# Prerequisites Almost essential Consumer Optimisation Useful but optional

Prerequisites Almost essential Consumer: Optimisation Useful, but optional Firm: Optimisation HOUSEHOLD DEMAND SUPPLY MICROECONOMICS Principles and Analysis Frank Cowell April 2018 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 April 2018 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 April 2018 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 U 1(x*) = mp 1 U 2(x*) = mp 2 … … … 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: x 1* = D 1(p, y) x 2* = D 2(p, y) … … … xn* = Dn(p, y) n ü ý þ S pi Di(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 April 2018 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 April 2018 Frank Cowell: Household Demand & Supply 5

How you would use this in practice § Consumer surveys • data on expenditure for each household • over a number of categories of goods • perhaps income, hours worked as well § Market data are available on prices § Given some assumptions about the structure of preferences • estimate household demand functions for commodities • from this recover information about utility functions April 2018 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 April 2018 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 April 2018 Frank Cowell: Household Demand & Supply 8

Effect of a change in income § Take the basic equilibrium § Suppose income rises x 2 §The effect of the income increase § Demand for each good does not fall if it is “normal” x** x* § But could the opposite happen? x 1 April 2018 Frank Cowell: Household Demand & Supply 9

An “inferior” good § Take same market data, but different preferences § Again suppose income rises §The effect of the income increase x 2 § 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? x 1 April 2018 Frank Cowell: Household Demand & Supply 10

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 April 2018 Frank Cowell: Household Demand & Supply 11

Effect of a change in price § Again take the basic equilibrium x 2 § 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 ° x* x** x 1 April 2018 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…) April 2018 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) April 2018 § From the comp. demand function § This is the Slutsky equation Frank Cowell: Household Demand & Supply 14

* detail on slide can only be seen if you run the slideshow The Slutsky equation Dji(p, y) = Hji(p, u) – xj* Dyi(p, y) § Gives fundamental breakdown of effects of a price change l l 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 ice cream” § Substitution effect: “When the price of jelly falls and I’m kept on the same utility level, I prefer to switch from ice cream for dessert” April 2018 Frank Cowell: Household Demand & Supply 15

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 April 2018 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 April 2018 Frank Cowell: Household Demand & Supply 17

Price fall: normal good p 1 ordinary demand curve D 1(p, y) § The initial equilibrium § price fall: substitution effect § total effect: normal good compensated (Hicksian) demand curve § income effect: normal good H 1(p, u) initial price level price fall §For normal good income effect must be positive or zero x*1 April 2018 x** 1 x 1 Frank Cowell: Household Demand & Supply 18

Price fall: inferior good p 1 § The initial equilibrium § price fall: substitution effect § total effect: inferior good ordinary demand curve § 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 April 2018 x** 1 x 1 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 April 2018 Frank Cowell: Household Demand & Supply 20

Overview Household Demand & Supply Response functions Extending the Slutsky analysis Slutsky equation Supply of factors Examples April 2018 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 • determined by value of resources you own § Analyse the effect of price changes • will get usual income and substitution effect • but an additional effect • arises from the impact of price on the valuation of income April 2018 Frank Cowell: Household Demand & Supply 22

Consumer equilibrium: another view x 2 § Type 2 budget constraint: fixed resource endowment §Budget constraint with endogenous income § Consumer's equilibrium §Its interpretation n n i=1 {x: Spi xi S pi Ri } so as to buy more good 2 l § Equilibrium is familiar: same FOCs as before x* consumer sells some of good 1 l April 2018 R x 1 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 2. The household’s supply curve: • The “mirror image” of household demand • Again the role of R is crucial April 2018 Frank Cowell: Household Demand & Supply 24

The offer curve x 2 § Take the consumer's equilibrium § Let the price of good 1 rise a bit more § Draw the locus of points l x*** § This path is the offer curve l x** l § Amount of good 1 that household supplies to the market x* l April 2018 R x 1 Frank Cowell: Household Demand & Supply 25

Household supply p 1 x 2 l l April 2018 R § This path is the household’s supply curve of good 1 x*** l l § Flip horizontally, to make supply clearer § Rescale the vertical axis to measure price of good 1 § Plot p 1 against x 1 x** x* 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 pj. Rj) §Income itself now depends on prices direct effect of pj on demand § Differentiate with respect to pj : §The indirect effect uses function * dxi dy -of-a-function rule again — = Dji(p, y) + Dyi(p, y) — dpj indirect effect of pj on demand via the impact = Dji(p, y) + Dyi(p, y) Rj on income § Now recall the Slutsky relation: Dji(p, y) = Hji(p, u) – xj* Dyi(p, y) § Use this to substitute for Dji: dxi* — = Hji(p, u) + [Rj – xj*] Dyi(p, y) dpj April 2018 §Just the same as on earlier slide §This is the modified Slutsky equation Frank Cowell: Household Demand & Supply 27

* detail on slide can only be seen if you run the slideshow 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 § The second term is just the same as before § The first term makes all the difference: • negative if the person is a net demander • positive if the person is a net supplier April 2018 Frank Cowell: Household Demand & Supply 28

Overview Household Demand & Supply Response functions Labour supply, savings… Slutsky equation Supply of factors Examples April 2018 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? April 2018 Frank Cowell: Household Demand & Supply 30

Subsistence agriculture x 2 § Resource endowment includes a lot of rice § Slope of budget constraint increases with price of rice § Consumer's equilibrium § x 1, x 2 are “rice” and “other goods” § Will the supply of rice to export rise with the world price…? l x* supply April 2018 l R x 1 Frank Cowell: Household Demand & Supply 31

The savings problem x 2 § Resource endowment is noninterest income profile § Slope of budget constraint increases with interest rate, r § Consumer's equilibrium § Its interpretation l § x 1, x 2 are consumption “today” and “tomorrow” § Determines time-profile of consumption § What happens to saving when the interest rate changes…? x* saving April 2018 l R 1+ r x 1 Frank Cowell: Household Demand & Supply 32

Labour supply x 2 § Endowment: total time & non-labour income § Slope of budget constraint is wage rate § Consumer's equilibrium § x 1, x 2 are leisure and consumption § Determines labour supply l labour supply x* § Will people work harder if their wage rate goes up? wage rate l R non-labour income x 1 April 2018 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 earnings 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 April 2018 §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: § Compensated (H) and ordinary (D) demand functions can be hooked together. § Slutsky equation breaks down effect of price i on demand for j § Endogenous income introduces a new twist when prices change April 2018 Frank Cowell: Household Demand & Supply 35

What next? § The welfare of the consumer § How to aggregate consumer behaviour in the market April 2018 Frank Cowell: Household Demand & Supply 36

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