EWU – ECON 457 - Natural Resource Economics
The n-period model revisited, using Solver.
The n-period model. Efficient intertemporal allocation revisited, using Solver.
Assume that we have a fixed reserve of a depletable resource to allocate between n periods. The marginal
willingness-to-pay in each period is given by the following equation: P = 8 - 0.4 Q, and the marginal cost
is constant at $2 per unit. Further assume that the reserve of the depletable resource is fixed at 40 units,
and the discount rate is 10%.
1. Recall from your handout on the simple mathematics of static and dynamic efficiency that the
present value of total net benefit in each period t is: PVTNB t 
b 2
q t  cq t
2
1  r t 1
aqt 
, where p  a  bq is
the equation of the demand curve for the depletable resource in period t and p  c is the equation
of its supply curve.
2. Also recall that the marginal user cost, which reflects a depletable resource increasing scarcity, is
the difference between the resource price ( a  bqt ) and its marginal extraction cost ( c ).Each
period t marginal user cost is thus: a  bqt  c .
3. In a Microsoft Excel file, rename Sheet1 "n period model". In the n period worksheet, organize
your data as indicated below:
1
2
3
4
5
6
7
8
9
A
B
C
n-period model
resource reserves
marginal social benefit curve
D
intercept
slope
marginal social cost of extraction
discount rate
year
4.
5.
6.
7.
consumption
reserves
PVNB
E
F
G
40
8
0.4
2
0.1
Total MC
MC of extraction
M user cost
In the year column, specify 21 years.
In the consumption column, specify 2 units for each year, from year 1 to year 20; and 0 for year 21
For year 1 reserves, specify the beginning of the year level of reserves as: = $E$2
For year t reserves (t = 2, …, 21), specify the beginning of the year level of reserves as being the
level of reserves at the beginning of the previous year minus the consumption level during that
year. For example, for year 2, in cell C11, you should have: = C10-B10. Copy this equation in the
remaining cells of this column.
8. For each year t, specify the present value of total net benefit (PVNB) by using the equation in (1).
In cell D10, you should have: =($E$3*B10-($E$4/2)*B10^2-$E$5*B10)/(1+$E$6)^(A10-1) Copy
this equation in the remaining cells of the PVNB column.
9. The Total Marginal Cost is the sum of the Marginal Cost of extraction and the Marginal user cost.
In cell E10, you should have: = F10+G10. Copy this equation in the remaining cells of this
column.
10. For each year t, the MC of extraction is constant: = $E$5.
11. For each year t, specify the marginal user cost by using the equation in (2). In cell G10, you
should have: =$E$3-$E$4*B10-$E$5. Copy this equation in the remaining cells of this column.
12. At the end of your PVNB column, add all PVNB over the above 21 years: =sum(D10:D29). Bold
this cell.
13. In the Tools menu, choose Solver. (If you do not have Solver on your computer, in the Tools
menu, choose Adds-in, select Solver Add-in, click ok, and close the window. Go back to
beginning of instruction in (13)). The cell you are going to target is the one you created in (12):
$D$31. Select the Max option, since you want to maximize the sum of the PVNB over a 20 yearperiod. You are going to do that by searching for optimum consumption level over the years, so by
changing cells $B$10:$B$29. You also have to specify two constraints: consumption level over
the 20 year-period has to be zero or positive, $B$10:$B$29 >= 0, and reserves level over the 21period has to be zero or positive, $C$10:$C$30 >= 0. You are ready to Solve!
14. Use your results to graph the consumption profile within which the resource is allocated. Select
the Chart Wizard icon on your toolbar (or select Chart under the Insert menu). Select the
XY(Scatter) chart type, and the scatter chart sub-type with data points connected by smoothed
lines. Select Next. Select data range $A$9:$B$18 (or type "= $A$9:$B$18" in the data range
window), and select series in columns. Select Next. Specify a chart title (how about: "YourName.
Consumption Profile"); "Year" should be the X axis label, and "Consumption (units)" can be the Y
axis one's. Select the legend tab and deselect Show legend. Select Next. I personally like to place
each of my new charts in a new worksheet; if you'd like to do that to, select As new sheet. Select
Finish.
15. Go back to the "n period model" worksheet by left-clicking on the tab "n period model". Use your
results to graph the marginal cost profile over the time horizon within which the resource is
allocated. Select the Chart Wizard icon on your toolbar (or select Chart under the Insert menu).
Select the XY(Scatter) chart type, and the scatter chart sub-type with data points connected by
smoothed lines. Select Next. Select data range $A$9:$A$18, $E$9:$F$18 (or type "= A$9:$A$18,
$E$9:$F$18" in the data range window), and select series in columns. Select Next. Select Show
legend on Top. Select the Title tab. Specify a chart title (how about: " YourName. Marginal-Cost
Profile"); "Year" should be the X axis label, and "Marginal Cost (dollars per unit extracted)" can
be the Y axis one's. Select Next. I personally like to place each of my new charts in a new
worksheet; if you'd like to do that to, select As new sheet. Select Finish.
The n-period model. Efficient intertemporal allocation with environmental costs.
Extraction of the above resource involves problem of pollution (dust, smoke, and reduced enjoyment from
vista). The marginal damage (health and property damages) is estimated at a constant $2 per unit of the
resource extracted. In light of this new information, revisit your consumption and marginal cost profiles.
HINT: Go back to the "n period model" worksheet by left-clicking on the tab "n period model". Use your
n-period model worksheet. All you have to do is: (1) change the value of your marginal cost of extraction,
(2) re-apply Solver, (3) create Chart 3 (Select data range $A$9:$B$21. Give the following title:
"YourName. Consumption Profile with environmental costs") and Chart 4 (Select data range
$A$9:$A$21, $E$9:$F$21. Give the following title: "YourName. Marginal Cost Profile with
environmental costs").
The n-period model. Transition to a renewable substitute.
So far we have discussed the allocation of a depletable resource when no substitute is available to take its
place. Now suppose that a substitute renewable resource is available at constant marginal cost of $6 per
unit. In light of this new information, revisit your consumption and marginal cost profiles. HINT: in this
case, you need to realize that the level of consumption in any year t can be a consumption of the
depletable resource, or a consumption which comprises a combination of depletable resource and the
substitute, or consumption of the substitute; i.e. qt = qd + qs, where qt, qd, qs ≥0.
16. Rename Sheet2 "n period model with substitute". Go back to the "n period model" worksheet,
select all by left clicking the upper left corner of the worksheet (between 1 and A), copy, and paste
in cell A1 of your "n period model with substitute" worksheet. At the top of your worksheet
change the title from "n period model" to "n period model with substitute".
17. Select cell A6 through cell G6, insert cells by shifting cells down. In cell A6, type "marginal cost
of production of substitute". In cell E6, type 6.
18. Select cell C10 through cell C32, insert cells by shifting cells to the right. Repeat. In cell C10, type
"depletable". In cell D10, type "substitute".
19. In cell B11, type "=C11+D11". Copy this equation in the remaining cells of this column. In cell
C11 through cell C30, enter 2. In cell C31, enter 0. In cell D11 through cell D31, enter 0. In cell
E12, change the equation to =E11-C11. Copy this new equation in the remaining cells of this
column.
20. Since qt = qd + qs, the equation for the PVTNBt has to be changed slightly:
PVTNB t 
aq d  q s  
b
q d  q s 2  cq d  sq s
2
1  r t 1
, where p  a  bq d  q s  is the equation of the demand
curve for the resource (depletable or substitute one) in period t, p  c is the marginal cost of
extraction of the depletable resource, and p  s is the marginal cost of production of the renewable
substitute. In cell F11, you should have:
=($E$3*B11-($E$4/2)*B11^2-$E$5*C11-$E$6*D11)/(1+$E$7)^(A11-1)
Copy this equation in the remaining cells of the PVNB column.
21. You are ready to Solve! Note that for the range of "Changing Cells" you need to consecutively
select cells C11 through C30, and cells D11 through D30 (or type $C$11:C$30,$D$11:$D$30 in
the "By Changing Cells" window). Additionally, you have now three constraints:
(1) $C$11:$C$30 >= 0, (2) $D$11:$D$30 >= 0, and (3) $E$11:$E$31 >= 0.
22. Use your results to graph the consumption profile within which the resource is allocated. Select
the Chart Wizard icon on your toolbar (or select Chart under the Insert menu). Select the
XY(Scatter) chart type, and the scatter chart sub-type with data points connected by smoothed
lines. Select Next. Select data range $A$10:$C$30 (or type "= $A$10:$C$30" in the data range
window), and select series in columns. Select Next. Specify a chart title (how about: "YourName.
Consumption Profile with substitute"); "Year" should be the X axis label, and "Consumption
(units)" can be the Y axis one's. Select the legend tab and select Show legend on Top. Select Next.
I personally like to place each of my new charts in a new worksheet; if you'd like to do that to,
select As new sheet. Select Finish.
23. Go back to the "n period model with substitute" worksheet by left-clicking on the tab "n period
model with substitute". Use your results to graph the marginal cost profile over the time horizon
within which the resource is allocated. Select the Chart Wizard icon on your toolbar (or select
Chart under the Insert menu). Select the XY(Scatter) chart type, and the scatter chart sub-type with
data points connected by smoothed lines. Select Next. Select data range $A$10:$A$30,
$G$10:$H$30 (or type "= $A$10:$A$30, $G$10:$H$30 " in the data range window), and select
series in columns. Go to the Series tab. Select the MC of Extraction series. Restrict the selected X
values from year 1 through year 6 (i.e. $A$11:$A$16). Restrict the selected Y values to the same
years (i.e. $H$11:$H$16). Select Next. Select Show legend on Top. Select the Titles tab. Specify a
chart title (how about: " YourName. Marginal-Cost Profile with substitute"); "Year" should be the
X axis label, and "Marginal Cost (dollars per unit extracted or produced)" can be the Y axis one's.
Select Next. I personally like to place each of my new charts in a new worksheet; if you'd like to
do that to, select As new sheet. Select Finish.
The n-period model. Efficient intertemporal allocation with increasing marginal extraction cost.
Consider a situation in which the marginal cost of extracting the depletable resource rises with the
cumulative amount extracted. This is commonly the case, for example, with minerals, where the highergrade ores are extracted first, followed by an increasing reliance on lower-grade ores (Tietenberg, 2002).
HINT: in this case, the marginal cost of extraction of the resource in year t is of the general form: MCt = c
MCt
+ f Qt, where Qt is cumulative extraction to date and c, f > 0.
 f , i.e. the marginal cost of
dQt
extraction of the resource is not constant, it varies with the level of cumulative extraction to date by a
factor f. In our example we let c=$2 and f=0.1
24. Rename Sheet3 “increasing MC of extraction”. Go back to the "n period model" worksheet, select
all by left clicking the upper left corner of the worksheet (between 1 and A), copy, and paste in
cell A1 of your "increasing MC of extraction" worksheet. At the top of your worksheet change the
title from "n period model" to "n period model with increasing marginal extraction cost".
25. Select cell A6 through cell G6, insert cells by shifting cells down. In cell D5, type c; in cell D6,
type f, and in cell E6, type 0.1.
26. Select cell C10 through cell C32, insert cells by shifting cells to the right. In cell C10, type
"cumulative". In cell C11, type 0. In cell C12, enter “=B11+C11”; copy this equation in the
remaining cells of this column.
27. Since MCt = c + f Qt, the equation for the PVTNBt has to be changed slightly:
b
aqt  qt2  cqt  fQt qt
2
, where p  a  bqt is the equation of the demand curve for
PVTNBt 
1  r t 1
the resource in period t, p  c  fQt is the marginal cost of extraction of the resource in period t.
In cell E11, you should have:
=($E$3*B11-($E$4/2)*B11^2-$E$5*B11-$E$6*C11)/(1+$E$7)^(A11-1)
Copy this equation in the remaining cells of the PVNB column.
28. Of course, since MCt = c + f Qt, the equation for MC of extraction has also to be changed slightly:
In cell $G$11, you should have: “=$E$5+$E$6*C11”. Copy this equation in the remaining cells of
this column.
29. You are ready to Solve, by changing cells $B$11 through $B$30, and adding the usual constraints
regarding consumption and reserves values: (1) $B$11:$B$30 >= 0, and (2) $D$11:$D$31 >= 0.
30. Use your results to graph the consumption profile within which the resource is allocated. Select
the Chart Wizard icon on your toolbar (or select Chart under the Insert menu). Select the
XY(Scatter) chart type, and the scatter chart sub-type with data points connected by smoothed
lines. Select Next. Select data range $A$10:$B$22 (or type "= $A$10:$B$22" in the data range
window), and select series in columns. Select Next. Specify a chart title (how about: "YourName.
Consumption Profile with increasing marginal extraction cost"); "Year" should be the X axis label,
and "Consumption (units)" can be the Y axis one's. Select the legend tab and deselect Show
legend. Select Next. I personally like to place each of my new charts in a new worksheet; if you'd
like to do that to, select As new sheet. Select Finish.
31. Go back to the "increasing MC of extraction" worksheet by left-clicking on the tab "increasing
MC of extraction". Use your results to graph the marginal cost profile over the time horizon within
which the resource is allocated. Select the Chart Wizard icon on your toolbar (or select Chart
under the Insert menu). Select the XY(Scatter) chart type, and the scatter chart sub-type with data
points connected by smoothed lines. Select Next. Select data range $A$10:$A$22, $F$10:$G$22
(or type "= $A$10:$A$22, $F$10:$G$22 " in the data range window), and select series in
columns. Select Next. Select Show legend on Top. Select the Titles tab. Specify a chart title (how
about: " YourName. Marginal-Cost Profile with increasing MC of extraction"); "Year" should be
the X axis label, and "Marginal Cost (dollars per unit extracted)" can be the Y axis one's. Select
Next. I personally like to place each of my new charts in a new worksheet; if you'd like to do that
to, select As new sheet. Select Finish.
Source: Jon M. Conrad, Resource Economics, Cambridge, 1999.