21st 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 guards. The follwoign table shows how the number of security guards affects the number of radios stolen each week:

# of guards # of radios stolen per wk
0 50
1 30
2 20
3 14
4 8
5 6

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

First, determine the marginal change in radios saved with each guard. Going from 0 to 1, the number of stolen radios goes from 50 to 30; ergo the marginal change is 20. Going from 1 to 2, the marginal change in radios is 10. Continue this with each additional guard.

a) the marginal number of radios saved would need to be 8 or greater.
b) the first security guard saves 20 radios, at $25 each, the firm would pay up to $500 for a guard.
c) take it from here.

The marginal number of radios saved would need to be 4 or greater.

To determine the optimal number of security guards that should be hired, we need to analyze the marginal change in the number of radios stolen with each additional guard. Let's calculate the marginal change for each increase in the number of guards:

From 0 to 1 guard: The number of stolen radios decreases from 50 to 30, resulting in a marginal change of 20 (50-30).
From 1 to 2 guards: The number of stolen radios decreases from 30 to 20, resulting in a marginal change of 10 (30-20).
From 2 to 3 guards: The number of stolen radios decreases from 20 to 14, resulting in a marginal change of 6 (20-14).
From 3 to 4 guards: The number of stolen radios decreases from 14 to 8, resulting in a marginal change of 6 (14-8).
From 4 to 5 guards: The number of stolen radios decreases from 8 to 6, resulting in a marginal change of 2 (8-6).

Now let's answer the given questions:

a) If each security guard is paid $200 per week and the cost of a stolen radio is $25, the firm should hire guards until the marginal number of radios saved is 8 or greater. Looking at the marginal changes we calculated, we can see that the marginal change drops below 8 after 2 guards (10->6), so the firm should hire 2 guards.

b) If the cost of a stolen radio is $25, the most the firm would be willing to pay to hire the first security guard is equal to the marginal change in radios saved. In this case, the marginal change from 0 to 1 guard is 20, so the firm would be willing to pay up to $500 (20 * $25) for the first guard.

c) If each security guard is paid $200 per week and the cost of a stolen radio is $50, we need to recalculate the marginal changes. Using the same method as before:

From 0 to 1 guard: The marginal change is still 20.
From 1 to 2 guards: The marginal change is still 10.
From 2 to 3 guards: The marginal change is now -6 (14-20).

Since the marginal change drops below 8 after 2 guards (10->-6), it would not be cost-effective to hire any more guards. Therefore, the firm should hire 2 guards.

I hope this helps! Let me know if you have any further questions.

To continue with part c), let's calculate the marginal change in radios saved with each additional guard.

Going from 2 to 3 guards, the number of stolen radios decreases from 20 to 14, which is a marginal change of 6.

Going from 3 to 4 guards, the marginal change is 6 as well (from 14 to 8).

Finally, going from 4 to 5 guards, the marginal change is 2 (from 8 to 6).

Now, let's compare the marginal benefit with the cost to determine how many guards should the firm hire.

The cost per security guard is $200 per week, regardless of the number of guards hired.

The cost of a stolen radio is $50.

For each additional guard hired, the firm saves a certain number of stolen radios. The marginal benefit (in terms of the cost of saving radios) can be calculated by dividing the cost per guard ($200) by the marginal change in radios saved.

For the first additional guard (2 guards to 3 guards), the marginal benefit is $200 / 6 = $33.33 per radio saved.

For the second additional guard (3 to 4 guards), the marginal benefit is $200 / 6 = $33.33 per radio saved.

For the third additional guard (4 to 5 guards), the marginal benefit is $200 / 2 = $100 per radio saved.

Now, let's consider the cost of a stolen radio being $50. The most the firm would be willing to pay to hire the first security guard (to save 20 radios) is the cost of those saved radios.

The maximum amount the firm would pay is $50 * 20 = $1000.

To summarize:

a) The firm should hire enough guards until the marginal change in radios saved is 8 or greater.
b) The most the firm would be willing to pay to hire the first security guard (to save 20 radios) is $1000.
c) With each security guard paid $200 a week and with the cost of a stolen radio at $50, the firm should hire enough guards until the marginal change in radios saved is 4 or greater.