in deciding what to buy to maximize utility

Learning Objectives

• Explain why maximizing utility requires that the terminal unit of each item purchased must have the same marginal utility per dollar
• Summate the utility-maximizing choice

The trouble of finding consumer equilibrium, that is, the combination of goods and services that will maximize an private's total utility, comes down to comparison the merchandise-offs between i affordable combination (shown past a bespeak on the budget line in Figure ane, below) with all the other affordable combinations.

Most people approach their utility-maximizing combination of choices in a stride-by-footstep manner. This step-by-footstep approach is based on looking at the tradeoffs, measured in terms of marginal utility, of consuming less of one adept and more than of some other. You can think of this step-by-step approach every bit the "biggest bang for the cadet" principle. For instance, say that José starts off thinking almost spending all his money on T-shirts and choosing point P, which corresponds to 4 T-shirts and no movies, as illustrated in Figure 1.

Figure 1. A Choice between Consumption Goods. José has income of $56. Movies cost $seven and T-shirts cost $14. The points on the budget constraint line show the combinations of movies and T-shirts that are affordable.

José chooses this starting indicate randomly; he has to showtime somewhere. Then he considers giving upwards the last T-shirt, the ane that provides him the least marginal utility, and using the coin he saves to buy ii movies instead. Table 1 tracks the step-by-step series of decisions José needs to make ( Cardinal : T-shirts toll $14, movies cost $vii, and Jose's income is $56).

Tabular array 1. A Step-by-Step Approach to Maximizing Utility

Effort	Which Has	Total Utility	Marginal Gain and Loss of Utility, Compared with Previous Selection	Conclusion
Choice 1: P	4 T-shirts and 0 movies	81 from iv T-shirts + 0 from 0 movies = 81	–	–
Choice 2: Q	3 T-shirts and two movies	63 from 3 T-shirts + 31 from 0 movies = 94	Loss of 18 from 1 less T-shirt, but gain of 31 from 2 more movies, for a internet utility gain of 13	Q is preferred over P
Pick 3: R	2 T-shirts and 4 movies	43 from 2 T-shirts + 58 from iv movies = 101	Loss of 20 from 1 less T-shirt, but gain of 27 from two more than movies for a net utility proceeds of vii	R is preferred over Q
Selection 4: S	1 T-shirt and 6 movies	22 from 1 T-shirt + 81 from 6 movies = 103	Loss of 21 from 1 less T-shirt, but gain of 23 from two more movies, for a internet utility proceeds of 2	S is preferred over R
Choice 5: T	0 T-shirts and 8 movies	0 from 0 T-shirts + 100 from 8 movies = 100	Loss of 22 from one less T-shirt, only proceeds of nineteen from two more movies, for a net utility loss of 3	Southward is preferred over T

Decision MAKING BY Comparing MARGINAL UTILITY

José could apply the following thought process (if he thought in utils) to make his decision regarding how many T-shirts and movies to purchase:

Step 1. From Tabular array one, José can run into that the marginal utility of the 4th T-shirt is 18. If José gives up the fourth T-shirt, so he loses 18 utils.

Footstep 2. Giving upward the quaternary T-shirt, nevertheless, frees up $14 (the price of a T-shirt), assuasive José to buy the showtime 2 movies (at $7 each).

Step iii. José knows that the marginal utility of the kickoff movie is 16 and the marginal utility of the 2nd movie is fifteen. Thus, if José moves from point P to bespeak Q, he gives upwardly 18 utils (from the T-shirt), but gains 31 utils (from the movies).

Footstep 4. Gaining 31 utils and losing 18 utils is a net gain of 13. This is just some other way of maxim that the total utility at Q (94 co-ordinate to the last column in Table 1) is 13 more than than the full utility at P (81).

Stride 5. So, for José, it makes sense to give up the fourth T-shirt in order to buy two movies.

José clearly prefers point Q to point P. At present repeat this step-by-footstep procedure of decision making with marginal utilities. José thinks nearly giving up the 3rd T-shirt and surrendering a marginal utility of twenty, in commutation for purchasing two more movies that promise a combined marginal utility of 27. José prefers point R to point Q. What if José thinks about going beyond R to bespeak S? Giving up the second T-shirt ways a marginal utility loss of 21, and the marginal utility gain from the fifth and sixth movies would combine to make a marginal utility gain of 23, so José prefers point Southward to R.

However, if José seeks to go beyond point South to betoken T, he finds that the loss of marginal utility from giving up the first T-shirt is 22, while the marginal utility gain from the last two movies is only a total of 19. If José were to choose betoken T, his utility would fall to 100. Through these stages of thinking about marginal tradeoffs, José over again concludes that Due south, with 1 T-shirt and 6 movies, is the choice that volition provide him with the highest level of full utility. This footstep-by-footstep approach will reach the aforementioned conclusion regardless of José's starting point.

This approach to finding consumer equilibrium is somewhat tedious. Next, we'll turn to a quicker and more intuitive approach.

Maximizing Utility Rule

This procedure of conclusion making described previously suggests a rule to follow when maximizing utility. Since the price of T-shirts is not the aforementioned as the price of movies, it'south not enough to just compare the marginal utility of T-shirts with the marginal utility of movies. Instead, we need to control for the prices of each product. We can do this past computing and comparison marginal utility per dollar of expenditure for each product.Marginal utility per dollar is the corporeality of additional utility José receives given the cost of the product.

[latex]\brainstorm{array}{rcl}\text{marginal utility per dollar}& =& \frac{\text{marginal utility}}{\text{price}}\end{assortment}[/latex]

For José'south T-shirts and movies, the marginal utility per dollar is shown in Table 2.

Table two. Marginal Utility per Dollar
Quantity of T-Shirts	Total Utility	Marginal Utility	Marginal Utility per Dollar	Quantity of Movies	Total Utility	Marginal Utility	Marginal Utility per Dollar
one	22	22	22/$fourteen=1.6	1	16	sixteen	xvi/$vii=2.3
2	43	21	21/$14=1.5	2	31	15	15/$7=two.14
3	63	20	twenty/$14=1.4	3	45	xiv	fourteen/$7=2
4	81	18	18/$14=one.3	4	58	13	13/$7=1.9
5	97	16	xvi/$14=1.i	five	70	12	12/$vii=i.seven
6	111	fourteen	xiv/$14=i	6	81	11	11/$seven=i.6
7	123	12	12/$14=0.86	vii	91	10	x/$seven=one.4

A Rule for maximizing Utility

If a consumer wants to maximize total utility, for every dollar that they spend, they should spend it on the detail which yields the greatest marginal utility per dollar of expenditure.

Applying the Rule

José'south start buy volition be a picture. Why? José's choices are to purchase either a T-shirt or a moving-picture show. Table ane shows that the marginal utility per dollar spent on the first T-shirt is i.6 compared with 2.5 for the first moving-picture show. Because the first movie gives José more marginal utility per dollar than the get-go T-shirt, and because the movie is inside his upkeep, he will purchase a flick first.

José volition go on to purchase the good which gives him the highest marginal utility per dollar until he exhausts the budget. José will keep purchasing movies because they give him a greater "blindside for the cadet" until the 6th picture is equivalent to a T-shirt purchase. José can afford to purchase that T-shirt. So José will choose to buy six movies and one T-shirt. That combination, six movies and one T-shirt, is his consumer equilibrium

Another Dominion for Maximizing Utility

Since the price of T-shirts is twice as high as the price of movies, to maximize utility the last T-shirt called needs to provide exactly twice the marginal utility (MU) of the last movie. If the concluding T-shirt provides less than twice the marginal utility of the concluding moving picture, then the T-shirt is providing less "blindside for the buck" (i.e., marginal utility per dollar spent) than if the aforementioned coin were spent on movies. If this is so, José should trade the T-shirt for more movies to increase his full utility. Marginal utility per dollar measures the additional utility that José will relish given what he has to pay for the good.

If the last T-shirt provides more twice the marginal utility of the last film, then the T-shirt is providing more than "bang for the buck" or marginal utility per dollar, than if the money were spent on movies. As a issue, José should buy more T-shirts. Notice that at José's optimal choice the marginal utility from the outset T-shirt, of 22 is exactly twice the marginal utility of the sixth motion picture, which is eleven. At this choice, the marginal utility per dollar is the same for both goods. This is a tell-tale bespeak that José has plant the point with highest total utility.

This argument tin be written every bit another rule: the utility-maximizing pick between consumption goods occurs where the marginal utility per dollar is the same for both goods, and the consumer has wearied his or her budget.

[latex]\displaystyle\frac{MU_1}{P_1}=\frac{MU_2}{P_2}[/latex]

A sensible economizer will pay twice every bit much for something simply if, in the marginal comparing, the detail confers twice as much utility. Find that the formula for the table above is

[latex]\displaystyle\frac{22}{fourteen}=\frac{11}{seven}[/latex]

[latex]one.6=1.6[/latex]

The following feature provides step-by-step guidance for this concept of utility-maximizing choices.

Another Rule for Maximizing Utility

The rule, [latex]\displaystyle\frac{MU_1}{P_1}=\frac{MU_2}{P_2}[/latex], means that the terminal dollar spent on each good provides exactly the same marginal utility. So:

Step 1. If we traded a dollar more of movies for a dollar more of T-shirts, the marginal utility gained from T-shirts would exactly first the marginal utility lost from fewer movies. In other words, the net gain would be zero.

Step ii. Products, however, usually toll more than a dollar, then we cannot trade a dollar'due south worth of movies. The all-time we can practise is trade 2 movies for some other T-shirt, since in this case T-shirts cost twice what a movie does.

Stride 3. If we trade two movies for 1 T-shirt, we would terminate upwardly at indicate R (2 T-shirts and four movies).

Stride four. Pick 4 in Table iii shows that if we movement to point S, nosotros would lose 21 utils from i less T-shirt, but gain 23 utils from two more than movies, and then we would end up with more total utility at signal Southward.

Table iii. A Step-by-Step Approach to Maximizing Utility
Try	Which Has	Total Utility	Marginal Gain and Loss of Utility, Compared with Previous Option	Conclusion
Selection ane: P	4 T-shirts and 0 movies	81 from four T-shirts + 0 from 0 movies = 81	–	–
Choice two: Q	3 T-shirts and 2 movies	63 from 3 T-shirts + 31 from 0 movies = 94	Loss of eighteen from one less T-shirt, but gain of 31 from 2 more movies, for a internet utility proceeds of 13	Q is preferred over P
Option 3: R	2 T-shirts and 4 movies	43 from ii T-shirts + 58 from 4 movies = 101	Loss of 20 from ane less T-shirt, simply gain of 27 from 2 more movies for a net utility gain of seven	R is preferred over Q
Selection 4: S	1 T-shirt and half-dozen movies	22 from one T-shirt + 81 from 6 movies = 103	Loss of 21 from 1 less T-shirt, but proceeds of 23 from ii more movies, for a net utility gain of 2	S is preferred over R
Choice 5: T	0 T-shirts and eight movies	0 from 0 T-shirts + 100 from 8 movies = 100	Loss of 22 from one less T-shirt, but gain of 19 from two more movies, for a internet utility loss of three	S is preferred over T

In short, the rule shows usa the utility-maximizing selection.

At that place is another, equivalent way to think about this. The dominion can too exist expressed as the ratio of the prices of the two appurtenances should exist equal to the ratio of the marginal utilities. When the toll of good 1 is divided by the cost of good ii, at the utility-maximizing signal this volition equal the marginal utility of practiced 1 divided by the marginal utility of skillful ii. This rule can be written in algebraic class:

[latex]\displaystyle\frac{P_1}{P_2}=\frac{MU_1}{MU_2}[/latex]

Forth the budget constraint, the total toll of the two goods remains the same, and so the ratio of the prices does non change. Still, the marginal utility of the two goods changes with the quantities consumed. At the optimal choice of one T-shirt and six movies, point S, the ratio of marginal utility to price for T-shirts (22:14) matches the ratio of marginal utility to price for movies (of 11:7).

Sentry Information technology

This video applies these same concepts to a graph. It demonstrates how in that location will come up a point when the marginal utility per dollar of one skilful goes down enough then that it makes more sense to purchase another skillful.

Measuring Utility with Numbers

This discussion of utility started off with an assumption that it is possible to identify numerical values on utility, an supposition that may seem questionable. You tin can buy a thermometer for measuring temperature at the hardware store, simply what store sells an "utilimometer" for measuring utility? However, while measuring utility with numbers is a convenient assumption to clarify the explanation, the key assumption is not that utility tin be measured by an outside party, but only that individuals tin can decide which of two alternatives they prefer.

To understand this point, call up back to the step-by-step procedure of finding the choice with highest full utility by comparing the marginal utility that is gained and lost from different choices along the budget constraint. As José compares each choice along his upkeep constraint to the previous choice, what matters is not the specific numbers that he places on his utility—or whether he uses any numbers at all—but only that he personally can place which choices he prefers.

In this style, the footstep-by-step procedure of choosing the highest level of utility resembles rather closely how many people brand consumption decisions. We think about what volition brand the states the happiest; we think nearly what things cost; nosotros think about buying a little more than of i item and giving up a little of something else; we choose what provides usa with the greatest level of satisfaction. The vocabulary of comparing the points along a budget constraint and total and marginal utility is only a set of tools for discussing this everyday process in a articulate and specific way. It is welcome news that specific utility numbers are not cardinal to the argument, since a good utilimometer is difficult to find. Do not worry—while we cannot measure out utils, past the finish of the next module, we volition have transformed our analysis into something nosotros can measure—need.

Try It

Try It

These questions allow y'all to get as much practice as y'all need, as you can click the link at the summit of the first question ("Effort another version of these questions") to become a new gear up of questions. Practice until you feel comfy doing the questions.

Try It

These questions allow yous to get as much exercise every bit you demand, equally you lot tin click the link at the summit of the first question ("Try some other version of these questions") to get a new ready of questions. Practice until you lot feel comfortable doing the questions.

Glossary

consumer equilibrium:

the combination of appurtenances and services that volition maximize an private's full utility

marginal utility per dollar:

the additional satisfaction gained from purchasing a good given the toll of the product; MU/Cost

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Source: https://courses.lumenlearning.com/wm-microeconomics/chapter/rules-for-maximizing-utility/

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