The Welfare Effects of Price Stabilization:A Simple Multi - Market Extension

The Welfare Effects of Price Stabilization: A Simple Multi - Market Extension*

The question of whether or not there are welfare gains from stabilizing fluctuations in prices has been a subject of much debate since it was first put forward in Waugh’s 1945 articles in the Quarterly Journal of Economics. This general question is of considerable interest to Aus- tralians, given the importance of agricultural goods whose prices are subject to random fluctuations, and given the existence of the Wool Board’s price stabilization scheme.

The bulk of the theoretical analysis undertaken thus far in the analysis of this question has been done in the framework of a single, independent market in which various furms of demand and supply relations have been assumed.l Independence is not very realistic for producers of rural commodities for whom this debate has perhaps most relevance. Therefore it is worthwhile to ask whether extending the analysis to a very simple short-run general equilibrium model changes the more important conclusions derived from the single market analysis.

Basically this paper extends the single market stochastic model examined by Massell [5] [6], to a twocommodity model, and considers the effect on Massell’s conclusions of the limited introduction of inter-commodity relations. Before expounding the larger model, we will h-st briefly review Massell’s model, his methodology, and his conclusions. Demand for and supply of the single commodity are assumed to lx linear functions of prices and the disturbances are additive. Thus:

A 0 [; I=[ O--B ][ ; I+[ y” ] (l)

where A , B are positive constants, and x, y are disturbances rdecting random factors such as climatic conditions, taste changes and income fluctuations; p denotes the price, S denotes the quantity supplied and

*The author is grateful for the comments of P. B. Dixon, S. J. Turnovsky and two anonymous referees. Any remaining errors are solely his own respon- sibility.

1 See Massell [ 5 ] [6], and Turnovsky [ 121 [ 131.

483

484 THE ECONOMIC RECORD DEC.

D is the quantity demanded of the commodity considered. The variance of the disturbance terms is denoted by us2 and ug2, and the covariance between x and y is zero on ,the ground that random factors affecting demand are generally unrelated to random factors af€ecting supply.

The market clearing price is given by:

and the market clearing price is stabilized2 at the statistically expected value of P, (denoted by Fa).

Surplus concepts are used to evaluate welfare changes. Expected producer surplus and expected consumer surplus measure producer and consumer well-being respectively. The weaknesses of the concept are well known, However, as pointed out by Turnovsky [13], while one would ideally like to carry out the analysis in terms of the underlying utility functions, this approach becomes much more difiicult if one wants to go beyond prices and consider the ultima'te random disturbances which are what random movements in price must be reflecting.3

Given the assumption of linearity, then the gain to consumers from stabilization, (measured by the difference between the expected consumer surplus when price is stabilized, and when there is no stabilization) can be shown to ber4

EGC = -tE[(p--p)(D(p)+D(p))l

- (2'4 +B)u;-BO: 2(A+B)2 *

EG, = t E [ ( P - - P ) ( ~ ( P ) S ~ ( P ) ) l

- ( A + 2 B ) 4 - A 4 - 2(A+B)a *

(2) -

Similarly, the gains to producers from stabilization are measured by:

(3)

The sum of welfare gains to consumers and producers is:

(4) 4 +4 2(A +B)'

E G = - Massell's principal conclusions are easily derived from (2), (3), and

(i) The sum of welfare gains is unambiguously positive, and is proportional to the sum of the supply and demand variances. Furthermore, increasing the elasticity of supply or demand (that is, increasing A or B ) decreases the total gains.

(ii) Two results, one due to Oi [9] [lo] and the other due to Waugh [14] [16] arise only as special cases. When only demand is

2 There are several possible ways this could be done, but the most commonly

3 Turnovsky [13, footnote 51. 4Note that equation (2) comes from the direct evaluation of equation

(4). These are:

discussed method is a buffer stock [Massell, Turnovsky].

(A.13) in Appendix 1.

1976 WELFARE AND PRICE STABILIZATION 485

unstable (i.e. uz = 0), producers lose unambiguously from stabilization, as claimed by Oi. (At the same time, in the case consumers unambiguously gain.) On the other hand, when only supply is unstable (thus ut = 0), consumers lose from stabilization as claimed by Waugh.

(iii) In general, however, the signs of expressions (2) and (3) are ambiguous and no conclusions can be drawn concerning the distribution of gains and losses from stabilization without further information as to the slope of the demand and supply functions, and the relative degree of variability of the disturbances.

A Two-Market System

The simplest possible extension to a mu1,ti-market system is considered. There are two goods only, which we shall arbitrarily call Beef and Wh0at and which are denoted by a subscript b and w respectively. Supply and demand in all cases are linear functions of price. As in the Massell model, all disturbances are additive, and fluctuations in supply are assumed mcorrelated with fluctuations in demand.

Producers are divided into three distinct and exhaustive groups: two 'specialist' p n p s , each of which produces only one of the corn- modities, and the 'joint production' group which produces both goods simultaneously.

Demand relations for the two commodities are:

(5 )

(6)

(7)

while supply relations are:

b b b w ab a w abS

The matrices, 6: 21, [b, bA, [ bb b w j, [nbl 0 bul J are the matrices

of coefficients of demand, aggregate supply, joint producer supply and specialist producer supply respectively; own-price coefficients (that is the elements of the principal diagonal) have the usual sign. Thus the diagonal terms of the demand coefficient matrix are negative and the diagonal terms of the supply matrices are positive. The signs of the remaining coefficients are left unspecified.

["'I, u w ["I, vw $I x w and p] Y w

are the vectors of the disturbances in demand, aggregate supply, joint producer supply and specialist producer supply respectively. The same convention as in the previous section is retained for denoting the variance and mean of disturbances.


Assuming both markets clear, and solving for p , gives:

(8) The market clearing price of wheat is obtained by equating D, and

(pw-bw)(ub-Yb)-(aw -aw)(uw-vw> pbc =

('b-ab)(Pw-bw)-(pb--bb)(ub-ab) *

S, and solving for pw to give:

When equation (10) is substituted into the equation for the supply and demand of beef, the resulting coefficient for P b allows for market clearing in wheat. Denote by Ssw and s d w these new coefficients of pa in the supply of and demand for beef. It is assumed that the equilibrating effects in the second market do not completely offset the sign of the own-price coefficients, thus :

ab(pw-bw)-aw(pb-bb) < 0

(10) 1 (Pw---bw)

( P w - b w )

ssb =

and : ub(/gw-bur)-uw(~b-bb) < 0.

sdb =

The corresponding term for the wheat market is: b b p w - bwflb s s w = s d w =

(Pw-bw) Ssw may be positive, negative or zero.6

The expected values of producer and consumer surplus are used to measure welfare exactly as in the Massell model. The general form of EG,, EG, and EG for this larger model are derived in Appendix 1.

Gains to Consumers The measure for gains to consumers is:6

(12')

5 For example, assuming the goods to be unrelated in demand CS, = a, = 0)

8See Appendix 1 for the derivation of equation (14). but substitutes in supply (b,,, a, < 0) then S,, is positive.


where the g, (i = 1 . . . 4 ) are complex expressions involving the stochastic and non-stochastic parameters. It should be noted that the g, cannot be signed without further information about the signs and relative magnitudes of all coefficients.

The first two terms of equation (12’) bear a strong resemblance to the gain from stabilization in the Massell model (see equation (2) ) . However, the coefficients S d b and S,, cannot be identified with B and A respectively, since the former coefficients incorporate the assumption that the wheat market clears, while no such assumption is made about A and B. These two coefEcients are more accurately identified with ab, and ab, respectively: this interpretation is used in Appendix 2 in com- paring the sum of gains from stabilization in the Massell model with the corresponding term from the extended model.

For simplicity in the subsequent discussion, it is assumed that both c9uW and 02v, are zero. Thus the only sources of instability in the system are fluctuations in the supply of and demand for beef, and the terms g, ( i = 1 ,..., 4) are all zero.

The assumption of a second related market allows for induced fluctuations in the market-clearing price of wheat. Stabilizing P b will fully stabilize pW (noting that in equation (9), u, and v, are non- stochastic by the assumption in the previous paragraph). This stabilization will further affect consumer well-being in addition to the gains coming from the stabilization in the beef market. The third term in equation (12’)’ represents this ‘induced‘ gain.

The first two terms of (12’) are each unambiguous in sign, given the assumptions made in the previous section about the signs of Ssb and S d b . However, the ‘induced gain’ term is ambiguous in sign, depending solely on the sign of ( p b - bb) S s b and therefore on the nature of commodity relations (i.e. complements or substitutes in production or consumption). As an aside, it is worth noting that the relative magnitudes of fluctuations in the supply or demand for beef are irrelevant in

48 8 THE ECONOMIC RECORD DEC.

this induced term, since &&, and 0 2 v b enter the expression with the same sign.

Consider the following case. Suppose beef and wheat are substitutes in consumption but are unrelated in supply. Thus aw, &, > 0 and b b , a, = 0. Then S,, is positive and the induced effect is ambiguously positive. This in turn implies that coefficient of 2 v b is unambiguously psitive,8 but that the coefficient of 2u may be either positive or negative. This ambiguity of sign Gu contrasts with the determinancy in the sign of the coefficient of 2% in equation (2). Clearly then, the Waugh effect9 may be reversed even in the case where the only fluctuations originate in the demand for beef.

Suppose now a case in which beef and wheat are substitutes in production but are unrelated in consumption. Then aw, P b = 0 and

S,, is negative and the induced effect reduces the welfare gain to consumers from price stabilization. Thus in this particular case the coefficient of 02, is unambiguously negative, and the Waugh result reappears. However, the sign of u2, is now ambiguous, contrasting the result obtained by Massell in equation (2), where the coefficient is positive.

When the commodities are related both in supply and demand, as would be expected, the sign of ,the 'induced' effeot term is ambiguous, creating ambiguity are to the signs of the coefficients on 2, and 2,. Thus, as in the Massell expression (equation (2) ) it is unclear as to whether consumers benefit or lose from the price stabilization scheme specified.

Gains to Specialist Producers

The gains from stabilization of beef prices to the specialist producers will be considered first. The term for the expected gains to specialist beef producers obtained from evaluating expression (A. 1 .vi) using equation (7).

b

b

b

b b < 0.

b

b

b b

BG%db) = 3E{(~b.-Pbe)(sb(b)( i jbs)+Sb(b)(Pbe))} a w - a w = u,"

= - b b l Var [ vb- (-1 vw] - h b l ( ?.=.-) a, - a,

+ ~ - ~ w ) ( ~ b l u ~ ~ u . + ( S d l - s b b ) u u W b . I (1 3)

W

s d b - s s b ( S d b - s 8 b ) a

(abluib +2(sdb-ssb>uu bzb)

( s d b - Ssb)' -3

E x J (Sdb- sab) '

8 Meaning that consumers gain from stabilizing the fluctuations originating in demand. Massell [S] also obtains the result as may be seen in equation (2).

9Which claimed that consumers lose from price stabilization at the expected value. Massell [S] showed this to be unambiguously true only when fluctuations originated in supply.


A noteworthy feature of (15) is that, with the exception of the covariance term :

aW-aw bl u u +(sdb-s8b)guwZ) (id(= w ( s d b 4 s b ) 2

the sign of each coefficient is relatively determinate, and is independent of the nature of the commodity relations.

More specifically, fluctuations in the demand for both goods are a source of loss to the specialist producer of beef. So are fluctuations originating in the supply of wheat. The only possible source of gain is from stabilizing fluctuations in price which originate in the specialist supply and in total supply of beef. However, the less the correlation between fluctuations in specialist supply and fluctuations in the industry suKply, the more likely it is that fluctuations originating in the supply of beef reduce welfare. All these results are independent of the nature of the commodity relations.

Consider the special case in which specialist supply is perfectly inelastic. Then specialist beef producers unambiguously gain from stabizing the price of beef at its expected value provided that a,

is positive and U, is zero. If in addition Specialist beef supply does

fluctuate, or if the total beef supply does not fluctuate, then specialist producers are unaffected by the price stabilization suggested.

In general, the sign of the right-hand side of equation (15) is ambiguous, depending principally on the size of fluctuations, and the origin of these fluctuations, as well as on the nature of the commodity relations.

The expression for the gain from stabilization to the specialist producers of wheat is much less determinate than equation (15). On evaluating (A.l.vii) using equation (S), the term for the expected gains is:

b b

w b

E G p ( W ) = 3 E [ ( p b , - p b e ) ( s w ( W > p , , ) + S ~ ( ~ ) ( p b e ) > l

b w b w

where :


The precise nature of the commodity relations is obviously important in evaluating EGp(w). A simple case to analyse is when two goods are unrelated in supply and demand, so that ( p b - b b ) is zero. Then specialist producers of wheat do not benefit-and do not lose-from stabilizing pbe at its expected value.

For simplicity, the covariance terms uU , uu and U, are

assumed to be zero. Then the following points may be made. First, all fluctuations originating in the beef market are a source of loss to specialist producers of wheat provided their supply is not perfectly inelastic. This conclusion is true regmd2ess of the nature of product relations. Second, whether or not the sign of hs (the term reflecting the effects of fluctuations in demand for wheat) is positive or negative depends on the sign of (aw - G) (Pa - ba). If the products are substitutes only in supply, or only in demand, then this term would reduce to %bt, and a,& r e s p v e l y , both of which would be positive. In such cases, specialist producers of wheat gain from the stabilization of fluctuations originating in the demand for wheat.

Finally, the effect of stabilizing fluctuations originating in the supply of wheat may be positive or negative.

A comparison of (14) and (13) makes it clear that even among producers, one group may gain while another may lose. Consider the case in which the only source of fluctuations in the system is from the supply of beef, and specialist supply of beef is perfectly inelastic. Then (15) reduces to :

b u r b w b w

(provided Ouch > 0) while (16) reduces to :

Furthermore, the gains and losses increase with increasing variability in the fluctuations. The existence of this case suggests the need for care in determining the distribution of all producer benefits in the assessment of any proposed stabilization scheme because even though a stabilization scheme may be intended primarily to help specialist producers of beef, the specialists in wheat may also be a policy relevant group and may lose from stabilization. Consequently, even though fluctuations may arise only in the supply of the good whose price is stabilized, it need not be true that the Massell result-claiming that stabilization in this particular case is desirable from the producers' viewpoint-is valid, once limited short run general equilibrium adjustment is permitted.

1976 WELFARE AND PRICE STABILIZATION 49 1 Joint Producer Gains

The term for the gains to joint producers is derived in much the same way as (13) and (14)-by evaluating (A.1.v) using equation (7) to get:

EGp(J) = fE( ( j b e - p b e ) [ s , (J ) (?be ) + s b ( J ) ( P b e )

-(-) BbVbb ( s ~ ( ~ ) ( ~ b . ) + ~ w ( ~ ) ~ b . ) ) ]

P w - b w

= -3 -f s'8b-u: [2(Sdb-s8b)'u Y +s'sbu:)

( S d b - ssb) (sdb-ssb)a +f(PJ * S ' ~ W .(ui +4) B -b ( S d b - S s b ) a '

+fi+fa+f+h. (15)

where bwol(Bb-bb)

S ' s b = S s b - a b l , s ' s w == s s w + ( P w - b w ) '

Equation (15) has some resemblance to equation (12) in that 'Massell terms' can be distinguished from other effects and that there is an induced effect term.

It should be noted that the sum of the right hand terms in (15) may be positive or negative and so joint producers may benefit or lose from


the suggested price stabilization scheme. Additional problems arise in this respect in that each of the functions fi , . . . , f4 is itself ambiguous in sign, so that even if there is only one source of fluctuation, the effect on joint producer welfare of stabilization is unclear.

Because of the similarity of (15) to (12), many of the comments made before about (12) apply to (15) and will not therefore be repeated.

Aggregate Gains The sum of the welfare gains from stabilization is given by:

EC = EG,(b)fEG,(w)+EG,(J) f E G ,

= * ( s s b - s ).* >o db Pb .

where u: is the variance ofp,,.

Thus there are unambiguous gains from price stabilization at the expected market clearing value. Furthermore, these gains are pro- portionate to the variance of price fluctuations, and may be greater or less than the result Massell obtains. The relationship between the Massell result in equation (4) and the result in equation (16) is not clear, principally because of the differences in meaning between A and B in equations ( 1 ) and S r b and Saa in the larger system.

However, it is possible to show that a sufficient, but not necessary, condition for the Massell expression to understate the actual gain given in equation (16) is that the two goods be substitutes in production and consumption and that u, + uU ,, is non-negative. If the latter term is non-positive, then the expression in ( 16) unambiguously exceeds the expression in (4) when the two goods are complements both in production and consumption.l0

be

b t a b w

Conclusion The extension of the MasselI model to the multi-market model

considered here is necessarily simple for two reasons. First, the intention was to retain the same methodology as used by Massell in the simplest model that incorporated several markets. Second, the formulae become even more complex when a larger number of markets is considered (in respect of this it should be reported that generally similar results applied in, a three market model initially examined).

While it is true that the special assumptions made, linearity, addi- tivity of disturbances and only two markets, will undoubtedly limit the usefulness of the results obained, the importance of Massell's isolated market assumption is clear. Even when only limited short-run general equilibrium effects are considered, the results differ from, and in some

10 See Appendix 2.

1976 WELFARE AND PRICE STABILIZATION 493 cases contradict, the results obtained by Massell. Whether these differences are important from a policy viewpoint is an empirical question; nevertheless, they exist at the theoretical level.

It is not clear who gains from stabilizing beef prices at the expected market clearing value in this extended model. However, the .total gain, obtained by summing the values of expected gains to each group, was unambiguously positive, and under certain conditions (some of which are derived in,Appendix 2) was shown to be greater than the corresponding gain in the Massell model.

An additional point was that once different producer groups were distinguished, it became clear that some producers could gain and others lose from stabilizing the price of a given commodity. While it is true that the model from which such results were deduced is extremely stylized, the conclusion points to the need to assess the distribution effects of short-run price stabilization schemes which stabilize price at the expected short-run market clearing value. This need is particularly important for the various producer groups since it is these whom the stabilization schemes are intended to assist.

MATTHEW BUTLIN Reserve Bank of Australia Date of Receipt of Final Typescript: July 1975

APPENDIX I It is assumed that each consumer acts to maximize utility subject to

his budget constraint. The familiar conditions are :

(i) %= Api where pi is the price of the i-th commodity, X is the

is the marginal utility from ax, marginal utility of income and good i,

ax,

(ii) M = pixi where M is the budget or income constraint. L

Defining Ri = pixi as the expenditure on good i,

Then a small change in utility’l is approximated by p,dXf = dR,-X,dp,.

= z (dR,-XdP,) = X(dM- XidpJ. (A. 1 .i)

The assumption that the second market always clears provides the

11 This derivation closely follows that of Burns [l].


relation between P b and pw given by (9). If it is further assumed that expenditure does not change, then the change in welfare is given by:

If, furthermore, the marginal utility of income is constant,1z then the expected change in consumer surplus between the stabilized regimes is measured by :13

(A. 1 .ii)

In the particular two market system considered, (A. 1 .ii) reduces to :

Producers are assumed to maximize the utility of total profit (which includes profit from all production activities). The utility-of-profits function is denoted by V and total profit is denoted by Profit from the i-th activity is:

where y,’ denotes the employment of factor j in the production of b. p j is the price of factor j . b is produced from m factors of production by means of a production function.

and each factor has diminishing positive marginal productivity. x t = &(y*b, * * - 3 yfm>

Necessary first order conditions require: tan - o for all i, j . vG-

Thus : an: - = 0 (since V’ >O). ay:

A small change in producer well-being is therefore approximated by : d V = V’drI

12 If the marginal utility of income is not constant, then the expected change in utility is:

(A.1 ,ii)’

where is a function of thep, and M . (A,l.ii)’ will not in general be equal to:

13 Assume for convenience that A = 1. 14 It is assumed = V‘ > 0. dlI


(A. 1 .iv)

Provided that in the time span of interest factor prices do not change, the measure of producer welfare is obtained by integration after using equation (9) to force a relation between pa and pw. As in the consumer case, Y' is assumed constant (and equal to one).

The expected gain to joint producers from stabilizing the price of beef is:

(A. 1 .v)

Expected gain to specialist producers of beef from stabilizing the E p'( Pbs sb(J)+ap,sw(q)dpb* aP b

price of beef is:

E E ; : St.(b)dPb. Finally, the gain to specialist producers of wheat is:

(A. 1 .vi)

(A. 1 .vii)

Expressions (A. 1 .+(A. 1 .vii) reduce to the expressions used in the text on being evaluated using equations (5)-(11).

APPENDIX 2 Proposition: The Massell expression15 understates the total gain from stabilization when the two goods are substitutes both in production and consumption provided:

Proof: As pointed out earlier, S,b and &b from the extended model do not have the same interpretation as A and B (in the Massell model) respectively. The A and B coefficients from the Massell model, since these coefficients do not incorporate the market-clearing adjustments in other markets, are better identified with ab and ab respectively, where a, and ab come from the extended model.

Now the assumption that the two goods are substitutes implies:

u u u +%, >o b w b w

aw, bb, -t%, -a, > 0. Thus :

S,, = a,-a, - -= ah(= A ) since a, (;:I::) - ' O , (Bw-bYI)

Similarly, s&, = a b - a , , , ( R ) >a, = B.

15 See equation (4).

496

Thus : THE ECONOMIC RECORD DEC.

4 +4? 4 fo, Lb<-. b b

A+B S s b - s d b Since we have assumed u,, ,, +a, , >0, the terms in (16) apart from

b w b w

4 +4 are non-negative, the proposition follows immediately.

When the two goods are complements in both production and consumption (thus a2, p1 < 0; u2, bl > 0) then an argument similar to the one above establishes that:

& b - s d b

.if +4 4 +d b b b b -<-.

A f B S s b - S d b However, assuming u,, +a, , > 0, the sum of the remaining terms

b w b w

in (18) other than: .“ + .:b s,,-s,,

is ambiguous in sign. If uU ,, +u, ,, GO, then the remaining terms are

non-negative, and the total welfare gains from stabilization in the extended model exceed the total welfare gains from stabilization in the Massell model.

b w b w

REFERENCES Bum, M. E., ‘A Note on the Concept and Measures of Consumen’ Surplus’, American Economic Review, 63, June 1973, pp. 335-44. Howell, L. D., ‘Does the Consumer Benefit from Price Stability?’, Quarterly Journal of Economics, 59, February 1945. Hueth, D. and A. Schmitz, ‘Trade in Intermediate and Finals Goods’, Quarterly Journal of Economics, 86, August 1972. Lovasy, G., ‘Further Comment’, Quarterly Journal of Economics, 59, February 1945. Massell, B. F., ‘Price Stabilization and Welfare’, Quarterly Journal of Economics, 83, May 1969.

, ‘Some Implications of International Price Stabilization’, Journal of Political Economy, 78, March/April 1970. McCall, J. J., ‘Probabilistic Microeconomics’, Bell Journal of Economics and Management. McKinnon, R. I., ‘Future Markets, Buffer Stocks and Income Stability for Primary Producers’, JournaZ of Political Economy, 75, December 1967. Oi, W. Y., T h e Desirability of Price Instability under Perfect Competition’, Econometrica, 29, January 1961.

Tisdell. C.. ‘Uncertainty, Instability and Expected Profit’, Econometrica. , ‘Rejoinder’, Econometrica, 3 1, J m ~ / A p r i l 1963.

31, Ja&@/April 1963; 1 Turnovsky, S. I., Price Expectations and the Welfare Gains from Stabilia-

tion, American Journal of Agricultural Economics, November 1974. , The Distribution of Welfare Gains from Price Stabiliza-

tion: The Case of Multiplicative Disturbances, ANU M h e o , March 1975. Waugh, F. V., ‘Does the Consumer Benefit from Price Instability?’, Quar- terly Journal of Economics, 58, August 1944.

‘Consumer Aspects of Price Instability’, Econometrica. 34, April 1986.

, ‘Reply’, Quarterly Journal of Economics, 59, February 1945.

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The Welfare Effects of Price Stabilization:A Simple Multi - Market Extension