3. Value^J Capital and Interest
Value, Capital and Interest Cost-Benefit Approach
Units
Importance of a common unit for evaluation
Measures attributes in a systematic way
Allows comparison of effects at different times
Unit Defined: A choice of numeraire that standardizes evaluations
Common Numeraire
Examples of possible numeraires: chocolates, land (ecological footprint)
Properties of a numeraire:
Maintains same relative prices at a point in time
Allows calculation of changes in values over time
Money is the most convenient choice despite challenges
Present Value
Necessary concept when comparing climate damage over time
Allows examination of values at different points in time
Enables comparison of future cash flows to present worth
Present Value & Weights
Value in numeraire at time t is denoted as vt
Comparison factors at time t are denoted as wt
Total value across different times calculated as:
V0 = Σt wt vt
This formula aggregates the weighted present values
Comparisons
Weights wt help convert values at time t to present value
Units are expressed as (numeraire at t)/(numeraire at reference point)
Typically uses present as the reference point (t=0)
Conditions:
wt tends to decrease over time
Formula: wt+1 wt = 1/(1+r) where r > 0 is constant
Thus, wt = w0/(1 + r)^t
Growth
Investment at rate r over t years grows as:
1 + rt for an initial investment of 1
For any invested value v:
Grows to v(1 + rt) at time t
Present value of an expected future value:
v/(1 + rt) at time 0
The Power of Compound Interest
For typical values of r, growth occurs rapidly
Example: Over 7 years at 10%, $1 grows to approximately $2
Long-term investment in stock market greatly increases wealth
Explains that understanding compound interest is crucial for financial success
A Lesson
Quote: “Those who understand compound interest are destined to earn it…”
Important lesson: Save and invest to earn r, irrespective of income level
Why is r Positive?
Three reasons based on theories from Eugen von Böhm-Bawerk:
Positive time preference—value of present consumption > future
Expected future wealth due to economic growth
Natural productivity of resources and savings
Rate of interest on savings and return on capital reflect these preferences
Ramsey’s Rule for Optimal Growth
Maximizes sequence of utility functions: σt=1∞ u(ct)/(1+ρ)^t
Relation to capital and production: r = f′(k) = ρ + ηg
Consumption rate of interest is tied to this outcome
Implicit Interest in Decisions
Economic decisions are influenced by implicit interest even among simple cases (e.g., Robinson Crusoe analogy)
N.B.
Utilizes a rate of interest on utility, termed rate of pure time preference
Emphasizes utility is cardinal, even if not directly observable
Money is the only observable interest and is defined by market prices
Level of r
Historical perspective on understanding the interest rate
Short to medium-term rates known from market observations
Long-term rates need theoretical assumptions and are critical for evaluating important issues like climate change
Infinite Series
Investment of V at rate r in an account produces continuous cash flow
Value of a continuous cash stream is calculated as:
V = Σt=1∞ (rV)/(1+r)^t
Value of a Perpetuity
Payment of 1 each year yields a value determined as:
V = 1/r for a constant stream of payments
The behavior of partial sums converges towards this perpetual formula
Why Bother?
Highlights the dynamic nature of financial evaluations over time
Importance in addressing long-term issues like climate change
Nicholas Stern vs William Nordhaus on the discount rate impacts decisions
Substitution of Investments at Margin
Securities such as bonds and stocks impact investment decisions based on returns
Inflation affects desirability and pricing of securities and commodities
Example 1: Housing Market
Connection between interest rates (r) and house pricing dynamics
Implications of sunk investments when r changes
How the Market “Works”
Buyers adjust budgets according to market interest rates
Competition and bidding behavior reflects these underlying economic principles
Example 2: Lottery Winnings
Comparison of immediate lump sum versus annuity illustrates present vs future value consideration
Complexity lies in variables like taxes and investment potential
Her Choice
Evaluating the implications of both choices requires accounting for various factors
Investments in the stock market as a potential alternative to consuming immediately
Wealth
Defined as the present value of future consumption growth
Relates to growth rates relevant to economic modeling
If growth rate g exceeds the rate of return r, wealth becomes infinite
The “One Per Cent”
Piketty's findings on interest rates exceeding growth rates reflect wealth distribution trends
Savers vs. Spenders
Economic sustainability relates to saving behaviors and investments based on optimal growth equations
Different economic systems yield different outcomes based on saving tendencies
Accumulation
Piketty's perspective on interest rates indicating overall secular stagnation versus growth fluctuations
Conclusion on Wealth Management
Quote from Shakespeare underscores the importance of proactive financial planning
Engaging with savings and investments lead to improved long-term outcomes
Dynamic Decisions
Discussions of discounted cash flow vs internal rate of return for investment evaluations
Emphasis on using DCF as a recommended approach
Uncertainty
Risk aversion influences choices between fixed interest yielding bonds and variable returns
Application of finance theory like CAPM in estimating appropriate returns
Appendix: Arithmetic
Basics of powers and roots defined mathematically for any positive value
More Arithmetic
Properties of multiplication of bases with varying powers clarified
Yet More
Convergence behavior of functions and definitions for specialized cases defined for clarity.