Calculus for Economics, Exercises (Nov. 13, 2017）

I Find the stationary points of the following functions and wheather the points are minimal or maximal.

(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

We are given postive constants $p,q,r>0$ and a production function $$ f(x,y)=x^{\frac 13}y^{\frac 12} $$ We consider the profit function $$ \pi(x,y)=rf(x,y)-px-qy $$ Find the stationary point of $\pi(x,y)$.

III

We consider the Cobb-Douglass production function $$ f(x,y)=x^\alpha y^\beta\quad (\alpha,\beta>0) $$

(1) Find $\det(H(f))$, $f_{xx}$

(2) In case $\alpha+\beta\not=1$, find the stationary point of the profit function $$ \pi(x,y):=pf(x,y)-qx-ry $$ Here $p,q,r>0$.

(3) Show that we have the unique $(x,y)$ maximazing the profit if $\alpha+\beta<1$.

IV

We considered the production function $f(x,y)=x^{\frac 13}y^{\frac 13}$ and have maximized $$ \pi(x,y)=pf(x,y)-qx-ry $$ to get the produnction element demand function. $$ x(p,q,r)=\frac {p^3}{27q^2r},\quad y(p,q,r)=\frac {p^3}{27qr^2} $$ We define the profit function and the product supply function by $$ \Pi(p,q,r)=\pi(x(p,q,r),y(p,q,r)) $$ $$ z(p,q,r)=f(x(p,q,r),y(p,q,r)) $$ Show the following identities \begin{align*} z(p,q,r)&=\frac {\partial \Pi}{\partial p}\\ x(p,q,r)&=-\frac {\partial \Pi}{\partial q}\\ y(p,q,r)&=-\frac {\partial \Pi}{\partial r} \end{align*}