Transcript Economics 1100

Ch. 5 Dynamic Efficiency and
Sustainable Development
(Fall 2011)
Introduction
• First, we will develop conceptual framework for
dynamic efficiency.
• Efficiency is not the only criterion.
• Fairness or justice is another criterion. We will
examine the relationship of dynamic efficiency
and fairness.
Dynamic Efficiency
• Time is often a very important factor affecting
the use of resources. e.g. pumping
exhaustible groundwater now implies less will
be available in the future; pollution can
accumulate over time.
• How do we compare the net benefit in one
period vs. another? We use present value.
• PV[Bn] = Bn/(1+r)n (present value of a one
time net benefit to be received n years from
now)
Dynamic Efficiency (cont.)
• present value of stream of benefits equals:
n
PV [ B0 ,..., Bn]  
i 0

Bi
(1 r )
i
The process of calculating the present value
is called discounting. The rate r is referred
to as the discount rate.
Dynamic Efficiency (cont.)
• Dynamic efficiency: allocate resource use over time so
that the present value of the net benefits are
maximized.
• We will start with a two period model:
• Assumptions:
– fixed supply of resource to allocate over 2 periods
– Demand (marginal willingness to pay) function is
constant, i.e., the inverse demand function, is given by : P
= 8 -0.4q
– marginal cost of extraction is constant $2/unit
– interest rate is 10%
Two Period Model
– Note if total supply is greater than 30, the efficient
allocation would be 15 units/period, regardless of
interest rate.
– In this case static efficiency is sufficient.
– One period’s consumption does not affect the other
period’s consumption. In this sense, they are
independent.
9.0
9.0
8.0
8.0
7.0
7.0
6.0
6.0
Price ($/unit)
Price ($/unit)
Fig. 5.1 The allocation of an abundant depletable resource
5.0
4.0
3.0
5.0
4.0
3.0
MC
2.0
1.0
1.0
0.0
0.0
0
5
10
15
Quantity
(a) Period 1
20
MC
2.0
25
0
5
10
15
Quantity
(b) Period 2
20
25
Two Period Model
– What if available supply is less than 30, say only 20?
– How should we allocate across the 2 periods?
– Try 15 in period 1 and 5 in period 2.
Fig. 5.1 (modified, not shown in text) An arbitrary allocation of
an limited depletable resource (q1=15, q2=5)
9.0
9.0
8.0
8.0
P.V. Net Benefits=
0.5*(6)*15=$45
7.0
7.0
P.V.=25/(1.10)=$22.73
6.0
Price ($/unit)
6.0
Price ($/unit)
Net Benefits=
0.5*(6+4)*5=$25
5.0
4.0
5.0
4.0
3.0
3.0
2.0
2.0
1.0
1.0
0.0
0.0
0
5
10
15
20
25
0
10
15
Quantity
Quantity
(a) Period 1
5
P.V. Total
Benefits = $67.73
(b) Period 2
20
25
Two Period Model
– How do we find the allocation that gives us maximum
present value net benefits? We could guess or have a
computer search iteratively.
– But the best way is use economic logic (and/or math)
– Dynamically efficient allocation requires that the present
value of the marginal net benefit (PVMNB1) in period one
equals the present value of the marginal net benefit in period
two (PVMNB2).
– Fig. 5.2 shows the PVMNB for each period. Period 1 is read
from left to right and period 2 is read from right to left. Note
that the intercept for period 1 is $6 ($8-$2), but intercept of
period 2 is $5.45 (=$6/1.1)
Fig. 5.2 The optimal allocation of a limited depletable resource
P.V.MNB1
P.V.MNB2
P.V.NB1 =
0.5*(6+1.905)*10.238
= $40.466
P.V.NB2 =
0.5*(5.45+1.905)*9.762=
$35.90
MUC
5.45
1.905
Quantity in
Period 1
20 19 18 17 16 15 14 13 12 11 10
9
8
7
6
5
q1=10.238, q2=9.762
4
3
2
1 0
Quantity in
Period 2
Two Period Model
– The solution set is q1=10.238 and q2=9.762. The Total PV of
the net benefits equals $40.47 in period 1 plus $35.90 in
period 2, i.e., $76.37. This is the maximum PVNB. Recall for
example, that in the q1=15 and q2=5 allocation, PVNB =
$67.73
– This solution occurs where the two PVMNB curves cross. The
vertical distance where they cross is referred to as the
marginal user cost.
– marginal user cost (MUC) is the present value of the
foregone opportunities at the margin.
– in this example the MUC is $1.905, indicating that use of the
resource today will reduce the present value of future net
benefits by $1.905
Two Period Model
– Thus, the PVMNB of the last unit used in period 1 should
be worth $1.905. If it is worth more in period 1, than we
should consume more in period 1 until the PVMNBs are
equal. If it is worth more in period 2, than we should
consume more in period 2 until the PVMNBs are equal.
– Price in period 1 should be $3.905 = 8-.4*10.238.
– Price in period 2 should be $4.095 = 8-.4*9.762.
– The actual marginal user cost rises at the rate of
interest. $1.905*(1+.1) = $2.095.
Two Period Model
– Important point: don’t get marginal net benefit and
total net benefit confused.
– We maximize present value of total net benefit
(PVNB) when present value of marginal net benefits
are equal (PVMNB1= PVMNB2).
– Is this clear?
Two Period Model
– The undiscounted marginal user cost rises at the rate of
interest. Why?
– because the present value of the marginal user costs are
equal across time periods:
– MUC2 = MUC1*(1+r)
– e.g. $2.095 = $1.905*(1.10)
– What happens if the discount rate increases?
– Allocate more to period 1 and less to period 2. The PVMNB
curve in period 2 rotates down. (See modified Fig. 5.2)
Fig. 5.2 (modified, not shown in book) The optimal allocation of a
limited depletable resource
P.V.MNB1
5.45
P.V.MNB2
MUC
interest
rate
increases
to 50%
1.60
Quantity in
Period 1
20 19 18 17 16 15 14 13 12 11 10 9
8 7
6
5
4
3
2
1 0 Quantity in
Period 2
Verify these solutions
w/ appendix method.
q1=11, q2=9
9.0
9.0
8.0
8.0
7.0
7.0
6.0
6.0
5.0
Price ($/unit)
Price ($/unit)
Fig. 5.3 The efficient market allocation of a depletable resource:
The Constant Marginal Cost Case (a) Period 1 (b) Period 2
P = $3.905
4.0
3.0
5.0
P = $4.095
4.0
3.0
MC
MC
2.0
2.0
1.0
1.0
10.238
0.0
9.762
0.0
0
5
10
15
Quantity
(a) Period 1
20
25
0
5
10
15
Quantity
(b) Period 2
20
25
Defining Intertemporal Fairness
• How can we be fair to all generations?
• John Rawls, A Theory of Justice: suggests that we
pretend we are behind a “veil of ignorance” and
had to predetermine the rules we wanted to live by
if we did not know when we would be born.
– his solution: sustainability criterion: future generations
should be left no worse off than current generations. It
does not say that present generations can not make
themselves better off.
– Does dynamic efficiency violate sustainability criterion?
Are Efficient Allocations Fair?
– Does dynamic efficiency violate sustainability criterion?
Not necessarily.
– It might seem so since the net benefits in period 1 are
$40.466 while they are only $39.512 in period 2.
– But if sharing takes place, the first generation could save
say $0.466 and invest it at 10% interest, growing to
$0.513, when added to $39.512, gives the second
generation net benefits of $40.025, more than the first
generation receives.
– demonstrates that dynamic efficiency can be consistent
with sustainability as long as gains are shared among the
generations.
– See Example 5.1 The Alaska Permanent Fund
Applying the Sustainability Criterion
– “Hartwick Rule”: easier to apply than Rawls’ sustainability rule
– Constant level of consumption can be maintained perpetually if all
scarcity rent is invested in capital.
– Total capital stock should not decline.
– Our endowment consists of natural capital (environment) and physical
capital (buildings, machines, etc.)
– How easily can these two forms of capital be substituted?
• “weak sustainability”: maintenance of total capital
• “strong sustainability”: maintenance of natural capital
• “environmental sustainability”: maintain physical flow of
individual resources
– See Example 5.1: Alaska’s permanent fund
– See Example 5.2 Nauru: Weak Sustainability in the Extreme
Appendix (Chs. 2 & 5)
– Pt = a - b qt (called inverse demand equation, i.e.,
marginal willingness to pay curve)
– Total benefit is given by area under demand curve
qt
Total Benefits 
 (a  bq )dq
0
b 2
 a qt  qt
2
Total Cost  c q t
Appendix (cont.)
– We want to maximize the net benefits of a fixed
amount of the resource Q over n time periods:
b

aq 2 q  cq 
Max

  Q   q 
q  (1 r )


2
n
t
i
i
i
i 1
i 1
n
i 1
i
The first order conditions:
a bq  c
(1 r )
i
i 1
 0
i  1,..., n
n


Q


0
q



i
i 1


Example: a  8, c  $2, b  0.4, Q  20, and r  010
.
8  0.4 q1  2    0
8  0.4 q 2  2
110
.
q q
1
2
 20
 0
Appendix (cont.)
– solve the above three equations simultaneously:
8  0.4 q1  2   
q
1
8  0.4 q 2  2
110
.
 5.4545.3636 q 2
 20  q 2
6  0.4(20  q 2)  5.4545.3636 q 2
6  8  5.45  .7636 q 2
 7.4545  .7636 q 2
q
q
2
1
 7.4545 / .7636  9.762
 20  9.762  10.238
  8  0.4 q1  2  8.4 * 10.238  2  $1.905
The End