Microeconomics 1

The staff of a large company is maintained as a stationary population by 500 new entrants

each year at exact age 20. One third of those reaching age 30 leave immediately. Of the

remainder, 1

4

of those attaining age 60 retire immediately and the survivors retire at age 65.

The only other decrement is death.

Calculate

(i) the number of staff,

(ii) the number of deaths in service each year.

Basis: English Life Table No. 12 - Males

A certain country's school system provides education for all children between the ages of 5

and 16 exactly. The country's population is stationary, there being 100,000 births per year,

uniformly distributed over the year. The population of the country is subject to the mortality

of English Life Table No. 12 - Males.

(i) Find the number of pupils at any given time.

(ii) The country's teacher training colleges are such that a constant flow of new entrants to

the profession is maintained. Teachers are recruited uniformly over the year, and the ratio of

pupils to teachers is 20 to 1. All teachers enter the profession at age 21 and retire at age 60,

there being no withdrawals. Find the annual number of new teachers recruited.

For many years a company has recruited, uniformly over each year, 200 employees on their

20th birthdays and a fixed number of additional employees on their 25th birthdays. Mortality has followed English Life Table No. 12 - Males. Employees may retire on their 60th

or 65th birthdays, and one third of employees reaching their 60th birthdays retire on that

date. Employees leave the company only through death or retirement, and the total number

of employees is 10,000.

Find the total number of new recruits each year.

The male population of a certain country has been stationary for many years, there being

100,000 male births per annum, spread uniformly over the year; the mortality of males follows

English Life Table No.12 - Males, and migration may be ignored.

(a) Calculate the size of the country's male population at any time.

(b) The government of the country has decided to introduce a social security plan, under

which all employed men between ages 15 and 65 must contribute a fixed sum every week, the

same sum being also payable by their employers. Men over age 65 will receive a pension of 100

units of currency per week, and those sick or unemployed between ages 15 and 65 also receive

this amount. If it may be assumed that at any time 95% of men between ages 15 and 65 are

employed, while the remaining 5% are sick or unemployed, calculate the weekly contribution

payable by each employed worker. The scheme is to be financed on a pay-as-you-go basis (no

fund is built up) and administrative costs are to be ignored.

A large manufacturing company has for many years staffed one of its divisions by the recruitment, uniformly over each year, of 1,000 staff at exact age 20. At the end of one year in the

job, new staff are examined for suitability, and 20% are dismissed. All employees are assessed

at age 35, and 50% are immediately moved out of the division. At age 40, all remaining employees are moved out of the division. Death is the only other reason for leaving the division.

Staff experience mortality according to English Life Table No. 12 - Males.

Calculate the number of staff in the division

Note the random variable U has a gamma distribution with parameters a = 9 and B = 1/2.Gammadistributionswithd = nIZandB = 1/2 arechisquare distributions with n degrees of freedom. Thus [I has a old-square distribution with 18 degrees of freedom. From tables of d.f.'s of chi-square distributions, we obtain yous = 23.869 and 30.01 = 34.805. an 2.5 Calculate the probability that the total cost in Example 2.5.3 will exceed 8,250,000 if the retention limit is a. 30,000 b. 50,000. Calculate the retention limit that minimizes the probability of the total cost in Example 2.5.3 exceeding 8,250,000. Assume that the limit is between 30,000 and 50,000. A re insurance compan)r covers 160 structures against re damage up to an amount stated in the contract. The numbers of contracts at the different con tract amounts are given below. Contract Amount Number of Contacts 10 000 30 20 000 35 30 one 25 50 000 15 100 NO 5 Assume that for each of the structures, the probability of one claim within a year is 0.04, and the probability of more than one claim is 0. Assume that res in the structures are mutually independent events- Furthermore, assume that the conditional distribution of the claim size, given that a claim has oc- curred, is uniformly distributed over the interval from 0 to the contract amount. Let N be the number of claims and let S be the amount of claims in a 1-year period. a. Calculate the mean and variance of N. b. Calculate the mean and variance of S. c. What relative security loading, 0, should be used so the company can collect an amount equal to the 99th peromlile of the distribution of total claims? (Use a normal approximation.) . Consider a portfolio of 32 policies. For each policy, the probability a} of a claim is 1! 6 and B, the benet amount given that there is a claim, has p.d.f. _ 2(1y) 0 4)