7.5 The MRS and the Price Ratio

Now that we’ve established the nature of the constraint faced by a consumer, we can turn our attention to the consumer’s constrained optimization problem. The process for solving the constrained maximization problem subject to a budget line is exactly the same as it was for maximizing the utility subject to a PPF: only now we want to maximize $u(x_1,x_2)$ subject to the parameterized linear constraint $p_1x_1 + p_2x_2 = m$.

While the process is identical mathematically, the economic intuition behind this optimization problem is a bit new. In particular, the interpretation of the conditions comparing the MRS to the slope of the constraint now takes on a more specific meaning.

The “gravitational pull” argument from Chapter 5 went someting like this:

• If the slope of the indifference curve (the MRS) is greater than the slope of the constraint (the MRT, or the opportunity cost of good 1), then the agent would want to move to the right along the constraint (devote more resources to good 1)
• Conversely, if the slope of the indifference curve is less than the slope of the constraint, the agent would want to devote more resources to good 2.

In the context of a consumer, the slope of the constraint is the price ratio $p_1/p_2$: so we can rephrase this as:

• If the MRS is greater than the price ratio at some point along a budget line, the consumer would do better by spending more money on good 1 and less on good 2.
• If the MRS is less than the price ratio at some point along a budget line, the consumer would do better by spending more money on good 2 and less on good 1.

Visually, we can see this in the following diagram. Try dragging the bundle $X$ to the right and left along the budget line, and see how the MRS and the price ratio compare to one another:

When the $MRS > p_1/p_2$ at $X$, there is an area to the right of $X$ that is both affordable and preferred to $X$, and vice versa.

This comparison of the MRS to the price ratio is critical for understanding consumer behavior, so let’s take some time to look at it in detail. First, let’s think about what it means for the MRS to be greater than the price ratio — e.g. at some point to the left of the optimal bundle. There are three ways we might think about this:

• Let’s first think about the general formulation \(MRS > {p_1 \over p_2}\)The MRS represents the amount of good 2 the consumer is willing to give up to get another unit of good 1, and the price ratio represents the amount of good 2 the market requires the consumer to give up to get another unit of good 1. Therefore, if the $MRS > p_1/p_2$ at some bundle, it means that the consumer is more willing to give up good 2 to get good 1 than the market requires; so they should spend more money on good 1.

• Next let’s substitute the mathematical definition of the MRS; that is, $MRS = MU_1/MU_2$. We can then rewrite the above condition as \({MU_1 \over MU_2} > {p_1 \over p_2}\) We can interpret this is as saying that the consumer receives more utility from good 1 relative to good 2 than it costs them to buy good 1 relative to good 2, and should therefore devote more of their money to good 1. Note that as above, the units of both sides of this inequality are “units of good 2 per units of good 1.”

• If we cross-multiply the above condition, bringing everything related to good 1 to the left-hand side and everything related to good 2 to the right-hand side, then we get \({MU_1 \over p_1} > {MU_2 \over p_2}\) Now the units are somewhat different: since $MU$ is measured in utils per unit of a good, and the price is measured in dollars per unit of the same good, each side of this equation is measured in utils per dollar. We sometimes refer to this as “bang for your buck”: that is, how much utility you would get from spending an additional dollar on the good. Therefore, this interpretation says that if you get more “bang for your buck” from good 1 than good 2, you should spend more money on good 1 and less on good 2.

Of course, the converse holds as well in all three interpretations: when $MRS < p_1/p_2$, the consumer is less willing to give up good 2 to get good 1 than the market requires; they derive less utility from good 1 than it costs, relative to good 2; and they got less “bang for the buck” from the last unit of good 1 they purchased than from the last unit of good 2; all of which would mean that they should shift their consumption from good 1 to good 2.