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Set Up a Household Budget

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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




October 2006
                               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
                                                                               about the primal
                               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 ]
Link to full
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.
 Link to
 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?
                               A glimpse ahead...
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”

                                                          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 be negative
                                       inferior goods.
                                  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?
                                  A neat result is available if we look at the special case
                                   where j = i.                                        back to the
                                                                                           maths
                                      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
                                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.
                               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
 Link to
                                   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
                                 — = Hji(p,u) + [Rj – xj*] Dyi(p,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: could ibe + 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...
                                Simple facts about labour supply
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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