6.1 Corner Solutions
The gravitational pull argument holds that if the $MRS > MRT$, Chuck can do better by moving to the right along his PPF. In the cases we’ve looked at thus far, Chuck has stopped when he reached a point of tangency, where $MRS = MRT$. But what happens if $MRS$ never gets as low as $MRT$? That is, even the last hour he spends producing fish (good 1) gives him more utility than if he’d spent that hour collecting coconuts?
In order for this to occur, it must be the case that the agent’s indifference curves cross the axes. Two examples we’ve seen of this kind of utility function are the perfect substitutes utility function, whose indifference curves are just straight lines (shown in the left diagram below), and the quasilinear utility function, whose indifference curves are parallel transforms of each other (shown in the right diagram below).
In the left graph, Chuck maximizes his utility by producing only coconuts (drag the bundle all the way to the left), and in the right-hand graph he maximizes his utility by producing only fish (drag the bundle all the way to the right). Let’s examine these two cases mathematically to see what’s going on.
In each case, we’ll assume that Chuck has 100 hours of labor, and plot his MRS and MRT along his PPF as $L_1$ increases from 0 to 100.
Example 1: Linear utility, curved PPF
In the first example, his production functions are \(x_1 = f_1(L_1) = L_1\) \(x_2 = f_2(L_2) = 10\sqrt{L_2}\) Therefore his MRT is \(MRT = \frac{MP_{L2}}{MP_{L1}} = \frac{5}{\sqrt{L_2}}\) This is equal to $\frac{1}{2}$ when $L_2 = 100$, and gets infinitely large as $L_2$ approaches 0. Now suppose Chuck’s utility function is \(u(x_1,x_2) = x_1 + 4x_2\) Then his MRS is constant at every point along his PPF: \(MRS = \frac{MU_1}{MU_2} = \frac{1}{4}\) Since his MRS is always less than his MRT, he always wants to move to the left along the PPF. If we plot MRS and MRT along the PPF in this case, the two curves never touch, and that MRT is always greater than MRS:
Example 2: Quasilinear utility, linear PPF
In the second example, his production functions are \(x_1 = f_1(L_1) = L_1\) \(x_2 = f_2(L_2) = \tfrac{1}{2}L_2\) Therefore his MRT is constant, and his PPF is linear: \(MRT = \frac{MP_{L2}}{MP_{L1}} = \frac{\frac{1}{2}}{1} = \frac{1}{2}\) His utility function here is the quasilinear utility function \(u(x_1,x_2) = 100 \ln x_1 + x_2\) which has the associated MRS \(MRS = \frac{MU_1}{MU_2} = \frac{\frac{100}{x_1}}{1} = \frac{100}{x_1}\) This is infinite if Chuck produces no fish ($x_1 = 0$), and 1 when he devotes all 100 hours to producing fish (resulting in $x_1 = 100$).
Since his MRS is always greater than his MRT, he always wants to move to the right along the PPF. If we plot MRS and MRT along the PPF in this case, the two curves never touch, and that MRS is always greater than MRT:
Conditions for an interior (non-corner) solution
From the two above examples we can derive one necessary condition for an interior solution to an optimization problem: specifically, that the \(MRS > MRT\) when $x_1 = 0$, and \(MRS < MRT\) when $x_2 = 0$.
We might notice that there are some utility functions that guarantee an interior solution. For example, a Cobb-Douglas utility function of the form $u(x_1,x_2) = a \ln x_1 + b \ln x_2$ has an MRS of \(MRS = \frac{ax_2}{bx_1}\) This is infinite when $x_1 = 0$, which must be greater than any finite MRT; and it’s zero when $x_2 = 0$, which must be less than any finite MRT. Therefore a Cobb-Douglas utility function will never yield a corner solution.
There are also conditions on production functions which can prevent us from ever optimizing at a boundary: specifically, if both production functions have an infinite $MP_L$ at $L = 0$, the MRT will be zero when $x_1 = 0$ and infinite when $x_2 = 0$.
In general, rather than relying on theoretical constructs, it’s safest to always check the MRS and MRT at the each of the corners to see whether there’s a potential corner solution. As long as $MRS > MRT$ when $x_1 = 0$ and $MRS < MRT$ when $x_2 = 0$, you know that the solution will be an interior solution.
However, just because an optimum is interior doesn’t mean it’s automatically characterized by a tangency condition. To ensure that we need to be sure that both the MRS and MRT are continuous — that is, that they don’t have any kinks.