Corner Solutions
The gravitational pull argument holds that if the $MRS > p_1/p_2$, a consumer can do better by moving to the right along her budget constraint, and vice versa. In some cases, this process leads the consumer to a point along the budget constraint where $MRS = p_1/p_2$, so the indifference curve passing through that point is tangent to the constraint. But what happens if $MRS$ never gets as low as the price ratio? That is, if even the last dollar spent on good 1 gives the consumer more utility than if she’d spent that dollar on good 2?
In order for this to occur, it must be the case that the consumer’s indifference curves cross the axes. Two examples we’ve seen of this kind of utility function are the quasilinear utility function, whose indifference curves are parallel transforms of each other (shown in the left diagram below), and the perfect substitutes utility function, whose indifference curves are just straight lines (shown in the right diagram below).
In the left graph, the consumer maximizes her utility by buying only good 1 (drag the bundle all the way to the right), and in the right-hand graph she maximizes her utility by buying only good 2 (drag the bundle all the way to the left). Let’s examine these two cases mathematically to see what’s going on. In each case, we’ll assume the consumer has $m = 100$ and faces prices $p_1 = 1$ and $p_2 = 2$, so their budget constraint is given by the equation \(x_1 + 2x_2 = 100\) and the price ratio is $p_1/p_2 = 1/2$.
Example 1: Quasilinear utility
Suppose the consumer’s preferences could be represented by 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 the consumer buys only good 2 ($x_1 = 0$), and 1 if she buys only good 1 (resulting in $x_1 = 100$).
Since her MRS is always greater than the price ratio, there is always an area of overlap between the preferred region and the budget set, and the consumer always wants to move to the right along the budget line — up to and including the point where they’re spending all their money on good 1! If we plot the utility (top right graph), we can see that her utility is increasing at every point along the budget line. And if we plot MRS and price ratio along the budget line (bottom right graph), the two curves never touch, and that MRS is always greater than price ratio:
For cases like this, it’s worth noting that you could try to solve using the tangency condition and budget constraint. As a reminder, this method finds the point along the budget constraint where the MRS is equal to the price ratio. But if we set the MRS equal to the price ratio, we get \(\begin{aligned} MRS &= {p_1 \over p_2}\\ {100 \over x_1} &= {1 \over 2}\\ x_1 &= 200 \end{aligned}\) In other words, the tangency condition is a vertical line at $x_1 = 100$. This intersects the equation of the budget line, $x_1 + 2x_2 = 100$, at the point $(200,-50)$, but while this makes mathematical sense it doesn’t make economic sense:
This illustrates an important point: a lot of times, the solution you get by plugging something into a mathematical formula may not be the actual solution to an economic problem. It’s important to check your work after you solve a problem to make sure that the solution you found is actually feasible! A key way to do this is to check if any values are negative that cannot be. In this case, the mathematical “solution” yields a negative value for good 2; since you can’t go into a store and buy a negative quantity of a good, that can’t be the economic solution.
Example 2: Perfect Substitutes
Let’s consider the same budget constraint as above \(x_1 + 2x_2 = 100\) but this time use the utility function \(u(x_1,x_2) = x_1 + 4x_2\) Note that the price ratio is 1/2, meaning that good 2 is twice as expensive as good 1. But with this utility function, each unit of good 1 brings you 1 util, while each unit of good 2 brings you 4 utils; so your MRS is \(MRS = {MU_1 \over MU_2} = {1 \over 4}\) That is, you enjoy each unit of good 2 four times as much as each unit of good 1, no matter how many of each good you have!
Logically, it follows that if you like each unit of good 2 four times as much as each unit of good 1, but it only costs twice as much, you’re not going to spend any of your money on good 1. And indeed that’s the case: the optimal bundle involves you spending all of your money on good 2:
In this case, the MRS is always less than the price ratio, so the consumer is always drawn to the left.
Note that you can’t even use the tangency condition here! Why not? The tangency condition finds the point along the budget line where the MRS equals the price ratio. But here, the MRS is always 1/4, and the price ratio is always 1/2. There is no point at which $1/4 = 1/2$, so the tangency condition is of no use to us.
Conditions for an interior (non-corner) solution
Notice that in the first example above, the MRS was greater than the price ratio even when the consumer was spending all their money on good 1: the utility function was always increasing along the budget line. By contrast, in the second example, the MRS was less than the price ratio even when the consumer was spending all their money on good 2, so the utility function was always decreasing along the budget line.
From these two examples we can derive one necessary condition for an interior solution to an optimization problem: specifically, that \(\begin{aligned} MRS > p_1/p_2 & \text{ when }x_1 = 0\\ MRS < p_1/p_2 & \text{ when }x_2 = 0\end{aligned}\) This means that at the left corner of the budget constraint, the consumer is being pulled to the right; and at the right corner of the budget constraint, the consumer is being pulled to the left.
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 price ratio; and it’s zero when $x_2 = 0$, which must be less than any finite price ratio. Therefore a Cobb-Douglas utility function will never yield a corner solution.
On the other hand, there are some utility functions which guarantee a corner solution. For example, a concave utility function of the form \(u(x_1,x_2) = ax_1^2 + bx_2^2\) has an MRS of \(MRS = \frac{ax_1}{bx_2}\) This is zero when $x_1 = 0$, and infinite when $x_2 = 0$.