Calculus for Economics, Exercises (Dec. 04,2017)

I

Let $p,q,I>0$. We maximize the utility function $$ u(x,y)=x^{\frac 13}y^{\frac 23} $$ subject to the budget constraint $$ I-px-qy=0 $$ Find the stationary point and show it the maximal point.

II

Let $p,q,I>0$. We maximize the utility function $$ u(x,y)=\frac 13\log x+\frac 23\log y $$ subject to the budget constraint $$ I-px-qy=0 $$ Find the stationary point and show it the maximal point.

III

We optimize the function $$z=x+y$$ subject to the constraint $$ x^2+2y^2-24=0 $$ Find the stationary points and check if they are maximal or minimal.

IV

We optimize the functionn $$z=xy$$ subject to the constraint $$ x^2+y^2-1=0 $$ Find the stationary points and check if they are maximal or minimal.

V

We are given a utility function $u(x,y)=x^{\frac 13}y^{\frac 13}$. We minimize $f(x,y)=px+qy$ subject to the constraint \begin{equation*} u(x,y)=\bar{u} \end{equation*} Find the stationary points.

VI

The compensated demand functions of Higgs type obtained in V are denoted by \begin{equation*} x^*(p,q,\bar{u}),\ y^*(p,q,\bar{u}) \end{equation*} We define the minimum const function by \begin{equation*} E(p,q,\bar{u})=px^*(p,q,\bar{u})+qy^*(p,q,\bar{u}) \end{equation*} Then show that the McKenzie's lemma \begin{equation*} \frac{\partial E}{\partial p}(p,q,\bar{u})=x^*(p,q,\bar{u}), \quad \frac{\partial E}{\partial q}(p,q,\bar{u})=y^*(p,q,\bar{u}) \end{equation*} holds.

VII

We maximize the utility function $u(x,y)=x^{\frac 13}y^{\frac 13}$ subject to the budget constraint $I-px-qy=0$ to get \begin{equation} x(p,q,I)=\frac I{2p},\ y(p,q,I)=\frac I{2q},\ \end{equation} Moreover we define the indirect utility function \begin{equation} v(p,q,I)=u(x(p,q,I),x(p,q,I)) \end{equation} Then show the following directly.

(1) \begin{equation} x^*(p,q,\bar{u})=x(p,q,E(p,q,\bar{u})) \end{equation}

(2) \begin{equation} x(p,q,I)=x^*(p,q,v(p,q,I)) \end{equation}

(3) \begin{equation} v(p,q,E(p,q,\bar{u}))=\bar{u} \end{equation}

(4) \begin{equation} E(p,q,v(p,q,I))=I \end{equation}

VIII

We express the curve $g(x,y):=x^2-xy+y^2-1$ explicitly in a neighborhood of $(1,1)$ by \begin{equation*} y=\varphi(x)=\frac {x+\sqrt{4-3x^2}}{2} \end{equation*} Find $\varphi''(1)$ by the 1st and second order partial derivatives of $g$.