# Household Demand and Supply

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Household Demand and Supply
Prerequisites Almost essential Consumer: Optimisation Useful, but optional Firm: Optimisation Household Demand and Supply MICROECONOMICS Principles and Analysis Frank Cowell March 2012

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 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 March 2012

Overview… The basics of the consumer demand system
Household Demand & Supply Response functions The basics of the consumer demand system Slutsky equation Supply of factors Examples March 2012

Solving the max-utility problem
The primal problem and its solution n max U(x) + m[ y – S pi xi ] i=1 Lagrangean for the max U problem U1(x*) = mp1 U2(x*) = mp2 … … … Un(x*) = mpn ü ý þ The n+1 first-order conditions, assuming all goods purchased n S pixi* = y i=1 Solve this set of equations: x1* = D1(p, y) x2* = D2(p, y) … … … xn* = Dn(p, y) Gives a set of demand functions, one for each good: functions of prices and incomes ü ý þ A restriction on the n equations. Follows from the budget constraint n S piDi(p, y) = y i=1 31 October 2012

The response function Spi Di(p, y) = y
The response function for the primal problem is demand for good i: xi* = Di(p,y) Should be treated as just one of a set of n equations The system of equations must have an “adding-up” property: n Spi Di(p, y) = y i=1 Reason? Follows immediately from the budget constraint: left-hand side is total expenditure Each equation in the system must be homogeneous of degree 0 in prices and income. For any t > 0: xi* = Di(p, y )= Di(tp, ty) Reason? Again follows from the budget constraint To make more progress we need to exploit the relationship between primal and dual approaches again… March 2012

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 March 2012

Overview… A fundamental decomposition of the effects of a price change
Household Demand & Supply Response functions A fundamental decomposition of the effects of a price change Slutsky equation Supply of factors Examples March 2012

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 March 2012

Effect of a change in income
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 March 2012

An “inferior” good x2 x1 Demand for good 1 rises, but…
Take same original prices, but different preferences x2 Again suppose income rises The effect of the income increase Demand for good 1 rises, but… Demand for “inferior” good 2 falls a little Can you think of any goods like this? How might it depend on the categorisation of goods? x* x** x1 March 2012

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… March 2012

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 income effect substitution effect x** x* x1 March 2012

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…) March 2012

A fundamental decomposition
compensated demand ordinary demand Take the two methods of writing xi*: Hi(p,u) = Di(p,y) Remember: they are two ways of representing the same thing Use cost function to substitute for y: Hi(p,u) = Di(p, C(p,u)) Gives us an implicit relation in prices and utility 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-a-function rule again Simplify : Hji(p,u) = Dji(p,y) + Dyi(p,y) Hj(p,u) Using cost function and Shephard’s Lemma From the comp. demand function = Dji(p,y) + Dyi(p,y) xj* And so we get: Dji(p,y) = Hji(p,u) – xj*Dyi(p,y) This is the Slutsky equation March 2012

The Slutsky equation Dji(p,y) = Hji(p,u) – xj*Dyi(p,y) x** x*
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** x* “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” March 2012

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 March 2012

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) – xi*Dyi(p,y) Own-price substitution effect must be negative Follows from the results on the firm – xi*  income effect is non-positive for normal goods Price increase means less disposable income So, if the demand for i does not decrease when y rises, then it must decrease when pi rises March 2012

Price fall: normal good
The initial equilibrium ordinary demand curve price fall: substitution effect compensated (Hicksian) demand curve total effect: normal good D1(p,y) income effect: normal good H1(p,u) initial price level For normal good income effect must be positive or zero price fall x1 x1 * ** x1 March 2012

Price fall: inferior good
The initial equilibrium price fall: substitution effect ordinary demand curve total effect: inferior good income effect: inferior good compensated demand curve Note relative slopes of these curves in inferior-good case initial price level For inferior good income effect must be negative price fall x1 x1 * ** x1 March 2012

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 March 2012

Overview… Extending the Slutsky analysis Household Demand & Supply
Response functions Extending the Slutsky analysis Slutsky equation Supply of factors Examples March 2012

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 March 2012

Consumer equilibrium: another view
x2 Type 2 budget constraint: fixed resource endowment Budget constraint with endogenous income Consumer's equilibrium Its interpretation n n {x: Spi xi  SpiRi } i= i=1 Equilibrium is familiar: same FOCs as before so as to buy more good 2 x* consumer sells some of good 1 R x1 March 2012

Two useful concepts From the analysis of the endogenous-income case derive two other tools: The offer curve: Path of equilibrium bundles mapped out by prices Depends on “pivot point” - the endowment vector R The household’s supply curve: The “mirror image” of household demand Again the role of R is crucial March 2012

The offer curve x*** x** x* R x2 x1 This path is the offer curve
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 March 2012

Household supply x*** x** x* R
Flip horizontally , to make supply clearer Rescale the vertical axis to measure price of good 1 Plot p1 against x1 R supply of good 1 x2 x* x*** x** supply of good 1 p1 This path is the household’s supply curve of good 1 Note that the curve “bends back” on itself Why? March 2012

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 : dxi* dy — = Dji(p, y) + Dyi(p, y) — dpj dpj = Dji(p, y) + Dyi(p, y) Rj The indirect effect uses function-of-a-function rule again indirect effect of pj on demand via the impact on income Now recall the Slutsky relation: Dji(p,y) = Hji(p,u) – xj* Dyi(p,y) Just the same as on earlier slide Use this to substitute for Dji in the above: dxi* — = Hji(p,u) + [Rj – xj*] Dyi(p,y) dpj This is the modified Slutsky equation March 2012

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 he is a net supplier Notice that it is very similar to what we had before If the R_j term were eliminated it would be just as we had it before Suppose you are looking at market demand in a country Good j is oil How does price of j affect demand for other stuff? If you are a country without oil? If you are a country with lots of oil? some examples March 2012

Overview… Labour supply, savings… Household Demand & Supply
Response functions Labour supply, savings… Slutsky equation Supply of factors Examples March 2012

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? March 2012

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 R x1 March 2012

The savings problem… x* R x2 x1
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 consumption What happens to saving when the interest rate changes…? x* saving R 1+r x1 March 2012

Labour supply… x* R x2 x1 x1,x2 are leisure and consumption
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? x* wage rate R labour supply non-labour income x1 March 2012

Modified Slutsky: labour supply
Take the modified Slutsky: dxi* — = Hii(p,u) + [Ri – xi*] Diy(p,y) dpi The general form. We are going to make a further simplifying assumption Assume that supply of good i is the only source of income (so y= pi[Ri – xi]) Then, for the effect of pi on xi* we get: dxi* y — = Hii(p,u) + — Diy(p,y) dpi pi 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 . Rearranging : pi dxi* pi y – —— — = – —— Hij(p,u) – —— Diy(p,y) Ri– xi* dpi Ri–xi* Ri–xi* Divide by labour supply; multiply by (-) wage rate . Write in elasticity form: etotal = esubst + eincome The Modified Slutsky equation in a simple form Estimate the whole demand system from family expenditure data… March 2012 34