8.3 Demand Functions for Perfect Complements

We can write a generic perfect complements utility function as \(u(x_1,x_2) = \min\left\{\frac{x_1}{a}, {x_2 \over b}\right\}\) As we’ve argued before, the optimal bundle for this sort of utility function will occur where the minimands are equalized: that is, \({x_1 \over a} = {x_2 \over b}\) or \(x_2 = {b \over a}x_1\) Plugging this back into the budget constraint, we can solve for $x_1$: \(\begin{aligned} p_1x_1 + p_2x_2 &= m\\ p_1x_1 + p_2\left[{b \over a}x_1\right] &= m\\ ap_1x_1 + bp_2x_1 &= am\\ x_1^\star(p_1,p_2) &= {am \over ap_1 + bp_2} \end{aligned}\) and therefore \(x_2^\star = {b \over a}x_1^\star = {bm \over ap_1 + bp_2}\) Intuitively, one way of thinking about this is that you want to always buy $a$ units of good 1 for every $b$ units of good 2 you buy. A bundle of $a$ units of good 1 and $b$ units of good 2 costs $ap_1 + bp_2$; therefore, the maximum number of such bundles you can buy with $m$ dollars of income is \(N = \text{Max bundles of (a units of good 1, b units of good 2)} = {m \over ap_1 + bp_2}\) Since each of those bundles has $a$ units of good 1, your demand for good 1 is \(x_1^\star = a \times N = a \times {m \over ap_1 + bp_2}\) since each bundle contains $b$ units of good 2, your demand for good 2 is \(x_2^\star = b \times N = b \times {m \over ap_1 + bp_2}\) Try playing around with the graph below to see how $a$, $b$, $p_1$, $p_2$, and $m$ affect the optimal choice: