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Prices vs. Quantities • Distributional Issues * Baumol and Oates (I believe) • Uncertainty 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 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 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. 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 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 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 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 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 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 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 * 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 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 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 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) = qr + (p - c’ - a) / c’’

The center chooses p … • The center chooses p to maximize expected net 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) / 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}/ {2 c” 2} • 0 = Dp. E[B-C] = p - c’ • E[B-C] • = b-c + 2/c” + {(b’’-c’’) 2}/ {2 c” 2}

Advantage of Prices over Quant. • • • Under price setting E[B-C] = b-c Advantage of Prices over Quant. • • • Under price setting E[B-C] = b-c + 2/c” + {(b’’-c’’) 2}/ {2 c” 2} Less E[B-C] under quantity: = b-c Advantage of price over quantity….

The advantage of prices over quantities The advantage of prices over quantities

Deadweight Loss using p*. +e Half the time each triangle is the DWL -e 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’