \right)^\frac 1{\rho} \end{equation} for postive constants $\alpha,\beta>0$, $\rho>0$.

(1) Deriavate partially $\log F(K,L)$ espectively by $K$ and $L$ to find $F_K(K,L)$ and $F_L(K,L)$.

(2) Sow that the function $F(K,L)$ satisfies the Euler's identity \begin{equation} K\cdot F_K(K,L)+L\cdot F_L(K,L)=F(K,L) \end{equation}

V

We define the Cobb-Douglass function by \begin{equation} Y=F(K,L)= AK^\alpha L^\beta \end{equation} for positive constants $\alpha,\beta>0, A>0$.

(1) Find $F_{KK}$, $F_{KL}$, $F_{LK}$, $F_{LL}$.

(2) Find $(\alpha,\beta)$ for which any point $(K,L)\in\mathbf{R}^2_{++}$ satisfies \begin{equation} F_{KK}(K,L)<0,\ \text{AND}\ \det(H(F)(K,L)>0 \end{equation}