Tom lives on an island and has 20 coconut trees in the current period, which currently produce 180 coconuts. Tom detests coconuts, but he can trade them with people on other neighboring islands for things that he wants. Further, Tom can borrow and lend coconuts with neighboring islands. In the coconut credit market, a loan of 1 coconut in the current period is repaid with 2 coconuts in the future period. Each period, Tom's trees produce, and then 10% of them die.
If Tom plants a coconut in the ground in the current period, it will grow into a productive coconut tree in the future period. At the end of the future period, Tom can sell any remaining coconut trees for 1 coconut each. When Tom plants coconuts in the current period, he plants them in successively less fertile ground, and the less fertile the ground, the less productive is the coconut tree. For convenience, we assume here that fractions of coconuts can be produced by trees. Output in the future period, for given numbers of trees in production in the future period, is given in the following table:
Trees in Production Future Output
in the Future of Coconuts
15 ............. 155
16 ............. 162
17 ............. 168
18 ............. 173
19 ............. 177
20 ............. 180
21 ............. 182
22 ............. 183.8
23 ............. 184.8
24 ............. 185.2
25 ............. 185.4
(a) Plot the level of output against the quantity of capital for the future period.
(b) Plot the marginal product of capital against the quantity of capital for the future period.
(c) Calculate Tom’s present value of profits given each quantity of future trees.
(d) Calculate the net marginal product of capital for each quantity of future trees.
(e) Determine Tom’s optimal quantity of investment, and explain your results.