1. Marginal analysis Discounted cash flow
Analytical Tools
Marginal analysis
Discounted cash flow

2. Marginal Analysis
Resources are limited, therefore we want the most “bang for our bucks” -- benefits from resources allocated to a project

3. Marginal Analysis Economic efficiency
Maximize “profit”, i.e. total revenue - total costs (TR - TC)
Slope of TR curve is MR
Profit
Slope of TC curve is MC

4. Marginal Analysis Economic efficiency
Profit maximized where marginal cost = marginal revenue (MC = MR)
Price (P)
Market equilibrium exists when,
MR = MC = ATC MC
ATC
P = MR
No “pure profit” (economic rent) to attract new firms to industry
Quantity (Q)

5. Marginal Analysis Price (P) Pure profit = P1*Q1 - P2*Q1, or
= (P1 – P2) *Q1 MC
ATC P1
Assumes perfect competition, i.e., P = MR P2
Quantity (Q) Q2 Q1

6. Advantage of Marginal Analysis
Don’t have to do a complete analysis of costs and revenues
Can estimate MC directly from market price by assuming a given profit percentage
Can estimate MR from market price and knowledge of market structure, i.e. perfectly competitive, monopolistic, or somewhere in-between.

7. Applications of Marginal Analysis
Financial maturity of individual tree
Minimum size tree to harvest
Break-even analysis

8. Biological Maturity Output (volume) Inflection point

9. Biological vs. Financial Maturity
Financial maturity is based on benefit from letting tree/stand grow for another time period compared to cost for doing so.
Would you expect biological and financial maturity to occur at the same point in time?

10. Financial Maturity of Tree
Age
Volume b.f.
Tree value given $/bf
Change in value
Return on investment 70
240
$0.5
n.a. 75
310
$66
66/120 = 55% 80
360
$48
48/186 = 26% 85
400
$26
26/234 = 11%

11. Financial Maturity of Tree
Compare return on investment with return from other 5-year investments
If rate of return on the alternative is 10% then don’t cut yet
If rate of return on the alternative is > 20%, then cut at age 80
More practical to compare with prevailing annual compound rates of interest?
How can we compute annual compound rate of interest for changes in value over a 5-year period?

12. Estimating Annual Compound Rate of Interest
Use basic compounding multiplier,
Vn = V0 (1+i)n, where,
Vn = value n years in future
V0 = value in year 0 (current time)
n = number of years
i = annual compound rate of interest

13. Before solving for I let’s review how compounding and discounting multipliers are used

14. Example of use of compounding multiplier
Buy $100 worth of stock today
If it increases in value at a rate of 18% each year, what will it be worth in 5 years?
V5 = ?, V0 = $100, i = 0.18, n = 5
V5 = $100 (1.18)5 = $100 x 2.29 = $229

15. Example of use of discounting multiplier
Solve compounding multiplier for V0 by dividing both sides by (1+i)n
Vn = V0 (1+i)n
V0 = Vn /(1+i)n = Vn * 1/ (1+i)n

16. Solve Compounding Multiplier for i
Vn = V0 (1+i)n
Vn/ V0 = (1+i)n
(Vn/ V0 )1/n = ((1+i)n)1/n = (1+i)n/n = 1+i
(Vn/ V0 )1/n – 1 = i

17. Calculate compound rate of interest for 5 year value increments
Age 70 to 75:
i = (186/120)1/5 – 1 = (1.55)0.2 –1 = 1.09 – 1 = 0.09
Age 75 to 80:
i = (234/186)1/5 – 1 = (1.26)0.2 –1 = 1.05 – 1 = 0.05
Age 80 to 85:
i = (260/234)1/5 – 1 = (1.11)0.2 –1 = 1.02 – 1 = 0.02

18. When should we cut? Age Volume b.f. Tree value given $/bf
Change in value
Return on investment 70
240
$0.5
n.a. 75
310
$66 9% 80
360
$48 5% 85
400
$26 2%

19. When should we cut?
Depends on what rate of return the owner is willing to accept.
We refer to this rate as the owner’s guiding rate or, alternative rate of return.
Rate is based on owner’s alternative uses for the capital tied-up in the trees.

20. When should we cut? If owner’s alternative rate of return is
10% - cut at age 70
7% - cut at age 75
5% - cut at age 75
1% - let grow to age 85

21. How does an owner select her alternative rate of return?
Borrowing rate – if she would have to borrow money if tree wasn’t get, she could use the interest rate she would have to pay on the loan, i.e. the “borrowing rate”
Lending rate – if owner could “lend” the revenue from cutting the tree now to someone else, she could use the rate she would get by making the “loan”

22. Minimum Size Tree to Cut
Logger buys cutting rights on 200 acre tract of pine pulpwood for lump sum amount of $40,000. Landowner placed no limits on what logger can cut. Logger wants to cut to maximize net revenue (profit). Should he give cutting crew a minimum size tree to cut? Answer with marginal approach.

23. Calculate Marginal Cost
Min. Cutting Dia. (inches)
Total Volume Deliverd to Mill (cords)
Total Cost
(dollars)
Change in Cost (dollars)
Change in Volume (cords)
Marginal Cost per Cord (dollars) 20
3,300
82,965 18
4,200
92,316
9,360
900
10.40 16
5,000
101,916
9,600
800
12.00 14
5,700
111,856
9,940
700
14.20 12
6,320
122,520
10,664
620
17.20 10
6,920
135,000
12,480
600
20.80 8
7,500
150,000
15,000
580
25.86 6
8,000
165,000
500
30.00 4
8,400
179,400
14,400
400
36.00

24. Compare MC and MB
If price per cord received by logger is $30, then shouldn’t cut any tree less than about 7 inches.
If price per cord increases to $35, then cut down to 5 inches.
If price per cord decreases to $25, then cut down to about 9 inches

25. Minimum diameter (q) for lump sum payment for stumpageTR TC
Stumpage cost
Fixed cost q
Declining cutting diameter

26. Pay as cut contract
Would minimum diameter change if logger paid for stumpage as trees were cut (log scale) instead of for lump sum amount in advance?
Yes, stumpage now a variable cost, not a fixed cost

27. Analytical Tools Discounted cash flow Net present value
Discount or compound all cash flows to same point in time
Calculate using an assumed interest rate
Internal rate of return
Interest rate (i) that makes NPV zero, i.e. equates PV of all costs and all benefits
“Calculate” the interest rate

28. Time line of benefit and cost flows
NPV and IRR
Time line of benefit and cost flows
$ Revenue (R2)
$ Revenue (R8)
C C C C C C C C C8
Cost in each year for 8 years plus “year zero”
All revenues and costs discounted to year zero

29. Formula for NPV for a given interest rate ( i )
- C0 - C1/(1+i)1 - C2/(1+i)2 - C3/(1+i) C8/(1+i)8 +
R2/(1+i)2 + R8/(1+i)8
Simplify by netting R’s and C’s for given year
C0 - C1/(1+i)1 +(R2 -C2)/(1+i)2 - C3/(1+i)
+ (R8 - C8)/(1+i)8

30. NPV Using Summation Notation
NPV = [ (Rt – Ct)/(1+i)t]
t=0
where,
NPV = unknown
n = number of years
Rt = revenue (income) in year t
Ct = cost (expense) in year t
t = index number for years
i = discount rate (alternative rate of return)

31. Internal Rate of Return Using Summation Notation
NPV = [ (Rt – Ct)/(1+i)t]
t=1
where,
NPV = 0
Rt = revenue (income) in year t
Ct = expense (cost) in year t
t = index number for years
i = unknown

32. Finding Internal Rate of Return
Calculate NPV using spreadsheet
Make “i” a variable referenced to one cell
Change “i” in that cell until NPV equals approximately zero