Dynamic Efficiency
Illustration
Dynamic efficiency is illustrated for the two period case.
The example is taken from "Environmental and Natural Resource Economics"
by Tom Tietenberg, fourth edition pages 2530. Choose one of the
options below to explore a mathematical and graphical analysis of dynamic
efficiency.
Assumptions and Variables
A Case of Static Efficiency
Dynamic Efficiency  Case I 
Equivalent to Static Efficiency
Dynamic Efficiency  Case II
Interactive Two Period Dynamic
Efficiency
Back to Main Page
Assumptions and Variables

In this two period example, demand and MOC are the same for both periods.

Demand  p = 8  0.4 q where p is price and q is quantity demanded

MOC = $2 constant marginal cost

discount rate is 0.1 or 10 percent
Back to top
Back
to Main Page
A Case of Static Efficiency

Static Efficiency  using the concepts previously defined, static efficiency
is found by setting the MOC curve equal to the demand curve. This
can be shown graphically as

As shown in the static efficient figure, the equilibrium point is a quantity
of 15 units. In this figure, the green area is consumer surplus and
the purple area is total cost. No producer surplus exists because
of the constant MOC curve. Mathematically, this point can be solved
by setting the MOC curve equal to demand:

MOC = D = 2 = 8  0.4 q
solving for q one obtains q = (8  2) / .4 = 15 units
Back to top
Back
to Main Page
Dynamic Efficiency
 Case I  Equivalent to Static Efficiency

30 units or more of the resource is available

In this case, static equilibrium works. The optimal solution is to
allocate 15 units to each time period.
Back to top
Back
to Main Page
Dynamic Efficiency
 Case II

Less than thirty units available  assume 20 units of the resource is available

From the condition previously mentioned,
we know that dynamic efficiency is achieved if the present value of the
marginal net benefits in each time period are equal. Therefore, we
must get the marginal net benefits (MNB), which are found by subtracting
MOC from demand.

Period One Present Value of Marginal Net Benefits (MNB_{0}) =
8  0.4 q_{0 } 2 = 6  0.4 q_{0}

Period Two Present Value of Marginal Net Benefits (MNB_{1}) =
(8  0.4 q_{ 1} 2) (1/(1 + 0.1))
= (6  0.4 q_{1}) 1/(1.1) = 5.45  .36 q_{ 1}
Key
Idea: The first time period is not discounted because it is considered
the current period. The second time period is discounted.

Optimal allocation is found by setting MNB_{0 }= PVMNB_{1}
and the resource amount constraint q_{ 1} = 20  q_{0}.
Substituting the resource constraint into the equilibrium conditions one
obtains
6  0.4 q_{0} = 5.45  .36 (20  q_{0})
Solving one obtains q_{0} = 10.2 and q_{1} =
9.8

Graphically, this can be illustrated using the following graph.

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

On the right hand side, marginal net benefits for the second period are
graphed in present value terms.

The bottom axis represents the amount of resource. Reading the bottom
axis from left to right, one obtains the amount of resource consumed in
the first time period. Reading from right to left, one obtains the
amount of resource consumed in the second time period. Note, the
bottom axis goes from zero to the total amount of resource available.

Why this is the efficient point can be shown as follows. Consider
allocating 15 units to the first time period as given by the static efficient
point. Five units are then available for the second time period.
The following graph illustrates this situation.

At this point, the green area gives the net benefits for the first time
period, whereas the blue area is the present value of net benefits for
the second time period. Recall, these are marginal curves, therefore,
the area underneath both curves represent total net benefits from consumption
in the two periods. At the allocation 15, 5 the red cross hatched
area is lost. Reallocation of the resource consumed by consuming
less in the first period and more in the second period one can capture
the red cross hatched area and increase total net benefits.

Any allocation to the right of the dynamic efficient point the marginal
net benefits for consuming in the second period (MNB_{1}) are greater
than the marginal net benefits associated with the consumption in the first
period. Consumption should be reallocated towards the second time
period.

Any allocation to the left of the dynamic efficient point the marginal
net benefits for consuming in the first (MNB_{1}) are greater than
the marginal net benefits associated with the consumption in the second
period. Consumption should be reallocated towards the first time
period.
Key
Idea: Considering dynamics may lead to a different equilibrium solution
than if only static efficiency is considered. For the above example,
static efficiency equilibrium is 15 and 5, dynamic efficiency is 16.2 and
9.8.
Back to top
Back to Main Page
Interactive Two
Period Dynamic Efficiency

The following interactive graph allows you, the viewer, to determine the
impact of various components of the dynamic efficiency model.
Legend:
Period 1 Demand
Supply(MOC) Marginal
Net Benefit
Period 2 Demand
Supply(MOC)
Marginal Net Benefit
Questions:

What is the effect on current and future consumption if the discount rate
is increased (decreased)?

What is the effect on current and future consumption if demand in the second
period is increased (decreased)? This can be illustrated by changing
either the intercept, slope, or both of the second period demand curve.

What is the effect on current and future consumption if cost in the second
period is increased (decreased)? This can be illustrated by changing
either the intercept, slope, or both of the second period MOC curve.
Back to top
Back
to Main Page