Profitable Speculation, Price Stability, and Welfare

. (#727 in RFD Series)

INTERNATIONAL FINANCE DISCUSSION PAPERS @

PROFITABLE SPECULATION, PRICE STABILITY, AND WELFARE

by

Stephen W. Salant

Discussion Paper No. 54, November 7, 1974

| Division of International Finance

Board of Governors: of the Federal Reserve System

The analysis and conclusions of this paper represent the views of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or its staff. Discussion papers in many cases are

circulated in preliminary form to stimulate discussion and

comment and are not to be cited or quoted without the permission of the author.

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" profitable Speculation, Price Stability, and Welfare —_ by Stephen Salant a) ; In 1953, Milton Friedman asserted that profitable speculation increases ‘price stability: ~'People who argue that speculation can be destabilizing seldom realize that this is largely equivalent to saying that speculators lose money, since speculation can be destabilizing in general enly if speculattors sell when the currency is low in price and buy when it is high." ‘This remark, repeated in 1971, has generated research by Telser, Farrell, Kemp, | Schimmler, and others. To analyze Friedman's assertion they had to give a precise interpretation to his words. By "speculation", they took him to mean inter-temporal

arbitrage under certainty. By "stability of prices", they interpreted him to mean

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the sum of squared deviations from the average price. His assertion, as interpreted by them, is: with pro ofitable, tnter-temporal carry-overs. under certainty, the sum of squared deviations from the average price is smaller than with no carry-overs.

Farrell, Kemp, and Schimmler analyzed this propositon, allowing the speculator

1/

to pursue any profitable strategy.— Telser, in contrast, required his speculator

to choose the profit-maximizing, monopolistic strategy. Farrell, Kemp, and Schimmler showed that the Friedman proposition is false unless there are only two periods or linear demand curves with the same slope... Telser showed that optimal speculation does

reduce price variability. However, since he used linear demand curves of the same

slope, his result was guaranteed without invoking profit-maximization. The impressicn

2/

gained by some from Telser's article™ is that by requiring that the monopolistic

Speculator behave optimally, the Friedman proposition can be saved. It cannot.

1/ The speculator is constrained to strategies which leave him with no inventory or

“~ debt at the completion of the game.

2/ See, for example, the first complete paragraph | on p. 68 of Flexible Exchanee Rates (revised edition), by Egon Sohmen, | .

‘addresses a question of no significance.

" distributions can be made.

The point of this paper is to show that the Friedman proposition, as interpretod

by previous researchers, is both wrong and uninteresting. Profitable speculation may . . ‘ increase price variability relative to the situation with no carry-overs. Presumably ,; the proposition that speculation reduces price variability was advanced in defense of speculation. This seems, however, to be the wrong defense. Profitable speculation improves welfare no matter what its consequence for price stability. The case on behalf of speculation is strictly analogous to the argument that trade is in a nation': interest. Whether the price vector with trade is more variable than the price vector with no trade is irrelevant. St ae ... Price instability in the sense of Telser-Kemp-Farrell-Sthimmler does not mean tha-

the model (or the world it portrays) will explode. It does not even mean that consuxe> are exposed to greater uncertainty, since there is no randomness in the model. It concerns an unimportant characteristic of the prices at which commodities are purchase: As positive economics, the Friedman proposition is wrong. As normative economics, it

I have not seen any examination of the Friedman propositon under uncertainty. Constructing a simple, general-equilibrium, carry-over model with stochastic endowment: is not difficuit.2/ From such a model, we can obtain the endogenous, stochastic proce:

of prices which clears markets over time. This process can be compared to the stochas-

tic process of prices that would emerge without speculation. In a T-period model, we

, Would have one joint probability distributien for T variables which would arise wither:

carry-overs and another joint distribution which would arise with optimal carry-overs.

The problem is not in solving such a model but in knowing how to compare these two

joint distributions. What does someone mean when he says that the prices generated by

one joint distribution are more unstable than the prices generated by the other? I ce-.-

mot resolve this question. Once this question is answered, the comparison of the two

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3/ See Robert Gustafson: "Carry-over Levels for Grains" (Dept. of Agriculture, 1955); Paul Samuelson: "Stechastic Speculative Price" (Vol. 3 of Collected Works, 1971); Stephen Salant; "The Telser Model Under Uncertainty" (unpublished); Robert Townsend: “Price Fixing Schemes and Optimal Buffer Stock Policies" (unpublished); and Michael Mussa's unpublished work on the subject. '

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In the first section of this paper, I place the convertional carry-over model in a general equilibrivm context. This makes the analysis mote familiar and leads easily to welfare comparisons. In the next two sections I show that the restrictions that speculdtion be profitable or optimal are inadequate to salvage the Friedman propusition. Section IV explores the welfare gains from trading with the speculator, I show that consumers gain even if the speculator increases price variability. In the fifth section, I consider extensions to a stochastic environment. By considering a twoperiod, carry-over model with uncertainty in only one period, I avoid the question of defining “price instability" in a multi-period model under- uncertainty. I. find that speculation expected to be profitable raises the welfare of the consumer but will (if th.

demand curve is concave) increase the variance of the randon price.

I. ©The Micro-Underpinnings of the Usual Speculation Model

The analyses of Farrell, Telser, Schimmler and others begin by assuming that the speculator faces a set of downward-sloping excess demand curves for the commodity which the speculator can transport over time. The excess demand for the commodity in pericd i is assumed to depend only on the "price" in that period. In the absence of intervention by the speculator, the price which eliminates excess demand will emerge. A sale by the speculator would reduce the price, generating excess demand by consumers equal to his sale; conversely, a purchase would raise the price, generating excess supply equal to his purchase. I would like, briefly, to specify the conventional model for studying inter-temporal choice and then limit it so that it is consistent with the usual assumptions of the speculation literature. i

In general, each consumer receives endowments of the two goods in each of T periods and is unable to carry either good from one period tto the next. Calling the’ two goods "soybeans" (x) and the "background good" (B) we can picture each consumer ‘as having his own preferences over all possible bundles of the 2T goods: ,

4/ The same modifications pertain to the exhaustible resource and random walk Literature.

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U(X, By »X_ Bo, seceey XpsBy)- If an auctioneer called out.2T-1 relative prices, each ‘consumer would choose the best bundle which he could acquire without spending more then the income from his endowments, valued at the auctioncer's prices. For each price vecte. -each cotsumer would have demands for the 27 goods. These could be added across indivic: to yleld- aggregate demand for each good. By deducting the aggregate endowments of each the goods, we would obtain 2T excess .demand curves. Each would, in general, depend on 27-1 relative prices. The auctioneer would choose the price vector so that the aggrepate demand for each of the 2T goods equalled the aggregate endowment (so that excess demand is eliminated). An individual in this model can make exchanges over time. I ce> gain soybeans later by selling someone a (possible different) quantity of soybeans nox.

, However, society as a whole cannot == without the "speculato::" -- increase aggregate sc: bean consumption later by reducing it earlier.

For any price vector specified by the auctioneer, there are 2T excess demands. in general, each excess demand curve will depend on 2T-l relative prices. In the speculétion literature, however, each excess demand curve depends en a single price. We can achieve this result only if we assume that each consumer ranxs bundles by a utility

£.(X.) + 5 B a/ Then thiMG a i’ =

. 1 i=1 optimal feasible choice by the consumer satisfies the following conditions (provided t:-

T function of the form UG, By» X5» Bo, eoeeey Kos By) = ,

solution is interior):

ae | : | | £7) = dq; i£=°1,T T x os T 3 - x eS - Xx.) + ea - B,) = 0, where qs is the price of commodity X in peric:

i, Xx; is the consumption of that good in that period, xX, is the endowment in that peric<

.and , is an undetermined multiplier.

The second line indicates that, since the consumer is indifferent between consucp-

tion of the same quantity of the background good at.any time, the price of the backgroeur

5/ Each consumer could, of course, rank bundles in the same way by using a monotonic

transformation of this utility function. Different consumers would have different Ei (+3 functions and different constants. In the exhaustible resorrce literature, an inter¢srate is introduced by assuming time-preference in the consuyption of the background ¢. :.

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good (ay) is driven to equality in all periods: q’. Hence, we can simplify the

equation and can solve them for consumption of soybcans in each period (X) and the

total consumption of the background good (r B,):

£°(X,) sq. ; ‘ ft. i=<1,T

T ay _ Tt _ ; -DY = (, - X,). +c (, - BL) = 0

witgp fF ge FF

The demand for ith period soybeans depends only on the ith period price (relative to ch.

ba:kground good): the desired result. We denote this th period relative price as P,.

-

There are two related consequences of the Marshallian assumption of a constant margine? utility good. First, all increases in income are spent on the background good; doubii-. the endowments will not, at fixed prices, alter the desired consumption of soybeans in any period. Second, the consumer's decision about consumption in the yeh period would ~ the Same no matter what relative price was called in the future; hence, uncertainty abe subsequent relative prices will not affect the current, optimal decision of the consi=: Each of these two consequences will be utilized later.

By assuming consumer preferences of the above form, we-create a general-equilibri.. context for the speculation models where excess demand in one period is assumed to de; . on the single (relative) price then prevailing.

Speculators, in this model, provide a service: they reallocate the aggregate

. . wy endowments of soybeans in different periods. Following the literature, I assume away

transport costs and interest charges. The speculator is assumed able to reallocate so:-

beans by buying in one period, storing the commodity, and selling it in another.” Denoting a speculative sale in the jth period by S, (a negative value indicates a pur~

chase), we assume the speculator is able to adopt any strategy (S} So. eecces S) which leaves him without debt or inventory at the end of the oth period (£ S; = 0). ; a . {#1

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He neither injects nor withdraws soybeans from the system. &/ The monopolist sets

prices in such a way that he creates offsetting excess supplics and demands for soybean: in different periods. He buys the excess in periods of over-supply and sells it in periods4of excess demand in exchange for the backgrcund good. If the resulting real

value of the excess demands for soybeans is positive (2 PS > 0), we know from Walras'

i

. Law that consumers will have supplied the speculator with a net amount of the backgrou:.

' good of equal magnitude.: The speculator takes this as profit from his carry-over

service _/ We now consider how his carry-overs affect price: variability. Il. Profitable Speculation and Price Variability

Denote the excéss demand curves faced by the speculator as E; = &, (P;); where BF <Q. The jth price is determined by the equilibrium conditions S; = E; = 8, (P;)-

The profits of the speculator are:

tT s T = Y iti

. and the variability of price is defined as:

The variance of prices (Po) without speculation Wis) is defined analogously.

6/ In making these assumptions, I follow the literature I am amplifying. However, it should be noted that there are several problems with these zessumptions. First, the speculator is assumed able to sell soybeans that he has not previously purchased and <c mot get from other participants-in the model. This problem can be circumvented by és: ing the speculator begins the game with a large inventory amd is required to end with ' game inventory or by adding stock-out constraints and utilizing Kuhn-Tucker conditicns. The latter procedure introduces intractable non-linearities when the model is extencec = multi-period, stochastic framework. Since this paper consists largely of counter-exe=: constructed so that the stock-out constraint is not violated, the problem can be ignore.

7/° It is also a poor implicit assumption that the speculater has no desire for the so:beans he transports: his demand for soybeans is zero. We could avoid this by giving <. speculator a utility function. Then two speculators faced with the same prices woulc =have in the same way if their behavior as producers were "separable" from their behavicr &S consumers. Under certainty, separability depends on the existence. of a market terms: trade for selling the transported good over time. Under certainty, however, separabiii > Gepends on the existence of a futures market.

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The proposition that Tf > 0 implies Vy < Vis is false, as the following example illustrates. Assume the excess demand curves are downward sloping lines of different

slopes and intercepts. That is, Ey = Fy - a,P 5. In constructing our example, we are

free to choose the slopes, intercepts and any feasible strategy {S;}. Can we find a

case where profitable speculation increases dispersion?

The following table summarizes numerical data of the example:

Period 1 . Period 2 Period 3 Pr $105 $145 $295 PF $100 $2000 $300

T = $1150 > 0.

ae AV =V. - Vi, = 6688 - 6666 = 22 50. | The speculator buys 10 units at the beginning, selling 5 at higher prices in each

of the next two periods. Profitable speculation has increased the ‘variance by 22.

Graphically, we may portray the situation as follows:

Period 2

SR A tt URN ENA i Ae ce Pine aa i aR RN mE Ree ere nerer ~ + £

____.Period 1

Period 3 ’

All three demand curves slope down, the slopes being -1/2, ~-11, and -1, respectively. The vertical intercepts are respectively 190, 200, 300.

The general approach here is to make the outer demand curves flat.and with very differert vertical intercepts. This guarantees that speculation causes profits and little change in two prices. By making the middle curve steep, we can get the movement in the remaining price we need to increase the variance.

It should be obvious that Friedman's proposition is true for the uninteresting case of two periods. Then, for any downward sloping excess demand curves, profitability implies redsnegsdincrgnahinn... Potaake profits, the speculator must buy low and sell high. The lower price rises and the higher one falls. The variance of two , numbers falls as they move toward each other; this idea, reflected in the opening quot:

from Friedman, does not generalize to more than two periods. -JII. Profit-Maximizing, Monopolistic Speculation and Price Wariability

_We restrict our ability to generate counter-examples to the Friedman propositicn if we require the speculator to choose not merely a profitable strategy but the one which maximizes monopoly profits. Telser considered the case of a profit-maximizing monopolistic speculator facing linear excess demand curves of the same slope. He founc that optimal speculation reduced price variability to one fourth -the amount that would have occurred without carry-overs. In this section I simplify his analysis and extend it to the case of linear excess demand curves of different slopes. In this case, inpe: ing optimality does rescue the Friedman proposition: specwlation quarters pricevariability as in the case examined by Telser. However, as I illustrate at the end of the section, even the imposition of optimality cannot, in general, salvage the “proposition that speculation reduces price variability. .*

In' the case of linear demand curves, imposing optimality rescues Friedman's resul: /

if he sells S,, he

The monopolistic speculator faces demand curves E, = F, - @.P,3 i i ii’ £ Ls - Grives the price down to where E, = S;- The price will then be a ~ . The speculator

T F,-S wishes to maximize £ s.{ ~ 1) by choosing {S35 he is comstrained to set zs, = 0 .8/ i=1 i ,

For the {S;} to be feasible, they must satisfy the constraimt; for them to be optinal, the marginal revenue in each market must be equated. Calling } the common marginal

43 . wo revenue, we obtain the necessary conditions

. : . . / . 7) . F,-2S ‘ ATi, $e21,T | ; ‘X T = S, =0. i=1 * roy :

These T + 1 conditions determine the T + 1 unknowns: RR, Sy> cesses Sie ‘The ‘ second order conditions are satisfied since, in each period, marginal revenue declines with increased sales,

' Summing the first T equations and substituting the last, we can solve for }

°

- 9 = ‘ uF, 2S r ta.ca a x va

th

The optimal sale in the i” market is, therefore,

1 fp ot pp} | | 2 {r, - Ea, uF Si° _

8/ At first glance, the requirement that the speculator comclude his transactions in the same position as he began seems unsatisfying. As obserwers, we might have difficu.evaluating the speculator's profits if he had unsold inventerries; but that is our pred: not his. However, the constraint may be viewed differently. Suppose the speculator maximized profits over a long horizon. As observers, we might choose to study his pro. and price variability over any sub-interval where we found his purchases and sales cancelled out. If his entire strategy were optimal, his behavior in each sub-interval we! likewise be optimal. In this view, T would be endogenous. ‘Since we are only interest«. in his behavior during periods where he neither accumulates nor decumulates soybeans, i: is legitimate to introduce the constraint.

\ | . - 10 -

\ 4

The price that will result from the sale S

1 is . FS, F, 8,0 Fy 1 PaO a, a. 2a Dea, ** 5° i i 745 i j -@

The average price in the T markets resulting. from optimal speculation is =P, F, =F. . T T za, 2a, °

The difference between the ith price and this mean ds: =P, F

: 1 F, P -j-i-i x).

i T 2a, 2T “\a.

a J Hence, the variance of prices with profit maximizing speculation is

vo bot - Me

This variance is one quarter as large as the variability that would occur withozt the speculator. Then the jth price would be:

ed = Fy F,/a,. =P; The average price would be 7 * FAs). The variability « of prices without the speculator would be:

-: F, 2,/a,) Vas 7 dz fe ~ Beer * AV ee

The foregoing model reduces to Telser's if all demand curves have the same slope

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To consider optimal speculation graphically, we must Sketch in the marginal

curves;

pg

MR

f

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Ne a

s {

ay \ t \ . t : | ' \ 1 ’ i to ! i % s * S, 7 So 2 53 3

' Period 1 - Period 2 © Period 3

We can consider {S,} equating the marginal revenues by drawing a horizontal line

with height equal to some common marginal revenue (}) and finding the Ss; in each market

ty

which will produce it. We then must check that our solution is feasible (rs; = 0).. L:

t

not, the height of our horizontal line must be adjusted. Once a feasible solution

equating the marginal revenues is found, the prices determined by the optimal strategy may be read from the excess demand curves. The prices which would occur without

speculation are the vertical intercepts of the excess demand curves. : . :

. . For each period, the total revenue function associated with the average and

marginal curves above will have the following shape: /

er

; . Total Revenue for Period i’

S,: sales by the speculator

of

Sales generate revenue; purchases require expenditure (negative revenue). Hence, the _revenue function for each period will be positive for positiwe S values and negetive for negative values. If each period's revenue function is strictly concave, their sum will be strictly concave. It follows that there will be a unigue, profit-maxiniziz: strategy for the monopolist. This strategy can be identified as the only feasibie one equating marginal revenues across time.

Because each revenue function slopes upward, passes through the origin, and is concave, the average rtvenue exceeds marginal revenue for positive 5S while the reverse is true for negative S;-

°

With this in mind, we can illustrate a case where profit-maximizing, monopolistic

--. Speculation increases price variability. The Telser result @oes not generalize, once

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the assumption of linear excess demand curves is dropped.

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Imagine we had a concave revenue function (Ry) for the first period with the

following characteristics:

Ry (0) = 100. oc Rj (-10) = 140 R, (-10) = - 1050 These values could be generated by an upward sloping, concawe function passing throus> the origin; we need hot spectty the wametinon amalyticahiv. oo eee A different concave revenue function (Ro) could generatte the following data for c-

kee

second period:

| R5 (0) 200 B5 (5) 725

R,(5) = ge

140

e

~~

The revenue function (R3) associated with the third period might have the followi- |

characteristics:

By (0) 300 140

R30) | 6)

1475

A profit-maximizing, monopolistic speculator would choese to purchase ten units iz the first period and to sell five units in the second and third period. The marginal

revenue in each period resulting from this strategy would then be equal (to 140) and

Sn re tee ence

|

the constraint would be satisfied.. Since each revenue function is concave, the strategy would generate the highest profits (Ry + Ry + R3 = 1150}.

In the absence of speculation, the prices for the threc periods would be 100, 202

,

>. . R.(S.,) and 300. With speculation the prices become 105, 145, and 295 Cs = +). . - . i

We have seen (p. 7) that the second set of prices is more variable by (22), Hence. not even the imposition of optimality can salvage the proposition that speculation reduces price variability. |

In general, without speculation, we have a set of different prices which we may arrange in order of increasing magnitude (to simplify the notation, assume the

arrangement by size is the same as the ordering over time);

i.) fo) oO (a) . Py < Po < Ps one ic Pr:

“

The common marginal revenue (}) which will occur in each market when the speculator optimizes must be below the highest price and above the lowest (otherwise he would buy or sell in all markets, violating the constraint). In all markets where the initial price (Po) is smaller than ), he buys (raising the marginal revenue in that market to 4); in all markets where the initial price exceeds 2, he sells (lowering the marginal

- revenue to }).

° 2

buys in markets to the left of , and sells in markets to the right. All prices below

“Hence, we can insert } in our series: PY < Pl < ie. hk Sees Po. The speculator

A vise toward it and all prices above it fall toward it. The dispersion from 1. de-

°

creases. That is about all imposing optimality and assuming concave profit functions

buys us. It cannot rescue the Friedman proposition about dispersion from the averace

“" : .

price. .o Bee IV. “The Gains From Trade Once Again," Once Again

. We have compared the price vector that would occur with and without carry-overs

and have found that, in general, nothing can be said about which one will be more

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variable. But why should we care? In other certainty models we do not care about the variability of prices but about the allocation of commodities which accompanies these prices. We should then ask: Would the community be better off with the bundle of soce available to it with no speculator or with the bundle available to it with the speculator? Consider the community of consumers as the "hore country" and the. speculator as the trading partner. We can now re~phrase our question: Should the hor: country prefer trade to autarky? |

Re-phrased in this manner, the question has already been answered. If the hore country consists of one ~resicent , trade <ie-neserSinrmini - Trade gives the lone resicethe opportunity to consume bundles not available to him under autarky without removin: the option.of consuming his endowment. If the lone resident chooses to trade with the speculator, he must prefer the bundle he acquires from the speculator to his own endowment since consuming the latter is also feasible.

If the home country contains more than one resident, we must consider distributicr problems. Like trade with another country, dealing with the speculator can harm sone while helping others. A resident with all his endowments in a period of high prices under autarky may well be injured by trade, since carry~overs would reduce the value c= his endowment. |

However, as Samuelson has shown for the case of international trade, all residex:: could be made better off by trade if, prior to trade, endowments were properly re-cis:: uted. Because of our Marshallian assumption of a constant marginal-utility good, pretrade re-distribution would not affect the excess demand curves faced by the speculatcr Hence, his monopolistic pricing strategy would not change. No matter how endownents woe re-distributed prior to trade, the community would acquire the same bundle of commodi =: from the speculator. If re-distribution occurred after trade instead of prior to it, 2:

community would acquire that bundle of goods and each resident could then be given the

same allocation as when re-distribution occurred prior to trade. Hence, for our

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Marshallian case, the timing of the re-distribution needed to make everyone better cl: is unimportant. .Consider two residents, X and Y. Suppose we leave man Yat the utility level

achievédd under autacky. Subject to this, how well off can we make man X by selling

society’ S aggregate endownents to the speculator (at the prices he sets) in exchange for a bundle to be consumed by our residents. If nothing is sold, we can achieve the

autarkic level for man X. But, almost always, we can do better. The best we can co

“

will be to equate the marginal rates of substitution to each other and to the "foreisy - rate of transformation" (the-prices set by the speculator). This solution will be ore

where society's endowments .are exchanged for a different bundle of equal value and th:.

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new bundle distributed between our two residents so as to make X better off and YX equally well-off compared to autarky. The market, combined with re-distribution,

Can reproduce this allocation when trade is opened even if the re-distribution occurs

9/ -

after trade with the speculator.= We have seen repeatediy that trade with the speculator may increase the variabili:

of prices relative to autarky. However, in every case, the bundle acquired by the ; \

community through trade makes it better off regardless of the larger price variabiii If there is a single resident in the community, he is made better off, If there are

many residents, the bundle acquired can be re-distributed after trade to everyone's advantage. Hence, the comparison of price variability with and without carry-overs é

.

seems to me unimportant and misleading.

9/ If we drop the Marshallian assumption, each excess demand curve would depend on many relative prices instead of one, as conventionally assumed. In this case, we could still analyze a carry-over model, The community would gain from trade if redistribution occurred prior to trade.

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V.. cars}-overs Under Uncertainty | os In this section, I consider speculation under uncertainty. The randomness result: from stochastic endowments. Without carry-overs, we can derive a joint probability distribution of prices. with carry-overs, we can derive a @ifferent joint distributic.. The price realizations which emerge in each setting depend on the endowment realization. since the optimal’ reactions of all participants are endogenfzed. Once price variabili-is defined, we can easily compare the two situations. However, to my knowledge, no on: has stated what he means by price variability in a multi-period, stochastic framework, Suppose the random price Maaal has T components: (Py5 eeeeery Pp We might cef:. variability te mean al ,£(P, - = *): the expected value of variability in the sense usvd in the certainty literature. We might define it to be EVar(P,), the sum of the unconditional, price variances. In the certainty case, competitive speculation leads to an equalization of prices

over time. Variability, as defined in the certainty case, vanishes (trivially) if

Speculation is perfectly competitive. By analogy, perhaps we’should choose our

definition under uncertainty so that perfectly competitive speculation eliminates

variability entirely. With competitive speculation, prices follow a martingale (a

generalized random walk): E(P.,,| realizations through t) = Pe i=1,2 ... . The price

' expected to prevail at any time in the future--conditional om all realizations up

through the current period--is equal to the current price. If we chose as our definit:: 2 of variability 5 E(P.,,| realizations through t) -P. | » competitive speculation would i=1 e reduce variability to zero. For any definition, we could ask whether any kind of

profitable speculation reduces variability. 1 doubt it woutd ,20/ However, it would

“appear foolish for me to select some strange definition of wariability and then use

it to present a counter-example to the Friedman proposition.

10/ One reason for doubt is that, as the endowment randomness vanishes in the limit, the uncertainty case reduces to the certainty case and some of these definitions reduce to the concept of variability we used to question the Friedman proposition in, the certainty case.

me - 18 -

Instead, I will present a very sinple model. with only one random price. ‘A natural definition of variability would be the variance of that price. Using this definition, we will see that carry-overs expected to be profitable improve welfare, but increase price variability.

To begin, consider the planning problem of a single res:ident with abundant, known - current endowments of two goods (X,B) and random future endowments (XB X°,B) which depend on the unknown weather (rain or shine). Our resident: can carry some of his initial endowments, K,, Ky into the period of uncertainty to augment his smaller, random endowments then. Each selection of Ky: Ky provides him with a lottery with

known current consumption and random future consumption:

aed =, =r —r ss =s &-K,B-K, ER +K, Pt+Ks +n, P+ KD

The consumer can rank these iotteries and choose the best. ‘Assume his preferences are

of the forn: U(K,) = £CK-K,) + aeB-R,) + Me K,) + aK) ) + mle Gan, + a(B4K,)],

where f(-) and g(+)} are concave, increasing functions and 1! is the objective probabili:of rain.

. Optimally, the consumer should pick K, and K, to satisfy the following equations:

£°(K - K,) = 1 eG + K).+-G-1 g GP +k)

a = Tle + (1-TI)a

The first equation defines the optimal amount of soybeans to carry over. The second

7

indicates that the consumer is indifferent about carry-overs of the background ood.

*

A NL A CN RS A RIA RRO tn RE: = eee TREN RE BERETS OE RIN ARR Se UR ON meme SRN NE RARE OR am me ER . " pe aes eae ILE T ETON a

The consumer chooses to carry enough soybeans to make his marginal utility

from current consumption equal to his expected marginal utility from future

consumption. Since the background good carry-over does met matter, we can portray

utility as a function of the soybeans transported:

Prd utility as a function of _ Carry-over

Bie t

autarkic utility

How would the market solve our planning problem if the speculator--not the consumer-~had carry-over facilities. The consumer would have a current demand for soybeans (x) which would depend on its price relative to the background good:

E°(x) _

-@ Pe

Next period, the consumer demand would depend on the price then:

By acquiring KX, soybeans, transporting them to the next period and selling then,

ow, the speculator can generate the current price

.- ~

eee een ~ poe ae ce comme thr, wae CN wremmewrameeran | nea RegpERORTIR -~ a" ‘ ws . oars | alo cera ana y ae ‘ f ates net sow we - wen eee saree + orga Pm nm eee a ERs ri featieal

mw £4 ™

a

and the future prices which depend on the weather realization: os r & (x + K,)

Po = —_— aT » with probability Tf . 7 , . ; i.

° ge + K). | nn yo Po = —_ > with probability (1-1). -

The expected profits of the speculator will be:

= r . Ss . ~ RK) = [1P,(K,) + C-™P,(K,) - Py (KOK On the following graph, I show the expected profit of the speculator and the expected

utility of the consumer for each carry-over decision.

M . ‘ Ce K. . ; K,.

ee A competitive sector of speculators would drive away profits in the scramble to

get them. The current price would be driven up and the expected future price would be

PRE Ae eR Se RE EI ROY ae Re ee Nn ge Se oe neti eg

feepeneimares me meter mame meg Ee ee ane RR EERE | tn RR RRR

6 = 2h

driven down to a point where they were equal. The competitive carry-over, denoted

C, results in

7 .

_ r _ s P) = EPS = WP, + (1-11)P,

or ’

f-G@-K) Tev® +K) (-mMeGt+K) x x + x a a a

=

Hence, the competitive solution maximizes consumer welfare. A monopolistic speculator

would equate current marginal revenue to the marginal revenue he expects next period.

He would carry less to maintain a gap between his buying price and expected selling

price. - -

In.our example, any positive carry-over which makes profit improves welfare. We

mow must examine the effect on price variance.

Since the constant carry-over is added to the consumer's random endowment, the variance of consumption in the second period is not affected. The variance of the random price, however, will be affected, unless the demand curve happens to be linear. If g°°°(+) is negative, the demand curve will be concave. The variance of the randon

price will then increase with larger carry~overs. . The following graph shows a ccnceve

....,COnsumer demand curve for the second period: “rama TE ; ~ .< Gem

-

oe P,: Second Period Price Contingent on State tae, ae

Consumer Demand Curve: Er)

“4 |

SR meron cermin an mene: nerzine gemma eRe RRR

o

r e

rt, ee t t

with zero carry-over, the price that will emerge with the endowments in state r or —r —s state s may be read from the graph above x” and x”. If the speculator carried over oe . =r , 4 =s : : K,. units, we would read the prices above x” + K and x” + K.- Since the slope becozes 2

increasingly steep, the difference between the two state dependent prices widens as the fixed gap between the state dependent endowments is moved to the right. Hence, any carry-over increases price variance; however, the consumer benefits from tradinz

with the speculator whose profitable carry-over causes the prices to be more uncertain.

VI. Concluding Remarks.dAbous Speculation

s

In-this paper, we have taken a further look at a question examined by other

researchers. Using the carry-over definition of speculation, we have seen that

:

profitable speculation does not reduce price variability, but that people gain fron trading with the speculator anyway. The Friedman proposition, as interpreted by

others, is wrong.

I do not know if he intended "price Stability" to be defined as we have done. Nor do I know if the comparison between the regimes of autarky and profitable speculation is the one he intended. In 1971, he repeated the substance of his earlier assertion without modifying it in light of the considerable research which had been

published on the subject:

“It is worth noting that, in general, speculation can destabilize exchange rates only if speculators buy spot to hold when prices are high and sell Spot out of inventories when prices are low. In that case, speculative transactions do make the swing in rates wider--but also speculators lose money. The belief that speculation is destabilizing is therefore largely equivalent to the belief that speculators on the whole lose money ... ."

- . This paper simplifies and extends the body of literature interpreting his proposition

in a uniform way.

There are now many general-equilibrium models where all agents behave optimally

Over time and states. It is easy to distinguish optimal from sub-optimal behavior.

TOTEER SM et rae een ean Renn paMiNn ee GL Aner s at eenmenmnnin een ay gece TO ea a mena ite ee tee em ee pe tem he ee Mmm gg BS TTT er mene cnmrgpenma © © esemerrmni 2 = ae ea RRR ORE,

oO - 23 -

Defining which behavior is "speculative", however, is more difficult. Meany differen: activities have been labellcd speculation: playing the futwres market, utilizing new information to advantage, carry-overs etc. 2 - . Why we need to define the concept at all is a legitimate question. Without a

definition, we can still predict how each optimizer in our wodels will react to any

exogenous change. My major reason for studying speculation is to evaluate the basis for the

=

widespread hatred of speculators among the general public. The definition of speculation we choose should, in my opinion, correspond to the behavior people

abhor. We should then examine the social merits of.:this behavior within our models and, eventually, within the real world. Perhaps the public maans something quite different from carry-overs by the term speculation. Or perhzps the public believes that, although speculators provide a service, they are monopslistic in the real world and should be made competitive.

Speculation, within the model we have considered, provides a service which is easy to overlook. The speculators appear to do nothing: they bry soybeans from the corral and sell the same amount back--at a profit. People may resent buying back from the speculator in times of scarcity what they sold him cheaply om an earlier occasion.

The carry-over service of the speculator might be ignored. If so, this analysis shoule be useful in clarifying the carry-over function of the specrfator and the irrelevance

of price stability.

Bibliography

Farrell, M.J., "Profitable Speculation," Economica, 33 (1966), 183-193.

Friedmaf, N., Essays in Positive Economics (Chicago: University of Chicago Press, 1953). .

,» "The Need for Futures Markets in Currencies," in "The Futures Market in Foreign Currencies," Chicago Mercantile Exchange (1971), P- 11.

Gustafson, Robert, "Garry-Over Levels for Grains," Technical Bulletin No. 1178, U.S. Dept. of Agriculture (19538).

Kemp, M.C., "Speculation, Profitability, and Price Stability," R.E. Stat., 45 (1963), 185- 189. -

Samuelson, P.A., "Stochastic Speculative Price,'' Proceedings of the National Aca adeny of Sciences, Vol. 68, No. 2, 335-7, February, 1971.

“Intertemporal Price Equilibrium: A Prologue to the Theory of Speculation," Waltwirtschafliches, Archiv, (Dec. 1957).

, "The Gains from International Trade Once Again," Economic Journal, December 1962. ‘

Schimmler, J., “Speculation, Profitability, and Price- “Stability - A Formal Appreeach," R.E. Stat., February 1973, 110-114.

Sohmen, Flexible Exchange Rates (Revised Edition), (Chicago: University of Chicago Press, 1969), p. 68.

Telser, L.G., "A Theory of Speculation Relating Profitability and Stability," R.E. Stat., 41 (1959), 295-301.

~

Townsend, R., "Price Fixing Schemes and Optimal Buffer Stock Policies," (unpublished).

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