(1) The production function of a firm is given by f (K, L) = AK15-410 where k is the amount of machines used by the firms and L is the above of labor. Further, the total number of people using each machine is 16. That means if you have 5 machines, you can only hire 10 people to operate on the machines. You have to pay a rental rate of r on the machines and you pay a wage w to the workers. The firm wants to rent machines and hire workers to maximize the total profit. Write down the optimization of the firm. What is the optimal number of machines used by the firm. The term A is a constant. Suppose o = 0.25, A = :, w = 1,r = 4. Price of each product is $1. (2) Suppose that there is economies of scale. That means the parameter A now depends on the number of machines that you install. And this dependence is given by A(k) = 3 1 What is the optimal number of machines to be rented to maximize profits. (3) Suppose now there is uncertainty. If you produce too much of the product, consumer demand crashes and you are not able to sell anything. Assume that the probability p(k) of demand crash is given by p(k) = 1-ek With the remaining probability (1 - p(k)), you expect to sell everything. As a manager, you hate the uncertainty. You want to maximize expected profits but you dislike the variance. The optimization problem you are looking at is k max E(II(@)) -V(II(K)) where E(.) is the expectation and V(.) is the variance. A is constant at 2. Calculate the total number of machines to use to maximize revenues