The Deacon Model of Forest Economics: Practice 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 Ã?¦ 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*âÂ?Â?f(t), where p = 100 and âÂ?Â?f(t) = f(90) – f(89).

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

Problem ETH2. The trees in Forest Ã?¦ 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.

Problem ETH3. Now assume that, in a parallel universe to that in Problems ETH1-2, all other things are equal, but the interest rate is different such that t = 89 is the exact optimal time to harvest the trees. What is the annual interest rate r? Assume that Imhotep can only make a decision to harvest the trees once per year.

Solution ETH3. At the optimal harvest time, r = âÂ?Â?f(t)/f(t). Here, âÂ?Â?f(t) = f(90) – f(89), so

r = (f(90) – f(89))/f(89) = (502686000 – 481177877)/481177877 = r = 0.0446989025

Problem ETH4. The trees in Forest Ã?¨ (another unusual forest – both biologically and in its nomenclature) grow such that the volume of timber at time t can be represented as f(t) = e^t – 2^t. The annual real interest rate is 0.07, and the per-year opportunity cost of using the forest land to grow timber is 36000 platinum hexagons (PH). Each unit of timber can be sold at a price of 900 PH. Tlaloc becomes the owner of the forest at t = 7 years. What is his marginal benefit of waiting another year to cut the trees?

Solution ETH4. MB(waiting) = p*âÂ?Â?f(t) = 900*(f(8) – f(7)) =

900(2724.95798704 – 968.633158429) = MB = 1,580,692.346 PH.

Problem ETH5. The trees in Forest Ã?¨ (another unusual forest – both biologically and in its nomenclature) grow such that the volume of timber at time t can be represented as f(t) = e^t – 2^t. The annual real interest rate is 0.07, and the per-year opportunity cost of using the forest land to grow timber is 36000 platinum hexagons (PH). Each unit of timber can be sold at a price of 900 PH. Tlaloc becomes the owner of the forest at t = 7 years. What is his marginal cost of waiting another year to cut the trees? Should he wait another year to cut down the trees?

Solution ETH5. Here, the per-period cost of waiting is rpf(t) + R, where R = 36000. Thus, MC(waiting) = 0.07*900*f(7) + 36000 = 0.07*900*968.633158429 + 36000 =

MC = 97023.88898 PH. Here, MB > MC, so Tlaloc should wait another year to harvest the trees.

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