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Note: These explanations are in the process of being adapted from my textbook.
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The “Gravitational Pull” Toward Optimality

The central constrained optimization problem for a consumer is to find the utility-maximizing bundle in their budget set.

For now, we’ll restrict ourselves to strictly monotonic preferences, which means that more of every good is always preferred. It follows logically from this that the optimal bundle for such preferences must always lie along the constraint, since for any bundle in the interior of the budget set, there must always be a strictly preferred bundle which is also affordable. Therefore, we can reduce our constrained optimization problem to finding the utility-maximizing bundle along their budget line.

To visualize this problem, we can think about plotting the budget line and the utility function in the same 3D graph. In the left-hand graph below, we visualize the constraint as a kind of “fence” over the utility function “hill.” In the right-hand graph, we show the utility you reach at different points along the budget constraint. You can drag the point left and right; the relevant bundle is shown below. Note that the left-hand side of this graph represents the vertical intercept of your budget constraint (where you’re spending all your money on good 2), and the right-hand side represents the horizontal intercept of your budget constraint (where you’re spending all your money on good 1).

The usual way we would find the peak of a curve like this would be to analyze it using (univariate) calculus. In particular, if we let $m_1$ be the amount of money spent on good 1, and ($m - m_1$) be the amount spent on good 2, then the utility as a function of the amount spent on good 1 is \(\hat u(m_1) = u(x_1(m_1),x_2(m_1))\) What happens if we spend a little more money on good 1? Mathematically, by the chain rule, we have \(\frac{d \hat u(m_1)}{dm_1} = \frac{\partial u}{\partial x_1} \times \frac{dx_1}{dm_1} + \frac{\partial u}{\partial x_2} \times \frac{dx_2}{dm_1}\) Since each dollar spent on good 1 increases the consumption of good 1 by $1/p_1$ units of good 1, and decreases consumption of good 2 by $1/p_2$ units of good 2, we can write this perhaps more simply as \(\frac{d \hat u(m_1)}{dm_1} = {MU_1 \over p_1} - {MU_2 \over p_2}\) We sometimes call the term $MU_i/p_i$ the “bang for your buck for good $i$:” that is, the additional utility you get from another dollar spent on good $i$. Thus the first term is the utility gain per each dollar spend on good 1, and the second term is the utility loss per dollar not spent on good 2.

If this is positive, then you get more “bang for your buck” from good 1 than good 2, so you increase your utility by moving to the right along the budget line; if it’s negative, the opposite holds. That is,

\({MU_1 \over p_1} > {MU_2 \over p_2} \iff \text{ buy more good 1, less good 2}\) \({MU_1 \over p_1} < {MU_2 \over p_2} \iff \text{ buy less good 1, more good 2}\)

Example

The graph above is built from the following example: suppose your preferences over apples (good 1) and bananas (good 2) may be represented by the utility function \(u(x_1,x_2) = x_1^{3 \over 4}x_2^{1 \over 4}\) For this utility function, the marginal utilities are given by \(\begin{aligned} MU_1(x_1,x_2) &= \tfrac{3}{4}x_1^{-{1 \over 4}}x_2^{1 \over 4}\\ MU_2(x_1,x_2) &= \tfrac{1}{4}x_1^{3 \over 4}x_2^{-{3 \over 4}} \end{aligned}\) For the budget constraint, let’s assume:

Now let’s examine the point at which you spend half your money on each good. This would mean buying the bundle $(6,12)$. At this point, your marginal utility from apples is \(MU_1(6,12) = \tfrac{3}{4}6^{-{1 \over 4}}12^{1 \over 4} \approx 0.9 \frac{\text{utils}}{\text{apple}}\)and your marginal utility from bananas is \(MU_2(6,12) = \tfrac{1}{4}6^{3 \over 4}12^{-{3 \over 4}} \approx 0.15 \frac{\text{utils}}{\text{banana}}\) If you spent one more dollar on apples, you could buy 1/4 more pounds of apples (since apples cost $p_1 = 4$ dollars/pound). This would increase your utility by approximately \(\text{Utility gain} = {1 \over 4} \text{ apples} \times 0.9 \frac{\text{utils}}{\text{apple}} = +0.225 \text{ utils}\) At the same time, if you spent one less dollar on bananas, you would have to buy 1/2 fewer pounds of bananas (since bananas cost $p_2 = 2$ dollars per pound). This would decrease your utility by approximately \(\text{Utility loss} = {1 \over 2} \text{ bananas} \times 0.15 \frac{\text{utils}}{\text{banana}} = -0.075 \text{ utils}\) Since the utility gain from buying a dollar more apples is greater than the utility loss from buying a dollar fewer bananas your utility at this point is increasing as you move to the right along the budget constraint.

The MRS and the Price Ratio

Recall that “Good 1 - Good 2 space,” any slope represents a tradeoff between good 1 and good 2, and is measured in units of good 2 per unit of good 1. We have two slopes which are of interest to us:

We found above that you should spend more money on good 1 if you got more “bang for you buck” from good 1 than good 2: \({MU_1 \over p_1} > {MU_2 \over p_2}\) Note that we can rearrange this to read \({MU_1 \over MU_2} > {p_1 \over p_2}\) But note that the left-hand side of this equation is just the MRS, while the right-hand side of this equation is the price ratio. Therefore, another way of thinking about the consumer problem is to say:

Visually, if the MRS is not equal to the price ratio at some bundle along the budget line, then the indifference curve passing through that point cuts through the budget line, meaning that there’s a region of affordable bundles which are strictly preferred to that point. If the MRS is greater than the price ratio, then that region must lie to the right of the point; if the MRS is less than the price ratio, then that region must lie to the left of that point:

Example, continued

In our example, the MRS at the point $(6, 12)$ would be \(MRS(6, 12) = {MU_1(6,12) \over MU_2(6,12)} = {0.9 \text{ utils/apple} \over 0.15 \text{ utils/banana}} = 6\ {\text{bananas} \over \text{apple}}\) Likewise, the price ratio was \({p_1 \over p_2} = {4 \text{ dollars/apple} \over 2 \text{ dollars/banana}} = 2\ {\text{bananas} \over \text{apple}}\)

Since you’re willing to give up 6 bananas per apple, and their market prices mean you only need to give up 2 bananas per apple, you’re better off buying more apples and fewer bananas.

Of course, the MRS changes as you move along the budget line: in particular, because preferences are “well behaved” (strictly monotonic and strictly convex), the MRS is decreasing as you increase $x_1$ and decrease $x_2$.

We can, in fact, plot the indifference curve/budget line diagram, the total utility along the budget line, and a new graph showing the MRS vs the price ratio along the budget line all together. Note that the ideal point here is at (9, 6). Everywhere to the left of that point, the MRS is greater than the price ratio drawing the consumer to the right; everywhere to the right of that point, the MRS is less than the price ratio, drawing the consumer to the left.

Key Takeaway

The “gravitational pull” argument outlined above applies to any point along any budget constraint. It tells you, relative to that point, which direction along the constraint corresponds to increasing utility.

In the example above, we can see that at the optimal point, the MRS equals the price ratio, meaning the indifference curve passing through the optimal bundle is tangent to the budget line at that bundle. In some cases, that will characterize the optimal bundle. However, in others, the MRS and the price ratio may be different, or may even not exist at all.

In a way, it might be helpful to go back to our initial exercise of plotting utility along the budget line. All we’ve found is how to characterize where utility is increasing or decreasing along a constraint. Just as finding the highest point of a curve isn’t quite as simple as setting $f’(x) = 0$, finding the optimal bundle is not always as simple as setting the MRS equal to the price ratio.

In short: the gravitational pull argument can tell us which way the optimum lies, but it hasn’t quite found it for us yet.

Next: Optimal Choices Characterized by a Tangency Condition
Copyright (c) Christopher Makler / econgraphs.org