Chapter 3 homework

Numbers 6, 7, 12

Review session: Monday 6:30-7:30

Thomas 324

Managerial Economics & Business Strategy

Chapter 3Quantitative Demand Analysis

P

Q Q

D D

Linear Log Linear

Graphical Representation of Linear and Log-Linear Demand

P

Let’s try some homework (number 2)

• The demand curve for a product is given by

• Where Pz = $400• What is the own price elasticity of demand when Px=$154? Is

demand elastic or inelastic? What happens to TR if it decided to charge a price below $154?

• What is the own price elasticity of demand when Px=$354? Is demand elastic or inelastic? What happens to TR if it decided to charge a price below $354?

• What is the cross price elasticity of demand when Px=$154? Are the goods compliments or substitutues?

zxdx PPQ 02.02000,1

X

Y

.

.

..

.

.

What does the demand curve look like?

Regression Line

Regression Analysis

• One use is for estimating demand functions.• Important terminology and concepts:

Least Squares Regression: Y = a + bX + e.

• e are the “dotted” lines from the dot to the regression line

• Called the error term (mean of zero)

• Best Fit Regression Line minimizes the sum of squared errors

• Best Fit XbaY ˆˆ

More terminology

Standard error• Measures how much each coefficient would varies from the mean of

the population Confidence Intervals

• Give you the upper and lower bounds for the “true” coefficient.

• Calculated coefficient lies in the middle of this interval

• 95% confidence intervals says that we are 95% sure that the “true” value will lie between these values

Rule of Thumb Confidence Interval Calculation

ba banda ˆˆ 2ˆ2ˆ

The is the standard error to the coefficient

Still more terminology

t-statistic• Ratio of parameter estimate to the standard error

• Large??? true value of the variable is NOT zero

• Rule of thumb if greater than or equal to the absolute value of 2 then the parameter estimate is statistically different from zero

How good does the line fit???• R-square or Coefficient of Determination

– Fraction of the total variation in the dependent variable explained by the independent variables

– Ratio of the Sum of Squared Errors to the total sum of squared errors

– Range is from 0 to 1

And more terminology

• Adjusted R-square Adjusts for the number of observations as well as number of

coefficients

• F-statistic Provides a measure of the total variation explained by the

regression relative to the total unexplained variation Look at the number and the significance of the value

• F-statistic = 23.94• Significance F = 0.0012• Says there is less than 0.12% chance that the data was fit by accident• If F-statistic significant at 5% or less then we can say that the

regression is statistically significant

An Example

• Using a spreadsheet we can estimate the following log-linear demand function.

0ln lnx x xQ P e

Summary Output

Regression StatisticsMultiple R 0.41R Square 0.17Adjusted R Square 0.15Standard Error 0.68Observations 41.00

ANOVAdf SS M S F Significance F

Regression 1.00 3.65 3.65 7.85 0.01Residual 39.00 18.13 0.46Total 40.00 21.78

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 7.58 1.43 5.29 0.000005 4.68 10.48ln(P) -0.84 0.30 -2.80 0.007868 -1.44 -0.23

Let’s try some homework (number 15)

• As the newly appointed “Energy Czar”, your goal is to reduce the total demand for residential heating fuel in your state. You must choose one of three legislative proposals designed to accomplish this goal: (a) a tax that would effectively increase the price of residential heating fuel by $2; (b) a subsidy that would effectively reduce the price of natural gas by $1; (c) a tax that would effectively increase the price of electricity (produced by hydroelectric facilities) by $5. To assist you in your decision, and economist in your office has estimated the demand for residential heating fuel using a linear demand specification. Based on this information, which proposal would you favor? Explain.

Summary OutputAdjusted R Square 0.49Standard Error 47.13Observations 25.00

ANOVAdf SS MS F Significance F

Regression 4.00 60936.56 15234.14 6.86 0.30Residual 20.00 44431.27 2221.56Total 24.00 105367.84

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 136.96 43.46 3.15 0.010000 50.60 223.32P heating fuel -91.69 29.09 -3.15 0.010000 -149.49 -33.89P Natural Gas 43.88 9.17 4.79 0.00 25.66 62.10P Electricity -11.92 8.35 -1.43 0.17 -28.51 4.67Income -0.50 0.35 -0.14 0.90 -0.75 0.65