Microeconomics theory and analysis
Lecture 1 & 2 Consumer Choice
“People choose the best bundle they can afford”
What can you afford?
- Budget constraint (Chapter 2)
What is the best bundle?
- Preferences (Chapter 3)
How to model preferences?
- Utility functions (Chapter 4)
What to choose?
- Choice (Chapter 5)
Budget constraint
● let us say there are two goods x1 and x2 with prices p1 and p2
● consumer has income m
● a consumption bundle is (x1, x2)
● the budget constraint defines what a consumer can afford:
○ p1x1 + p2x2 ≤ m
● everything that is affordable lies within the budget set
● the set of consumption bundles that cost exactly m form the
budget line
○ p1x1 + p2x2 = m
What happens if the income decreases from m to m’?
note: the slope does not change since prices are the same
What happens if the price of good 1 increases?
,To sum up
● a change in income causes a parallel shift of the budget line → the slope does not
change
● if the price of one good changes, this changes the slope of the budget line
𝑝1
● slope is: − 𝑝2
● If p1 ↑ → slope increases If p1 ↓ → slope decreases
● If p2 ↑→ slope decreases If p2 ↓ → slope increases
Preferences
● consumers choose between consumption bundles
● consider two bundles X = (x1, x2) and Y = (y1, y2)
● if X is strictly preferred to Y, we write:
○ (x1, x2) ≻ (y1, y2)
● if the consumer is indifferent between X and Y, we write:
○ (𝑥1, 𝑥2) ∼ (𝑦1, 𝑦2)
● if x is weakly preferred to Y, we write:
○ (𝑥1, 𝑥2) ≽ (𝑦1, 𝑦2)
Three usual assumptions about preferences
● complete. Any two bundles can be compared
○ (x1, x2) ≽(y1, y2) or (y1, y2) ≽ (x1, x2) or both
● reflexive. Any bundle is at least as good as itself
○ (x1,x2) ≽ (x1, x2)
● transitive. If (x1, x2) ≽ (y1, y2) and (y1, y2) ≽ (z1, z2)
then (x1, x2) ≽ (z1, z2)
Preferences can graphically be described by indifference curves
All bundles the lie on an indifference curve are equally good (the consumer is indifferent)
indifferences curves can have many shapes but they cannot cross
why?
1. X ~ Z
2. Z ~ Y
● by transitivity it should be that: X ~ Y
● but X and Y are not on the same indifference curve so
the consumer is not indifferent between them!
Perfect substitutes
● two goods are substitutes if the consumer is willing to
substitute one good for the other at a constant rate
○ for me consuming two apples is the same as consuming one orange
● two goods are perfect substitutes if the consumer is willing to substitute the goods
on a one-to-one basis
○ for me, consuming one apple is the same as consuming one orange. So i
am indifferent between consuming two apples, two oranges, and one apple
and one orange
,Perfect substitutes
● the consumer prefers bundles on higher indifference curves to
bundles on lower indifference curves
● that is, the consumer prefers changes in the direction of the
arrows
● slope: -1 (straight line because of the constant rate)
Perfect complements
● complements are goods that are always consumed in fixed
proportions
● perfect complements are goods that are always consumed
one of each
● they ‘complement’ each other
● example: right shoe and left shoe
Bads
● a bad is a commodity that the consumer does not like
● in this example good 2 is a bad
Preferences
● of it is convenient to assume that preferences are
‘well-behaved’
● in this case, we make two further assumptions
● monotonicity. more is better
○ if 𝑥1 ≥ 𝑦1 𝑎𝑛𝑑 𝑥2 > 𝑦2, 𝑡ℎ𝑒𝑛 (𝑥1, 𝑥2) ≻ (𝑦1, 𝑦2)
● convexity. averages are preferred (to extremes)
○ if (x1, x2) ~ (y1, y2) then 𝑡𝑥1 + (1 − 𝑡)𝑦1 ≽ (𝑥1, 𝑥2)
○ for any t between 0 and 1
If preferences are monotonic:
● more of both goods means a better bundle
● less of both goods mean a worse bundle
● indifference curves have a negative slope
● indifference curves further from the origin represent
better bundles
If preferences are convex:
● averages are preferred to extremes
● all of the weighted averages of X and Y are
weakly preferred to X and Y
,Utility
● preferences can be described by a utility function
● a utility function assigns a number to a bundle such that better bundles get higher
numbers (i.e. a higher utility)
○ (x1, x2) ~ (y1, y2) if and only if u(x1, x2) = u(y1, y2)
● and
○ (𝑥1, 𝑥2) ≻ (𝑦1, 𝑦2) 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑢(𝑥1, 𝑥2) > 𝑢(𝑦1, 𝑦2)
● for a utility function to represent preferences, the only thing that matters is the order
of the bundles
● the value themselves have no meaning apart from the order.
● for example, if u(A) = 1 and u(B) = 2, it does not mean that B is twice as good as A. It
only means that the consumer prefers B to A
● this is called ordinal utility
● this also implies that there are many utility functions that represent the same
preferences
● in fact, any (positive) monotonic transformation of the utility function is allowed
● a monotonic transformation transforms each number such that the order is preserved
● examples:
○ adding or subtracting a number
■ 𝑓1(𝑢1) = 𝑢1 + 10
○ multiplication by a positive number
■ 𝑓2(𝑢1) = 2𝑢1
○ raising each number by an odd power
3
■ 𝑓3(𝑢1) = (𝑢1)
● all of these functions represent the same preferences as u1
● on an indifference curve, all bundles are valued equally
● hence the utility of all the bundles on an indifference curve
must be the same
● e.g. suppose u(x1, x2) = x1x2 = k
● preferences determine the shape of the indifference curve
● this means that any monotonic transformation of a utility function will not change the
shape of the indifference curves
● example: u(x1, x2) = 2x1x2 = 2k
● a widely used type of utility functions are Cobb-Douglas utility
functions
𝑐 𝑑
○ 𝑢(𝑥1, 𝑥2) = 𝑥1 𝑥2 where c and d are positive numbers
● they describe well-behaved preferences and have more convenient
properties
, ● with quasilinear preferences, utility is linear in one good and
nonlinear in the other good:
○ 𝑢(𝑥1, 𝑥2) = 𝑣1(𝑥1) + 𝑥2
𝑢(𝑥1, 𝑥2) = 𝑥1 + 𝑥2
● in this case, indifference curves are vertically shifted versions of
each other
● How does utility change if you receive a little more of good 1?
● this rate of change is called the marginal utility of good 1:
δ𝑢(𝑥1, 𝑥2)
○ 𝑀𝑈1 = δ𝑥1
● so, the change in utility associated with a change in consumption of good 1 is
○ 𝑑𝑈 = 𝑀𝑈1𝑑𝑥1
● the marginal rate of substitution (MRS) measure the slope of an
indifference curve
● what does this mean?
● the MRS indicates the rate at which a consumer is willing to
substitute two goods
Some remarks regarding 𝑀𝑅𝑆𝑥 𝑥
1 2
● the x1x2 under the MRS means marginal rate of substitution of x1 for x2
● it is the amount of x2 which a consumer can exchange for one unit of x1 to stay on
the same indifference curve
● for example, if 𝑀𝑅𝑆𝑥 𝑥 = 2 means
1 2
𝑥1
○ 𝑥2
= 2 ⇒ 𝑥1 = 2𝑥2
● that is, 1 unit of x1 is worth 2 units of x2 to this consumer
● the consumer will give up 2 units of x2 to obtain 1 additional unit of x1 and will stay
on the same indifference curve
In the subscript of MRS, always put the good on the horizontal axis first!
● the MRS is equal to the ratio of the marginal utilities:
𝑀𝑈1
○ 𝑀𝑅𝑆𝑥 𝑥 = 𝑀𝑈2
1 2
