Prices vs. Quantities
• Distributional Issues
 Baumol and Oates (I believe)
• Uncertainty
 Weitzman, Martin. “Prices vs. Quantities.”
Review of Economic Studies. Oct 1974 61(4):
477-491
– Simplify: make benefits deterministic
(c) 1998 by Peter Berck
Tax
Before regulation profits are
dark green and purple areas
mc
If, instead, tax T=mc-mcf at
reg Q: Q is still Reg Q, green
area is tax take and only purple
remains as profit
When regulation reduces Q
Profits are the purple plus
green areas (mcf > mr as drawn)
mcf
mcp
Reg Q
Unreg. Q
The Uncertainty Problem
• A private producer needs to be motivated to
produce a good that is not sold in a market.
• The government does not know the costs of
producing the goods.
• In particular it does not know a, a mean
zero variance 2 element of the cost
function
Quantity Regulation
• The firm can be told to produce a quantity
certain, qr.
• The level of benefits will be certain, since
qr is certain, but
• the level of costs isn’t known so the
government will accept the uncertainty in
the cost to be paid.
Price Motivation
• Or, the Government can offer to pay a price,
p for any units produced.
 The firm will observe which cost they incur and
react to the the true supply curve and set p=mc
correctly,
 but the level of production and level of benefits
will be variable
Which to choose?
• Professor Weitzman (to the best of my
ancient memory) gave the example of
medicine to be delivered to wartime
Nicaragua.




Too little and people die
Too much not worth anything more
cost doesn’t matter that much
so, choose qr and get the right amount there for
certain
In quantity mode,
• the regulator chooses a quantity, qr,
• then the state of nature becomes known,
• then the firm produces and costs are
incurred and benefits received.
• B(q) is benefits and B’ is marginal benefit.
• C(q,a) is cost and is a function of the state
of nature, a.
B’ = MC
 qr = argmaxq E( B - C).
• Gives the optimal choice of qr.
• Of course, E[B’ - Cq] = 0 at qr.
Approximate About qr
• Approximate B and C about qr.
• Note that the uncertainty in marginal cost is all in a,
which is just a parallel shift in mc. Could also have a
change in slope.
 C(q,a) = c +( c’ + a) (q-qr)
+ .5 c’’ (q-qr)2
 B(q) =b + b’ (q-qr) + .5 b’’ (q-qr)2
• b and c are benefits and costs at qr
Obvious algebra.
• mc = c’ + a+ c’’ (q-qr)
• marginal cost
 E[mc(qr,a)] = c’ + E[a] = c’
• mb = b’ + b’’ (q- qr)
• marginal benefit
 E[B’(qr) ] = b’
• FOC for qr implies b’=c’
A picture.
•mc = c’ + a+ c’’ (q-qr); here a takes on the
values of
+/- e with equal probability
qr
.
c’+e + c’’ (q-qr)
c’-e + c’’ (q-qr)
c’ + c’’ (q-qr)
B’
Deadweight Loss using qr.
+e
Half the time each triangle is
the DWL
qr
-e
As the slope of B’
approaches vertical
DWL goes down
B’
The Supply Curve
• The firm sees the price, p, and maximizes
its profits after it knows a, so
• p = mc
• p = c’ + a + c’’ (q-qr)
• Solving gives the supply curve:
• h(p,a) = q = qr + (p - c’ - a) / c’’
The center chooses p …
• The center chooses p to maximize expected
net benefits:
• p* = argmaxp E[ B(h(p,a) - C(h(p,a))]





B-C = b-c +(b’-c’- a)(q-qr) + (b’’-c’’).5(q-qr)2
substitute q-qr = (p - c’ - a) / c’’
= b-c - a (p - c’ - a) / c’’
+ (b’’-c’’).5 ((p - c’ - a) / c’’ )2
Zero by FOC for qr
Take Expectations
 B-C = b-c - a (p - c’ - a) / c’’

+ (b’’-c’’).5 ((p - c’ - a) / c’’ )2
• E[B-C] = b-c + 2/c” +
 (b’’-c’’) {(p-c’)2 + 2}/ {2c”2}
• 0 = DpE[B-C] = p - c’
• E[B-C]
• = b-c + 2/c” + {(b’’-c’’) 2}/ {2c”2}
Advantage of Prices over Quant.
•
•
•
•
•
Under price setting
E[B-C]
= b-c + 2/c” + {(b’’-c’’) 2}/ {2c”2}
Less E[B-C] under quantity: = b-c
Advantage of price over quantity….
The advantage of prices over
quantities
b' ' 
c' ' 
+
2
2
2 c' '
2 c' '
2
2
Deadweight Loss using p*.
+e
Half the time each triangle is
the DWL
-e
P*
As the slope of B’
approaches vertical
DWL goes up
B’