WRITTEN REPORT
IN BASIC MICROECONOMICS
CONSUMER CHOICE AND UTILITY MAXIMIZATION
GROUP 3
Pamantasan ng Lungsod ng Muntinlupa, College of Business Administration
BSBA2OCT-3
Torres, Rochena Julifer
Epanis, Dicerie Nicole
Radan, Marive
Roxas, Marcon
Espino, Frederick Loren
Mr. Mhandy Asiado
February, 2023
We as Consumers, We, have different wants and demands, We, differ like and dislike
standard. However, our behavior as a consumer is hard to determine and measure.
From an Economic stand point, our objective as consumers is to maximize our
satisfaction given our limited budget or income.
Here in our report we will learn how we, as consumers behave in order to
maximize our satisfaction on the goods and services that we consume given our limited
income. Before we proceed of consumer behavior, We, have to know who is a
consumer? Consumer is one who demands and consume goods and services,
Consumer is the king pin of the market. In a market economy, prices of the commodities
in the market are affected by the forces of demand and supply which depends upon the
purchasing power of the consumer that generates open competition in the market.
Therefore, consumers are also known as sovereign in the market. In general terms, if
we as consumers, demand more of a good or service, then more of it will be supplied.
Now the producers simply obey the wishes and desires as well as needs and wants of
consumer. Nevertheless, our freedom to satisfy our human wants is not completely
unlimited. For the good of society and the individual consumers, the government
restricts consumer sovereignty. For Example, the government prohibits the use of
dangerous drugs and substances and regulates the use of products that are health
hazards like alcoholic beverages and cigarettes. It also regulates products that are
destructive to the environment like the use of leaded gasoline.
CONSUMER BEHAVIOR
The Consumer behavior involves the use and disposal of products as well as the
study of how they are purchased. This talks about how these individuals or group, as a
consumers deal with what to buy, where they buy goods and services, how they buy
and how much they buy them, when they buy and the reason or why they buy.
According to the economist Alfred Marshall The theory of consumer behavior is based
on the assumption that an individual, as rational buyer, has the perfect information
about the market and is fully aware of his needs and can determine full satisfaction.
This theory asserts that the consumer as a rational person spend his or her income on
goods and services that yield or would give, the highest level of satisfaction. Here in this
topic it is important to study the consumers and how they behave to help firms sustain
and improve their operation and production. The ability to understand the consumers
would allow them to produce better goods and services that can help create better
economic conditions for consumer as well.
GOOD AND SERVICES
In Good and Services it is important to clarify what goods and services are. So
Goods it refers to anything that provides satisfaction of the consumer like needs, wants
and desires that they can be any tangible economic products, For example is Cars,
Clothes, or any gadgets. Now in services on the other hand are any intangible economic
activities for example Hairdressing, telecommunications etc, that likewise contribute
directly or indirectly to the satisfaction of human wants.
It says Tangible Goods can be classified according to, but not limited to, the following
are:
CONSUMER GOODS
These are the goods that yield satisfaction directly to any consumer. These
Goods are primarily sold for consumption, and not to be used for further processing or
as an input or raw material needed in producing another good. Examples of consumer
goods Food, personal-care items, entertainment, and clothing. it also includes services
like dry cleaning, auto repairs and so on. All goods that are durable or nondurable
maintained for consumption in any form but not reproduction or reselling for any other
intention.
ESSENTIAL OR NECESSITY GOODS VS. LUXURY GOODS.
As we study consumer behavior it’s important to reiterate what these goods are,
Essential or necessity goods are goods that satisfy the basic needs of man. In other
words, these are goods that are necessary in our daily existence as human beings.
These are also goods that we cannot live without such as food, water, shelter, clothing,
electricity etc.
Conversely, luxury goods is that which is both useful and scarce. It has value
attached to it and a price has to be paid for its use. If a good is so Abundant that there is
enough of it to satisfy everyone’s needs without anybody paying for it, that good is free
Water from our faucet is an economic good because we are not utilizing it for
free, we have to pay its distributor. The air that we breathe and the sunlight coming from
the sun are examples of free goods
TASTE AND PREFERENCES
The Consumers have various tastes and preferences. Generally taste and
preferences are determined by age, income, education, gender, occupation customs,
and tradition as well as culture. Preferences are the choices made by us consumers as
to which product or services to consume. The strength of our preferences will determine
which products to buy given our limited disposable income thus the demand of products
to buy. We us consumers also express preferences to which particular brand of a
product to purchase. Even the choice of food, clothing, shelter, for instance, we differ in
our choices and preferences. Some prefer bread than rice other like fish and vegetable
than meat. In fact we can generalize that no two consumers have exactly the same likes
and dislikes.
MASLOW’S HIERARCHY OF NEEDS
The original hierarchy of needs five-stage model includes:
Maslow (1943, 1954) stated that people are motivated to achieve certain needs and that
some needs take precedence over others.
Our most basic need is for physical survival, and this will be the first thing that motivates
our behavior. Once that level is fulfilled the next level up is what motivates us, and so
on.
1. Physiological needs – these are biological requirements for human survival,
e.g. air, food, drink, shelter, clothing, warmth, sex, sleep.
If these needs are not satisfied the human body cannot function optimally. Maslow
considered physiological needs the most important as all the other needs become
secondary until these needs are met.
2. Safety needs – once an individual’s physiological needs are satisfied, the needs
for security and safety become salient. People want to experience order,
predictability and control in their lives. These needs can be fulfilled by the family
and society (e.g. police, schools, business and medical care).
For example, emotional security, financial security (e.g. employment, social welfare),
law and order, freedom from fear, social stability, property, health and wellbeing (e.g.
safety against accidents and injury).
3. Love and belongingness needs – after physiological and safety needs have
been fulfilled, the third level of human needs is social and involves feelings of
belongingness. Belongingness, refers to a human emotional need for
interpersonal relationships, affiliating, connectedness, and being part of a group.
Examples of belongingness needs include friendship, intimacy, trust, and acceptance,
receiving and giving affection, and love.
4. Esteem needs – are the fourth level in Maslow’s hierarchy and include selfworth, accomplishment and respect. Maslow classified esteem needs into two
categories: (i) esteem for oneself (dignity, achievement, mastery, independence)
and (ii) the desire for reputation or respect from others (e.g., status, prestige).
Maslow indicated that the need for respect or reputation is most important for children
and adolescents and precedes real self-esteem or dignity.
5. Self-actualization – needs are the highest level in Maslow’s hierarchy, and refer
to the realization of a person’s potential, self-fulfillment, seeking personal growth
and peak experiences. Maslow (1943) describes this level as the desire to
accomplish everything that one can, to become the most that one can be.
Individuals may perceive or focus on this need very specifically. For example, one
individual may have a strong desire to become an ideal parent. In another, the
desire may be expressed economically, academically or athletically. For others, it
may be expressed creatively, in paintings, pictures, or inventions.
THE ECONOMICS OF SATISFACTION
You may be wondering by now how economics can explain the behavior of consumers
in order to attain maximum level of satisfaction on the goods and services that they
consume. In this section, we’re going to explain how consumers attain maximum
satisfaction level on the many goods and services available for their consumption. But,
we have to remember that every consumer has different preferences on attaining their
satisfaction to their highest level so every one of us has different likes and dislikes. This
section will discuss to you some of the theories that economists have devised to explain
how consumers are able to attain maximum level of satisfaction when consuming a
particular good or services
UTILITY THEORY
Utility, in economics, refers to the satisfaction or pleasure that an individual or consumer
gets from the consumption of a good or service that he or she purchases. For purposes
of economic analysis, utility is also measured by how much a consumer is willing to pay
for particular goods or services that means that utility is a satisfaction or pleasure of a
consumer so whatever the price of a certain products or services as long as the
customer is satisfied then he/she will buy the products or services.
The utility function is based on a consumer's satisfaction. It assesses how much one
loves purchasing something. A utility is a measure of how much one loves a certain
movie, meal, or other item. It fluctuates according on the level of desire. The following
conclusions can be drawn:
•
•
•
The utility of a good varies from one customer to the next.
It changes for the same customer as the number of desires changes.
It should not be confused with its usefulness.
Utility Characteristics
•
•
•
•
•
It is determined by human desires.
That is incalculable.
A utility is a subjective concept.
It is determined by knowledge.
The utility of anything is determined by how it is used.
•
•
It is entirely subjective.
It is determined by ownership.
Measurement of utility
The measurement of a utility aids in the analysis of a customer’s demand behavior. It is
assessed in two ways.
CARDINAL APPROACH
This technique assumes that it is
quantifiable. Cardinal numbers, or
quantitative numbers such as 1, 2, 3,
and so on, can be used to indicate one’s
contentment. It reveals a customer’s
preference in cardinal measurement. It
is calculated in utils.
ORDINAL APPROACH
This technique implies that it is similar.
Ranking allows one to express
satisfaction. One can compare goods and assign them ranks such as first,
second, tenth, and so forth. It displays the preferred order. An ordinal approach is
a qualitative method for calculating utility.
Price represents
the utility or the
satisfaction of the
consumer
Quality demand is
represents how much
the consumer,
consumes
Table 4.1 presents a hypothetical demand schedule for siopao. The chart shows that
the amount of money you are willing to pay for an additional unit of siopao decreases.
What is the reasoning for this? As you may have seen, the more siopao you eat, the
more full you get, to the point that you are unwilling to spend more money on the next
siopao you want to consume. In other words, the enjoyment or utility gained from
consuming an extra siopao decreases as you eat more and more of it. The hypothetical
example that we have illustrated is what the utility theory is all about, It simply tries to
explain how our satisfaction or utility as consumers decline when we try to consume
more and more of the same good at a particular point in time.
They’re 2 important concepts that should be explain before we fully understand the
utility theory
1. Marginal utility
2. Total utility
MARGINAL UTILITY
Marginal utility is defined as the additional satisfaction that an individual derives from
consuming an extra unit of a good or service. Marginal means 'additional' or 'extra'.
In economics, we use marginal analysis to investigate the impact of adding one more
unit to, or subtracting one unit from, some economic variable. We are interested in the
incremental or extra utility received from greater use of a commodity for this purpose.
So, the marginal utility of a commodity is the gain in overall utility or satisfaction
resulting from the consumption of an additional or extra unit of such commodity; it is the
loss of utility or satisfaction resulting from the consumption of one unit less. In other
words, it is the change in total utility caused by an increase in the quantity of a good
consumed by one unit.
In other words the marginal utility is the added satisfaction that a consumer gets form
having a more units of goods or services
TOTAL UTILITY
Total utility, is the total satisfaction that a consumer derives from the consumption of a
given quantity of a good or service in a particular time period. Total utility may also be
defined as the whole benefit derived by a person through the consumption of an item or
service. Overall utility is proportional to the quantity of the good consumed; more
consumption typically results in higher total utility. As a result, our overall utility often
grows when we use more and more of an item or service, albeit at a slower or falling
pace. This means that when we get saturated with the commodity or service we are
consuming, each additional unit eaten contributes less and less marginal benefit than
the prior units spent.
Let us illustrate this using the hypothetical utility schedule presented in Table 4.2
Assume that at the end of our class, you are too hungry so that you went directly to the
cafeteria. In the cafeteria, you bought and consumed one siopao for your merienda. In
this case, your total and marginal utilities are 40 utils. Assume further that you
consumed another siopao because you are too hungry after the class. Your total utility
now increases to 90 utils so that marginal utility increases by 50 utils. Let us now
assume that you have consumed five siopao. Take note in that table that your total
utility for the fifth unit is 350 utils. However what is more important is the marginal utility.
As we can observe, marginal utility has declined to only 80 utils. Why is this so? This is
because of the Law of Diminishing Marginal Utility.
DIMINISHING MARGINAL UTILITY.
This Law states that as a consumer gets more satisfaction in the long run, he or she
experiences a decline in his satisfaction for goods and services.
This means that the more we consume a goods or services the more we get a less
marginal utility or an extra satisfaction or utilities.
MATHEMATICAL DERIVATION OF MARGINAL UTILITY
How do we derive marginal utility? Marginal utility is simply the change in total utility
divided by the change in quantity. Thus,
MU (marginal utility) =
Delta TU (total utility)
divided by Delta Q
(quality consumer)
Where:
Applying the formula, we can derive the marginal utility for the total utility presented in
Table 4.2. Thus, if we want to determine the marginal utility from the consumption of two
pieces of siopao to three pieces, we can simply apply the formula presented above.
MU = 170-90
3-2
MU = 80
GRAPHICAL ILLUSTRATION OF TU AND MU
We have already discussed the concepts behind total utility and marginal utility. Now,
we are ready to analyze total utility and marginal utility in graphical form. Figures 4.1
and 4.2 are graphical presentations of the total and marginal utility concepts
We mentioned this before. As we have previously discussed, when we use more and
more of an item or service, our overall utility rises but at a decreasing pace.
Figure 4.1 illustrates this notion. Our total utility curve, as you may have seen, is
concave from the origin. The curve, as illustrated by the dotted lines, shows that the
initial consumption of 1 unit of Q leading to 2 units of Q has a greater angle than the
subsequent consumption (from 2 units to 3 units). As a result, we may conclude that the
consumption of 1 unit of Q to 2 units of Q has a total utility of 10 utils, or a 5 utils
incremental gain, While the next consumption only grew to 11 utils, or an additional 1
util, the previous consumption only increased to 11 utils.
The marginal utility curve illustrated in Figure 4.2 is a convex curve. This is because
marginal utility starts at high level but as consumption of the same good increases,
marginal utility declines. Like I've said the more that you consume a good or services
the more that it added a marginal utility but it also declines because we are already
satisfied of what we have consumed
MAXIMIZING TOTAL UTILITY
When do we stop consuming a goods or services? that’s when maximizing Total Utility
came, we are given the opportunity to maximizes our satisfaction by continuously
consuming more units of asserting good until our satisfaction falls down to zero, when
we reach the peak of our satisfaction that’s when the marginal utility falls down to 0
(1) allocate the entire available budget, (it is to maximize the total utility)
(2) make the marginal utility per peso spent the same for all goods. (It means that
we have to spend our satisfaction in the same goods or service)
CONSUMER SURPLUS
- Measure of the welfare we gain firm the consumption
of goods and services or a measure of benefits that we derived from the exchange of
goods.
Ex. Supposed you are interested in buying a new pair of maong pants. So you went to
the mall and look for the pants you wanted so much. Your budget is P3,000.00. So you
went to the boutique you found, but then the pants you like most cost only P2,500.00.
Immediately you bought the pants.
Because Consumer surplus, also known as buyer’s surplus, is the economic measure of
a customer’s excess benefit. It is calculated by analyzing the difference between the
consumer’s willingness to pay for a product and the actual price they pay, also known
as the equilibrium price. A surplus occurs when the consumer’s willingness to pay for a
product is greater than its market price.
While taking into consideration the demand and supply curves, the formula for
consumer surplus is CS = ½ (base) (height). In our example, CS = ½ (2) (3000-2500) =
500.
Consumer surplus for a product is zero when the demand for the product is perfectly
elastic. This is because consumers are willing to match the price of the product. When
demand is perfectly inelastic, consumer surplus is infinite because a change in the price
of the product does not affect its demand. This includes products that are basic
necessities such as milk, water, etc.
Demand curves are usually downward sloping because the demand for a product is
usually affected by its price. With inelastic demand, consumer surplus is high because
the demand is not affected by a change in the price, and consumers are willing to pay
more for a product.
In such an instance, sellers will increase their prices to convert the consumer surplus to
a producer surplus. Alternatively, with elastic demand, a small change in price will result
in a large change in demand. It will result in a low consumer surplus as customers are
no longer willing to buy as much of the product or service with a change in price.
INDIFFERENCE CURVE
An indifference curve is a chart showing various combinations of two goods or
commodities that leave the consumer equally well off or equally satisfied—hence
indifferent—when it comes to having any combination between the two items that is
shown along the curve.
For instance, if you like both hamburgers and pizza, you may be indifferent to buying
either 20 hamburgers and no pizza, 35 pizza and no hamburgers, or some combination
of the two—for example, 4 hamburgers and 2 pizza (see point “A” in the chart below).
Either combination provides the same utility.
KEY TAKEAWAYS
⚫
⚫
⚫
⚫
An indifference curve shows a combination of two goods in various quantities that
provides equal satisfaction (utility) to an individual.
It is used in economics to describe the point where individuals have no particular
preference for either one good or another based on their relative quantities.
Along the curve, a consumer thus has an equal preference for the various
combinations of goods shown.
Typically, indifference curves are shown convex to the origin, and no two indifference
curves ever intersect
Understanding Indifference Curves
Standard indifference curve analysis operates using a simple two-dimensional chart.
Each axis represents one type of economic good. Along the indifference curve, the
consumer is indifferent between any of the combinations of goods represented by points
on the curve because the combination of goods on an indifference curve provides the
same level of utility to the consumer.
Indifference curves are heuristic devices used in contemporary microeconomics to
demonstrate consumer preference and the limitations of a budget. Economists have
adopted the principles of indifference curves in the study of welfare economics.
indifference Curve Analysis
Indifference curves operate under many assumptions; for example, each indifference
curve is typically convex to the origin, and no two indifference curves ever intersect.
Consumers are always assumed to be more satisfied when achieving bundles of goods
on indifference curves that are farther from the origin.
As income increases, an individual will typically shift their consumption level because
they can afford more commodities, with the result that they will end up on an
indifference curve that is farther from the origin—hence better off.
Many core principles of microeconomics appear in indifference curve analysis, including
individual choice, marginal utility theory, income, substitution effects, and the subjective
theory of value. Indifference curve analysis emphasizes marginal rates of substitution
(MRS) and opportunity costs. Indifference curve analysis typically assumes that all other
variables are constant or stable.
Most economic textbooks build upon indifference curves to introduce the optimal choice
of goods for any consumer based on that consumer’s income. Classic analysis
suggests that the optimal consumption bundle takes place at the point where a
consumer’s indifference curve is tangent with their budget constraint.
The indifference Curve maps indicates the several sets of indifference curves consumer
prefers. The IC map provides corresponding to different level of satisfaction
MARGINAL RATE OF SUBSTITUTION (MRS)
-
In economics, the marginal rate of substitution (MRS) is the amount of a good
that a consumer is willing to consume compared to another good, as long as the
new good is equally satisfying. MRS is used in indifference theory to analyze
consumer behavior. When someone is indifferent to substituting one item for
another, their marginal utility for substitution is zero since they neither gain nor
lose any satisfaction from the trade.
FORMULA AND CALCULATION OF THE MARGINAL RATE OF SUBSTITUTION
(MRS)
The marginal rate of substitution (MRS) formula is:
MRS xy= dx/dy = MUx/MUy
where:
x,y =two different goods
dx/dy =derivative of y with respect to x
MU =marginal utility of good x, y
The marginal rate of substitution is a term used in economics that refers to the
amount of one good that is substitutable for another and is used to analyze consumer
behaviors for a variety of purposes. MRS is calculated between two goods placed on an
indifference curve, displaying a frontier of utility for each combination of "good X" and
"good Y." The slope of this curve represents quantities of good X and good Y that you
would be happy substituting for one another. MRS is a critical component for
businesses to understand when analyzing consumption trends or for government
entities to understand when setting public policy. Consider an example of a government
wanting to analyze how offering electric vehicle incentives may spur more
environmentally-friendly purchases. Understanding how MRS is impacted before and
after a tax incentive can allow for the government to analyze the financial implications of
the plan.
BUDGET OR CONSUMPTION-POSSIBILITY LINE
-
A budget or consumption-possibility line shows the various combinations of two
products that can be purchased by the consumer with his or her income, given
the prices of the products.
PURPOSE OF BUDGET
-
Do not spend more than what you have.
It tracks the incoming and outgoing monies
Individuals, households, businesses. and even government use budgets so that
they have at least a small amount of money left (savings) at the end of the month
For most of us, the idea of scarcity and trade-offs is something we experience in a very
real way when it comes to our own budget constraints. Most of us have a limited
amount of money to spend on the things we need and want. Another kind of budget
constraint is time. For instance, as a student, you only have twenty-four hours in the day
to study, eat, sleep, and check Facebook. An hour spent studying economics is an hour
that can’t be used for sleep or play (or something else). As a result, you have to make
choices and trade-offs.
BUDGET CONSTRAINT
-
refers to all possible combinations of goods that someone can afford, given the
prices of goods, when all income (or time) is spent.
CHANGE IN CONSUMER INCOME
-
If the income of the consumer increases, the consumer has the ability to buy
more or higher combinations of goods. This will cause the budget line to shift to
the right. Conversely, if there is a decrease in income, budget line will shift to the
left
TOPICS: INCOME AND SUBSTITUTION EFFECT, ENGEL CURVE, ENGEL'S LAW
AND INCOME CONSUMPTION CURVE
*INCOME AND SUBSTITUTION EFFECT*
The income effect indicates that when the price of a good falls, the income of the
person who buys that good increases. The substitution effect states that when the price
of a good decreases, consumers will replace the relatively more expensive good with
the cheaper one. The Income Effect is the effect due to the change in real income. For
example, when the price goes up the consumer is not able to buy as many bundles that
she could purchase before. This means that in real terms she has become worse off.
The Substitution Effect is the effect due only to the relative price change, controlling for
the change in real income. In order to compute it we ask what is the bundle that would
make the consumer just as happy as before the price change, but if they had to make
their choice faced with the new prices.
*ENGLE CURVE*
In Engel Curve, There are two types of Engel curves. Budget Share The Engel line
describes how the proportion of household income spent on goods varies with income.
In addition, the Engel curve can also describe how real spending varies with household
income. Engel Curves are the locus of all points representing the quantities demanded
of the goods at various levels of income, when prices and preferences are held constant
*ENGEL'S LAW*
Engel's Law is a 19th-century observation that as household income increases,
the percentage of income that a household spends on food will decline. In part, this is
because the amount and quality of food that a family can consume is fairly limited. As
food spending declines on a relative basis, households spend a greater portion of their
income on other things, such as education and recreation. Engel found that the poorer
the group, the greater the percentage of their budget that went to food, while a lesser
percentage went, for example, to clothing and education.
*INCOME CONSUMPTION CURVE*
It is also called income expansion path and income offer curve. The income
consumption curve is a curve in a graph in which the quantities of two goods are plotted
on two axes; the curve is the locus of the points representing the selected set of
consumption at each different level of income. Income consumption curve traces out
the income effect on the quantity consumed of the goods. Income effect can either be
positive or negative.
REFERENCES
Consumer-surplus Pierce. J. W. (2014). Investopia.Indifference
https://www.investopedia.com/terms/i/indifferencecurve.asp
Corporate Finance Institute. (2022, December 20). Engel’s Law.
https://corporatefinanceinstitute.com/resources/economics/engels-law/
Definition of Total Utility in Economics, With Example. (2022, June 21). Investopedia.
https://www.investopedia.com/terms/t/totalutility.asp
Hayes, Adam (2022) MRS in Economics: What It Is and the Formula for Calculating It.
(2022, December 30). Investopedia.
https://www.investopedia.com/terms/m/marginal_rate_substitution.asp
Income Effect vs. Substitution Effect: What’s the Difference? (2022, April 15).
Investopedia. https://www.investopedia.com/ask/answers/041415/whatsdifference-between-income-effect-and-substitution-effect.asp
Lumen Learning. (n.d.). Reading: Budget Constraints and Choices | Microeconomics.
https://courses.lumenlearning.com/suny-microeconomics/chapter/readingbudget-constraints-and-choices/
Marginal Utilities: Definition, Types, Examples, and History. (2023, January 13).
Investopedia. https://www.investopedia.com/terms/m/marginalutility.asp
Mcleod, S., PhD. (2023, March 2). Maslow's Hierarchy of Needs Theory. Simply
Psychology - Study Guides for Psychology Students.
https://simplypsychology.org/maslow.html
Reed. N. T. (2017). Corporate Finance Instutute. Economics, Curve asp
https://corporatefinanceinstitute.com/resources/economics/consumer-surplus/
,, S., & Hallam, J. A. (2017). Engel Curve. https://www.sciencedirect.com/.
https://www.sciencedirect.com/topics/economics-econometrics-andfinance/engel-curve
Theory of Consumer Behavior Marginal Utility Analysis. (n.d.). https://www.toppr.com.
https://www.toppr.com/guides/business-economics/theory-of-consumerbehavior/marginal-utility-analysis/
Theory of Consumer Behavior Meaning and Concept of Utility. (n.d.).
https://www.toppr.com/guides/business-economics-cs/theory-of-consumerbehavior/meaning-and-concept-ofutility/#:~:text=The%20satisfaction%20of%20a%20consumer,with%20the%20am
ount%20of%20desire.
Trinidad, C. (2022, December 4). Utility Maximization. Corporate Finance Institute.
https://corporatefinanceinstitute.com/resources/economics/utility-maximization/
Viray, Maldeva (2016). Microeconomics Simplified
Wikipedia contributors. (2022, November 29). Engel curve. Wikipedia.
https://en.m.wikipedia.org/wiki/Engel_curve
Wikipedia contributors. (2023, February 7). Engel’s law. Wikipedia.
https://en.m.wikipedia.org/wiki/Engel’s_law