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# Managerial Economics: Appalachian Coal Mining, Vanguard Corp

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Managerial Economics

(1) Appalachian coal mining believes that it can increase labor productivity and, therefore, net revenues by reducing air pollution in its mines. It estimates that the marginal cost function for reducing pollution by installing additional capital equipment is
MC= 40P

Where P represents a reduction of one unit of pollution in the mines. It also feels that for every unit of pollution the marginal increase in revenue ( MR) is

MR= 1,000- 10p

How much pollution reduction should Appalachian Coal Mining undertake?

(2) Twenty first century electronics has discovered a theft problem at its warehouse and has decided to hire security guards. The firm wants to hire the optimal number of security guards. The following table shows how the number of security guards affects the number of radios stolen per week.

Number of security guards-------------- Number of radios stolen per week

0--------------------------------------50
1---------------------------------------30
2----------------------------------------20
3----------------------------------------14
4-----------------------------------------8
5-----------------------------------------6

(a) If each security guard is paid \$200 a week and the cost of a stolen radio is \$25, how many security guards should the firm hire?

(b) If the cost of a stolen radio is \$25, what is the most firm would be willing to pay to hire the first security guard?

(c) If each security guard is paid \$200 a week and the cost of a stolen radio is \$50, how many security guards should the firm hire?

(3) The director of Marketing at Vanguard Corporation believes that sales of the company's bright side laundry detergent(s) are related to Vanguard's own advertising expenditure (A), as well as the combined advertising expenditures of its three biggest rival detergents ®. The marketing director collects 36 weekly observations on S, A, and R to estimate the following multiple regression equation:
S= a + bA + cR

Where S, A and R are measured in dollars per week. Vanguard's marketing director is comfortable using parameter estimates that are statistically significant at the 10 percent level or better.

(a) What sign does the marketing director expect a, b, and c to have?
(b) Interpret the coefficients a, b, and c.

The regression output from the computer is as follows:

Dependent Variable: S-------- R- Square F- Ratio p-Value on F
OBSERVATIONS: 36 0.2247 4.781 0.0150

PARAMETER STANDARD
VARIABLE ESTIMATE ERROR T-RATIO P-VALUE
INTERCEPT 175086.0 63821.0 2.74 0.0098
A 0.8550 0.3250 2.63 0.0128
R - 0.284 0.164 -1.73 0.0927

(c) Does Vanguard's advertising expenditure have a statistically significant effect on the sales of bright side detergent? Explain, using the appropriate p-value.
(d) Does advertising by its three largest rivals affect sales of bright side detergent in a statistically significant way? Explain, using the appropriate p-value
(e) What fraction of the total variation in sale of bright side remains unexplained? What can the marketing director do to increase the explanatory power of the sales equation? What other explanatory variables might be added to this equation?
(f) What is the expected level of sales each week when Vanguard spends \$40,000 per week and the combined advertising expenditure for the three rivals are \$100,000 per week?

#### Solution Preview

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Managerial Economics

(1) Appalachian coal mining believes that it can increase labor productivity and, therefore, net revenues by reducing air pollution in its mines. It estimates that the marginal cost function for reducing pollution by installing additional capital equipment is
MC= 40P

Where P represents a reduction of one unit of pollution in the mines. It also feels that for every unit of pollution the marginal increase in revenue ( MR) is

MR= 1,000- 10p

How much pollution reduction should Appalachian Coal Mining undertake?

The installation of additional capital equipment will reduce the pollution and thereby increases the labor productivity. But we should take into account the additional cost of such activity. We should see to that the cost is not offsetting the benefit.
Thus we should fix the level of pollution reduction in an optimal manner. This optimal value of P (Pollution reduction) is given by the following condition.
Marginal Cost = Marginal Revenue
In the given problem we have Marginal cost = MC = 40P
and Marginal Revenue = MR = 1,000 - 10P
Thus to get the optimal value we should have: MC = MR
40P = 1,000 - 10P
40P + 10P = 1,000
50P = 1,000
P = 1,000 / 50
P = 20
Thus the required unit of reduction in the pollution is 20 units.

(2) Twenty first century ...

#### Solution Summary

This solution is comprised of detailed explanation for these managerial economics questions:

(1) Appalachian coal mining believes that it can increase labor productivity and, therefore, net revenues by reducing air pollution in its mines. It estimates that the marginal cost function for reducing pollution by installing additional capital equipment is
MC= 40P

Where P represents a reduction of one unit of pollution in the mines. It also feels that for every unit of pollution the marginal increase in revenue ( MR) is

MR= 1,000- 10p

How much pollution reduction should Appalachian Coal Mining undertake?

(2) Twenty first century electronics has discovered a theft problem at its warehouse and has decided to hire security guards. The firm wants to hire the optimal number of security guards. The following table shows how the number of security guards affects the number of radios stolen per week.

Number of security guards-------------- Number of radios stolen per week

0--------------------------------------50
1---------------------------------------30
2----------------------------------------20
3----------------------------------------14
4-----------------------------------------8
5-----------------------------------------6

(a) If each security guard is paid \$200 a week and the cost of a stolen radio is \$25, how many security guards should the firm hire?

(b) If the cost of a stolen radio is \$25, what is the most firm would be willing to pay to hire the first security guard?

(c) If each security guard is paid \$200 a week and the cost of a stolen radio is \$50, how many security guards should the firm hire?

(3) The director of Marketing at Vanguard Corporation believes that sales of the company's bright side laundry detergent(s) are related to Vanguard's own advertising expenditure (A), as well as the combined advertising expenditures of its three biggest rival detergents ®. The marketing director collects 36 weekly observations on S, A, and R to estimate the following multiple regression equation:
S= a + bA + cR

Where S, A and R are measured in dollars per week. Vanguard's marketing director is comfortable using parameter estimates that are statistically significant at the 10 percent level or better.

(a) What sign does the marketing director expect a, b, and c to have?
(b) Interpret the coefficients a, b, and c.

The regression output from the computer is as follows:

Dependent Variable: S-------- R- Square F- Ratio p-Value on F
OBSERVATIONS: 36 0.2247 4.781 0.0150

PARAMETER STANDARD
VARIABLE ESTIMATE ERROR T-RATIO P-VALUE
INTERCEPT 175086.0 63821.0 2.74 0.0098
A 0.8550 0.3250 2.63 0.0128
R - 0.284 0.164 -1.73 0.0927

(c) Does Vanguard's advertising expenditure have a statistically significant effect on the sales of bright side detergent? Explain, using the appropriate p-value.
(d) Does advertising by its three largest rivals affect sales of bright side detergent in a statistically significant way? Explain, using the appropriate p-value
(e) What fraction of the total variation in sale of bright side remains unexplained? What can the marketing director do to increase the explanatory power of the sales equation? What other explanatory variables might be added to this equation?
(f) What is the expected level of sales each week when Vanguard spends \$40,000 per week and the combined advertising expenditure for the three rivals are \$100,000 per week?

\$2.19