Consumer optimisation

```Prerequisites
Almost essential
Firm: Optimisation
Consumption: Basics
CONSUMER OPTIMISATION
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2012
Frank Cowell: Consumer Optimisation
1
What we’re going to do:
 We’ll solve the consumer's optimisation problem…
 …using methods that we've already introduced
 This enables us to re-cycle old techniques and results
 A tip:
• Run the presentation for firm optimisation…
• look for the points of comparison…
• and try to find as many reinterpretations as possible
March 2012
Frank Cowell: Consumer Optimisation
2
The problem
 Maximise consumer’s utility
U(x)
U assumed to satisfy the
standard “shape” axioms
 Subject to feasibility constraint
Assume consumption set X is the
non-negative orthant
 and to the budget constraint
The version with fixed money
income
x X
n
S pixi ≤ y
i=1
March 2012
Frank Cowell: Consumer Optimisation
3
Overview…
Consumer:
Optimisation
Two fundamental
views of consumer
optimisation
Primal and
Dual problems
Lessons from
the Firm
Primal and
Dual again
March 2012
Frank Cowell: Consumer Optimisation
4
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…
March 2012
Frank Cowell: Consumer Optimisation
5
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
March 2012
Frank Cowell: Consumer Optimisation
6
A five-point plan
The primal
problem
 Set out the basic consumer optimisation problem
 Show that the solution is equivalent to another
The dual
problem
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…
March 2012
Frank Cowell: Consumer Optimisation
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The primal problem
x2
 The consumer aims to
maximise utility…
 Subject to budget constraint
Contours of
objective function
 Defines the primal problem
 Solution to primal problem
Constraint
set
max U(x) subject to

n
S pixi y
x*
i=1
x1
March 2012
But there's another way
of looking at this
Frank Cowell: Consumer Optimisation
8
The dual problem
z2x2
 Alternatively the consumer
could aim to minimise cost…
 Subject to utility constraint
q
u
Constraint
set
 Defines the dual problem
 Solution to the problem
 Cost minimisation by the firm
minimise
n
S pixi


x*
z*
i=1
subject to U(x)  u
z1
x1
March 2012
But where have we seen
the dual problem before?
Frank Cowell: Consumer Optimisation
9
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
March 2012
Frank Cowell: Consumer Optimisation
10
Overview…
Consumer:
Optimisation
Reusing results
on optimisation
Primal and
Dual problems
Lessons from
the Firm
Primal and
Dual again
March 2012
Frank Cowell: Consumer Optimisation
11
A lesson from the firm
 Compare cost-minimisation
for the firm…
…and for the consumer
x2 u
z2 q

 So their
solution functions
and response
functions must be
the same

x*
z*
z1
March 2012
 The difference
is only in notation
x1
Frank Cowell: Consumer Optimisation
Run through
formal stuff
12
Cost-minimisation: strictly quasiconcave U
 Minimise
Lagrange
multiplier
n
S pi xi
i=1
 Use the objective function
…and output constraint
…to build the Lagrangean
+ λ[u 
– U(x)]
U(x)
 Because of strict quasiconcavity we
have an interior solution
 Differentiate w.r.t. x1, …, xn and set
equal to 0
 … and w.r.t l
 Denote cost minimising values with a *
 A set of n+1 First-Order Conditions
l* U1 (x* ) = p1
one for
each good
l* U2 (x* ) = p2
… … …
l* Un (x* ) = pn



u = U(x* )
March 2012
utility
constraint
Frank Cowell: Consumer Optimisation
13
If ICs can touch the axes…
 Minimise
n
Spixi
i=1
+ l[u – U(x)]
 Now there is the possibility of corner solutions
 A set of n+1 First-Order Conditions
l*U1 (x*)  p1
l*U2 (x*)  p2
… … …
l*Un(x*)  pn
u = U(x*)
March 2012



Interpretation
Can get “<” if optimal
value of this good is 0
Frank Cowell: Consumer Optimisation
14
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
March 2012
Frank Cowell: Consumer Optimisation
15
The solution…
 Solving the FOC, you get a cost-minimising value for each
good…
xi* = Hi(p, u)
 …for the Lagrange multiplier
l* = l*(p, u)
 …and for the minimised value of cost itself
 The consumer’s cost function or expenditure function is defined
as
C(p, u) := min S pi xi
{U(x) u}
vector of
goods prices
March 2012
Specified
utility level
Frank Cowell: Consumer Optimisation
16
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 u
 Concave in p
 Homogeneous of degree 1 in all prices p
 Shephard's lemma
March 2012
Frank Cowell: Consumer Optimisation
17
Other results follow
 Shephard's Lemma gives
H is the “compensated” or
demand as a function of prices conditional demand function
and utility
Hi(p, u) = Ci(p, u)

Downward-sloping with respect
Properties of the solution
function determine behaviour to its own price, etc…
of response functions

For example rationing
“Short-run” results can be
used to model side constraints
March 2012
Frank Cowell: Consumer Optimisation
18
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:
Consumer
Firm
m
n
min Swizi + l[q – f(z)]
min Spixi + l[u – U(x)]
 Solution
function:
C(w, q)
C(p, u)
 Response
function:
zi* = Hi(w, q)
xi* = Hi(p, u)
 Problem:
z
March 2012
i=1
x
i=1
Frank Cowell: Consumer Optimisation
19
Overview…
Consumer:
Optimisation
Exploiting the
two approaches
Primal and
Dual problems
Lessons from
the Firm
Primal and
Dual again
March 2012
Frank Cowell: Consumer Optimisation
20
The Primal and the Dual…
 There’s an attractive symmetry
about the two approaches to the
problem
 In both cases the ps are given
and you choose the xs But…
 …constraint in the primal
becomes objective in the dual…
n
S pixi+ l[u – U(x)]
i=1
n
U(x) + m y – S pi xi
[
]
i=1
 …and vice versa
March 2012
Frank Cowell: Consumer Optimisation
21
A neat connection
 Compare the primal
problem of the consumer…
…with the dual problem
x2 u
x2
 The two are
equivalent
 So we can link up
their solution
functions and
response functions


x*
x*
x1
March 2012
x1
Run through
the primal
Frank Cowell: Consumer Optimisation
22
Utility maximisation
 Maximise
Lagrange
multiplier
nn
– Spi xi
U(x) + μ y 
[
i=1
i=1
 Use the objective function
…and budget constraint
…to build the Lagrangean
]
 If U is strictly quasiconcave we have
an interior solution
 A set of n+1 First-Order
Conditions
one for
each good
U1(x* ) = m* p1
If U not strictly
U2(x* ) = m* p2
quasiconcave then
… … …
replace “=” by “”
budget
Un(x* ) = m* pn
constraint
n
y = S pi xi*



 Differentiate w.r.t. x1, …, xn and
set equal to 0
 … and w.r.t m
 Denote utility maximising
values with a *
Interpretation
i=1
March 2012
Frank Cowell: Consumer Optimisation
23
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
(same as before)
 “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
March 2012
Frank Cowell: Consumer Optimisation
24
The solution…
 Solving the FOC, you get a utility-maximising value for each
good…
xi* = Di(p, y)
 …for the Lagrange multiplier
m* = m*(p, y)
 …and for the maximised value of utility itself
 The indirect utility function is defined as
V(p, y) := max U(x)
{S pixi y}
vector of
goods prices
March 2012
money
income
Frank Cowell: Consumer Optimisation
25
A useful connection
 The indirect utility function
maps prices and budget into
maximal utility
The indirect utility function works
like an "inverse" to the cost
function
u= V(p, y)
 The cost function maps prices The two solution functions have
and utility into minimal budget to be consistent with each other.
y = C(p, u)
 Therefore we have:
u= V(p, C(p, u))
y = C(p, V(p, y))
March 2012
Two sides of the same coin
Odd-looking identities like these
can be useful
Frank Cowell: Consumer Optimisation
26
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)
 Roy's Identity
March 2012
But what’s
this…?
Frank Cowell: Consumer Optimisation
27
Roy's Identity
u = V(p, y)= V(p, C(p,u))
“function-of-afunction” rule
0 = Vi(p,C(p,u)) + Vy(p,C(p,u)) Ci(p,u)
 Use the definition of the
optimum
 Differentiate w.r.t. pi
 Use Shephard’s Lemma
 Rearrange to get…
0 = Vi(p, y)
+ Vy(p, y)
xi*
 So we also have…
Marginal disutility
of price i
Vi(p, y)
xi* = – ————
Vy(p, y)
Marginal utility of
money income
Ordinary demand
function
xi* = –Vi(p, y)/Vy(p, y) = Di(p, y)
March 2012
Frank Cowell: Consumer Optimisation
28
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
[
 Problem: max U(x) + μ y –
x
 Solution
function:
V(p, y)
 Response x * = Di(p, y)
function: i
March 2012
n
Spi xi ]
i=1
n
min S pixi + l[u – U(x)]
x
i=1
C(p, u)
xi* = Hi(p, u)
Frank Cowell: Consumer Optimisation
29
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
March 2012
Frank Cowell: Consumer Optimisation
30
1. Cost minimisation: two applications
 THE FIRM
 THE CONSUMER
 min cost of inputs
 min budget
 subject to output
target
 subject to utility
target
 Solution is of the
form C(w,q)
 Solution is of the
form C(p,u)
March 2012
Frank Cowell: Consumer Optimisation
31
2. Consumer: equivalent approaches

PRIMAL
 DUAL

max utility
 min budget
subject to budget
constraint
 subject to utility
constraint
Solution is a
function of (p,y)
 Solution is a
function of (p,u)


March 2012
Frank Cowell: Consumer Optimisation
32
Basic functional relations
Utility

C(p,u)

Compensated
i
H (p,u)
Review

V(p, y)
indirect utility
Review

Di(p, y)
ordinary demand for
input i
Review
Review
cost (expenditure)
H is also known as
"Hicksian" demand
demand for good I
money
income
March 2012
Frank Cowell: Consumer Optimisation
33
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
March 2012
Frank Cowell: Consumer Optimisation
34
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