8/13/2019 Weitzman Theorem Proof

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Proof of Weitzman Theorem

Let MB = a-bq -- (1)

MC = uq where u~ (0, 2 ) -- (2)

Regulator maximises W = E[Net social benefit] = q

dquqMCqMBE

0

, --(3)

Procedure: (i) Obtain optimal values of q* and p* substtg these successively into(3) get EWGquota, EWGtax ( EWG = expected welfare gain)

(ii) Compare the two, EWGtaxEWGquotato measure the expected netbenefit of one policy over another.

Optimal quota: Choose q* to max E ( NSB ) FOC - *q

NSBE

= 0

E MCMB = 0 E uqbqa ** = 0 a-bq* = *q

or q* =

b

a

So, EWGquota= E

b

aq

o

*

uqbqa dq

= E *

2

2

q

o

q

bqa

=

ba 2

21 --(4)

Optimal tax (P*) : derive the firm's reaction function, i.e., the q that is produced by anyP- MC (q,u) = P - firms treat u as certain since they know their MC -

- invert MC - q = h (P,u) q = h (P,u) gives the qty abated (a randomvariable) as a function of P -- this is the firm's reaction function, i.e.,

the q produced by any P.

p

O Q

MC

8/13/2019 Weitzman Theorem Proof

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To pick optimal tax, set P to maximise NSB

FOC 0)()()(

P

hNSB

h

E

P

qNSB

q

ENSB

P

E

= E

0,,,

P

huuphMC

q

uphMB

Note, MB (q) = MB (h (p,u) ) -MB is a function of u because the level of abatement isuncertain under a tax ( unlike the case w/ quotas ) e.g., set tax = p - you think you'll get

to E on MB but actually you get to F

MCe

E MCa

MBF

Abatement

Now, since MC (q,u ) = Puq

Solve for q, q = uPhuP ,1

Substitute for q in (5): E 01**

uuPuP

ba

Or,

a

bp*

1

p

h

Optimal tax (P*):

b

aP

*

1

recall E(u) = 0 --(6)

So, EWGtax = E

dquqMCqMBuPhq

*,

0

,

Aside : h (P*,u) =

upup *

1*

1

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from (6) h ( P*,u) =

u

b

a

So, EWGtax= E

b

a

dquqMCqMB

0

,

= E

u

b

a

dquqbqa0

= E

u

b

a

qbuqqa

0

2

2

Skipping a few steps--- Recall E(u 2 )

=

b

b

a

b

a.

2.

2

12

2

2

222

=

b

b

a2

2.

2

12

22

same as (4) EWGquota

So, EWGtax - EWGquota = This is the key result. If > b then

tax better & vice-verse. If b EWGtax= EWGquota

Note that E(u2

) = 2 affects the magnitude of the difference between EWG under the

two policies ( i.e., it reflects the distance b/w MACe & MACt), but it does not affect the

choice of the policy instrument.

b 2

2

2