Finally, with a fundamental model of consumer preference and its representation - utility - along with a model of consumer budget, we are able to motivate the conditions for optimal consumer choice. At its core, economists at all levels motivate their model of optimal consumer choice with the same phrase, from introductory college courses through PhD courses: a consumer aims to maximize her utility subject to her budget constraint. The idea, of course, is that given the limited amount of resources the consumer has (her budget constraint), a rational consumer should choose the bundle which provides her with the highest level of satisfaction - that is, maximize her utility.
Figure2.5.1.Visualizing the utility maximization problem for consumers. Left: each bundle gives a level of utility to the consumer. Right: The consumer’s problem is to choose the bundle with the highest utility value within her (marked in red) budget set.
So, how will we be able to identify an \((x, y)\) bundle which maximizes a consumer’s utility subject to their budget? We will need to combine two mathematical conditions. An optimal bundle, which we will denote by \((x^*, y^*)\text{,}\) will satisfy both of the conditions, while a suboptimal bundle will fail to satisfy at least one of the conditions. What are our two optimal bundle conditions?
Condition 1 is all about affordability: the budget constraint. An optimal bundle \((x^*, y^*)\) should be an affordable bundle, which means, at the very least, \(P_x x^* + P_y y^* \leq I\text{.}\) But let’s take this one step further. Would it be optimal for a consumer to spend less than her income and choose a bundle in the interior of her budget set? Well, under the assumption of monotonicity (A3), if having more cannot make the consumer worse off, then the consumer should make sure to spend all of her income, because those additional goods will give her at least as much satisfaction as she had before - if not more. This means the consumer should choose a bundle along the boundary of the budget set: our first optimal bundle condition is that at \((x^*, y^*)\text{,}\)
\begin{equation*}
P_x x^* + P_y y^* = I
\end{equation*}
Figure2.5.2.Condition 1: choose a bundle along the boundary of the budget set, where \(P_x x^* + P_y y^* = I\text{.}\)
Since the first condition is purely about affordability, surely the second condition must incorporate the consumer’s preferences. If our consumer is indeed rational, she would ask: Out of all of the bundles along my budget constraint, which one do I prefer the most? If her budget constraint is \(x + 2y = 4\text{,}\) should she choose \((4, 0)\text{?}\)\((2, 1)\text{?}\)\((3, \frac{1}{2})\text{?}\) Conceptually, the second condition must help us identify her most preferred affordable choice. But how can we capture this with a mathematical condition?
We will frame our analysis by looking at tradeoffs: when would our consumer be better off trading one unit of \(x\) for one unit of \(y\text{,}\) moving from \((4, 0)\) to \((3, 1)\text{,}\) for example? In part, we will want to examine the marginal utility of each good. How much additional utility will she gain from that additional unit of \(y\text{?}\) How much utility will she lose by trading away one more unit of \(x\text{?}\) But remember - trades here occur at a rate determined by the prices of the goods! While the consumer might gain more utility from trading 1 \(x\) for 1 \(y\text{,}\) the trade might not be worth it if \(P_y\) is very high and units of \(y\) are relatively expensive compared to units of \(x\text{!}\)
With this in mind, Condition 2 balances the consumer’s willingness to trade \(x\) for \(y\text{,}\) considering both her marginal utility of each good and the relative prices, by defining the marginal utility per dollar of a good, defined as \(\frac{MU_x}{P_x}\text{.}\) While the marginal utility of good \(x\) gives the added utility the consumer receives from consuming one more unit of good \(x\text{,}\) the marginal utility per dollar of good \(x\) gives the added utility the consumer receives from spending one more dollar on good \(x\). In this way, the expression balances the tradeoff between preferences and budget.
then the marginal utility for spending another dollar on \(x\) is larger than the marginal utility for spending another dollar on \(y\text{.}\) This means if the consumer reallocated her budget and spent one more dollar on \(x\) and one less dollar on \(y\text{,}\) then (1) her utility would go down from the decrease in \(y\text{,}\) (2) her utility would go up from the increase in \(x\text{,}\) and (3) she would gain more than she’d lose - due to the \(>\) sign. This means the reallocation would make her better off, invalidating the current bundle as an optimal bundle. In general, if she would gain more utility than she’d lose from a reallocation of her dollars, her current choice cannot be optimal!
Based on this rationale, we can fully characterize our Condition 2: at an optimal bundle \((x^*, y^*)\text{,}\) the marginal utility per dollar of \(x\) should be exactly equal to the marginal utility per dollar of \(y\text{.}\) Formally, Condition 2 can be written as
Recall from Section 2.3 that marginal utility is the change in utility when a consumer changes consumption. Another way to see this is as the utility from the last unit in a bundle. Use the table of \(MU\) for donuts below to answer the questions which follow.
Table2.5.4.Marginal utility for donuts
number of donuts
marginal utility
1
35
2
25
3
18
4
12
5
8
1. If the price of donuts is $1.50, what is the marginal utility per dollar of the 3rd donut? the 4th?
2. Explain in words what this means if a consumer chooses to buy a bundle that is on her budget constraint containing four donuts.
Answer.
1. Marginal utility per dollar is just \(MU/P\text{,}\) so for Q=3 she gets \(\frac{18}{1.5}=12\) and for Q=4 she gets \(\frac{12}{1.5}=8\)
2. If she is choosing to buy a bundle with four donuts, the last dollar she spent on donuts gave her 8 units of utility. If this were her optimal bundle, all other goods in the bundle would also have given 8 units of utility per dollar. Otherwise, she could have increased her utility by spending a dollar on a good or service that brought her more additional utility (higher \(\frac{MU}{P}\)).
The theory of optimal choice for consumers weighs tradeoffs across goods, identifying any bundle \((x^*, y^*)\) which satisfies
as a bundle which will maximize the consumer’s utility subject to her budget. To see how this plays out, let’s look at an example.
Consider a consumer choosing bundles of donuts (\(x\)) and coffee (\(y\)). Let the price of a donut as \(P_x = 1\text{,}\) the price of a cup of coffee as \(P_y = 2\text{,}\) and consumer income as \(I = 12\text{.}\) Let \(MU_x = \frac{1}{x}\) and \(MU_y = \frac{1}{y}\text{;}\) as we have seen above, these functions satisfy both A3 and A4. Given this information, can we determine which bundle \((x^*, y^*)\) is optimal for our consumer? Condition 1 says that it should be a bundle where the consumer spends all of her income:
At this point, we have a problem with two equations and two unknowns. Such problems can be solves in a multitude of ways, but the most reliable approach is to use the substitution method; it substitutes one of the two equations into the other and allows us to solve for one of our two variables. Let’s start with condition 2. From the equation
From here, we can substitute \(x = 2y\) into our budget constraint (condition 1) and solve for \(y\text{:}\)
\begin{equation*}
x + 2y = 12
\end{equation*}
\begin{equation*}
(2y) + 2y = 12
\end{equation*}
\begin{equation*}
4y = 12
\end{equation*}
\begin{equation*}
y^* = 3
\end{equation*}
Finally, since \(x = 2y\) and \(y^* = 3\text{,}\) we can plug back in to get \(x^* = 6\text{.}\) This makes the consumer’s optimal bundle \((x^*, y^*) = (6, 3)\text{:}\) 6 donuts and 3 coffees. Now that we have done the math, we can interpret our result. Why would it be optimal for our consumer to buy twice as many donuts as coffee? Notice first that since \(MU_x = \frac{1}{x}\) and \(MU_y = \frac{1}{y}\text{,}\) the consumer has identical preferences for the two goods. The first donut and the first coffee are worth the same; so are the fourth donut and the fourth coffee. So, why not optimally buy the same number of each? Because \(P_y\) is twice as high as \(P_x\text{!}\) Since \(y\) is twice as expensive as \(x\text{,}\) the consumer ideally wants twice as many units of \(x\) in her optimal bundle!
To see a second example, consider a choice between pens \(x\) and pencils \(y\text{.}\) Let \(P_x = 1\text{,}\)\(P_y = 1\text{,}\) and \(I = 10\text{.}\) Here, let \(MU_x = 6 - x\) and \(MU_y = 8 - y\text{.}\) For this problem, the two goods have the same price, but the consumer values them differently. The first pen adds utility of 5, while the first pencil adds utility of 7. This means our consumer has a stronger preference for pencils than pens: she values her third pencil the same as her first pen. What is the optimal bundle?
Now, through substitution, the budget constraint becomes
\begin{equation*}
x + y = 10
\end{equation*}
\begin{equation*}
x + (2 + x) = 10
\end{equation*}
\begin{equation*}
2x = 8
\end{equation*}
\begin{equation*}
x^* = 4
\end{equation*}
And, by substitution into \(y = 2 + x\text{,}\)\(y^* = 6\text{.}\) The consumer’s optimal bundle therefore is \((x^*, y^*) = (4, 6)\) This consumer chooses more pencils than pens despite them having the same price, because she has a stronger preference for pencils!
At this point, it is worth reflecting on the assumptions of monotonicity (A3) and diminishing marginal utility (A4). As we have discussed, neither of these assumptions is required under the assumption of rationality. Moreover, there are clear examples where each of these assumptions does not hold true. If this is the case, why would we ever rely on assumptions of monotonicity or diminishing marginal utility when studying consumer choice?
First, let’s look at monotonicity. A consumer preference can fail to satisfy monotonicity if additional units of a good make a consumer worse off, bending the consumer’s utility function downward. Since we have seen examples of situations where additional units clearly make consumers worse off (such as buying an excessive quantity of a good you have no room to store), don’t we lose some realism by assuming monotonicity?
With the utility maximization problem in mind, consider: would it ever be optimal for a rational consumer to choose a point where her utility function is decreasing? In other words, would a rational consumer ever willingly spend money on a unit of a good which will make her worse off? She certainly would not! Given the consumer’s budget constraint, spending a dollar to receive a lower level of utility is not rational - the consumer would be better off taking the dollar spent on the good whose utility is decreasing and re-allocating that dollar to a good with positive marginal utility. Therefore, if some quantity of a good is on a section of utility where monotonicity fails, this quantity cannot be part of a utility-maximizing bundle.
A rational consumer will never choose optimally on the far right of the utility function.
The importance of the assumption of monotonicity here depends heavily on the objective of the model to which it is applied. If the primary purpose of the model is to explore realistic characteristics of consumers preferences, then allowing for non-monotonic preferences can be important. However, if the primary purpose of the model is to understand consumer choice and utility maximizing bundle selection, we can be confident that we are not missing any viable options by assuming monotonicity.
What about diminishing marginal utility? Concave utility captures goods for which additional units add less and less utility as more units are acquired. But there are examples, such as addictive goods or collectibles, where additional units add more utility as more are acquired. What does diminishing marginal utility get us?
The consequences of this result cannot be fully analyzed in an introductory-level course, but Assumption A4 gets us the following: if a consumer’s utility function is (strictly) concave, then there is at most one optimal bundle for the consumer to choose. While we will leave the mathematical details to more advanced course 2 , this result is about ensuring there is only one solution to the problem. The assumption of diminishing marginal utility plays a role in both describing consumer behavior and clarifying the nature of the predictions of consumer choice models.
