Question

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 - 0.1(N1)^2 and

Q2 = 16N2 - 0.4(N2)^2,

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 need to solve the equation in this step.

3. The first order condition (which you did not solve yet) 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).

Answer 1

The problem of the firm s to maximize

The FOC for profit maximization is

Simplifying the above equation

......................... (1)

Then the implicit function is defined as

The implicit function theorem states that the partial derivatives for implicit variables are

Where

Therefore, the partial derivatives are

From equation (1) solving for N_1^*

The SOC is satisfied if and only if

Now,

Then the SOC is satisfied. N1* is the optimum value of N1.