Transcript Slide 1

Prerequisites
Almost essential
Firm: Optimisation
Consumption: Basics
Frank Cowell: Microeconomics
October 2006
Consumer Optimisation
MICROECONOMICS
Principles and Analysis
Frank Cowell
What we're going to do:
Frank Cowell: Microeconomics
Jump to
“Firm”

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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.
The problem
Frank Cowell: Microeconomics

Maximise consumer’s utility
U(x)

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

U assumed to satisfy the
standard “shape” axioms
n
S pixi
i=1
≤y
Overview...
Frank Cowell: Microeconomics
Two fundamental
views of
consumer
optimisation
Consumer:
Optimisation
Primal and
Dual problems
Lessons from
the Firm
Primal and
Dual again
An obvious approach?
Frank Cowell: Microeconomics

We now have the elements of a standard
constrained optimisation problem:
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The next steps might seem obvious:

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the constraints on the consumer.
the objective function.
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...
Frank Cowell: Microeconomics
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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
Frank Cowell: Microeconomics
The primal
problem



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
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
Frank Cowell: Microeconomics
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
But there's another way
at looking at this
The dual problem
Frank Cowell: Microeconomics
xz2
 Alternatively the consumer
could aim to minimise cost...
 Subject to utility constraint
uq
Constraint
set
 Defines the dual problem.
 Solution to the problem
 Cost minimisation by the
firm
minimise
n
S pixi


i=1
x*
z*
Contours of
objective function
subject to U(x)  u
xz1
But where have we seen
the dual problem before?
Two types of cost minimisation
Frank Cowell: Microeconomics
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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
Overview...
Frank Cowell: Microeconomics
Reusing results
on optimisation
Consumer:
Optimisation
Primal and
Dual problems
Lessons from
the Firm
Primal and
Dual again
A lesson from the firm
Frank Cowell: Microeconomics
 Compare cost-minimisation
for the firm...
...and for the consumer
z2
x2
q

u
 The difference
is only in notation
 So their
solution functions
and response
functions must be
the same

x*
z*
z1
x1
Run through
formal stuff
Frank Cowell: Microeconomics
Cost-minimisation: strictly
quasiconcave U

Minimise
Lagrange
multiplier
n
S pi xi
i=1
+ l[u
U(x)]
u – U(x)
Because of strict quasiconcavity we
have an interior solution.
 A set of n+1 First-Order Conditions

l* U1 (x* ) = p1
l* U2 (x* ) = p2
… … …
l* Un (x* ) = pn
u = U(x* )



one for
each good
utility
constraint
 Use the objective function
...and output constraint
...to build the Lagrangean
 Differentiate w.r.t. x1, ..., xn and
set equal to 0.
 ... and w.r.t l
 Denote cost minimising
values with a * .
If ICs can touch the axes...
Frank Cowell: Microeconomics

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*)



Can get “<” if optimal
value of this good is 0
Interpretation
From the FOC
Frank Cowell: Microeconomics
If both goods i and j are purchased
and MRS is defined then...

Ui(x*)
pi
———
= —
*
Uj(x )
pj

MRS = price ratio

If good i could be zero then...

MRSji  price ratio
 “implicit” price = market price
Ui(x*)
pi
———
 —
*
Uj(x )
pj
 “implicit” price  market price
Solution
The solution...
Frank Cowell: Microeconomics
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
Specified
utility level
Frank Cowell: Microeconomics
Jump to
“Firm”
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.

Other results follow
Frank Cowell: Microeconomics
Shephard's Lemma gives demand
as a function of prices and utility
H is the “compensated” or
conditional demand function.

Properties of the solution
function determine behaviour of
response functions.
Downward-sloping with respect
to its own price, etc…
“Short-run” results can be used
to model side constraints
For example rationing.

Hi(p, u) = Ci(p, u)
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Comparing firm and consumer
Frank Cowell: Microeconomics
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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
 Problem: min
z
 Solution
function:
S wizi + l[q – f(z)]
i=1
C(w, q)
 Response z * = Hi(w, q)
function: i
n
min
x
S pixi + l[u – U(x)]
i=1
C(p, u)
xi* = Hi(p, u)
Overview...
Frank Cowell: Microeconomics
Exploiting the
two approaches
Consumer:
Optimisation
Primal and
Dual problems
Lessons from
the Firm
Primal and
Dual again
The Primal and the Dual…
Frank Cowell: Microeconomics
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…
n
S pixi+ l[u – U(x)]
i=1

…constraint in the primal
becomes objective in the dual…


…and vice versa.
n
U(x) + m y – S pi xi
[
i=1
]
A neat connection
Frank Cowell: Microeconomics
 Compare the primal problem
of the consumer...
...with the dual problem
x2
x2
u
 The two are
equivalent
 So we can link up
their solution
functions and
response functions


x*
x*
x1
x1
Run through
the primal
Utility maximisation
Frank Cowell: Microeconomics

Maximise
Lagrange
multiplier
n
U(x) + m y – S pii xii
[
i=1
]
If U is strictly quasiconcave we have
an interior solution.
 A set of n+1 First-Order Conditions
If U not strictly
U1(x* ) = m* p1
one for
quasiconcave then
each good
U2(x* ) = m* p2
replace “=” by “”
… … …
Un(x* ) = m* pn

budget
constraint
i=1
 Differentiate w.r.t. x1, ..., xn and
set equal to 0.
 ... and w.r.t m
 Denote utility maximising
values with a * .



n
y =S
 Use the objective function
...and budget constraint
...to build the Lagrangean
pi xi*
Interpretation
From the FOC
Frank Cowell: Microeconomics
If both goods i and j are purchased
and MRS is defined then...

Ui(x*)
pi
———
= —
*
Uj(x )
pj

MRS = price ratio

If good i could be zero then...

MRSji  price ratio
(same as before)
 “implicit” price = market price
Ui(x*)
pi
———
 —
*
Uj(x )
pj
 “implicit” price  market price
Solution
The solution...
Frank Cowell: Microeconomics
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
money
income
A useful connection
Frank Cowell: Microeconomics
The indirect utility function maps
prices and budget into maximal utility
The indirect utility function works
like an "inverse" to the cost
function
The cost function maps prices and
utility into minimal budget
The two solution functions have
to be consistent with each other.
Two sides of the same coin

u = V(p, y)

y = C(p, u)

Therefore we have:
u = V(p, C(p,u))
y = C(p, V(p, y))
Odd-looking identities like these
can be useful
Frank Cowell: Microeconomics
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
But what’s
this…?
Roy's Identity
Frank Cowell: Microeconomics
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)
Utility and expenditure
Frank Cowell: Microeconomics
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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
[ S px]
 Problem: max U(x) + m y –
x
 Solution
function:
V(p, y)
 Response x * = Di(p, y)
function: i
i=1
i i
min
x
S pixi + l[u – U(x)]
i=1
C(p, u)
xi* = Hi(p, u)
Summary
Frank Cowell: Microeconomics

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:


2.
cost minimisation is reinterpreted from output targets
to utility targets.
The primal-dual insight:

utility maximisation subject to budget is equivalent to
cost minimisation subject to utility.
1. Cost minimisation: two applications
Frank Cowell: Microeconomics

THE FIRM

THE CONSUMER

min cost of inputs

min budget


subject to output
target
Solution is of the
form C(w,q)


subject to utility
target
Solution is of the
form C(p,u)
2. Consumer: equivalent approaches
Frank Cowell: Microeconomics

PRIMAL

DUAL

max utility

min budget


subject to budget
constraint
Solution is a
function of (p,y)


subject to utility
constraint
Solution is a
function of (p,u)
Basic functional relations
Frank Cowell: Microeconomics
Utility

C(p,u)
Review

Compensated demand
i
H (p,u) for good i
Review

V(p, y)
indirect utility

Di(p,
ordinary demand for
input i
Review
Review
y)
cost (expenditure)
money
income
H is also known as
"Hicksian" demand.
What next?
Frank Cowell: Microeconomics
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
