Valuation 2: Environmental Demand

Theory• Why valuation?• Theory of Consumer Demand• Restricted Expenditure and

Demand Functions for the Consumer

Uses of Economic Valuation

• Find optimum: Marginal benefit equals marginal cost– Ex ante, e.g., Pigou tax– Ex post, e.g., evaluation of policy

• Demonstrate value of environment• Extend national accounts

Pollution Damage(billions of US$)

Netherlands

Germany USA*

Air 0.5-0.8 19.3-21.5 21.7

Water 0.1-0.3 3.0 4.8

Noise 0.0 11.6 n.a.

Total 0.6-1.1 33.9 26.5

%GDP 0.3-0.5 2.9 0.8

* Damage avoided

Consumer Demand Theory

• Consider a consumer who maximises the utility of a bundle of goods q, with prices p, and income Y

• This solves to the ordinary or Marshallian demand function

• And to the indirect utility function v(p,Y); this gives you the highest level of utility attainable, given prices p and income Y; it is an optimal value function

max ( ) s.t. ' ; 0q

u q p q Y q

* ( , )q x p Y

Consumer Demand Theory -2

• Roy‘s identity relates x and v:

• That is, the derivative of indirect utility with respect to the ith price yields the ith demand function, after normalising by the marginal utility of income

( , )

( , )( , )pv p Y

x p Yv p Y Y

Consumer Demand Theory -3

• The dual of this problem:

• This solves to the compensated or Hicksian demand function

• This gives the quantity demanded as a function of price and utility. Income is of no consequence; as prices change, expenditures are adjusted to maintain constant utility.

min ' s.t. ( ) ; 0q

p q u q U q

* ( , )q h p U

Consumer Demand Theory -4

• The expenditure function:

• Defines the minimum expenditure needed to achieve utility U at prices p

• Shephard‘s lemma:

• Demand for the ith commodity is the derivative of the expenditure function to the price of i

( , ) ' ( , )e p U p h p U

( , ) ( , )ph p U e p U

Consumer Demand Theory -5

• We derived ordinary and compensated demand functions

• Ordinary demand functions bundle income and price effects together

• Compensated demand function do not have this problem, but look at price effects alone

• Typically, economists estimate ordinary demand functions, as utility cannot be observed

Consumer Demand Theory -6

• For all prices p, it must be true that

* ( *, *) ( *, *)q x p Y h p U

( , *) ( , ( , *))i ih p U x p e p U

( , *) ( , ) ( , ) ( , *)i i i

j j j

h p U x p Y x p Y e p Up p Y p

( *, *) ( *, *) ( *, *)

*i i ij

j j

h p U x p Y x p Yx

p p Y

* ( *, *)j jx e p U p

Consumer Demand Theory -7

• Now suppose we know the compensated demand function h; the expenditure function e follows from:

• According to Frobenius theorem this is true of the Slutsky matrix is symmetric, that is,

• We can derive the utility function u if and only if the Slutsky matrix is symmetric and negative semi-definite; if not, demand is inconsistent with utility maximisation

( , ) ( , )pe p U h p U

phj ji ih p h p

Consumer Demand Theory -8

• Now suppose we know the ordinary demand function x; the indirect utility function v follows from:

• Rewrite U=v(p,Y) as Y=e(p,U)• Hicksian demand follows from

• The first step is not trivial!

( , )( , )

( , )pv p Y

x p Yv p Y Y

( , ) ( , )ph p U e p U

Restricted Demand

• Demand for environmental commodities is only indirectly observed. People change their behaviour in response to changes cq differences in the environment, but do not purchase environmental quality directly.

• We‘ll repeat the analysis above, but now assume that only n-1 goods are directly traded; the nth good is the environmental commodity of interest.

Restricted Demand -2

• This leads to restricted ordinary demand functions and a restricted indirect utility function

• Roy‘s identity implies

• The duality between prices and quantities

1

1 21

max ( , ,..., ) s.t. ; 0n

n

n ni iq i

u q q q pq Y q

* ( , , )n n n nq x p q Y( , , )n nv p q Y

( , , )( , , ) ; 1,..., 1

( , , )n n i

n nin n

v p q Y px p q Y i n

v p q Y Y

( , , )( , , )

( , , )n n n

n n nn n

v p q Y qp p q Y

v p q Y Y

Restricted Demand -3

• This leads to restricted compensated demand functions and a restricted expenditure function

• If the consumer were to pay for qn

• The first order condition is

• Shephard‘s lemma

min ' s.t. ( ) ; 0n

n n nq

p q u q U q

* ( , , )n n n nq h p q U( , , )n ne p q U

min +e( , ,U) s.t. 0n

n n n n nq

p q p q q

( , , )n n n np e p q U q

( , , ) ( , , )nn n n p n nh p q U e p q U

Restricted Demand -4

1

2 11

2 1

( , ,..., , , )d

( , ,..., , , ) ( , ,U)

a

nnp

n n nn

ex p p q U x

p

e a p p q U e p q

1

2

2 11

2 1

( , ,..., , , )d

( , ,..., , , ) ( , ,U)

a

nnnp

n n nnn n

ex p p q U x

p q

e ea p p q U p q

q q

Restricted Demand -5

1

2

2 11

2 1

( , ,..., , , )d

( , ,..., , , ) ( , ,U)

a

nnnp

n n nnn n

ex p p q U x

p q

e ea p p q U p q

q q

1

12 1

2 1

( , ,..., , , )d

( , ,..., , , )

a

nnnp

n nnn

hx p p q U x

q

ea p p q U p

q

( , , )n n n np e p q U q ( , , ) ( , , )nn n n p n nh p q U e p q U

Restricted Demand -6

• So, if it can be determined how expenditures change with qn for some price a of good 1 (ordering goods at will), then this equation implicitly defines a demand function for qn

• This is not trivial, unless we assume weak complimentarity

1

12 1

2 1

( , ,..., , , )d

( , ,..., , , )

a

nnnp

n nnn

hx p p q U x

q

ea p p q U p

q

Restricted Demand -7

• Weak complimentarity:

• That is, if demand for good i drops to zero, then demand for good n goes to to zero, and marginal changes in n no longer affect expenditure

• For example, if swimming is too expensive, water quality is irrelevant

1 1 1 1( ,..., , , ,..., , , ) 0ni i i nh p p a p p q U

1 1 1 1( ,..., , , ,..., , , ) 0ni i nn

ep p a p p q U

q

Restricted Demand -8

2 1( , ,..., , , ) 0nnn

ea p p q U

q

1

12 1

2 1

( , ,..., , , )d

( , ,..., , , )

a

nnnp

n nnn

hx p p q U x

q

ea p p q U p

q

1

12 1

( , , )( , ,..., , , )d

an n

n nnn np

e p q U hp x p p q U x

q q

Restricted Demand -9

• So, assuming weak complimentarity between a market good and an environmental good, one can derive the compensated demand curve of the environmental good from the restricted compensated demand curve of the market good

• Without weak complimentarity, one has to use the restricted expenditure function

• The expenditure function is derived from all restricted compensated demand curves, using a complicated integral function

Summary

• We observe ordinary demand functions, but we are interested in compensated demand function – the latter can be derived from the former if agents are rational, and even then it involves many steps including integration

• We can deduce demand for non-traded goods from demand for traded goods, but this requires either a complete description of consumer behaviour or the assumption of weak complimentarity