This document contains all the notes from the synchronous and asynchronous lectures throughout the year and is split up lecture by lecture. It includes diagrams where relevant, as well as mathematical notation for completeness. It contains everything you need to secure a high mark in the module.
Necessary and Sufficient Conditions: The statement “A implies B” is logically correct unless it
is possible to simultaneously have A is true and B is false. A ⇒ B , B ⇐ A
If the statement A is always false then “A implies B” holds trivially since the empty set is a
subset of every set.
To show a statement of the form A ⇒ B is false, you need to find a counter example where
A is true but B is false. If no such example exists, then the statement is true.
When we have both A ⇒ B and B⇒ A , we write A ⇔ B and say that A and B are equivalent
to each other. Or, in other words, A happens if and only if B happens.
The Contrapositive: of the statement A ⇒ B is not B⇒ not A . These two statements are
equivalent to each other. I.e. if one is true, then the other must be true; and if one is false
then the other must be false too.
If we take the contrapositive of the contrapositive then we must get back to the original
statement itself.
By definition, if we take the contrapositive of any true statement we get another true
statement. E.g. x ≥ 4 ⇒ x ≥ 3 has contrapositive x <3 ⇒ x< 4
When trying to prove or disprove a statement of the form A ⇒ B we could equally try to
prove or disprove the contrapositive not B⇒ not A .
∀ , ∃∧negation: ∀ means for all or for every.
∃means there exists and the symbol ∄ means there does not exist
Example: Consider the statement ∀ a , b ∈ R ,with a< b ,∃ c ∈( a ,b)
This translates as: for any two real numbers a and b, there exists a third real number c which
lies between a and b.
Compound Statements: Take the statement “for all natural numbers n, if x n > y n then x n <10.
∀ n ∈ N , xn > y n ⇒ x n <10
Taking its negation we get “there exists some natural number n satisfying x n > y n but not
x n <10. So we get: ∃n ∈ N , x n > y n∧x n ≥10
Negating this we get that an equivalent way of expressing equation is “there does not exist a
natural number n satisfying both x n > y n and x n ≥ 10
∄n ∈ N , s . t . xn > y n∧x n ≥ 10
To say this another way, “for any natural number n, at least one of x n > y n∧x n ≥10 must be
violated.” Thus if one holds, the other cannot. Equation can be seen as saying if the first
holds, then the second does not. But alternatively we could have if x n ≥ 10 holds then x n > y n
does not. So we get: ∀ n ∈ N , xn ≥ 10⇒ xn ≤ y n
Note that the last two equations are contrapositives of each other
Proving and disproving statements: To prove a statement of the form A ⇒ B: not sufficient
to find an example where A is true and show that B also holds. You need to show that B
holds in every example where A is true. Alternatively by the contrapositive, you can show
that if B is violated then A must be too. This is closely linked to “proof by contradiction”
where you assume A is true and B is false and show that it generates a logical contradiction.
A Recap of Consumer Theory: The consumption set is defined as the set of commodities
consumers can consume. Let there be J goods the consumption set is X =RJ≥0 .
,Notice the consumption set is unbounded. That there is no upper bound to how much of
any one good can be consumed. Also note that the consumption set is infinitely divisible:
that is any Real number of any good can be consumed, not just integer amounts.
The budget constraint equation is p1 x 1 +…+ p j x j ≤ M
The Walrasian budget set is the set of bundles our consumer can choose between:
J
{( x 1 , … . x j)∈ R ≥0∨ p1 x1 + …+ p j x j ≤ M ¿}
Preferences: The preference relation ≥ defines weak preference over bundles. For any two
bundles of goods in X, ^x ≥ x translates as ^x is weakly preferred to x
Given ≥ we define: The strict preference relation: ^x > x ⇔ ^x ≥ xand not ^x ≤ x
The indifference relation: ^x x ⇔ ^x ≥ x∧ x^ ≤ x
Completeness: For any two bundles of goods in X, at least one of ^x ≥ x∨^x ≤ x must hold. If
both hold then ^x x . This says that for any two bundles in our consumption set, we can
compare them and say which is better.
Transitivity: If we prefer bundle 1 to bundle 2 and bundle 2 to bundle 3, then we should also
prefer bundle 1 to bundle 3.
Typically, we call a consumer rational if they satisfy Completeness and Transitivity.
Transitivity of ≥ implies that w y ∧w x ⇒ x y
Convexity says that consumers prefer averages of goods to extremes: For every x ∈ X , the
upper contour set { x^ ∈ X ∨^x ≥ x } is a convex set. In other words, if ^x ≥ x and ^x ≥ x then
α x^ + ( 1−α ) x ≥ x for any α ∈[ 0,1]. Perfect substitutes satisfy convexity
Strictly convex preferences: For every x ∈ X ,if ^x ≥ x and x ≥ x , with ^x ≠ x then for any
α ∈ ( 0,1 ) , α ^x + ( 1−α ) x > x . Perfect substitutes does not satisfy strong convexity
Monotonicity: Strong monotonicity: if you increase the amount you consume of any good
without decreasing the consumption of any other good, then you are strictly better off.
Weaker property of monotonicity: Preferences are monotone if for every x, ^x ∈ X , if xj <
x j ∀ j∈ J then ^x > x
^
In turn, monotonicity can be weakened to local non-satiation. Preferences are locally non
satiated if for every x ∈ X , ∀ ∈>0 , ∃ ^x ∈ X such that ||x− x^||< ε and ^x > x
In words, starting from any consumption bundle, we can always find another bundle “close
to it” that the consumer strictly prefers. This holds regardless of how small the distance
must be to be considered “close to”. Monotonicity implies local non satiation.
Note that local non-satiation allows one or more commodities to be “bads” although we
can’t have all commodities be bads as then 0 ∈ X would be a point of satiation.
The preference relation ≥can be represented by a utility function u: X → R if for every pair
of bundles x , x^ ∈ X , ^x ≥ x ⇔ u ( x^ ) ≥u (x)
If preferences are represented by a utility function u then any increasing transformation of u
will also represent these preferences. Therefore, we only care about whether one bundle
has higher utility than another; not about the exact numerical amounts – this is utility
functions are ordinal not cardinal.
A function u: X → R is quasi concave if its upper level sets, { x ∈ X :u ( x ) ≥ c }, are convex for
every c ∈ R . Suppose preferences can be represented by u: X -> R . Then preferences are
convex iff u is quasi concave.
Consumer Optimisation: This is finding the best bundle the consumer can afford. In other
words our consumer wants to solve: max u(x )s . t . p1 x 1 +…+ p j x j ≤ M
x∈X
,Marginal utility of good i is the rate at which utility of consumer increases with good i:
du
M U i=
d xi
Marginal rate of substitution is the slope of the indifference curve, the rate at which the
M U1
consumer is willing to substitute good 1 for good 2. MR S1,2 =
M U2
α β
A utility function of the form u ( x 1 , x 2 )=x 1 x 2 is called Cobb Douglas. This is a common type
of utility function and has the property that at the optimal bundle the consumer will spend
( )
α M
α + β p1
of their income on good 1 and ( )
β M
α + β p2
of their income on good 2.
dL dL dL
Lagrangian method is solved by setting = = =0. This is an alternate way of
d x 1 d x 2 dλ
finding a bundle that exhausts the budget while setting slope of budget line equal to slope
of indifference curve (MRS equal to price ratio).
However, note that using the Lagrangian or setting MRS equal to price ratio will not always
work as: the utility function may not be differentiable; the optimal bundle might not be
interior, which can happen for two reasons – preferences are not convex, preferences are
convex but the MRS is either always steeper or always shallower than the budget line.
An example of non-differentiability is with perfect compliments where the point of
optimisation occurs where the consumer exhausts her budget and where we are at a kink of
an indifference curve.
An example of non-convexity is where sometimes using the Lagrangian gets us minimum
utility and we need to use the corner solutions to find max utility. You can find these corner
M Ui M U j
solutions by using the bang per buck method ≤ – finding the good which is
pi pj
worth more to the consumer per pound spent.
Useful results to help find optimal bundles: If preferences satisfy local non-satiation then the
consumer must expend all her budget to maximise utility.
If preferences satisfy local non satiation and are represented by a differentiable, quasi-
concave utility function (so that preferences are convex) then x ∈ X is an optimal bundle if
and only if satisfies: The consumer expends all her budget; If x i=0 then ∃ j∈ J , j≠ i such
M Ui M U j M Ui M U j
that ≤ ; If x i> 0 then ≥ ∀ j∈J
pi pj pi pj
If additionally the utility function is strictly quasi concave (so that preferences are strictly
convex) then the optimal bundle is unique.
Note that the solution to the utility maximisation problem should be: Homogeneous of
degree 0 in terms of income and prices since if all prices and income double the budget set
remains unchanged; Invariant to increasing transformations of the utility function, since
they all represent the same preferences, with the same indifference curves.
Lecture 2: Walrasian Equilibrium in Pure Exchange Economies: In our economy, with a set I
of agents and a set J of different goods as commodities, we take two things as exogeneous
parameters:
The initial allocation of each agent: For each agent i∈ I ,we let e i=(e i1 , ei 2 … , eiJ ) be
consumer I’s initial allocation or endowment of the different goods, 1 to j.
, The preferences of each agent: For each agent i∈ I , we let ≥ be the preferences of
agent i.
Our aim in finding a Walrasian Equilibrium is to evaluate the price of every good and the
final allocation after trade as a function of these two things.
Our first assumption when doing this is that there is only exchange in our economy.
Second assumption is that there are well defined property rights.
Third assumption is that there is a market for every commodity. In particular, there is a price
that consumers can buy or sell units of every commodity.
Fourth assumption is that everyone is a price taker. In other words the price at which people
can buy or sell units of a commodity is the same for everybody.
Fifth assumption is that goods are rivalrous: this means that each unit of every good can be
consumed by only one individual.
Sixth assumption is that an agents utility depends only upon what goods he consumers.
Seventh assumption is that there are no transaction costs to trade.
Eighth assumption is that every individual has rational (complete and transitive) preferences
and chooses his optimal bundle subject to budget constraint.
Ninth assumption is that markets clear.
2.2: Walrasian Equilibrium: An individual i∈ I with endowment e i=(e i1 , ei 2 … , eiJ ) at a given
price vector p ∈ R≥J 0 has an effective wealth of: p . ei= p 1 e i 1+ …+ pJ e iJ
And so faces budget constraint: p . x i < p . ei ⟺ p 1 x i 1+ …+ pJ x iJ ≤ p 1 e i 1+ …+ p j e ij
The idea of Walrasian equilibrium is to have a price vector such that at this vector, when
everyone maximises utility subject to budget constraint, markets will clear.
Each consumer is subject to the following Utility Maximisation Problem (UMP):
max ui ( x i ) s . t . p . x i ≤ p . ei
xi ∈ X
¿ ¿ ¿ J
Walrasian Equilibrium is defined as: A price vector p =( p1 , … p J )∈ R≥0 and an allocation
x ¿= ( x ¿i )i ∈J ∈ R I≥0× J constitute a Walrasian Equilibrium if:
¿
i) For each consumer i∈ I ,bundle x i solves the consumer’s UMP at prices p*
All markets clear: ∑ x i =∑ e i
¿
ii) i∈ J i ∈J
p* is a Walrasian Equilibrium price vector and x* is a Walrasian equilibrium allocation.
For consumer i∈ I we define x i ( p ) ⊆ RJ≥0 as the set of optimal bundles (solutions to UMP)
given p. This is often called the demand correspondence.
When there is always a unique optimal bundle, so that it is single valued as it will be under
certain assumptions it is called the demand function
For simplicity, it is assumed that the demand correspondence is single valued and summing
across all consumers we get:
The aggregate demand function ∑ x i ( p)∈ R≥0 is the total demanded by consumers as a
J
i∈ J
function of prices. This is a J dimensional vector whose jth element is ∑
i∈ J
x ij ( p)
We can compare how much is demanded of each good to how much is supplied of each
good to calculate the excess demand:
The excess demand function is z(p) = ∑ x i ( p)=∑ ei ∈ R .
J
i∈ J i∈ J
When zj(p) > 0 we say there is excess demand of good j. When zj(p) < 0 we say there is
excess supply of good j.
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