Option Values and Accounting Prices

Karl-Göran MälerBeijer Institute of Ecological

Economics

Option values and accounting pricesArrow and Fisher

• Two periods• An irreversible change is contemplated in the

first period.• Information on the net benefits from the

change is not known with certainty• More information will be forthcoming in the

second period

Option values

Option value is defined as the expected value of the forthcoming information in the second period, when no irreversible change is made in the first period

Accounting price(Dasgupta and Mäler)

• A change in a capital asset is contemplated now

• The present value of future benefits from a marginal change of the asset now is estimated

• This present value is the accounting price of the asset.

Accounting price

( ),

,

( )ti t

i tt

U Cp e d

K

Sustainable development

… development that meets the needs of the present without compromising the ability of the future to meet their own needs. (Our Common Future, 1987)

Forrunners

Allen Kneese and Bob AyresAndIUCN

Sustainable development

• Social welfare should not deteriorate• But change in social welfare in period t is

• Thus the criterion for sustainable development is that this expression in non-negative

, , 1 ,1

( )n

i t i t i t t

dVp K K v

dt

The shallow lakeRunoff of phosphorous from a farm using fertilizers reach a lakeAlgae start to growEventually, the lake becomes so murky that sunlight will not reach the bottom and green plants will dieBottom becomes unstable and will release more phosphorous – positive feedback

Non convex dynamics• Dynamics of ecosystems are

usually non-convex• This means that the right hand

side of the differential equation is non convex in the state variable

The dynamics of eutrophication

𝑋 (𝑡+1 )− 𝑋 (𝑡 )=𝑎 (𝑡 )−𝑏𝑋 (𝑡 )+𝑟 𝑋 2(𝑡)1+𝑋 (𝑡)2

The lake in equilibrium

Phosphorousinput

Equilibrium points

Stable partsFlip point

Algae Unstable part

Consequences

• Multiple steady states• Flips from one state to

another• Valuation difficult• Use of taxes to achieve

environmental improvements may be impossible

The accounting price of the stock of phosphorous

• Extremely difficult problem• Very important problem: It has policy implications and

implications for valuation

Salinity

• In south eastern Australia , there is a serious problem with salinity in the ground water

Ground water

2 m

6 m

Saline ground water

RESILIENCE

Root zone

Accounting price for resilience

1 1 1

2 2 2

1 2

( ( ), ( ))

( ( ), ( ))

( ) ( )

U V C t K t Utility with no flip

U V C t K t Utility after flip

U t U t

The accounting price

• F(X0, s) cumulative probability distribution for flipping in time interval (t0, s), given the initial stock of resilience Xo

• U1(s) benefits at time s if a flip has not taken place

• U2(s) benefits after a flip

0

0

( )0 00 1 2

0 0

( ( )) (( , )( ) ( ) ( ) r s t

t

E W s F X sq t U s U s e ds

X X

Accounting price vs option value

Compare this last expression with Fisher’s expression for option values

Superficially, these are very similar, indicating that there may be a deeper connection between these two concepts