# Calculating quantity demanded, price elasticity, income elasticity of demand, and statistical significance

1. A firm has estimated the following demand function for its product:

Q = 100 - 5 P + 5 I + 15 A

Where Q is quantity demanded per month in thousands, P is product price, I is an index of consumer income, and A is advertising expenditures per month in thousands. Assume that P=$200, I=150, and A=30. Use the point formulas to complete the elasticity calculations indicated below.

(i) Calculate quantity demanded.

(ii) Calculate the price elasticity for demand. Is demand elastic, inelastic, or unit elastic? (Hint: use equation (3-8), page 100, the price elasticity is

(-5)(200/300)= -3.33. The demand is elastic.

(iii) Calculate the income elasticity of demand. Is the good normal or inferior? Is it a necessity or a luxury?

2. A firm has kept track of the quantity demanded of its output, Good X, during three time periods. The price of X and the price of the other good, Good Y, were also recorded for each time period. The information is provided in the table that follows. Use it to calculate the own-price arc elasticity of demand and the cross-price elasticity of demand. Determine whether Good Y is a complement or a substitute for Good X.

Time Period 1 2 3

Quantity of X 220 80 250

Price of X 15 25 15

Price of Y 10 10 5

3. Just The Fax, Inc. (JTF) has hired you as a consultant to analyze the demand for its line of telecommunications devices in 35 different market areas. The available data set includes observations on the number of thousands of units sold by JTF per month (QX), the price per unit charged by JTF (PX), the average unit price of competing brands (PZ), monthly advertising expenditures by JTF (A), and average gross sales (in $1,000) of businesses in the market area (I). The results of a regression analysis (with standard errors in parenthesis) are given below.

Q X = 300 - 6 PX + 2 PZ + 0.04 A + 0.01 I

(200) (1.8) (0.8) (0.03) (0.004)

R2 = 0.91 S.E.E. = 3.6

Evaluate the statistical significance of each of its coefficients. (Hint: the t ratio for Px is -3.33 (=-6/1.8), and significant)

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Problem

1. A firm has estimated the following demand function for its product:

Q = 100 - 5 P + 5 I + 15 A

Where Q is quantity demanded per month in thousands, P is product price, I is an index of consumer income, and A is advertising expenditures per month in thousands. Assume that P=$200, I=150, and A=30. Use the point formulas to complete the elasticity calculations indicated below.

(i) Calculate quantity demanded.

Q=100-5*200+5*150+15*30=300

Q=300,000 per month

(ii) Calculate the price elasticity for demand. Is demand elastic, inelastic, or unit elastic? (Hint: use equation (3-8), page 100, the price elasticity is

(-5)(200/300)= -3.33. The demand is elastic.

dQ/dP=-5

Price elasticity = dQ/dP*P/Q = -5*200/300=-3.33

Since absolute value of elasticity is >1.0, ...

#### Solution Summary

This solution provides step by step equations for calculating quantity demanded, price elasticity, income elasticity of demand, and statistical significance.

Integrating Problem Elasticity

Starting with the date from Problem 6 and the data on the price of a related commodity for the years 1986 to 2005 given below, we estimated the regression for the quantity demanded of a commodity (which we now relabel Q Ì?_x), on the price commodity which we now label P_x, consumer income ( which we now label Y), and the price related commodity (P_(z) ), and we obtain the following results.

Year 1986 1987 1988 1989 1990

Pz ($) 14 15 15 16 17

Year 1991 1992 1993 1994 1995

Pz ($) 18 17 18 19 20

Year 1996 1997 1998 1999 2000

Pz($) 20 19 21 21 22

Year 2001 2002 2003 2004 2005

Pz($) 23 23 24 25 25

Q Ì?x=121.86-9P_x+0.04Y-2.21P_z

(-5.12) (2.18) (-0.68)

R^(2 )= .9633 F=167.33 D-W=2.38

(a) Explain why you think we have chosen to include the price commodity Z in the above regression. (b) Evaluate the above regression results. (c) What type of commodity is Z? Can you be sure?

**15(b) is to evaluate the above regression results in terms of the signs of

the coefficients, the statistical significance of the coefficients and the

explanatory power of the regression (R2) The number in parentheses

below the estimated slope coefficients refer to the estimated t values.

The rule of thumb for testing the significance of the coefficients is if the

absolute t value is greater than 2, the coefficient is significant, which

means the coefficient is significantly different from zero. For example,

the absolute t value for Px is 5.12 which is greater than 2, therefore, the

coefficient of Px, (-9.50) is significant. In order words, Px does affect

Qx. If the price of the commodity X increases by $1, the quantity

demanded (Qx) will decrease by 9.50 units.

15(c) X and Z are complementary or substitutes?

INFORMATION FROM PROBLEM 6

. From the following table giving the quantity demanded of a commodity (Y), its price (X1), and the consumers income (X2) from 1986 to 2005, a) estimate the regression equation of Y on X1 and X2, b) test at the 5 % level for the statistical significance of the slope parameters, c) find the unadjusted and the adjusted coefficients of determination and d) test at the 5 % level for the overall statistical significance of the regression. Show all your results to three decimal places.

Year Y X1 X2

1986 72 $10 $2,000

1987 81 9 2100

1988 90 10 2,210

1989 99 9 2,305

1990 108 8 2,407

1991 126 7 2,500

1992 117 7 2,610

1993 117 9 2,698

1994 135 6 2,801

1995 135 6 2,921

1996 144 6 3,000

1997 180 4 3,099

1998 162 5 3201

1999 171 4 3,308

2000 153 5 3,397

2001 180 4 3,501

2002 171 5 3,689

2003 180 4 3,800

2004 198 4 3,896

2005 189 4 3,989

Solution:

The regression equation is given as follows:

y=-9.470x1+0.029x2+114.074

where x_1 represents the independent variable 'price'

x_2 represents the independent variable 'consumer^' s income^'

and,y represents the dependent variable 'quantity demanded^'

Let us test the significance for the slope parameters of the independent variables.

The null hypothesis,H_0 states that there is no significant correlation, or the correlation coefficient ?=0.

Decision Rule:

Reject H_0 if the p-value < 0.05 (significance level,alpha)

Independent Variables p- value Null Hypothesis Decision Conclusion

x_1 0.00007 ?_1=0 Reject H_0 x_1 has a significant contribution in the regression model

x_2 0.0003 ?_1=0 Reject H_0 x_2 has a significant contribution in the regression model

The unadjusted and the adjusted coefficients of determination are given as 0.984 and 0.964 respectively.

The null hypothesis,H_0 states that there is no significant correlation, or the correlation coefficient ?=0.

Significance Level, ? = 0.05

Decision Rule:

Reject H_0 if the Significance F( p-value) < 0.05 (significance level,alpha)

From the ANOVA table, we find that the Significance F=1.856Ã-ã?-10ã?-^(-13) , which is much less than 0.05. Therefore, we reject the null hypothesis that there is no significant correlation and conclude that, according to the overall test of significance, the regression model is valid.

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