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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

(c) 1998 by Peter Berck

dark green and purple areas

When regulation reduces Q

Profits are the purple plus

green areas (mcf > mr as drawn)

Taxmc

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

- 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

- 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,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 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
- 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 the values of
- +/- e with equal probability
- .

qr

c’+e + c’’ (q-qr)

c’-e + c’’ (q-qr)

c’ + c’’ (q-qr)

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 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 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) / 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.

- 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….

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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