Consumer optimisation
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Transcript 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
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Frank Cowell: Consumer Optimisation
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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
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Overview…
Consumer:
Optimisation
Two fundamental
views of consumer
optimisation
Primal and
Dual problems
Lessons from
the Firm
Primal and
Dual again
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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…
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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
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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…
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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
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But there's another way
of looking at this
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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
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But where have we seen
the dual problem before?
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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
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Overview…
Consumer:
Optimisation
Reusing results
on optimisation
Primal and
Dual problems
Lessons from
the Firm
Primal and
Dual again
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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
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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* )
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utility
constraint
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If ICs can touch the axes…
Minimise
n
Spixi
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*)
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Interpretation
Can get “<” if optimal
value of this good is 0
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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
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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
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Specified
utility level
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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
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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
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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 Swizi + l[q – f(z)]
min Spixi + 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
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Overview…
Consumer:
Optimisation
Exploiting the
two approaches
Primal and
Dual problems
Lessons from
the Firm
Primal and
Dual again
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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
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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
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x1
Run through
the primal
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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
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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
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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
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money
income
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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))
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Two sides of the same coin
Odd-looking identities like these
can be useful
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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
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But what’s
this…?
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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)
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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)
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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
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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)
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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)
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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
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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
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