Theoretical and Empirical Issues in Demand Analysis

1. Theoretical and Empirical Issues in Demand Analysis By Anna Rapoport under the supervision of professor Yakar Kannai

2. Consumer’s problem Given price p and wealth w, choose consumption bundle x from Bpw = {x ≥ 0: p·x ≤ w}

3. Indifference curves x2 increasing preference x1 The UMP • The consumer aims to maximise utility... • Subject to the budget constraint max u(x) subject to n Spi xi £ w i=1 • Defines the UMP • Solution to the problem Budget set • x*

4. Comparative Statics: Wealth Effects • Definition 1: For fixed prices p*, the function of wealth x(p*,w) is called the consumers Engel function. • Definition 2: At any (p,w), the derivative xm(p,w)/w is known as the wealth (income) effect for the m-th good. The wealth effects in matrix notation

5. Effect of a change in income x2 • Take the basic equilibrium • What happens if income rises…? • Equilibrium shifts from x* to x** • Demand for each good does not fall if it is “normal” • …but could the opposite happen? • x** • x* x1

6. An “inferior” good x2 • The same original prices, but different preferences... • Again, let income rise... • The new equilibrium demand for “inferior” good 2 falls a little as income rises • x* • x** x1

7. Normal and Inferior Goods • Definition 1: A commodity m is normal at (p,w) if xm(p,w)/w ≥ 0, that is demand is nondecreasing in wealth. If commodity m’s wealth effect is instead negative, then it's called inferior in (p,w). • Definition 2: If every commodity is normal at all (p,w), then we say that demand is normal.

8. Comparative Statics: Price Effects • Definition 1: The derivative xm(p,w)/pk is known as the price effect of pk , the price of good k, on the demand for good m. • Definition 2: Good m is said to be Giffen good at (p,w) if xm(p,w)/pm > 0. The price effects in matrix notation

9. Effect of a change in price x2 • Again take the basic equilibrium • ...and let the price of good 1 fall • The effect of the price fall... • The “journey” from x* to x** can be (imaginarily) broken into two parts: • A substitution effect Income effect • An income effect • x** • x* x Substitution effect x1

10. Effect of a change in price for Giffen Good x2 • Again take the basic equilibrium • ...and let the price of good 1 fall • The effect of the price fall... • A substitution effect • An income effect Income effect • x** • x* x Substitution effect x1

11. The Slutsky equation • Gives fundamental breakdown of effects of a price change • Income effect • Substitution effect

12. Slutsky Matrix and Substitution Effects Slutsky matrix Negative semidefinite and symmetric where Substitution effects

13. The good can be Giffen at (p,w) only if it is inferior!!!

14. Mean market demand • Economy consists of a continuum consumers, which all have the same demand function f(p,w) but differ by w. •  - the density of distribution of w with finite mean: • Mean market demand: in order to shorten notation we will write F(p).

15. On the “Law of Demand”

16. The Necessary Condition for Giffen Good ≤0 Mean (average) income effect term If then ≥0 <0

17. Some Assumptions

18. Sufficient condition for a negative mean income effect term

19. Shochu and Special Grade Sake Rich consumers Special grade sake Shochu Poor consumers

20. Inferiority and Giffen Effect (intuition) Market Prices 1Sake+2Shochu=3 or 0Sake+12Shochu=12… Poor consumer 1Sake+6Shochu=7 or 0Sake+12Shochu=12… 1x+ 10x =11 1x =50¥ 1x =5¥ Wealth = 100¥ = 1x +10x =11 Wealth = 100¥ Giffen effect: Inferiority: Price of Shochu 5 ¥  8 ¥ Wealth 100 ¥  60 ¥ Buy 12xShochu Buy 12xShochu WealthDemand Shochu Price Shochu Demand Shochu

21. Data on Shochu and Sake suggests: • Special grade sake is a normal good. • Shochu is an inferior good – (UM) and (ID) hold. • Need to examine the movements of prices and quantities consumed of Shochu and Sake. • Time series data. • Supply-and-demand model  simultaneity.

22. Demand-and-Supply Model Demand function: QtD=0+1Pt+2dec +u1t Supply function: QtS=0+1Pt+2dec +3int +u2t Equilibrium condition: QtD =QtS Demand function: QtD=0+1Pt+2dec +u1t Supply function: QtS=0+1Pt+2dec +u2t Equilibrium condition: QtD =QtS Q D1 S1 In this case we cannot distinguish between demand and supply We need shift in supply curve in order to determine demand - int S2 D2 S3 S4 P

23. Simultaneous Equation Model Demand function: Qt=0+1Pt+2dec +u1t Supply function: Qt=0+1Pt+2dec +3int +u2t Structural form equation Endogenous (dependent) variables Q  P Exogenous (determined outside the model) variables

24. Problem: Why not OLS? Pt=0+1Qt+2dec +uP Qt=0+1Pt+2dec +3int +uQ E(Qt uP)≠0 E(Pt uQ)≠0 Pt=0+1{ 0+1Pt+2dec +3int +uQ}+2dec +uP Qt=0+1{0+1Qt+2dec +uP }+2dec +3int +uQ That is a violation of classical regression model (Gauss-Markov condition )  OLS coefficients biased and not consistent. Pt={0+10+ (12+2)dec + 13int + 1 uQ +uP}/(1- 11) Qt={0+10+ (12+2)dec +3int + 1uP +uQ}/(1- 11)

25. What can we do? Demand function: Qt=0+1Pt+2dec +u1t Supply function: Qt=0+1Pt+2dec +3int +u2t Structural form equation Using equilibrium condition obtain: We CAN estimate these equations using OLS since all the RHS variables are exogenous Qt=10+ 11dec + 12int +v1t Pt= 20+ 21dec + 22int +v2t Reduced form equation But… Can we obtain from ’s ’s and ’s?

26. Identification Problem We could have three possible situations for the equation: • Underidentified: We cannot get the structural coefficients from the reduced form estimates. • Exactly (just) identified: Can get unique structural form coefficient estimates. • Overidentified: More than one set of structural coefficients could be obtained from the reduced form.

27. Order and Rank Conditions The order condition(necessary): • Let G denote the number of structural equations. An equation is just identified if the number of variables excluded from an equation is G-1. • If more than G-1 are absent, it is overidentified. If less than G-1 are absent, it is underidentified. Demand function: Qt=0+1Pt+2dec +u1t Supply function: Qt=0+1Pt+2dec +3int +u2t The rank condition (necessary and sufficient): Used in practice G-1=1  just identified G-1=1>0  underidentified

28. 2SLS Stage 1: • Estimating the reduced-form equation forP: Pt*= 0+ 1dec + 2int +vt • Stage 2: • In structural equation, regress Q on P* and exogenous variables: Qt= 0+ 1 Pt*+ 2dec + ut

29. The results of 2SLS: Shochu is a Giffen good

30. Yt= 0+ 1Xt+ ut 0 and 1 are estimated using OLS Statistical significance: t-test: t= (i*-i)/(i*)against Student’s. If time series are stationary the t statistic will falsely reject H0  5% when evaluated against the Student’s t dist at p≤ 0.05 Simple Regression Model H0: 1=0

31. Stationary and Nonstationary Time Series • Definition: A stochastic process Ytisstationaryif • E[Yt]= is independent of t; • Var[Yt]=E(Yt-)2 = Y2 is independent of t; • Cov[Yt, Ys ] = E[(Yt-)(Ys-)] is a function of t-s but not of t. • Otherwise the stochastic process is callednonstationary.

32. Examples of Stationary Time Series • White noise: {ut}t=(-,+) such that • E[ut]=0; • Var[ut]= u2; • Cov[ut, us ]=0 for all s≠t. • AR(1) process:Yt= Yt-1 + ut , -1 <  < 1 and ut is a white noise:

33. Nonatationary: Random Walk • The condition –1< <1 was crucial for stationarity. If  = 1  is a nonstationary process known as a random walk. Yt=Yt-1 + ut Yt=Y0 + u1 +… + ut E[Yt]= E[Y0] + E[u1]+… + E[ut]=Y0 Var[Yt]= tu2 is increasing witht

34. More Examples of Nonstationary Time Series • Random walk with drift: • E[Yt]=Y0 + t depends on t ; • Time series with time trend: • E[Yt]=  + t depends on t ; • Random walk with drift and linear time trend: Yt=  + Yt-1 + ut Yt=  + t + ut Yt=  + t+Yt-1 + ut

35. Random walk with drift Yt = 0.2 + Yt-1 + ut Stationary process: Yt = 0.7 Yt-1 + ut Random walk: Yt = Yt-1 + ut All three series are generated with the same set of disturbances

36. Trend-Stationarity • Definition:A trend-stationary model is one that can be made stationary by removing a deterministic trend. Example: Series with linear time trend Yt=  + t + ut Yt*=  + t Stationary Yt’=Yt – Yt*=ut • By contrast: Yt= t + Y0 + u1 +…+ut Zt=Yt- t = Y0 + u1 +…+ut; Var[Zt]= tu2

37. Difference-Stationarity • Definition: If a nonstationary process can be transformed into a stationary one by differencing, it is said to be difference-stationary. Example: Random walk with or without drift Yt=  + Yt-1 + ut I(1) Zt = Yt= (Yt –Yt-1) =  + ut I(0) Many economic time series are I(1).

38. Spurious regression Xt Yt Yt= 0+ 1Xt+ ut Granger and Newbold in a Monte Carlo experiment fitted the model where Yt and Xt were independently-generated random walks.

39. Results: • Obviously, a regression of one random walk on another ought not to yield significant results except as a matter of Type I error. The true slope coefficient is 0, because Y was generated independently of X. • However, performing the experiment with 100 pairs of random walks, Granger and Newbold found that the null hypothesis of a 0 slope coefficient was rejected 77 times (5%) and 70 times (1%). They found that in this case instead of t-critical value=2 (5%) one should use t =11.2.

40. Why? Yt= 0+ 1Xt+ ut • ut has the same autocorrelation properties as Yt which is nonstationary (or at best highly autocorrelated), but ut is white noise  standard t, F statistics will fail. • Low Durbin-Watson statistic will show that the regression is misspecified. H0: 1=0 ut = Yt - 0

41. Unit Root Test If H0 is true the OLS estimator is biased downward and conventional t and F tests will tend incorrectly to reject H0. Dickey and Fuller revised set of critical values (based on MC) Yt= Yt-1 + ut H0: =1 against H1:  < 1 Yt= Yt-1 + ut H0: =0 against H1:  < 0

42. Dickey-Fuller unit root test

43. Cointegration Yt= 0+ 1Xt+ ut • Definition: If  0 and 1 such that ut is stationary Xt and Yt are called cointegrated processes. • Thus Y and X could both be I(1), and yet, if the model is correctly specified, one would expect u to be I(0). • A requirement for cointegration is that all the variables in the relationship should be subject to the same degree of integration. • Example: PDI (personal disposable income) & PCE (personal consumption expenditure) ut=Yt- 0- 1Xt Nonstationary

44. Empirical Results • First step: analysis of all presented time series for stationarity and order of integration use DF. I(1) I(0) I(0) I(1) I(0) I(0) I(1) I(0)

45. Empirical Results (continuation) Model was correctly specified.

46. The End

47. Glossary • Consumer preferences: rationality, desirability, convexity, continuity. • Utility function: representation, properties, UMP. • The Walrasian demand function: definition, properties, assumptions. • WARP & compensated law of demand.

48. Consumer Preferences: Rationality • Rationality: a preference relation ≥ is rational if it is possesses: • Completeness • Transitivity

49. Consumer Preferences: Desirability • Monotonicity: ≥ is monotone on X if, for x,y  X, y>>x implies y > x. • Strong monotonicity: ≥ is strongly monotone if y≥x and y≠x imply that y > x. • Local nonsatiation: ≥ is locally nonsatiated if for every x  X and  > 0  y  Xs.t. ||y-x||≤  and y > x.

50. Consumer Preferences: Convexity • Convexity: The preference relation ≥ is convex if, x  X, the upper contour set {y  X: y ≥ x} is convex, that is if y ≥ x and z ≥ x, then ty+(1-t)z ≥ x for all 0<t<1. • Strict convexity: The preference relation ≥ is strictly convexif, x,y,z  X, y ≥ x, z ≥ x, and y ≠ z implies ty+(1-t)z > x for all 0<t<1.