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 B pw = {x ≥ 0: p·x ≤ w}

3. x2x2 x1x1 l x* increasing preference Indifference curves  The consumer aims to maximise utility...  Subject to the budget constraint The UMP max u(x) subject to n  p i x i  w i=1 max u(x) subject to n  p i x i  w i=1  Defines the UMP Budget set Budget set  Solution to the problem

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 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  x m (p,w )/  w is known as the wealth (income) effect for the m-th good. Definition 2: At any (p,w), the derivative  x m (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 l x* l x** x1x1 x2x2  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?

6. x1x1 x2x2 l x* l x**  The same original prices, but different preferences...  Again, let income rise... An “inferior” good  The new equilibrium demand for “inferior” good 2 falls a little as income rises demand for “inferior” good 2 falls a little as income rises

7. Normal and Inferior Goods Definition 1: A commodity m is normal at (p,w) if  x m (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 1: A commodity m is normal at (p,w) if  x m (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. 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  x m (p,w )/  p k is known as the price effect of p k, the price of good k, on the demand for good m. Definition 1: The derivative  x m (p,w )/  p k is known as the price effect of p k, 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  x m (p,w )/  p m > 0. Definition 2: Good m is said to be Giffen good at (p,w) if  x m (p,w )/  p m > 0. The price effects in matrix notation

9. l x* l x** x1x1 x2x2 Income effect Substitution effect  Again take the basic equilibrium ...and let the price of good 1 fall Effect of a change in price  The effect of the price fall...  The “journey” from x* to x** can be (imaginarily) broken into two parts: o An income effect o A substitution effect x

10. l x* l x** l x* l x**  The income effect: “how do demands respond to changes in the cost of living?” Close up….  The substitution effect: “at a given utility level how do demands respond to relative prices?” l xl x

11. l x* l x** x1x1 x2x2 Income effect Substitution effect  Again take the basic equilibrium ...and let the price of good 1 fall Effect of a change in price for Giffen Good  The effect of the price fall... o An income effect o A substitution effect x

12.  Gives fundamental breakdown of effects of a price change The Slutsky equation  Income effect  Substitution effect

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

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

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

16. On the “Law of Demand”

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

18. Some Assumptions

19. Sufficient condition for a negative mean income effect term

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

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

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

23. Demand function: Q t D =  0 +  1 P t +  2 dec +u 1t Supply function: Q t S =  0 +  1 P t +  2 dec +u 2t Equilibrium condition: Q t D =Q t S Demand function: Q t D =  0 +  1 P t +  2 dec +u 1t Supply function: Q t S =  0 +  1 P t +  2 dec +  3 int +u 2t Equilibrium condition: Q t D =Q t S Demand-and-Supply Model In this case we cannot distinguish between demand and supply P Q S1S1 D1D1 S2S2 D2D2 S3S3 S4S4 We need shift in supply curve in order to determine demand - int

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

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

26. What can we do? Demand function: Q t =  0 +  1 P t +  2 dec +u 1t Supply function: Q t =  0 +  1 P t +  2 dec +  3 int +u 2t Structural form equation Using equilibrium condition obtain: Q t =  10 +  11 dec +  12 int +v 1t P t =  20 +  21 dec +  22 int +v 2t Reduced form equation We CAN estimate these equations using OLS since all the RHS variables are exogenous  ’s  ’s and  ’s? But… Can we obtain from  ’s  ’s and  ’s?

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

28. 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. 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. If more than G-1 are absent, it is overidentified. If less than G- 1 are absent, it is underidentified. The rank condition (necessary and sufficient) : Used in practice Demand function: Q t =  0 +  1 P t +  2 dec +u 1t Supply function: Q t =  0 +  1 P t +  2 dec +  3 int +u 2t G-1=1  just identified G-1=1>0  underidentified

29. 2SLS P t *=  0 +  1 dec +  2 int +v t P t *=  0 +  1 dec +  2 int +v t Stage 1: Estimating the reduced-form equation for P: Estimating the reduced-form equation for P: Stage 2: P* In structural equation, regress Q on P* and exogenous variables: Q t =  0 +  1 P t *+  2 dec + u t Q t =  0 +  1 P t *+  2 dec + u t

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

31. Simple Regression Model Y t =  0 +  1 X t + u t  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 H 0  5% when evaluated against the Student’s t dist at p≤ 0.05  1 =0 H 0 :  1 =0

32. Stationary and Nonstationary Time Series Definition: A stochastic process Y t is stationary if E[Y t ]=  is independent of t;  Y Var[Y t ]=E(Y t -  ) 2 =  Y 2 is independent of t; Cov[Y t, Y s ] = E[(Y t -  )(Y s -  )] is a function of t-s but not of t. Otherwise the stochastic process is called nonstationary.

33. Examples of Stationary Time Series White noise: {u t } t=(- ,+  ) such that White noise: {u t } t=(- ,+  ) such that E[u t ]=0;  u Var[u t ]=  u 2 ; Cov[u t, u s ]=0 for all s≠t. AR(1) process: u t, -1 < < 1 and u t is a white noise: AR(1) process: Y t =  Y t-1 + u t, -1 <  < 1 and u t is a white noise:

34. Nonatationary: Random Walk The condition –1<  <1 was crucial for stationarity. If  = 1  is a nonstationary process known as a random walk. u t Y t = Y t-1 + u t u 1 + … + u t Y t = Y 0 + u 1 + … + u t u 1 ] + … + E[u t ]= E[Y t ]= E[Y 0 ] + E[u 1 ] + … + E[u t ]=Y 0  u Var[Y t ]= t  u 2 is increasing with t

35. More Examples of Nonstationary Time Series Random walk with drift: Random walk with drift: E[Y t ]=Y 0 +  t depends on t ; E[Y t ]=Y 0 +  t depends on t ; Time series with time trend: Time series with time trend: E[Y t ]=  +  t depends on t ; E[Y t ]=  +  t depends on t ; Random walk with drift and linear time trend: Random walk with drift and linear time trend: u t Y t =  + Y t-1 + u t u t Y t =  +  t + u t u t Y t =  +  t + Y t-1 + u t

36. Random walk with drift Y t = 0.2 + Y t-1 + u t Stationary process: Y t = 0.7 Y t-1 + u t Random walk: Y t = Y t-1 + u t All three series are generated with the same set of disturbances

37. Difference between RW and LTT u t = RWD: Y t =  + Y t-1 + u t =  t + Y 0 + u 1 +…+u t u t LTT: Y t =  +  t + u t Random walk with drift Linear time trend The divergence from the trend line is random walk and the variance around the trend increases without limit. The deviations from the trend are short-lived. The series sticks to its trend in the long run.

38. Trend-Stationarity Definition: Definition: A trend-stationary model is one that can be made stationary by removing a deterministic trend. u t Y t =  +  t + u t Y t * =  +  t u t Y t ’ = Y t – Y t * = u t Example: Series with linear time trend Stationary  By contrast: Y t =  t + Y 0 + u 1 +…+u t Z t =Y t -  t = Y 0 + u 1 +…+u t ;  u Var[Z t ]= t  u 2

39. 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 u t Y t =  + Y t-1 + u t u t Z t =  Y t = (Y t –Y t-1 ) =  + u t I(1) I(0) Many economic time series are I(1).

40. Spurious regression Granger and Newbold in a Monte Carlo experiment fitted the model where Y t and X t were independently-generated random walks. XtXt YtYt Y t =  0 +  1 X t + u t

41. 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.

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

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

44. Dickey-Fuller unit root test

45. Cointegration Definition: If   0 and  1 such that u t is stationary X t and Y t are called cointegrated processes. Definition: If   0 and  1 such that u t is stationary X t and Y t 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) Y t =  0 +  1 X t + u t Nonstationary u t =Y t -  0 -  1 X t

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

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

48. The End

49. Glossary Consumer preferences: rationality, desirability, convexity, continuity. Consumer preferences: rationality, desirability, convexity, continuity.rationalitydesirability convexitycontinuityrationalitydesirability convexitycontinuity Utility function: representation, properties, UMP. Utility function: representation, properties, UMP.representationproperties UMPrepresentationproperties UMP The Walrasian demand function: definition, properties, assumptions. The Walrasian demand function: definition, properties, assumptions.definition propertiesassumptionsdefinition propertiesassumptions WARP & compensated law of demand. WARP & compensated law of demand. WARP & compensated law of demand WARP & compensated law of demand

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

51. Consumer Preferences: Desirability Monotonicity: ≥ is monotone on X if, for x,y  X, y>>x implies y > x. 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. 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  X s.t. ||y-x||≤  and y > x. Local nonsatiation: ≥ is locally nonsatiated if for every x  X and  > 0  y  X s.t. ||y-x||≤  and y > x.

52. 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. 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 convex if,  x,y,z  X, y ≥ x, z ≥ x, and y ≠ z implies ty+(1-t)z > x for all 0 x for all 0<t<1.

53. Consumer Preferences: Continuity Lexicographic (strict) preference ordering L is not continuous. For all x,y  X=  L +, xLy if x 1 >y 1, or if x 1 =y 1 and x 2 >y 2, or if x i =y i for i = 1, …, k-1 y k.

54. Utility function representation Definition: The utility function u: X   represents ≥ if Definition: The utility function u: X   represents ≥ if  x,y  X, u(x)  u(y)  x ≥ y. Theorem (Debreu, 1954): Suppose that the rational preference ordering ≥ on  L + is continuous. Then it can be represented by a continuous utility function u:  L + . Theorem (Debreu, 1954): Suppose that the rational preference ordering ≥ on  L + is continuous. Then it can be represented by a continuous utility function u:  L + .

55. A utility function u 0 u(x 1,x 2 ) x2x2 x1x1 Utility Function indifference curve

56. Properties of Utility Function For any strictly increasing function f:  For any strictly increasing function f:  v(x)= f(u(x)) is a new utility function representing the same preferences as u v(x)= f(u(x)) is a new utility function representing the same preferences as u Convex preferences  quasiconcave utility function. Convex preferences  quasiconcave utility function.

57. The Utility Maximization problem We assume that the consumer has a rational, continuous and locally nonsatiated ≥ and u(x) is a continuous utility function. We assume that the consumer has a rational, continuous and locally nonsatiated ≥ and u(x) is a continuous utility function. UMP: UMP: Formalization of consumer’s problem If p>>0 and u(·) continuous => UMP has a solution. If p>>0 and u(·) continuous => UMP has a solution.

58. The Walrasian Demand Function Definition: The consumer’s Walrasian demand correspondence x(p,w) assigns a set of optimal consumption vectors in UMP to each price- wealth pair (p,w)>>0. Definition: The consumer’s Walrasian demand correspondence x(p,w) assigns a set of optimal consumption vectors in UMP to each price- wealth pair (p,w)>>0. In principle this correspondence can be multivalued. When x(p,w) is single-valued, we refer to it as a demand function. In principle this correspondence can be multivalued. When x(p,w) is single-valued, we refer to it as a demand function.

59. Properties of the Demand Function Theorem: u(·) is a continuous utility function representing a locally nonsatiated preference relation ≥ defined on the consumption set X=  L +. Then the x(p,w) possesses the following properties: Theorem: u(·) is a continuous utility function representing a locally nonsatiated preference relation ≥ defined on the consumption set X=  L +. Then the x(p,w) possesses the following properties: Homogeneity of degree zero in (p,w): Homogeneity of degree zero in (p,w): Walras' law: Walras' law: Convexity / uniqueness. Convexity / uniqueness.

60. Assumptions on x(p,w) x(p,w) is always single-valued. x(p,w) is always single-valued. When convenient, we assume x(p,w) to be When convenient, we assume x(p,w) to be continuous continuous differentiable. differentiable.

61. The WARP and the Law of Demand

62. Proof of the Theorem 2.3.1