Unformatted text preview: EEP 101/ECON 125
EEP 101/ECON 125
Lecture 14: Natural Resources
Deepak Rajagopal
Energy Biosciences Institute
UC Berkeley Natural Resources
Natural Resources • We distinguish between nonrenewable resources and renewable resources. • Nonrenewable resources: mineral deposits – gold, copper, coal, oil • Renewable resources: Flora (species of trees etc.) and Fauna (species of fish etc.) are examples of renewable resources, since they can be selfreplenishing. • What about water? Arctic ice sheet? Old growth redwoods? Natural Resource Economics
Natural Resource Economics
• Natural Resource Economics addresses the allocation of resources over time. – How much oil should we extract today vs. how much should we extract tomorrow?
– How much water should we release from the dam? – How much fish should we catch per month? Intertemporal Resource Intertemporal Resource Allocation – TwoPeriod Case
Period 1 Period 2 P P 8 8 MC 2
15
Q MC 2 20 15
Q Total extractable stock = 35
What if total extractable stock = 20? 20 Market Allocation of Natural Market Allocation of Natural Resources • Market allocation may lead to depletion of nonrenewable natural resources too quickly or too slowly. • or cause renewable resource use to not be sustainable over time (such as when species extinction occurs or a ground water aquifer is depleted). Natural Resource Economics (Cont.)
Natural Resource Economics (Cont.)
• Natural resource economics suggests policy intervention in situations where markets fail to maximize social welfare over time. • Natural resource economics also investigates how natural resources are allocated under alternative economic institutions. Key Elements of Dynamics: Key Elements of Dynamics: Interest Rate
• One of the basic assumptions of dynamic analysis is that individuals are impatient. • They would like to consume the goods and services that they own today, rather than saving for the future or lending to another individual. • Individuals will lend their goods and services to others only if they are compensated for delaying their own consumption. The Interest Rate
The Interest Rate
• The interest rate (often called the discount rate in resource contexts) is the fraction of the value of a borrowed resource paid by the borrower to the lender to induce the lender to delay her own consumption in order to make the loan. • The interest rate is the result of negotiation between the lender and the borrower. • The higher the desire of the lender to consume her resources today rather than to wait, and/or the higher the desire of the borrower to get the loan, the higher the resulting interest rate. • In this sense, the interest rate is an equilibrium outcome, like the price level in a competitive market. Example
Example
• Suppose X owns a resource and would like to consume the resource today. • Y would like to borrow X’s resource for one year. • X agrees to loan John the resource for one year if Y will pay X an amount to compensate her for the cost of delaying consumption for one year. • The amount loaned is called the principal. • The payment from John to Mary in compensation for Mary's delayed consumption is called the interest on the loan. Example (Cont.)
Example (Cont.)
• Suppose X’s resource is $100 in cash. • Suppose the interest amount agreed to by X and Y is $10. • Then, at the end of the year of the loan, Y repays X the principal plus the interest, or $110:
Principal + Interest = $100 + $10 = $110 Example (Cont.)
Example (Cont.)
• The (simple) interest rate of the loan, denoted r, can be found by solving the following equation for r:
– Principal + Interest = (1 + r) Principal • For this example: $110 = (1 + r) $100 • So, we find: r = 10/100 or 10% • Hence, the interest rate on the loan was 10%. Example (Cont.)
Example (Cont.)
• Generally, we can find the interest rate by noting that:
B1 = B0 + r B0 = (1 + r) B0 • where B0 = Benefit today, and B1 = benefit tomorrow The Interest Rate is an Equilibrium of Outcome
The Interest Rate is an Equilibrium of Outcome •
• C1 = consumption in period 1
C2 = consumption in period 2 The Interest Rate is an Equilibrium of The Interest Rate is an Equilibrium of Outcome (Cont.) • Delay of consumption (saving) in period 1 reduces current utility but increases utility in period 2. • The intertemporal production possibilities curve (IPP) denotes the technological possibilities for tradingoff present vs. future consumption. • The curve S, is an indifference curve showing individual preferences between consumption today and consumption in the future. • Any point along a particular indifference curve leads to the same level of utility. • Utility maximization occurs at point A, where S is tangent to the IPP. • The interest rate, r, that is implied by this equilibrium outcome, can be found by solving either of the following two equations for r:
– slope of S at point A = (1 + r)
– slope of IPP at point A = (1 + r) Consumption
Consumption
• Even an isolated individual must decide how much of his resources to consume today and how much to save for consumption in the future. • In this situation, a single individual acts as both the lender and the borrower. • The choices made by the individual reflect the individual's implicit interest rate of trading off consumption today for consumption tomorrow. The Interest Rate is an Equilibrium of The Interest Rate is an Equilibrium of Outcome (Cont.)
• Therefore, if we can determine the slope of either S or IPP at tangency point A, then we can calculate the interest rate, r. This is often done by solving the following individual optimization problem where I is the total income available over the two periods: max { (C1, C 2)}
U
æ1 ö
subject to C1 + ç
÷C 2 £ I
è1 + r ø The Interest Rate is an Equilibrium of The Interest Rate is an Equilibrium of Outcome (Cont.) • which can be written as: 1 L = Υ (Χ1, Χ 2) + λ Ι − Χ1 − Χ2 1+ ρ ΦΟΧΣ : Υ Χ1 = λ Υ Χ1 = 1+ ρ
λ⇒
ΥΧ 2
ΥΧ 2 = 1 + ρ The Indifference Curve
The Indifference Curve
• The indifference curve is found by setting: UC 1 dC 1 + UC 2 dC 2 = 0 Þ dC 2  UC 1
=
=  (1 + r)
dC 1 UC 2 • The indifference curve simply indicates that the equilibrium occurs where an individual cannot improve her intertemporal utility at the margin by changing the amount consumed today and tomorrow, within the constraints of her budget. The Components of Interest Rate
The Components of Interest Rate
• Interest rates can be decomposed into several elements:
• Real interest rate r • Rate of inflation IR
• Transaction costs TC
• Risk factor SR • The interest rate that banks pay to the government (i.e., to the Federal Reserve) is the sum r + IR. – This is the nominal interest rate. • The interest rate that lowrisk firms pay tobanks is the sum r + IR + TC + SR, where TC and SR are minimum transactions costs and risk costs, respectively. – This interest rate is called the prime rate. The Components of Interest Rate The Components of Interest Rate (Cont.)
• Lenders (banks) analyze projects proposed by entrepreneurs before financing them. • They do this to assess the riskiness of the projects and to determine SR. • Creditrating services and other devices are used by lenders (and borrowers) to lower TC. Some Numerical Examples
Some Numerical Examples
(1) If the real interest rate is 3% and the inflation rate is 4%, then the nominal interest rate is 7%.
(2) If the real interest rate is 3%, the inflation rate is 4% and TC and SR are each 1%, then the prime rate is 9%. $ Today vs. $ Future
$ Today vs. $ Future
For an individual, say: $ 1000 today ~ $ 1000 (1+δ) a year from now
δ – personal discount rate Discounting
Discounting
• Discounting is a mechanism used to compare streams of net benefits generated by alternative allocations of resources over time. • There are two types of discounting, depending on how time is measured. • If time is measured as a discrete variable (say, in days, months or years), discretetime discounting formulas are used, and the appropriate real interest rate is the "simple real interest rate". • If time is measured as a continuous variable, then continuoustime formulas are used, and the appropriate real interest rate is the "instantaneous real interest rate". • We will use discretetime discounting in this course. • Hence, we will use discretetime discounting formulas, and the real interest rate we refer to is the simple real interest rate, r. • Unless stated otherwise, assume that r represents the simple real interest rate. Lender’s Perspective
Lender’s Perspective
• From a lender's perspective, $10 received at the beginning of the current time period is worth more than $10 received at the beginning of the next time period. • That's because the lender could lend the $10 received today to someone else and earn interest during the current time period. • In fact, $10 received at the beginning of the current time period would be worth $10(1 + r) at the beginning of the next period, where r is the interest rate that the lender could earn on a loan. A Different Perspective & A Different Perspective & Discounting (Cont.)
• Viewed from a different perspective, if $10 were received at the beginning of the next time period, it would be equivalent to receiving only $10/(1 + r) at the beginning of the current time period. • The value of $10 received in the next time period is discounted by multiplying it by 1/(1+r). • Discounting is a central concept in natural resource economics. • So, if $10 received at the beginning of the next period is only worth $10/(1 + r) at the beginning of the current period, how much is $10 received two periods from now worth? 2 Present Value
Present Value
• In general, the value today of $B received t periods from now is $B/(1 t
+ r) . • The value today of an amount received in the future is called the present value of the amount. • The concept of present value applies to amounts paid in the future as well as to amounts received. • For example, the value today of $C paid t periods from now is $C/(1 + t
r) . • Note that if the interest rate increases, the value today of anamount received in the future declines. • Similarly, if the interest rate increases, then the value today of an amount paid in the future declines. You Win the Lottery!
You Win the Lottery!
• You are awarded aftertax income of $1M. However, this is not handed to you all at once, but at $100K/year for 10 years. If the interest rate is, r = 10%, net present value:
• NPV = 100K+(1/1.1)100K+(1/1.1)2100K + (1/1.1)3100K + … + (1/1.1)9100K.
= $675,900
• The value of the last payment received is: NPV = (1/1.1)9100K = $42,410. • That is, if you are able to invest money at r = 10%, you would be indifferent between receiving the flow of $1M over 10 years and $675,900 today or between receiving a one time payment of $100K 10 years from now and $42,410 today. The Present Value of an The Present Value of an Annuity
• An annuity is a type of financial property (in the same way that stocks and bonds are financial property) that specifies that some individual or firm will pay the owner of the annuity a specified amount of money at each time period in the future, forever! • Although it may seem as if the holder of an annuity will receive an infinite amount of money, the Present Value of the stream of payments received over time is actually finite. • In fact, it is equal to the periodic payment divided by the interest rate r (this is the sum of an infinite geometric series). Annuity (Cont.)
Annuity (Cont.)
• Let’s consider an example where you own an annuity that specifies that Megafirm will pay you $1000 per year forever. • Question: What is the present value of the annuity? • We know that NPV = $1000/r. Suppose r = 0.1 then the present value of your annuity is $1000/0.1 = $10,000. • That is a lot of money, but far less than an infinite amount. • Notice that if r decreases, then the present value of the annuity increases. • Similarly, if r increases, then the present value of the annuity decreases. • For example, you can show that a 50% decline in the interest rate will double the value of an annuity. Transition from Flow to Stock
Transition from Flow to Stock
• If a resource is generating $20,000/year for the foreseeable future and the discount rate is 4%, the price of the resource should be $500,000 • If a resource generates $24K annually and is sold for $720K, the implied discount rate is 24/720 = 1/30 = 3.333%. The Impact of Price Expectation
The Impact of Price Expectation
• If the real price of the resource (oil) is expected to go up by 2% and the real discount rate is 4% • What is the value of an oil well which provides for the for seeable 5000 barrel annually, and each barrel earns 30$ (assume zero extraction costs)? – 1. Is It (A) $3.750K (B) $7.500K ?
– 2. If the discount rate is 7%, will you pay $2 millions for the well?
– 3. What is your answer to 1. If inflation is 1%? Answers
Answers
5000 * 30 5000 * 30 * (1 + i) 5000 * 30 * (1 + i) 2
+
+
+ ...
2
3
1+ r
(1 + r)
(1 + r)
5000 * 30
=
( r  i)
5000 * 30
= 7, 500, 000
0.02
5000 * 30
2)
= 3, 000, 000 > 200, 000 Ans : yes
0.05
5000 * 30
3)
= 5, 000, 000
0.03 1) Uncertainty and Interest Rates
Uncertainty and Interest Rates
• Lenders face the risk that borrowers may go bankrupt and not be able to repay the loan. To manage this risk, lenders may take several types of actions:
• Limit the size of loans.
• Demand collateral or cosigners.
• Charge highrisk borrowers higher interest rates. (Alternatively, different institutions are used to provide loans of varying degrees of risk.) RiskYield Tradeoffs
RiskYield Tradeoffs
• Investments vary in their degree of risk. • Generally, higher risk investments also tend to entail higher expected benefits (i.e., high yields). • If they did not, no one would invest money in the higher risk investments. • For this reason, lenders often charge higher interest rates on loans to highrisk borrowers, while large, low
risk, firms can borrow at the prime rate. Criteria for Evaluating Alternative Criteria for Evaluating Alternative Allocations of Resources Over Time • Net present value (NPV) is the sum of the present values of the net benefits accruing from an investment or project. • Net benefit in time period t is Bt Ct, where Bt is the total benefit in time period t and Ct is the Total Cost in time period t.
– The discrete time formula for N time periods with Ν
constant r:
(Βτ − Χτ )
NPV = ∑
τ.
τ= 0 (1 + ρ) NFV and IRR
NFV and IRR • Net future value (NFV) is the sum of compounded differences between project benefits and project costs. – The discrete time formula for N time periods with Ν
constant r:
NFV = ∑ (Βτ − Χτ ) ⋅ (1 + ρ)Ν − τ
τ= 0 • Internal rate of return (IRR) is the interest rate that is associated with zero net present value of a project. IRR is the x that solves the equation:
Ν
(Βτ − Χτ )
0= ∑
τ
τ= 0 (1 + ξ) The Relationship Between IRR and The Relationship Between IRR and NPV
• If r < IRR, then the project has a positive NPV • If r > IRR, then the project has a negative NPV • It is not worthwhile to invest in a project if you can get a better rate of return on an alternate investment. Familiarizing Ourselves with the Familiarizing Ourselves with the Previous Concept
• Two period model: If we invest $I today, and receive $B next year in returns on this investment, the NPV of the investment is: $I + $B/(1 + r). Notice that the NPV declines as the interest rate r increases, and vice versa. • Suppose you are considering an investment which costs you $100 now but which will pay you $150 next year.
–
–
– If r = 10%, then the NPV is: 100 + 150/1.1 = $36.36 If r = 20%, then the NPV is: 100 + 150/1.2 = $25
If r = 50%, then the NPV is: 100 + 150/1.5 = $0 BenefitCost Analysis
BenefitCost Analysis
• Benefitcost analysis is a pragmatic method of economic decisionmaking. The procedure consists of the following two steps: • Step 1: Estimate the economic impacts (costs and benefits) that will occur in the current time period and in each future time period. • Step 2: Use interest rate to compute net present value or compute internal rate of return of the project/investment. – Use internal rate of return only in cases in which net benefits switches sign once, meaning that investment costs occur first and future cash of flows results only in net benefits. Intertemporal Resource Intertemporal Resource allocation – TwoPeriod Case
Period 1 Period 2 P P 8 8 MC 2
15
Q MC 2 20 15 20 Q Total extractable stock = 20. How much should be consumed in each period if discount rate is 10% Dynamically Efficient allocation
Dynamically Efficient allocation
NB2 (q2 )
max NB1 (q1 ) +
, such that q1 + q2 = Q
q1 , q 2
1+ r
where, NBt = Benefitst  Costst
NB2 (q2 )
L = NB1 (q1) +
+ l (Q  q1  q2 )
1+ r
¶NB1
1 ¶NB2
FOC :
= l ... (1) &
= l ... (2)
¶q1
1 + r ¶q2
¶NB1
1 ¶NB2
Equating 1 and 2 Þ
=
¶q1 1 + r ¶q2
In other words, for dynamically efficient allocation, PV of marginal net benefit of consumption is equal across periods Example
Example
Discount rate =10%
P
6 5.45
PV of marginal net benefit in period 2 in $ Marginal net benefit in period 1 in $ 0 5 10 20 15 10 15 20 5
0
10.238, 9.762 Example
Example Discount rate =10% Period 1 Period 2 P P 8 8
4.095 3.905
MC 2
10.238
Q 20 MC 2
9.762
Q 20 Unlike static efficiency when P = MC, P > MC in every period under dynamically efficient allocation. P MC is referred to as the marginal user cost or scarcity rent. Are Efficient Allocations Fair?
Are Efficient Allocations Fair? Allocation with Sharing Across Allocation with Sharing Across Generations
• If the generations living in the two periods did • • not share, then the secondgeneration loses whenever the firstgeneration discounts the future.
If the first generation consumes only 40 units and saves 0.465 which is invested at a rate of interest equal to discount rate, say 10%, the second generation receives $39.51 + 1.1*0.465 = $40.025.
But what about resources which cannot be replaced? BenefitCost Analysis (Cont.)
BenefitCost Analysis (Cont.)
• A key assumption of benefitcost analysis is the notion of potential welfare improvement. That is, a project with a positive NPV has the potential to improve welfare, because utility rises with NPV. • Some issues in benefitcost analysis to consider include:
– How discount rates affect outcomes of benefitcost analysis. – When discount rates are low, more investments are likely to be justified. – Accounting for public rate of discount vs. private rate of discount.
– Incorporating nonmarket environmental benefits in benefitcost analysis.
– Incorporating price changes because of market interaction in benefitcost analysis.
– Incorporating uncertainty considerations in benefitcost analysis. The Social Discount Rate
The Social Discount Rate
• The social discount rate is the interest rate used to make decisions regarding public projects. It may be different from the prevailing interest rate in the private market. Some reasons are:
• Differences between private and public risk preferences — the public overall may be less risk averse than a particular individual due to pooling of individual risk.
• Externalities: In private choices we consider only benefits to the individuals; in public choices we consider benefits to everyone in society. • It is argued that the social discount rate is lower than the private discount rate. In evaluating public projects, the lower social discount rate should be used when it is appropriate. Next Time
Next Time
• Allocation under different institutional settings and the role of policies • Application to specific resources ...
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 Spring '09
 ZELBERMAN
 Interest Rates, BFC Siófok

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