static efficiency, dynamic efficiency and sustainability

Static Efficiency, Dynamic Static Efficiency, Dynamic Efficiency and Efficiency and SustainabilitySustainability

Wednesday, January 25Wednesday, January 25

Represent the demand for a resource as: P = 8 – 0.4 q

Demand = marginal willingness to pay = Marginal Benefit (MB)

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Demand

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Quantity

Demand

$

P = 8 - 0.4qP = 8 - 0.4qqq PP

00 88

55 66

1010 44

1515 22

2020 00

Represent the demand for a resource as: P = 8 – 0.4 q

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Demand

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(5,6)

(15,2)

Assume a constant marginal cost of extraction = $2.00(Marginal cost = supply)

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Demand

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Quantity

MB

$

MC

Efficient allocation occurs where MB = MC, q = 15 units

Static EfficiencyStatic Efficiency

MB = MCMB = MC Criteria for allocation in a given time Criteria for allocation in a given time

period, with no consideration of period, with no consideration of future time periodsfuture time periods

Efficiency: no one can be made Efficiency: no one can be made better off without making someone better off without making someone else worse offelse worse off

$

Q

MC

MB

MB>MC

MC>MB

MB=MC

What are the net benefits of the efficient allocation?

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Demand

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MB

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MC

Efficient allocation occurs where MB = MC, q = 15 units

TB (area under MB curve)=½(6x15) + (2x15)=45+30=75

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MC

MB

Quantity

$

TC (area under MC curve) = (2x15) = 30

NB = TB – TC = ½(6x15) = 45

NB

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Quantity

$

MNB

Total NB (area under MNB curve) = ½(6x15) = 45

This graph illustrates marginal net benefits:

MB-MC = (8-0.4q)-2 = 6-0.4q = MNB

Dynamic EfficiencyDynamic Efficiency

When the concern is efficient When the concern is efficient allocation of a nonrenewable allocation of a nonrenewable resource over multiple time periodsresource over multiple time periods

MNBMNB00 = PV MNB = PV MNB11 = PV MNB = PV MNB22 = … = PV = … = PV MNBMNBtt

t represents time periodt represents time period

With only 20 units of the resource available, what is the present value of total net benefits if effective demand is met in the first period, with no consideration of the second period?

Only two time periods in this Only two time periods in this exampleexample

For present value calculations, r=.10For present value calculations, r=.10

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Quantity

Period t0

MB

MC

$

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Period t0

MB

NB = Area = ½(6x15) = 45

MC

$

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$

Quantity

Period t1

MB

MC

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Quantity

Period t1

MB

NB = Area = ½(2x5) + (4x5) = 25

MC

NB for t0 = $45

Present Value of NB for t1 = 25/(1+r) =

25/1.1= $22.73

PV Total net benefit for two periods =

$45 + $22.73 = $67.73

With only 20 units of the resource available, what is the present value of total net benefits if the resource is allocated equally across two time periods?

(q0 = q1)

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Quantity

Period t0

MB

MC

$

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Period t0

MB

NB = Area = ½(4x10) + (2x10) = 40

MC

$

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Period t1

MB

NB = Area = ½(4x10) + (2x10) = 40

MC

$

NB for t0 = $40

Present Value of NB for t1 = 40/(1+r) =

40/1.1= $36.36

PV Total net benefit for two periods =

= $40 + $36.36 = $76.36

Recall, for dynamic efficiency (to maximize PV of total net benefits), MNB0 = PV MNB1

Find the dynamically efficient quantities for q0 and q1.

Find the efficient allocation of the resource over the two periods (dynamic efficiency).

1)1) MNBMNB00 = PV MNB = PV MNB11

2) MNB = MB - MC 3) MB = 8 – 0.4q4) MB – MC = (8 – 0.4q) – 2 = 6 – 0.4q5) MNB = 6 – 0.4q

1)1) MNBMNB00 = PV MNB = PV MNB11

2) 6 - .4q0 = (6 - .4q1)/1.1

3) q0 + q1 = 20

4) 6 - .4q0 = (6 - .4[20-q0])/1.1

5) 1.1(6 - .4q0)= (6-8+.4q0)

6) 6.6-.44q0 = (-2 +.4q0)

7) 8.6=.84q0

8) q0 = 10.238

9) q1 = 9.762

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Quantity

$

MNB

This graph illustrates marginal net benefits: MB-MC=MNB

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Quantity

$

MNB0

Period t0

MNB = MB – MC = 6 – 0.4q

Period t1

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Quantity

$

PV MNB1

5.45

Present value calculation: 6/1.1 = 5.45

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MNB0

51015

MNB1

5.45

q0=10.238

q1=9.762

t0

t1

MNBMNB00 = 6 – 0.4(10.238) = 1.9048 = 6 – 0.4(10.238) = 1.9048

MNBMNB11 = [6 – 0.4(9.762)]/1.1 = [6 – 0.4(9.762)]/1.1

= 2.9052/1.1 = 1.9048= 2.9052/1.1 = 1.9048

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Quantity

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MNB0

51015

MNB1

5.45

q0

q1

t0

t1

MNB=1.9048

To calculate total benefits, total costs, and net benefits:

P0 = 8 - .4q0

P0 = 8 - .4(10.238)

P0 = 3.905P1 = 8 - .4q1

P1 = 8 - .4(9.762)

P1 = 4.095

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Quantity

Period t0

MB

MC

$

10.238

NB = ½(4.095x10.238) + (1.905x10.238) = 40.46

3.905

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Period t1

MB

MC

9.762

NB = ½(3.905x9.762) + (2.095x9.762) = 39.51

4.095

NB for t0 = $40.46

Present Value of NB for t1 = 39.51/(1+r)

= 39.51/1.1

= $35.92

Total net benefit for two periods = $40.46+35.92

= $76.38

Comparing allocations:Comparing allocations:

Maximize NB to period 0Maximize NB to period 0 TNB = $67.73TNB = $67.73

qq00 = q = q11

TNB = $76.36TNB = $76.36 Dynamically efficient allocationDynamically efficient allocation

TNB = $76.38TNB = $76.38

SustainabilitySustainability Environmental sustainabilityEnvironmental sustainability

Do not reduce total stock of natural capitalDo not reduce total stock of natural capital Strong sustainabilityStrong sustainability

Do not reduce productivity (value) of Do not reduce productivity (value) of natural capital stocknatural capital stock

One type of natural capital may substitute One type of natural capital may substitute for anotherfor another

Weak sustainabilityWeak sustainability Do not reduce productivity of capitalDo not reduce productivity of capital May substitute manufactured capital for May substitute manufactured capital for

natural capitalnatural capital

With equal distribution NB0 = $40 NB1 = $40

With efficient distribution NB0 = $40.46 NB1 = $39.51

With sharing, keep NB0 = $40, invest $.46 @ 10%, send to t1 .46(1.1) = .506 NB1 = $39.51 + .51 = $40.02

Marginal User CostMarginal User Cost MNBMNB00 = 6 – 0.4(10.238) = 1.905 = 6 – 0.4(10.238) = 1.905

MNBMNB11 = [6 – 0.4(9.762)]/1.1 = [6 – 0.4(9.762)]/1.1 = 2.0952/1.1 = 1.905= 2.0952/1.1 = 1.905

The value of the last unit extracted in tThe value of the last unit extracted in t00 Foregone benefit for tForegone benefit for t11

Opportunity cost of choosing to extract the last Opportunity cost of choosing to extract the last unit used in tunit used in t00

P = MEC + MUCP = MEC + MUC $3.905 = $2.00 + 1.905$3.905 = $2.00 + 1.905

User Cost and Natural Resource User Cost and Natural Resource Rent Rent

Quantity

Period t0

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MB

MC

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10.238

3.905

Rent

Wages, etc.

User Cost

P = MEC + MUCP = MEC + MUC $4.095 = $2.00 + 2.095$4.095 = $2.00 + 2.095 MUC increases at the rate of discountMUC increases at the rate of discount 2.095 = 1.1(1.905)2.095 = 1.1(1.905)

Period t1

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$

Quantity

MBMC

9.762

4.095

Rent

User Cost

Reading for Wed. Feb. 2:

Hartwick and Olewiler, on ANGEL

and

Field, Ch. 6