ECO 310: Empirical Industrial Organization Lecture 2 - Estimation of Demand and Supply

Transcription

1 ECO 310: Empirical Industrial Organization Lecture 2 - Estimation of Demand and Supply Dimitri Dimitropoulos Fall 2014 UToronto 1 / 55

2 References RW Section 3. Wooldridge, J. (2008). Introductory Econometrics: A Modern Approach, 4th Edition. South-Western College Publishers. Epple, D. and B. McCallum (2006). "Simultaneous Equation Econometrics: The Missing Example." Economic Inquiry, Vol. 44(2), pp / 55

3 Motivation We now begin our study of the New Empirical Industrial Organization New Empirical IO is characterized by a close connection between economic theory, econometric methods, and data. New Empirical IO is based on Structural Models We model consumer and firm behavior explicitly, where consumers are utility maximizers and firms are profit maximizers. As a result, the parameters of the model are Structural, in the sense that they have clear interpretations in terms of consumer preferences, production technology, or institutional constraints Parameters are estimated using the Principle of Revealed Preference The economic outcomes we observe in data are choices made by consumers and firms in markets As such, the data contains information regarding consumer preferences, firm production technology, and markets constraints. This information is used to "back-out" the underlying structural parameters. 3 / 55

4 Motivation We will consider an important application of the New Empirical IO - Estimating Demand, Supply and Market Power To do this, we will consider an ever complexing model 1. Estimating Demand using Experimental Data 2. Estimating Demand and Supply using Observational Data 3. Estimating Demand, Supply & Market Power using Obs. Data 4 / 55

5 Estimating Demand using Experimental Data 5 / 55

6 Estimating Demand using Experimental Data There are many important questions in economics which answers depend crucially on reliable estimates of Market Demand Demand systems are the major tool for comparative static analysis of any change in a market that does not have an immediate impact in costs For example, estimation of price elasticities are important for Marketers in designing pricing policies introducing new goods Policy Offi cials Etc. in designing tax schemes in evaluating mergers 6 / 55

7 Experimental Data To empirically estimate Market Demand, the first thing we need is data. Ideally, this data will have come from a Controlled Experiment This way we are assured that our data is a Random Sample We conduct the following experiment to collect demand data on Good-Q 1. We random sampled N consumers from the population 2. For each consumer i, we confront them with a randomly selected price P i 3. We then observe consumer i s buying behavior Q i The experiment will result in a Random Sample of data {Q i, P i : i = 1, 2,...N} 7 / 55

8 A Structural Model: The Economic Model Once we have collected data, we need an Econometric Model through which we can obtain estimates of Market Demand. With the New Empirical IO, our Econometric Models are Structural: Begin with an Economic Model derived from theory Then translate this into a Statistical Model for use in empirical application 8 / 55

9 A Structural Model: The Economic Model Consider a consumer who buys only two goods: Q and Z We are interested in the demand for Good-Q Good-Z is interpreted as a composite good (or "numeraire"). The consumer has Y dollars of income to spend. The price of Good-Q is P. The price of Good-Z is $1 Consumer s preferences given by Utility Function U = A 0 Q A 1 Q 2 /2 + Z Economic Theory tells us the problem of the consumer is to choose the bundle of goods from his budget set which yields him highest utility The consumer s Demand Function Q(P) for Good-Q gives the utility maximizing quantity of Good-Q in his budget set. max Q,Z A 0Q A 1 Q Z s.t. PQ + Z = Y 9 / 55

10 A Structural Model: The Economic Model If we substitute the budget constraint into the objective function, we can equivalently write the consumer s choice problem as [ max A 0 Q A ] 1 Q 2 Q 2 PQ + Y The FOC for this problem gives the Inverse Demand Function for Good-Q P = A 0 A 1 Q and implies sets Marginal Willingness to Pay equal to Price. Solving for the consumer s Demand Function for Good-Q Q = A 0 A 1 1 A 1 P = Q = α 0 + α 1 P 10 / 55

11 A Structural Model: The Econometric Model Our Economic Model is a a Theoretical model of Demand. To derive an Econometric Model, we translate this into a Statistical Model The quantities Q i that we observe in the data are Random Variables We can decompose a random variable into: (1) a Systematic part µ, and (2) an Unobservable Error ε d i Q i = µ + ε d i The systematic part of Qi is its expected value µ = E [Q i ] The unobservable error ε d i is the stochastic part of Q i E [ε d i ] = 0 and var(ε d i ) = σ 2 11 / 55

12 A Structural Model: The Econometric Model We use Economic Theory to give us our predications about Q i. E[Q i ] = α 0 + α 1 P Error ε d i is an Independent and Identically Distributed "demand shock" Captures how consumer decisions differ from predictions of economic model Thus, our Econometric Model for Demand Q i = α 0 + α 1 P i + ε d i ε d i IID(0, σ 2 ) Note, the parameters of this model are structural - they have clear interpretations in terms of the underlying demand primitives. In particular: α = (α0, α 1) are the parameters from the consumer utility function Moreover, it is straight forward to show that the elasticity of demand is η = α 0 α 1 P P 12 / 55

13 A Structural Model: The Econometric Model Given the random sample nature of our data {Q i, P i : i = 1,..., N}, our Econometric Model has certain useful properties. Q i = α 0 + α 1 P i + ε d i ε d i IID(0, σ 2 ) Assumption 1 - Zero Mean: E[ε d i ] = 0 Assumption 2 - Exogeneity: E[P i ε d i ] = 0 By assumption, Pi is "randomly assigned" Thus, Pi is independent ε d i 13 / 55

14 Aside: Statistical Theory We need to establish some Statistical Theory before we continue Let M(θ) be an Econometric Model on the parameters θ An Estimator θn is a rule (a function) that, when provided with n observations of data provides us with an estimate of the parameter θ An estimator θn is said to be Consistent if it yields an unbiased estimate of the parameters when provided with "enough" data. We say the parameter θ is Identified when it can be consistently estimated θ n θ 14 / 55

15 Aside: Statistical Theory Critical to statistical theory is the Law of Large Numbers Let X be a random variable with mean E [X ] = µ Suppose X1, X 2,..., X n is random sample of nrandom draws of X Then, the sample mean of X1, X 2,..., X n X n = 1 n i=1 X i is a consistent estimator of the mean of X X n µ 15 / 55

16 Estimating Demand: OLS Given the random sample {Q i, P i : i = 1, 2,...N}, we want to derive an estimator of the parameters α = (α 0, α 1 ) of our econometric model Q i = α 0 + α 1 P i + ε d i ε d i IID(0, σ 2 ) The most common of all estimators is Ordinary Least Squares We do not observe ε d i However, given an estimate α of the parameters, we can construct estimates of ε d i, which we call "residuals" ê i = Q i ( α 0 + α 1P i ) OLS estimator α is estimate that minimizes the sum of squared residuals min α 0, α 1 N { Q i α 0 α 1P i } 2 i=1 16 / 55

17 Estimating Demand: OLS In this Simple Linear Regression model, the OLS estimator or α 1 is N i=1 α 1 = (Q i Q)(P i P) N i=1 (P i P) 2 What is the statistical interpretation of the estimator? α 1 = cov(q, P) var(p) 17 / 55

18 A More General Model of Demand Our model assumes that demand only depends on prices But demand may very well depend on other variables as well - in particular, demand may depend on income We conduct the following experiment to collect demand data on Good-Q 1. We randomly sampled N consumers from the population 2. For each consumer i, we measure the consumer s income Y i, and then confront the consumer with a randomly selected price P i 3. We then observe consumer i s buying behavior Q i Our random sample of data is now {Q i, P i, Y i : i = 1, 2,...N} We can augment our econometric model to take into account income as well. Q i = α 0 + α 1 Y i + α 2 P i + ε d i ε d i IID(0, σ 2 ) 18 / 55

19 A More General Model of Demand We can augment our econometric model to take into account income as well. Q i = α 0 + α 1 Y i + α 2 P i + ε d i ε d i IID(0, σ 2 ) It will be useful to write this model using matrix notation. Our model holds for each consumer in the sample Q 1 = α 0 + α 1Y 1 + α 2P 1 + ε d 1 Q 2 = α 0 + α 1Y 2 + α 2P 2 + ε d 2. Q N = α 0 + α 1Y N + α 2P N + ε d N Define Q 1 Q 2 q =. Q N 1 Y 1 P 1 1 Y 2 P 2 X =... 1 Y N P N ε d 1 ε d 2 ε =.. ε d 2 19 / 55

20 A More General Model of Demand We can augment our econometric model to take into account income as well. Q i = α 0 + α 1 Y i + α 2 P i + ε d i ε d i IID(0, σ 2 ) It will be useful to write this model using matrix notation. Define Q 1 Q 2 q =. Q N 1 Y 1 P 1 1 Y 2 P 2 X =... 1 Y N P N ε d 1 ε d 2 ε =.. ε d 2 Then, we can write the model compactly as q = Xα + ε ε IID(0, σ 2 ) where the vector of parameters is α = (α 0, α 1, α 2) 20 / 55

21 A More General Model of Demand Given the random sample nature of our data, our econometric model q = Xα + ε ε IID(0, σ 2 ) has very important properties Assumptions: 1. Zero Mean: E [ε d i ] = 0 2. Exogeneity: Income Y i is an exogenous variable - it is determined in labor/income markets outside the model. Thus, Y i is independent ε d i E [Y i ε d i ] = 0 3. Exogeneity: By assumption, P i is "randomly assigned" by the researcher. Thus, P i is independent ε d i E [P i ε d i ] = 0 21 / 55

22 A More General Model of Demand Given the random sample nature of our data, our econometric model q = Xα + ε ε IID(0, σ 2 ) has very important properties Assumptions: These assumptions can be written in vector form as ε d i 1 E Y i ε d i = E Y i ε d P i ε d i = 0 i P i or more compactly where x i is a row of the matrix X E [x i ε i ] = 0 22 / 55

23 A More General Model of Demand As before we can use Ordinary Least Squares to derive an estimator of α The OLS Estimator in matrix form α = (X X) 1 X q Interpretation? or ( α = XX ) 1 XQ α = VAR(X d i ) 1 COV(X i, Q i ) 23 / 55

24 A More General Model of Demand It is straightforward to show that the OLS estimator is consistent Note that the OLS estimator α = (X X) 1 X q can be manipulated, by substituting in for q using Xα + ε, into α = (X X) 1 X (Xα + ε) = (X X) 1 X Xα + (X X) 1 X ε = α + (X X) 1 X ε or α = α + ( ) 1 XX Xε 24 / 55

25 A More General Model of Demand It is straightforward to show that the OLS estimator is consistent Now, by the Law of Large Numbers 1 E [x i ε i ] = 0 N Xε Thus α = α + ( ) 1 XX Xε α + ( XX ) 1 {0} = α i.e. α α and α is identified by OLS Note the importance of having a random sample It is the random sampling assumption through which we can assert exogneity E [x i ε i ] = 0 And it is this assumption which gives allows us to identify α by OLS 25 / 55

26 Estimating Demand & Supply using Observational Data 26 / 55

27 Estimating Demand and Supply Typically, in Economics we do not have access to experimental data. Moreover, our data is often not at the individual level Rather, the data we use is Observational Data Moreover, the unit of measurement is usually at the Market Level. Suppose we collected data using the following method. 1. We randomply sampled M markets from Canada. 2. For each market i, we observe the price P i and quantity Q i of the good traded in that market, as well as the prevailing market income Y i. Data collection will result in the following sample of data {Q i, P i, Y i : i = 1, 2,...M} 27 / 55

28 Estimating Demand and Supply With observational data, price is no longer an Exogenous Variable Instead, price is Endogenous Variable, determined by the interaction of market demand and supply. The prices and quantities in our data set are not from a random sample. Rather, they are equilibrium data. To solve this Endogeneity Issue, we will need to take supply into consideration when we estimate demand Otherwise, our estimates of the demand parameters will be biased 28 / 55

29 Estimating Demand and Supply Modeling Supply Suppose that the Industry for good Q is Perfectly Competitive. There are many firms, each with marginal costs given by MC = β 0 + β 1Q The goal of each firm in the industry is to maximize profits. Each perfect competitor takes price as given, and chooses output such that max PQ C (Q) and so will set price equal to marginal cost. So, the supply function P = β 0 + β 1Q 29 / 55

30 Estimating Demand and Supply Modeling Supply The supply function from our economic theory describes average firm behaviour - it is best thought of as model of expected supply E [P i ] = β 0 + β 1Q i To translate this into an econometric model we introduce a "supply shock" ε s i ε s i captures how firm behaviour differ from that predicted by economic theory Thus, the econometric model of Market Supply P i = β 0 + β 1Q i + ε s i ε s i IID(0, σ 2 s ) 30 / 55

31 Estimating Demand and Supply What happens if we ignore Supply? Suppose we plan to estimate demand using our observational data {Q i, P i, Y i : i = 1, 2,...M} and the following econometric model of Market Demand Q i = α 0 + α 1Y i + α 2P i + ε d i ε d i IID(0, σ 2 d ) Is the parameter vector α = (α0, α 1, α 2) identified by OLS? The answer is no! This is because, with observational data, we have E [P i ε d i ] 0 31 / 55

32 Estimating Demand and Supply The issue is that P i is Endogenous - E[P i ε d i ] > 0 32 / 55

33 Estimating Demand and Supply The issue is that P i is Endogenous - E[P i ε d i ] > 0 Consider two markets with identical incomes Yi that face same supply curve The fact that the two markets have the same level of Yi means that, on average, the two markets should have the same demand curve. Nevertheless, different markets will have different demand shocks the market with higher ε d i will have a higher demand curve. Demand Effect: ε d i Q i And, since price is determined by the intersection of demand and supply, the market with higher ε d i will also have a higher price. Equilibrium Effect: ε d i P i But demand shocks are unobservable all we see in our sample of data is higher quantities being associated with higher quantities Thus, the only conclusion that OLS can make is that Qi is caused by P i 33 / 55

34 Estimating Demand and Supply The issue is that P i is Endogenous - E[P i ε d i ] > 0 As a result, the parameter estimate for the effect of Pi is too big That is, α2 will not be "negative enough", and maybe even "positive" By ignoring supply, we fail to account that prices are not randomly assigned, but are rather equilibrium prices deternubed by supply and demand That is, we erroneously assume that prices are Exogenous, when in fact the market prices we observed are Endogenous 34 / 55

35 Solving the Identification Problem We overcome identification problem through use of Instrumental Variables The idea is that shifts in the supply curve will trace out the demand curve 35 / 55

36 Solving the Identification Problem We overcome identification problem through use of Instrumental Variables The idea is that shifts in the supply curve will trace out the demand curve Consider two markets with identical income Yi but different wage rates W i The fact that the two markets have the same level of Yi means that, on average, these two markets will have the same demand curve However, because the markets have different wage rates, there will be a different supply curve in each market each market s supply curve being a vertical translation of the other s Thus, by comparing equlibrium (Pi, Q i ) outcomes in these two markets with shifted supply curves we can sketch out the demand curve. 36 / 55

37 Solving the Identification Problem Consider first the solution to the identification problem of Demand. For a variable Zi to be a valid instrument in demand 1. Significant First Stage: Z i is correlated with the endogenous variables i.e. Z i aids in the determination of equilibrium prices. E [Z i P i ] 0 2. Exclusion Restriction: Z i is not correlated with the demand error i.e. it is not a demand-side variable E [Z i ε d i ] = 0 More plainly, an instrument for the demand function is a variable that aids determine equilbrium prices BUT does not belong in the demand equation. A good example of this would be a Cost Shifter. 37 / 55

38 Solving the Identification Problem Suppose we considered the following system of Demand and Supply Q i = α 0 + α 1 Y i + α 2 P i + ε d i ε d i IID(0, σ 2 d ) P i = β 0 + β 1 W i + β 2 Q i + ε s i ε s i IID(0, σ 2 s ) And, we collected the following sample of data where {Q i, P i, W i, Y i : i = 1, 2,...M} We first randomply sample M markets from Canada. For each market i, we observe the prevailing market price Pi and quantity Q i Then, for each market i, we collecte data on the prevailing wage rate Wi and prevailing level of consumer income Y i. 38 / 55

39 Solving the Identification Problem The wage rate W i is an exogenous variable - it s determined in labor market, not in the market for Good-Q However W i will alter the supply equation to P i = β 0 + β 1 W i + β 2 Q i + ε s i ε s i IID(0, σ 2 s ) Thus, W i can serve as a valid instrument for P i in demand Clearly, Wi will affect equilibrium prices, through its effect on supply First Stage: E [W i P i ] 0 Moreover, since Wi does not enter the demand equation, it is uncorrelated with demand shocks Exclusion Restriction: E [W i ε d i ] = 0 39 / 55

40 Estimating Demand and Supply How can we use W i to estimate Demand? With OLS, we define the matrix of regressors as 1 Y 1 P 1 X d 1 Y 2 P 2 =... 1 Y N P N However, we know that ε d i E Y i ε d i 0 P i ε d i so we cannot define a conistent estimator using the matrix X d 40 / 55

41 Estimating Demand and Supply How can we use W i to estimate Demand? Now, define the matrix of Instruments Z by substituting in Wi for P i 1 Y 1 W 1 1 Y 2 W 2 Z =... 1 Y N W N We have or ε d i E Y i ε d i = 0 W i ε d i E [z i ε d i ] = 0 so we can define a conistent estimator using the matrix Z 41 / 55

42 Estimating Demand and Supply The Instrumental Variables (IV) estimator for the Demand equation α IV = (Z X d ) 1 Z q Interpretation? α IV ( ZXd ) 1 = ZQ or α IV = COV(Z i, X d i ) 1 COV(Z i, Q i ) 42 / 55

43 Estimating Demand and Supply The Instrumental Variables estimator α IV = (Z X d ) 1 Z q is commonly referred to as the Two Stage Least Squares (2SLS) estimator A 2-Stage method can be use to calculate IV estimates In the 1st stage, the endogenous variable is regressed on all of the exogenous variables, including both exogenous explanatory variables and the instruments. The predicted value of the endogenous variable is obtained. In the 2ns stage, the regression of interest is estimated as usual, except the endogenous covariate is replaced with its predicted values from 1st stage. 43 / 55

44 Estimating Demand and Supply The Instrumental Variables estimator α IV = (Z X d ) 1 Z q is commonly referred to as the Two Stage Least Squares (2SLS) estimator A 2-Stage method can be use to calculate IV estimates 1. Estimate the First Stage: Regress P i on Y i and W i using OLS P i = π o + π 1Y i + π 2W i + ζ i and obtain the predicted values P i = p o + p 1Y i + p 2W i 2. Estimate the Model: Regress Q i on Y i and P i using OLS Q i = α 0 + α 1Y i + α 2 P + ε d i 44 / 55

45 Estimating Demand and Supply In STATA, the syntax for the 2SLS estimator is ivregress 2sls y x 1 x 2... x j (x k = z 1 z 2... z l ) where x1,..., x j is the list of exogenous explanatory variables xk is the endogenous explanatory variable z1,..., z l are the insturmental variables Recall, to estimate demand using OLS we would use regress Q Y P But, to estimate our demand model using 2SLS we would use ivregress 2sls Q Y (P = W) 45 / 55

46 Estimating Demand and Supply It is straight forward to show that the Demand 2SLS estimator is consistent First, re-write the 2SLS estimator as α IV = (Z X d ) 1 Z q = α + (Z X d ) 1 Z ε d or α IV ( ZXd ) 1 = α + Zε By the Law of Large Numbers 1 N Zε E [z i ε d i ] = 0 And since Z is a valid instrument ( ) 1 α IV α + (0) ZX d 46 / 55

47 Estimating Demand and Supply It is straight forward to show that the Demand 2SLS estimator is consistent Note the importance of the IV conditions We need the Exclusion Restriction otherwise E [W i ε D i ] = 0 1 N Zε 0 We need a Significant First Stage E [W i P i ] 0 otherwise ( ) 1 ZD 47 / 55

48 Solving the Identification Problem We can use the same idea on the supply identification problem. P i = β 0 + β 1 W i + β 2 Q i + ε s i ε s i IID(0, σ 2 s ) Income Yi can serve as a valid instrument for Q i in supply Clearly, Y i will affect equilibrium quantities, through its effect on demand E [Y i Q i ] 0 Moreover, since Y i does not enter the supply equation, it will be uncorrelated with random supply shocks E [Y i ε S i ] = 0 48 / 55

49 Estimating Demand and Supply How can we use Y i to estimate Supply? p = X s β + ε s The matrix of supply regressors is 1 W 1 Q 1 X s 1 W 2 Q 2 =... 1 W N Q N But since Qi is endogenously determined ε s i E W i ε s i 0 Q i ε s i so the regressors in X s cannot be used to estimate the supply estimation 49 / 55

50 Estimating Demand and Supply How can we use Y i to estimate Supply? p = X s β + ε s Now, define the matrix of Instruments Z by substituting Yi for Q i 1 W 1 Y 1 1 W 2 Y 2 Z =... 1 W N Y N We have E [z i ε s i ] = E ε s i W i ε s i Y i ε s i = 0 The IV (or 2SLS) estimator for Supply β IV = (Z X s ) 1 Z p 50 / 55

51 Epple and McCallum (2006) - Motivation Epple and McCallum (2006) "Simultaneous Equation Econometrics: The Missing Example." ABSRACT: For introductory presentation of issues involving simultaneous equation systems, a natural vehicle consists of supply and demand relationships for a single good. One would expect to find in econometrics textbooks a supply-demand example featuring actual data in which structural estimation methods yield more satisfactory results than does ordinary least squares. But a search of 26 existing textbooks finds no example with actual data in which all crucial parameter estimates are of the proper sign and are statistically significant. The present article accordingly develops a simple but satisfying example, for broiler chickens, based on U.S. annual data from 1960 to / 55

52 Epple and McCallum (2006) - Motivation Epple and McCallum (2006) "Simultaneous Equation Econometrics: The Missing Example." Pg 375. "Specifically, we develop and estimate a simple demand-supply system involving annual U.S. time series data from 1960 to 1999 for chicken. Our specification of the demand and supply functions attempts to be theoretically sensible, and our two-stage least squares estimation yields statistically significant estimates of all structural parameters, each of which is of the appropriate sign and is plausible in magnitude. Moreover, these estimates are more satisfactory than ones obtained by application of ordinary least squares to the structural equations." 52 / 55

53 Epple and McCallum (2006) - Model Broiler chickens are chickens bred specifically for meat production. Epple and McCallum use time serires data to study the aggregatve demand for broiler chickens in the U.S. between 1960 and 1999 The specifications of demand 1 and supply 2 ln Q t = α 0 + α 1 ln Y t + α 2 ln P t + α 3 ln PB t + ε d t where ln Q A t = β 1 + β 2 ln P t + β 3 ln PF t + β 4 time + β 5 ln Q A t 1 + ε s t Q t per capita chicken cons. Qt A agg. output of chicken Y t per capita income PF t price of chicen feed P t price of chicken time trend PB t price of beef Q A t 1 lagged agg. output of chicken 1 Note: Demand is estimated in first differences to eliminate serial correlation. 2 Note: Supply includes the previous period s value of output Q A t 1 to reflect adjustment costs in the production process. 53 / 55

54 Epple and McCallum (2006) - Results Demand OLS 2SLS Eq. (6) Eq. (12) Supply OLS 2SLS Eq. (9) Eq. (13) Y (0.150) (0.142) P (0.052) (0.106) P (0.058) (0.086) PF (0.032) (0.052) PB (0.068) (0.093) time (0.004) (0.006) R T Q A t (0.108) (0.125) R T / 55

55 Epple and McCallum (2006) - Results Pg OLS yields estimates "that are of the wrong algebraic sign and not statistically insignificant. By contrast, the [simultaneous equation model] with two-stage least squares has coeffi cients of the correct sign and statistically significant." Pg. 380 "The estimated demand coeffi cients imply an own-price elasticity of 0.40, an income elasticity of 0.84, and a cross-price elasticity with respect to the substitute good (beef) of These are of the expected algebraic signs and strike us as being quite reasonable in magnitude. The own-price elasticity of supply is 0.22, and the elasticity of supply with respect to the price of the primary input (feed) is Again, these are of the expected algebraic signs and seem to be quite plausible in magnitude." 55 / 55