The Deacon Model of Forest Economics: Practice Problems and Solutions

Mathematical Economics Problems and Solutions - Section 8

See Mr. Stolyarov's complete list ofMathematical Economics Problems and Solutions.

These problems use as their basis the economic model developed in Robert T. Deacon's paper, "The Simple Analytics of Forest Economics."

Problem ETH1. The trees in Forest Φ grow such that the volume of timber at time t can be represented as f(t) = 900t - 50t² + 60t³ + 7t⁴. (Note that these trees do not follow a typical biological growth function). The annual real interest rate is 0.19, and each unit of timber can be sold at a price of 100 Yap pieces of stone (YPS) net of harvest costs. Assume there is no opportunity cost to using the land to grow timber. At t = 89 years from the beginning of the trees' growth, Imhotep obtained sole ownership of the forest through a surprise bequest. What is the marginal benefit Imhotep would get by waiting to cut the trees for another year?

Solution ETH1. MB(waiting) = p*∆f(t), where p = 100 and ∆f(t) = f(90) - f(89).

Thus, MB = 100(502686000 - 481177877) = MB = 2,150,812,300 YPS

Problem ETH2. The trees in Forest Φ grow such that the volume of timber at time t can be represented as f(t) = 900t - 50t² + 60t³ + 7t⁴. (Note that these trees do not follow a typical biological growth function). The annual real interest rate is 0.19, and each unit of timber can be sold at a price of 100 Yap pieces of stone (YPS) net of harvest costs. Assume there is no opportunity cost to using the land to grow timber. At t = 89 years from the beginning of the trees' growth, Imhotep obtained sole ownership of the forest through a surprise bequest. What is the marginal cost he would incur by waiting to cut the trees for another year? Should he wait another year to cut down the trees?

Solution ETH2. MC(waiting) = r*p*f(t) = 0.19*100*f(89) = MC = 9,142,379,663 YPS

We note that 9,142,379,663 > 2,150,812,300, so MC > MB, and the owner would incur a much higher cost by waiting another year to cut down and sell the timber than he would gain in terms of marginal benefits. Thus, the owner should not wait another year to cut down the trees.