(c) 1998 by Peter Berck

Prices vs. QuantitiesPrices 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

TaxTax

mcf

mcp

mc

Unreg. QReg Q

Before regulation profits aredark green and purple areas

When regulation reduces QProfits are the purple plusgreen areas (mcf > mr as drawn)

If, instead, tax T=mc-mcf atreg Q: Q is still Reg Q, greenarea is tax take and only purpleremains as profit

The Uncertainty ProblemThe 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 mean zero variance 2 element of the cost function

Quantity RegulationQuantity 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 MotivationPrice 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?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,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,) is cost and is a function of the state of nature,.

B’ = MCB’ = MC

qr = argmaxq E( B - C). • Gives the optimal choice of qr.

• Of course, E[B’ - Cq] = 0 at qr.

Approximate About qrApproximate About qr

• Approximate B and C about qr. • Note that the uncertainty in marginal cost is all in

which is just a parallel shift in mc. Could also have a change in slope.

C(q,) = c +( c’ + ) (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. Obvious algebra.

• mc = c’ + c’’ (q-qr)• marginal cost

E[mc(qr,)] = c’ + E[] = c’

• mb = b’ + b’’ (q- qr)• marginal benefit

E[B’(qr) ] = b’

• FOC for qr implies b’=c’

A picture.A picture.

B’

•mc = c’ + c’’ (q-qr); here takes on the values of+/- e with equal probability

.

c’ + c’’ (q-qr)

c’+e + c’’ (q-qr)

c’-e + c’’ (q-qr)

qr

As the slope of B’ approaches verticalDWL goes down

Deadweight Loss using qr.Deadweight Loss using qr.

B’

-e

+e

qr

Half the time each triangle is the DWL

The Supply Curve The Supply Curve

• The firm sees the price, p, and maximizes its profits after it knows , so

• p = mc• p = c’ + + c’’ (q-qr) • Solving gives the supply curve: • h(p,) = q = qr + (p - c’ - ) / c’’

The center chooses p …The center chooses p …

• The center chooses p to maximize expected net benefits:

• p* = argmaxp E[ B(h(p,) - C(h(p,))] B-C = b-c +(b’-c’- (q-qr) + (b’’-c’’).5(q-qr)2

substitute q-qr = (p - c’ - ) / c’’ = b-c - (p - c’ - ) / c’’ + (b’’-c’’).5 ((p - c’ - ) / c’’ )2

Zero by FOC for qr

Take ExpectationsTake Expectations

B-C = b-c - (p - c’ - ) / c’’ + (b’’-c’’).5 ((p - c’ - ) / 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.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 The advantage of prices over quantitiesquantities

2

2

2

2

''2

''

''2

''

c

c

c

b

Deadweight Loss using p*.Deadweight Loss using p*.

B’

-e

+e

P*

Half the time each triangle is the DWL

As the slope of B’ approaches verticalDWL goes up