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T1 P1: Consumer Theory: Differences in behaviour can emerge from different tastes and
different circumstances, such as different levels of income
T1 P2: Budget Constraints: Income and prices affect the quantity consumers demand.
Income can be determined exogenously (set outside the model) as an amount, M.
Endogenous income means that it is determined inside the model
The slope of the budget constraint measures the rate the market ‘substitutes’ good 1 for
good 2: it is the opportunity cost of consuming good 1. The slope of the budget constraint is
given by the price ratio.
If we consume more good 1, ∆ x 1 , by how much must good 2 change to continue to satisfy
the budget constraint?
p1 x 1 + p2 x 2=m Original budget constraint
p1 ( x1 + ∆ x 1 ) + p2 ( x 2+ ∆ x2 ) =m
Subtract (1) from (2): p1 ∆ x1 + p2 ∆ x 2=0
∆ x 2 − p1
Thus, =
∆ x1 p2
This expression states therefore, that the rate at which you have
to substitute between two goods is equal to the price ratio.
T1 P3: Preferences: Specifying preferences tells us something about a consumer’s choice.
Strict preference >
Weak preference ≥
Indifference
We only care about ordinal relations – the ordering of the preferences of the goods.
Cardinal relations is the opposite, we care about the amount of happiness you get from one
good compared to the other (magnitude of preference).
Properties of Preferences: These are properties we assume consumers have, which gives us
rational consumer behaviour (the first 3). The other 2 are properties that are not required to
give rational behaviour, but they give us “well behaved preferences”.
Completeness: The consumer can always compare/rank bundles. E.g. X >Y
Transitivity: If X ≥ Y ∧Y ≥ Z then X ≥ Z . Any mathematical function automatically
satisfies transitivity.
Continuous: If is preferred to Y, and there is a third bundle Z which lies within a
small radius (i.e. very similar) of Y, then X will be preferred to Z. Tiny
changes in bundles will not change preference ordering.
Monotonicity (non-satiation): More is better – you will never reach the
satiation point (where you do not want any more).
Convexity: Averages are better than extremes. Strict convexity is averages
are definitely better than extremes, while weak convexity is where
averages are not worse than extremes. If averages are not better than
extremes the indifference curves would be backward sloping
as any point on the chord would take you to a lower indifference curve This is where there is a satiation
z=(t x 1+ ( 1−t ) y 1 , t x 2+ ( 1−t ) y 2) ≥(x 1 , x 2) point – there is an overall best
bundle. Relaxing, monotonicity
assumption
,T1 P4: Indifference Curves: All points on the same indifference curve give
the consumer the same utility. Bundles on I3 will be preferred to bundles
on I2 as more is better (monotonicity).
Indifference curves are continuous due to our properties of monotonicity,
continuous and convex
Indifference curves cannot cross – due to transitivity.
Most peoples’ indifference curves are convex to the origin – as if you pick
extreme points on the indifference curve, an average is preferred to an
extreme. This can be seen by if you draw a chord between two extreme points on I 1, any
bundle on that chord is strictly preferred to any point on I 1.
Indifference curves are downward sloping as you have to reduce the quantity of good y to
increase the quantity of good x, otherwise your utility would increase and you would be on
another indifference curve.
‘Bads’ – a good that you dislike
Neutral goods – a good that you do not care about
Satiation – an overall best bundle: too much AND too little is worse.
T1 P5: Utility Functions: describe preferences, assigning higher numbers to more preferred
bundles. All combinations of two goods that give the same level of utility lie on the same
indifference curve.
The further from the origin, the higher the utility
Bundle ( x 1 , x 2 ) > ( y 1 , y 2) iff u( x 1 , x 2)>u ( y 1 , y 2)
This says that you prefer bundle x to bundle y if and only if you get more utility from bundle
x than bundle y
Perfect substitutes – goods with a constant rate of substitution – therefore the thing
that matters for your happiness is the total sum of the two goods. Therefore in perfect
substitutes, U(X, Y) = aX + bY. Where a and b are constants which refer to the rate of
substitution.
Perfect complements – goods that are always consumed together.
Therefore, U(x, y) = min(ax, by)
Cobb-Douglas utility function is a set of different utility functions that lie between
the two extremes: Imperfect Substitutes. The exact format of the function will tell us the
degree of substitutability between the two goods. General equation of a Cobb-Douglas
α 1−α
function: U ( x 1 , x 2 )=x 1 x 2
T1 P6: Monotonic Transformations: Applying monotonic transformations to a utility
function creates a new function but with the same preferences. This is due to the fact we
care about ordinal preferences not cardinal.
However you cannot work backwards from optimal
demand to the equation of the indifference curve as
every indifference curve can be described by an infinite
number of utility functions.
,T1 P7: Revealed Preferences: Revealed preference is the way in which we observe
consumers choosing between given bundles and can make inferences about the ordering of
bundles based on what they choose.
Factors which affect utility: Psychological attitudes, peer group pressures, personal
experiences, the general cultural environment.
We hold these assumptions to allows us to make other judgements of one variable on
utility: Only consider choices among quantifiable options; hold constant other things that
affect our behaviour (ceteris paribus).
Do the axioms (assumptions) always hold? Are consumers truly rational?
Too many choices/too much information; how choices are framed
Loss aversion: the disutility of giving up an object is greater than the utility
associated with acquiring it (cognitive bias)
Bounded rationality:
Behaviour is influenced by our environment and the information we have
Behaviour can be influenced to become more rational through incentives and nudges.
( X1 , X2)
If an optimising consumer chooses , when these bundles are different ,
( Y 1 ,Y 2)
It must be that: P1 X 1+ P 2 X 2 =M ∧P1 Y 1 + P2 Y 2 ≤ M
P 1 X 1+ P 2 X 2 ≥ P 1 Y 1 + P 2 Y 2 as bundle X is preferred it must be on a higher
indifference curve.
If this inequality is satisfied, then (X1, X2) is directly revealed preferred: ( X 1 , X 2 ) >(Y 1 ,Y 2 )
WARP (weak axiom of revealed preferences) – if you prefer A to B you always prefer A to B
(and therefore will always purchase A over B unless A becomes unaffordable)
SARP (strong “” “” “’ “”) – WARP + satisfies transitivity – if you prefer A to B and B to C
you should always prefer A to C
T1 P8: Marginal Rate of Substitution: If we reduce consumption of good 1 by ∆ x 1 … how
much good 2, ∆ x 2 , is needed to put Joe back on the same indifference curve?
∆ x2
The rate at which a consumer is willing to substitute one good for the other: = MRS
∆ x1
This is also the slope of an indifference curve at a particular point and will always be
negative (we know this from monotonicity).
∆ x2 M U1
MRS = and MRS =
∆ x1 M U2
∂ u(x 1 , x 2 )
d x2 ∂ x1
= =MRS
d x 1 ∂ u(x 1 , x 2 )
∂ x2
MRS therefore equals ratio of the marginal utilities which can be calculated by taking these
partial derivatives.
, T1 P9: Applying the MRS: For perfect substitutes the indifference curves are linear. This
M U1
means that MRS is constant all along the indifference curve. MRS =
M U2
For Perfect Compliments (diagram): U ( S R , S L ) =min { a S R , b S L }
To get our MRS value we would normally have to differentiate the utility
function, however, the utility function here is not differentiable. But, we
know a different way of calculating our MRS. Where our MRS is the slope of
the indifference curve i.e.. the change in y/change in x.
Along the vertical portion of the graph the gradient is infinity.
Along the horizontal portion of the graph the gradient is zero.
At the kink, the MRS is undefined.
α 1−α
For Cobb-Douglas Preferences: U ( x 1 , x 2 )=x 1 x 2
M Ux1 α x2
MRS is: = .
M Ux2 1−α x 1
MRS does not change when there is a monotonic transformation.
As we move down the indifference curve, MRS diminishes in absolute value.
This is due to diminishing marginal utility.
T1 P10: Indifference Maps: Homothetic tastes: It implies that if we were to
draw a ray from the origin through the indifference maps, the MRS on that ray
on any indifference curve is going to be identical. The relative quantity of
each good remains constant along any ray from the origin.
( x 1 , x 2 ) > ( y 1 , y 2) → ( t x 1 , t x 2 ) >(t y 1 ,t y 2)
When income rises, demand rises by the same proportion – the ratio with
which you will trade between the two goods is not going to change.
Doubling of income = doubling of consumption between the two goods.
Quasilinear tastes: Tastes are linear in one good, but may not be in the
other good: u ( x 1 , x 2 )=v ( x 1 ) + x 2
With quasilinear tastes, indifference curves are vertical translates of one
another
MRS is constant along any vertical line from the x – axis.
E.g. salt, as my income goes up my demand for the good does not change.
With a quasilinear good, my demand for the good is independent of income.
Quasilinear goods are borderline goods between the set of normal and the set
of inferior goods.
T1 P11: Elasticity of Substitution: the degree of substitutability measures how responsive
the bundle of goods along an indifference curve is to changes in the MRS. The elasticity of
x2
substitution is defined as:
| |
σ=
%∆
% ∆ MRS
( )
x1
The less curved the indifference curve, the greater is the %∆ x 2 / x 1 required for the MRS to
have changed by 1%, which implies a higher elasticity of substitution
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