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AGEC/FNR 406 LECTURE 12
Static vs. Dynamic Efficiency
Static efficiency is obtained when a single period’s net benefits are maximized.
Dynamic efficiency is obtained when the present values of net benefits are equal for all periods in a multi-period problem.
Dynamic efficiency DOES NOT mean an equal allocation across all periods.
Q
8
4
Example, continued
Q10 10
8
22
4
2. NB = 0.5*(8-4)*10 + 10*(4-2) = 40
1. P = 8 - .4*Q = 4
3. NPV = 40 + 40 /1.05 = 40 + 38 = 78
8
An alternative approach for a two period model
Step 1: Graph demand in period 1
20 20
8 . 1.05
22
-0.40
Step 2: Modify period 2 demand and graph it BACKWARDS!
-0.40 1.05
8
Step 3: Overlay the graphs
2020
8 . 1.05
22
-0.40 -0.40 1.05
Efficient allocation
Efficient allocation is where demand curves intersect.
P1 P2
Q1
Q2
10••
=10.12
Important concepts
2020
2 MC = 2 = MEC
Q
P1 P2 MUC = P - MEC
Scarcity rent
Numerical Approach
Step 1: write down NB in each period
NB1 = benefits - costs = (a-(bQ/2))Q - cQ= 8Q1-.2Q1
2 - 2Q1
= 6Q1-.2Q12
NB2 = benefits - costs = (a-(bQ/2))Q - cQ
= 8 Q2 -.2 Q2 2 - 2 Q2
= 6 Q2 -.2 Q2 2
Numerical Approach
Step 2: Get rid of Q2 in NB2
If Q1 + Q2 = Q, then Q2 = Q - Q1 or Q2 = 20 - Q1
so NB2 = 6(20 - Q1)-.2(20 - Q1)(20 - Q1) = 120 - 6 Q1 - 80 + 8 Q1 -.2 Q1 2
= 40 +2 Q1 -.2 Q1 2
Numerical Approach
Step 3: Calculate present value of 2-period benefit
NPV = NB1/(1+r)0 + NB2/(1+r)1
= 6Q1-.2Q12 + (40 +2Q1 -.2Q1
2 )/1.05
= 6Q1-.2Q12 + 38.095 + 1.905*Q1 - 0.1905*Q1
2
= 7.905Q1 - .3905 Q12 + 38.095
Numerical Approach
Step 4: Differentiate to find where NPV reaches max
Max where dNPV/dQ1 = 0
or 7.905 - 0.781Q1 = 0
Q1 = 10.12Q2 = 20 - Q1 = 9.88
P1= 8 - .4*10.12 = 3.95P2 = 8 - .4*9.88 = 4.05
Numerical Approach
NPV = 0.5*(8-P1)*Q1 + (P1 - 2)*Q1
+[ 0.5*(8-P2)*Q2 + (P2 - 2)*Q2 ] /(1+r)
= 0.5*(8- 3.95)*10.12 + (3.95 - 2)*10.12
+[ 0.5*(8-4.05)*9.88 + (4.05-2)*9.88 ] /(1.05)
= 78.10
Key Points
1. Price = MEC + MUC
MUC = 3.95 - 2 = 1.95
2. Under dynamic efficiency Q1 is lower than under static efficiency, and P1 is higher.
3. Under dynamic efficiency P2 is higher than P1 and Q2 is lower than Q1 due to discounting.
Extensions
1. Higher discount rate: allocate more to present.
2. Lower discount rate: allocate more to future
3. With zero discount rate, allocations are equal inboth periods
4. Greater scarcity reduces allocations in both periods.
Without scarcity, MUC = 0
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