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Consider the decision facing a land owner who has the opportunity to convert a parcel of undeveloped land into farmland. There are two periods, period 0 and period. The conversion is irreversible but can be deferred.
If the land is not converted and remains undeveloped in period 0, then the land owner earns a present value of U0 = 40 in period 0 from the collection of maple syrup and other forest products. If the land is left undeveloped in period 1, then the land owner earns a profit with a present value U1 = 110.
If the land were converted to farmland in period 0, the land would produce a profit in period 0 with a present value of F0 = 60. However, in period 0, the present value of the profit obtained from farmland in period 1 is uncertain due to the uncertainty in the prices of farm produce. With probability 1/2, the period 1 profit would have a high present value, FH = 180. With probability 1/2, the profit would have a low present value of FL = 40.
For simplicity, we assume that the cost of converting the land to farmland is zero.
The land owner has three possible decisions in period 0. (1) The land owner can choose to abandon the conversion opportunity in period 0. (2) The land owner can choose to convert the land to farmland in period 0. (3) Finally, the land owner can choose to defer conversion, waiting until the uncertainty in the profitability of farm produce is resolved at the beginning of period 1 and then deciding whether or not to convert the land. By deferring the decision to convert the land, the farm owner learns whether he or she will receive FH or FL in period 1 before the owner decides whether or not to convert the land in period 1. Of course, if the land owner defers the decision to convert the land until period 1, then the land owner receives only the present value U0 rather than the larger present value F0 in period 0.
The land owner is assumed to be risk neutral and wishes to maximise the expected net present value obtained from the parcel of land.
Please show work.
(i) Draw a decision tree representing the decision problem facing the land owner.
(ii).What is the optimal policy for the land owner to follow regarding the conversion of his or her land? What is the expected net present value produced by the land when the land owner follows the optimal policy? Carefully explain your answers.
(iii) Suppose that the opportunity to defer the conversion decision until period 1 were not available. What would be the optimal choice for the land owner in this case? Briefly explain your answer.
(iv) Suppose now that, in period 0, the land owner must pay an amount D in order to keep available the option to defer the conversion decision to period 1. What is the maximum amount that the land owner should be willing to pay to keep available the opportunity to defer conversion? Explain your answer.
(v) Suppose now that, as in questions (i) to (iii), D = 0. Suppose also that, before any decision has been made in period 0, the value of FL falls so that FL < 40. By how much, if at all, is this fall in the net present value of farm produce likely to reduce the expected net present value obtained from the land when the land owner uses the optimal conversion policy? Explain your answer.
(vi) In a few paragraphs, explain why uncertainty may be increase the value of an investment when there is an option to defer the start of the investment.
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