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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


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.


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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