Consumption basics

Frank Cowell: Microeconomics

Consumption Basics

MICROECONOMICS
Principles and Analysis
Frank Cowell

October 2006

Overview... Consumption:
Basics

The setting

The environment
for the basic Budget sets
consumer
optimisation
problem. Revealed
Preference

Axiomatic
Approach

Notation

 Quantities
a ―basket
xi of goods •amount of commodity i
x = (x1, x2 , ..., xn) •commodity vector

X •consumption set
x X denotes
feasibility
 Prices
pi •price of commodity i
p = (p1 , p2 ,..., pn) •price vector

y •income

Things that shape the consumer's
problem

 The set X and the number y are both important.
 But they are associated with two distinct types of
constraint.
 We'll save y for later and handle X now.
 (And we haven't said anything yet about
objectives...)

The consumption set

 The set X describes the basic entities of the
consumption problem.
 Not a description of the consumer’s opportunities.
 That comes later.
 Use it to make clear the type of choice problem we
are dealing with; for example:
 Discrete versus continuous choice (refrigerators vs.
contents of refrigerators)
 Is negative consumption ruled out?
 ―x X ‖ means ―x belongs the set of logically
feasible baskets.‖

The set X: standard assumptions

Axes indicate quantities of
x2 the two goods x1 and x2.
Usually assume that X
consists of the whole non-
negative orthant.
Zero consumptions make
good economic sense
But negative consumptions
ruled out by definition

no points  Consumption goods are
here…
(theoretically) divisible…
 …and indefinitely
x1 extendable…
…or here
 But only in the ++
direction

Rules out this case...

Consumption set X
x2 consists of a countable
number of points

 Conventional assumption
does not allow for
indivisible objects.
x1  But suitably modified
assumptions may be
appropriate

... and this

Consumption set X has
x2 holes in it

x1

... and this

Consumption set X has
x2 ˉ
the restriction x1 < x

 Conventional assumption
does not allow for physical
upper bounds
x1  But there are several
ˉ
x economic applications
where this is relevant

Basics

The setting

Budget
constraints: Budget sets
prices, incomes
and resources
Revealed
Preference

Axiomatic
Approach

The budget constraint

The budget constraint
x2 typically looks like this
Slope is determined by
price ratio.
“Distance out” of budget
line fixed by income or
resources

Two important subcases
determined by
1. … amount of money
income y.
p1
– __
p2 2. …vector of resources R

x1 Let’s see

Case 1: fixed nominal income

y  Budget constraint
__
.

x2 .

p2 determined by the two end-
points
 Examine the effect of
 changing p1 by “swinging”
the boundary thus…

 Budget constraint is
n
pixi ≤ y
i=1
y
__
.

.

p1

 x1

Case 2: fixed resource endowment

 Budget constraint
x2 determined by location of
“resources” endowment R.
 Examine the effect of
changing p1 by “swinging”
the boundary thus…

n  Budget constraint is
y= piRi n n
i=1 pixi ≤ piRi
i=1 i=1
R

x1

Budget constraint: Key points

 Slope of the budget constraint given by price ratio.
 There is more than one way of specifying
―income‖:
 Determined exogenously as an amount y.
 Determined endogenously from resources.
 The exact specification can affect behaviour when
prices change.
 Take care when income is endogenous.
 Value of income is determined by prices.

Basics

The setting

Deducing
preference from Budget sets
market
behaviour?
Revealed
Preference

Axiomatic
Approach

A basic problem

 In the case of the firm we have an observable
constraint set (input requirement set)…
 …and we can reasonably assume an obvious
objective function (profits)
 But, for the consumer it is more difficult.
 We have an observable constraint set (budget
set)…
 But what objective function?

The Axiomatic Approach

 We could ―invent‖ an objective function.
 This is more reasonable than it may sound:
 It is the standard approach.
 See later in this presentation.
 But some argue that we should only use what we
can observe:
 Test from market data?
 The ―revealed preference‖ approach.
 Deal with this now.
 Could we develop a coherent theory on this basis
alone?

Using observables only

 Model the opportunities faced by a consumer
 Observe the choices made
 Introduce some minimal ―consistency‖ axioms
 Use them to derive testable predictions about
consumer behaviour

―Revealed Preference‖

 Let market prices
x2 determine a person's budget
constraint..
Suppose the person
chooses bundle x...
x is example x is
For revealed
preferred to all
revealed  Use this to introduce
these points.x′
preferred to Revealed Preference

 x′
x

x1

Axioms of Revealed Preference

 Axiom of Rational Choice Essential if observations are to
have meaning
the consumer always makes a
choice, and selects the most
preferred bundle that is available.
Weak Axiom of Revealed If x was chosen when x' was
Preference (WARP) available then x' can never be
chosen whenever x is available
If x RP x' then x' not-RP x.

WARP is more powerful than might be thought

WARP in the market

Suppose that x is chosen when
prices are p.
If x' is also affordable at p then:

Now suppose x' is chosen at
prices p'
This must mean that x is not
affordable at p':

Otherwise it would graphical
violate WARP interpretation

WARP in action

 Take the original equilibrium
x2
 Now let the prices change...
Could we have chosen x
on Monday? x violates
WARP rules out some points
WARP; x does not.
as possible solutions

Tuesday's choice:
On Monday we could have
x afforded Tuesday’s bundle

Clearly WARP
 x′ induces a kind of
Monday's negative substitution
choice:
effect
 x
 But could we extend
x1 this idea...?

Trying to Extend WARP

Take the basic idea of
x2 revealed preference
x″ is revealed
preferred to all  Invoke revealed preference
these points. again
 Invoke revealed preference
yet again
 x'' x' is revealed  Draw the “envelope”
preferred to all
these points.

 x'
x is revealed  Is this an “indifference
preferred to all
these points. curve”...?
 x No. Why?
x1

Limitations of WARP

WARP rules out this
pattern
...but not this

x x′

 WARP does not rule out
cycles of preference
 You need an extra axiom
to progress further on this:
x″′ x″ the strong axiom of
revealed preference.

Revealed Preference: is it useful?

 You can get a lot from just a little:
 You can even work out substitution effects.
 WARP provides a simple consistency test:
 Useful when considering consumers en masse.
 WARP will be used in this way later on.
 You do not need any special assumptions
about consumer's motives:
 But that's what we're going to try right now.
 It’s time to look at the mainstream modelling of
preferences.

Basics

The setting

Standard
approach to Budget sets
modelling
preferences
Revealed
Preference

Axiomatic
Approach

The Axiomatic Approach

 Useful for setting out a priori what we mean
by consumer preferences
 But, be careful...
 ...axioms can't be ―right‖ or ―wrong,‖...
 ... although they could be inappropriate or
over-restrictive
 That depends on what you want to model
 Let's start with the basic relation...

The (weak) preference relation

 The basic weak-preference "Basket x is regarded as at
relation: least as good as basket x' ..."

x < x'
 From this we can derive the “ x < x' ” and “ x' < x. ”
indifference relation.

x v x'
 …and the strict preference “ x < x' ” and not “ x' < x. ”
relation…

x Â x'

Fundamental preference axioms

 Completeness For every x, x' X either x<x' is true, or
x'<x is true, or both statements are true
 Transitivity
 Continuity
 Greed
 (Strict) Quasi-concavity
 Smoothness


 Completeness
 Transitivity For all x, x' , x″ X if x<x' and x'<x″
then x<x'″.
 Continuity
 Greed
 Smoothness


 Completeness
 Transitivity
 Continuity For all x' X the not-better-than-x' set and
the not-worse-than-x' set are closed in X
 Greed
 Smoothness

Continuity: an example

Take consumption bundle x .
x2  Construct two other
bundles, xL with Less than
x , xM with More
Better  There is a set of points like
than x ? xL, and a set like xM
do we jump straight from
 Draw a path joining xL , xM.
a point marked ―better‖ to
M
 x
one marked ―worse"?  If there’s no “jump”…
 x but what about the
boundary points
between the two?
The indifference
 xL curve

Worse
than x ?
x1

Axioms 1 to 3 are crucial ...

completeness

transitivity
The utility
continuity function

A continuous utility function then
represents preferences...

x < x' U(x) U(x')

Tricks with utility functions

 U-functions represent preference
orderings.
 So the utility scales don’t matter.
 And you can transform the U-function in
any (monotonic) way you want...

Irrelevance of cardinalisation

 U(x1, x2,..., xn)  So take any utility function...
 This transformation
represents the same
preferences...
 log( U(x1, x2,..., xn) )  …and so do both of these
 And, for any monotone
increasing φ, this represents
 exp( U(x1, x2,..., xn) ) the same preferences.

 ( U(x1, x2,..., xn) )  U is defined up to a
monotonic transformation
 φ( U(x1, x2,..., xn) ) Each of these forms will
generate the same
contours.
Let’s view this graphically.

A utility function

 Take a slice at given utility level
 Project down to get contours

U(x1,x2)

The indifference
curve
x2
0

Another utility function

 By construction U* = φ(U)
 Again take a slice…
U*(x1,x2)  Project down …

The same
indifference curve
x2
0

Assumptions to give the U-function
shape

 Completeness
 Transitivity
 Continuity
 Greed
 Smoothness

The greed axiom

Pick any consumption
x2 bundle in X.
Greed implies that these
bundles are preferred to x'.
Gives a clear “North-East”
direction of preference.

 Bliss!
B What can happen if
consumers are not greedy

 Greed: utility function is
monotonic
x' 

x1

A key mathematical concept

 We’ve previously used the concept of concavity:
 Shape of the production function.
 But here simple concavity is inappropriate:
 The U-function is defined only up to a monotonic transformation.
 U may be concave and U2 non-concave even though they represent
Review
Example
the same preferences.
 So we use the concept of ―quasi-concavity‖:
 ―Quasi-concave‖ is equivalently known as ―concave contoured‖.
 A concave-contoured function has the same contours as a concave
function (the above example).
 Somewhat confusingly, when you draw the IC in (x1,x2)-space,
common parlance describes these as ―convex to the origin.‖
 It’s important to get your head round this:
 Some examples of ICs coming up…

Conventionally shaped indifference
curves

Slope well-defined
x2 everywhere
Pick two points on the
same indifference curve.
Draw the line joining them.
A  Any interior point must line
on a higher indifference
curve

 C  ICs are smooth
…and strictly concaved-
contoured
B
(-)I.e. strictly quasiconcave
Slope is the Marginal
Rate of Substitution
sometimes these
x1 U1(x)
—— ..
assumptions can
U2be relaxed
(x) .

Other types of IC: Kinks

Strictly quasiconcave
x2
But not everywhere smooth

A

 C
MRS not
defined here

B

x1

Other types of IC: not strictly
quasiconcave

Slope well-defined
x2 everywhere
Not quasiconcave
Quasiconcave but not
strictly quasiconcave
utility here lower
than at A or B
A

 C
B

Indifference curves
Indifference curve
follows axis here with flat sections make
sense
x1 But may be a little
harder to work with...

Summary: why preferences can be a
problem

 Unlike firms there is no ―obvious‖ objective
function.
 Unlike firms there is no observable
objective function.
 And who is to say what constitutes a ―good‖
assumption about preferences...?

Review: basic concepts

Review
 Consumer’s environment
Review
 How budget sets work
Review
 WARP and its meaning
Review  Axioms that give you a utility function
Review
 Axioms that determine its shape

What next?

 Setting up consumer’s optimisation problem
 Comparison with that of the firm
 Solution concepts.

Prerequisites

Almost essential
Firm: Optimisation
Consumption: Basics

Consumer Optimisation

MICROECONOMICS
Principles and Analysis
Frank Cowell

October 2006

The problem

 Maximise consumer’s utility U assumed to satisfy the
U(x) standard “shape” axioms

 Subject to feasibility constraint Assume consumption set X is
x X the non-negative orthant.

 and to the budget constraint The version with fixed money
n income
pixi ≤ y
i=1

Overview... Consumer:
Optimisation

Primal and
Two fundamental Dual problems
views of
consumer Lessons from
optimisation the Firm

Primal and
Dual again

An obvious approach?

 We now have the elements of a standard
constrained optimisation problem:
 the constraints on the consumer.
 the objective function.
 The next steps might seem obvious:
 set up a standard Lagrangean.
 solve it.
 interpret the solution.
 But the obvious approach is not always the most
insightful.
 We’re going to try something a little sneakier…

Think laterally...

 In microeconomics an optimisation problem can
often be represented in more than one form.
 Which form you use depends on the information
you want to get from the solution.
 This applies here.
 The same consumer optimisation problem can be
seen in two different ways.
 I’ve used the labels ―primal‖ and ―dual‖ that have
become standard in the literature.

A five-point plan
The primal

problem
 Set out the basic consumer optimisation
problem. The dual
problem
 Show that the solution is equivalent to
another problem.
 Show that this equivalent problem is
identical to that of the firm. The primal
problem again
 Write down the solution.
 Go back to the problem we first thought of...

The primal problem

Contours of  The consumer aims to
x2 maximise utility...
objective function
 Subject to budget constraint
 Defines the primal problem.
 Solution to primal problem

Constraint
set
max U(x) subject to
n
 x* pixi y
i=1

But there's another way
x1 at looking at this

The dual problem

 Alternatively the consumer
x2
z
q could aim to minimise cost...
Constraint  Subject to utility constraint
set  Defines the dual problem.
 Solution to the problem
 Cost minimisation by the
firm

minimise
n
pixi

 i=1
x*
z* subject to U(x)

Contours of x1
z
But where have we seen
objective function
the dual problem before?

Two types of cost minimisation

 The similarity between the two problems is not
just a curiosity.
 We can use it to save ourselves work.
 All the results that we had for the firm's ―stage 1‖
problem can be used.
 We just need to ―translate‖ them intelligently
 Swap over the symbols
 Swap over the terminology
 Relabel the theorems

Optimisation

Primal and
Reusing results Dual problems
on optimisation
Lessons from
the Firm

Primal and
Dual again

A lesson from the firm

 Compare cost-minimisation
for the firm...
...and for the consumer

z2 q x2  The difference
is only in notation

 So their
solution functions
and response
functions must be
  the same
z* x*
Run through
z1 x1 formal stuff

Cost-minimisation: strictly
quasiconcave U

 Use the objective function
 Minimise Lagrange ...and output constraint
n multiplier ...to build the Lagrangean
pi xi + [ – U(x)]
U(x)  Differentiate w.r.t. x1, ..., xn and
i=1 set equal to 0.
 ... and w.r.t
 Because of strict quasiconcavity we  Denote cost minimising
have an interior solution. values with a * .

 A set of n+1 First-Order Conditions

U1 (x ) = p1 one for
U2 (x ) = p2 each good

… … …
Un (x ) = pn
= U(x ) utility
constraint

If ICs can touch the axes...

 Minimise
n
pixi + [ – U(x)]
i=1
 Now there is the possibility of corner
solutions.

U1 (x ) p1
U2 (x ) p2
… … …
Un(x ) pn
Interpretation
= U(x ) Can get ―<‖ if optimal
value of this good is 0

From the FOC

If both goods i and j are purchased
and MRS is defined then...
Ui(x ) pi
——— = —
Uj(x ) pj
 MRS = price ratio  “implicit” price = market price

 If good i could be zero then...
Ui(x ) pi
——— —
Uj(x ) pj
 MRSji price ratio  “implicit” price market price
Solution

The solution...
Solving the FOC, you get a cost-minimising value for



each good...

xi* = Hi(p, )

 ...for the Lagrange multiplier
* = *(p, )
 ...and for the minimised value of cost itself.

 The consumer’s cost function or expenditure function is

defined as
C(p, ) := min pi xi
{U(x) }

vector of
goods prices Specified
utility level

The cost function has the same
properties as for the firm

 Non-decreasing in every price. Increasing in
at least one price
 Increasing in utility .
 Concave in p
Jump to
“Firm”
 Homogeneous of degree 1 in all prices p.
 Shephard's lemma.

Other results follow

 Shephard's Lemma gives demand H is the “compensated” or
as a function of prices and utility conditional demand function.
Hi(p, ) = Ci(p, )

 Properties of the solution Downward-sloping with respect
function determine behaviour of to its own price, etc…
response functions.

 ―Short-run‖ results can be used For example rationing.
to model side constraints

Comparing firm and consumer

 Cost-minimisation by the firm...
 ...and expenditure-minimisation by the consumer
 ...are effectively identical problems.
 So the solution and response functions are the same:

Firm Consumer
m n
 Problem: min wizi + [q – (z)] min pixi + [ – U(x)]
z i=1 x i=1
 Solution
function:
C(w, q) C(p, )
 Response z * = Hi(w, q) xi* = Hi(p, )
function: i

Optimisation

Primal and
Exploiting the Dual problems
two approaches
Lessons from
the Firm

Primal and
Dual again

The Primal and the Dual…

There’s an attractive symmetry
about the two approaches to the n
problem pixi+ [ – U(x)]
i=1
In both cases the ps are given
and you choose the xs. But… n
U(x) + [y– pi xi ]
…constraint in the primal i=1
becomes objective in the dual…

 …and vice versa.

A neat connection

 Compare the primal problem
of the consumer...
...with the dual problem

x2 x2  The two are
equivalent

 So we can link up
their solution
functions and
response functions
 x* 
x*
Run through
x1 x1 the primal

Utility maximisation

Lagrange  Use the objective function
 Maximise multiplier ...and budget constraint
n ...to build the Lagrangean
U(x) + [ y – i=1 p x ] ii ii
 Differentiate w.r.t. x1, ..., xn and
set equal to 0.
i=1
 ... and w.r.t
 If U is strictly quasiconcave we have  Denote utility maximising
an interior solution. values with a * .


U1(x ) = p1 If U not strictly
one for
quasiconcave then
U2(x ) = p2 each good
replace ―=‖ by ― ‖
… … …
Un(x ) = pn
budget n
constraint Interpretation
y = pi xi
i=1

From the FOC

If both goods i and j are purchased
and MRS is defined then...
Ui(x ) pi (same as before)
——— = —
Uj(x ) pj
 MRS = price ratio  “implicit” price = market price

 If good i could be zero then...
Ui(x ) pi
——— —
Uj(x ) pj
 MRSji price ratio  “implicit” price market price
Solution

The solution...

 Solving the FOC, you get a utility-maximising value for
each good...
xi* = Di(p, y)
 ...for the Lagrange multiplier
* = *(p, y)
 ...and for the maximised value of utility itself.
 The indirect utility function is defined as

V(p, y) := max U(x)
{ pixi y}
vector of money
goods prices income

A useful connection

 The indirect utility function maps The indirect utility function works
prices and budget into maximal utility like an "inverse" to the cost
= V(p, y) function

 The cost function maps prices and The two solution functions have
utility into minimal budget to be consistent with each other.
y = C(p, ) Two sides of the same coin

 Therefore we have:
Odd-looking identities like these
= V(p, C(p, )) can be useful
y = C(p, V(p, y))

The Indirect Utility Function has
some familiar properties...

(All of these can be established using the known
properties of the cost function)

 Non-increasing in every price. Decreasing in at
least one price
 Increasing in income y.
 quasi-convex in prices p
 Homogeneous of degree zero in (p, y)
But what’s
this…?
 Roy's Identity

Roy's Identity

= V(p, y)= V(p, C(p, )) ―function-of-a-  Use the definition of the
function‖ rule optimum
 Differentiate w.r.t. pi .
0 = Vi(p,C(p, )) + Vy(p,C(p, )) Ci(p, )  Use Shephard’s Lemma
 Rearrange to get…
 So we also have…
0 = Vi(p, y) + Vy(p, y) xi*
Marginal disutility
of price i
Vi(p, y) Marginal utility of
xi* = – ———— money income
Vy(p, y)
Ordinary demand
function
xi* = –Vi(p, y)/Vy(p, y) = Di(p, y)

Utility and expenditure

 Utility maximisation
 ...and expenditure-minimisation by the consumer
 ...are effectively two aspects of the same problem.
 So their solution and response functions are closely connected:

Primal Dual
n n
 Problem: max U(x) + [y – pixi ] min
x i=1
pixi + [ – U(x)]
x i=1
 Solution
function:
V(p, y) C(p, )
 Response x * = Di(p, y) xi* = Hi(p, )
function: i

Summary

 A lot of the basic results of the consumer theory
can be found without too much hard work.
 We need two ―tricks‖:

1. A simple relabelling exercise:
 cost minimisation is reinterpreted from output targets
to utility targets.
2. The primal-dual insight:
 utility maximisation subject to budget is equivalent to
cost minimisation subject to utility.

1. Cost minimisation: two applications

 THE FIRM  THE CONSUMER

 min cost of inputs  min budget

 subject to output  subject to utility
target target

 Solution is of the  Solution is of the
form C(w,q) form C(p, )

2. Consumer: equivalent approaches

 PRIMAL  DUAL

 max utility  min budget

 subject to budget  subject to utility
constraint constraint

 Solution is a  Solution is a
function of (p,y) function of (p, )

Basic functional relations

Utility

Review
 C(p, ) cost (expenditure) H is also known as
"Hicksian" demand.

Compensated demand
Review  Hi(p, ) for good i

Review  V(p, y) indirect utility

ordinary demand for
Review
 Di(p, y) input i
money
income

What next?

 Examine the response of consumer demand
to changes in prices and incomes.
 Household supply of goods to the market.
 Develop the concept of consumer welfare

Consumption basics

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