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Note: These explanations are in the process of being adapted from my textbook.
I'm trying to make them each a "standalone" treatment of a concept, but there may still
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# Demand with Quasilinear Utility Functions

With a quasilinear utility function of the form $$u(x_1,x_2) = v(x_1) + x_2$$ the marginal rate of substitution is just $v^\prime(x_1)$. If we assume $v^\prime(x_1)$ is continuous and exhibits diminishing marginal utility, there is some point at which the MRS equals the price ratio. However, that point is not guaranteed to be in the first quadrant (i.e. have positive quantities of both good 1 and good 2), so corner solutions are possible. In particular, with a quasilinear utility function, it may be the case that there is an interior solution characterized by a tangency condition for certain values of $p_1$, $p_2$, and $m$, and a corner solution for other values.

For example, consider the utility function $$u(x_1,x_2) = a \ln x_1 + x_2$$ For simplicity, let’s suppose that good 2 is “dollars spent on other goods;” this is a convenient way to analyze a generic tradeoff between “good 1” and “all other goods.” Since good 2 is measured in dollars, $p_2$ is just 1 (more specifically, 1 dollar per dollar), so the price ratio is $p_2 = 1$. This means that the MRS is $$MRS = {MU_1 \over MU_2} = {a/x_1 \over 1} = {a \over x_1}$$ Because this is measured in “units of good 2 per units of good 1” as always, and because “units of good 2” is just dollars, we can interpret this MRS as the consumer’s marginal willingness to pay for the $x_1^\text{th}$ unit of good 1. The tangency condition just says that she will buy units of good 1 up until the point where her marginal willingness to pay is equal to the price. We can write this as the “Lagrange solution” $x_1^\mathcal{L}$: \begin{aligned} MRS &= p_1\\ {a \over x_1} &= p_1\\ x_1^\mathcal{L} &= {a \over p_1} \end{aligned} Because $p_2 = 1$, the budget line has the equation $$p_1x_1 + x_2 = m$$ Plugging $x_1^\mathcal{L}$ into that and solving for $x_2$ gives us the Lagrange solution for good 2 of $$x_2^\mathcal{L} = m - a$$ However, note that this is positive only if $m \ge a$; if $m < a$, then the consumer doesn’t have enough money to afford $x_1^\mathcal{L}$, so the optimum occurs at a corner solution in which she simply spends all her money on good 1: that is $x_1 = m/p_1$, $x_2 = 0$.

Because the optimal behavior changes according to income level, the demand functions must be defined in a piecewise manner: \begin{aligned} x_1^\star(p_1,p_2,m) &= \begin{cases} {a \over p_1} & \text{ if }m \ge a\\ \\ {m \over p_1} & \text{ if }m \le a \end{cases}\\ \\ x_2^\star(p_1,p_2,m) &= \begin{cases} m - a & \text{ if }m \ge a\\ \\ 0 & \text{ if }m \le a \end{cases} \end{aligned}

Try playing around with the graph below to see how $a$, $p_1$, and $m$ affect the optimal choice. In particular, try lowering $m$ until it is less than $a$, to see how the solution shifts from being an interior solution to a corner solution:

Note that this is not a “general” solution for all quasilinear utility functions; quasilinear utility functions cover a broad range of possible functions $v(x_1)$, each of which will have its own unique demand function. But because the tangency condition is just a value of $x_1$, they all share the characteristic that the solution will sometimes be a corner solution, and sometimes not.

Copyright (c) Christopher Makler / econgraphs.org