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Chapter 5

Dynamic Efficiency and Sustainable

Development

© 2012 Pearson Education, Inc. All rights reserved. 5-2

Chapter 5: Dynamic Efficiency and Sustainable Development

• Introduction• A Two-Period Model• Defining Intertemporal Fairness• Are Efficient Allocations Fair?• Applying the Sustainability Criterion• Implications for Environmental Policy


Introduction

• Chapter 5 deals with the treatment of future generations.

• Topics covered include– How to allocate resources efficiently over time– The concern of fairness in the allocation of

a resource over time;– The compatibility of equity and efficiency;– Sustainable development.


Introduction

• We have developed two specific means for identifying environmental problems:– Static Efficiency: Evaluate circumstances when time

is not a crucial aspect of the allocation problem.• Solar energy; Wind energy

– Dynamic Efficiency: Suitable for circumstances when time is a crucial aspect of allocation problem.• Depletable resources like oil, minerals etc.


Fair allocation over time

• Our focus is:– How should we allocate the depletable resource

over time?– How can we treat the future generation fairly in

regards to environmental resource use?


A Two-Period Model

• Dynamic efficiency balances present and future uses of a depletable resource by maximizing the PV of net benefits derived from its use.

• This implies a particular allocation of the resource across time. We can investigate the properties of this allocation through the two-period model. (intertemporal decisions)


A Two-Period Model

• The goal of the two-period model is to illustrate how to allocate a depletableresource over two time periods for which total demand is greater than the amount available.

– Let’s first start with a case when demand and supply are equal.


A Two-Period Model

• Assumptions– Fixed supply of certain depletable resource– Consider two time periods only (period 1;

Period 2)– Total supply available is 30 units– Demand (marginal WTP) is constant:


A Two-Period Model

• Efficient Allocation: P=MC [MB = MC]

8 – 0.4q = 2Þ q = 15 units

ÞTotal supply available = 30 unitsÞTherefore, you can use q1 = 15 units in period

1 and q2 =15 units in period 2.


FIGURE 5.1 The Allocation of an Abundant Depletable Resource: (a) Period 1 (b) Period 2


A Two-Period Model

• In the previous example, thirty units is sufficient to cover the demand in both periods.– Therefore, the consumption in period 1 does not

reduce the consumption in period 2.

• This is a static efficiency criterion the allocation are time independent.


• If supply is sufficient to meet demand, then a static efficient solution will provide the optimal allocations over time, regardless of the discount rate.

• But,•What if the total supply is 20 and not 30?

A Two-Period Model


• What happens when the supply is less than 30? Suppose it equals 20 i.e., (D>S). – How do we determine the efficient allocation?

• If supply is not sufficient we must determine the optimal allocation using the dynamic efficiency criterion: maximize the present value of net benefits.

• The present value for a two-period model is the sum of the present values in each of the two years.

A Two-Period Model


• If the total supply is 20,– The first period would have 15 units– The second period would have 5 units

• Is this an efficient allocation? => look at if it maximizes present value of net benefits.

• How do we compute the present value of this allocation?– The present value of net benefits for a two-

period model is equal to the net benefit in Period 1 plus the present value of the net benefit in Period 2.

A Two-Period Model


• The net benefit in each period is the area under the demand curve and above the supply curve up to the allocated quantity (Figure 5.1).

– Net benefit = MB – MC

– The net benefit for Period 2 must be divided by (1+r) to calculate the present value.

A Two-Period Model


A Two-Period Model(We decide to allocate 15 units in the first period and 5 in the second period).

The net benefits will be the area under the MB curve and above the MC curve.



The net benefit in period 2 is A + B in the second figure.



We know the present value of net benefits for the two years is $67.73.But, which allocation maximizes the net benefit?


• The dynamically efficient allocation will satisfy the condition that:– The present value of the marginal net benefit

from the last unit in period 1 equals the present value of the marginal net benefit in period 2.

• A two period model can be illustrated graphically by flipping the graph of period 2 such that the zero axis for the period 2 net benefits is on the right side, rather than the left. – The size of the box represents the resource constraint. – Any point on the horizontal axis sums to the amount of

the resource constraint (=20 units).

A Two-Period Model


FIGURE 5.2 The Dynamically Efficient Allocation


FIGURE 5.2 The Dynamically Efficient Allocation

• The optimal allocation will be the intersection of the Period 1 marginal net benefit curve with the curve representing the present value of marginal net benefit in Period 2.

• On the left hand side are the marginal net benefits for the first time period

• On the right hand side are the marginal benefits for the second period graphed in present value terms. It is lower because of the discount rate.


The Dynamically Efficient Allocation(period 1)


The Dynamically Efficient Allocation(period 2)

Key Idea: The first time period is not discounted because it is the current period. The second time period is discounted.


The Dynamically Efficient Allocation

• Period 1 marginal net benefit (MNB1): • 8 – 0.4q1 – 2 => 6 – 0.4q1

• Period 2 marginal net benefit (MNB2):• (8 – 0.4q2 – 2)/(1.1) => 5.45 – 0.36q2

• Optimal allocation is found by setting MNB1 = PVMNB2 & the resource amount constraint (q1+q2=20)• 6 – 0.4q1 = 5.45 – 0.36q2

Or, 6 – 0.4q1 = 5.45 – 0.36 (20 - q1)Solving for q1 gives q1* = 10.2. Plugging q1 into (q1+q2=20) equation

given q2* = 9.8.


The dynamic efficient point



• What is the present value of net benefit in the dynamic efficient allocation?– Is it greater than $67.73 that we found earlier?

• It should be.

• Net benefit:– Period 1 = green area in the last page figure– Period 2 = blue area in the last page figure



• Present Value of Net benefit:– Period 1:

(1.905*10.238)+[0.5*(4.095)*(10.238)]

– Period 2: (2.095*9.762)+[0.5*(3.905)*(9.762)]


What happens if we increase the discount rate?(r = 10%)


What happens if we increase the discount rate?(r = 20%)


What happens if we increase the discount rate?

• A higher discount rate will favor the present. The amount allocated to the second period falls as the discount rate rises.

• The larger the discount rate, the greater the amount of rotation required.

• Higher discount rates tend to skew resource extraction towards the present because they give the future less weight.


• With any efficient allocation, you must take scarcity into account. When resources are scarce, greater current use diminishes future opportunities.

• The opportunity cost caused by intertemporal scarcity is called the marginal user cost (MUC).– The marginal user cost is the present value of these

forgone opportunities at the margin. – In the absence of scarcity, the marginal user cost is zero.– When there are 30 or more units available in our previous

example, both periods would get 15 units each and there would be no marginal user cost.

A Two-Period Model:Marginal User Cost


FIGURE 5.3 The Efficient Market Allocation of a Depletable Resource: The Constant-Marginal-Cost Case: (a) Period 1 and (b) Period 2

• To calculate MUC, substitue optimal q1* = 10.2 and q2* = 9.8 into p.

– P1 = 8 – 0.4q1 = $3.905– P2 = 8 - 0.4q2 = $4.095

– MUC1 = P1 – MC = 3.905 -2 =$1.905

– MUC2 = P2 – MC = 4.905-2 = $2.095

• At the static allocation, price = $2 in both period and quantity = $15.

• With scarcity, price increases in 2nd period as scarcity rises.


Marginal User Cost

• No Scarcity:P = marginal cost (or marginal extraction cost)

• With Scarcity:P = marginal extraction cost + MUC

An efficient market would have to consider not only the marginal cost of extraction for this resource but also the marginal user cost.

Marginal user cost will rise over time at the rate of discount causing prices to rise over time.


Defining Intertemporal Fairness

• How much should we leave for future generations? What is the appropriate rate of discount?

– A Theory of Justice by John Rawls

• One possibility:• Sustainability criterion—at a minimum, future

generations should be left no worse off than current generations.– Allocation that impoverish future generation in order to

enrich current generations are unfair.


Are Efficient Allocations Fair?

• A dynamic efficient allocation will not automatically satisfy the sustainability criterion, but can be consistent with sustainability.

– With a discount rate greater than zero, an economically efficient allocation will allocate more of a resource to the first period than the second. • Net benefits will be greater in the first period than the

second. (sustainable criterion fails here)

– The sustainability criterion can still be met if the first period sets aside sufficient net benefits for the second period.


Period 1 Period 2

4.09523.9048

10.238 9.762

• NB1 = $40.466• NB2 = $39.512


Are Efficient Allocations Fair?

– Suppose the first generation spends $40 and saves $0.46 for future.

– At 10% interest rate, this would grow to• [(0.466)*(0.10)] + (0.466) = $0.513• $39.51+0.513 = $40.025

– Future better of accepting the dynamically efficient allocation.

– Alaska Permanent Fund (Alaska pipeline construction): Alaska voters approved a constitutional amendment that authorized the establishment of a dedicated fund.

• >= 25% of the rent from state’s oil is invested in capital markets to be shared with future generation.

• As of 2012: $42.12 billion


Applying the Sustainability Criterion

• It is very difficult to implement because it requires to know the preferences of the future generation.

• A more operational version of sustainablity criterion is “Hartwick’s Rule”. (Weak Sustainability)– A constant level of consumption can be maintained perpetually

from an environmental endowment if all scarcity rent is invested in capital.

– If all scarcity rent is invested in capital, the value of the total capital stock will not decline.

– If the principal or the value of total capital is declining, the allocation is not sustainable.



• Example,– Receive inheritance of $10,000– Put it in bank with 10% interest => $1,000– If you spend exactly $1000 per year, than the

principle will never decline.– Spending $1,000 (from interest) is sustainable.



• Current generation has been given an endowment (total capital)– Total capital = natural + physical.– Sustainable use implies we keep the principle

(endowment) and live off of the flow of service provided.– Ensure that the value of the total capital stock is

maintained.

• Example: Government Pension Fund of Norway. ASA invested its surplus profit from petroleum into a portfolio now worth over $325 billion.– The fund allows for long-lasting income for the population in exchange

for a finite resource, actually increasing the total capital available for Norway above the original levels.


• Weak sustainability—the maintenance of total capital.

• Strong sustainability—the maintenance of the value of the stock of natural capital– Strong sustainability assumes that there is little or no

substitution between physical and natural capital and emphasizes preserving natural capital as opposed to total capital. Again, the focus is on preserving value and on preserving an aggregate of natural capital.



Implications for Environmental Policy

• Efficiency = concerned with eliminating waste in the use of resource. (Maximize net benefits to the society).

• Sustainability = fair treatment to all parties. (fair allocation of resources across all generations).– WCED (1987): “Development is sustainable when it

meets the needs of the present without compromising the ability of future generations to meet theirs”.


Implications for Environmental Policy

• Not all efficient allocations are sustainable and not all sustainable allocations are efficient.

• Market allocations may be either efficient or inefficient and either sustainable or unsustainable.

• Given a number of sustainable allocation possibilities, choose the one that maximizes either static or dynamic efficiency (net benefits or the present value of net benefits), e.g., maximize wealth, subject to a sustainability criterion.

• Policy changes that can produce win-win situations because by correcting an inefficiency, net benefits are increased.

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