Budget Constraint
ECON201- MICROECONOMICS
FALL 2025
Consumer Theory
• We want a model that will answer the question: how do consumers
choose from a set of available options?
• The economic model of the consumer is built on an extremely simple
set of assumptions.
• Classical economic theory assumes "that consumers choose the best
bundle of goods they can afford." (Varian p. 20)
Consumer Theory
• We consider a rational consumer.
• We address the following two questions:
• Which combinations of goods is the consumer able to buy?
• Among the combinations of goods that the consumer is able to buy, which
combination(s) does the consumer prefer to buy?
Motivating Example
• Suppose your weekly income is $100 and you consume only food and
clothing (in their respective units).
• Let the price of food per unit be $2 and the price of clothing be $5.
• Which combinations of food and clothing are available for your
consumption?
• What about the combinations of food and clothing that you can
afford?
Consumption Choice Sets
• We start from the assumption that there is a set of goods from which
consumers can choose.
• A consumption choice set is the collection of all consumption choices
available to the consumer.
• What constrains consumption choice?
• Budgetary, time, and other resource limitations.
• How to define a consumption choice set?
The Consumption Bundle
• A consumption bundle is a particular combination of two or more
goods.
• A bundle containing x1 units of good 1 and x2 units of good 2 is
denoted by the vector (x1, x2).
• In our models, we're usually only going to talk about two arbitrary
goods (call them good 1 and good 2), because this lets us work with
two dimensions figures and highlights the fact that the two goods
can be any two goods we want.
The Price of a Bundle
• We assume that the price of each good is fixed and known by the consumer.
• Commodity prices are denoted by the vector ( 𝑝1 , 𝑝2 ).
• So for a given bundle (𝑥1 , 𝑥2 ) at prices ( 𝑝1 , 𝑝2 ), the total cost of purchasing the
bundle (𝑥1 , 𝑥2 ) is 𝑝1 𝑥1 + 𝑝2 𝑥2
• This is just the cost of purchasing 𝑥1 units of good 1 and 𝑥2 units of good 2.
• Sometimes 𝑝1 𝑥1 + 𝑝2 𝑥2 is called the expenditure on the bundle (𝑥1 , 𝑥2 ).
Naturally, 𝑝1 𝑥1 is the expenditure on good 𝑥1 , and 𝑝2 𝑥2 is called the
expenditure on good 𝑥2 .
Budget Constraints
• When is a bundle (x1, x2) affordable at prices (p1, p2)?
• We assume that a consumer has some amount of money, 𝑚 that
can be spent on goods 1 and 2.
• Then, the set of affordable bundles are all the pairs (x1, x2) for which
the total expense does not exceed consumer’s income:
𝑝1 𝑥1 + 𝑝2 𝑥2 ≤ 𝑚
• This is known as the budget set.
Budget Set
• The consumer’s budget set is the set of all affordable bundles
𝐵 𝑝1 , 𝑝2 , 𝑚 = (𝑥1 , 𝑥2 ) 𝑥1 ≥ 0, 𝑥2 ≥ 0 𝑎𝑛𝑑 𝑝1 𝑥1 + 𝑝2 𝑥2 ≤ 𝑚}
• As long as 𝑝1 , 𝑝2 and 𝑚 are positive, the budget set, on the 𝑥1 − 𝑥2
plane, corresponds to the triangle area bounded by the two axes and
the budget constraint
𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚
as the upper boundary of the budget set:
Budget Constraint
• The bundles that are only just affordable form the consumer’s
budget constraint. This is the set:
(𝑥1 , 𝑥2 ) 𝑥1 ≥ 0, 𝑥2 ≥ 0 𝑎𝑛𝑑 𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚}
Budget Set and Constraint for n=2
Budget Set and Constraint for n=2
Budget Set and Constraint for n=2
x2
m/p2
Budget constraint is
p1x1 + p2x2 = m (slope is -p1 /p2).
The collection of
all affordable bundles
m/p1
x1
Budget Constraints
• For n = 2 and x1 on the horizontal axis, the constraint’s slope is
–p1/p2. What does it mean?
x2 = −
p1
m
x1 +
p2
p2
• Increasing x1 by 1 must reduce x2 by p1/p2.
• The opportunity cost of an extra unit of commodity 1 is p1/p2 units foregone of
commodity 2.
• Thus, the budget constraint tells you the rate at which the market will convert
good 1 into good 2.
• Price ratio p1/p2 interpretations: price of x1 in terms of x2, the market value of x1
in terms of x2, how much x2 you need (get) to buy (from selling) one unit of x1.
Budget Constraints
Generalizing the Model
• While strictly speaking, the model we've developed so far only
accounts for two goods, a little clever reinterpretation suggests that
this isn't really a problem.
• If we think of good 1 as representing something specific, like your
favorite football team tickets, and good 2 as representing everything
else, then the model looks a lot more general.
• Good 2 is just the dollars you have left over for other stuff after you
buy your football team tickets.
• This also simplifies things because we need only worry about one
price, i.e. 𝑝2 = 1 since the price of a dollar is $1. So now the budget
set looks like:
(𝑥1 , 𝑥2 ) 𝑥1 ≥ 0, 𝑥2 ≥ 0 𝑎𝑛𝑑 𝑝1 𝑥1 + 𝑥2 ≤ 𝑚}
A Composite Consumption Good
• When we adopt this version of the model, we call good 2 a composite
consumption good (or composite good).
• In this case, we can think about questions like: how does an increase
in the price of my favorite football team tickets impact the rest of my
consumption decisions?
Income and Price Changes
• The budget constraint and budget set depend upon prices and
income.
➢ What happens as prices or income change?
How Do the Budget Set & Constraint Change as Income m Changes?
Higher Income Gives More Choice
Lower Income Shrinks the Budget Set
Budget Constraints: Income Changes
• Increases in income m shift the budget constraint outward in a
parallel manner, thereby enlarging the budget set and improving
choice.
• Decreases in income m shift the budget constraint inward in a
parallel manner, thereby shrinking the budget set and reducing
choice.
Budget Lines: Income Changes
• Your income increases while the prices remain constant. Are you
better off or worse off?
Budget Lines: Income Changes
• Your income increases while the prices remain constant. Are you
better off or worse off?
• No original choice is lost and new choices are added when income
increases, so higher income cannot make a consumer worse off.
• An income decrease may (typically will) make the consumer worse
off.
Budget Lines: Price Changes
• What happens if just one price decreases?
• Suppose p1 decreases.
How Do the Budget Set & Constraint Change as p1 Decreases
from p1ʹ to p1ʺ?
How Do the Budget Set & Constraint Change as p1 Decreases
from p1ʹ to p1ʺ?
Budget Constraints: Price Changes
• The price of the first good decreases while your income and the
price of the other good remain constant. Are you better off or
worse off?
Budget Constraints: Price Changes
• The price of the first good decreases while your income and the
price of the other good remain constant. Are you better off or
worse off?
• Reducing the price of one commodity pivots the constraint
outward. No old choice is lost and new choices are added, so
reducing one price cannot make the consumer worse off.
• Similarly, increasing one price pivots the constraint inward, reduces
choice and may (typically will) make the consumer worse off.
Uniform Ad Valorem Sales Taxes
• An ad valorem sales tax levied at a rate of 5% increases all prices by
5%, from p to (1 + 005)p = 105p.
• An ad valorem sales tax levied at a rate of t increases all prices by tp
from p to (1+ t)p. (Subsidies just work in the opposite direction, i.e.,
and ad valorem subsidy of 𝜎% would decrease prices by 𝑝𝜎).
• Ad valorem taxes are known as value taxes, i.e., a tax on the value –
the price – of a good.
• A uniform sales tax is applied uniformly to all commodities.
Uniform Ad Valorem Sales Taxes
A uniform sales tax levied at rate t changes the constraint from
to
Or equivalently,
Uniform Ad Valorem Sales Taxes
Uniform Ad Valorem Sales Taxes
Shapes of Budget Constraints
• What makes a budget constraint a straight line?
➢ A straight line has a constant slope and the budget constraint is
So, if prices are constant, then the budget constraint is a straight line.
Shapes of Budget Constraints
• But what if prices are not constants?
➢ For example, bulk buying discounts, or price penalties for buying
“too much.”
➢ Then constraints will be curved.
Application: Quantity Discounts
• Suppose p2 is constant at $1 but that p1 = $2 for 0  x1  20 and p1 =
$1 for x1 > 20. Then the constraint’s slope is:
for
for
and
Application: Quantity Discounts
Application: Quantity Discounts
Application: Quantity Penalty
Budget Constraints: Generalization to n goods
• A consumption bundle for n-goods case: (x1, x2, . . . , xn).
• The price vector is (p1, p2, . . . , pn).
Budget Constraints: Generalization to n goods
• A consumption bundle for n-goods case: (x1, x2, . . . , xn).
• The price vector is (p1, p2, . . . , pn).
• When is a bundle (x1, . . . , xn) affordable at prices (p1, . . . , pn)?
• When
• What is the consumer’s budget line?
and
• What about the consumer’s budget set?
and
Remarks
• What happens to the budget set if you change the “unit of
account”?
• What happens if you apply a “negative tax”?
• Choices are usually constrained by more than a budget; e.g., time
constraints and other resources constraints.
• A bundle is affordable only if it meets every constraint.
• How does a budget constraint look like if the price of one good is
positive while the price of the other good is negative?