Lecture notes: Demand estimation introduction 11 Why demand analysis/estimation?• Estimation of demand functions is an important empirical endeavor. Why?• Fundamental empircial question: how much market power do firms have?– Market power: ability to raise prices profitably. (What market power doprice-taking firms have?)– Market power measured by markup:p−mcp.– Problem: mc not observed!– Motivates empirical methodology in IO.– For example, you observe high prices in an industry. Is this due to marketpower, or due to high costs? Cannot answer this question directly, becausewe don’t observe costs.• Indirect approach: obtain estimate of firms’ markups by estimating firms’ de-mand functions.• Intuition is most easily seen in monopoly example:– maxppq(p) − C(q(p)), where q(p) is demand curve.– FOC: q(p) + pq′(p) = C′(q(p))q′(p)– At optimal price p∗, Inverse Elasticity Property holds:(p∗− MC(q(p∗))) = −q(p∗)q′(p∗)orp∗− mc (q(p∗))p∗= −1ǫ(p∗),where ǫ(p∗) is q′(p∗)p∗q(p∗), the price elasticity of demand.– Hence, if we can estimate ǫ(p∗), we can infer what the markupp∗−mc(q(p∗))p∗is, even when we don’t observe the marginal cost mc (q(p∗)).1Lecture notes: Demand estimation introduction 2– Caveat: validity of exer cise depends crucially on using the right supply-sidemodel (in this case: monopoly without entry possibility).If costs were observed: markup could be estimated directly, and we couldtest for vaalidity of monopoly pricing model (ie. test whether markup=−1ǫ).• Start by reviewing some econometrics. (No attempt to be exhaustive.)2 Primer: Least-squares estimation• Observe data points {yi, xi} for i = 1, . . . n. What is linear relationship betweeny and x?• Graph. What linear function of x – that is, α + βx – fits y the best?• Ordinary least squares (OLS) regression:minα,βXi[yi− α − βxi]2.• In multivariate case: Xiand~β are both K − dimensionalvectors. Thenminα,~βXi[yi− α − X′i~β]2.To analyze properties of OLS r egression, consider a closely-related statistical problemof Best Linear Prediction,Consider two random variables X and Y . What is the “best” predictor o f Y , amongall the possible linear functions of X?“Best” linear predictor minimizes the mean squared error of prediction:minα,βE(Y − α − βX)2. (1)2Lecture notes: Demand estimation introduction 3(Recall: expectation is linear operator, so that E(A + B) = EA + EB)The first-order conditions are:For α: 2α − 2EY + 2βEX = 0For β: 2βEX2− 2EXY + 2αEX = 0.Solving:β∗=Cov(X, Y )V Xα∗= EY − β∗EX(2)whereCov(X, Y ) = E[(X − EX) (Y − EY )] = E(XY ) − EX · EYandV X = E[(X − EX)2] = E(X2) − (EX)2.Additional implications of b.l.p.: LetˆY ≡ α∗+ β∗X denote a “fitted value” ofY , and U ≡ Y −ˆY denote the “residual” or prediction error:• EU = 0• VˆY = (β∗)2V X = (Cov(X, Y ))2/V X = ρ2XYV Y• V U = V Y + (β∗)2V X − 2β∗Cov(X, Y ) = V Y − (Cov(X, Y ))2/V X = (1 −ρ2XY)V YHence, the b.l.p. accounts for a ρ2XYproportion of the variance in Y ; in this sense,the correlation measures the linear relationship between Y and X.3Lecture notes: Demand estimation introduction 4Also note thatCov(ˆY , U) = Cov(ˆY , Y −ˆY )= E[(ˆY − EˆY )(Y −ˆY − EY + EˆY )]= E[(ˆY − EˆY )(Y − EY ) − (ˆY − EˆY )(ˆY − EˆY )]= Cov(ˆY , Y ) − VˆY= E[(α∗+ β∗X − α∗− β∗EX)(Y − EY )] − VˆY= β∗E[(X − EX)(Y − EY )] − VˆY= β∗Cov(X, Y ) − VˆY= Cov2(X , Y )/V X − Cov2(X , Y )/V X= 0.(3)Hence, for any random variable X, the random variable Y can be written as the sumof a part which is a linear function o f X, and a part which is uncorrelated with X.Also,Cov(X, U) = 0 . (4)Note: in practice, with a finite sample of Y, X, the minimization problem (1) isinfeasible. In practice, we minimize the sample counterpartminα,βXi(Yi− α − βXi)2(5)which is the objective function in ordinary least squares regression. The OLS valuesfor α and β are the finite-sample versions of Eq. (2).(In “sample” version, expectations are replaced by sample averages. eg. mean Ex isreplaced by sample average from n observations¯Xn≡1nPiXi. Law of la r ge numberssay this approximation should not be bad, especially for large n.)4Lecture notes: Demand estimation introduction 5Next we can see some intuition of least-squares regression. Assume that the “true”model describing the generation of the Y process is:Y = α + βX + ǫ, Eǫ = 0. (6)What we mean by true model is that this is a causal model in the sense that a one-unit increase in X would raise Y by β units. (In the previous section, we just assumethat Y, X move jo intly together, so there is no sense in which changes in X “cause”changes in Y .)Question: under what assumptions does doing least-squares on Y, X (as in Eqs. (1)or (5) above) recover the true model; ie. α∗= α, and β∗= β?• For α∗:α∗= EY − β∗EX= α + βEX + Eǫ − β∗EXwhich is equal to α if β = β∗.• For β∗:β∗=Cov(α + βX + ǫ, X)V arX=1V arX· {E[X(α + βX + ǫ)] − EX · E[α + βX + ǫ]}=1V arX·αEX + βEX2+ E[ǫX] − αEX − β[EX]2− EXEǫ=1V arX·β[EX2− (EX)2] + E[ǫX]which is equal to β ifE[ǫX] = 0. (7)This is an “exogeneity” assumption, that (roughly) X and the disturbance term ǫ areuncorrelated. Under this assumption, the best linear predictors from the infeasible5Lecture notes: Demand estimation introduction 6problem (1)) coincide with the true values of α, β. Correspondingly, it turns out thatthe feasible finite-sample least-squares estimates from (5) are “g ood” (in some sense)estimators for α, β.Note that the orthogonality condition (7) differs from the zero covariance property(4), which is a feature of the b.l.p.When there is more than one X variable, then we use multivariate regression. Inmatrix notation, true model is:Yn×1= Xn×kβk×1+ ǫn×1.The least-squares estimator for β isβOLS= (X′X)−1X′Y.Next we consider est imating demand functions, where exogeneity is usually violated.3 Demand estimationLinear demand-supply model:Demand: qdt= γ1pt+ x′t1β1+ ut1Supply: pt= γ2qst+ x′t2β2+ ut2Equilibrium: qdt= qstDemand function summarizes consumer preferences; supply functio n summarizesfirms’ cost structureFirst, focus on estimating demand function:Demand: qt= γ1pt+ x′t1β1+ ut1If