# Finding the optimal activity levels

1. Appalachian Coal Mining believes that it can increase labor productivity and, therefore, net revenue by reducing air pollution in its mines. Its estimates that the marginal cost function for reducing pollution by installing additional capital equipment is: MC = 40P where P represent a reduction of one unit of pollution in the mines. It also feels that for every unit of pollution reduction 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. If:

Number of Number of

security guards stolen radios 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 the stolen radio is $25, what is the most the 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?

https://brainmass.com/economics/price-levels/finding-the-optimal-activity-levels-468221

#### Solution Preview

Please refer attached file for better clarity of tables.

Problem 1

Net revenue can be maximized by reducing the pollution level such that

MR=MC

1000-10P=40P

50P=1000

P=20

20 units of pollution should be reduced.

Problem 2

Number, N Number of radios stolen per week, Q Cost of Radio, P Total Loss, TS=P*Q Marginal Savings*

0 50 25 1250

1 30 25 750 500

2 20 25 500 250

3 14 ...

#### Solution Summary

There are two problems. Solutions to these problems explain the methodology to find the optimal activity levels.

Find total benefit.

Suppose a firm is considering two different activities, activities X and Y, which yield

the following schedule of total benefits shown below. The price of X is $2 per unit and the price of Y is $10 per unit.

Total benefit of Total benefit of

Level of Activity Activity X Activity Y

_________________________________________________________

0 0 0

1 $30 $100

2 54 190

3 72 270

4 84 340

5 92 400

6 98 450

A. The firm places a budget constraint of $26 on expenditures on activities X and

Y. What is the level of activity that maximizes the total benefit subject tot he budget constraint? What is the total benefit if your solution is implemented?

B. Suppose that the budget increases to $58. What is the optimal level of activity

X and Y now? What is the total benefit when the budget is $58?