Optimization Problem Assignment | Homework for You
You own two lakes rich in fish. The quantity of fish caught in each lake depends on the number of persons who fish in each, according to
Q1 = 10N1 – 0.1N12 and
Q2 = 16N2 – 0.4N22,
where N1 and N2 denote the number of fishers at each lake. In all, there are N fishers working for you, each of them is paid a wage w, and the price of fish is P.
1. Write down your optimization problem as an optimization problem with a single variable, N1. Hint: find N2 from the formula N = N1 + N2, and replace it in the objective function.
2. Write down the first order condition for this problem. Do not solve the equation yet!
3. The first order condition (which you did not solve yet, right?) implicitly defines the optimal number of fishers in lake 1, N1* as a function of P, w, and N. Use the implicit function theorem to calculate all three partial derivatives of N1*.
4. Solve the first order condition to calculate N1*.
5. Write down and check the second order condition(s).
7. Write the optimization problem you are faced in the previous section (questions 1 — 5), as a constrained optimization problem with 2 variables.
8. Write down the first order conditions.
9. Write down the second order condition(s) and check whether it/they are satisfied.
10. Find the optimal numbers of fishers in each lake.
On a popular air route, an airline offers two classes of service: business class (B) and economy class (E). The respective inverse demands are given by
PB = 540 – 0.5QB and PE = 380 – QE.
Because of ticketing regulations, business passengers cannot take advantage of economy’s low fares.
Each flight has a capacity of 200 passengers, The cost per full flight is $20,000, so you may assume the marginal cost per passenger is $100.
11. Write down the optimization problem this airline is faced with. Set it up as an unconstrained optimization problem with two variables.
12. Write down the first order condition(s).
13. Solve the first order condition(s) to find out the optimal number of flights.
14. Write down the second order condition(s) and check whether it/they is/are satisfied.
End of assignment.