MANAGERIAL ECONOMICS 12th Edition By Mark Hirschey

Demand Estimation Chapter 5

Chapter 5 OVERVIEW Interview and Experimental Methods Simple Demand Curve Estimation Simple Market Demand Curve Estimation Identification Problem Regression Analysis Measuring Regression Model Significance Measures of Individual Variable Significance

Interview and Experimental Methods Consumer Interviews Interviews can solicit useful information when market data is scarce. Consumer opinions can differ from behavior. Market Experiments Controlled experiments can generate useful insight. Experiments can be expensive.

Simple Demand Curve Estimation Simple Linear Demand Curves The best estimation method balances marginal costs and marginal benefits. Simple linear relations are often useful for demand estimation. Using Simple Linear Demand Curves Straight-line relations can give useful approximations.

Simple Demand Curve Estimation 𝑃=𝑎+𝑏𝑄 Consider the linear demand: when P=$12, Q=3,200 and P=$10, Q=4,000 12 =𝑎+𝑏 3200 12 =𝑎+𝑏 3200 minus 10 =𝑎+𝑏 4000 12 =𝑎−0.0025 3200 12 =𝑎−8 2 =−800𝑏 𝑎=20 b =−0.0025 𝑃=𝑎+𝑏𝑄 is 𝑃=20−0.0025𝑄

Demand Curve and TR Maximization 𝑇𝑅=𝑃∙𝑄 = $20−$0.0025𝑄 𝑄 =$20𝑄−$0.0025 𝑄 2 𝑀𝑅= 𝜕𝑇𝑅 𝜕𝑄 =$20−$0.005𝑄 $20−$0.005𝑄=0 $0.005𝑄=$20 𝑄=4000 and 𝑃=20−0.0025 4000 =$10 max 𝑇𝑅 =𝑃∙𝑄=4000∙$10=$40,000

Market Demand Curve Estimation Shows total quantity customers are willing to buy at various prices under current market conditions. Graphing the Market Demand Curve Market demand is the sum of individual demand quantities, Q1 + Q2 = Q1+2. Add quantities, not prices!

Market Demand Curve Estimation 𝑃 𝐷 =$100−$0.001 𝑄 𝐷 Domestic Demand 𝑃 𝐹 =$80−$0.004 𝑄 𝐹 Foreign Demand 𝑇𝐶=$1,200,000+$24𝑄 Total Cost 𝑃 𝐷 =$100−$0.001 𝑄 𝐷 𝑃 𝐹 =$80−$0.004 𝑄 𝐹 0.001 𝑄 𝐷 =100− 𝑃 𝐷 0.004 𝑄 𝐹 =80− 𝑃 𝐹 𝑄 𝐷 =100,000− 1000𝑃 𝐷 𝑄 𝐹 =20,000− 250𝑃 𝐹

Market Demand Curve Estimation 𝑄= 𝑄 𝐷 + 𝑄 𝐹 𝑄=100,000− 1000𝑃 𝐷 + 20,000− 250𝑃 𝐹 𝑄=120,000−1,250𝑃 1,250𝑃=120,000−𝑄 𝑃=$96−$0.0008𝑄 Market Demand

Market Demand Curve Estimation 𝑇𝑅=𝑃∙𝑄 = $96−$0.0008𝑄 𝑄 =$96𝑄−$0.0008 𝑄 2 𝑀𝑅= 𝜕𝑇𝑅 𝜕𝑄 =$96−$0.0016𝑄 𝑇𝐶=$1,200,000+$24𝑄 𝑀𝐶= 𝜕𝑇𝐶 𝜕𝑄 =$24

Market Demand Curve Estimation 𝑀𝑅=𝑀𝐶 at profit maximization $96−$0.0016𝑄=$24 $0.0016𝑄=$72 𝑄=45,000 𝑃=$96−$0.0008 45,000 =$60 𝜋=𝑇𝑅−𝑇𝐶 𝜋=$96𝑄−$0.0008 𝑄 2 −($1,200,000+$24𝑄) 𝜋=−$0.0008 𝑄 2 +$72𝑄−$1,200,000 𝜋=−$0.0008 45,000 2 +$72 45,000 −$1,200,000 𝜋=$420,000

Market Demand Curve Estimation Domestic Demand Market Demand Foreign Demand

Identification Problem Changing Nature of Demand Relations Demand relations are dynamic. Interplay of Demand and Supply Economic conditions affect demand and supply. Shifts in Demand and Supply Curve shifts can be estimated. Simultaneous Relations Quantity and price are jointly determined.

Identification Problem The demand curve “D” does not exist the data are for three shifting demand curves All else equal w.r.t. demand determinants S1 D1 The problem: “D” has higher elasticity than “D1”. P S1 P Q D D S2 S2 D2 P1 Q1 P1 Q1 S3 D3 S3 P2 Q2 P2 Q2 P3 Q3 P3 Q3 Q

Regression Analysis What Is a Statistical Relation? A statistical relation exists metrics are related A deterministic relation is true by definition. Specifying the Regression Model Dependent variable Y is caused by X. X variables are independently determined from Y. Least Squares Method Minimize sum of squared residuals.

Regression Analysis . . . Direct Relation Inverse Relation No Relation Unit Sales X . X . Unit Sales X . Unit Sales Advertising Price Orbit of Jupiter

Regression Analysis . . . . . . . . . . . . 𝑚𝑖𝑛 𝑌− 𝑌 2 𝑚𝑖𝑛 𝑌− 𝑌 2 Regression Analysis fits a “Sum of Least Squares” line to the data Y . Estimate or 𝑌 . 𝑏= 𝑛 𝑋𝑌 − 𝑋 𝑌 𝑛 𝑋 2 − 𝑋 2 . . . . . Error or 𝑌− 𝑌 . . . . Observation or 𝑌 𝑎= 𝑌 −𝑏 𝑋 𝑛 . 𝑌 =𝑎+𝑏𝑋 𝑌 =0.136047+0.005962𝑋 X

Regression Analysis 𝑋 𝑌 24 78 43 100 86 34 82 36 38 84 22 75 23 80 30 83 33 91 𝑋𝑌 𝑋 2 1872 576 4300 1849 2064 2788 1156 3096 1296 3192 1444 1650 484 1840 529 2490 900 3003 1089 𝑌 𝑈 79.50 -1.50 93.67 6.33 6.50 86.96 -4.96 88.45 -2.45 89.94 -5.94 78.01 -3.01 78.76 1.24 83.98 -0.98 86.21 4.79 𝑏= 𝑛 𝑋𝑌 − 𝑋 𝑌 𝑛 𝑋 2 − 𝑋 2 = 10 26295 − 307 845 10 9899 − 307 2 =0.745623 𝑎= 𝑌 −𝑏 𝑋 𝑛 = 856−0.745623∙307 10 =61.60937 𝑌 =61.60937+0.745623𝑋

Regression Analysis Regression equations can take on any functional form 𝑄=𝑎+𝑏𝑃 𝑄=𝑎+ 𝑏 𝑃 𝑃+ 𝑏 𝐴 𝐴+ 𝑏 𝐼 𝐼 𝑄= 𝑏 0 𝑃 𝑏 𝑃 𝐴 𝑏 𝐴 𝐼 𝑏 𝐼 The above multiplicative form is popular among economist because the exponents “bP” is the constant elasticity of the variable. 𝑄=4 𝑃 −0.4 𝐴 0.2 𝐼 0.003 has the constant price elasticity of – 0.4

Measuring Regression Significance Standard Error of the Estimate (SEE) reflects degree of scatter about the regression line. Y X 𝑌 =𝑎+𝑏𝑋 Upper 95% confidence bound: + 1.96 standard error of the estimate 𝑏 = Slope of curve 𝑌 𝑋 𝑎 Lower 95% confidence bound: - 1.96 standard error of the estimate

Goodness of Fit Correlation shows degree of concurrence. r = 1 means perfect correlation. r = 0 means no correlation. Coefficient of determination, R2. R2 = 100% means perfect fit. R2 = 0% means no relation. Corrected coefficient of determination Adjusts R2 downward for small samples.

Regression Analysis 𝑅 2 = 𝑌 − 𝑌 2 𝑌− 𝑌 2 𝑅 2 = 𝑌 − 𝑌 2 𝑌− 𝑌 2 Regression Analysis fits a “Sum of Least Squares” line to the data Y X 𝑌 𝑌 Unexplained variation 𝑌− 𝑌 Total variation 𝑌− 𝑌 Explained variation 𝑌 − 𝑌 𝑌 𝑎𝑑𝑗− 𝑅 2 =1−(1− 𝑅 2 ) 𝑛−1 𝑛−𝑝−1

F statistic Tells if R2 is statistically significant Goodness of Fit Test Do not Reject H0 Reject H0 F  = 0.05 F = 2.69 Critical Value  = 0.1 F = 2.14

Judging Variable Significance t statistics compare sample characteristics to the standard deviation of that characteristic. t > 1.645 implies a strong effect of X on Y (90% conf.). t > 1.96 implies an even stronger effect of X on Y (95% conf.) Two-tail t Tests Tests of effect. One-Tail t Tests Tests of magnitude or direction.

Judging Variable Significance t – statistic H0: b=0 Ha: b≠0 Reject H0 Reject H0 Do not Reject H0  / 2 = 0.025  / 2 = 0.025 z – 1.96 1.96  / 2 = 0.025 Critical Value = ± 1.96 Confidence Level = 95%  / 2 = 0.05 Critical Value = ± 1.645 Confidence Level = 90%

Multiple Regression Example SUMMARY OUTPUT   Regression Statistics Multiple R 0.939811 R Square 0.883244 Adjusted R Square 0.836542 Standard Error 2.929296 Observations 15 ANOVA df SS MS F Significance F Regression 4 649.1256 162.2814 18.91221 0.000118 Residual 10 85.80775 8.580775 Total 14 734.9333 Coefficients t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 23.30213 17.55563 1.327331 0.213904 -15.8142 62.41851 P -5.96115 2.928719 -2.03541 0.069176 -12.4867 0.564444 Ps 6.50636 3.888925 1.673049 0.125263 -2.1587 15.17142 Pc -1.09766 3.416276 -0.3213 0.754595 -8.7096 6.514275 I -1.3E-05 0.000116 -0.10919 0.915211 -0.00027 0.000245 Q P Ps Pc I 20 2 1.5 3 38000 30 2.5 2.3 22000 10 1.7 3.2 40000 15 1.9 2.8 25000 12 1.4 3.5 28000 28 2.7 2.2 17000 17 2.4 1.8 3.1 35000 14 3.8 2.1 2.9 20000 22 34000 32 25 36000 1.6 2.6 30000