8.2 Demand Functions for Cobb-Douglas Utility Functions

For a generic Cobb-Douglas utility function \(u(x_1,x_2) = x_1^a x_2^b\) or equivalently, \(u(x_1,x_2) = a \ln x_1 + b \ln x_2\) the MRS is \(MRS = {ax_2 \over bx_1}\) It’s easy to see that all the conditions for using the Lagrange method are met: the MRS is infinite when $x_1 = 0$, zero when $x_2 = 0$, and smoothly descends along any budget line. Therefore, to find the optimal bundle, we will set the MRS equal to the price ratio and plug the result back into the budget constraint.

Setting the MRS equal to the price ratio $p_1/p_2$ gives us the tangency condition \(\begin{aligned} MRS &= {p_1 \over p_2}\\ {ax_2 \over bx_1} &= {p_1 \over p_2}\\ x_2 &= {b \over a}{p_1 \over p_2}x_1 \end{aligned}\) Plugging this back into the budget constraint, we can solve for $x_1$: \(\begin{aligned} p_1x_1 + p_2x_2 &= m\\ p_1x_1 + p_2\left[{b \over a}{p_1 \over p_2}x_1\right] &= m\\ ap_1x_1 + bp_1x_1 &= am\\ p_1x_1 &= {a \over a + b}m\\ x_1^\star(p_1,p_2) &= {a \over a + b}{m \over p_1} \end{aligned}\) and therefore \(x_2^\star = {b \over a}{p_1 \over p_2}x_1^\star = {b \over a + b}{m \over p_2}\) Intuitively, this means you spend fraction ${a \over a+b}$ of your income on good 1, and fraction ${b \over a + b}$ on good 2.

Try playing around with the graph below to see how $a$, $b$, $p_1$, $p_2$, and $m$ affect the optimal choice: