The Internet Economy

Econ 407: Applications of Exponential and logarithmic functions in economics

Assignment: To be completed in class--work will be graded.

1. Actuaries use the functions f(t) = l^2*t*e^(-lt) to model the probability f(t) that a person will die "near" age t (in years). The constant l (0<l <e) represents a particular person’s occupational and health hazards.

For l = 0.02, compute f(10), f(60), f(70), and f(100).

For l = 0.02, find the maximum of f(t).

Sketch f(t) for l = 0.02.

 

 

2. Certain office machinery constantly depreciates (decays in value) at an instantaneous rate of 12 percent per year. How much will a $500 typewriter be worth in 10 years?

3. Many banks now offer interest at rate r "compounded continuously" under the principle that money should grow at a rate proportional to the present balance. That is, if  P(t) is the value (balance) of an account at time t (in years), then S = p*e^(r*t), where r is the advertised rate of interest. Find the value at time t = 1 (year) of $100 at 5 percent interest compounded continuously.

Find the effective annual rate of 5 percent interest compounded continuously.

4. An in investor needs $10,000 for a down payment on some property. She wishes to accumulate this amount in 1 year by making quarterly deposits in equal amounts drawing 8 percent interest (compounded continuously). How much must she deposit each quarter to accomplish this?

5. Assuming the population model P(t) = p*e^(k*t), use the fact that the U.S. population grew from 5 million to 200 million in its first 200 years, and show that P(t) = 5(1.0186)^(t).

6. The Dutchman Peter Minuit purchased Manhattan Island in 1626 for $24. If this $24 was then invested at 5 percent annual interest, compounded continuously, what would this balance have been in 1980?