the relationship between wholesale and retail prices …

The Economic Studies QuarterlyVol. 37, No. 1, March 1986

THE RELATIONSHIP BETWEEN WHOLESALE

AND RETAIL PRICES AND SPECULATION:

THE CASE OF A MIDDLEMAN ECONOMY*

By KOICHI MASHIYAMA

1. Introduction

This study considers the impact of the activities of a market-maker acting on his own behalf

upon the operating characteristics of a product market when the product is storable. Specifically,

we explore the role of the retailer-wholesaler (middleman) and his impact on movements in

wholesale and retail prices.

As such, this study relates to the literature on inventory behavior. Recent studies on inven

torybehavior include Abel [1], Amihud and Mendelsen [3], Blinder [5], Irvine [8], Reagan [14],

and Schutte [16]. Most of those models analyze the optimal inventory decisions of individual

producing firms, and examine the relationship between prices, production, and inventory.However, this conventional approach to inventory problems is not entirely satisfactory for

explaining the movements in wholesale and retail prices. It most often relies upon a rudimen

taryformalization of the inventory decision in which producers hold inventory only because the

production process is subject to a fixed delay. In other words, it ignores the specific role ofmarket-makers.1) To circumvent the less plausible implications of the traditional theory, the

importance of market organization should be emphasized and the explicit role of market-makers

incorporated into the associated models. Towards this end, we particularly consider the market

organization in which competitive markets for inventories of finished goods are organized by

wholesalers.

We often observe that competitive markets for commodity stocks have been developed for

agricultural products and raw or intermediate products such as steel bars, textile products and IC

chips. Product stock markets are interpreted as a sort of"inside markets"among wholesalers. In

such a market organization, exchanges of products take place in four sub-markets: the"inside

market"of product stocks among wholesalers; the producer-wholesaler market; the wholesaler

- retailer market; and the retailer-consumer market. To make the model manageable, two

markets consisting of the"inside market"and the wholesaler-retailer market are consolidated

* The author would like to thank Professors K. Hamada, K. Otani, M. Otsuki, an anonymous referee,and the Editor for their helpful comments. Improvements on English are due to Mr. Julian Morison,graduate student of economics and management science.

1) Exceptions are Irvine [8] and Schutte [16]. Irvine examined the pricing rule of an isolated retailerwho holds inventory. Schutte examined the price movements under the presence of the cost of distribution.See also Blanchard [4].

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K. Mashiyama: The Relationship between Wholesale and Retail Prices

into a single market, hereafter called a commodity stock market. The group of middlemen is

assumed to be composed of wholesalers and retailers. Therefore, exchange is performed in

three types of markets: a producer-middleman market, in which producer's prices are deter

mined;a competitive commodity stock market, where wholesale prices are determined; and local

(physically separated) retail markets dominated by monopolistic middlemen. It is also assumedthat there are consumers that are devoid of search capabilities, and speculators executing spot

transactions at the commodity stock market.

In Section 2, the basic framework of the middleman economy is outlined, where the com

moditystock market and local retail markets are defined in more detail and the objectives of

middlemen and pure speculators are formalized. Section 3 is devoted to the derivation of the

rational behavior of middlemen and speculators. The middleman's rational behavior is charac

terizedas follows: (i) the demand for inventory is a decreasing function of the current wholesale

price; and (ii) the inventory demand and the pricing rule (for retail price) are increasing functionsof the forecast of consumption demand for the next period. In Section 4, the dynamic equations

governing the time evolution of market prices are derived and the typical price response isexamined. In the absence of pure speculators, the response pattern of wholesale prices is charac

terizedby a monotonically damping response and follows the first-order Markov process; as

does the response pattern of retail prices, although its fluctuations are moderated. An explana

tionfor mark-up pricing as a rational response to market conditions is also provided. In Section

5, the effects of the presence of pure speculation are discussed, and in particular it is shown that

the presence of pure speculation leads to the movement of market prices as a second-order

Markov process. The fact that pure speculation causes price dynamics to be more complicated is

indeed curious. This also sheds new light on the issue concerning the"stickiness"or"persist

ence"of prices currently being debated in the rational expectations literature.2) The final section

concludes the paper.

2. The Basic Framework of a Middleman Economy Model

As seen in the Introduction, there exist three types of markets in the present model: a pro

ducer-middleman market, a commodity stock market, and local retail markets. Consequentlythere invole three types of prices: producer's price, wholesale price, and retail price. First, con

siderhow the producer's price is determined. Suppose that producers deliver their products to

the warehouses of middlemen where trade takes place between the producers and middlemen.We assume the following price contract between producers and middlemen: for one unit of prod

uctdelivered, middlemen pay the wholesale price determined in the commodity stock market

minus, the transaction fee agreed among them. Although this transaction fee might be adjusted

according to market conditions, we assume that the transaction fee is fixed.3) Therefore, the pro

ducer-middlemen market does not play an explicit role in the model.

2) See, for example, McCallum [10].3) In some specific product markets, investigation into the working of markets between producers and

middlemen would be very important and interesting. It requires the appropriate modelling of the varioustransaction costs and asymmetric information, as Williamson [18] and Carlton [6] have suggested.

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The Economic Studies Quarterly

We now consider the microstructure of the commodity stock market. The collective activity of

the market participants can be characterized as a stochastic flow of market sell and buy orders,

which are the sum of discrepancies between individual agent's actual and desired holdings of

inventory. In general, there exist bid and ask prices in a stock market, and market participants

adjust their bid-ask prices in response to temporal discrepancies between market sell and buy

orders. Therefore, the spread between the bid and ask prices is adjusted to market conditions.4)

In the present model, we assume that the induced change in the price spread is negligible and the

average of bid and ask prices is adjusted in response to market disequilibrium. We call this

average price a wholesale price. In other words, we are assuming (i) that the temporal discrep

ancybetween market sell and buy orders in the commodity stock market is cleared by the move

mentof the wholesale prices and (ii) that the commodity stock market is characterized by the

stock equilibrium.5)

We assume that local retail markets are physically separated. Each middleman owns a retail

shop in one of the local markets. Retail price in the local market is charged by the middleman,

whose store is located in this market. Each consumer visits the middleman's store and buys the

product from him. The posted retail price might be distributed across local markets due to the

traveling costs consumers have to bear.6)

Denote by qdjt the demand for the commodity in the jth local market at time period t. To

simplify our analysis, we postulate the following linear demand function;

(1) q

djt=ƒ¿djt-ƒÀdjpcjt, where pcjt is the retail price charged by the jth middleman at time period t,ƒÀdj is a positive con

stant,and ƒ¿djt is a random disturbance, which might consist of market-wide disturbance and local

shock specific to the local market. Even if the linearity assumption is relaxed, the middleman's

behavior would not be changed. For example, the demand function with constant elasticity

leads to the same behavior of the middleman.

Since the commodity stock market is competitive, producers have no incentive to hold inven

tory.However, producers speculating on the commodity stock market are called pure specula

tor.We consider a producing firm that hires input factors in period t and delivers his output in

period t+1. At period t, when the production plan must be made, the period t+1's market

price is unknown. The producer must predict the market price for period t+1 on the basis of

his own information. The objective of the producer is to choose his output level which maxi

mizesthe discounted present value of the expected profit. There is no adjustment cost, and so

the producer's behavior is myopic, i,e., the producer maximizes his one-period profit. Therefore,

4) The formulation of the bid and ask prices in a security market has been explored by Amihud and

Mendelsen [2]. Their modelling approach could be applicable to the formation of the bid and ask prices

in the commodity stock market if the retail price is assumed constant. However, the story is not so

simple in the present model.

5) A stock equilibrium model of a commodity stock market can be formulated as the Hawtrey-Otani

model. See Otani [13]. The issue concerning economic effects of speculative storage has been inten

sivelyexplored in the speculation literature. See Turnovsky [17] and Newbery and Stiglitz [12].

b) This assumption is not so restrictive because our purpose is not to explore the retail price dispersion

across local markets but to explore the relationship between the wholesale price and the retail price. The

relaxation of the assumption would require a modification of the middleman's pricing behavior.

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fore, it is plausible to postulate the following linear supply function:

(2) qst=ƒ¿st+ƒÀsE[pt|It-1], ƒÀs>0,

where qst is the amount of commodity made by the producer at the beginning of period t, pt is

wholesale price in period t,It-1 is information available to the producer in period t-1, and E is

the mathematical operator of taking expectations.7) The first term ƒ¿st is the supply disturbance

caused by external shocks such as weather conditions or shortage of raw materials.

The middleman's store will maintain the desired level of commodity stocks to make efficient

his"middleman services"or transactions. The exchange process of commodity stocks is assumed

to be completed during the beginning of each period. After the completion of this exchange the

middleman opens his retail store and sells the commodity to customers. At the start of period t,

when he decides what amount of stocks to hold and which retail price to quote, he does not have

complete knowledge about forthcoming demand conditions or about future wholesale prices.

Therefore, the middleman must predict the future economic conditions based on his incomplete

information. Let Jt denote information available to the middleman at the beginning of period t,

and define

(3a) tqdƒÑ=E[qdƒÑ|Jt]=E[ƒ¿dƒÑ|Jt]-ƒÀdpcƒÑ,

and

(3b) tpƒÑ=E[pƒÑ|Jt].

We drop the index for individual middlemen here. Define the forecasting error tuƒÑ by

(4) tuƒÑ=qdƒÑ-tqdƒÑ.

We assume that the forecasting error u has a density function f(u) and the distribution of u is

invariant over time. Let yt be the holding of inventory when the middleman enters period t and

xt the net amount of commodity stocks purchased on the commodity stock market at period t.

The level of inventory at period t is yt+xt when he opens his store. Then the amount of inven

torywhen he enters period t+1 is

(5) yt+1=zt-qdt,

where

(6) zt=yt+xt.

The expected gross revenue of the middleman when retail price is pct is given by pcttqdt. When

the middleman carries inventory, an inventory-holding cost is incurred. In addition, if the

inventory is exhausted, there is a cost associated with unsatisfied current demand. Let w(•E)

denote the inventory-holding cost function and h(•E) the shortage penalty cost function.8)

The objective of the middleman is to maximize the discounted present value of expected

profits from the"middleman services". That is, the middleman selects a sequence of inventory-

7) Since the transaction fee determined by the price contract between producers and middlemen is

assumed to be constant, introducing the transaction fee explicitly into the producer's model changes

only the value of the parameter ƒÀs.

8) The shortage cost function utilized here has similar economic interpretations as Amihud and

Mendelsen [3]'s and Blinder [5]'s have. However, the form of the cost function is different due to the

economic role of the middleman. In general the shortage cost depends not only on the amount of

inventory but also on the other variables such as prices. We assume away the latter factors for technical

simplicity but we can show that the dependency of the shortage cost on the prices does not change the

key behavioral features of the middleman.

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holdings and retail prices that maximize the discounted present value:

(7) ‡”•‡ƒÑ=t(1/1+r)ƒÑ[pcƒÑtqdƒÑ-{•çzƒÑ-tqdƒÑ-•‡w(zƒÑ-tqdƒÑ-u)f(u)du The last term of the profit function expresses the cost of inventory purchase. The parameter r is

the rate of time discount, which is assumed constant and identical across middlemen. To be able

to solve the middleman's optimization problem, we have to impose the concavity conditions on

the profit function:

(8) h'>0, h">0, w'>0, and w">0,

where primes refer to the derivative of the corresponding variables.

The only kinds of transactions open to the pure speculator are spot buying and selling of com

moditystocks. The speculator engages only in speculative trading, i.e., he purchases stocks with

the intention to sell those later at a profit on the commodity stock market but he enjoys no con

venienceyeilds of the stock like the middleman.

We assume that the variable cost of storage increases monotonically and the marginal cost also

increases with inventory-holding. Denote by g the variable cost function:

(9) g">0, g"(z)>0, for z•¬0, and g(0)=0.

At the beginning of period t the speculator observes the stock price pt but he cannot know the

price for the next period with certainty. He must predict the future prices based on his own

information. Let Kt be the information the speculator possesses at period t. The objective of

the speculator is to choose a sequence of optimal levels of storage, z0i, i•¬t, which maximizes

the present value:

E[‡”•‡i=t(1/1+r)i{(y0i-z0i)pi-g(z0i)}|Kt], where y0i is the level of his stock as he enters period i and z0i is his carry-over inventory from

period i to i+1. The discounted present value can be rewritten as

(10)

‡”•‡i=t(1/1+r){(y0i-z0i)E[pi|Kt]-g(z0i)}. The price expectations defined in (2), (3b), and (10) are different if producers, middlemen, and

speculators have different information. However, we suppress this difference when we analyzethe characteristics of the market price movements.

3. The Rational Behavior of the Middleman and Speculator

In this section we derive the optimal pricing rule and optimal inventory-holding of the middle

man,and optimal storage rule of the speculator. We start with the middleman's behavior. We

rewrite the discounted value as

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(11)

‡”•‡i=t(1/1+r)i[pci tqdi(pci)-{W(zi-tqdi)+H(zi-tqdi)}-tpixi], whereW(zi-tqdi)=•çzi-tqdi-•‡w(zi-tqdi-u)f(u)du,

H(zi-tqdi)=•ç•‡zi-tqdih(u+tqdi-zi)f(u)du. W(•E) is interpreted as the expected holding-cost of inventory and H(•E) the expected average

shortage cost of inventory. It should be noted that the total cost of inventory can be defined to

be the sum T=W+H.9) The corresponding Lagrangian Lt is given by

(12) Lt=•ç•‡i=t(1/1+r)i[pci tqdi-W(zi-tqdi)-H(zi-tqdi)

-tpixi-ƒÉi{yi+1-yi+tqdi-xi}], where ƒÉi is the associated Lagrange multiplier. The first-order conditions are

(13) tpi+W'(tyi+1)=ƒÉi-H'(tyi+1),

(14)

pci•Ýtqdi/•Ýpci+tqdi=-{W'(tyi+1)+H'(tyi+1)-ƒÉi}•Ýtqdi/•Ýpci, (15) ƒÉi+1=(1+r)ƒÉi+W'(tyi+2)+H'(tyi+2),

where

tyi+2=zi+1-tqdi+1.The variable ƒÉi is interpreted as the expected shadow price of inventory. The first relationship

states that the expected shadow price plus the marginal shortage cost saved by adding one unit of

inventory equals the expected market price plus the marginal inventory-holding cost. If we

introduce the notion of the price elasticity of demand ƒÅ:

ƒÅ=-pci/tqdi•Ýtqdi/•Ýpci, the second equation can be simplified into

(16)

pci(1-1/ƒÅ)=ƒÉi-W'(tyi+1)-H'(tyi+1)=tpi. The relationship (16) implies that the marginal revenue equals the sum of marginal inventory-

holding and shortage costs and the expected shadow price of inventory. The marginal revenue

also equals the expected wholesale price. This equation is interpreted as the mark-up pricing

rule for the retail price. The retail price depends on the wholesale price and the price elasticity of

demand. Substituting (13) into (15) yields

9) As suggested in the footnote8), we have been assuming that the cost function T(y) of inventoryachieves its minimum at some point y*, where y* is called the efficient level. The efficient level y*depends on the middleman's techniques of distribution, which gives rise to the fact that y* is greaterthan zero.

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(17) ƒÉt=tpt+1/(1+r).

This means that the expected shadow price equals the discounted value of the future market

price. Substituting (17) back into (13) leads to the following equation:

(18) pt+W'(tyt+1)=

1/1+rtpt+1-H'(tyt+1). This relationship implies that the difference between the discounted expected price and the

current price equals the net marginal cost of inventory. It is clear that if there is no discrepancy

between the current wholesale price and the shadow price of inventory, we have

(19) W'(tyt+1)+H'(tyt+1)=0.We find here that the optimal holding of inventory minimizes the total cost function T of

inventory.

To derive the behavioral equations, we take the total variation of (13) and (14), which yields

(20) [ƒÐ1 ƒÐ2 ƒÐ2 ƒÐ3]•E[dzt dpct]=[-1 0]dpt+1/1+r[1 ƒÀd]dtpt+1+[ƒÐ1 ƒÐ2+1]dtƒ¿dt,

where

ƒÐ1=H"+W">0,ƒÐ2=ƒÀdƒÐ1>0,ƒÐ3=2ƒÀd+(ƒÀd)2ƒÐ1>0. Define

ƒ¢=ƒÐ1ƒÐ3-(ƒÐ2)2=2ƒÀdƒÐ1>0.

Solving equation (20) for dzt and pct, we have(21)

[dzt dpct]=ƒ¢-1[[-ƒÐ3 ƒÐ2](-)(+)dpt+1/1+r[ƒÐ3-ƒÀdƒÐ2 0](+)dtpt+1+[ƒ¢-ƒÐ2 ƒÐ1](+)(+)dtƒ¿dt]•E This shows that an increase in the current market price has a negative influence on the demand

for inventory but a positive effect on the retail price. The second term represents the"specula

tive"component. An increase in the expected wholesale price for the next period raises inven

torydemand but leaves the retail price intact. The last term expresses the effect of changes in

the expectation of the consumption demand level. When the middleman expects forthcoming

consumption to rise, he raises inventory demand and at the same time, quotes a higher retail

price. This means that the mark-up ratio for the retail price is increased by the expectation of

higher consumption demand.

Consequently we can postulate the following behavioral equations:

(22a) zt=z(pt,tpt+1,tƒ¿dt),

(-) (+) (+) (22b) pct=pc(pt,tƒ¿dt)

(+) (+)

(22a) is the inventory demand function and (22b) is the retail pricing rule. This retail pricing rule

provides the theoretical foundation for the so-called mark-up pricing rule in which the mark-up

ratio depends on the forecast of consumption demand.

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K. Mashiyama: The Relationship between wholesale and Retail Prices

Now we consider the rational behavior of the speculator. The Lagrangian Lt for the speculator

is

(23) Lt

=‡”•‡i=t(1/1+r)i[(y0i-z0i)tpi-g(z0i)+ƒÉi(z0i-y0i+1)], where ƒÉi is the associated Lagrange multiplier. The first-order conditions are

(i) tpi+1=(1+r)ƒÉi,

(ii) tpi+g'(z0i)=ƒÉi, for z0i>0,

tpi+g'(z0i)>ƒÉi, for z0i=0. Hence we have

(1/1+r)tpt+1=pt+g'(z0t) if pt+g'(0)<(1/1+r)tpt+1,(24)zt=0 if pt+g'(0)•¬(1/1+r)tpt+1. This optimal policy shows that the speculator's inventory-holding depends only on his expecta

tionof the next period's wholesale price. That is, speculation is definitely a myopic activity. In

addition, we observe that the speculator holds no stocks if the market price is expected to fall next

period. These are key characteristics of the speculator's behavior. For more rigorous analyses,

see Kohn[9] and Samuelson[15].

Consequently we can define the speculative demand for inventory as a function of the current

price and the expected price for the next period:

(25)

z0t=z0(1/1+rtpt+1-pt).z0(•E) is the function that satisfies the optimal policy rule (24). Function (25) has the property

that an increase in the expected price for the next period raises the speculative inventory

demand. However, it should be noted that the function z0(•E) has a kink at the price such that

p*+g'(0)=1/1+rtpt+1. The inventory demand z0 is zero for prices below this point, given the expectation tpt+1.

4. Price Movements in the Absence of Pure Speculators

In this section, we analyze the movements of wholesale prices in the commodity stock market.

The total supply of commodity stocks at the beginning of period t consists of the aggregate

amount Yt+Z0t-1 of inventory carried-over by middlemen and speculators from the last period

plus the aggregate inflow Qs of commodity production. The arbitrage activity leads to themarket-clearing condition:

(26) Yt+Z0t-1+Qst=Zt+Z0t,

where the uppercase letters refer to the aggregate quantities of the corresponding variables. Fluc

tuationin the aggregate level of inventory is governed by

(27) Yt+1=Yt-Qdt+Qst.

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If there exists no middleman in the economy, as in the purely speculative model, the market

clearingcondition turns out to be

(28) Z0t-1+Qst=Qdt+Z0t.

In this case, as Muth[11] and others have shown, the larger presence of speculative storage stabil

izesprice movements. In our model, however, even if there exists no speculator, speculation-

like activity emerges and so we expect the middleman's activity to have a stabilizing influence on

the price movements. To provide the answer to this problem, we shall explore the time pattern

of price movements.

We set Z0t=0 for all t. The market-clearing condition reduces to

(29) Yt+Qst=Zt.

Then we must have

Yt+Qs(ƒ¿st,t-1pt)=Z(pt,tpt+1,rƒ¿dt).

(+) (+) (-) (+) (+)

Solving this equation for pt yields the following price dynamics:

(30) pt=P(Yt,tpt+1,t-1pt,tƒ¿dt, ƒ¿st).(-) (+) (-) (+) (-)

We observe the following phenomena: An increase in the economy-wide inventory level induces

a decline in the current market price. When the middleman expects a rise in the next period's

market price, the current price will increase. This is the effect of evening out price fluctuations,

which is usually called the stabilizing influence of speculation in the speculation literature. The

lagged price expectation formed by producers has a negative effect on the current price since the

output level is positively related to the lagged price expectation. When the middleman forecasts

that forthcoming consumption will increase, for example, due to an increase in real money

balances, the current market price will rise. The supply disturbance has a negative influence. It

should be reiterated that these are impact effects.

We also observe that there exists an intertemporal transmission channel from the exogenous

disturbance to the variation in market price. This transmission path is the typical one at work

through the commodity stock market, which is attributable to the existence of positive storage.

To examine the intertemporal relationship of prices, we need to specify the formation of price

expectations. We utilize the linearized system of the model and assume that the public forms

rational expectations. The assumption of rational expectations is very strong but it provides a

short-cut route for the investigation of the time pattern of price movements. The linearized

system can be given by

(31a) pt=-ƒ¢/ƒÐ3Yt+1/(1+r)ƒÐ3(ƒÐ3-ƒÀdƒÐ2)tpt+1

-ƒ¢/ƒÐ3ƒÀst-1pt-ƒ¢/ƒÐ3ƒ¿st+1/ƒÐ3(ƒ¢-ƒÐ2)tƒ¿dt,(31b)

Yt+1=Yt+(ƒ¿st-ƒ¿dt)+ƒÀst-1pt+ƒÀdpt+ƒÁdtƒ¿dt, where

ƒÀd=ƒ¢-1ƒÐ2ƒÀd, ƒÁd=ƒ¢-1ƒÐ1.

Combining (31a) and (31b) yields the following equation for price dynamics:

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(32) pt+1=(1-ƒ¢ƒÐ-13ƒÀd)pt-ƒ¢ƒÐ-13ƒÀdtpt+1+(1/1+r)ƒÐ-13(ƒÐ3-ƒÀdƒÐ2)(t+1pt+2-tpt+1)

-ƒ¢ƒÐ-13ƒ¿st+1+ƒÐ-13(ƒ¢-ƒÐ2)t+1ƒ¿dt+1-ƒÐ-13(ƒ¢-ƒÐ2)tƒ¿dt+ƒ¢ƒÐ-13ƒ¿dt-ƒ¢ƒÐ-13ƒÁdtƒ¿dt. To analyze the response pattern of prices to demand shocks, we posit the general form of the

solution:

(33) pt

=‡”•‡i=taiƒ¿dt-i, where ai's are undetermined coefficients. Here we have suppressed the supply disturbance.10)

In general, the ƒ¿dt's are governed by the stochastic process in which the exogenous driving

shock is white noise. We consider only the case:

(34) ƒ¿dt=ƒÃt,

where the ƒÃt's is are independent and identically distributed. We assume that ƒÃt-N(0,ƒÐƒÃ).

In this case the public will believe that the shock perceived is transitory. Since the public

forms the rational expectations, the expected price is the expected value of pt conditional on the

public's information in period t. Therefore

(35)

tpt+1=‡”•‡i=taiƒÃt+1-i, tƒ¿dt=0. Substituting (35) into (32), we have

(36)

a1=ƒÓ5/1-ƒÉƒÓ3,, ai+1=ƒÉiai, i=1,2,•c where ƒÉ is the real root with modulus less than one of the characteristic equation:

A(ƒÉ)=ƒÓ3ƒÉ2-(1+ƒÓ2+ƒÓ3)ƒÉ+ƒÓ1=0,

and

ƒÓ1=1-ƒ¢ƒÐ-13ƒÀd>0,ƒÓ2=ƒ¢ƒÐ-13ƒÀs>0,ƒÓ3=1/1+rƒÐ-13(ƒÐ3-ƒÀdƒÐ2)=2/1+rƒÐ-13ƒÀd>0,ƒÓ4=ƒÐ-13(ƒ¢-ƒÐ2)=ƒÐ-13ƒÀdƒÐ1>0,ƒÓ5=ƒ¢ƒÐ-13>0. Hence the solution to (32) is

(37) pt=‡”•‡i=1a1ƒÉi-1ƒÃt-i.

The time pattern of price movements depends crucially on the values of a1 and ƒÉ, which are

functions of the model's parameters.

Since ƒÓ3<1 and ƒÓ5>0, the last period's demand shock exerts a positive effect on the current

market price with the value of a1. The pressumption that |ƒÉ|<1 implies that the strength of

the demand effect monotonically declines over time. We can derive the condition for the

10) Since the response pattern of prices to a supply shock can be similarly analyzed and is similar to that

under the demand shock, we omit the case of the supply shock.

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uniqueness and existence of the real root with modulus less than one of the characteristic

equation (37). Since A(1)=-1, we must have

(38) A(-1)=1+ƒÓ1+ƒÓ2+2ƒÓ3>0,

in order to have exactly one real root with modulus less than one. Note that the inequality (38)

can be rewritten as

2+ƒ¢ƒÐ-13(ƒÀs-ƒÀd)+4/1+rƒÐ-13ƒÀd>0. This inequality is automatically satisfied, so that the uniqueness of the solution is proved. (See

Futia [7] for a more rigorous treatment.)

The sign of ƒÉ is positive if and only if A(0)>0. That is,

(39) ƒÓ1>0.

Since ƒÓ1>0, we can assert that ƒÉ is always positive. Therefore, the typical time pattern of the

market price to the demand shock is characterized by a monotonically decreasing response.

From (37), we have

(40) pt+1=ƒÉpt+a1ƒÃt.

It is clear that the price movement follows the first-order autoregressive process. Using the root

locus analysis, we find that it is likely that a decline in the interest rate increases the value of ƒÉ,

through an increase in inventory demand. Therefore, the middleman's inventory-holding

strengthens the effect of the last period's price on the current price. In other words, as the

middleman's holding of inventory becomes larger due to a change in the cost function, the serial

correlation of price movements is higher and the impact effect is also raised.

From (21), we have

pct=1/2pt+1/2ƒÀdtƒ¿dt, which implies that fluctuations of the retail price is smaller in magnitude than those of the whole

saleprice. Substituting this relationship into equation (40) yields the first-order Markov Process:

pct+1=ƒÉpct+1/2a1ƒÃt. Hence the impact effect of a demand shock on the retail price is less than on the wholesale price.

Such a response of the retail price is obtained because the commodity stock market plays the

role of a buffer, which absorbes exogenous shocks. This result is intuitively clear.

5. Price Movements in the Presence of Pure Speculators

In this section we discuss how the presence of pure speculators alters the time pattern of price

movements. From (25) and (26), we have

(41) Yt+Qs(ƒ¿st,t-1pt)=Z(pt,tpt+1,tƒ¿dt)+Z0(1/1+rtpt+1-pt)-Z0(1/1+rt-1pt-pt-1). The speculative demand function Z0(•E) is characterized by a kinked curve and so it cannot be

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differentiated at the kink. The speculative demand function Z0 cannot be linearized but we

approximate it by a linear function:

Z0(1/1+rtpt+1-pt)=ƒÌ(1/1+rtpt+1-pt), where ƒÌ is a positive constant. This approximation is necessary as without it we cannot have a

manageable model on which economically meaningful propositions are established. The cost

incurred from the approximation is that our analysis ignores the asymmetric property of specula

tion.The costs and benefits arising from such a approximation must be considered but we

believe that the present approximation is justifiable in these terms.

Thus we have the linearized system:

(42) pt=-ƒ¢ƒÐ-13Yt+ƒÂƒÐ-13(ƒÐ3-ƒÀdƒÐ2)tpt+1-ƒ¢ƒÐ-13ƒÀst-1pt

+ƒ¢ƒÐ-13ƒÌ[ƒÂtpt+1-pt-(ƒÂt-1pt-pt-1)]-ƒ¢ƒÐ-13ƒ¿st+ƒÐ-13(ƒ¢-ƒÐ2)tƒ¿dt, where

ƒÂ =1/(1+r).

Combining (42) and (31b) yields the following price equation:

(43) P

t+1=ƒÓ1pt-ƒÓ2tpt+1+ƒÓ3(t+1pt+2-tpt+1)+ƒÓ[(ƒÓt+1pt+2-pt+1)-2(ƒÂtpt+1-pt)+(ƒÂt-1pt-pt-1)]+ƒÓ5ƒ¿dt, where ƒÓ=ƒ¢ƒÐ-13ƒÌ, and the supply shock is suppressed. The difference between (32) and (43) is

given by the ƒÓ-term. Looking at the ƒÓ-term, we fmd that the equation for price dynamics turns

out to be a third order difference equation. Therefore the introduction of pure speculation

makes price dynamics more complicated. This result is very different from the result claimed in

the conventional model of purely speculative markets, and this difference originates from the fact

that the middleman carries a positive amount of inventory over all periods. In purely speculative

markets it is shown that the larger the speculative demand, the more likely the price movement

follows a random walk. However, in the present model, the presence of pure speculators causes

the market price to follow an oscillatory behavior. To see this, we have to solve the equation

(43).

Generally, the stationary solution of (43) can be expressed as

(44) pt={c+Ld(L)}ƒÃt-1,

where c is a constant to be determined, L is the lag operator, and d(L) is a fixed analytic function

of the lag operator. Substituting the expression (44) into (43) yields the following equation:

(45) pt+1={c+Ld(L)}ƒÃt

=[ƒÓ1L{c+Ld(L)}-ƒÓ2Ld(L)+ƒÓ3(1-L)d(L)

+ƒÓ(1-L)2{-c+(ƒÂ-L)d(L)}]ƒÃt+ƒÓ5ƒÃt. Taking the z-transform of both sides, we obtain

(46) A(z)d(z)=B(z)c+ƒÓ5

where

A(z)=ƒÓ(1-z)2(z-ƒÂ)-{ƒÓ3-(1+ƒÓ2+ƒÓ3)z+ƒÓ1z2},

B(z)=-1+ƒÓ1z-ƒÓ(1-z)2.

Note that the equation (43) has a solution if and only if we can find a real number c and ananalytic

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function d(z) solving (46) for all value |z|<1. Therefore, if the equation A(z)=0 hasexactly one real root with modulus less than one, there exists a solution of (43).

The root locus analysis leads to the observation that the equation A(z)=0 has exactly one realroot with modulus less than one and two roots with modulus greater than one. The latter two

roots are complex and have a positive real part except for sufficiently small ƒÓ. When ƒÓ is

extremely small, these roots are real and greater than one. Let ƒÉ* be the real root with modulus

less than one. Substituting z=ƒÉ* into (46) yields the equation for c:

B(ƒÉ*)c+ƒÓ5=0.

Solving this equation we can compute the value of c. Then, we have the analytic function d(z)

of the form:

d(z)=(t4+t5z)/(t1+t2z+t3z2),where the ti's are constants. Hence the price movement can be expressed as

pt=cƒÃt-1+t4+t5L/t1+t2L+t3L2ƒÃt-2. This equation describes the time pattern of price response in the presence of pure speculation.

We do not intend to compute the exact values of the parameter ti since they would not provide

any significant economic implications.

Now looking at the form of the price dynamics, we can note the following: When the

equation

(47) t1+t2z+t3z2=0

has complex roots, the price response must be oscillatory. Although it is difficult to clarify the

relationship between the roots of (47) and the amount of speculative storage, ƒÌ it is clear that the

large amount of speculative storage does not necessarily induce the random walk property of

price movements. We might say that the presence of pure speculation leads to price dynamics

characterized by the Markov process with an order higher than second. This phenomenon

occurs because the presence of pure speculators introduces dynamic inventory fluctuations, in

addition to the middleman's inventory dynamics. The dynamics of pure speculators' inventory-

holdings are not synchronized with those of the middlemen because pure speculators and

middlemen are different economic agents.

We can see that we do not have the somewhat clear-cut result concerning the effect of pure

speculation. However, unlike Muth's result, in the present model, a large amount of speculative

storage does not induce the random walk property. Even if we utilize the original kinked

demand curve rather than the linear demand curve, the essential feature could not be eliminated.

The modification to be required is to take into account the asymmetric nature of speculation and

how this asymmetry exerts its effect on the magnitude of price fluctuations.

6. Concluding Remarks

The present study has formulated a simple model of the product market characterized by the

existence of commodity stock markets to explore the characteristics of price movements. In par

ticular,we have developed a theoretical model of the middleman, in which we derive the rational

inventory-holding and pricing rule. Since the summary of the derived behavioral equatoins are

provided in the Introduction, we do not reproduce them here. One point we wish to reiterate is

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that the retail pricing rule provides the theoretical explanation for the spread between wholesale

and retail prices.

Also we have developed a model of price movements, in which the wholesale price is shown tofollow a first-order Markov process in the absence of pure speculation. The middleman's spe

culativestorage arising from the cost-saving incentive has the role to increase the impact effect of

a demand shock and the serial correlation of prices. The retail price also follows a first-order

Markov process but fluctuations of the retail price are smaller in magnitude than those of the

wholesale price. In other words, the wholesale price is more sensitive to a demand shock than

the retail price. This shows the role of the commodity stock market as a type of shock absorber.

In Section 5, we attempted to clarify the effects of pure speculation on price movements but we

have not been successful in providing a clear-cut result due to the complexity of the model. It

should be mentioned, however, that the effect of pure speculation in our model is very different

from those established in the speculation literature.

(Toyohashi University of Technology)First draft received January 19, 1985; final draft accepted June 21, 1985.

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the relationship between wholesale and retail prices …

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