dolution

An entrepreneur is engaged in tree growing. He purchases a seedling for 4 dollars, incurs a cultivation cost flow at a rate of G(t) = 0.4t dollars per year during the life of a tree, and sells the tree at t = T for R(T) = 4 + 8T – T2 dollars. The market rate of interest is 0.20. Determine an optimal length for his cultivation period, T. Apply the appropriate second-order condition to verify that your solution is a maximum.
Entrepreneur plans for a one-machine horizon. He purchases the machine for 500 dollars. Its scrap value at time T is S(T) = 500 – 40T. The rate of interest is 0.05. The machine yields a quasi-rent flow at the rate Zt = 85 – 4t dollars per year. When should the entrepreneur retire this machine?
Let two duopsonists have production functions q1 = 13x – 0.2x²1 and q2 = 12×2 – 0.1x²2 where x1, x2 are the input levels employed by the duopsonists. Assume that the input supply function is r = 2 + 0.1(x1 + x2) where r is the supply price of the input, and that q1 and q2 are sold in competitive markets for prices p1 = 2 and p2 = 3. (a) Find the input reaction functions. (b) Determine the Cournot equilibrium values for x1, x2, and q1, q2, p1,p2