🔬 Tutorial problems theta \theta

🔬 Tutorial problems theta \(\theta\)#

Note

This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left out are for additional practice on your own.

\(\theta\).1#

Consider the maximization problem

\[ \max_{c_1, c_2} ( \sqrt c_1 + \beta \sqrt{c_2}) \]

subject to \(c_1 \geq 0\), \(c_2 \geq 0\) and \(p_1 c_1 + p_2 c_2 \leq m\). Here \(p_1, p_2\) and \(m\) are nonnegative constants, and \(\beta \in (0, 1)\).

Show that this problem has a solution if and only if \(p_1\) and \(p_2\) are both strictly positive.

Solve the problem by substitution and using the tangency (relative slope) condition. Discuss, which solution approach is easier.

To answer the first part of the question, review facts of existence of optima.

\(\theta\).2#

Solve the following constrained maximization problem using the Lagrange method, including the second order conditions.

\[\begin{split} f(x,y) = \frac{x^3}{3}-3y^2+2x \to \max_{x,y} \\ \text { subject to } \\ 4x = y^3,\\ x,y \in \mathbb{R} \end{split}\]

Follow standard algorithm of Lagrange method.

\(\theta\).3#

Find the maxima and minima of the function

\[ f(x,y) = xy \text{ subject to } x^2+y^2=2a^2, a>0 \]

Check both first and second order conditions.

Follow standard algorithm of Lagrange method.

\(\theta\).4#

Find the maxima and minima of the function

\[ f(x,y) = \tfrac{1}{x} + \tfrac{1}{y} \]

subject to

\[ (\tfrac{1}{x})^2+(\tfrac{1}{y})^2=(\tfrac{1}{a})^2, \]

where \(a>0\).

Follow standard algorithm of Lagrange method.

\(\theta\).5#

Solve the following maximization problem treating the last constraint as binding

\[\begin{split} xy^{\tfrac{1}{2}}z^{\tfrac{1}{3}} \longrightarrow max_{x,y,z} \\ \text{ subject to }\quad\quad\\ x \ge 0, y \ge 0 ,z \ge 0,\\ 3x + 2y + z \le 10\\ \end{split}\]

Follow standard algorithm of Lagrange method realizing that the inequality constraints have to be satisfied with equality.

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