1 / 36

Household Demand and Supply

Prerequisites. Almost essential Consumer: Optimisation Useful, but optional Firm: Optimisation. Household Demand and Supply. MICROECONOMICS Principles and Analysis Frank Cowell. Working out consumer responses. The analysis of consumer optimisation gives us some powerful tools:

lorne
Download Presentation

Household Demand and Supply

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related