Problem 2
A consumer lives on an island where she produces two goods, \(x\) and \(y,\) according to the production possibility frontier \(x^{2}+y^{2} \leq 200,\) and she consumes all the goods herself. Her utility function is \(U=x y^{3}\) The consumer also faces an environmental constraint on her total output of both goods. The environmental constraint is given by \(x+y \leq 20\) (a) Write out the Kuhn-Tucker first-order conditions. (b) Find the consumer's optimal \(x\) and \(y\). Identify which constraints are binding.
Problem 2
Is the Kuhn-Tucker sufficiency theorem applicable to: (a) Maximize \(\quad \pi=x_{1}\) \\[ \text { subject to } \quad x_{1}^{2}+x_{3}^{2} \leq 1 \\] and \\[ x_{1}, x_{2} \geq 0 \\] (b) Minimize \(\quad C=\left(x_{1}-3\right)^{2}+\left(x_{2}-4\right)^{2}\) \\[ \text { subject to } \quad x_{1}+x_{2} \geq 4 \\] and \\[ x_{1}, x_{2} \geq 0 \\] (c) Minimize \(\quad C=2 x_{1}+x_{2}\) \\[ \text { subject to } \quad x_{1}^{2}-4 x_{1}+x_{2} \geq 0 \\] and \\[ x_{1}, x_{2} \geq 0 \\]
Problem 3
Minimize \\[ \begin{array}{l} C=x_{1} \\ x_{1}^{2}-x_{2} \geq 0 \end{array} \\] subject to and \\[ x_{1}, x_{2} \geq 0 \\] Solve graphically. Does the optimal solution occur at a cusp? Check whether the optimal solution satisfies \((a)\) the constraint qualification and \((b)\) the Kuhn-Tucker minimum conditions.
Problem 3
An electric company is setting up a power plant in a foreign country, and it has to plan its capacity. The peak-period demand for power is given by \(P_{1}=400-Q_{1}\) and the off-peak demand is given by \(P_{2}=380-\mathrm{Q}_{2}\). The variable cost is 20 per unit (paid in both mar. kets) and capacity costs 10 per unit which is only paid once and is used in both periods. (a) Write out the Lagrangian and Kuhn-Tucker conditions for this problem. (b) Find the optimal outputs and capacity for this problem. (c) How much of the capacity is paid for by each market (i.e., what are the values of \(\lambda\) ) and \(\lambda_{2}\) )? (d) Now suppose capacity cost is 30 cents per unit (paid only once). Find quantities, capacity, and how much of the \epsilonapacity is paid for by each market (i.e., \(\lambda_{1}\) and \(\lambda_{2}\) ).
Problem 4
\\[ \begin{array}{ll} \text { Minimize } & C=x_{1} \\ \text { subject to } & -x_{2}-\left(1-x_{1}\right)^{3} \geq 0 \end{array} \\] and \\[ x_{1}, x_{2} \geq 0 \\] Show that \((a)\) the optimal solution \(\left(x_{1}^{*}, x_{2}^{*}\right)=(1,0)\) does not satisfy the Kuhn-Tucker conditions, but \((b)\) by introducing a new muttiplier \(\lambda_{0} \geq 0,\) and modifying the Lagrangian function (13.15) to the form \\[ Z_{0}=\lambda_{0} f\left(x_{1}, x_{2}, \ldots, x_{n}\right)+\sum_{j=1}^{m} \lambda_{1}\left[r_{i}-g^{j}\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] \\] the Kuhn-Tucker conditions can be satisfied at (1,0) . (Note: The Kuhn-Tucker conditions on the multipliers extend to only \(\left.\dot{\lambda}_{1}, \ldots, \lambda_{m}, \text { but not to } \lambda_{0} .\right)\)
