Hildreth, Clifford

Kislev, Yoav

English

Research Papers in Economics

Staff Papers Series
Staff Paper I?80-28 December 5, 1980
A Model of Yield Variability and Price Effect
Clifford Hildreth and
Departmentof Agriculturaland AppliedEconomical
Uniwrsity of Minnesotu
[nstitutc of Agriculture, Forestry and Home Economics
St. Paul. Minnesota 55108A Model of Yield Variability and Price Effect
Clifford Hildreth and Yoav Kisle.v*
1. Introduction
Suppose a crop can be produced by either of two sets of cultural
practices. One promises a higher yield if growing conditions turn out
to be favorable; yield under the second will be higher if unfavorable
conditions are encountered. An expected utility maximizing, risk
averse producer is allocating his acreage between the two regimes. We
are particularly interested in how the allotted acreages respond to
changes in the price of the crop.
Questions of this kind sometimes arise in developing economies.
If the less variable technology is in common use, but the more variable
technology is associated with higher average yield, then partial
adoption of the more variable system increases supply over time and
exerts downward pressure on prices. Will lower prices create additional
incentives to adopt the more variable system or will lower prices reduce
incentives to change? This was the primary question which led to the
present investigation. One example of alternative technologies is the
choice between chemical and organic fertilizers. With ample rainfall
chemical fertilizer should be more productive. If there is little rain,
organic will turn out better.
In this paper price of output is regarded as known when the
production decision is made. This corresponds to forward contracting
*Respectively, University of Minnesota and the Hebrew University,
Rehovot, Israel. Part of the work on this study was conducted while
Yoav Kislev was a visitor at the Department of Agricultural and Applied
Economics of the University of Minnesota at St. Paul. Work done in Israel
was supported by a grant from the Ford Foundation through the Israel
Foundation Trustees.-L-
or to a controlled price announced in advance. Models involving a random
price will be explored later.
The following notation is used:
W represents yield per acre from the more variable technology.
V represents yield per acre from the more stable technology.
W and V are random variables on an underlying probability space
(0,~’~~j P) where u c O is a possible sequence of growing conditions.
y is price of output
~ is the decision maker’s utility of gain function
b is total acreage on the farm in question ~~’~t-ll a acres cultivated
in the more variable way and (b - a) in the less variable way.
~ is the expected utility function and the decision problem is
(1.1) max n(a;y) = E~[yQ(a)] y > 0 where
a~[o, b]
Q(a) is a random variable representing prospective total yield when
a acres are cultivated by the more variable method.
(1.2) Q(a) =aW+(b-a)V=a(W- V)+bV.
The semicolon separating the arguments of n is
that a is the decision variable while y is regarded
inserted to recognize
as a parameter by the
decision maker. The following assumptions are used:
(1.3) (a) $’ >0, $’’<0,+’” >0
(b) limo’(x) = o
X*
(c) Let Z=W-V. For all.asR, y > 0 the following random
22’ 2
variables have finite means W, V, W , V , Zt’(yaZ), Z 4“(yaZ).-3-
(a) and (b) have been commonly assumed by economists, +’ > 0 expresses
preference for larger gains rather than smaller, 4“ < 0 represents risk
aversion, ~’” > 0 is implied by (but weaker Chan) decreasing absolute risk
aversion, (b) is more difficult to assess since it involves what would
happen as potential gains became arbitrarily large and, therefore, outside
our normal experience and contemplation. It has, however, seemed reasonable
to other analysts (e.g. Leland [6], Bertsekas [2]) as well as the present
authors.
(c) also seems reasonable and is assumed for mathematical tractability.
It will hold providing tails of the distributions of W, V are not too fat;
for example if W, V are bounded.
If n(~) > n(a) for all real numbers a, then ~ will be called an
unrestricted maximizer of n. If O ~ ~ f b and n(~) ~ n(a) for all a c [o, b]
than $will be called a restricted maximizer of rI.
Assumptions (1.3), which are maintained throughout this paper, have
the following implications:
(1.4) (i) continuous first and second derivatives of n exist and may
be obtained by differentiation under the expectation.
(ii) n is strictly concave in a and has, for each y, a unique
unrestricted maximizer.
(iii) ~<0+~ =O;~>b+;=b;O~~ ~b++=;.
(iv) (~ - a) ~ Dan(a;y) acR.
,,:,,
means ‘hgrees in sign
with”.
A proof of (i) is indicated in the third section of this paper.
(ii) is proved in Hildreth [4], pp. 9, 10 (note that p(W - V ‘ 0)>0,
P(w - V<o)> o). (iii), (iv) are simple consequences of the fact that,-4-
for given y, ~ is single peaked and everywhere strictly concave.~’ Thus
the contemplated decision problem (summarized in 1.1) has a unique
solution ~ and the associated unrestricted problem has a unique solution
a. It will be convenient to examine the relation between ~ and y and to
write
(1.5) a = a(y) y>o.
Assentation (iii) above connects ~ with ~.
In the next section a particular specification of a possible relation
between W and V is introduced. It is shown that, under the specification,
response of more variable acreage to a change in price has sign opposite
to the derivative of the decision makerts relative risk aversion function.
This is the paper’s main result and it adds to the importance of finding
empirical evidence on how decision makers’ , and particularly entrepreneurs’ ,
relative risk aversion responds to increases in gain.
One expects that per acre costs of the two regimes will differ in
most applications. It is convenient to think of gain as being measured
after deducting the costs of cultivating the entire acreage
way is more expensfve. One regards these cultivation costs
when the decision
be added to gross
for lower costs.
is contemplated. Any savings in per acre
in whichever
as prepaid
costs can then
receipts under the less expensive procedure to allow
This enables one to regard W, V, Q as non-negative
random variables and does not change the interpretation of the model.
Before introducing the specification, it is perhaps worth reminding
some readers that if risk neutrality were assumed we could write O(x) = x
——-
~/ If (iii), (iv) are not immediately obvious, the reader can draw
a peaked, smooth, strictly concave function and compare a, 2.,~ for various
values of a.-5-
and the decision problem (1.1) would have the following trivial solution.
(1.6) EW>EV+~=+=b
EW<EV+~=-m, :=()
EW = EV =)~, ~ indeterminate.
2. Derivation of Principal Result
The notion that one procedure will be advantageous if favorable
circumstances are encountered while another will be better if developments
in the decision maker’s environment are adverse can be made more precise
in many ways. Such notions are relevant in many ,f’!’lsxts--risky versus
less risky securities, uninsured versus insured ventures, etc. We hope
the specification (s) below represents some production problems reasonably
well. It is assumed that for some intermediate events, yields under the
two regimes are equal (W = V = k). If experience is better, acreage
under the less variable technology benefits somewhat but acreage under
the other technology benefits more. Conversely, under less favorable
experience yield on acreage under the more variable technology declines
more. This can be stated
(s) w-v ~V-k.
Since ~ is differentiable and strictly concave, 2 may be obtained
for given y by solving Dan(a; Y) = 0. As shown in the final section this
yields
Dan(a; y) = y E(W - V) $’[ya(W - V) - ybV] = O
Not much can be said about ~ without further assumptions. One may note-6-
(2.1) ~an(o; Y) = YE(W - V) $’(ybV)<yf(W - V) $’(ybk) + yj(w - V) ~’(ybk)
w>v W<v
= y 0’(ybk) E(W - V) ~ E(W - V).
Thus, by (1.4 - iii),
(2,2) E(W - V) LO+~<Oand~=O.
No acreage will be devoted to the more variable method unless its average
yield is higher, and one clearly could have ~ < 0, ~ = O even though EW
were somewhat larger than EV. More precise statements would depend on
making further assumptions about ~, W, V.
The main interest here is in the response of ~ and ~ to changes in
Y. It will be convenient to discuss the relation (1.5) & = a(y) and let
the reader note possible responses of ~ using (1.4 - iii).
Since iIEC2 (is twice continuously differentiable), the implicit
function theorem may be applied to investigate the sign of Dya, namely
(2.3) Dya = -(Daa~(L3; y))-l D@yq(~; y).
Differentiation under the expectation yields
(2.4) Daa~(~; y) = y2 E(W - V)2 ~“(yQ(~)) < 0
so, using (2.3),
(2.5) Dya ~ DayV(~; y).
Again differentiating under the expectation,
(2.6) Dayn(~; y) = E(W - V) $’(yQ(~)) + E(W
The first term on the right equals y‘1 Da(~; y)
of a. Let
- V) yQ(5) v“(yQ(~)).
and vanishes by definition-7-
(2.7)
l)” (x)
r(x) = - —————
+’ (x)
R(x) = -
xl)” (x)
4’ (x)
be the Pratt-Arrow indices of
(2.8) Dayq(5; y) = -E(W -
absolute and relative risk aversion. Then
V) ~’(yQ(~)) R(yQ(~))
= j-(v_ W) 0’(yQ(2)) R(yQ(~)) + J(V - W) v’(yQ(~)) R(YQ(A))
W>v W<v
If $J< 0 or ~> b, then small changes in y will not affect ~ so assume
o~~<b, Then under (S), Q(2) > Q(ybk) in the first integral and
Q(a) < Q(ybk) in the second integral. Thus, if R is an increasing
function,
(2.9) Dayn(~; Y) < J(V - W) ~’(yQ(~)) R(ybk) + J(V - W) 4’(yQ(~)) R(ybk)
W>v W<v
= R(ybk) f(V - W) ~’(yQ(~)) = R(ybk) [-y-l Da~(~; y)] = O
If R is a decreasing function, the inequality in (2.9) is reversed. Clearly
R constant implies Dayrl(~; Y) = O. Putting these together
(2.10) Dya(y) ~ D ~(~; y) ~ - DXR(X).
ay
Thus the acreage response to price turns on how, for the relevant decision
maker, relative risk aversion responds to an increase in gain. If relative
risk aversion increases with gaj.n,as initially suggested by Arrow [1],
then acreage using the more variable technology declines as price increases.
If relative risk aversion is a decreasing function of gain than the
indicated acreage moves in the same direction as price.-8-
However, existing evidence on relative risk aversion is
clear (see Stiglitz [7] and [81). There are also many other
not entirely
contexts,
e.g. Diamond and Stiglitz [3], in which optimal choice depends on the
behavior of relative risk aversion. The present result adds to our need
to obtain firmer empirical evidence on this point.
One fragment of recent evidence is furnished by a study of farmer’s
utility functions by Hildreth and Knowles [5]. Utility functions were
fitted to responses of 13 Minnesota farmers to a series of questions on
hypothetical decisions under uncertainty. Of several forms of utility
functions tried, substantially the best fits were obtained using
-Alx -a2x
(2.11) v(x) = - e - Be A
1’ ‘2’
~>o,
‘1>A2”
For such utility of gain functions, differentiation and rearrangement
of terms and factors yields
(kl+A2)x
($’)2e A;
(2.12) R’(x) = (Al + ~2) + + ‘l;’x - .2X
ala26 A2BJX
where u =
‘1
.-A2>0.
Since coeff R’(x) > 0
~2fielJx
A2+2
(2.13) R’(x) = Y(X) = (xl + 12) + - lJ2x .
A2f3eux al
The first three terms of y(x) are positive. Clearly y(x) > 0 for
Al + A2
x<
2
and y(x) > 0 for x sufficiently large. For some parameter
u
sets (11, A2, 6), Y(x) > 0 for all x therefore R’(x) > 0 for all x.-9-
However, in the study cited, 11 of 13 of the estimated utility functions
showed an intermediate interval of x for which relative risk aversion was
decreasing (R’ < O). Also, if we look at “typical” parameter values obtained
in this study by taking approximate medians of the 13 estimated values,
these approximate medians are xl = .!, ~ = .003, E = 50. For a utility
2
function with these parameters, R(x) is decreasing for 15.2 < x < 43.5 and
increasing for x < 15.2, x > 43.5 (X was measured in thousands of dollars).
This suggests that it may
about R’(x) and that this
studied in each empirical
be difficult to establish broad generalizations
important property may have to be carefully
application.
3. Differentiability and Continuity
Application of the implicit function theorem in Section 2 required
tl(a;y) to have continuous first and second
differentiation under the expectation. Two
derivatives are involved. Arguments in the
similar that it does not seem worthwhile to
we shall illustrate proofs by showing that,
(i) Da n(a; Y) = yEZ lj’(yaZ+ ybV)
(ii) Day n(a; Y) = EX 4’(yaZ + ybV)
+ yaEZ(aZ + bV) ~“(yaZ + ybV)
(iii) Dayn(a; y) is continuous.
~. Recall n(a; y) = E@(yaZ + ybv). If Da
derivatives obtainable by
first and three second
several cases are sufficiently
present all of them. Instead
under Assumptions (1.3),
n exists. then
(3.1) Darl= lim#n(a + h; Y) - n(a,y)]
h-@
= lim~[EV(yaZ + ybV + yhZ) - E 4(yaZ + ybV)]
h+o-1o-
By the mean value theorem
(3.2) 4(yaZ + ybV + yhz) = $(yaZ +ybV) +YZ $’(yaZ +ybV+hGyZ)
where O < G < 1. --
Substituting into (3.1) yields
(3.3) Dan = y lim EZ ~’(ya Z -tybV i-hGyZ).
h+o
Without loss of generality assume Ihl < 1.
Let Z+ = max {Z, O}, Z- = max {-Z, O}. Then, since~’ is a decreasing
function and V > 0,
(3.4) lZ~’(yaZi- ybV+hGyZ)l = lZ+~’(yaZ +i-ybV+hGyZ+)
_Z-~l(-YaZ- + ybV - hGyZ-l < \Z+~’((y-l)a Z+l
The right side is integrable by (1.3c) so, by the dominated convergence
theorem, the order of lim and E may be interchanged in (3.3) producing
(3.5) Dan = yE[lim Z~’(yaZ + ybV + hGyZ)] = yEZ~’(yaZ + ybV).
h+o
(ii). If DayrIexists then
(3.6) Day~ = lim~ [(y + h) EZ ~’(y +h)aZ + (y +b)bV)
h+()
- yEZ$’(yaZ -t- ybV)]
By the mean value theorem
(3.7) ~’(yaZ+ ybV+h(aZi-bV)) = 4’(yaZ+ybV)
+ h(aZ -tbV) ~“(yaZ -tybV + hG(aZ + bV))
where ()<G<~ -_(3.8)
-11-
Substituting from (3.7) to (3.6),
Day~ = limj$hEZ$’(yaZ
h+Q
+ hyEZ(aZ + bV) ~“(yaZ
+ h2EZ(aZ + bV) ~“(yaZ
= EZ$’(yaZ + ybV) + ya
+ ybV)
+ ybV 1-hG(aZ
+ y~V -thG(aZ
lim EZ2~’’(yaZ
-I- bV))
+ bV))]
+ ybV +hG(aZ + bV))
h+O
+ yb lim EZV ~“(yaZ + ybV + hG(aZ + bV))
h-+()
+ a lim hEZ2 ~“(yaZ + ybV + hG(aZ + bV))
h+o
+ b lim hEZV ~“(yaZ + ybV f-hG(aZ + bV)).
h+()
Without loss of generality let lhl < 6 < y. By (1.3a) ~“ is negative
and increasing so
(3.9) \Z2$’’(yaZi-ybV+hG(aZ+bV))l < l(Z+)2~’’(y-6) aZ+ I
+ ~(Z-)2$’’(-(yi-6)aZ-)1.
The right side of (3.9) is integrable by (1.3c) so, again using the
dominated convergence theorem, lim may be taken before E in the second
*ls~/
and fourth terms of the final expression for Dayrlin (3.8).
(3.10) lZV~’’(yaZ+ybV+hG(aZ+bV)l < \Z+Vv’’((y- d)aZ+)l
—
2_/ Note that EZ2$” integrable implies EZVY” integrable.-12-
so lim and E may be reordered in the third and final terms, Simplifying
then gives
(3.11) Dayn = 13Z~’(yaZ+ ybV) + yaEZ2v’’(yaZ + ybV)
+ ybEZV~’’(yaZ + ybV)
= EZv’(yaZ -t- ybV) + yEZ (aZ + bV)~’’(yaZ + ybV).
(iii) Let (yn, an) + (y, a). Without loss of generality assume
[Yn-Y\ ‘~<mi-n{y, a, l}and Ian-al <~.
(3.12) Dayn(an; yn) = EZt’(ynanZ + ynbV)
+ ynanEZ2$’’(ynanZ + ynbV) + ynbEZV~’’(ynanZ + ynbV).
Because of the continuity of products it is sufficient to show continuity
of the three expectations. Note
(3.13) lZ$’(ynanZ-tynbV)\ < lZ+~’(yn -d) (a -6) Z+)l
n
lZ2t’’(ya Z+ynbV)l < l(Z+)21J’’(yn -6) (an-d) Z+l
nn
+ l(z-)2 $“(-(yn + 6) (an + 6)Z)I
]W’’(ynanz+ynbv)l < lZ+WJ’’((Yn- 6) (an- 6) Z+l
-+ IZ-VIJJ”(- (yni- 6) (an+ 6)Z-)1.
Again, the dominating random variables are integrable by (1.3c) so lim
as (yn, an) + (y, a) of each expectation is the expectation of its limit
and the sum of these with appropriate coefficients is D
ay
q(a; y).1.
2.
3.
4.
5.
6.
7.
8.
References
Arrow, Kenneth J., “The Theory of Risk Aversion,” Essay 3 in Essays
in the Theory of Risk-Bearing, Markham, Chicago, 1971.
Bertsekas, Dimitri P., “Necessary and Sufficient Conditions for the
Existence of an Optimal Portfolio,” Journal of Economic Theory, v. 8,
n. 2, 1974.
Diamond, Peter A. and Joseph E. Stiglitz, “Increases in Risk and Risk
Aversion,” Journal of Economic Theory, v. 8, n. 3, 1974.
Hildreth, Clifford, “Expected Utility of Uncertain Ventures,”
Journal of the American Statistical Association, v. 69, n. 1, 1974.
Hildreth, Clifford and Glenn J. Knowles, ’’SomeHints about Farmers’
Utility Functions’ University of Minnesota preliminary report,
(unpublished). A more complete report is ir’1)1I’aration.
Leland, Hayne E., “On the Existence of Optimal Policies under Uncertainty,”
Journal of Economic Theory, v. 4, n. 1, 1972.
Stiglitz, Joseph E., “The Effects of Income, Wealth, and Capital Gains
Taxation on Risk Taking,” Quarterly Journal of Economics, May, 1969,
pp. 263-83.
Stiglitz, Joseph E., “Review of Arrow, Aspects of the Theory of Risk
Bearing,’’Econometrica, 37:4 October, 1969, PP. 742-3.

A MODEL OF YIELD VARIABILITY AND PRICE EFFECT

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