MICROECONOMICS Principles and Analysis Frank Cowell

Transcription

1 Prerequisites Almost essential Consumer: Optimisation Useful, but optional Firm: Optimisation HOUSEHOLD DEMAND AND SUPPLY MICROECONOMICS Principles and Analysis Frank Cowell Note: the detail in slides marked * can only be seen if you run the slideshow July 2017 Frank Cowell: Household Demand & Supply 1

2 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 July 2017 Frank Cowell: Household Demand & Supply 2

3 Overview Household Demand & Supply The basics of the consumer demand system Response functions Slutsky equation Supply of factors Examples July 2017 Frank Cowell: Household Demand & Supply 3

4 Solving the max-utility problem The primal problem and its solution n Lagrangean for the max U problem max U(x) + µ[y Σ p i x i ] i=1 U 1 (x * ) = µp 1 U 2 (x * ) = µp 2 U n (x * ) = µp n n Σ p i x i* = y i=1 x 1* = D 1 (p, y) x 2* = D 2 (p, y) x n* = D n (p, y) n Σ p i D i (p, y) = y i=1 Solve this set of equations: The n + 1 first-order conditions, assuming all goods purchased 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 July 2017 Frank Cowell: Household Demand & Supply 4

5 The response function The response function for the primal problem is demand for good i: x i* = D i (p,y) The system of equations must have an adding-up property: n Σp i D i (p, y) = y i=1 Each equation in the system must be homogeneous of degree 0 in prices and income. For any t > 0: x i * = D i (p, y )= D i (tp, ty) Should be treated as just one of a set of n equations Reason? Follows immediately from the budget constraint: lefthand side is total expenditure Reason? Again follows from the budget constraint To make more progress we need to exploit the relationship between primal and dual approaches again July 2017 Frank Cowell: Household Demand & Supply 5

6 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 July 2017 Frank Cowell: Household Demand & Supply 6

7 Overview Household Demand & Supply A fundamental decomposition of the effects of a price change Response functions Slutsky equation Supply of factors Examples July 2017 Frank Cowell: Household Demand & Supply 7

8 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 July 2017 Frank Cowell: Household Demand & Supply 8

9 Effect of a change in income x 2 Take the basic equilibrium Suppose income rises 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 July 2017 Frank Cowell: Household Demand & Supply 9

10 An inferior good x 2 Take same market data, 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 on the categorisation of goods? x 1 July 2017 Frank Cowell: Household Demand & Supply 10

11 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 July 2017 Frank Cowell: Household Demand & Supply 11

12 Effect of a change in price x 2 Again take the basic equilibrium Allow price of good 1 to fall The effect of the price fall The journey from x* to x** broken into two parts x * income substitution effect effect x ** x 1 July 2017 Frank Cowell: Household Demand & Supply 12

13 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 ) July 2017 Frank Cowell: Household Demand & Supply 13

14 A fundamental decomposition compensated demand ordinary demand Take the two methods of writing x i* : H i (p,υ) = D i (p,y) Use cost function to substitute for y: H i (p,υ) = D i (p, C(p,υ)) Differentiate with respect to p j : H ji (p,υ) = D ji (p,y) + D yi (p,y)c j (p,υ) Simplify : H ji (p,υ) = D ji (p,y) + D yi (p,y) H j (p,υ) Remember: they are two ways of representing the same thing Gives us an implicit relation in prices and utility Uses y = C(p,υ) and function-of-afunction rule again Using cost function and Shephard s Lemma = D ji (p,y) + D yi (p,y) x j * From the comp. demand function And so we get: D ji (p,y) = H ji (p,υ) x j* D yi (p,y) This is the Slutsky equation July 2017 Frank Cowell: Household Demand & Supply 14

15 *The Slutsky equation D ji (p,y) = H ji (p,υ) x j * D yi (p,y) Gives fundamental breakdown of effects of a price change 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 July 2017 Frank Cowell: Household Demand & Supply 15

16 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 July 2017 Frank Cowell: Household Demand & Supply 16

17 The Slutsky equation: own-price Set j = i to get the effect of the price of ice-cream on the demand for ice-cream D ii (p,y) = H ii (p,υ) x i * D yi (p,y) Own-price substitution effect must be negative This is non-negative for normal goods So the income effect of a price rise must be non-positive for normal goods Important special case Follows from the results on the firm Price increase means less disposable income Theorem: if the demand for i does not decrease when y rises, then it must decrease when p i rises July 2017 Frank Cowell: Household Demand & Supply 17

18 Price fall: normal good initial price level p 1 ordinary demand curve D 1 (p,y) compensated (Hicksian) demand curve H 1 (p,υ) The initial equilibrium price fall: substitution effect total effect: normal good income effect: normal good For normal good income effect must be positive or zero price fall x* 1 x 1 ** x 1 July 2017 Frank Cowell: Household Demand & Supply 18

19 Price fall: inferior good p 1 ordinary demand curve The initial equilibrium price fall: substitution effect total effect: inferior good income effect: inferior good initial price level compensated demand curve Note relative slopes of these curves in inferiorgood case price fall For inferior good income effect must be negative x* 1 x 1 ** x 1 July 2017 Frank Cowell: Household Demand & Supply 19

20 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 July 2017 Frank Cowell: Household Demand & Supply 20

21 Overview Household Demand & Supply Extending the Slutsky analysis Response functions Slutsky equation Supply of factors Examples July 2017 Frank Cowell: Household Demand & Supply 21

22 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 substation effect but an additional effect arises from the impact of price on the valuation of income July 2017 Frank Cowell: Household Demand & Supply 22

23 Consumer equilibrium: another view x 2 Type 2 budget constraint: fixed resource endowment Budget constraint with endogenous income Consumer's equilibrium Its interpretation n {x: Σp i x i Σ p i R i } i=1 i=1 n so as to buy more good 2 x * consumer sells some of good 1 R Equilibrium is familiar: same FOCs as before x 1 July 2017 Frank Cowell: Household Demand & Supply 23

24 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 July 2017 Frank Cowell: Household Demand & Supply 24

25 The offer curve x 2 Take the consumer's equilibrium Let the price of good 1 rise Let the price of good 1 rise a bit more x ** x *** x * Draw the locus of points This path is the offer curve Amount of good 1 that household supplies to the market R x 1 July 2017 Frank Cowell: Household Demand & Supply 25

26 Household supply x 2 p 1 Flip horizontally, to make supply clearer Rescale the vertical axis to measure price of good 1 Plot p 1 against x 1 x *** x ** This path is the household s supply curve of good 1 R x * supply of good 1 supply of good 1 Note that the curve bends back on itself Why? July 2017 Frank Cowell: Household Demand & Supply 26

27 Decomposition another look Take ordinary demand for good i: x i* = D i (p,y) Substitute in for y : x i* = D i (p, Σ j p j R j ) direct effect of p j on demand Differentiate with respect to p j : dx * i dy = D ji (p, y) + D yi (p, y) dp j dp j = D ji (p, y) + D yi (p, y) R j Now recall the Slutsky relation: D ji (p,y) = H ji (p,υ) x j* D yi (p,y) Use this to substitute for D ji : dx * i = H ji (p,υ) + [R j x j* ] D yi (p,y) dp j Function of prices and income Income itself now depends on prices The indirect effect uses function-of-a-function rule again indirect effect of p j on demand via the impact on income Just the same as on earlier slide This is the modified Slutsky equation July 2017 Frank Cowell: Household Demand & Supply 27

28 *The modified Slutsky equation: dx i * = H ji (p, υ) + [R j x j* ] D yi (p,y) dp j 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 July 2017 Frank Cowell: Household Demand & Supply 28

29 Overview Household Demand & Supply Labour supply, savings Response functions Slutsky equation Supply of factors Examples July 2017 Frank Cowell: Household Demand & Supply 29

30 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? July 2017 Frank Cowell: Household Demand & Supply 30

31 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? x * supply R x 1 July 2017 Frank Cowell: Household Demand & Supply 31

32 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 x * x 1,x 2 are consumption today and tomorrow Determines time-profile of consumption What happens to saving when the interest rate changes? saving R 1+ r x 1 July 2017 Frank Cowell: Household Demand & Supply 32

33 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 labour supply x * wage rate R non-labour income Will people work harder if their wage rate goes up? x 1 July 2017 Frank Cowell: Household Demand & Supply 33

34 Modified Slutsky: labour supply Take the modified Slutsky: dx * i = H i i(p,υ) + [R i x i* ] D dp i y(p,y) i Let l:=[r i x i* ] be the supply of labour and s the share of earnings in total income y, Then we get: dx * i = H i i(p,υ) + l D dp i y(p,y) i Rearranging : p i dx i * p i y = H i i(p,υ) s D i y(p,y) l dp i l l Write as elasticities of labour supply: ε total = ε subst + s ε income The general form. We are going to make a further simplifying assumption Suppose good i is labour time; then R i x i is the labour you sell in the market (leisure time not consumed);. p i is the wage rate Divide by labour. supply; multiply by (-) wage rate The Modified Slutsky equation in a simple form July 2017 Frank Cowell: Household Demand & Supply 34

35 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 July 2017 Frank Cowell: Household Demand & Supply 35

36 What next? The welfare of the consumer How to aggregate consumer behaviour in the market July 2017 Frank Cowell: Household Demand & Supply 36