# The Deacon Model of Forest Economics: Practice Problems and Solutions

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**Mathematical 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 Ã?Â¦ grow such that the volume of timber at time t can be represented as f(t) = 900t – 50t^{2} + 60t^{3} + 7t^{4}. (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*Ã¢Â?Â?f(t), where p = 100 and Ã¢Â?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?Â¦ grow such that the volume of timber at time t can be represented as f(t) = 900t – 50t^{2} + 60t^{3} + 7t^{4}. (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 = Ã¢Â?Â?f(t)/f(t). Here, Ã¢Â?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?Â¨ (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*Ã¢Â?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?Â¨ (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 of****Mathematical Economics Problems and Solutions.**