Household Demand and Supply

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

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

3. Overview Household Demand & Supply Response functions The basics of the consumer demand system Slutsky equation Supply of factors Examples

4. 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 Spixi* = 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 pi Di(p, y) = y i=1

5. The response function • 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 • Eachequation 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

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

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

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

9. 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

10. An “inferior” good • Take same market data, 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

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

12. 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

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…)

14. 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

15. * 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 • 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… • x** • …if the price change makes me better off then I buy more normal goods, such as ice cream” • x* • 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”

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

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 • Important special case Dii(p,y) = Hii(p,u) – 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

18. Price fall: normal good p1 • The initial equilibrium ordinary demand curve • price fall: substitution effect • total effect: normal good compensated (Hicksian) demand curve • income effect: normal good D1(p,y) H1(p,u) initial price level • For normal good income effect must be positive or zero price fall x1 x1 * ** x1

19. Price fall: inferior good p1 • 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

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

21. Overview Household Demand & Supply Response functions Extending the Slutsky analysis Slutsky equation Supply of factors Examples

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 substitution effect • but an additional effect • arises from the impact of price on the valuation of income

23. 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:SpixiS pi Ri} i=1i=1 • Equilibrium is familiar: same FOCs as before so as to buy more good 2 • x* consumer sells some of good 1 • R x1

24. 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

25. The offer curve 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*** • This path is the offer curve • x** • Amount of good 1 that household supplies to the market • x* • R x1

26. p1 x2 • x*** • x** • x* supply of good 1 supply of good 1 • R Household supply • Flip horizontally, to make supply clearer • Rescale the vertical axis to measure price of good 1 • Plot p1against x1 • This path is the household’s supply curve of good 1 • Note that the curve “bends back” on itself • Why?

27. 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,SjpjRj) • 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) — dpjdpj = Dji(p, y) + Dyi(p, y) Rj • The indirect effect uses function-of-a-function rule again indirect effect of pjon 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: dxi* • — = Hji(p,u) + [Rj – xj*] Dyi(p,y) dpj • This is the modified Slutsky equation

28. * 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

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

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?

31. Subsistence agriculture x2 • Resource endowment includes a lot of rice • Slope of budget constraint increases with price of rice • Consumer's equilibrium • x1,x2are “rice” and “other goods” • Will the supply of rice to export rise with the world price…? • x* • R supply x1

32. The savings problem x2 • Resource endowment is non-interest income profile • Slope of budget constraint increases with interest rate, r • Consumer's equilibrium • Its interpretation • x1,x2are consumption “today” and “tomorrow” • Determines time-profile of consumption • What happens to saving when the interest rate changes…? • x* • R 1+ r saving x1

33. Labour supply x2 • 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

34. 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 • 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 • Suppose good i is labour time; then Ri – xiis the labour you sell in the market (leisure time not consumed); piis the wage rate . • Rearranging : pi dxi* piy • – — — = – — Hii(p,u) – s — Diy(p,y) • ℓdpiℓℓ • Divide by labour supply; multiply by (-) wage rate . • Write as elasticities of labour supply: • etotal = esubst + seincome • The Modified Slutsky equation in a simple form

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 ion demand for j Endogenous income introduces a new twist when prices change

36. What next? The welfare of the consumer How to aggregate consumer behaviour in the market