Uniqueness of Non-Walrasian Equilibrium in the Macroeconomic Model with Decreasing or Increasing...

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Uniqueness of Non-Walrasian Equilibrium in the Macroeconomic Model with Decreasing orIncreasing ReturnsAuthor(s): Paul MaddenSource: The Review of Economic Studies, Vol. 52, No. 4 (Oct., 1985), pp. 703-713Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2297741 .

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Review of Economic Studies (1985) LII, 703-713 0034-6527/85/00480703$02.00

? 1985 The Society for Economic Analysis Limited

Uniqueness of Non-Wairasian

Equilibrium in the Macroeconomic

Model with Decreasing or

Increasing Returns PAUL MADDEN

University of Manchester

A class of three good (output, labour, money), one consumer private ownership economies is studied. With an exogenous number of identical, increasing or decreasing returns firms necessary and sufficient conditions for globally unique, non-trivial, uniform rationing non-Walrasian equilibria are obtained in terms of suitably defined "marginal propensities". Following Weddepohl a non-Walrasian equilibrium at a set of parameters with m firms producing is stable if there is no equilibrium at those parameters with m + 1 firms producing. Under the global uniqueness conditions only trivial stable equilibria occur under decreasing returns; under increasing returns multiple, non-trivial stable equilibria can occur.

1. INTRODUCTION

Following the studies of Barro and Grossman (1976) and Malinvaud (1977), Laroque (1978) has studied a macroeconomic model with three goods (output, labour services and money), a single firm producing output from labour under decreasing returns and a single consumer who receives the firms profits. Assuming parametric prices, "fixed-price" or "non-Walrasian" equilibria (NW) of this economy can be studied where imposition of quantity constraints on demand or supply (but not both) of non-money goods generates compatible desired transactions. The Walrasian equilibrium (W) is a special NW where no quantity constraints are needed to attain compatibility. For Laroque's macroeconomic model there is a unique price leading to W and Laroque (1978,1981) and Madden (1982a) provide local studies of the set of NW near W. In particular if the marginal propensity to consume is less than one and the marginal propensity to leisure is less than one, then at any price close to the W price there is just one NW allocation close to the W allocation (local uniqueness). Moreover under a regularity condition this "marginal propensities" assumption is necessary for local uniqueness. We show here that under a slightly different definition of these marginal propensities, their values being less than one is sufficient and (almost) necessary for global uniqueness (i.e. at all prices there is exactly one NW). This is a strong result; Schultz (1983) has provided sufficient conditions for global uniqueness in general exchange economies which use "marginal propensity" types of assumptions. Given the interest in "disequilibrium macroeconomics" our stronger result seems worth knowing. We show furthermore that the analysis can be easily applied to an increasing returns macroeconomic model as well. Here when marginal propensities are less than one there will typically be either one NW (a trivial equilibrium involving zero trade) or two NW (one trivial and one non-trivial). There is a "bootstrap" feature to the increasing returns model. Weddepohl (1982) has analysed a three good increasing returns

703



704 REVIEW OF ECONOMIC STUDIES

macroeconomic model which differs from ours in a number of aspects including the profit distribution assumption. Again our sufficient and almost necessary assumptions seem better than the corresponding sufficient results of Weddepohl. We conclude with some preliminary remarks on how the stories change when there are m identical firms. The increasing returns case seems to hold some particularly interesting potential here.

2. THE ECONOMY

There are three goods, money (good 0), output (1), and labour services (2) with prices po = 1, P, and P2 which are assumed strictly positive; P = (Pi, P2). There is one firm in the economy which produces output from labour services according to the twice differentiable production function f(l) where I denotes the employment level of labour services. We assume throughout that f(O) = 0 and f'(1) > 0 for 1 > 0. We are interested in two models, the decreasing returns model (DR) and the increasing returns model (IR) defined by the following alternative, additional assumptions on f(l):

(DR) f'(l) < O for I > 0

(IR) AP'(l) >0for1>0where AP(l)=f(l)/l. Also f'(O)=O.

So the increasing returns model is defined by increasing average products: the additional assumption that f'(0) =0 is for expositional convenience only and is not essential.

Both models are completed with a consumption side which consists of a single ("representative") consumer/worker who is endowed with money (w0> 0) and leisure (w2>0) and has no endowment of output (w1 = 0). The consumer's consumption set is X = {(q0, ql, q2)Iqo, q1 _ 0, w2_ q2? 0} and the consumer's preferences are represented by a utility function u(qo, ql, q2) which is strictly increasing, strictly quasi-concave, twice continuously differentiable on X and which exhibits non-zero Gaussian curvature on X. The consumer receives the firm's profits.

A non-Walrasian equilibrium (NW) at p is defined by a set of quantity constraints which constitute upper bounds on output demand, output supply, labour demand and labour supply with the properties:

(i) Given p and the quantity constraints, agents' desired transactions are compatible. (ii) If the output transaction equals the output demand quantity constraint then the

output supply quantity constraint exceeds (in absolute value) the output transaction; and vice versa. Similarly for the labour transaction.

A more precise statement of this definition is given in Madden (1982a, p. 662) if the reader wishes. The assumption that profits are paid out to the consumer is very fruitful since in any NW the firm must then have zero net trade in money. Compatibility of transactions in money then ensures that in any NW the consumer will end up with the money endowment (w0). Moreover if I is the employment level in a NW we know that production will be f(l) since f'(l) >0 and the consumer will end up with (q0, ql, q2) =

(Wo f(l), W2 - 1). Hence I is sufficient to describe any NW. It is this feature which formally allows the strong "sufficient and almost necessary" uniqueness results derived here.

The following notation for the consumers "marginal rates of substitution" on the production function with the money endowment is used frequently in what follows:

LTh(1) U( uWO f( l), w2- 1)' h = 1, 2.

Uh(*) denotes the partial derivative of u( - ) with respect to qh, h = 0, 1, 2.



MADDEN UNIQUE NON-WALRASIAN EQUILIBRIA 705

The structure of the NW is uncovered via study of the isoemployment curves, I(l):

I(1) = {(P1, P2)I3 a NW at p with employment level 1}.

3. THE DECREASING RETURNS MODEL

Walrasian equilibrium is a special NW where all quantity constraints exceed the transaction. For the decreasing returns model there is a unique Walrasian equilibrium employment level which we denote 1*: we assume zero trade is inefficient so that 1* > 0. Straightforward extensions of the arguments of Madden (1982a) gives the following isoemployment curve properties:

Lemma 1. (a) For 0 < < W2, I(l) = {(Pl, P2) If(l) P2/P1, P2 0_2(l) Pl _o (l) and at least one inequality holds with equality}

(b) I(0) = {(Pl, P2)If'(0) -P2/P1 or P2? 02(0) orP1 p o1(0)} (C) I(W2) = {(Pi, P2)lf(w2) -P2/P1, P2 o-2(W2) Pl '-(W2) (d) If l* < W2, I(l*) = (O.-(l*), 0.2(1*)); if 1* = W2, I(l*) = I(w2) in (c). (e) If I>1*,I(1)=0.

I(1*) are Walrasian equilibrium prices, unique if 1* < w2. Figure 1 illustrates the isoemployment curves implied by (a), (b) and (c) assuming nonemptiness: the labels KU, CU, RI in (a) are Keynesian unemployment, classical unemployment and repressed inflation in the Malinvaud (1977) terminology. Notice from Lemma 1(e) that if I(w2) ? 0 then 1*= w2 and Figure l(c) depicts the set of Walrasian equilibrium prices. Thus I(l*)

(a) (b) (c) P2 p~~~~~~~~~~2 P2

'p

p~~~~~~~~~~~~~~~~~~~~~~~

2(0)~~~~~~~~~~~~~~~~~~~~~~~~~~~

-Q t'(w2)

CU KU

(2 (w )~==-~?-

(1 pi 11 C1 w2 1i

FIGURE 1.

is either a single point, (ufi(l*), o-2(1*)) or the shaded region in Figure l(c). As l falls from 1*, I(l) becomes a non-degenerate triangle typically as shown in Figure l(a), for 0 < I < 1*, culminating in the shaded region I(0) of Figure 1(b) when I = 0. Since f'(l) < 0 in this model, the XZ sides of the triangles in Figure 1 increase in slope as I falls from 1* to 0. Clearly the I(l) sets will be non-intersecting if we assume:

(Gi) oA(1l)<0,forall IG(0, 1*)

(G2) o(1) > 0, for all 1 c (0, 1*).

Hence (Gl) and (G2) are sufficient to ensure global uniqueness of NW: i.e. (Gl) and (G2) imply there is exactly one NW allocation at all p. Moreover if we assume:




Assumption 1. oI(l) $O, a'(1) l) ?, l E (O, I*)

then (Gi) and (G2) are necessary for global uniqueness. Visually if (Gi) and (G2) hold then as I falls from l* the Y corners of the triangles move to the south-east. If either (Gi) or (G2) is reversed the Y corners move so as to generate intersecting triangles and non-uniqueness. Hence

Lemma 2. Under Assumption 1, (Gi) and (G2) are necessary and sufficientfor global uniqueness of NW.

(Gi) and (G2) are not very informative. In Laroque (1978), Madden (1982a) it was seen that (Gi) and (G2) hold at 1* if and only if, on certain definitions of the terms, the marginal propensity to consume (MPC) and the marginal propensity to leisure (MPL) are less than one giving rise to necessary and sufficient conditions for local uniqueness of NW close to the Walrasian equilibrium. Holding profits and prices at their Walrasian levels, quantity constrain the consumer's supply of labour to equal I say. Demand for output becomes a function of this ration and the MPC concept used in the local studies Laroque (1978), Madden (1982a) is the derivative of the demand for output with respect to supply of output (f( I)) as this ration changes. In value terms this is the change in desired expenditure on output as income (f(l)) changes, which ties in with the usual usage of the MPC term. MPL measures the change in labour supply with respect to labour demand as the demandfor output is quantity constrained away from its Walrasian levels, prices and profits remaining at their Walrasian values. We now show that interesting global characterisations of (Gi) and (G2) exist in terms of slightly different MPC and MPL concepts.

Consider any borderline KU/RI equilibrium: i.e. the economy is at a point like Y in Figure 1(a) with employment level 1I (0, 1*). Let p be the prices. In this equilibrium the consumer is constrained on neither market. Now keep p fixed and quantity constrain the consumer to a labour supply equal to 1, near 1. Assume that the consumer anticipates profits consistent with this ration: i.e. plf( 1) -P2 1. Then output demand as a function of the ration q1( 1) solves:

max u(qo, q1, w2- 1)

subject to wO+P2l+ (p1f(l) -P2l) = qo+plql

and

(qo, q1, w2- E)X.

Then the new MPC definition is aq1( l)/af( 1) = qj( I)/f'( I) evaluated at F= 1. We write this MPC (1).

Analogously keep p fixed, quantity constrained output demand to c1 nearf(l), assume profits consistent with this ration (i.e. Plqi P2f -1(l)) and l(q1) solves:

max u(qo, q1, W2- 1)

subject to wO+p21+ (pi'h -P2f 1(41)) =qo+p

and

(qo, 41, w2-l)EX.

The new MPL definition is l(q1)/af1(q1)=f'[f1(q1)]. l'(q1) evaluated at 41=f(l), written MPL (1).




Lemma3. (a) (G1)4.'MPC(1)<1, lz(0, 1*), (b) (G2)4.'MPL(1)<1, l:(O,1*).

Proof (a) For I close enough to 1 e (0, 1*), q1( 1) is defined by:

u1[wo+plf(l)-p1ql, ql, W2- 1]

uo[wo + plf( I )-P1 ql, ql, W2-

Differentiating with respect to 1, substituting I = I and rearranging gives

q1(l) dul a ul a u1 a u1

aq, uo aqo uo

where all right-hand side derivatives are evaluated at (wo, f(l), W2 - 1). The denominator on the right-hand side is negative since Pi = o-r(l) and the denominator is then the (1, 1) element of the Antonelli matrix at (wo, f(l), W2 - 1). Under our assumptions this matrix is negative definite (see Katzner (1970)). Rearranging, MPC (1) < 1 if and only if

f'(I) -a u a < aq, Uo aq2 UO

derivatives evaluated at (wo, f( 1), W2 - 1). Straightforwardly we find

a zau ul aq, uo aq2 UO'

derivatives evaluated at (wo,f(l), w2-1). Hence MPC (1) <1 if and only if o'(l) <0 as required. The proof of (b) is similar.

Hence we have:

Theorem 1. Under Assumption 1, the NW are globally unique if and only if MPC (1) < 1 and MPL (I) < 1 for I E (O, 1*).

Figure 2 illustrates the typical decomposition of NW into regimes if the hypothesis of Theorem 1 are satisfied and 1* < w2.

P2 -,2 22 ~ ~ ~ ~ ~ - -,/ p2f'(o)

pi 1 )

Trivial B

CU KU Trivial

RI

2 '' / I II TrivialI

F I G (R ) .P

FIGURE 2.




4. THE INCREASING RETURNS MODEL

Things are quite different under increasing returns. To illustrate this most forcefully we make use of our expositional simplification in this case that f'(0) = 0. As in Theorem 1 of Madden (1984) we then find that there exists a trivial equilibrium at all positive prices.

Lemma 4. I(0) D R ++.

Proof Since f'(0) =0 for any strictly positive price there are non-zero quantity constraints on labour demand and output supply such that the firm's desired transaction is zero. No profits are then made and constraining labour supply and output demand to zero generates zero trade as a NW. ||

A non-trivial equilibrium now becomes a "multiple equilibrium". There is a "bootstrap" feature of the increasing returns model. Firms need to anticipate certain minimum levels of labour supply and output demand to overcome the increasing returns. If their expectations fall short of this the economy is stuck at no trade.

The following result whose proof is straightforward provides the required insight into non-trivial equilibria:

Lemma 5. (a) For W2> 1>0,I(I) = {(P, P2)IAP(l) -P2/P1 P2~-02(l),P1 _i ?1(I) and either P2 = 02(1) or p1 = o1(1)1-

(b) I(W2) = {(Pl, P2)IAP(l) _P2/P1 P2 _02(l), Pi _ c01(1)}

Figure 3 illustrates Lemma 5 assuming I(1) 0: CU is not now possible as the firm must be constrained under increasing returns.

The condition for I(1) $ 0 is AP(I) _ r02(1)/01o(l). The set of I values which satisfies this condition does not seem to have any simple features. In general it will be a sequence of intervals [11, 12], [13, 14], . . ., [II-,, II], where 1i ' l+l

(a) (b) P2 P2

p I P 7/ p-=AP(,) |,, - =AP(w2)

- (J) KU aT (w )------ 2 2~~~~~~~~

2 V)RI /

p1 " a(W2) Pi FIGURE 3.

As I decreases AP(1) decreases and from Figure 3 sufficient conditions for at most one non-trivial equilibrium at any p are o-'(l) < 0, o-'(1) > 0 for I in the interiors of the above intervals. With Assumption 1 these conditions become sufficient and Lemma 3's characterisation of them remains intact, giving:




Theorem2. Under Assumption 1, there is at most one non-trivial NW at any positive price vector if and only if MPC (I) < 1 and MPL (1) < 1 for I E (11, 12) U(13, 14) U U(II-1, II).

Figure 4 is the analogue of Figure 2 for increasing returns assuming just two "non-trivial regions" and I(w2) =0.

P2 KU

RI*

Trivial everywhere

pi FIGURE 4.

Remark. The analysis presented here differs significantly from that of Weddepohl: (i) Weddepohl requires non-binding quantity constraints to be at +00 which generates fewer trivial equilibria (ii) we have sufficient and almost necessary conditions for global uniqueness of non-trivial equilibria whereas Weddepohl has only sufficiency (iii) these conditions are consistent with multiple "non-trivial regions" as shown in Figure 4; Weddepohl rules this out with an ad hoc assumption regarding concavity of a demand function and yet their possibility seems interesting (iv) we assume profits are distributed (v) Weddepohl's primitive data describing the consumption side of the economy are quantity constrained demands and supplies.

5. THE MODELS WITH MANY IDENTICAL FIRMS

We now extend the previous models to the case where there are m firms each facing the same production function. Moreover rationing on both the labour market and output market is uniform so all firms will behave identically in all equilibria. An interesting question is how the set of equilibria change with m. In particular let

Q = {P1, P213 a non-trivial equilibrium at (Pl, P2) with exactly m firms making positive profits}.

For the decreasing returns model, Q1 is the interior of the triangle labelled ABC in Figure 2. With m decreasing returns firms it is easy to check that the story unfolds in the manner of m = 1 if we replace the production function f(l) with the "aggregate production function" m f(l/m). The derivative of this at 1 = 0 is still f'(O). Moreover defining now:

OI) Uh, (wOmf(l/m), w2 -1) h = 1 2 u0(wO, mf(l/m), w2- I)




we see that o- (O) and 0-2(0) are invariant with m. So if we assume that for all m, the marginal propensities to consume and to leisure are less than one we generate pictures like Figure 2 for each m and the triangle ABC does not change with m. Hence

Theorem 3. For the decreasing returns model where all relevant marginal propensities are less than one, Q = Q2= ... = Qm=

Remark. Within ABC the decomposition into KU, CU and RI will change with m. In fact as m -> oo the aggregate production function converges to the constant returns function 1 f'(0). The CU region then shrinks to the line AB as m -> oo.

Following Weddepohl we define: A NW at p with exactly m firms making positive profits is stable if there is no NW

at p with m + 1 firms making positive profits. Clearly from Theorem 3 there is no stable NW in the decreasing returns model. This

is merely a statement of the old idea that under decreasing returns with "free entry" the number of firms is indeterminate. In the increasing returns model things are more interesting. Weddepohl shows that under his conditions Q1 Q2D Q3 ... D Qm D

A result along these lines can be found in our model as follows. As in the decreasing returns model the structure of the set of NW with m increasing

returns firms can be found via the 1 firm procedure replacing the production function with the aggregate mf(l/ m). Assuming relevant marginal propensities are less than one we generate pictures like Figure 4 for each m. The upper and lower boundaries of the cigar shaped regions of Figure 4 are then defined respectively by the equations:

f(I/M) P29 pi=yi (wO,mf(fI9w2lW2) (U) I/im Pi m

f(lM) P29 P2 = cr2( wo, mf(-I),w21 (L) I/im Pi m

These boundaries of the non-trivial regions are not now invariant with m however. Straightforward differentiation and use of the marginal propensities less than one condition gives

Lemma 6. For the increasing returns model, assume all relevant marginal propensities are less than one. Then:

(a) On the upper boundaries of the non-trivial regions dp2/ dm < 0 if and only if:

a0-1>f(l1/m ) ao- aq2 1/im aq1

where dp2/dm is the "Pi constant" derivative of P2 w.r.t.m. along (U). (b) On the lower boundaries of the non-trivial regions, dp1/dm > 0 if and only if

ao-2 > 1/ m ao2

aq1 f(l/m) aq2

where dp1/dm is the"p2 constant" derivative ofp1 w.r.t.m. along (L), and where all derivatives of o-h in -(a) and (b) are evaluated at (w0, mf(l/m), W2-l)-

When dp2/ dm < 0 along the upper and dp1/ dm > 0 along the lower boundary we get the Weddepohl result that Ql D Q2 D Q3 ... -D Qm D * - - . The conditions given in Lemma




6 for this boundary behaviour do not follow from assumptions so far. Suppose however that aol/aq2 ?0 and ao-2/aqC 0. Then it is easy to check that "the marginal propensity to consume less than one" implies the condition in Lemma 6(a) while "the marginal propensity to leisure less than one" implies the condition in Lemma 6(b). Hence

Theorem 4. For the increasing returns model where all relevant marginal propensities are less than one and where al1/aq2 - 0, ao-2/aq1?0 it follows that QlDQ2 v QmD

Remark 1. The conditions aol/aq2 C0 and ao-2/aq ?-0 allow the weak separability case, aol/aq2 = aOf21aq, = 0.

Remark 2. When the conditions of Theorem 4 hold at all prices there is a unique number of firms consistent with stable equilibrium. Figure 5 illustrates a case where

=Q3= Q4= ... and Ql Q2.

p1

FIGURE 5

In Q2 there is stable equilibrium only with 2 firms producing. In Ql r Q2 there is stable equilibrium only with 1 firm producing. In Q1 there is only zero trade stable equilibrium. The number of firms has become determinate.

Remark 3. On the other hand if we assume only that all relevant marginal propensities are less than one then there are cases where p c Qm+' but piX Qm. Figure 6 illustrates Ql and Q2 ( = Q3= Q4= ...) when

u(qO, ql, q2)=2q1/2(q2+3)1/2+2q1/2

f(l)= 13, w 3=3, w2=2.

Here there is just one non-trivial region for both m = 1 and 2 (1 c [0, 2] and 1 c [3,2] respectively) and the "cigar" for m = 1 is flat at one end since I(2) $ 0 and open at the other since o-, (1) -* oo as 1 -* 0. The stability criterion does not now determine the number of firms. At A in Figure 6 there is stable equilibrium only with 2 firms producing. However at B there are 2 stable equilibria with none and 2 firms producing. At C there is stable equilibrium only with zero trade. Such multiple stable equilibria gives rise to hysteresis possibilities. Start at A with 2 firms producing. Move to B and suppose 2 firms still produce. Moving to C produces shutdown of both firms. Moving back to B however we may stay in this stable shutdown situation and we have to go back to A to pick the




P2 boundary of Q' boundary of Q2

16

I \ I \ I \

c

1 1 1 4P 8 4 2 p 1

FIGURE 6.

economy up again. An ad hoc story is told in Madden (1982b) which generates the even more severe conclusion that Q' c c ... c Qm for a many output economy by assuming that whenever an industry shuts down frustrated demand for that industry spills over into "leakages" (savings).

6. ADDITIVE SEPARABILITY AND MANY SECTORS

The marginal propensity conditions which ensure global uniqueness depend on both production and consumption characteristics in general. It is worth noting that if u(q0, ql, q2)= h[2=0 vi(qj)] where h is monotone increasing and each vi function has strictly negative second derivatives (and so is strictly concave) then this guarantees o(i() <0, o-'(1) >0 and hence global uniqueness in all the models so far, provided only that f is increasing. More interestingly define now a many-sector economy as follows:

Definition. A many-sector economy is characterised by: 1. n outputs, ql ..., qn; n types of leisure, q , ... qn and so n types of labour, 11, . .. in. 2. Money, qo. 3. n firms, the ith firm producing qi from 1i according to the production functionf ( i). 4. One consumer who receives all profits, has endowments of w0 and w2,... 2,

consumption set

{qO; ql,..* q ; q2 ...

q2 jqo>-? qlj-0,i =1 .. n

and w'- q O,i=1 .. n}

and utility function

h[vo(qO)+En71 v'4(q')+E` P'(qD)]

where h is monotone increasing and each v function has strictly negative second derivatives.

Non-Walrasian equilibrium for this 2n + 1 good economy is defined in the obvious way (money not rationed) and (11,..., ln) fully characterises a NW at (pl .. .

pn;




p') analogous to the n = 1 case with q' =f (li), i = 1 ... n and qo = wo. Suppose there is just one decreasing returns firm in each sector. Then, straightforwardly, (11, . . . ), 0< < w2, i = 1 ... n, is a NW employment vector at (pl ... p'; p2 * p ) if and only if, for each i= 1 .. *n;

_i(li) ,_; a4l/aql[f;(li)]> , . av4/1aq'(w1 - 0i) pi1 a v/laqo(wo) - avo/aqo(wo) -

with at least one equality. But these are exactly the conditions for 1i c (0, w') to be a NW of the 3 good economy consisting only of the ith sector (production function fJ, utility function h[vo(q0) + v4(q') + v4(q')]) for i = 1, . . ., n. In other words the many sector result is found by combining earlier results for each sector in isolation. This decomposition principle is true generally with the additively separable formulation and many-sector economies (some increasing, some decreasing returns) which satisfy the assumption can be analysed using the results of Sections 2-5, treating each sector in isolation.

With one sector, many firms and increasing returns, additive separability implies Q1 D Q2 = Q3 ... (see Theorem 4, Remark 1). So in the many sector setting the additive utility assumption rules the phenomenon of Figure 6 in any sector. A more general analysis of the many sector case is called for.

First version received May 1984; final version accepted May 1985 (eds.) I am most grateful to the editor and two referees for helpful comments and suggestions: one referee

effectively provided the example of Section 5 which was originally merely conjectured. However, all errors and shortcomings are the author's responsibility.

REFERENCES BARRO, R. and GROSSMAN, H. (1976), Money, Employment and Inflation (Cambridge: Cambridge University

Press). KATZNER, D. W. (1970), Static Demand Theory (New York: MacMillan). LAROQUE, G. (1978), "The Fixed-price Equilibria: Some Results in Local Comparative Statics", Econometrica,

46, 1127-1154. LAROQUE, G. (1981), "On the Local Uniqueness of the Fixed Price Equilibria,", Review of Economic Studies,

48, 113-129. MALINVAUD, E. (1977), Theory of Unemployment Reconsidered (Oxford: Basil Blackwell). MADDEN, P. (1982a), "Catastrophic Walrasian Equilibrium from the Non-Walrasian Viewpoint: a Three-good

Macroeconomic Example", Review of Economic Studies, 49, 661-667. MADDEN, P. (1984), "Existence of Dreze Equilibrium under Set-up Costs", Journal of Economic Theory, 33,

275-288. MADDEN, P. (1982b), 'Macroeconomic Shutdown under Increasing Returns and Set-up Costs" (Discussion

paper ES125, Department of Econometrics, University of Manchester). SCHULTZ, N. (1983), "On the Global Uniqueness of Fix-price Equilibria", Econometrica, 51, 47-68. WEDDOPOHL, C. (1982), "Equilibria with Rationing in an Economy with Increasing Returns", Journal of

Economic Theory, 26, 143-163.



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