Measuring the Benefits from Futures Markets: Conceptual Issues

Measuring the Benefits from Futures Markets: Conceptual Issues

Donald Lien*

Department of Economics, University of Texas at San Antonio, U.S.A. James Quirk

Emeritus Professor, California Institute of Technology, U.S.A.

Abstract

This paper illustrates that the usual consumer surplus approach to evaluation of the bene-fits of a futures market fails because of certain unobserved benebene-fits. In particular, when fu-tures markets provide benefits in the form of a reduced variability of future spot prices, the usual consumer surplus approach will systematically understate the benefits of futures mar-kets.

Key words: futures markets; price variability; consumer surplus JEL classification: D6; G0

1. Introduction

Over the past decade or so, there has been an impressive expansion in the number of commodity and financial futures markets throughout the world, and in the variety of such markets. It is widely believed that, in a world of incomplete markets, the introduction of new financial markets will improve social welfare. For example, futures markets provide benefits to participating traders and through them, to con-sumers, by reducing transaction costs, by providing a more efficient flow of infor-mation among traders, and by shifting risks among them. Of course, any new mar-kets can be beneficial to the overall economy as well, although external effects might negate these benefits [see Hart (1975), Newbery and Stiglitz (1984), Geana-koplos and Polemarchakis (1986), Elul (1995), and Duffie and Rahi (1995)].

A specific benefit of futures markets is that they can act to reduce risks for market participants. They do this in part by providing a mechanism for shifting risks from more risk averse to less risk averse traders. They also do this by improving the flow of information among market participants. In addition, under certain conditions at least, futures markets can reduce the perceived variability of future spot prices.

Received July 12, 2001, accepted August 22, 2001.

*_{Correspondence to: Department of Economics, University of Texas at San Antonio, 6900 North Loop} 1604 West, San Antonio, TX 78249, U.S.A. Email: dlien@utsa.edu. An anonymous referee provided helpful comments and suggestions.

For the most general case, there is little agreement on the effect the futures markets have on the underlying spot market; see Mayhew (2000) for a comprehensive survey.

From a public policy point of view, these potential benefits of futures markets must be weighed against the potential costs in evaluating regulatory proposals con-cerning such markets. The appropriate way to measure the benefits of futures mar-kets is through a consumer surplus approach. However, there are basic conceptual problems with applying consumer surplus to the evaluation of the benefits of a fu-tures market in a world with an incomplete set of markets. As will be shown here, the assumption of rational behaviour by consumers in the face of uncertainty about future spot prices of commodities leads to the conclusion that the usual consumer surplus approach to evaluation of the benefits of a futures market fails because of certain unobserved benefits. In particular, when futures markets provide benefits in the form of a reduced variability of future spot prices, the usual consumer surplus approach will systematically understate the benefits of futures markets.

In what follows, we examine only the conceptual problems associated with consumer surplus in measuring the benefits of a futures market associated with a reduction in the variability of future spot prices for a commodity. However, it is ob-vious that the same problems arise in more inclusive studies of the benefits of futures markets.

2. The Analytical Framework

For simplicity of notation, we use a two-period, two-commodity framework to derive our measure of the value of a futures market. An extension of the result to the n-period, m-commodity case is straightforward. We consider a pure exchange economy, in which a typical consumer is provided with known endowments of the two goods, x and y , in each of the two periods. At time t=1, prices p and 1

1

q for the goods x and y , are known with certainty, but prices in the second pe-riod are random variables. At time t=2, second period prices of the goods are known before the consumer makes his consumption choices. It is assumed that the market rate of interest r is known with certainty.

Let x and t y denote the consumption levels of goods x and y in period t

t, 2t=1, , respectively. In each period, prices of the two goods are p and t q , t respectively, whereas the consumer has endowments of the two goods, ω and xt

yt

ω , respectively. Thus, the present value of income for the consumer is ) ( ) 1 ( 1 _{2 2 2 2} 1 1 1 1 x q y r p x q y p V = ω + ω + + − ω + ω

.

(1)

Let β denote the personal time discount rate. Suppose that the consumer has an additive, separable inter-temporal utility function. His expected utility is then

∫ ∫

∞ ∞ − ∞ ∞ − + =u₁(x₁,y₁) u₂(x₂,y₂)f(p₂,q₂)dp₂dq₂ EU β

,

(2)

where (.,.)ut is the utility function and f(.,.) is the joint density function of the second period prices. Based upon the information available, the consumer will choose optimal consumption levels in the two periods to maximize his expected util-ity. Because of price information updating, the problem is to be analyzed by the usual dynamic programming approach; that is, a backward induction procedure is adopted.

At time t=2, when the choices x and 1 y have already been made and 1 prices p and 2 q are known, the amount of money available to be spent is 2

[

V px q y

](

r

)

V2= −( 1 1+ 1 1) 1+

.

(3)

We assume away the default problem, so that it is simply assumed that V₂>0. The consumer's problem is to maximize u₂(x₂,y₂) subjecttotheconstraintthat

2 2 2 2

2x q y V

p + =

.

This results in the demand functions x =₂ x₂(p₂, q₂;V₂) and ) ; , ( _{2 2 2} 2 2 y p q V y =

.

Returning to the first period problem, the consumer chooses x and 1 y to 1 maximize

∫ ∫

∞ ∞ − ∞ ∞ − + =u₁(x₁,y₁) u₂[x₂(p₂,q₂;V₂),y₂(p₂,q₂;V₂)]f(p₂,q₂)dp₂dq₂ EU β

.

(4)

The first order conditions are as follows: 0 ) 1 ( 2 2 1 1 1 1 = ∂ ∂ + − ∂ ∂ = ∂ ∂ V Eu r p x u x EU _β

_,

(4a) 0 ) 1 ( 2 2 1 1 1 1 = ∂ ∂ + − ∂ ∂ = ∂ ∂ V Eu r q y u y EU _β

_.

(4b) Now, assume that there exists a shift parameter α appearing in the probability density function f(.,.) such that an increase in α results in a decrease in the variability of p . For example, an increase in 2 α decreases the variance of a nor-mally distributed p . A futures market is assumed to be represented by an increase 2 in α .

3. The Effect of a Futures Market

We wish to measure the consumer surplus that results from an increase in α . Since we are concerned with a compensated demand curve, changes resulting from an increase in α must be such that d(EU)=0

,

where

dV V EU d EU d dq q EU d dp p EU d dy y EU d dx x EU EU d ∂ ∂ +       ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = α α α α α α1 ₁ 1 ₁ 1 ₁ 1 1 ) (

.

(5)

Note that, in an equilibrium framework, both p and 1 q are determined endoge-1 nously (although individuals take them as given when making their decisions). Consequently, both are functions of α . By the first order conditions,

0 1 1 = ∂ ∂ = ∂ ∂ y EU x EU

_,

(6) whereas 0dq1/dα = by the usual partial equilibrium assumption. Equation (5) im-plies α α α d EU d dp p EU dV V EU       ∂ ∂ + ∂ ∂ − = ∂ ∂ ₁ 1

.

(7) Moreover, 2 2 1 1 ) 1 ( V Eu r x p EU ∂ ∂ + − = ∂ ∂ _β (8) and 2 2 ) 1 ( V Eu r V EU ∂ ∂ + = ∂ ∂ _β

_.

(9) We derive the following equation:

α α β α d EU V Eu r d dp x dV         ∂ ∂       ∂ ∂ + + − = − −1 2 2 1 1 (1 )

.

(10)

Assume that introducing a futures market changes the value of α from 0

=

α to α =α*, and that associated with that change in α is a change in p 1 from )p1(0 to p1(α*). The measure of consumer surplus due to the introduction of the futures market is then given by

α α β α d EU V Eu r dp q p x W p p

∫

∫

∞ − ∂ ∂       ∂ ∂ + + − = 0 1 2 2 ) ( ) 0 ( 1 1 1 1 (1 ) 1 ) , ( * 1 1

,

(11) with 2 2 2 2 2 2 2 2 2 2 2 2 2 ) ; , ( )] ; , ( ), ; , ( [x p q V y p q V f p q dp dq u EU

∫ ∫

∞ ∞ − ∞ ∞ − ∂ ∂ = ∂ ∂ α α β α

.

(12)

) (q₂

f_q where f_p(..) denotes the conditional density of p given 2 q and 2 (.)

q

f is the marginal density of q . Furthermore, 2 u2(.,.) is monotonically de-creasing in p and 2 q . By the Quirk-Saposnik (1962) result concerning the 2 first-degree stochastic dominance, it follows that if the conditional distribution func-tion f_p(p₂q₂;α) is an increasing function of α, then ∂EU/∂α >0. More gen-erally, assume that u2(.,.) is monotonically decreasing and concave in p and 2

2

q . By the Hadar-Russell (1969) result, if an increase in α leads to a distribution )

; (p₂q₂ α

f_p that is stochastically dominated in the second degree, then .

0 /∂ >

∂EU α In particular, if f_p(p₂q₂;α) is normal and if there is a mean-preserving decrease in the variance resulting from an increase in α, then

0 /∂ > ∂EU α .

Thus, under any of the above conditions, if a futures market leads to stabiliza-tion of future spot prices, then ∂EU/∂α>0. As a result, the second integral of equation (11) is positive, but not observable. The integral measures the benefits to a consumer from a decrease in the variability of p even if this has no effect on 2 p . 1 Also, note that the consumer surplus measure depends only on the first period demand for x ; the backward oriented dynamic programming routine incorporates all second period benefits into the first period measure. Equation (10) points out a difficulty in measuring the contribution to the consumer surplus by a futures market. To accurately measure the benefits of a futures market, information on consumers’ inter-temporal preferences must be gathered. Lacking this information, the benefits of futures markets tend to be underestimated when the futures market stabilizes spot prices.

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