(c) 1998 by Peter Berck Prices vs. Quantities Distributional Issues  Baumol and Oates (I believe) Uncertainty  Weitzman, Martin. “Prices vs. Quantities.” Review of Economic Studies. Oct (4): –Simplify: make benefits deterministic

Tax mcf mcp mc Unreg. Q Reg Q Before regulation profits are dark green and purple areas When regulation reduces Q Profits are the purple plus green areas (mcf > mr as drawn) 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

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 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,  ) is cost and is a function of the state of nature, .

B’ = MC  qr = argmax q E( B - C). Gives the optimal choice of qr. Of course, E[B’ - C q ] = 0 at qr.

Approximate 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. 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. 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 vertical DWL goes down Deadweight Loss using qr. B’ -e +e qr Half the time each triangle is the DWL

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 to maximize expected net benefits: p* = argmax p 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 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 = D p E[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

Deadweight Loss using p*. B’ -e +e P* Half the time each triangle is the DWL As the slope of B’ approaches vertical DWL goes up