Static Efficiency, Dynamic
Efficiency and Sustainability
Wednesday, January 25
Represent the demand for a resource as:
P = 8 – 0.4 q
$
9
8
7
6
5
Demand
Demand
4
3
2
1
0
0
5
10
15
Quantity
20
Demand = marginal willingness to pay = Marginal Benefit (MB)
P = 8 - 0.4q
q
P
0
8
5
6
10
4
15
2
20
0
Represent the demand for a resource as:
P = 8 – 0.4 q
$
9
8
7
(5,6)
6
5
Demand
Demand
4
3
(15,2)
2
1
0
0
5
10
15
Quantity
20
Assume a constant marginal cost of
extraction = $2.00 (Marginal cost = supply)
$
9
8
7
6
5
MB
DemandMC
4
3
2
1
0
0
5
10
15
Quantity
20
Efficient allocation occurs where MB = MC, q = 15 units
Static Efficiency



MB = MC
Criteria for allocation in a given time
period, with no consideration of future
time periods
Efficiency: no one can be made better off
without making someone else worse off
$
MC
MB>MC
MC>MB
MB
MB=MC
Q
What are the net benefits of the efficient
allocation?
$
9
8
7
6
5
MB
DemandMC
4
3
2
1
0
0
5
10
15
Quantity
20
Efficient allocation occurs where MB = MC, q = 15 units
TB (area under MB curve)=½(6x15) + (2x15)=45+30=75
TC (area under MC curve) = (2x15) = 30
$
9
8
NB
7
6
5
MB
4
3
MC
2
1
0
0
5
10
15
Quantity
NB = TB – TC = ½(6x15) = 45
20
This graph illustrates marginal net benefits:
MB-MC = (8-0.4q)-2 = 6-0.4q = MNB
$
7
6
5
4
MNB
3
2
1
0
0
5
10
15
Quantity
Total NB (area under MNB curve) = ½(6x15) = 45
Dynamic Efficiency


When the concern is efficient allocation of
a nonrenewable resource over multiple
time periods
MNB0 = PV MNB1 = PV MNB2 = … = PV MNBt

t 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 example
For present value calculations, r=.10
Period t0
$
9
8
7
6
MB
5
4
3
MC
2
1
0
0
5
10
15
Quantity
20
Period t0
$
9
8
7
6
MB
5
4
3
MC
2
1
0
0
5
10
15
Quantity
NB = Area = ½(6x15) = 45
20
Period t1
$
9
8
7
6
5
MB
4
MC
3
2
1
0
0
5
10
15
Quantity
20
Period t1
$
9
8
7
6
5
MB
4
MC
3
2
1
0
0
5
10
15
Quantity
NB = Area = ½(2x5) + (4x5) = 25
20
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)
Period t0
$
9
8
7
6
5
MB
4
3
MC
2
1
0
0
5
10
15
Quantity
20
Period t0
$
9
8
7
6
5
MB
4
3
MC
2
1
0
0
5
10
15
Quantity
NB = Area = ½(4x10) + (2x10) = 40
20
Period t1
$
9
8
7
6
5
MB
4
3
MC
2
1
0
0
5
10
15
Quantity
NB = Area = ½(4x10) + (2x10) = 40
20
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
Find the efficient allocation of the resource
over the two periods (dynamic efficiency).
Find the dynamically efficient
quantities for q0 and q1.
Recall, for dynamic efficiency (to maximize PV
of total net benefits), MNB0 = PV MNB1
1)
2)
3)
4)
5)
MNB0 = PV MNB1
MNB = MB - MC
MB = 8 – 0.4q
MB – MC = (8 – 0.4q) – 2 = 6 – 0.4q
MNB = 6 – 0.4q
1)
2)
3)
4)
5)
6)
7)
8)
9)
MNB0 = PV MNB1
6 - .4q0 = (6 - .4q1)/1.1
q0 + q1 = 20
6 - .4q0 = (6 - .4[20-q0])/1.1
1.1(6 - .4q0)= (6-8+.4q0)
6.6-.44q0 = (-2 +.4q0)
8.6=.84q0
q0 = 10.238
q1 = 9.762
This graph illustrates marginal net benefits: MB-MC=MNB
$
7
6
5
4
MNB
3
2
1
0
0
5
10
15
Quantity
Period t0
$
7
6
5
4
MNB0
3
2
1
0
0
5
10
15
Quantity
MNB = MB – MC = 6 – 0.4q
Period t1
$
7
6
5.45
5
4
PV MNB1
3
2
1
0
0
5
10
15
Quantity
Present value calculation: 6/1.1 = 5.45
$
7
6
5.45
5
MNB0
4
MNB1
3
2
1
t0
0
5
10
15
15
10
5
Quantity
q0=10.238
q1=9.762
0
t1


MNB0 = 6 – 0.4(10.238) = 1.9048
MNB1 = [6 – 0.4(9.762)]/1.1
= 2.9052/1.1 = 1.9048
$
7
6
5.45
5
MNB0
4
MNB1
3
MNB=1.9048
2
1
t0
0
5
10
15
15
10
5
q0
Quantity
q1
0
t1
To calculate total benefits, total costs, and
net benefits:
P0 = 8 - .4q0
P0 = 8 - .4(10.238)
P0 = 3.905
P1 = 8 - .4q1
P1 = 8 - .4(9.762)
P1 = 4.095
Period t0
$
9
8
7
6
MB
5
3.905
4
3
MC
2
1
0
0
5
10
15
10.238
Quantity
NB = ½(4.095x10.238) + (1.905x10.238) = 40.46
20
Period t1
$
9
8
7
6
5
4.095
MB
4
MC
3
2
1
0
0
5
10
9.762
15
Quantity
NB = ½(3.905x9.762) + (2.095x9.762) = 39.51
20
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:

Maximize NB to period 0


q 0 = q1


TNB = $67.73
TNB = $76.36
Dynamically efficient allocation

TNB = $76.38
Sustainability

Environmental sustainability


Strong sustainability



Do not reduce total stock of natural capital
Do not reduce productivity (value) of natural
capital stock
One type of natural capital may substitute for
another
Weak sustainability


Do not reduce productivity of capital
May substitute manufactured capital for natural
capital

With equal distribution



With efficient distribution



NB0 = $40
NB1 = $40
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 Cost



MNB0 = 6 – 0.4(10.238) = 1.905
MNB1 = [6 – 0.4(9.762)]/1.1
= 2.0952/1.1 = 1.905
The value of the last unit extracted in t0


Foregone benefit for t1
Opportunity cost of choosing to extract the last unit
used in t0
User Cost and Natural Resource Rent


P = MEC + MUC
$3.905 = $2.00 + 1.905
$
Period t0
9
8
Rent
7
6
MB
5
Wages, etc.
3.905
4
User Cost
3
MC
2
1
0
0
5
10
15
20
Quantity
10.238




P = MEC + MUC
$4.095 = $2.00 + 2.095
MUC increases at the rate of discount
2.095 = 1.1(1.905)
$
Rent
Period t1
9
8
7
6
5
4.095 4
MB
User Cost
3
MC
2
1
0
0
5
10
9.762
15
Quantity
20
Reading for Wed. Feb. 2:
Hartwick and Olewiler, on ANGEL
and
Field, Ch. 6