Question

10. Consider a firm that uses 2 inputs and has the production function $f(x_1, x_2) = 12x_1^{1/3}x_2^{1/2}$. Let the firm face the input prices $(p_1, p_2)$ and the output price $q$. Find the input combination for maximizing the firm's profit. 11. A student is preparing for exams in two subjects. She estimates that the grades she will obtain in each subject as a function of amount of time spent working on them are: $g_1 =$ $20 + 20(t_1^{1/2})$ and $g_2 = -80 + 3t_2$, where $g_1$ and $g_2$ are grades in each subject and $t_1$ and $t_2$ are the number of hours per week spent in studying the subjects. She wishes to maximize her grade average $(g_1 + g_2)/2$. She can not spend in total more than 60 hours studying in the week. Formulate the problem as a constrained optimization problem and find the optimal values of $t_1$, $t_2$ and the Lagrange multiplier. Interpret the Lagrange multiplier. 12. Maximize $Y = x_1^{0.25}x_2^{0.75}$, subject to $100 - 2x_1 - 4x_2 = 0$. Check the relevant second order sufficiency condition.

          10. Consider a firm that uses 2 inputs and has the production function $f(x_1, x_2) = 12x_1^{1/3}x_2^{1/2}$.
Let the firm face the input prices $(p_1, p_2)$ and the output price $q$. Find the input combination
for maximizing the firm's profit.

11. A student is preparing for exams in two subjects. She estimates that the grades she will
obtain in each subject as a function of amount of time spent working on them are: $g_1 =$
$20 + 20(t_1^{1/2})$ and $g_2 = -80 + 3t_2$, where $g_1$ and $g_2$ are grades in each subject and $t_1$ and $t_2$ are
the number of hours per week spent in studying the subjects. She wishes to maximize her
grade average $(g_1 + g_2)/2$. She can not spend in total more than 60 hours studying in the
week. Formulate the problem as a constrained optimization problem and find the optimal
values of $t_1$, $t_2$ and the Lagrange multiplier. Interpret the Lagrange multiplier.

12. Maximize $Y = x_1^{0.25}x_2^{0.75}$, subject to $100 - 2x_1 - 4x_2 = 0$. Check the relevant second
order sufficiency condition.

10. Consider a firm that uses 2 inputs and has the production function f(x1, x2) = 12x1^1/3x2^1/2.
Let the firm face the input prices (p1, p2) and the output price q. Find the input combination
for maximizing the firm's profit.

11. A student is preparing for exams in two subjects. She estimates that the grades she will
obtain in each subject as a function of amount of time spent working on them are: g1 =
20 + 20(t1^1/2) and g2 = -80 + 3t2, where g1 and g2 are grades in each subject and t1 and t2 are
the number of hours per week spent in studying the subjects. She wishes to maximize her
grade average (g1 + g2)/2. She can not spend in total more than 60 hours studying in the
week. Formulate the problem as a constrained optimization problem and find the optimal
values of t1, t2 and the Lagrange multiplier. Interpret the Lagrange multiplier.

12. Maximize Y = x1^0.25x2^0.75, subject to 100 - 2x1 - 4x2 = 0. Check the relevant second
order sufficiency condition.

Added by Lourdes L.

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