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