7.1.b. Harvesting Trees

Decisions about when to harvest trees involve a number of factors, one of these being opportunity cost. Suppose it is 2012. We have a stand of trees we must harvest by 2013, but we could harvest it now in 2012 . Imagine if we ignored opportunity costs and allowed trees to grow another year so long as the tree would actually increase in size and bring in higher accounting profits.

The table below shows the expected tree growth between 2012 and 2013. Allowing another year of growth adds 1,000 tons per acre of wood that can be harvested. If each ton provides $0.50 when harvested, then letting the trees grow until 2013 increases its value from $7,500 to $8,000. Waiting another year increases accounting profits attributable to trees by $500. If that was all that mattered, let the trees grow. But why would we concern ourselves only with money from the trees? Doesn’t money earned elsewhere matter also?

Now suppose we do consider opportunity cost, specifically, the fact that instead of letting the trees grow another year we could have harvested the trees in 2012, investing the profits at a real, risk-free rate of 8% (that 8% is an assumption I give you). To do this, we would harvest in 2012 and collect the $7,500 per acre of profits. This $7,500 then earns 8% interest between 2012 and 2013, thereby increasing our wealth from $7,500 to $8,100.

Clearly, it is better to harvest now, in 2012, as we prefer $8,100 in 2013 to $8,000.

Remember that opportunity cost is the value of the next best alternative. Letting the trees grow until 2013 would increase our accounting profits attributed to trees, but it ignores the opportunity cost of earning money elsewhere, and prevents us from maximizing our wealth from all potential sources.

Table 1—Tree harvesting decision

Column A	Column B	Column C	Column D	Column E
Year	Age of stand in years	Tons per acre, if harvested	Accounting profits, if harvested this year (profits = $0.50 per ton)	Accounting profits, if harvested last year (profits = $0.50 per ton of profits last year invested at a real, risk-free interest rate at a return of 8%)
2012	20	15,000	($0.5 per ton)(15,000 tons per acre) = $7,500 per acre	————
2013	21	16,000	($0.5 per ton)(16,000 tons per acre) = $8,000 per acre	($7,500 from last year’s profits)(1.08) = $8,100
We would rather our total wealth in 2013 be $8,100 than $8,000, so we decide to harvest the trees in 2012 and invest the profits at 8%.

By allowing the stand to grow from 20 to 21 years of age the value of the stand increases by ($8,000 - $7,500) = $500. There is an opportunity cost to letting the stand age. We could have harvested the trees at age 20 and invested it, earning an 8% interest rate. The total interest earned would be $7,500(0.08) = $600. Since the opportunity cost of letting the tree grow ($600) is greater than the increase in the value of the stand ($500), we maximize our wealth by harvesting the trees when the stand is 20 years old.

Now consider a more complex question. The basic question is the same, but now we are tasked when determining when to harvest the trees over a larger range of ages. You answer it just the same. Start at the youngest age of 28 years, in 2012. Ask yourself whether you would be richer in 2013 by letting the stand age or harvesting it now and investing it at the interest rate. If you let the stand age, then go to year 2013 and ask yourself whether you will be richer in 2014 by letting the stand age or harvesting the trees now and investing it at the interest rate.

As soon as it becomes more profitable to harvest the trees, you do it then and don’t worry about the latter years. If it is most profitable to harvest in, say, 2015, you never have to wonder whether it would have been better to wait until 2017, 2018, or later years.

Table 2—Another tree harvesting decision

Column A	Column B	Column C	Column D	Column E
Year	Age of stand in years	Tons per acre, if harvested	Accounting profits, if harvested this year (profits = $0.15 per ton)	Accounting profits, if harvested last year (profits = $0.15 per ton of profits last year invested at a real, risk-free interest rate at a return of 3%)
2012	28	12,000	($0.15 per ton)(12,000 tons per acre) = $1,800 per acre	————
2013	29	14,800	$2,200 per acre	($1,800 from last year’s profits)(1.03) = $1,854
2014	30	16,000	$2,400 per acre	$2,266
2015	31	16,800	$2,520 per acre	$2,472
2016	32	17,000	$2,550 per acre	$2,596
You keep letting the trees grow another year so long as the additional value of the stand is greater than the interest one could earn by harvesting the trees now. In year 30, letting the trees grow another year adds ($2,520 - $2,400 = ) $120, whereas harvesting them then and invested them at 3% only increases wealth by $2,400(0.03) = $72. So let them grow to be at least 31 years of age. In year 31, letting the trees grow another year only adds ($2,550 - $2,520 = ) $30, whereas harvesting them then and invested them at 3% only increases wealth by $2,520(0.03) = $76. So harvest the trees in year 31 and invest the money at a 3% interest rate.