Managerial Economics Estimation of Demand How to estimate a demand equation? How to calculate demand elasticities?

Is this how it’s done? Last This % Year Year Change Change Price $4 $5 $1 +25% Q 8 10 2 +25% Ep = +25% / +25% = + 1 Questions: 1) Is Ep = + 1? 2) What’s the right way to do it?

Estimation of Demand I. The Direct Methods II. The indirect Methods Interviews and Surveys Market Experimentations and Simulations II. The indirect Methods Regression Estimation of Consumer Demand

Interviews and Surveys Ask buyers or potential buyers about their likely reactions to a change in each of the demand determinants. Practical Issues: Randomness of the sample Interviewer bias The best-of-intentions problem Confusing questions and confusing answers

Simulation and Market Experimentations Simulation of consumer responses in constructed market situations Test-market experiments in selected markets

Regression Analysis of Consumer Demand A statistical technique that attempts to "explain" or "predict" movements in one economic variable, the dependent variable, as a function of the movements of a set of independent (explanatory) variables.

Desirable Characteristics It shows explicitly the association between the dependent variable and the independent variables. It also provides statistical reliability allowing the researchers to measure the reliability of the prediction. In Econometrics, the economists call it: ‘BLUE’ property.

Procedure of Regression Analysis Specifying the variables Obtaining data on the variables Specifying the form of the estimation equation Estimate the regression parameters using the method of least squares.

Demand and Elasticity Estimation Step 1 - The model: Q = f(A, Px, and Pq), where Q = Number of 2-year contracts sold A = Advertising expenditures (in dollars) Px = Price of 1-year contract (in dollars) Pq = Price of 2-year contract (in dollars)

The Data Time-series data for 1986-97 and the effect of inflation Time-series data and the effect of serial correlation N=12, K=4, 8 degrees of freedom

Estimation of 2 Equations The 1st Equation - Linear Q = a + b1 A + b2 Px + b3 Pq The 2nd Equation - multiplicative Q = aAb1Pxb2 Pqb3

The Estimated 1st Equation Q = 4,589.08 + 0.01015 A + 16.40334 Px - 10.82852 Pq s.e. (0.00410) (5.35702) (3.77846) t stat 2.47576 3.06202 -2.86586 p-value 0.00384 0.01553 0.02096 Adj R2 = 0.69127, F=9.20976 (p=0.00566) SEE=52.98, N=12, K=4

Evaluation of Regression Estimation 1. The coefficient of determination (R2) - How well does the regression line fit the data? 2. The F-Test - Does the estimated equation have sufficient explanatory power? 3. The t-Test - Is each of the independent variables statistically significant? 4. The Standard Error of the Estimate (SEE) - Can the confidence interval of the predicted value for the dependent variable be estimated?

The Standard Error of the Estimate (SEE) How accurate is the predicted sales? The SEE can be used to construct prediction intervals If SEE = 53, then and approximate 95% prediction interval for the sales of 2-year contracts is equal to Q’ + 2(53)

Uses of the 1st Equation The demand curve for explaining demand relationships and for predicting demand Estimation of the arc elasticities of demand using Q1-Q2 Pq1 - Pq2 Ep = ----------- --------------- Q1+Q2 Pq1+ Pq2 Estimation of the point elasticities of demand using ep = (dQ/dPq)(Pq/Q)

The Multiplicative Equation Q = aAb1Pxb2Pqb3 Two key questions: What are the advantages of the multiplicative form over the linear form How to go about estimating this non-linear equation? See pp. 154-56 of McGuigan/Moyer/Harris, 10th ed. for details

The Advantages of the Multiplicative Form Q = a Ab1Pxb2 Pqb3 dQ/dPq = (b3) (aAb1Pxb2 Pqb3 -1) Since epq = (dQ/dPq)(Pq/Q), epq = (b3)(aAb1Pxb2 Pqb3 -1)(Pq/Q) = (b3)[(aAb1Pxb2 Pqb3)/Q] = b3

How to estimate the parameters of the multiplicative equation? Converting the multiplicative equation Q = a Ab1Pxb2 Pqb3 into the natural-log form, we have: Ln Q = Ln a + b1 Ln A + b2 Ln Px + b3 Ln Pq