13.4 Demand Functions for Perfect Substitutes

We can write a generic perfect substitutes utility function as \(u(x_1,x_2) = ax_1 + bx_2\) This will have a constant MRS of \(MRS = {MU_1 \over MU_2} = {a \over b}\) Since the MRS is constant and the price ratio is constant, one of the following three conditions must hold:

• If the MRS is greater than the price ratio $(a/b > p_1/p_2)$, the utility-maximizing choice will be to buy only good 1
• If the MRS is less than the price ratio $(a/b < p_1/p_2)$, the utility-maximizing choice will be to buy only good 2
• If the MRS is exactly equal to the price ratio $(a/b = p_1/p_2)$, all bundles along the budget constraint will give the same amount of utility

We therefore need to express the optimal bundle as a piecewise function, to delineate what happens in each of those three cases: \(\begin{aligned} x_1^\star(p_1,p_2,m) &= \begin{cases} {m \over p_1} & \text{ if }{a \over b} > {p_1 \over p_2 }\\ \\ \left[0, {m \over p_1}\right] & \text{ if }{a \over b} = {p_1 \over p_2 }\\ \\ 0 & \text{ if }{a \over b} < {p_1 \over p_2 } \end{cases}\\ \\ x_2^\star(p_1,p_2,m) &= \begin{cases} 0 & \text{ if }{a \over b} > {p_1 \over p_2 }\\ \\ \left[0, {m \over p_2}\right] & \text{ if }{a \over b} = {p_1 \over p_2 }\\ \\ {m \over p_2} & \text{ if }{a \over b} < {p_1 \over p_2 } \end{cases} \end{aligned}\) Try playing around with the graph below to see how $a$, $b$, $p_1$, $p_2$, and $m$ affect the optimal choice: