ECO424 Homework Assignment-Key
Chapters 10-12
Chapter 10
1. How can we reconcile the theoretical results we obtained concerning increases in the prices of
nonrenewable resources with the declines in actual prices that have occurred in the past?
Suggested answer:
The primary “Hotelling” type results of the simple model, that production is tilted toward the
present, price is tilted toward the future, and rent rises at the rate of interest, are based on the
assumption that extraction cost and demand functions are stable. Also see our notes:
Extraction Economics for a Known Stock—
The two-period example: Present value of net benefits (PVNB) = Net benefits in year 0 +
[1/(1+r)] (Net benefits in year 1)
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( p1  MC1 )
1 r
Conclusion 4: price rises at the rate of discount r over time, or, price is tilted toward the future—
this is the famous “Hotelling Rule”. The assumption is that extraction cost and demand functions
are stable.
p0  MC0 
If, on the other hand, extraction costs are expected to decrease (so that supply increases), and
demand to increase (but increase in demand is smaller than increase in supply), the time paths of
quantity and price can be reversed, as they have been in reality. Also see our notes: Reductions
in extraction costs (due to technological changes) have occurred faster that demand increase,
driving prices down.
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2. Consider a three-period model of nonrenewable resource extraction. Lay out the maximization
problem in a way analogous to the two-period model of our lecture notes, and determine the
explicit expression for user cost. (optional; 10 extra points!)
Suggested answer:
The three-period model:
Present value of net benefits (PVNB) = Net benefits in year 0 + [1/(1+r)] (Net benefits in year 1)
+ [1/(1+r)2] (Net benefits in year 2)
PVNB = B0 – C0 + [1/(1+r)](B1 – C1) + [1/(1+r)2](B2 – C2)
Solving for intertemporal efficiency means to find a serial of extraction levels (q0*, q1*, and q2*)
in different years which can maximize PVNB; PVNB is a function of q0, q1, and q2. So,
d ( PVNB) 
 ( PVNB)
 ( PVNB)
 ( PVNB)
dq0 
dq1 
dq2  0
q0
q1
q2
2
d ( PVNB) 


[ B0  C0 
1
1
( B1  C1 ) 
( B2  C2 )]
1 r
(1  r ) 2
dq0
q0
[ B0  C0 
1
1
( B1  C1 ) 
( B2  C2 )]
1 r
(1  r ) 2
dq1
q1
[ B0  C0 
1
1
( B1  C1 ) 
( B2  C2 )]
1 r
(1  r ) 2
dq2  0
q2
( p0  MC0 )dq0 
1
1
( p1  MC1 )dq1 
( p2  MC 2 )dq2  0
1 r
(1  r ) 2
q0 + q1 + q2 = q, and q is a given number. Thus, q1 = q – q0 – q2, and dq1 = d(q – q0 – q2); q2 = q –
q0 – q1, and dq2 = d(q – q0– q1)
( p0  MC0 )dq0 
( p0  MC 0 )
1
1
( p1  MC1 )d (q  q0  q2 ) 
( p2  MC 2 )d (q  q0  q1 )  0
1 r
(1  r ) 2
dq0
d ( q  q0  q 2 )
d (q  q0  q1 )
1
1

( p1  MC1 )

( p2  MC 2 )
0
2
dq0 1  r
dq0
(1  r )
dq0
p0  MC0 
1
1
( p1  MC1 )( 1) 
( p2  MC 2 )( 1)  0
1 r
(1  r ) 2
p0  MC0 
User cost =
1
1
( p1  MC1 ) 
( p2  MC 2 )
1 r
(1  r ) 2
1
1
( p1  MC1 ) 
( p2  MC2 )
1 r
(1  r ) 2
Chapter 11
1. The most dominant source of global energy is fossil fuels which include coal, petroleum, and
natural gas. In the U.S., in 2005, fossil fuel made up 86% of total energy consumption. For
petroleum, percentage of consumption that was imported rose from 25% in 1970 to 71% in 2005
due to the boom in automobile transportation.
2. With energy subsidies, the market prices of energy are too low as compared to prices that
would be socially efficient; energy consumption is too high. Please draw a graph first and then
explain.
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Suggested answer:
With energy subsidies, the supply of energy curve shifts to the right. Compared with socially
efficient price and quantity, market price is too low and market quantity is too high.
3. What are the arguments of people who support replacing conventional gasoline with ethanol?
Suggested answer: Although ethanol involves burning carbon-based plant materials, the carbon
is contemporary carbon in the sense that is was recently absorbed from the atmosphere as the
plants grew.
Replacing conventional gasoline with ethanol is seen by its advocates as a way of moving to
lower greenhouse gases as well as toward greater energy independence.
4. What is energy intensity? If energy consumption is 100.6 quads, GDP is $13 trillion, how
much is energy intensity?
Suggested answer: Energy intensity …
(100.6 *1015 Btus) / ($13 * 1012) = (100.6 * 103) / 13 = 100600 / 13 = 7,738.26 Btus per dollar
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Chapter 12
1. Using Table 12-1 (page 222) to explain maximum sustainable yield.
Suggested answer: Table 12-1 tells us that, if we cut at 60 years when the average yield is the
highest, it would yield the largest volume of any cycle (over 600 years, this would yield 16,600
cubic feet, as against a 100-year cycle which would yield 12,540 cubic feet). The highest average
volume, 27.7 cu ft/year, is the maximum sustainable yield (MSY) for the forest.
In addition, if we refer to Dr. David’s notes on pages 3-4, we also know that:
slope of a ray = average volume;
slope of a tangent = annual increase or marginal growth.
Through the origin, draw a ray that is tangent to the growth function. It has the maximum
average volume (27.7 cu ft/yr) which equals marginal growth (30 cu ft/yr), in year 60. (MSY)
Thus, in order to harvest the MSY, we should cut the stand of trees when marginal growth equals
average growth of the stand. Where this condition is met, the average growth is tangent to the
growth function. In Biology, MSY is known as Maximizing the Mean Annual Increment.
2. How should we identify the optimal rotation interval t*? (Please use the formula and graph to
explain.)
Suggested answer: To find the optimal rotation interval, we need to maximize the present value
of net benefits of the forest with respect to the time period of harvest.
We select any two consecutive years.
V0: the monetary value of the wood that would result if the forest were harvested this year
V1: the monetary value of the wood that would be produced if the harvest is delayed one year
C: harvest costs, the monetary costs of felling the trees and getting them to market
S: the present value of the vacant site after the trees have been harvested
Net proceeds in year 0 = V0 – C + S
Net proceeds in year 1 = V1 – C + S = V0 + ΔV – C + S
where ΔV is the value of 1-year growth
When the forest is young and ΔV is relatively large, we should wait to harvest until next year
since (V0 + ΔV – C + S)/(1+r) > (V0 – C + S)
As the forest gets older and ΔV is small, it is time to harvest the trees! At this time,
(V0 + ΔV – C + S)/(1+r) = (V0 – C + S)
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V0 + ΔV – C + S = (V0 – C + S) (1+r)
V0 + ΔV – C + S = V0 – C + S + (V0 – C + S)r
ΔV = (V0 – C + S)r
ΔV = (V0 – C)r + Sr
MB of waiting (value of new growth) = MC of waiting (lost interest on total revenue or net
benefits)
Figure 12-3, page 225: the optimal rotation is identified by the intersection of two functions,
labeled as t* on the horizontal axis.
(I did not draw the graph; please refer to page 225!)
3. If the timber is harvested and converted into building materials, the carbon stays sequestered
until the materials decay. Suppose that these building materials did not decay at all. If there were
a market for carbon sequestration services, the market values of timber would reflect its value as
building materials and its value for carbon sequestration. How will this situation (increasing the
market price of timber by including the timber’s value for carbon sequestration) affect the
optimal rotation interval t*?
Suggested answer: This higher timber price will increase ΔV, V0, and S and shift the ΔV function
and the (V0 – C + S)r function outward. The optimal rotation interval could increase, decrease, or
remain unchanged.
(I did not draw three situations. Refer to our extra notes in class—three graphs!)
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