# Multivariable Calculus: Lagrange function

Q: Suppose that all you do in a day is work, play and sleep. Let $$x_1$$ be the number of hours per day you spend playing, $$x_2$$ be the number of hours you spend sleeping, and $$x_3$$ be the number of hours you spend working. Suppose that sleeping is free, but playing costs you $17 an hour. Furthermore, you spend all the money you earn from working on playing. The utility you get from sleeping and playing is given by a Cobb-Douglas utility function: $$U = x_1^{a_1} \times x_2^{a_2}$$, where $$a_1+a_2=1$$. If $$a_1 = \frac 34$$, and your hourly wage is $$w$$, find the number of hours you should work a day ($$x_3$$) in order to maximize your utility as a function of $$w$$. I know Lagrange multiplier can be applied to this question. So I write: $$L(x_1,x_2,x_3,λ_1,λ_2) = x_1^{a_1} \times x_2^{a_2} + λ_1(17x_1 - w x_3) - λ_2(x_1 + x_2 + x_3 -24),$$ where the first function is $$U$$, the second one is the assumption that you spend all the money you earn from working on playing. Solving gives: $$6x_1 = x_2, x_1 = w \frac{x_3}{17}$$, when $$U$$ is maximized. Thus, $$x_3 = \frac{408}{7w+17}$$. However, this is wrong. So, is my Lagrange function incorrect? • You've made a typo in the Lag-mult function?$\lambda_1(17x_1-wx_3)\dots\$ – Parcly Taxel Oct 15 '18 at 11:56
• Thank you. I've edited it. – lamhei Oct 15 '18 at 12:18

$$L(x,\lambda) = x_1^{\frac 34}x_2^{\frac 14}+\lambda_1(w x_3-17 x_1)+\lambda_2(24-x_1-x_2-x_3)$$
$$\nabla L = \left\{\begin{array}{rcl} \frac{3 \sqrt[4]{\text{x2}}}{4 \sqrt[4]{\text{x1}}}-17 \lambda_1-\lambda_2&=&0 \\ \frac{\text{x1}^{3/4}}{4 \text{x2}^{3/4}}-\lambda_2&=&0 \\ \lambda_1 w-\lambda_2&=&0 \\ w x_3-17 x_1&=&0 \\ 24-x_1-x_2-x_3&=&0 \\ \end{array}\right.$$
$$\left[ \begin{array}{cccccc} x_1 & x_2 & x_3 & \lambda_1 & \lambda_2 & U\\ \frac{18 w}{w+17} & 6 & \frac{306}{w+17} & -\frac{3^{3/4}}{4 \sqrt[4]{w} \sqrt[4]{(w+17)^3}} & -\frac{3^{3/4} w^{3/4}}{4 \sqrt[4]{(w+17)^3}} & 6\ 3^{3/4} \left(\frac{w}{w+17}\right)^{3/4} \end{array} \right]$$