Consider a farmer that owns a unit of land with attributes θ . Farm income in each period is given by z(θ). In each period, instead of farming the farmer can choose to sell his land to a developer at a price, p. The developer develops the land for residential, commercial or industrial purposes and it is assumed that it cannot be restored for use in farming. That is, the development decision is irreversible. The value of the land in development varies randomly over time according to a cumulative probability distribution F(P, θ) = prob(p <= P|θ). In each period the farmer must decide between continuing to farm and selling the land for development. The farmer’s objective is to maximize the expected discounted sum of returns over time. The discount factor is δ=1+ρ1 where ρ is the discount rate.
a. Formulate the farmer’s problem as a stochastic dynamic programming problem and write down the fundamental recursion relation for the problem (the Bellman equation).
b. There exists a unique reservation land value P*(θ) such that the farmer continues to farm if p < P*(θ) and sells his land if p > P*(θ). What is the equation that implicitly defines the reservation land value? Express this equation strictly in terms of the primitive elements of the model, i.e., the value function and its derivatives should not appear in the equation.
c. Let land attributes be such that an increase in θ is associated with higher quality land and increased farm income. In addition, F is decreasing in so that F(P,θ1) F(P,θ2) for θ2 >θ1: Intuitively, this means that an increase in θ is associated with an uniformly improved distribution of development values. Formally, this means that an increase in θ is associated with a first-order stochastic dominance shift in F: Using your answer to part (b) determine the effect of an increase in θ on the reservation land value that governs the farmer’s decision.
