# Microeconomics Principles and Tools

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```					Prerequisites

Almost essential
Consumner: Optimisation

Useful, but optional
Frank Cowell: Microeconomics

Firm: Optimisation

Household Demand and Supply

MICROECONOMICS
Principles and Analysis
Frank Cowell

March 2011
Working out consumer responses
Frank Cowell: Microeconomics

   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.
   The work we have done can map out the
consumer's responses
   to changes in prices
   to changes in income
what we know
Overview...         Household
Demand & Supply
Frank Cowell: Microeconomics

Response
functions
The basics of the
consumer               Slutsky
demand system.         equation

Supply of
factors

Examples
Solving the max-utility problem
Frank Cowell: Microeconomics

   The primal problem and its solution
n
max U(x) + m[ y – S pi xi ]
discussion                                                                  The Lagrangean for the max U
i=1             problem
U1      = mp1
(x*)
U2(x*) = mp2            
... ... ...                      The n+1 first-order conditions,
Un(x*) = mpn
n
         assuming all goods purchased.
S pixi* = y
i=1

   Solve this set of equations:
x1* = D1(p, y)
x2* = D2(p, y)
         Gives a set of demand functions,
... ... ...                      one for each good. Functions of

xn* = Dn(p, y)
n
         prices and incomes.

A restriction on the n equations.
S     piDi(p,   y) = y            Follows from the budget constraint
i=1
The response function
The response function for the primal   Should be treated as just one
Frank Cowell: Microeconomics

problem is demand for good i:           of a set of n equations.
xi* = Di(p,y)
The system of equations must have      Reason? This follows
an ―adding-up‖ property:                immediately from the budget
n                                   constraint: left-hand side is
S pi Di(p, y) = y                    total expenditure.
i=1

Each equation in the system must be     Reason? Again follows
homogeneous of degree 0 in prices and   immediately from the
income. For any t > 0:                  budget constraint.
xi* = Di(p, y )= Di(tp, ty)

To make more progress we need to exploit the relationship
between primal and dual approaches again...
How you would use this in practice...
Frank Cowell: Microeconomics

   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.
Overview...        Household
Demand & Supply
Frank Cowell: Microeconomics

Response
functions
A fundamental
decomposition of      Slutsky
the effects of a      equation
price change.
Supply of
factors

Examples
Consumer’s demand responses
Frank Cowell: Microeconomics

   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.
budget
   Begin with a type 1 (exogenous income) budget
constraint
constraint.
Effect of a change in income
Frank Cowell: Microeconomics

 Take the basic equilibrium
x2
 Suppose income rises
The effect of the income
increase.

 Demand for each
good does not fall if it
is “normal”

 x**
 But could the
                   opposite happen?
x*

x1
An ―inferior‖ good
Frank Cowell: Microeconomics

 Take same original prices,
x2                         but different preferences
 Again suppose income rises
The effect of the income
increase.

 Demand for good 1
rises, but…
 Demand for
“inferior” good 2 falls
a little

x*        x**
 Can you think of
any goods like this?
How might it depend
x1      on the categorisation
of goods?
Frank Cowell: Microeconomics

 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...
Effect of a change in price
Frank Cowell: Microeconomics

 Again take the basic
x2                                      equilibrium
 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
And now let’s look at it in maths
Frank Cowell: Microeconomics

 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...)
A fundamental decomposition
Frank Cowell: Microeconomics

compensated                     ordinary
demand                          demand
Take the two methods of writing xi*:  Remember: they are two ways of
Hi(p,u) = Di(p,y)                   representing the same thing
Use cost function to substitute for y: Gives us an implicit relation in
Hi(p,u) = Di(p, C(p,u))               prices and utility.

Differentiate with respect to pj :         Uses function-of-a-function rule
Hji(p,u) = Dji(p,y) + Dyi(p,y)Cj(p,u)      again. Remember y=C(p,u)

Simplify :                                Using cost function and
Hji(p,u) = Dji(p,y) + Dyi(p,y) Hj(p,u)     Shephard’s Lemma again
= Dji(p,y) + Dyi(p,y) xj*          From the comp. demand function

And so we get:
 This is the Slutsky equation
Dji(p,y) = Hji(p,u) – xj*Dyi(p,y)
The Slutsky equation
Frank Cowell: Microeconomics

Dji(p,y) = Hji(p,u) – xj*Dyi(p,y)
 Gives fundamental breakdown
of effects of a price change

 Income effect: “I'm better off if
the price of jelly falls, so I buy
more things, including icecream.
I’m worse off if the price of jelly
rises, so I buy less icecream”
   x**
 “Substitution effect: When the
   x*                  price of jelly falls and I’m kept on
the same utility level, I prefer to
switch from icecream for dessert”
Slutsky: Points to watch
Frank Cowell: Microeconomics

   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
back to the
   net complements                                    maths
   gin and tonic?
   Neat result is available if we look at special case where j = i
The Slutsky equation: own-price
Frank Cowell: Microeconomics

Set j = i to get the effect of the price of
icecream on the demand for icecream

Dii(p,y) = Hii(p,u) – xi*Dyi(p,y)

Link to                          Own-price substitution effect                  Follows from the results on
firm’s input
demand                            must be negative                               the firm

Income effect of price increase is             Price increase means less
non-positive for normal goods                  disposable income

So, if the demand for i does not decrease
when y rises, then it must decrease when pi rises.
Price fall: normal good
Frank Cowell: Microeconomics

p1                                             The initial equilibrium
ordinary                              price fall: substitution effect
demand                                total effect: normal good
curve    compensated                income effect: normal good
D1(p,y)            (Hicksian)
demand curve
H1(p,u)
initial price
level
For normal good
income effect must be
positive or zero
price

Compensating
fall

Variation

x*
1          x**
1
x1
Price fall: inferior good
Frank Cowell: Microeconomics

p1                              The initial equilibrium
ordinary               price fall: substitution effect
demand                 total effect: inferior good
curve                income effect: inferior good

compensated
initial price     demand curve                        Note relative slopes of
level                                                 these curves in
inferior-good case.

For inferior good
price

Compensating
income effect must be
fall

Variation
negative

x*
1   x**
1    x1
Features of demand functions
Frank Cowell: Microeconomics

 Homogeneous of degree zero
 Symmetric substitution effects
 Negative own-price substitution effects
 Income effects could be positive or
negative:
   in fact they are nearly always a pain.
Overview...         Household
Demand & Supply
Frank Cowell: Microeconomics

Response
functions
Extending the
Slutsky analysis.      Slutsky
equation

Supply of
factors

Examples
Consumer demand: alternative
approach
Frank Cowell: Microeconomics

 Now for an alternative way of modelling
consumer responses.
budget
constraint
 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
Consumer equilibrium: another view
Frank Cowell: Microeconomics

x2
 Type 2 budget constraint:
fixed resource endowment
Budget constraint with
endogenous income
 Consumer's equilibrium
Its interpretation

n             n
{x: S pi xi  S piRi }
i=1            i=1
 Equilibrium is
so as to                                                               familiar: same
buy more                         x*                                   FOCs as before.
good 2
consumer sells some
of good 1..
   R
x1
Two useful concepts
Frank Cowell: Microeconomics

    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
2.   The household’s supply curve:
   The ―mirror image‖ of household demand.
   Again the role of R is crucial.
The offer curve
Frank Cowell: Microeconomics

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***

   x**
 This path is the
offer curve.
   x*
 Amount of good 1
   R            that household
x1     supplies to the
market
Household supply
Frank Cowell: Microeconomics

 Flip horizontally , to make
supply clearer
 Rescale the vertical axis to
measure price of good 1.
 Plot p1 against x1 .

x2                              p1

 This path is the
household’s supply
   x***                                      curve of good 1.
   x**
 Note that the curve
“bends back” on itself.
   x*
supply of
Why?
    R              supply of        good 1
good 1
Decomposition – another look
Take ordinary demand for good i:                   Function of prices and income
Frank Cowell: Microeconomics

xi* = Di(p,y)
Substitute in for y :                                  Income itself now depends on
xi* = Di(p, Sj pjRj)                                  pj on
indirect effect of prices
demand via the impact
direct effect of         on income
Differentiateon demand
pj with respect  to pj :                The indirect effect uses
dxi*                      dy                    function-of-a-function rule again
— = Dji(p, y) + Dyi(p, y) —
dpj                       dpj
= Dji(p, y) + Dyi(p, y) Rj
Now recall the Slutsky relation:                   Just the same as on earlier
Dji(p,y) = Hji(p,u) – xj* Dyi(p,y)                slide

Use this to substitute for Dji in the above:

dxi*                                      This is the modified Slutsky
— = Hj   i(p,u) + [R – x *] D i(p,y)
j   j     y          equation
dpj
The modified Slutsky equation:
Frank Cowell: Microeconomics

dxi*
── = Hji(p,u) + [Rj – xj* ] Dyi(p,y)
dpj

Substitution effect has same interpretation as before.
Income effect has two terms.
This term is just the same as before.
This term makes all the difference:
oNegative if the person is a net demander.
oPositive if he is a net supplier.
some
examples
Overview...      Household
Demand & Supply
Frank Cowell: Microeconomics

Response
functions
Labour supply,
savings…            Slutsky
equation

Supply of
factors

Examples
Some examples
Frank Cowell: Microeconomics

   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?
Subsistence agriculture...
Frank Cowell: Microeconomics

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”
   x*
 Will the supply
supply..      R           of rice to export
x1    rise with the world
price...?.
The savings problem...
Frank Cowell: Microeconomics

x2
 Resource endowment is
non-interest 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
   x*                                     consumption

 What happens
saving..      R   1+r            to saving when
x1     the interest rate
changes...?.
Labour supply...
Frank Cowell: Microeconomics

x2
 Endowment is total time
available & non-labour income.
 Slope of budget constraint is
the wage rate
 Consumer's equilibrium

 x1,x2 are leisure
and “consumption”

 Determines
labour supply
   x*   wage
rate

labour            R                             Will people work
supply.            non-labour income.           harder if their
x1    wage rate goes
up?.
Modified Slutsky: labour supply
Take the modified Slutsky:                              The general form. We are
Frank Cowell: Microeconomics

dxi*                                                  going to make a further
— = Hij(p,u) + [Rj – xj*] Diy(p,y)                    simplifying assumption
dpj
Assume that supply of good i is the                     Suppose good i is labour time;
only source of income (so y= pi[Ri – xi]).               then Ri – xi is the labour you
sell in the market (I.e. leisure
Then, for the effect of pi on xi* we get:                time not consumed); pi is the
.
dxi*               y                                  wage rate
— = Hi  i (p,u) + — Di (p,y)
y
dpi                pi
Rearranging :                                           Divide by labour supply;
.
multiply by (-) wage rate
pi dxi*            pi              y
– —— * H
– —— — =labour supply ij(p,u) – ——* Diy(p,y)
Ri–xi* dpi
Total
Ri–x must –
elasticity: couldibe + orbe
Ri–xi
negative if leisure
Write in elasticity form:
positive
(backward-bending)         is a normal good
The Modified Slutsky equation

etotal = esubst + eincome                             in a simple form

Estimate the whole demand system from family expenditure data...
Frank Cowell: Microeconomics

 The estimated elasticities...

 Men's labour supply is
Source: Blundell and Walker (Economic Journal, 1982)        backward bending!

 Leisure is a "normal good" for
everyone

 Children tie down women's
substitution effect...
total        subst     income

Men:                  –0.23         +0.13 −0.36
Women:
No children           +0.43         +0.65 −0.22
One child          +0.10         +0.32 −0.22
Two          –0.19         +0.03 −0.22
children
Summary
Frank Cowell: Microeconomics

   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.
What next?
Frank Cowell: Microeconomics

 The welfare of the consumer
 How to aggregate consumer behaviour in
the market.

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