Fixed-Job Cost Minimizer

B. FIVE RATIONAL ACTOR MODELS

2. Fixed-Job Cost Minimizer

Assume that the government has a certain fixed job to do and is structured in such a way as to minimize the costs of doing the job.

⁴⁸

For example. the railroad might have its schedule fixed at ten trips per day and might be instructed to minimize the cost of making those ten trips per day.

A Fixed-Job Cost Minimizer will respond only to internal costs, and not to the social costs of its actions; if immune, the railroad will choose the level of spark emissions that makes running the railroad as inexpensive as possible,

and with contributory negligence,

TR = pqq.

The producer will presumably be found contributorily negligent if and only if the value of the marginal productivity of capital or labor (in accident prevention activities) is greater than the cost of capital or labor. In other words,

!17) PqqK(a) > r.

or

(l~l pqqJal > ~.

Sublemma. The private producer will always engage capital and labor to the point where the values of their marginal productivities are equal, respectively, to the interest and v.age rates

!pqqK(o) = rand pqqLCal = w.)

Proof of ~ublemma: [Case 1) Assume that the private producer has decided to be contributor- ily negligent-he does not count on being compensated. Then he will maximize a profit function which is the difference between the revenues from his sales and his costs due to hiring input~.

(Profit

=

^{rr =} pqq( o) - rK - wL.) To maximize his profit he must use inputs to the point that the value of their marginal productivity is equal to their cost. !pqqK(a)

=

rand pqqL(a) = w.)

[Ca~e ~)If the producer decides to be nonnegligent, then he will maximize the same prof11 function with damages added. ("TT= pqq(a) - rK - wL + (pqq(O) - pqq(a))). This will lead the producer to u~e inputs to the point that the value of their marginal productivity. given that the ----~--~~~~~-~~~~. -.l.!,inucte~rf.,l...e~r.,l..in!.!1gL!..'te,_..,c:..!.'h'.'.)n.='.o~log) is unu~ed, equalstheir cost. By Definition 15, supra, this condition is

· · equivalenHo hiring inputs until the value of.their marginalproductiYity (with.techlloJog~· at1ts true leYel) equals their cost. Thus. by definition 15,

and

arr

aL = p'qqL(O) - w = 0 = pqqL(a) - w.

Bt:cause the sublemma is true under either case (pqqK(a) = rand PqqL(a)

=

w). the sublemma is proYed.i

By substituting pqqK(a) for rand pqqL(a) for win equilibrium equations (14!, ( i 5). and ( 16).

the production efficiency conditions are obtained, thereby proYing Lemma 3.U

Combining Lemma 2 ( .. only if .. ) and Lemma 3 ("if .. ) results in an "if 'Ind only if ..

proposition, and adding Lemma I ( .. production possibilities curYe .. ). and a judicious choice of the government's budget, Theorem 3 is proved. The choice of budget is needed because the maximum of W lies on the production possibilities curve, but its exact location is not specified.

By controlling the goYernment's production (through its budget), society may reach the mo:.t desirable production point. Q. E.D.

Since assumption 3. note 40 supra, mandates the choice of the liability rule which leads to an optimal state of society, Theorem 3 prescribes suability.

48. An example of this model is a subcontractor with a lump-sum payment for services rendered.

without regard to the value of the Jost crops. If immunity is denied, the government railroad will be forced through the judgment process to inter- nalize the social costs that it imposes.

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Once the government takes these social costs into account, it will adjust its behavior in such a manner that society will be better ,

⁵⁰

i.e. , the citizens will be able to consume more satisfying amounts of goods and services (wheat and train rides).

⁵¹

The ref ore.

a Fixed-Job Cost Minimizer should be suable.

⁵²

3. Profit Maximizer

Profit maximization, a commonly used model of private behavior, can be used lo describe governmental motivation. The term "Profit Maximizer"

⁵³

is used because the government seeks to make the difference between its costs and revenues as large as possible. In the context of a railroad, the assumption is that the person in charge of the entity will sell tickets to citizens, drive the trains, and choose the level of spark emissions, with the single goal of making money.

If a profit-maximizing governmental entity is granted immunity from tort suits, it will ignore the social costs of its actions when making internal production decisions. The railroad will not care about the acres of wheat that are Jost in fires it has caused and will therefore select the level of spark emission at which internal production costs are minimized. Denying the

49. See note 45 supra.

50. For an explanation of the reasoning behind this step. see note 52 infra.

---~~~--~--~--~. S-l.,...-l=echnic;ally,B-Fede-f.inition-of.the-f.ixed-iob.ma.y.be-needed.to.achie..v,c..stric.t..[es.ults_J:.C>,,___ ___________________ ~

ane-X.planaiioOoftnisfei:fuir-eme·ncsti riote 52-infra~- -· · · - --- 52. For the general assumptions and defmitinns. see note 40 supra

FIXEL>-JOB COST MINIMIZER

The formal economic analysis of this model 1s so similar to the analysis of the Product- Maximizing Budget Consumer that it will not be fully presented. See note 4'7 supra. Instead. the model will be briefly described.

Specific Definitions and AssumptionJ

All of the assumptions and definitions used in the Product-Maximizing Budget Consumer are also used here, with the exception that the governmental bureau is assumed IJ have a fi'o:ed job to dn !f

= f₀), and its costs (depending upon the rule of immunity) are either the cost of inpu1~ or tht ccst of inputs plus the cost of tort judgments. (rK + wL, or rK + wL - pqq(o) + pqq<O).) In either cw.e.

the government minimizes the cost of production while producing the fixed amount (/.1: .. The Lagrangian expression is either

t =

^rK + wL + >-.(f - f0}or

t =

^{rK +} wL - pqq(o) + pqq(O) • /\! f - f₀).) Following differentiation, the analysis of the Fixed-Job Cost Maximizer is, line for hne.

virtually identical with that of the Product-Maximizing Budget Consumer. The following theorem is proved by that analysis:

THEOREM 4 In a society consisting of one Fixed-Job Cosr Minimiurand one pri1·ate ind11·idu,1!.

society will product on the production possibilities cun-'t if and only if the government is suable in tort. (Some judicious choice of the job constraint must be made in order to insure that max1mali1) will result.)

53. For a description of the circumstances in which a bureau might be expected :o act as a Profit Maximizer, see W. NISKANEN, JR., supra note 34, at 33-35

3t

government immunity forces it to consider the social costs caused by its production methods.

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After considering these social costs, the government would decrease the level of spark emissions, thus permitting citizens to consume more satisfying combinations of goods and services (wheat and train rides). Society will be better of f.3

⁵

To achieve this happy result, however, a Profit Maximizer must be suable in tort.

⁵⁶

54. Set note 45 supra. This result is a well-known theorem in law and economics. Su Brown. supra note I.

SS. The logical step at this point is provided by some complex economic analysis. Su note 56 infra.

Technically, a redistribution of wealth may be necessary, following a change in the immunity rule, to achieve these strict results. Changing the immunity rule is, in itself. a wealth redistribu- tion. It is possible that, following a change in the immunity rule, wealth holdings will have been so allered that the competitive system would not produce a better state of society. With an appropriate redistribution of wealth, however, any Pareto-optimal state of society may be achieved. J. QUIRK & R. SAPOSNIK. INTRODUCTION TO GENERAL EQUILIBRIUM THEORY AND WELFARE. ECONOMICS 149 (1968). This is sufficient to guarantee that with an appropriate wealth redistribution society will be better off if the government is suable. Su note 56 infra.

56. For general definitions and assumptions, su note Ml supra.

PROFIT MAXIMIZER MODEL Specific Definitions and Assumptions

Every definition and assumption used in the analysis of the Product-Maximiz.ing Budget Consumer will also be used here el'.cepl that here the government is assumed to act as if its primary goal were profit maximiz.ation (This theorem reproduces, in some respects, results contained in Brown. supra note I.)

THEOREM 5 In a society consisting of two profit maximiurs (an individual and a govtrnmental entity), society will product on the production possibilitits curve if and only if tht govtrnment is suable in tort.

-~---P~oof. Using theHresul,ts. on production efficiencyco111:lirit:Jl'l~<1n:d:--t;e·1111'11a=-t:--found-in-th~~~~~~~~~~~

analysis of the Product- Ma:idmiz.ing Budget Consumer in note 47 supra, consider the two cases immunity and suability.

Case /: Assume that the government is immune from tori suit. It will maximiz.e profit without regard to tort judgments and will use the interfering technology to the point where the marginal productivity of the interfering technology is zero. This, in turn, means that society is not producing on the production possibilities curve. (Profit

=

^1T

=

p1f - rK - wL where p¹

=

price off. ^1T0

=

P1f0

=

0 implies f0

=:

0.)

Case 2: Assume that the government is suable in tort. The measure of damages will be assumed to be the difference between the revenues which the private firm would have had in the absence of interfering technology and the revenue which it actually did have (holding price constant). (1.t., Damages = pqq(o}_ pqq(Q).) [On the measure of damages and the need for Definition 15, ste note 47 supra.]

The government will maximize profits while taking damages into account. (Profit = 1T = p₁f(a) - rK- wL - pqq(O) + pqq(a).) This will lead to a use of interfering technology to the point that the marginal value of increasing the technology to the government is equal to the marginal value of the decrease in private production. (p₁f0 + pqq(a) = 0.) The normal input conditions are also obtained. (p1f ^{L =} w, p1f K = r.)

That the private producer will use inputs to the point that the value of their marginal productivity equals their cost (pqqK = r, pqqL = w), follows from the results in the Product·

Maximiz.ing Budget Consumer. [Su note 47 supra.) Q.E.D.

Herc it may be necessary to redistribute initial wealth holdings to guarantee optimality.