Prerequisites
Almost essential
Consumer: Optimisation
Useful, but optional
Firm: Optimisation
HOUSEHOLD DEMAND
AND SUPPLY
MICROECONOMICS
Principles and Analysis
Frank Cowell
Note: the detail in slides marked “ * ” can
only be seen if you run the slideshow
October 2015
Frank Cowell: Household Demand & Supply
1
Working out consumer responses
 The analysis of consumer optimisation gives us some
powerful tools:
• The primal problem of the consumer is what we are really
interested in
• Related dual problem can help us understand it
• The analogy with the firm helps solve the dual
 Use earlier work to map out the consumer's responses
• to changes in prices
• to changes in income
what we know
about the primal
October 2015
Frank Cowell: Household Demand & Supply
2
Overview
Household
Demand & Supply
Response
functions
The basics of the
consumer demand
system
Slutsky
equation
Supply of
factors
Examples
October 2015
Frank Cowell: Household Demand & Supply
3
Solving the max-utility problem

The primal problem and its solution
n
max U(x) + m[ y – S pi xi ]
Lagrangean for the max U problem
i=1
U1(x*) = mp1
U2(x*) = mp2
… … …
Un(x*) = mpn



 The n+1 first-order conditions,
assuming all goods purchased
S pixi* = y
n
i=1

Solve this set of equations:
x1* = D1(p, y)
x2* = D2(p, y)
… … …
xn* = Dn(p, y)
n
S piDi(p, y) = y



Gives a set of demand functions, one
for each good: functions of prices and
incomes
A restriction on the n equations. Follows
from the budget constraint
i=1
October 2015
Frank Cowell: Household Demand & Supply
4
The response function
 The response function for the primal
problem is demand for good i:
xi* = Di(p,y)
 The system of equations must have
an “adding-up” property:
n
Spi Di(p, y) = y
Should be treated as just one
of a set of n equations
Reason? Follows immediately
from the budget constraint: lefthand side is total expenditure
i=1
 Each equation in the system must be
 Reason? Again follows from
homogeneous of degree 0 in prices and the budget constraint
income. For any t > 0:
xi* = Di(p, y )= Di(tp, ty)
To make more progress we need to exploit the relationship
between primal and dual approaches again…
October 2015
Frank Cowell: Household Demand & Supply
5
How you would use this in practice…
 Consumer surveys give data on expenditure for each
household over a number of categories…
 …and perhaps income, hours worked etc as well
 Market data are available on prices
 Given some assumptions about the structure of
preferences…
 …we can estimate household demand functions for
commodities
 From this we can recover information about utility
functions
October 2015
Frank Cowell: Household Demand & Supply
6
Overview
Household
Demand & Supply
Response
functions
A fundamental
decomposition of
the effects of a
price change
Slutsky
equation
Supply of
factors
Examples
October 2015
Frank Cowell: Household Demand & Supply
7
Consumer’s demand responses
 What’s the effect of a budget change on demand?
 Depends on the type of budget constraint
• Fixed income?
• Income endogenously determined?
 And on the type of budget change
• Income alone?
• Price in primal type problem?
• Price in dual type problem?
 So let’s tackle the question in stages
 Begin with a type 1 (exogenous income) budget
constraint
October 2015
Frank Cowell: Household Demand & Supply
8
Effect of a change in income
 Take the basic equilibrium
 Suppose income rises
x2
The effect of the income increase
 Demand for each good does not
fall if it is “normal”
 x**
x*
 But could the opposite happen?

x1
October 2015
Frank Cowell: Household Demand & Supply
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An “inferior” good
 Take same original prices, but
different preferences
 Again suppose income rises
x2
The effect of the income increase
 Demand for good 1 rises,
but…
 Demand for “inferior” good 2
falls a little
x*

 Can you think of any goods
like this?
 x**
How might it depend on the
categorisation of goods?
x1
October 2015
Frank Cowell: Household Demand & Supply
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A glimpse ahead…
 We can use the idea of an “income effect” in many applications
 Basic to an understanding of the effects of prices on the
consumer
 Because a price cut makes a person better off, as would an
income increase
October 2015
Frank Cowell: Household Demand & Supply
11
Effect of a change in price
 Again take the basic equilibrium
x2
 Allow price of good 1 to fall
The effect of the price fall
The “journey” from x* to x**
broken into two parts
incomesubstitution
effect effect
°
x*


x**
x1
October 2015
Frank Cowell: Household Demand & Supply
12
And now let’s look at it in maths
 We want to take both primal and dual aspects of the problem…
 …and work out the relationship between the response
functions…
 … using properties of the solution functions
 (Yes, it’s time for Shephard’s lemma again…)
October 2015
Frank Cowell: Household Demand & Supply
13
A fundamental decomposition
compensated
demand
ordinary
demand
 Take the two methods of writing xi*:  Remember: they are two ways of
representing the same thing
Hi(p,u) = Di(p,y)
 Use cost function to substitute for y: Gives us an implicit relation in
prices and utility
Hi(p,u) = Di(p, C(p,u))
 Differentiate with respect to pj :
Hji(p,u) = Dji(p,y) + Dyi(p,y)Cj(p,u)
 Uses y = C(p,u) and function-of-afunction rule again
 Simplify :
Hji(p,u) = Dji(p,y) + Dyi(p,y) Hj(p,u)
Using cost function and Shephard’s
Lemma
= Dji(p,y) + Dyi(p,y) xj*
 And so we get:
Dji(p,y) = Hji(p,u) – xj*Dyi(p,y)
October 2015
 From the comp. demand function
 This is the Slutsky equation
Frank Cowell: Household Demand & Supply
14
*The Slutsky equation
Dji(p,y) = Hji(p,u) – xj* Dyi(p,y)
 Gives fundamental breakdown
of effects of a price change


x*
x**
 Income effect: “I'm better off if
the price of jelly falls; I’m worse
off if the price of jelly rises. The
size of the effect depends on
how much jelly I am buying…
 …if the price change makes me
better off then I buy more normal
goods, such as icecream”
 Substitution effect: “When the
price of jelly falls and I’m kept on
the same utility level, I prefer to
switch from icecream for dessert”
October 2015
Frank Cowell: Household Demand & Supply
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Slutsky: Points to watch
 Income effects for some goods may have “wrong” sign
• for inferior goods…
• …get opposite effect to that on previous slide
 For n > 2 the substitution effect for some pairs of goods could
be positive…
• net substitutes
• apples and bananas?
 … while that for others could be negative
• net complements
• gin and tonic?
 Neat result is available if we look at special case where j = i
back to the
maths
October 2015
Frank Cowell: Household Demand & Supply
16
The Slutsky equation: own-price
 Set j = i to get the effect of the price of
ice-cream on the demand for ice-cream
Dii(p,y) = Hii(p,u)
 Important special case
– xi* Dyi(p,y)
 Own-price substitution effect must be
negative
 Follows from the results on
the firm
 This is non-negative for normal goods
 Price increase means less
disposable income
 So the income effect of a price rise must
be non-positive for normal goods
 Theorem: if the demand for i does not
decrease when y rises, then it must
decrease when pi rises
October 2015
Frank Cowell: Household Demand & Supply
17
Price fall: normal good
p1
 The initial equilibrium
ordinary
demand curve
D1(p,y)
 price fall: substitution effect
 total effect: normal good
compensated
(Hicksian)
demand curve
 income effect: normal good
H1(p,u)
initial price
level
price
fall
For normal good income effect
must be positive or zero
x*1
October 2015
x**
1
x1
Frank Cowell: Household Demand & Supply
18
Price fall: inferior good
 The initial equilibrium
p1
 price fall: substitution effect
ordinary
demand curve
 total effect: inferior good
 income effect: inferior good
Note relative slopes of
these curves in inferiorgood case
For inferior good income
effect must be negative
price
fall
initial price
level
compensated
demand curve
x*1
October 2015
x**
1
x1
Frank Cowell: Household Demand & Supply
19
Features of demand functions
 Homogeneous of degree zero
 Satisfy the “adding-up” constraint
 Symmetric substitution effects
 Negative own-price substitution effects
 Income effects could be positive or negative:
• in fact they are nearly always a pain
October 2015
Frank Cowell: Household Demand & Supply
20
Overview
Household
Demand & Supply
Response
functions
Extending the
Slutsky analysis
Slutsky
equation
Supply of
factors
Examples
October 2015
Frank Cowell: Household Demand & Supply
21
Consumer demand: alternative approach
 Now for an alternative way of modelling consumer responses
 Take a type-2 budget constraint (endogenous income)
 Analyse the effect of price changes…
 …allowing for the impact of price on the valuation of income
October 2015
Frank Cowell: Household Demand & Supply
22
Consumer equilibrium: another view
x2
 Type 2 budget constraint:
fixed resource endowment
Budget constraint with
endogenous income
 Consumer's equilibrium
Its interpretation
n
n
i=1
i=1
{x: Spi xi  SpiRi }
so as to
buy more
good 2

 Equilibrium is familiar:
same FOCs as before
x*
consumer sells
some of good 1

R
x1
October 2015
Frank Cowell: Household Demand & Supply
23
Two useful concepts

From the analysis of the endogenous-income case
derive two other tools:
1. The offer curve:
•
•
Path of equilibrium bundles mapped out by prices
Depends on “pivot point” - the endowment vector R
The household’s supply curve:
2.
•
•
October 2015
The “mirror image” of household demand
Again the role of R is crucial
Frank Cowell: Household Demand & Supply
24
The offer curve
x2
 Take the consumer's equilibrium
 Let the price of good 1 rise
 Let the price of good 1 rise a bit more
 Draw the locus of points

x***
 This path is the offer curve

x**

 Amount of good 1 that household
supplies to the market
x*

R
x1
October 2015
Frank Cowell: Household Demand & Supply
25
Household supply
 Flip horizontally , to make
supply clearer
 Rescale the vertical axis to
measure price of good 1
p1
x2


October 2015
R
 This path is the
household’s supply curve
of good 1
x***


 Plot p1 against x1
x**
x*
supply of
good 1
supply of
good 1
 Note that the curve
“bends back” on itself
Why?
Frank Cowell: Household Demand & Supply
26
Decomposition – another look
 Take ordinary demand for good i:
xi* = Di(p,y)
Function of prices and income
 Substitute in for y :
xi* = Di(p, Sj pjRj)
Income itself now depends on
prices
direct effect of
pj on demand
 Differentiate with respect to pj :
The indirect effect uses
*
dxi
dy
function-of-a-function rule again
i
i
— = Dj (p, y) + Dy (p, y) —
dpj
dpj indirect effect of pj on
= Dji(p, y) + Dyi(p, y) Rj
 Now recall the Slutsky relation:
Dji(p,y) = Hji(p,u) – xj* Dyi(p,y)
demand via the impact
on income
Just the same as on earlier
slide
 Use this to substitute for Dji in the above:

*
dxi
This is the modified Slutsky
— = Hji(p,u) + [Rj – xj*] Dyi(p,y)
equation
dpj
October 2015
Frank Cowell: Household Demand & Supply
27
*The modified Slutsky equation:
dxi*
── = Hji(p, u) + [Rj – xj*] Dyi(p,y)
dpj
 Substitution effect has same interpretation as before
 Two terms to consider when interpreting the income effect
 This is just the same as before
 This term makes all the difference:
• Negative if the person is a net demander
• Positive if the person is a net supplier
some
examples
October 2015
Frank Cowell: Household Demand & Supply
28
Overview
Household
Demand & Supply
Response
functions
Labour supply,
savings…
Slutsky
equation
Supply of
factors
Examples
October 2015
Frank Cowell: Household Demand & Supply
29
Some examples
 Many important economic issues fit this type of model :
• Subsistence farming
• Saving
• Labour supply
 It's important to identify the components of the model
• How are the goods to be interpreted?
• How are prices to be interpreted?
• What fixes the resource endowment?
 To see how key questions can be addressed
• How does the agent respond to a price change?
• Does this depend on the type of resource endowment?
October 2015
Frank Cowell: Household Demand & Supply
30
Subsistence agriculture…
x2
 Resource endowment
includes a lot of rice
 Slope of budget constraint
increases with price of rice
 Consumer's equilibrium
 x1,x2 are “rice” and “other goods”
 Will the supply of rice to export
rise with the world price…?

x*
supply
October 2015

R
x1
Frank Cowell: Household Demand & Supply
31
The savings problem…
x2
 Resource endowment is noninterest income profile
 Slope of budget constraint
increases with interest rate, r
 Consumer's equilibrium
 Its interpretation
 x1,x2 are consumption “today”
and “tomorrow”

 Determines time-profile of
consumption
 What happens to saving
when the interest rate
changes…?
x*
saving

R
1+r
x1
October 2015
Frank Cowell: Household Demand & Supply
32
Labour supply…
x2
 Endowment: total time & non-labour
income
 Slope of budget constraint is wage rate
 Consumer's equilibrium
 x1,x2 are leisure and consumption
 Determines labour supply
 Will people work harder if their wage
rate goes up?

labour
supply
x*
wage
rate

R
non-labour income
x1
October 2015
Frank Cowell: Household Demand & Supply
33
Modified Slutsky: labour supply
 Take the modified Slutsky:
dxi*
— = Hii(p,u) + [Ri – xi*] Diy(p,y)
dpi
 Let ℓ:=[Ri – xi*] be the supply of labour
and s the share of labour in total income
y, Then we get:
dxi*
— = Hii(p,u) + ℓ Diy(p,y)
dpi
 Rearranging :
pi dxi*
p
y
– — — = – —i Hii(p,u) – s — Diy(p,y)
ℓ dpi
ℓ
ℓ
 Write as elasticities of labour supply:
etotal = esubst + s eincome
October 2015
The general form. We are
going to make a further
simplifying assumption
Suppose good i is labour time;
then Ri – xi is the labour you sell
in the market (leisure time not
consumed);
pi is the wage rate
.
Divide by labour
. supply;
multiply by (-) wage rate
The Modified Slutsky equation
in a simple form
Frank Cowell: Household Demand & Supply
34
Summary
 How it all fits together:
Review
 Compensated (H) and ordinary (D) demand functions can be
hooked together.
Review  Slutsky equation breaks down effect of price i on demand for j
Review  Endogenous income introduces a new twist when prices change
October 2015
Frank Cowell: Household Demand & Supply
35
What next?
 The welfare of the consumer
 How to aggregate consumer behaviour in the market
October 2015
Frank Cowell: Household Demand & Supply
36