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The competition for rationed resources
JOURNALOF Journal of Economic Behavior and Organization Vol. 25 (1994) 53-71 -
ELSEVIER
EconomicBehavior & Organization
The competition for rationed resources * Stephen Department
M. Goldfeld,
Richard
E. Quandt
*
of Economics, Princeton Uniuersity, Princeton, NJ 085444-1021,
USA
Received May 1993, final version received September 1993
Abstract The paper considers a model in which firms face an output target and can purchase the input on two dates. On the first date, unlimited purchases are possible, but a carrying cost has to be paid. On the second date the input is rationed, but by spending specialized managerial resources, the firm can influence the rationing density to its own advantage. It also faces a quadratic penalty for missing the target and is assumed to minimize expected net costs. In an n-firm industry, a Cournot-Nash equilibrium in expenditures on the specialized managerial labor is defined and the comparative statics of the model are analyzed. We also find that under collusion the firms’ expected costs, the total expenditure on the otherwise unproductive managerial labor and the variance of the output are smaller than in the Cournot-Nash equilibrium. This provides an insight into why economic planners have tended to prefer to deal with a few very large firms rather than many small ones. Keywords:
Rationing; Cournot-Nash equilibrium; Collusive equilibrium; Planning
JEL classification:
D2; P2
1. Introduction Kornai
(1979,
ing shortages
1980, 1985) has noted
in a planned
economy
that one of the consequences
is that managers
attempt
of anticipat-
to the hoard
inputs
*We are indebted to L&&t Ambrus-Lakatos for comments. As usual, we bear the responsibility for errors. * Corresponding author. 0167-2681/94/$07.00
0 1994 Elsevier Science B.V. All rights reserved
SSDI 0167.2681(94)00050-O
54
S.M. Goldfeld, R.E. Quandi/J.
of Economic Behavior & Org. 25 (1994) 53-71
by making anticipatory purchases in those periods in which an input is freely available. Goldfeld and Quandt (1990a, 199Ob) formalized this notion in a series of simple models. Activity in these models takes place on two consecutive dates. On the first date, t = 0, the manager has the option of purchasing some amount of an input y, denoted by y,. At time t = 1, she may order an additional amount y,. At this later time the amount of the input that can be obtained is rationed and the manager obtains Y,+Y,
y, +x
if y,
if y1 >x,
where x is a random variable with density function h(x), which is the rationing density. Production takes place at time t = 1 after all input deliveries have been made and the manager’s objective is to minimize expected costs. Costs in this model are incurred from two sources: an inventory cost is incurred on the input units purchased at time t = 0 and a quadratic penalty is incurred for failing to meet production targets. Once one posits a production function f(y), and a production target y*, one can easily define the expected cost function E(C). ’ Several variants of models of this general type have been explored. The formulation we describe above represent a case of nonmanipulable rationing in that the manager’s demand has no effect on the amount delivered. This can be relaxed and the model can then be recast in a framework of manipulable rationing. It is also possible to introduce into the model the notion that some unknown fraction of the inputs ordered will turn out to be defective and thus unusable. If inputs can be defective, it is further possible to have the manager learn from his first-period experience and perform Bayesian updating of his estimate of the fraction of defectives. In all variants concerned so far, the enterprise was assumed to act in isolation. The purpose of the present paper is to introduce possible strategic interdependence when an enterprise has to compete against some other enterprise for the scarce inputs allocated by the policy makers. The possibility of lobbying for favorable treatment, which is explored in some detail, creates a connection between the present paper and the pathbreaking papers of Tullock (1967) Krueger (1974) Bhagwati and Srinivasan (1980) and Bhagwati (1983) which deal with various aspects of monopoly seeking, revenue seeking, lobbying, and related unproductive activities. The present paper examines the potential policy conflicts of regulators and sheds some light on the frequently observed phenomenon that socialist
’ In this formulation the manager was not required actually to pay for the amounts of inputs used. This assumption is a simplification that is loosely compatible with the notion of the soft budget constraint in the sense that any amount of actual input cost represents a wash transaction for the manager. The present paper relaxes this assumption.
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ofEconomic Behavior & Org. 25 (1994) 53-71
55
planners preferred to deal with highly concentrated industries. In Section 2 we pose the basic elements of the model and analyze the behavior of the individual firm. In Section 3 we discuss the equilibrium in an industry. Section 4 contains some numerical results, while Section 5 contains a brief summary and conclusions.
2. The basic model We model the competition for scarce resources by assuming that the manager is able to influence the amount of the input that is made available to her by employing a specialized form of managerial labor for the purpose of persuading the policy makers to favor her enterprise over that of another manager. The quantity of this labor employed by the ith firm is denoted by zi. This type of ‘whining’ is not unlike the seeking of bailouts that characterizes our models dealing directly with the soft budget constraint (Goldfeld and Quandt, 1988, 199Oc, 1990d). The principal difference is that in the soft budget constraint models the specialized managerial activity is directed at securing cash bailouts, whereas here it is directed at securing more favorable treatment when inputs are being allocated among firms. In general, we shall deal with an n-firm industry competing for the same resource; although, when convenient, we shall specialize the treatment to the 2-firm case. It is helpful to define n
czi,
z-
(1)
j=l
and ZCi)
=
czj. j#i
(2)
Clearly, the dependence of the amount of inputs allocated to the enterprise i must, other things being equal, increase with the amount of zi, decrease with zcij, and the fraction of the total amount of the input x available in the economy must be bounded below by 0 and above by 1. We further assume that the managers competing for x can never, in the aggregate, obtain the entire quantity x, because the policy makers always retain some (variable) fraction of the total x present in the economy, either for precautionary purposes or, possibly, because of corruption among the policy makers. This specification is satisfied by Assumption 1. The amount of input made available to Firm i is (z,/(Z + v)>x, where v is a positive constant set by the policy makers. Hence, ‘nature’ still generates a random x as before; this x is then divided up among the enterprises according to the values of the fraction given in Assumption 1. It should be noted that this formulation also has the realistic feature that a proportionate increase in all firms’ z-values leads to an increase in the proportion of x received by each firm; hence both the absolute level of zs and the ith firm’s proportionate share in
56
S.M. Goldfeld, R.E. Quandt/J.
of Economic Behavior & Org. 25 (1994) 53-71
the aggregate of zs has an effect. The zis represent a kind of rent-seeking by managers and the formulation in Assumption 1 has the effect that the fraction of the total available input retained by the policy makers diminishes as total rent-seeking increases. 2 The next assumption is made for convenience; without it the analysis generally becomes much more difficult. Assumption 2. The production function is linear in the single input. This assumption also allows us to define a net price for the output which is the sales price minus the input cost per unit. We then make Assumption 3. Managers sell the output and purchase inputs at fixed prices; the net price is denoted by p. We also make the following additional assumptions. Assumption 4. (a) The random variable x has density function h(x) over the range (0, a>, with mean p and variance (T *; (b) The unit cost of carrying inventories is the constant cl; (c) The cost of a unit of squared deviation between actual output and target output is the constant c2. None of the essential elements of the analysis is altered by assuming that h(x) has finite support. In addition to the other assumptions of the basic model outlined in Section 1, we also assume that the manager incurs constant unit costs for the employment of zi. This is expressed by Assumption 5. The costs associated with employing the specialized labor are cjzi. It should be noted that throughout most of the analysis we assume that the cost parameters faced by the various firms are identical; hence cl, c2, cg are not indexed by the subscript i. The ith Firm’s Optimization Problem. We consider a general case in which the manager is penalized not only for underproducing, but also for overproducing. The rationale for this is that the overproducing firm deprives other firms of needed inputs; hence such practice is to be discouraged. 3 Let yoi and y* denote the ith firm’s ‘early purchases’ and input target, respectively. The firm’s costs come from (1) inventorying yoi, (2) hiring zi, and
’ For a similar formulation of rent-seeking without a v parameter see Milgrom and Roberts (1992, pp. 573-4) ’ This assumes that if, after the resolution of uncertainty, the firm has obtained inputs that yield y* or greater, the firm produces this amount, even though it will be penalized for overproduction. This is optimal behavior if the policy makers know the technology and penalize the firm for excessive input purchases, whether or not they are actually used for production. Of course, as shown below, if p is sufficiently high, it will be optimal for the firm to assure itself of having sufficient inputs to produce more than y*. More generally, an alternative approach would permit the firm to return excess inputs without penalty, should the firm not wish to use the additional inputs. While this is a seemingly small difference in model specification, the model behaves quite differently, as we plan to show in subsequent work.
SM. Goldfeld, R.E. Quandt/J.
of Economic Behauior & Org. 25 (1994) 53-71
(3) exceeding or falling short of the target. The operating output has expected value
(
57
profit from selling the
‘i
P YOi+-
1’
z+l?
and works to offset the costs incurred for the three reasons above. Hence, it follows immediately that under the assumptions the net expected cost function of the firm is m
E(C) =cly,,+c,zi+c,
o j[
I2h(X)dx
&x
Ytx-yT+
(4) Performing
the integration
in (4) yields
E(C) = (ClpP)YlJi+c~zi-P&P+c2 -&+
+2(yl.X-yT) The first order conditions
(YO,-Yi*)’ [ (&)‘(p2iCr2)
for a minimum
aE(c)
----cc,-p+2c,
are
y,,-yL*+
WC) (Z(i) +v> [
aYOi
= cj +
aZi
2c,
(z+
V)” I
1.
1
&p
=o,
YOi_Y:: - &
2
(6)
I
IJ+ -&$+o2)
= 0.
1
(7)
We can then state Proposition
1. The second order conditions
Proof Obvious from differentiating evaluating the signs of the principal From Eq. (6) we can obtain Cl -P YOi =Y? - ---/_I,. 2c2
Substituting variable:
(6) and (7) with respect to yol and zi and minors of the relevant matrix. n
‘i
(8)
ZflJ
from (8) into (7) yields the ith firm’s reaction function
2c2( =(I, + v) c3+
are satisfied.
(z+vy
‘i
-&+-U [
2
Zfv
*
1
=o.
for its zi
(9)
58
S.M. Goldfeld,R.E. Quandt/ J. of Economic Behavior & Org. 25 (I 994) 53-71
It is clear that corner solutions are possible at which the first-order conditions (6) and (7) are not satisfied. Thus, for example, if the cost of failing to meet the target, c2, is very small relative to the inventory cost, cl, and c, >p or if p is very large (i.e., rationing is unlikely to occur), then (8) might yield a negative yoi and the appropriate solution is yoi = 0. Alternatively, writing (9) as
( zi +
z(i)
f
v)” =
2c2(z(i)
+
‘>
C3
c1
[
+++
GP(‘i
v) -zicr2
)
1
2
(10)
the left hand side has value equal to 0 at zi = -(zcij + V) and has value (zcI, + v>~ at zi = 0. Furthermore, the right hand side of (10) is linear in z, and is positive for either zi = 0 or zi = - ( zcrj + v). Hence, if a positive root exists, it is guaranteed to be unique; moreover, a positive root will exist if and only if This will tend to be the case if c1 or p are large or if c3, the ‘(i) + v < c1 P/‘3. cost of the specialized labor, or zcij + v is small. From (8) it is also clear that if a unique positive root for z, exists, the solution for yoi is also unique. We also note from (8) that if c, > p then yoi < y” ; that is, early purchases will always be less than the required amount of inputs to produce y*. However, if c1
2. (a)
dy,i/dc,
< 0; (b)
dy,j/dc,
(d) dy,,/dyT = 1; (e) dyoi/dz~,) > 0; (f) dY,i/dg2 > 0; (i) dy,i/‘dp > 0; (j) dz,/dc, 0; (m) dzi/‘dy: = 0, (n) dz,/dp > 0; Pro05
> 0 if c, >p;
(c)
dy,/dc,
> 0;
dyoi/dv> 0; (g) dy,,,/dp < o; (h) > 0; (k) dzi/dc2 < 0; (1) dzi/dc, < (0) dz,/da2 < 0; (p) dz,/dp = 0.
The proof is immediate from differentiating the first-order conditions totally and solving for the relevant derivatives. n Discussion. First, consider the derivatives of the variables with respect to the cost parameters. As expected, an increase in the inventory carrying cost decreases Y,,~, but this has to be compensated by an increase in the rent-seeking activity zi. Conversely, an increase in the cost of this activity diminishes its volume, but this has to be compensated by an increase in yoi. It is interesting that an increase in c2 diminishes the optimal zi. The reason for this is that the variance of the output of the ith firm increases with z, (see (13) below); thus, an increased zi increases the risk of missing the target, which becomes more expensive when c2 is higher. Hence zi has to decrease. What is also interesting is that the effect of increasing c2 is not unambiguous; a sufficient condition for dy,,/dc, > 0 is c, > p. From (8) we see that c, > p assures that y,,[ < y,*. When p > cl, it is possible for yoi > yf and for dy,,/dc, to be negative. We illustrate this with some numerical results for the case of industry equilibrium below.
SM. Goldfeld, R.E. Quandt/J.
of Economic Behavior & Org. 25 (1994) 53-71
59
We next observe that dz,/dy: = 0 is a direct consequence of the assumption that the production function is linear, so that the derivatives of the first-order conditions with respect to Y,,~ and y’ are identical (except for sign); it follows then from (8) that the derivative of yOi with respect to y’ has to be 1. Y,,~ increases with p, but again because of the linearity of the production function, it accounts for the entire necessary adjustment, leaving zi unchanged. The response of zi to changes in o2 is as expected: as the uncertainty increases, rent-seeking behavior declines; also, the positive sign of dy,,/da2 is the counterpart of this result. An increase in II, which indicates a reduction in the severity of rationing, leads to a decrease in yai, but an increase in zi. Finally, an increase in V, which measures the policy makers’ proclivity to withhold some X, or in zCi), increases yOi as one would expect, but the effect on zi is ambiguous. Also ambiguous is the effect of zCr) on zi, i.e., the slope of the reaction function. A fuller discussion of the reaction function will be postponed to Section 3 on industry equilibrium. The signs of the derivatives are also compatible with the discussion of the conditions that are likely to produce comer solutions, i.e., solutions at which yOi or zi are zero. In that discussion we noted that the optimal zi is likely to be zero when ci is small or cg is large. This is compatible with the signs of the derivatives dzJdc, and dz,/dc, in Proposition 2. Similarly, the optimal value of ye, is likely to be zero if c1 is high or c2 is small; this again agrees with the predictions of the relevant derivatives in Proposition 2. The Moments of Output. The output, qi, produced by the firms is qi = yoi +
Lx
z+v
(11)
’
and hence expected output is
E(qi)
=YlJi+
$P=Y* -
Cl
-P
2c3
(12)
2
by substituting Eq. (8). Thus the expected output of the firm diminishes with ci, increases with c2 and p, and does not depend on cg. 4 It also follows immediately that the variance of output is 2
var(qi)
=
& i
1
02.
(13)
Hence, the variance of output increases with the amount of zi employed and with the underlying intrinsic variance and diminishes with the quantity employed by other firms and with v.
4 It is clear that expected output generally differs from y:, which need not be the policy makers’ true objective. Indeed, policy makers can always set the value of y* so that their true objective is achieved (in the expected value sense).
S.M. Goldfeld,R.E. Quandt/J.
60
3. Industry
of Economic Behavior & Org. 25 (1994) 53-71
equilibrium
In the present section we consider two types of equilibria: Coumot-Nash equilibria and collusive equilibria. Cournot-Nash Equilibrium. If all firms face the same costs and have the same output target imposed on them, and if a positive yai and zi solves the ith firm’s optimization problem, then those solution values are common to all firms and a Cournot-Nash equilibrium has the feature that yoi =y,
and zi = z Vi.
The solution is obtained by making the above substitutions in the first-order conditions (6) and (7) and solving. The most natural assumption is that each firm’s output target is set at (l/nlth of an aggregate output target y*; thus y* = y*/n. It is also reasonable to consider that the value of c2 may depend on the number of firms in the industry, since in an industry with more firms, each one has to produce less. With an unchanged value of c2, a given aggregate deviation from target output would be penalized less heavily, because it is divided among more firms. Thus it is convenient to denote the value of c2 in an n-firm industry as cZII. The first-order conditions (6) and (7) evaluated at zi = z and yai = y0 then are c,-p+2c,,
1
Y*
y,--+
&P
2n
(nz+
V)”
Y* y” -
[i
(14)
=o>
c,+2c wl)z:4 [
n
P -
~+
2c*, -1
1
z ,,(p2+rr2)
= 0. The equivalent +
c 3
(15) of (9) becomes
2c2n((n-
l)z+ (nz+
Z
4
v)’
-g&L+-, [
2 2n
nz+
v
1.
(16)
=f-J
It is easy to show that the derivative of (16) with respect to z is positive in the neighborhood of a positive root, if one exists. Hence, if a positive root exists, it is unique. Moreover, a necessary and sufficient condition for the existence of a positive root is that (16) have a negative value at the origin, i.e., that c3 - cr p/ < 0. This is essentially the same condition as in the discussion following Eq. (10) for the individual firm, since at the origin zcij = 0. It is relevant to ask whether we can be certain that E(C)OPt < E(CloPt 1*=’ in all cases. It is straightforward to show that
qC)“p’=E(C)~pt’~=O
+
c-z+
f2!& - s!Q+ 24 nz + [
1 .
v
It is also easy to show that the difference between the two E(C) values is negative. Alternatively, one may note that the positive solution for (16) is unique;
S.M. Goldfeld, R.E. Quandt/J.
of Economic Behal,ior & Org. 25 (1994) 53-71
61
hence, even if the expected-cost function is not globally convex, the expected cost corresponding to the solution of (16) must be a global minimum. It is obvious that the expected output of the ith firm is E(q,) = y*/n - (c, and aggregate expected output, obtained by summing over all firms, is P)/2% 5E(qJ i=
=y*
-
4Cl -P> 2%
1
(17) .
Thus, to keep expected industry output invariant with respect to the number of firms, the penalty for missing the target must be so adjusted that c*,,/n = constant}. In particular, if c2 is the value of c2 in an industry with a single firm, then we must have c2,, = q. Comparative Statics for the Cow-not-Nash Equilibrium. Comparably to Proposition 2, we can obtain
ProPosition3.(a) d.~,/dc,
dy,/dc,,
> 0
ifc, >p;
(c) dy,/&,
> 0; (d)
dY”/dY,! = l/n; (e) dy,/dv > 0; (f) dy,,/dp < 0; (g) dy,/d~ 2 > 0; (h) dy,/dp > 0; (i) dz/dc, > 0; (j) dz/dc,. < 0; (k) dz/dc, < 0; (I) dz/dyT = o, (m) dz/dr.L > 0; (n) dz/da2 < 0; (0) dz/dp = 0. Prooj
Letting all yOi have the common value y,, and letting all zi have the common value z in the first-order conditions (6) and (71, differentiating totally and solving for the derivatives yields the resu1t.m These results are identical to the ones in Proposition 2, but note that the derivative of y, with respect to y * is l/n rather than 1. It should be noted that the derivative dz/dv is ambiguous in sign. What one can show (see Appendix) is that the elasticity vdz/zdv < 1, which will play a role in the proof of Proposition 6. Computational results reported below indicate that for small values of V, the derivative is positive, whereas for large values of v it is negative. Hence, if the starting level of v is reasonably modest and policy makers withhold more of x because of corruption, rent-seeking increases by everybody. This phenomenon can then be labelled as ‘waste begets more waste.’ If v is already large, and it increases further, firms respond by reducing waste. It should also be noted that what constitutes ‘small’ or ‘large’ values of Y depends on the values of the other parameters. It is worth noting that if v is zero, (16) permits an explicit solution for z, which is 2c,,(n
- 1)
Cl P
z= n2c3
[z-f12].
This is meaningful only if it yields a nonnegative solution, which is the case when cr p/2c2,, 2 u2. Hence, rent seeking behavior will occur only if the
62
S.M. Goldfeld,R.E. Quandt/ .I. of Economic Behavior & Org. 25 (1994) 53-71
variance of the aggregate available input is not too large; otherwise it simply is not worthwhile to engage in such behavior. Since c2,, = nc2, as n increases, the condition for a nonnegative solution will eventually fail and the corner solution z = 0 will be obtained. Hence, when v = 0, rent seeking behavior disappears if it is large enough. This is not the case when v > 0, as we can see by examining the limiting behavior of nz, ny, and &Z(C). For this we have Proposition
4.
(a) lim, +cclzz = constant; = constant.
lim n ~ ,nE(C)
(b)
lim,,,ny,
= constant;
(c)
Proof
(a) The behavior of nz can be analyzed by examining Eq. (16). Multiplying through by (nz + v)~, replacing nz by the symbol w and cZn by IZC*, and simplifying, leads to the following equation: Cl k
3c,v-c,/_L+
c,w3+
-
2c,&
II
[
Cl
w2
+2c,u2-n
PI,
1
I
(18)
+ 3c,u2 - 2c, /_&V + + 2c,a=v w + c3v3 - clv/.L=o. n [ If the conditions for a positive (unique) z solution hold, there will be a positive and unique solution for w for any value of n; moreover, the coefficients of the equation converge as n + 00; hence nz is a constant in the limit. (b) From (14), the solution for y, is Y* ” = T
Cl -P --5’“. 2nc,
-
(19)
Multiplying by n shows that ny, depends only on nz, which converges to a limit. (c) Replacing in (5) c2 by ncz, y,, by y,, y* by y*/n, and multiplying by n shows that nE(C) also depends on n only through nz and hence converges to a constant. n Proposition 4 thus shows that the aggregate amount of rentseeking, as well as the aggregate amount of early purchases and the aggregate cost incurred all stabilize as n becomes large and thus total waste remains bounded. Analogously to (13), the variance of aggregate output can be written as 2
var(q)
=
i
2
1
(20)
a2.
From Proposition 4 we see that the variance converges to a constant as n gets large. We now examine in more detail how z, nz and the variance of q depend on n. We have Proposition
5. (a) dz/dn
< 0; (b) d(nz)/dn
> 0; (c) dvar(q)/dn
> 0.
S.M. Goldfeld, RX. Quandt/J.
of Economic Behavior & Org. 25 (1994) 53-71
63
Proof For proof, see the Appendix. We thus note that an increase in the number of firms (with compensating adjustment in cZn) reduces the amount of z employed by each firm, but increases the aggregate amount employed, and increases the variance of aggregate output. Letting
+nz
(21)
nz+ v’
be the firms’ share of the rationed input, we have a!s -= dv
n (nz+
1
dz v----z. v)’ [ dv
(22)
As noted above, it is proved in the Appendix that the bracket in (22) is negative. Hence, dS/dv < 0, even though dz/dv is ambiguous in sign. From (20) it follows that
(23) which is therefore also negative. Proposition
6. (a> dvar(q)/dv
This proves (a) of < 0; (b) aE(CYp’/av
> 0.
For the proof of (b), see the Appendix. To interpret these results, we may consider the parameter v to be exogenously given or to be set by a policy maker. In the latter case, policy makers face a conflict in that increasing the value of v reduces the variance but increases expected costs. There may be a further conflict in that, depending on the value of v, rent-seeking behavior may either rise or fall. Hence the policy makers’ optimal action will depend on the precise form of their loss function (which we do not specify in the present paper). Collusive Equilibrium. The procedure in the Cournot-Nash case was to find the first order conditions for the individual firm and then set zi = z and yoi = ya. In the presence of collusion, the objective function is written from the beginning in terms of common z and y, variables and is then differentiated with respect to these. The expected cost function is *
Z E(C)
=
(Cl
-P)Yo
+c,z-P-
P+cc,” nz+
+(Yo-~)--&‘+(-$)*w+~*~].
v
2
[( ) Yo-$
(24)
64
S.M. Goldfeld, R.E. Quandt/J.
The derivative becomes
dE(C)/dy,
WC)
___
is the same as (14). However, the derivative
= cg + 2Czn
az
of Economic Behavior & Org. 25 (1994) 53-71
K
ZU
+(nz+ u)
Y*
P
LIE(C)/&
IJv
y"- L- - 23, i (nz+ v)"
3(/.L2+cT2) =o.
(25)
1
We note that it can be routinely verified that the second order conditions satisfied. Substituting from (14) into (25) yields Z
2c2nv c3
+Lf---~
+ (nz+
are
v)”
i
2 2n
nz+
Y
This contrasts with (16) in the Cournot-Nash
(26)
=o. i
equilibrium.
This leads to
Proposition 7. The solution for z under collusion is smaller than for the CournotNash equilibrium. Prooj At z = 0, (16) and (26) have identical values. For any other positive z-value, (12), plotted as a function of z, lies above (16), because subtracting (16) from (26) yields -
2c,,(n
- 1)z --&f-&o2
(nz + V)’
[
2n
1
>o.
Hence (26) must cross the z-axis to the left of (16). n The result suggests that rent-seeking activities are lessened by allowing collusion, i.e., by letting all firms behave as one. It also follows from (20) that the variance of output under collusion is less than in the Cournot-Nash equilibrium, even though expected output is the same. Both of these observations are consistent with the results of Proposition 5 and, in turn, suggest that other things being equal, policy makers who wish to reduce rent-seeking behavior as well as output variance would find it desirable to encourage the formation of monopolies. This then appears to be an economic justification for the ‘gigantism’ encountered in the industrial structure of planned economies. Comparison of Expected Costs. We next compare the level of optimal expected costs under Cournot-Nash equilibrium and collusion. The optimal value under Cournot-Nash equilibrium is obtained by replacing in (5) zi by z, yoi by yi, y* by y*/n and substituting for y, the optimal solution from (19). The optimal expected cost under collusion is obtained from (24) into which we substitute from (19). Denoting by k = c,cn the collusive and Cournot-Nash cases respectively, it
SM. Goldfeld, R.E. Quandt/J.
is easily seen that the expected cost functions identical in form and can be written as
E(Ck) = (Cl
65
of Economic Behavior & Org. 25 (1994) 53-71
-p)f - (Cl4c--P>’+c
condensed
with respect to y, are
CIWk+ C2ng2(zk)2
Zk_
3
nzk+
2n
1,
2 (nzk+
.
v)
(27) Hence the relative magnitudes of the optimal expected costs will depend only on the last three terms in (271, which can be written as 2 r=c,z-
Cl PZ nz+ u
C2ng +(
nz
2 Z
+
v)”
’
where we have omitted the superscript
ar --cc,+ 8Z
Cl
2C2nY
P
-(nz
+
v)”
[
2C2n
k. Differentiating, Z +
-CT nz+v
2 I
’
we obtain
(29)
which is identical to (26) and thus must equal zero in collusive equilibrium in order to satisfy the appropriate first order condition. Since the second order conditions are satisfied and Proposition 7 implies that zc” > zc, we have proved Proposition
8. E(CC) < E(F).
This then provides another reason why policy makers, intent on minimizing aggregate costs, might prefer monopoly firms to less concentrated industries. Comparative Statics under Collusion. Analogously to Proposition 3, we have Proposition 9. (a> dyO/dc, < 0; (b) dy,/dc,, > 0 ifc, >p; (c) dy,/dc, > 0; (d) dy,/dy* = l/n; (e) dy,/dp > 0; (f) dy,/dp < 0; (g> dy,/du2 > 0; (h) dz/dc, > 0; (i> dz/dc,, < 0; (j> dz/dc, < 0; (k) dz/dy* = 0; (E) dz/dp = 0; (m) dz/dp > 0; (n) dz/du* > 0. ProofI The results follow directly from total differentiation of the first order conditions. Hence the signs of the derivatives under collusion are identical to the signs under Cournot-Nash equilibrium. Analogously to Proposition 4, we finally examine the behavior of nz, ny, and nE(C). We start with Equ. (26) in which we replace c2,, by nc2. It is immediately obvious that this yields a cubic in (nz + v), with none of the coefficients depending on n at all; hence nz is a constant itself. Using (141 and (24) shows that ny, and nE(C) depend on n only through nz, which is a constant. We thus have proved Proposition to n.
10. In the collusive case, nz, ny, and nE(C) are invariant with respect
66
SM. Goldfeld, R.E. Quandt/J.
4. Numerical
of Economic Behavior & Org. 25 (1994) 53-71
results
While most of the comparative statics results are unambiguous, in a small number of instances we were not able to derive unambiguous conclusions. This is the case for dy,/dc,,, in Proposition 3, where the sign is unambiguous only if c1 > p, and is also the case for dz/dv in Cournot-Nash equilibrium, where there does not appear to exist a simple condition that allows the derivative to be signed. Finally, we have not dealt at all in the preceding sections with the asymmetric case in which the costs pertaining to different firms are different, because the interpretation of the formulas becomes exceedingly difficult. The objective of the present section is to shed some more light on these questions by performing computations or plotting the reaction functions for particular values of the parameters. We first consider the case of the optimal y, in a two-firm industry as czn varies. For the example that follows we assume that c1 = 20, cg = 10, v = 1, y* = 100, p = 15 and CT’ = 25. Table 1 displays the Cournot-Nash equilibrium values of y, for various values of p and czn. All the p-values are substantially greater than cl, which (b) of Proposition 3 suggests might be needed to cause an increase in czn to result in a decrease in y,, yet for p = 90, y, still increases. When p = 125 or p = 150, y, first decreases, but for high enough c1 it increases again. It takes a p of 200 before y0 uniformly decreases within the Table’s range of czn variation. It is also worth noting that the four instances in which y0 > 50 represent cases in which the total amount ordered ‘up front’ by the two firms is more than the industry target y*; this phenomenon also appears to require large values of p. In order to examine the variation in industry equilibrium values of z when v varies, we display in Figure 1 three pairs of reaction functions corresponding to the case when c1 = 20, c2,, = 10, c3 = 10, /_L= 15, (T* = 25 (The value of p and of y* are irrelevant because these parameters do not enter Equ. (161.1 The values of v are 1, 7, and 13 respectively, and the three equilibria correspond to the intersections of the appropriate reaction functions. It is obvious from the figure that the equilibrium value of z increases as v moves from 1 to 7, but decreases again when v increases further to 13. Other calculations suggested that it was a general phenomenon that dz/dv > 0 for small v and negative for large values. The last case we examine is the asymmetric one in which p = 30, u* = 10, v = 1, Firm 1 has the cost parameters c1 = 10, c2. = 3, cg = 6 and in which Firm 2
Table 1 Values of y,
%?n
p = 90
p=
10 20 30
47.3 47.7 48.4
49.1 48.6 48.9
125
p = 150
p = 200
50.3 49.2 49.4
52.8 50.4 50.2
S.M. Goldfeld, R.E. Quandt/J.
ofEconomic Behauior & Org. 25 (1994) 53-71
67
c1 = 20, c2 = 10, c3 = 10, p = 15, 02 = 25
Fig. 1.
has the same c1 and c2 as Firm 1, but has a ca that is alternately set at 3, 6, or 12. We are thus looking for the intersection of Firm l’s reaction function with one of Firm 2’s three possible reaction functions. It is clear that the optimal value of z2 declines monotonically as Firm 2’s cg increases; however, Firm l’s optimal zr first increases and then declines. It is clear that in the asymmetric case a variety of such reversals are possible. 5. Concluding
comments
The paper presents an analysis of a model in which an input is rationed, firms can expend resources in order to make the rationing less stringent, and can also try
68
SM. Goldfeld, R.E. Quandt / J.
ofEconomic Behavior Xr Org. 25 (1994) 53-71
50 ;I Fig. 2.
to avoid the bite of rationing by buying the rationed good at an earlier time when rationing is absent. While the model produces a number of reasonable conclusions, it is to be taken only as a first approximation and a number of relevant and realistic extensions remain for future research. 1. The model posits that firms employ all the inputs that they have acquired, even if this causes them to produce more than the target and even if such overproduction is penalized. In an alternative version, the firm might be able to return the excess input at no cost. While this appears to be a minor modification, it appears to have major effects on the analytics of the model. 2. The entire discussion in the present paper is based on the assumption that the production function is linear. The special nature of this assumption has had some beneficial effects in that it makes the model more tractable. However, a
SM. Goldfeld, R.E. Quandt/J.
of Economic Behavior & Org. 25 (1994) 53-71
69
number of specific effects clearly result from linearity (see, for example, the discussion of Proposition 2), and it would be desirable to relax this assumption, to see what impact this has on the conclusions. Losses accrued by the firm are borne entirely by the firm, but persistent losses are not compatible with survival; hence it would be of interest to see what effect bailouts by policy makers have on the results. Throughout the analysis, the parameter v is assumed to be given and known with certainty by the firms. In reality, v may not be known with certainty and individual firms may have prior densities over the value of V. The effect of such an assumption on the results would be of considerable interest. Some of the analysis has indicated that policy makers may face a conflict in setting the value of v or of II, the number of firms in the industry, or indeed whether to discourage collusion. We know that the variance of output falls, but expected costs rise as v increases; moreover, the variance of output rises with II, but expected costs are smaller if the industry acts as a monopoly. For these reasons it is of relevance to examine whether the introduction of an explicit loss function for the policy makers can shed some additional light on these conflicts. Finally, to the extent that such a loss function exists and is known to be employed, new incentives will be created for the managers and their behavior may change; conversely, firms may act as followers and policy makers as Stackelberg leaders. We hope to explore these rich possibilities in further work.
Appendix Proof of Proposition A=-
5. (a) We first start by defining
&+-, 2n
Z nz+v
2
(A.11
’
which is the bracket in Eq. (16) and is therefore negative;
we also define
B=nA.
(‘4.2)
We start with Eq. (16) in which c2” has been replaced totally with respect to n and solving for dz/dn yields dz
-=
dn
z(nz+
v)((n-2)z+
v)B-
vzu2((n-
by nc2. Differentiating
l)z+
V)
-(nz+v)[n(n-1)z+(n+1)v]B+a2((n-1)~++)~~’ (A.3)
It follows immediately that the derivative is negative. (b) Treating dz/dn as a fraction, we see immediately that the sign of d(nz)/dn depends on the sign of ndz + zdn, since the denominator of dz/dn is positive.
70
SM. Goldfeld, R.E. Quandt / J. of Economic Behavior & Org. 25 (1994) 53-71
Multiplying the numerator by n, the denominator by -z(nz + v)‘B < 0, as claimed. (c) Differentiating (18) yields after some rearrangement
d-4 q) dn
d( nz) -
2u*nzu =
(nz+
V)’
z and
adding
yields
>o,
dn
as claimed. W Proof
of vdz/dv
- z < 0. Differentiating
(16) totally
and solving
of dz/dv
yields nz + v)* + 2(nz + v)((n - 1)~ + v)]A + za2((n
;=I[-(
X{[(m+
v)*(n - 1) - 2n(nz + v)((n
- 1)~ + v)]A
- 1)~ + v)} + va’((n
- 1)~ + v)}-‘. (A.41
The coefficient of A in the denominator of (A.4) simplifies to (nz + v>[ - n( n l>z - v(n + l)] and hence the denominator is positive. Considering dz/dv as a fraction, the sign of vdz/dv - z depends on the sign of vdz - zdv. The numerator of dz/dv times v minus z times the denominator of this derivative simplifies to (nz + v)*((n - l)z + VIA, which is negative, thus proving the claim. n Proof
of Proposition
which the superscript
6. The optimal expected costs are given by Equ. (25) (in k may be ignored). Hence 2C2nV (nz+
where
1
A
dt_
v)”
dV
A is given
%nZ A,
(nz+
v)’
(A.51
by (A.l). Now substitute into (AS) from (A.4), considering as a fraction; i.e., denoting by dz the numerator and by dv the denominator of the right hand side of (A.4). After tedious algebra, the expression simplifies to dz/dv
c,dz +
2c2,A (nzf
v)’
[vdz-zdv].
Substituting for the bracket noting from (16) that cg = -
2c,,((n-
l)z+ (nz+
(A.6)
in (A.6) from the derivation
= lb
-2c,,((n
(A.4)
and
v)A
v)’
’
we obtain
aq cypt
following
- l)z+
v)A
(nzf v)’1, dz
-A
(A.7)
S.M. Goldfeld, R.E. Quandt/J.
of Economic Behavior & Org. 25 (1994) 53-71
and hence we need the sign of the bracket in (A.7). Multiplying becomes - 2(nz + v)zA + zu’((n - 1)~ + v) > 0, and since A > 0 as claimed.
71
out, the bracket
< 0,E(C)"P'/tb
References Bhagwati, J.N., 1983, Directly unproductive, profit-seeking (DUP) activities, The Theory of Commercial Policy 1, ed. by R.C. Feenstra, MIT Press. Bhagwati, J.N. and Srinivasan, T.N., 1980, Revenue-seeking: A generalization of the theory of tariffs, Journal of Political Economy 88, 1069-1087. Goldfeld, S.M. and Quandt, R.E., 1988, Budget constraints, bailouts, and the firm under central planning, Journal of Comparative Economics 12, 502-520. Goldfeld, SM. and Quandt, R.E., 1990a, Output targets, input rationing, and inventories, in: Optimal decisions in markets and planned economies, ed. by R.E. Quandt and D. Tfiska, Westview Press. Goldfeld, S.M. and Quandt, R.E., 1990b, Rationing, defective inputs and Bayesian updates under central planning, Economics of Planning 23, 161-174. Goldfeld, S.M. and Quandt, R.E., 199Oc, Output targets, the soft budget constraint and the firm under central planning, Journal of Economic Behavior and Organization 14, 205-222. Goldfeld, S.M. and Quandt, R.E., 1990d, Input rationing and bailouts under central planning, Jahrbuch der Wirtschaft Osteuropas 14, 17-37. Kornai, J., 1979, Resource-constrained versus demand-constrained, Econometrica 47, 801-820. Kornai, J., 1980, The economics of shortage, Amsterdam: North-Holland. Kornai, J., 1982, Growth, shortage and efficiency, Oxford: Blackwell. Kornai, J., 1986, The soft budget constraint, Kyklos 39, 3-30. Kornai, J. and Weibull, J.W., 1983, Paternalism, buyers’ and sellers’ market, Math. Social Sciences 6, 153-169. Krueger, A.O., 1974, The political economy of a rent-seeking society, American Economic Review 64, 291-303. Milgrom, P. and Roberts, J., 1992, Economics, organization and management, Englewood Cliffs, NJ: Prentice Hall. Tullock, G., 1967, The welfare cost of tariffs, monopolies and theft, Western Economic Journal 5, 224-232.
ELSEVIER
EconomicBehavior & Organization
The competition for rationed resources * Stephen Department
M. Goldfeld,
Richard
E. Quandt
*
of Economics, Princeton Uniuersity, Princeton, NJ 085444-1021,
USA
Received May 1993, final version received September 1993
Abstract The paper considers a model in which firms face an output target and can purchase the input on two dates. On the first date, unlimited purchases are possible, but a carrying cost has to be paid. On the second date the input is rationed, but by spending specialized managerial resources, the firm can influence the rationing density to its own advantage. It also faces a quadratic penalty for missing the target and is assumed to minimize expected net costs. In an n-firm industry, a Cournot-Nash equilibrium in expenditures on the specialized managerial labor is defined and the comparative statics of the model are analyzed. We also find that under collusion the firms’ expected costs, the total expenditure on the otherwise unproductive managerial labor and the variance of the output are smaller than in the Cournot-Nash equilibrium. This provides an insight into why economic planners have tended to prefer to deal with a few very large firms rather than many small ones. Keywords:
Rationing; Cournot-Nash equilibrium; Collusive equilibrium; Planning
JEL classification:
D2; P2
1. Introduction Kornai
(1979,
ing shortages
1980, 1985) has noted
in a planned
economy
that one of the consequences
is that managers
attempt
of anticipat-
to the hoard
inputs
*We are indebted to L&&t Ambrus-Lakatos for comments. As usual, we bear the responsibility for errors. * Corresponding author. 0167-2681/94/$07.00
0 1994 Elsevier Science B.V. All rights reserved
SSDI 0167.2681(94)00050-O
54
S.M. Goldfeld, R.E. Quandi/J.
of Economic Behavior & Org. 25 (1994) 53-71
by making anticipatory purchases in those periods in which an input is freely available. Goldfeld and Quandt (1990a, 199Ob) formalized this notion in a series of simple models. Activity in these models takes place on two consecutive dates. On the first date, t = 0, the manager has the option of purchasing some amount of an input y, denoted by y,. At time t = 1, she may order an additional amount y,. At this later time the amount of the input that can be obtained is rationed and the manager obtains Y,+Y,
y, +x
if y,
if y1 >x,
where x is a random variable with density function h(x), which is the rationing density. Production takes place at time t = 1 after all input deliveries have been made and the manager’s objective is to minimize expected costs. Costs in this model are incurred from two sources: an inventory cost is incurred on the input units purchased at time t = 0 and a quadratic penalty is incurred for failing to meet production targets. Once one posits a production function f(y), and a production target y*, one can easily define the expected cost function E(C). ’ Several variants of models of this general type have been explored. The formulation we describe above represent a case of nonmanipulable rationing in that the manager’s demand has no effect on the amount delivered. This can be relaxed and the model can then be recast in a framework of manipulable rationing. It is also possible to introduce into the model the notion that some unknown fraction of the inputs ordered will turn out to be defective and thus unusable. If inputs can be defective, it is further possible to have the manager learn from his first-period experience and perform Bayesian updating of his estimate of the fraction of defectives. In all variants concerned so far, the enterprise was assumed to act in isolation. The purpose of the present paper is to introduce possible strategic interdependence when an enterprise has to compete against some other enterprise for the scarce inputs allocated by the policy makers. The possibility of lobbying for favorable treatment, which is explored in some detail, creates a connection between the present paper and the pathbreaking papers of Tullock (1967) Krueger (1974) Bhagwati and Srinivasan (1980) and Bhagwati (1983) which deal with various aspects of monopoly seeking, revenue seeking, lobbying, and related unproductive activities. The present paper examines the potential policy conflicts of regulators and sheds some light on the frequently observed phenomenon that socialist
’ In this formulation the manager was not required actually to pay for the amounts of inputs used. This assumption is a simplification that is loosely compatible with the notion of the soft budget constraint in the sense that any amount of actual input cost represents a wash transaction for the manager. The present paper relaxes this assumption.
S.M. Goldfeld, R.E. Qunndt/J.
ofEconomic Behavior & Org. 25 (1994) 53-71
55
planners preferred to deal with highly concentrated industries. In Section 2 we pose the basic elements of the model and analyze the behavior of the individual firm. In Section 3 we discuss the equilibrium in an industry. Section 4 contains some numerical results, while Section 5 contains a brief summary and conclusions.
2. The basic model We model the competition for scarce resources by assuming that the manager is able to influence the amount of the input that is made available to her by employing a specialized form of managerial labor for the purpose of persuading the policy makers to favor her enterprise over that of another manager. The quantity of this labor employed by the ith firm is denoted by zi. This type of ‘whining’ is not unlike the seeking of bailouts that characterizes our models dealing directly with the soft budget constraint (Goldfeld and Quandt, 1988, 199Oc, 1990d). The principal difference is that in the soft budget constraint models the specialized managerial activity is directed at securing cash bailouts, whereas here it is directed at securing more favorable treatment when inputs are being allocated among firms. In general, we shall deal with an n-firm industry competing for the same resource; although, when convenient, we shall specialize the treatment to the 2-firm case. It is helpful to define n
czi,
z-
(1)
j=l
and ZCi)
=
czj. j#i
(2)
Clearly, the dependence of the amount of inputs allocated to the enterprise i must, other things being equal, increase with the amount of zi, decrease with zcij, and the fraction of the total amount of the input x available in the economy must be bounded below by 0 and above by 1. We further assume that the managers competing for x can never, in the aggregate, obtain the entire quantity x, because the policy makers always retain some (variable) fraction of the total x present in the economy, either for precautionary purposes or, possibly, because of corruption among the policy makers. This specification is satisfied by Assumption 1. The amount of input made available to Firm i is (z,/(Z + v)>x, where v is a positive constant set by the policy makers. Hence, ‘nature’ still generates a random x as before; this x is then divided up among the enterprises according to the values of the fraction given in Assumption 1. It should be noted that this formulation also has the realistic feature that a proportionate increase in all firms’ z-values leads to an increase in the proportion of x received by each firm; hence both the absolute level of zs and the ith firm’s proportionate share in
56
S.M. Goldfeld, R.E. Quandt/J.
of Economic Behavior & Org. 25 (1994) 53-71
the aggregate of zs has an effect. The zis represent a kind of rent-seeking by managers and the formulation in Assumption 1 has the effect that the fraction of the total available input retained by the policy makers diminishes as total rent-seeking increases. 2 The next assumption is made for convenience; without it the analysis generally becomes much more difficult. Assumption 2. The production function is linear in the single input. This assumption also allows us to define a net price for the output which is the sales price minus the input cost per unit. We then make Assumption 3. Managers sell the output and purchase inputs at fixed prices; the net price is denoted by p. We also make the following additional assumptions. Assumption 4. (a) The random variable x has density function h(x) over the range (0, a>, with mean p and variance (T *; (b) The unit cost of carrying inventories is the constant cl; (c) The cost of a unit of squared deviation between actual output and target output is the constant c2. None of the essential elements of the analysis is altered by assuming that h(x) has finite support. In addition to the other assumptions of the basic model outlined in Section 1, we also assume that the manager incurs constant unit costs for the employment of zi. This is expressed by Assumption 5. The costs associated with employing the specialized labor are cjzi. It should be noted that throughout most of the analysis we assume that the cost parameters faced by the various firms are identical; hence cl, c2, cg are not indexed by the subscript i. The ith Firm’s Optimization Problem. We consider a general case in which the manager is penalized not only for underproducing, but also for overproducing. The rationale for this is that the overproducing firm deprives other firms of needed inputs; hence such practice is to be discouraged. 3 Let yoi and y* denote the ith firm’s ‘early purchases’ and input target, respectively. The firm’s costs come from (1) inventorying yoi, (2) hiring zi, and
’ For a similar formulation of rent-seeking without a v parameter see Milgrom and Roberts (1992, pp. 573-4) ’ This assumes that if, after the resolution of uncertainty, the firm has obtained inputs that yield y* or greater, the firm produces this amount, even though it will be penalized for overproduction. This is optimal behavior if the policy makers know the technology and penalize the firm for excessive input purchases, whether or not they are actually used for production. Of course, as shown below, if p is sufficiently high, it will be optimal for the firm to assure itself of having sufficient inputs to produce more than y*. More generally, an alternative approach would permit the firm to return excess inputs without penalty, should the firm not wish to use the additional inputs. While this is a seemingly small difference in model specification, the model behaves quite differently, as we plan to show in subsequent work.
SM. Goldfeld, R.E. Quandt/J.
of Economic Behauior & Org. 25 (1994) 53-71
(3) exceeding or falling short of the target. The operating output has expected value
(
57
profit from selling the
‘i
P YOi+-
1’
z+l?
and works to offset the costs incurred for the three reasons above. Hence, it follows immediately that under the assumptions the net expected cost function of the firm is m
E(C) =cly,,+c,zi+c,
o j[
I2h(X)dx
&x
Ytx-yT+
(4) Performing
the integration
in (4) yields
E(C) = (ClpP)YlJi+c~zi-P&P+c2 -&+
+2(yl.X-yT) The first order conditions
(YO,-Yi*)’ [ (&)‘(p2iCr2)
for a minimum
aE(c)
----cc,-p+2c,
are
y,,-yL*+
WC) (Z(i) +v> [
aYOi
= cj +
aZi
2c,
(z+
V)” I
1.
1
&p
=o,
YOi_Y:: - &
2
(6)
I
IJ+ -&$+o2)
= 0.
1
(7)
We can then state Proposition
1. The second order conditions
Proof Obvious from differentiating evaluating the signs of the principal From Eq. (6) we can obtain Cl -P YOi =Y? - ---/_I,. 2c2
Substituting variable:
(6) and (7) with respect to yol and zi and minors of the relevant matrix. n
‘i
(8)
ZflJ
from (8) into (7) yields the ith firm’s reaction function
2c2( =(I, + v) c3+
are satisfied.
(z+vy
‘i
-&+-U [
2
Zfv
*
1
=o.
for its zi
(9)
58
S.M. Goldfeld,R.E. Quandt/ J. of Economic Behavior & Org. 25 (I 994) 53-71
It is clear that corner solutions are possible at which the first-order conditions (6) and (7) are not satisfied. Thus, for example, if the cost of failing to meet the target, c2, is very small relative to the inventory cost, cl, and c, >p or if p is very large (i.e., rationing is unlikely to occur), then (8) might yield a negative yoi and the appropriate solution is yoi = 0. Alternatively, writing (9) as
( zi +
z(i)
f
v)” =
2c2(z(i)
+
‘>
C3
c1
[
+++
GP(‘i
v) -zicr2
)
1
2
(10)
the left hand side has value equal to 0 at zi = -(zcij + V) and has value (zcI, + v>~ at zi = 0. Furthermore, the right hand side of (10) is linear in z, and is positive for either zi = 0 or zi = - ( zcrj + v). Hence, if a positive root exists, it is guaranteed to be unique; moreover, a positive root will exist if and only if This will tend to be the case if c1 or p are large or if c3, the ‘(i) + v < c1 P/‘3. cost of the specialized labor, or zcij + v is small. From (8) it is also clear that if a unique positive root for z, exists, the solution for yoi is also unique. We also note from (8) that if c, > p then yoi < y” ; that is, early purchases will always be less than the required amount of inputs to produce y*. However, if c1
2. (a)
dy,i/dc,
< 0; (b)
dy,j/dc,
(d) dy,,/dyT = 1; (e) dyoi/dz~,) > 0; (f) dY,i/dg2 > 0; (i) dy,i/‘dp > 0; (j) dz,/dc, 0; (m) dzi/‘dy: = 0, (n) dz,/dp > 0; Pro05
> 0 if c, >p;
(c)
dy,/dc,
> 0;
dyoi/dv> 0; (g) dy,,,/dp < o; (h) > 0; (k) dzi/dc2 < 0; (1) dzi/dc, < (0) dz,/da2 < 0; (p) dz,/dp = 0.
The proof is immediate from differentiating the first-order conditions totally and solving for the relevant derivatives. n Discussion. First, consider the derivatives of the variables with respect to the cost parameters. As expected, an increase in the inventory carrying cost decreases Y,,~, but this has to be compensated by an increase in the rent-seeking activity zi. Conversely, an increase in the cost of this activity diminishes its volume, but this has to be compensated by an increase in yoi. It is interesting that an increase in c2 diminishes the optimal zi. The reason for this is that the variance of the output of the ith firm increases with z, (see (13) below); thus, an increased zi increases the risk of missing the target, which becomes more expensive when c2 is higher. Hence zi has to decrease. What is also interesting is that the effect of increasing c2 is not unambiguous; a sufficient condition for dy,,/dc, > 0 is c, > p. From (8) we see that c, > p assures that y,,[ < y,*. When p > cl, it is possible for yoi > yf and for dy,,/dc, to be negative. We illustrate this with some numerical results for the case of industry equilibrium below.
SM. Goldfeld, R.E. Quandt/J.
of Economic Behavior & Org. 25 (1994) 53-71
59
We next observe that dz,/dy: = 0 is a direct consequence of the assumption that the production function is linear, so that the derivatives of the first-order conditions with respect to Y,,~ and y’ are identical (except for sign); it follows then from (8) that the derivative of yOi with respect to y’ has to be 1. Y,,~ increases with p, but again because of the linearity of the production function, it accounts for the entire necessary adjustment, leaving zi unchanged. The response of zi to changes in o2 is as expected: as the uncertainty increases, rent-seeking behavior declines; also, the positive sign of dy,,/da2 is the counterpart of this result. An increase in II, which indicates a reduction in the severity of rationing, leads to a decrease in yai, but an increase in zi. Finally, an increase in V, which measures the policy makers’ proclivity to withhold some X, or in zCi), increases yOi as one would expect, but the effect on zi is ambiguous. Also ambiguous is the effect of zCr) on zi, i.e., the slope of the reaction function. A fuller discussion of the reaction function will be postponed to Section 3 on industry equilibrium. The signs of the derivatives are also compatible with the discussion of the conditions that are likely to produce comer solutions, i.e., solutions at which yOi or zi are zero. In that discussion we noted that the optimal zi is likely to be zero when ci is small or cg is large. This is compatible with the signs of the derivatives dzJdc, and dz,/dc, in Proposition 2. Similarly, the optimal value of ye, is likely to be zero if c1 is high or c2 is small; this again agrees with the predictions of the relevant derivatives in Proposition 2. The Moments of Output. The output, qi, produced by the firms is qi = yoi +
Lx
z+v
(11)
’
and hence expected output is
E(qi)
=YlJi+
$P=Y* -
Cl
-P
2c3
(12)
2
by substituting Eq. (8). Thus the expected output of the firm diminishes with ci, increases with c2 and p, and does not depend on cg. 4 It also follows immediately that the variance of output is 2
var(qi)
=
& i
1
02.
(13)
Hence, the variance of output increases with the amount of zi employed and with the underlying intrinsic variance and diminishes with the quantity employed by other firms and with v.
4 It is clear that expected output generally differs from y:, which need not be the policy makers’ true objective. Indeed, policy makers can always set the value of y* so that their true objective is achieved (in the expected value sense).
S.M. Goldfeld,R.E. Quandt/J.
60
3. Industry
of Economic Behavior & Org. 25 (1994) 53-71
equilibrium
In the present section we consider two types of equilibria: Coumot-Nash equilibria and collusive equilibria. Cournot-Nash Equilibrium. If all firms face the same costs and have the same output target imposed on them, and if a positive yai and zi solves the ith firm’s optimization problem, then those solution values are common to all firms and a Cournot-Nash equilibrium has the feature that yoi =y,
and zi = z Vi.
The solution is obtained by making the above substitutions in the first-order conditions (6) and (7) and solving. The most natural assumption is that each firm’s output target is set at (l/nlth of an aggregate output target y*; thus y* = y*/n. It is also reasonable to consider that the value of c2 may depend on the number of firms in the industry, since in an industry with more firms, each one has to produce less. With an unchanged value of c2, a given aggregate deviation from target output would be penalized less heavily, because it is divided among more firms. Thus it is convenient to denote the value of c2 in an n-firm industry as cZII. The first-order conditions (6) and (7) evaluated at zi = z and yai = y0 then are c,-p+2c,,
1
Y*
y,--+
&P
2n
(nz+
V)”
Y* y” -
[i
(14)
=o>
c,+2c wl)z:4 [
n
P -
~+
2c*, -1
1
z ,,(p2+rr2)
= 0. The equivalent +
c 3
(15) of (9) becomes
2c2n((n-
l)z+ (nz+
Z
4
v)’
-g&L+-, [
2 2n
nz+
v
1.
(16)
=f-J
It is easy to show that the derivative of (16) with respect to z is positive in the neighborhood of a positive root, if one exists. Hence, if a positive root exists, it is unique. Moreover, a necessary and sufficient condition for the existence of a positive root is that (16) have a negative value at the origin, i.e., that c3 - cr p/ < 0. This is essentially the same condition as in the discussion following Eq. (10) for the individual firm, since at the origin zcij = 0. It is relevant to ask whether we can be certain that E(C)OPt < E(CloPt 1*=’ in all cases. It is straightforward to show that
qC)“p’=E(C)~pt’~=O
+
c-z+
f2!& - s!Q+ 24 nz + [
1 .
v
It is also easy to show that the difference between the two E(C) values is negative. Alternatively, one may note that the positive solution for (16) is unique;
S.M. Goldfeld, R.E. Quandt/J.
of Economic Behal,ior & Org. 25 (1994) 53-71
61
hence, even if the expected-cost function is not globally convex, the expected cost corresponding to the solution of (16) must be a global minimum. It is obvious that the expected output of the ith firm is E(q,) = y*/n - (c, and aggregate expected output, obtained by summing over all firms, is P)/2% 5E(qJ i=
=y*
-
4Cl -P> 2%
1
(17) .
Thus, to keep expected industry output invariant with respect to the number of firms, the penalty for missing the target must be so adjusted that c*,,/n = constant}. In particular, if c2 is the value of c2 in an industry with a single firm, then we must have c2,, = q. Comparative Statics for the Cow-not-Nash Equilibrium. Comparably to Proposition 2, we can obtain
ProPosition3.(a) d.~,/dc,
dy,/dc,,
> 0
ifc, >p;
(c) dy,/&,
> 0; (d)
dY”/dY,! = l/n; (e) dy,/dv > 0; (f) dy,,/dp < 0; (g) dy,/d~ 2 > 0; (h) dy,/dp > 0; (i) dz/dc, > 0; (j) dz/dc,. < 0; (k) dz/dc, < 0; (I) dz/dyT = o, (m) dz/dr.L > 0; (n) dz/da2 < 0; (0) dz/dp = 0. Prooj
Letting all yOi have the common value y,, and letting all zi have the common value z in the first-order conditions (6) and (71, differentiating totally and solving for the derivatives yields the resu1t.m These results are identical to the ones in Proposition 2, but note that the derivative of y, with respect to y * is l/n rather than 1. It should be noted that the derivative dz/dv is ambiguous in sign. What one can show (see Appendix) is that the elasticity vdz/zdv < 1, which will play a role in the proof of Proposition 6. Computational results reported below indicate that for small values of V, the derivative is positive, whereas for large values of v it is negative. Hence, if the starting level of v is reasonably modest and policy makers withhold more of x because of corruption, rent-seeking increases by everybody. This phenomenon can then be labelled as ‘waste begets more waste.’ If v is already large, and it increases further, firms respond by reducing waste. It should also be noted that what constitutes ‘small’ or ‘large’ values of Y depends on the values of the other parameters. It is worth noting that if v is zero, (16) permits an explicit solution for z, which is 2c,,(n
- 1)
Cl P
z= n2c3
[z-f12].
This is meaningful only if it yields a nonnegative solution, which is the case when cr p/2c2,, 2 u2. Hence, rent seeking behavior will occur only if the
62
S.M. Goldfeld,R.E. Quandt/ .I. of Economic Behavior & Org. 25 (1994) 53-71
variance of the aggregate available input is not too large; otherwise it simply is not worthwhile to engage in such behavior. Since c2,, = nc2, as n increases, the condition for a nonnegative solution will eventually fail and the corner solution z = 0 will be obtained. Hence, when v = 0, rent seeking behavior disappears if it is large enough. This is not the case when v > 0, as we can see by examining the limiting behavior of nz, ny, and &Z(C). For this we have Proposition
4.
(a) lim, +cclzz = constant; = constant.
lim n ~ ,nE(C)
(b)
lim,,,ny,
= constant;
(c)
Proof
(a) The behavior of nz can be analyzed by examining Eq. (16). Multiplying through by (nz + v)~, replacing nz by the symbol w and cZn by IZC*, and simplifying, leads to the following equation: Cl k
3c,v-c,/_L+
c,w3+
-
2c,&
II
[
Cl
w2
+2c,u2-n
PI,
1
I
(18)
+ 3c,u2 - 2c, /_&V + + 2c,a=v w + c3v3 - clv/.L=o. n [ If the conditions for a positive (unique) z solution hold, there will be a positive and unique solution for w for any value of n; moreover, the coefficients of the equation converge as n + 00; hence nz is a constant in the limit. (b) From (14), the solution for y, is Y* ” = T
Cl -P --5’“. 2nc,
-
(19)
Multiplying by n shows that ny, depends only on nz, which converges to a limit. (c) Replacing in (5) c2 by ncz, y,, by y,, y* by y*/n, and multiplying by n shows that nE(C) also depends on n only through nz and hence converges to a constant. n Proposition 4 thus shows that the aggregate amount of rentseeking, as well as the aggregate amount of early purchases and the aggregate cost incurred all stabilize as n becomes large and thus total waste remains bounded. Analogously to (13), the variance of aggregate output can be written as 2
var(q)
=
i
2
1
(20)
a2.
From Proposition 4 we see that the variance converges to a constant as n gets large. We now examine in more detail how z, nz and the variance of q depend on n. We have Proposition
5. (a) dz/dn
< 0; (b) d(nz)/dn
> 0; (c) dvar(q)/dn
> 0.
S.M. Goldfeld, RX. Quandt/J.
of Economic Behavior & Org. 25 (1994) 53-71
63
Proof For proof, see the Appendix. We thus note that an increase in the number of firms (with compensating adjustment in cZn) reduces the amount of z employed by each firm, but increases the aggregate amount employed, and increases the variance of aggregate output. Letting
+nz
(21)
nz+ v’
be the firms’ share of the rationed input, we have a!s -= dv
n (nz+
1
dz v----z. v)’ [ dv
(22)
As noted above, it is proved in the Appendix that the bracket in (22) is negative. Hence, dS/dv < 0, even though dz/dv is ambiguous in sign. From (20) it follows that
(23) which is therefore also negative. Proposition
6. (a> dvar(q)/dv
This proves (a) of < 0; (b) aE(CYp’/av
> 0.
For the proof of (b), see the Appendix. To interpret these results, we may consider the parameter v to be exogenously given or to be set by a policy maker. In the latter case, policy makers face a conflict in that increasing the value of v reduces the variance but increases expected costs. There may be a further conflict in that, depending on the value of v, rent-seeking behavior may either rise or fall. Hence the policy makers’ optimal action will depend on the precise form of their loss function (which we do not specify in the present paper). Collusive Equilibrium. The procedure in the Cournot-Nash case was to find the first order conditions for the individual firm and then set zi = z and yoi = ya. In the presence of collusion, the objective function is written from the beginning in terms of common z and y, variables and is then differentiated with respect to these. The expected cost function is *
Z E(C)
=
(Cl
-P)Yo
+c,z-P-
P+cc,” nz+
+(Yo-~)--&‘+(-$)*w+~*~].
v
2
[( ) Yo-$
(24)
64
S.M. Goldfeld, R.E. Quandt/J.
The derivative becomes
dE(C)/dy,
WC)
___
is the same as (14). However, the derivative
= cg + 2Czn
az
of Economic Behavior & Org. 25 (1994) 53-71
K
ZU
+(nz+ u)
Y*
P
LIE(C)/&
IJv
y"- L- - 23, i (nz+ v)"
3(/.L2+cT2) =o.
(25)
1
We note that it can be routinely verified that the second order conditions satisfied. Substituting from (14) into (25) yields Z
2c2nv c3
+Lf---~
+ (nz+
are
v)”
i
2 2n
nz+
Y
This contrasts with (16) in the Cournot-Nash
(26)
=o. i
equilibrium.
This leads to
Proposition 7. The solution for z under collusion is smaller than for the CournotNash equilibrium. Prooj At z = 0, (16) and (26) have identical values. For any other positive z-value, (12), plotted as a function of z, lies above (16), because subtracting (16) from (26) yields -
2c,,(n
- 1)z --&f-&o2
(nz + V)’
[
2n
1
>o.
Hence (26) must cross the z-axis to the left of (16). n The result suggests that rent-seeking activities are lessened by allowing collusion, i.e., by letting all firms behave as one. It also follows from (20) that the variance of output under collusion is less than in the Cournot-Nash equilibrium, even though expected output is the same. Both of these observations are consistent with the results of Proposition 5 and, in turn, suggest that other things being equal, policy makers who wish to reduce rent-seeking behavior as well as output variance would find it desirable to encourage the formation of monopolies. This then appears to be an economic justification for the ‘gigantism’ encountered in the industrial structure of planned economies. Comparison of Expected Costs. We next compare the level of optimal expected costs under Cournot-Nash equilibrium and collusion. The optimal value under Cournot-Nash equilibrium is obtained by replacing in (5) zi by z, yoi by yi, y* by y*/n and substituting for y, the optimal solution from (19). The optimal expected cost under collusion is obtained from (24) into which we substitute from (19). Denoting by k = c,cn the collusive and Cournot-Nash cases respectively, it
SM. Goldfeld, R.E. Quandt/J.
is easily seen that the expected cost functions identical in form and can be written as
E(Ck) = (Cl
65
of Economic Behavior & Org. 25 (1994) 53-71
-p)f - (Cl4c--P>’+c
condensed
with respect to y, are
CIWk+ C2ng2(zk)2
Zk_
3
nzk+
2n
1,
2 (nzk+
.
v)
(27) Hence the relative magnitudes of the optimal expected costs will depend only on the last three terms in (271, which can be written as 2 r=c,z-
Cl PZ nz+ u
C2ng +(
nz
2 Z
+
v)”
’
where we have omitted the superscript
ar --cc,+ 8Z
Cl
2C2nY
P
-(nz
+
v)”
[
2C2n
k. Differentiating, Z +
-CT nz+v
2 I
’
we obtain
(29)
which is identical to (26) and thus must equal zero in collusive equilibrium in order to satisfy the appropriate first order condition. Since the second order conditions are satisfied and Proposition 7 implies that zc” > zc, we have proved Proposition
8. E(CC) < E(F).
This then provides another reason why policy makers, intent on minimizing aggregate costs, might prefer monopoly firms to less concentrated industries. Comparative Statics under Collusion. Analogously to Proposition 3, we have Proposition 9. (a> dyO/dc, < 0; (b) dy,/dc,, > 0 ifc, >p; (c) dy,/dc, > 0; (d) dy,/dy* = l/n; (e) dy,/dp > 0; (f) dy,/dp < 0; (g> dy,/du2 > 0; (h) dz/dc, > 0; (i> dz/dc,, < 0; (j> dz/dc, < 0; (k) dz/dy* = 0; (E) dz/dp = 0; (m) dz/dp > 0; (n) dz/du* > 0. ProofI The results follow directly from total differentiation of the first order conditions. Hence the signs of the derivatives under collusion are identical to the signs under Cournot-Nash equilibrium. Analogously to Proposition 4, we finally examine the behavior of nz, ny, and nE(C). We start with Equ. (26) in which we replace c2,, by nc2. It is immediately obvious that this yields a cubic in (nz + v), with none of the coefficients depending on n at all; hence nz is a constant itself. Using (141 and (24) shows that ny, and nE(C) depend on n only through nz, which is a constant. We thus have proved Proposition to n.
10. In the collusive case, nz, ny, and nE(C) are invariant with respect
66
SM. Goldfeld, R.E. Quandt/J.
4. Numerical
of Economic Behavior & Org. 25 (1994) 53-71
results
While most of the comparative statics results are unambiguous, in a small number of instances we were not able to derive unambiguous conclusions. This is the case for dy,/dc,,, in Proposition 3, where the sign is unambiguous only if c1 > p, and is also the case for dz/dv in Cournot-Nash equilibrium, where there does not appear to exist a simple condition that allows the derivative to be signed. Finally, we have not dealt at all in the preceding sections with the asymmetric case in which the costs pertaining to different firms are different, because the interpretation of the formulas becomes exceedingly difficult. The objective of the present section is to shed some more light on these questions by performing computations or plotting the reaction functions for particular values of the parameters. We first consider the case of the optimal y, in a two-firm industry as czn varies. For the example that follows we assume that c1 = 20, cg = 10, v = 1, y* = 100, p = 15 and CT’ = 25. Table 1 displays the Cournot-Nash equilibrium values of y, for various values of p and czn. All the p-values are substantially greater than cl, which (b) of Proposition 3 suggests might be needed to cause an increase in czn to result in a decrease in y,, yet for p = 90, y, still increases. When p = 125 or p = 150, y, first decreases, but for high enough c1 it increases again. It takes a p of 200 before y0 uniformly decreases within the Table’s range of czn variation. It is also worth noting that the four instances in which y0 > 50 represent cases in which the total amount ordered ‘up front’ by the two firms is more than the industry target y*; this phenomenon also appears to require large values of p. In order to examine the variation in industry equilibrium values of z when v varies, we display in Figure 1 three pairs of reaction functions corresponding to the case when c1 = 20, c2,, = 10, c3 = 10, /_L= 15, (T* = 25 (The value of p and of y* are irrelevant because these parameters do not enter Equ. (161.1 The values of v are 1, 7, and 13 respectively, and the three equilibria correspond to the intersections of the appropriate reaction functions. It is obvious from the figure that the equilibrium value of z increases as v moves from 1 to 7, but decreases again when v increases further to 13. Other calculations suggested that it was a general phenomenon that dz/dv > 0 for small v and negative for large values. The last case we examine is the asymmetric one in which p = 30, u* = 10, v = 1, Firm 1 has the cost parameters c1 = 10, c2. = 3, cg = 6 and in which Firm 2
Table 1 Values of y,
%?n
p = 90
p=
10 20 30
47.3 47.7 48.4
49.1 48.6 48.9
125
p = 150
p = 200
50.3 49.2 49.4
52.8 50.4 50.2
S.M. Goldfeld, R.E. Quandt/J.
ofEconomic Behauior & Org. 25 (1994) 53-71
67
c1 = 20, c2 = 10, c3 = 10, p = 15, 02 = 25
Fig. 1.
has the same c1 and c2 as Firm 1, but has a ca that is alternately set at 3, 6, or 12. We are thus looking for the intersection of Firm l’s reaction function with one of Firm 2’s three possible reaction functions. It is clear that the optimal value of z2 declines monotonically as Firm 2’s cg increases; however, Firm l’s optimal zr first increases and then declines. It is clear that in the asymmetric case a variety of such reversals are possible. 5. Concluding
comments
The paper presents an analysis of a model in which an input is rationed, firms can expend resources in order to make the rationing less stringent, and can also try
68
SM. Goldfeld, R.E. Quandt / J.
ofEconomic Behavior Xr Org. 25 (1994) 53-71
50 ;I Fig. 2.
to avoid the bite of rationing by buying the rationed good at an earlier time when rationing is absent. While the model produces a number of reasonable conclusions, it is to be taken only as a first approximation and a number of relevant and realistic extensions remain for future research. 1. The model posits that firms employ all the inputs that they have acquired, even if this causes them to produce more than the target and even if such overproduction is penalized. In an alternative version, the firm might be able to return the excess input at no cost. While this appears to be a minor modification, it appears to have major effects on the analytics of the model. 2. The entire discussion in the present paper is based on the assumption that the production function is linear. The special nature of this assumption has had some beneficial effects in that it makes the model more tractable. However, a
SM. Goldfeld, R.E. Quandt/J.
of Economic Behavior & Org. 25 (1994) 53-71
69
number of specific effects clearly result from linearity (see, for example, the discussion of Proposition 2), and it would be desirable to relax this assumption, to see what impact this has on the conclusions. Losses accrued by the firm are borne entirely by the firm, but persistent losses are not compatible with survival; hence it would be of interest to see what effect bailouts by policy makers have on the results. Throughout the analysis, the parameter v is assumed to be given and known with certainty by the firms. In reality, v may not be known with certainty and individual firms may have prior densities over the value of V. The effect of such an assumption on the results would be of considerable interest. Some of the analysis has indicated that policy makers may face a conflict in setting the value of v or of II, the number of firms in the industry, or indeed whether to discourage collusion. We know that the variance of output falls, but expected costs rise as v increases; moreover, the variance of output rises with II, but expected costs are smaller if the industry acts as a monopoly. For these reasons it is of relevance to examine whether the introduction of an explicit loss function for the policy makers can shed some additional light on these conflicts. Finally, to the extent that such a loss function exists and is known to be employed, new incentives will be created for the managers and their behavior may change; conversely, firms may act as followers and policy makers as Stackelberg leaders. We hope to explore these rich possibilities in further work.
Appendix Proof of Proposition A=-
5. (a) We first start by defining
&+-, 2n
Z nz+v
2
(A.11
’
which is the bracket in Eq. (16) and is therefore negative;
we also define
B=nA.
(‘4.2)
We start with Eq. (16) in which c2” has been replaced totally with respect to n and solving for dz/dn yields dz
-=
dn
z(nz+
v)((n-2)z+
v)B-
vzu2((n-
by nc2. Differentiating
l)z+
V)
-(nz+v)[n(n-1)z+(n+1)v]B+a2((n-1)~++)~~’ (A.3)
It follows immediately that the derivative is negative. (b) Treating dz/dn as a fraction, we see immediately that the sign of d(nz)/dn depends on the sign of ndz + zdn, since the denominator of dz/dn is positive.
70
SM. Goldfeld, R.E. Quandt / J. of Economic Behavior & Org. 25 (1994) 53-71
Multiplying the numerator by n, the denominator by -z(nz + v)‘B < 0, as claimed. (c) Differentiating (18) yields after some rearrangement
d-4 q) dn
d( nz) -
2u*nzu =
(nz+
V)’
z and
adding
yields
>o,
dn
as claimed. W Proof
of vdz/dv
- z < 0. Differentiating
(16) totally
and solving
of dz/dv
yields nz + v)* + 2(nz + v)((n - 1)~ + v)]A + za2((n
;=I[-(
X{[(m+
v)*(n - 1) - 2n(nz + v)((n
- 1)~ + v)]A
- 1)~ + v)} + va’((n
- 1)~ + v)}-‘. (A.41
The coefficient of A in the denominator of (A.4) simplifies to (nz + v>[ - n( n l>z - v(n + l)] and hence the denominator is positive. Considering dz/dv as a fraction, the sign of vdz/dv - z depends on the sign of vdz - zdv. The numerator of dz/dv times v minus z times the denominator of this derivative simplifies to (nz + v)*((n - l)z + VIA, which is negative, thus proving the claim. n Proof
of Proposition
which the superscript
6. The optimal expected costs are given by Equ. (25) (in k may be ignored). Hence 2C2nV (nz+
where
1
A
dt_
v)”
dV
A is given
%nZ A,
(nz+
v)’
(A.51
by (A.l). Now substitute into (AS) from (A.4), considering as a fraction; i.e., denoting by dz the numerator and by dv the denominator of the right hand side of (A.4). After tedious algebra, the expression simplifies to dz/dv
c,dz +
2c2,A (nzf
v)’
[vdz-zdv].
Substituting for the bracket noting from (16) that cg = -
2c,,((n-
l)z+ (nz+
(A.6)
in (A.6) from the derivation
= lb
-2c,,((n
(A.4)
and
v)A
v)’
’
we obtain
aq cypt
following
- l)z+
v)A
(nzf v)’1, dz
-A
(A.7)
S.M. Goldfeld, R.E. Quandt/J.
of Economic Behavior & Org. 25 (1994) 53-71
and hence we need the sign of the bracket in (A.7). Multiplying becomes - 2(nz + v)zA + zu’((n - 1)~ + v) > 0, and since A > 0 as claimed.
71
out, the bracket
< 0,E(C)"P'/tb
References Bhagwati, J.N., 1983, Directly unproductive, profit-seeking (DUP) activities, The Theory of Commercial Policy 1, ed. by R.C. Feenstra, MIT Press. Bhagwati, J.N. and Srinivasan, T.N., 1980, Revenue-seeking: A generalization of the theory of tariffs, Journal of Political Economy 88, 1069-1087. Goldfeld, S.M. and Quandt, R.E., 1988, Budget constraints, bailouts, and the firm under central planning, Journal of Comparative Economics 12, 502-520. Goldfeld, SM. and Quandt, R.E., 1990a, Output targets, input rationing, and inventories, in: Optimal decisions in markets and planned economies, ed. by R.E. Quandt and D. Tfiska, Westview Press. Goldfeld, S.M. and Quandt, R.E., 1990b, Rationing, defective inputs and Bayesian updates under central planning, Economics of Planning 23, 161-174. Goldfeld, S.M. and Quandt, R.E., 199Oc, Output targets, the soft budget constraint and the firm under central planning, Journal of Economic Behavior and Organization 14, 205-222. Goldfeld, S.M. and Quandt, R.E., 1990d, Input rationing and bailouts under central planning, Jahrbuch der Wirtschaft Osteuropas 14, 17-37. Kornai, J., 1979, Resource-constrained versus demand-constrained, Econometrica 47, 801-820. Kornai, J., 1980, The economics of shortage, Amsterdam: North-Holland. Kornai, J., 1982, Growth, shortage and efficiency, Oxford: Blackwell. Kornai, J., 1986, The soft budget constraint, Kyklos 39, 3-30. Kornai, J. and Weibull, J.W., 1983, Paternalism, buyers’ and sellers’ market, Math. Social Sciences 6, 153-169. Krueger, A.O., 1974, The political economy of a rent-seeking society, American Economic Review 64, 291-303. Milgrom, P. and Roberts, J., 1992, Economics, organization and management, Englewood Cliffs, NJ: Prentice Hall. Tullock, G., 1967, The welfare cost of tariffs, monopolies and theft, Western Economic Journal 5, 224-232.