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 25-30.  Choose one of the options below to explore a mathematical and graphical analysis of dynamic efficiency.

Assumptions and Variables
• In this two period example, demand and MOC are the same for both periods.
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• 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
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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:
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• MOC = D = 2 = 8 - 0.4 q

• solving for q one obtains  q = (8 - 2) / .4 =  15 units
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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.
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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.
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• Period One Present Value of Marginal Net Benefits (MNB0) =

• 8 - 0.4 q0 - 2  = 6 - 0.4 q0
• Period Two Present Value of Marginal Net Benefits (MNB1) =

• (8 - 0.4 q 1- 2) (1/(1 + 0.1))
= (6 - 0.4 q1) 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 MNB0 = PVMNB1 and the resource amount constraint q 1 =  20 -  q0.  Substituting the resource constraint into the equilibrium conditions one obtains
•
6 - 0.4 q0 = 5.45 - .36 (20 - q0)
Solving one obtains q0 =  10.2 and q1 = 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.
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• Any allocation to the right of the dynamic efficient point the marginal net benefits for consuming in the second period (MNB1) are greater than the marginal net benefits associated with the consumption in the first period.  Consumption should be reallocated towards the second time period.
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• Any allocation to the left of the dynamic efficient point the marginal net benefits for consuming in the first (MNB1) 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.
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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
alt="Your browser understands the <APPLET>tag but isn't running the applet, for some reason."Your browser is completely ignoring the <APPLET>tag!nbsp; 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.
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