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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.