10.2 The "Gravitational Pull" Towards Optimality

While this is a chapter on solving an optimization problem, it’s important to think about the forces leading to optimal behavior. In your Principles of Econ class, for example, you probably solved for the equilibrium market price by finding the point at which supply and demand curves met. But even more important than that result were the forces that would pull the market price towards that equilibrium: that if the price were too low, shortages of goods would cause buyers to bid the price up, while if the price were too high, surpluses would cause sellers to lower their price to reduce excess inventory. In the same way, it’s important to understand how utility-maximizing agents might be drawn to their optimal choices. This is especially important because, as we’ll see later, the optimum is not always characterized by a neat mathematical solution — but the “gravitational forces” we analyze here are always applicable.

Thinking on the margin: Could Chuck do better by reallocating his time?

Let’s think about our castaway Chuck’s optimization problem intuitively. Chuck has to think about how to divide up his labor between producing fish (good 1) and coconuts (good 2). Intuitively, if Chuck is optimizing, he couldn’t reallocate his time in a way that would make him better.

Knowing this, before Chuck starts work, suppose he performs the following thought experiment: what would happen if he chose some arbitrary split between the two, and then asked himself: What would happen if I spent another hour fishing? In that hour he would produce some more fish, which would bring him utility; but on the other hand, it’s one hour less producing coconuts, so he’d forego the utility from those coconuts.

We can break down this decision using his production and utility functions. In particular, we have four concepts from the previous lectures which will be useful:

• Marginal product of fish $(MP_{L1})$: additional fish per hour spent fishing

• Marginal product of coconuts ($MP_{L2}$): additional coconuts per hour spent collecting coconuts

• Marginal utility of fish $(MU_1)$: additional utility per fish

• Marginal utility of coconuts $(MU_2)$: additional utility per coconut

If Chuck spends an additional hour fishing, he will therefore produce about ⊕ We say “about” because $MP_L$ and $MU$ are defined in calculus terms, for infinitesimally small changing in labor, not full hours. $MP_{L1}$ fish, each of which will bring him about $MU_1$ utils of utility; so his increase in utility from spending another hour fishing is approximately \(\frac{\text{utils}}{\text{hour fishing}} \approx MP_{L1} \frac{\text{fish}}{\text{hour fishing}} \times MU_1 \frac{\text{utils}}{\text{fish}}\) Because that hour spent fishing is one less hour collecting coconuts, in that hour he forgoes $MP_{L2}$ coconuts, each of which would have brought him about $MU_2$ utils of utiliity; so his decrease in utility from spending one less hour collecting coconuts is \(\frac{\text{utils}}{\text{hour collecting coconuts}}\approx MP_{L2} \frac{\text{coconuts}}{\text{hour collecting coconuts}} \times MU_2 \frac{\text{utils}}{\text{coconut}}\) He should therefore devote more time to fishing, and less to collecting coconuts, when \(MP_{L1} \times MU_1 > MP_{L2} \times MU_2\) If we cross multiply, we can see that this is the same as the condition \(\frac{MU_1}{MU_2} > \frac{1/MP_{L1}}{1/MP_{L2}}\) or \(MRS > MRT\) Conversely, Chuck should devote less time to fishing, and more to collecting coconuts, when $MRS < MRT$.

Comparing MRS and MRT intuitively

What does it mean to compare $MRS$ and $MRT$ in this way?

Recall our definitions of these two terms, as they relate to Chuck’s problem:

• Marginal Rate of Substitution ($MRS$): The number of coconuts Chuck is willing to give up in order to get another fish, based on his utility function; in other words, Chuck’s willingness to “pay” for fish in terms of coconuts.

• Marginal Rate of Transformation ($MRT$): The number of coconuts Chuck needs to give up in order to get another fish, based on his production functions; in other words, the opportunity cost of fish in terms of coconuts.

Note that both of these are measured in coconuts per fish — that is, units of good 2 per units of good 1.

Therefore, if $MRS > MRT$, Chuck’s opportunity cost of another fish is less than the number of coconuts he’d be willing to “pay” for a fish; so he could increase his utility by spending more time fishing and less time collecting coconuts. Conversely, if $MRS < MRT$, Chuck’s opportunity cost of another fish is more than the number of coconuts he’d be willing to “pay” for a fish; so he could increase his utility by spending less time fishing and more time collecting coconuts.

Comparing MRS and MRT graphically

If we think of what this looks like in a diagram in “good 1 — good 2 space,” we can plot Chuck’s PPF and his indifference map in the same diagram. Let’s think about the curved PPF derived in Week 2, in which Chuck had $\overline L = 100$ hours of available labor, and his production functions were given by \(x_1 = f_1(L_1) = 10\sqrt{L_1}\) \(x_2 = f_2(L_2) = 6\sqrt{L_2}\) For these production functions, we found that Chuck’s MRT was given by \(MRT = \frac{9x_1}{25x_2}\) Let’s suppose now that Chuck’s preferences over fish and coconuts may be represented by the Cobb-Douglas utility function \(u(x_1,x_2) = 16\ln x_1 + 9 \ln x_2\) which has the MRS \(MRS = \frac{16x_2}{9x_1}\)

The following graph allows you to experiment with different amounts of labor to devote to fish ($L_1$, which implies $L_2 = 100 - L_1$). This effectively drags Chuck’s production choice along his PPF. The top graph shows his PPF and feasible set, the bundle $X$ representing his choice, the indifference curve passing through $X$, and the set of bundles he prefers to $X$. the bottom graph shows his $MRS$ and $MRT$ as functions of $L_1$. Play around with it and see how the $MRS$ and $MRT$ change as you move along his PPF:

[ See interactive graph online at https://www.econgraphs.org/graphs/scarcity/autarky/mrs_vs_mrt ]

As you can see, at every point except the optimum at $L_1 = 64$, there is an area over overlap between Chuck’s feasible set and the set of bundles he prefers to $X$. This implies that there are feasible bundles that Chuck would prefer to $X$; so $X$ cannot be his optimal choice. Furthermore:

• When $L_1 < 64$, his $MRS > MRT$, so the area of overlap occurs to the right of $X$; so Chuck can improve his utility by moving to the right along his PPF (devoting more time to fish, and less to coconuts).

• When $L_1 > 64$, his $MRS < MRT$, and the area of overlap occurs to the left of $X$; so Chuck can improve his utility by moving to the left along his PPF (devoting less time to fish, and more to coconuts).

This is what we mean by a “gravitational force” — an imbalance between costs and benefits that “pulls” chuck towards his optimal choice. In this case, that optimal choice (at $L_1 = 64$) is characterized by a tangency condition between the PPF and Chuck’s indifference curve. Let’s now look at how we can use that tangency condition to solve for Chuck’s optimal bundle mathematically.