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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. Before regulation profits are dark green and purple areas.

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prices vs quantities
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

slide2

Before regulation profits are

dark green and purple areas

When regulation reduces Q

Profits are the purple plus

green areas (mcf > mr as drawn)

Tax

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

mcf

mcp

Unreg. Q

Reg Q

the uncertainty problem
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 2element of the cost function
quantity regulation
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
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
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,a) is cost and is a function of the state of nature, a.
slide8
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 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
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

B’

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)

deadweight loss using qr

B’

As the slope of B’

approaches vertical

DWL goes down

Deadweight Loss using qr.

+e

Half the time each triangle is

the DWL

-e

qr

the supply curve
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 …
  • 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
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
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….
deadweight loss using p

As the slope of B’

approaches vertical

DWL goes up

B’

Deadweight Loss using p*.

+e

Half the time each triangle is

the DWL

-e

P*

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