On the maximizing behavior of a monopoly planner

ECONOMICS OF PLANNING Vol. 10, No. 3, 1970 Printed in Norway

On the maximizing behavior of a monopoly planner j%dith Thornton

Donn L. Leber University of Washington

t The beginning student of general equilibrium economics soon reaches the saddening conclusion that ‘everything depends on everything else.’ Later on, with advanced study and maturity, comes the more sophisticated reali- zation that ‘everything depends on everything else in two ways.‘))

Robert M. Solow [l]

The theorists of market socialism take at least as much care as do other economists to remind their readers that, in general equilibrium theory, “everything depends on everything else in two ways” -not only does the output of goods depend on the supply of productive factors, but factor supplies, in turn, depend on real factor payments.

The substitution of state preferences for consumer preferences in an economy will normally result in the production of a different mix of goods than would have been produced in response to consumer demands, a mix yielding a different level of real consumer satisfaction. Except in the special case where factors are supplied totally inelastically with respect to the mix of goods produced, the productive capacity of the economy will not be independent of that mix.

Nevertheless, most static planning models assume either that productive capacity is fixed or that planners can influence the size of productive capacity solely through their saving and investment decisions. Benjamin Ward’s “The Planners’ Choice Variables,” [Z], is one of the few models that attempts to treat the way in which the supply of productive factors is affected by the substitution of state preferences for consumer preferences. Ward considers a single variable factor case in which the planners maximize their welfare, which is a function of commodities consumed by the state, subject to a factor supply equation in which the supply of labor depends on the commodities consumed by labor.

160 JUDITH THORNTON AND DONN L. LEBER

In the discussion that follows, we describe the basic Ward model and demonstrate that the equilibrium conditions of the model differ from those originally derived by Ward. Then the Ward model is extended to consider the general case where commodities consumed by the state and commodities consumed by the population are both arguments in the planners’ welfare function, but where the supply of inputs remains a function of commodities consumed by labor.

The models considered here yield some interesting results. Assuming no trade, a single variable input, and a divergence between the arguments of the planners’ welfare function and the input supply function, we find that the necessary conditions for the planners’ optimum involve the exercise of monopoly power by the planners. So, when these assump- tions hold, the planners’ maximum is non-optimal with reference to usual market socialist welfare criteria. In such an economy, the relevant resource constraint for the planners is not a single production possibilities function but the “planners’ feasibility constraint” developed here.

THE MONOPOLY PLANNER

The basic Ward model treats a case of two outputs produced by a single-variable input. The planners’ welfare function has the output of state goods - call them monuments - as its sole argument.

(1) W = w(M); dW/dM > 0, d2W/dC2 < 0.

The planners maximize their welfare subject to the constraints imposed by the (short-run) production function in each industry and by a labor supply equation.

(2) C = c(Vl); dC/dVl > 0, d2C/dV12 5 0

(3) M = m(V,); dM/dV, > 0, d”M/dV,” 5 0

(4) VI + V, = V, = v(C); dV,/dC > 0, d2Vs/dC2 < 0,

VI 2 0, v, 2 0

where C is the output of consumer goods. M is the output of state goods, and Vs is the total quantity of labor supplied consisting of VI allocated to the production of consumer goods V, allocated to the production of state goods.

MAXIMIZING BEHAVIOR OF A MONOPOLY PLANNER 161

Writing (2) in inverse form yields the labor cost of C and allows us to combine the three constraints into a single expression, which we call the planners’ feasibility function

(5) M = m( Vz) = m( v, - V,) = m[v(C) - c-‘(C)]

Then, substituting (5) into (1) and solving yields the necessary condition for the planners’ optimum:

The planners choose that rate of output of C that maximizes the possible output of M. At the optimum, the marginal labor generated by an

dv additional unit of consumer goods, -,

dC just equals the marginal labor

de-1 cost of that unit, -,

dC and the net labor surplus, or V,, available for the

production of M is maximized.

THE PLANNERS’ FEASIBILITY FUNCTION

When the labor supply is not fixed, the relevant constraint for the planners is not a single production possibilities curve, but rather the feasibility function. The planners’ feasibility function completely de- scribes the mixes of the two commodities that are feasible given the production functions and factor supply conditions. The decision variable for planners is the output of C. Each possible output of C projected by the planners generates a total labor supply, VS, and costs the planners a total labor cost, VI, that cannot exceed the total labor supply. The re- mainder of the labor force, if any, is available for the production of M.

The feasibility constraint is demonstrated in Figure 1. Combinations of state and consumer goods are measured in the first quadrant, production functions for M and C in the second and fourth quadrants. A given quantity of the variable input, labor, is represented as a factor constraint in the third quadrant, and any point on the factor constraint represents an allocation of the total variable factor between the two outputs. The labor supply equation, V,, appears in the fourth quadrant.

The maximum possible output of monuments occurs at that output of consumer goods where the vertical difference between the total labor

2


‘\ 5. I I ’ I I I I I I I I I I I

I I \ I VT\ “2 ;I

y I 0 ‘CW

\ \

\ \

\ \

\

Figure 1

supply function and the labor cost of consumer goods is maximized, at Cg. The reader can verify by similar construction that each alternative output of C calls forth a residual labor supply, V, - VI, that can be used to produce some quantity of monuments. These combinations trace the planners’ feasibility frontier, OBC T. This frontier differs from the usual production possibilities frontier, such as M&T which is defined for a given total labor supply, VT.

The slope of the feasibility function at any point, (7), differs from the slope of the production possibilities curve through that point, (S), by a term that represents the effect of variable labor supply.


(7)

(8)

k--l dv

dM dC dc --

z = - dm-1 + &-1 -- dM dM

dc-l

dM dC

dC =-dm-1 - ; v(C) = v

de-.1 dm-1

dM

The ratio F / dlM measures the ratio of the marginal labor cost

of the two goods in production. In the range where

the output of C may be viewed as an input into the production of M. Ex- pansion of consumer goods output produces a net surplus of the inter- mediate good, labor, which is available for the production of M. In the

range where g dv > z , C and M become substitutes, and the variable

labor supply term acts like an “external benefit” to the planners, reducing the planners’ full marginal cost of C in terms of M below the marginal labor cost of C.

MONOPOLY EQUILIBRIUM

Figure 2 demonstrates the peculiar sense in which the planners act as monopoly suppliers of consumer goods. They maximize their total labor surplus by equating the marginal labor (revenue) generated by consumer goods and the marginal labor cost of these consumer goods. The marginal labor cost of consumer goods is dV,/dC. The marginal and average labor revenue generated are dV,/dC and VJC. The relationship between marginal and average revenue yields:

VS dC c-1 C where q = - ___ . --- .


I I

G c

Figure 2


So the maximizing condition in (6) can be written:

dv dc-1 --- dC dC C

At equilibrium, the marginal product of labor in consumer goods exceeds the average real wage in consumer goods.

(11)

Comparison of the necessary condition for optimization presented here with the original Ward equilibrium condition demonstrates why it is logically important to think of the planners as monopoly suppliers of consumer goods rather than simply as monopoly buyers of labor.

Maximizing the planners’ welfare with respect Ward derives :

(12) dV

g = G(l + -k) where E = d

In terms of our notation, the Ward equilibrium condition requires:

to labor [2, p. 1361,

(12’) dc dv-l ----r dVl dV,

0

But this clearly yields a different and inferior solution to the necessary condition we present in equations (6) and (10). The Ward condition would tell the planner to treat the quantity of labor as the relevant decision variable and to act as monopsony buyers of labor. The Ward solution would yield a total labor supply VW in Figure 1, which is the quantity of labor that could maximize the surplus of consumer goods available. But, in the absence of foreign trade, VW is clearly non-optimal. The planners would produce Cw of consumer goods and Mw of monuments, which is less than &ZB.~


PRICING OF OUTPUT

The planners could call forth their desired mix of output in an environ- ment of market oriented Lange-Lerner type socialist firms simply by setting the relative prices of the two outputs paid to socialist firms equal to relative marginal labor costs. Taking the wage of labor as numeraire, ( =l), the marginal labor cost of the two goods, qe and qm, is simply:

(13)

(14)

dV, Qc = z

dV2 -- ‘m- dill

and the total value of output at marginal cost prices is:

But if labor is hired in the market and consumer goods are allocated in the market, then equilibrium market prices require that the value of the wages bill must equal the retail value of the output of consumer goods:

(16)&G = WV, = V s where w, the wage, is numeraire and pC is the market price of consumer goods.

Substituting our market price conditions back into (lo), we see that at equilibrium, the market price of consumer goods exceeds the marginal cost of consumer goods.2

But this implies, further, that if the total value of output at retail prices is to equal the total value at marginal cost prices, then the retail price of monuments must be less than its marginal cost.

( > c c 1

qm-pm= PC-qc -=---PC M Mr


THE GENERAL CASE: PLANNING SUBJECT TO A PLANNERS’ FEASIBILITY FUNCTION

The simple Ward model can be extended to treat the maximization of any planners’ welfare function subject to a feasibility constraint. In place of equation (l), we write :

aw aw w = w(C,M); dC > 0, m > 0

Equations (2), (3), and (4) are unchanged.

Maximizing (la) subject to (5) yields the Lagrangian expression:

(19 L = w(C,M) + A(M - m[u(C) - c-‘(C)] ).

Solving, we have:

(2Oa) .!&=&-~~[$--?]=O

Wb) aL aw

--=m+LO aiw

Combining (20a) and (20b) yields:

(21)

aw a0 av ac dCac ---=-- __ aw am-1 + am-1 -- -- aiw aM dM

The left-hand side of (21) measures the slope of the planners’ indif- ference curve and the right-hand side the sIope of the feasibility function.

av am-1 So long as dC / am - # 0 the planners marginal rate of substitution be-

tween C and M will not equal the marginal rate of transformation in production, measured as the ratio of marginal costs. The change in labor supply can be viewed as an external effect. The true marginal cost to the planners of a unit of consumer goods equals the marginal labor cost reduced by the amount of additional labor force called forth by that unit of consumer goods.

Depending on the form of the planners’ welfare function, the planners’


optimum will lie on the feasibility frontier at or between 3 and CT.

When g = 0, the planners maximize at B. When the planners’ valuation

of consumer goods is positive, equilibrium shifts toward CT. When

g = 0, only C will be produced and welfare will be maximized at CT

where the total labor cost of consumer goods just equals the total labor supplied in response to that quantity of consumer goods.

CONCLUSIONS

The model developed here leads to some useful conclusions. If we consider an economy without trade where the supply of inputs varies in response to the level of current real consumption paid to inputs, then the relevant social input constraint for the planners is not a single production possibilities locus, but the planners feasibility constraint developed here. This constraint traces the locus of points on a family of production pos- sibility surfaces that would be generated by each alternative quantity of inputs. The rate of product transformation along the planners’ feasibility function will differ from the rate of product transformation along the production possibilities locus by an amount that reflects the marginal input supply.

In such an economy, whenever the arguments of the planners’ welfare function are not identical with the arguments of the input supply function, the necessary conditions for the planners’ optimum will be non-optimal with reference to usual market socialist welfare criteria. At equilibrium, the marginal rate of product substitution in the planners’ welfare function will not equal the marginal rate of product transformation on the production possibilities function.

In addition, when the tastes of planners and of consumers diverge, the necessary conditions for the planners’ optimum imply the exercise of some monopoly power by the planners. For given input supply conditions and given production functions, the magnitude of the monopoly effect increases to a limiting value as the diverge of tastes between planners and consumers increases.


NOTES :

i The Ward equilibrium condition is the correct necessary condition for a different, single product model. Suppose the planners’ welfare function has as its sole argument the consumer goods consumed by planners, while the labor supply function depends, as before, on the consumer goods consumed by labor:

A. W = w(C,) B. Cp + Cu = C = c(V) c. v = v(G)

Substituting in A. and solving, we have:

D.

But the expression in brackets, in fact, turns out to be the Ward equilibrium condition. In this case, the planner maximizes his share of consumption by exercising monopsony power in the labor market.

2 This expression can be contrasted with Ward, [2, p. 139].

SOURCES

[l] Robert M. Solow, “Capital, Labor, and Income in Manufacturing,” in The Behavior of Income Shares (Princeton: National Bureau of Economic Research, 1964), pp. 101-41.

[2] Benjamin Ward, “The Planners’ Choice Variables,” Value and PZan, ed. Gregory Grossman (Berkeley: University of California Press, 1960), pp. 132-51.

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On the maximizing behavior of a monopoly planner