2 Consumer Theory
Consumers use their preferences to decide which bundle of goods they should allocate
their constrained income to.
It is possible that consumers select bundles totally at random, but the data collected by
economists appears to suggest that consumers do act broadly in-line with the assump-
tion of utility-maximisation subject to budgetary constraint.
2.1 Utility
2.1.1 Transformations
Utility is not of cardinal significance but can be used to make ordinal preference-ranking
comparisons between bundles of goods entered into the function. Since utility is an
ordinal concept, we can perform valid positive monotonic transformations to utility
functions
1. f = k + u.
2. f = exp(u).
3. f = log (u) for u > 0.
4. f = ku for k > 0.
5. f = uk where u > 0 or k is an odd number.
2.1.2 Indi↵erence Curves
Definition 9. An indifference curve is a set of all the bundles which share a particular utility-
value.
2.1.3 Utility Axioms
Preferences have thought to have the following properties, informally defined
Definition 10. Completeness says that for any given pair of bundles, one and only one of the
following is true: the consumer prefers one bundle over the other, the consumer is indifferent
between the two bundles.
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, Definition 11. Transitivity says that if a consumer prefers bundle x to bundle y and bundle
y to bundle z then she prefers bundle x to bundle z.1
Definition 12. Non-satiation says that more of any good is better than less of it i.e. x 0 >
x =) x 0 ⌫ x.
Definition 13. Continuity says that if a consumer prefers bundle a to bundle b and bundle c
is very close to bundle a then the consumer will prefer bundle c to bundle a. This allows us to
graph smooth indifference curves without volatile changes in preference.
Definition 14. Strict convexity says that if a and b are preferred to c, then ba + (1 b)b
(for 0 b 1) must also be preferred to c.
Some preferences have the property of being homothetic.
Definition 15. A consumer’s preferences are called homothetic if they can be represented by
a utility function which is homogeneous of degree 1.
2.1.4 Consumption Constraint
The fundamental constraint facing a consumer is that
C M. (16)
This says the total value of consumption is less than or equal to the total value of
income. A consumer choosing her consumption of two goods ( x1 , x2 ) has a budget
set
{ x1 , x2 : p1 x1 + p2 x2 m }. (17)
The frontier of this budget set is called the budget constraint which can be re-arranged
to
m p1
x2 = x1 . (18)
p2 p2
In the real world, these budget constraints are not linear. Non-linear budget constraints
are called ‘kinked’ budget constraints and may result from:
1. Taxation of labour income.
2. Interest rate spreads.
3. Bulk discounting.
We can think of a ‘kinked’ budget constraint as a minimum function of multiple linear
budget constraints at different prices.
1 Transitivity makes it impossible that indifference curves cross.
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Consumers use their preferences to decide which bundle of goods they should allocate
their constrained income to.
It is possible that consumers select bundles totally at random, but the data collected by
economists appears to suggest that consumers do act broadly in-line with the assump-
tion of utility-maximisation subject to budgetary constraint.
2.1 Utility
2.1.1 Transformations
Utility is not of cardinal significance but can be used to make ordinal preference-ranking
comparisons between bundles of goods entered into the function. Since utility is an
ordinal concept, we can perform valid positive monotonic transformations to utility
functions
1. f = k + u.
2. f = exp(u).
3. f = log (u) for u > 0.
4. f = ku for k > 0.
5. f = uk where u > 0 or k is an odd number.
2.1.2 Indi↵erence Curves
Definition 9. An indifference curve is a set of all the bundles which share a particular utility-
value.
2.1.3 Utility Axioms
Preferences have thought to have the following properties, informally defined
Definition 10. Completeness says that for any given pair of bundles, one and only one of the
following is true: the consumer prefers one bundle over the other, the consumer is indifferent
between the two bundles.
12
, Definition 11. Transitivity says that if a consumer prefers bundle x to bundle y and bundle
y to bundle z then she prefers bundle x to bundle z.1
Definition 12. Non-satiation says that more of any good is better than less of it i.e. x 0 >
x =) x 0 ⌫ x.
Definition 13. Continuity says that if a consumer prefers bundle a to bundle b and bundle c
is very close to bundle a then the consumer will prefer bundle c to bundle a. This allows us to
graph smooth indifference curves without volatile changes in preference.
Definition 14. Strict convexity says that if a and b are preferred to c, then ba + (1 b)b
(for 0 b 1) must also be preferred to c.
Some preferences have the property of being homothetic.
Definition 15. A consumer’s preferences are called homothetic if they can be represented by
a utility function which is homogeneous of degree 1.
2.1.4 Consumption Constraint
The fundamental constraint facing a consumer is that
C M. (16)
This says the total value of consumption is less than or equal to the total value of
income. A consumer choosing her consumption of two goods ( x1 , x2 ) has a budget
set
{ x1 , x2 : p1 x1 + p2 x2 m }. (17)
The frontier of this budget set is called the budget constraint which can be re-arranged
to
m p1
x2 = x1 . (18)
p2 p2
In the real world, these budget constraints are not linear. Non-linear budget constraints
are called ‘kinked’ budget constraints and may result from:
1. Taxation of labour income.
2. Interest rate spreads.
3. Bulk discounting.
We can think of a ‘kinked’ budget constraint as a minimum function of multiple linear
budget constraints at different prices.
1 Transitivity makes it impossible that indifference curves cross.
13
