Li, Bingbing; Long, Yan
Wo king Pape
Ou pu unce ain y mi iga ion in compe i i e ma ke s
Cen e o Ma hema ical Economics Wo king Pape s, No. 698
P o ided in Coope a ion wi h:
Cen e o Ma hema ical Economics (IMW), Biele eld Uni e si y
Sugges ed Ci a ion: Li, Bingbing; Long, Yan (2024) : Ou pu unce ain y mi iga ion in compe i i e
ma ke s, Cen e o Ma hema ical Economics Wo king Pape s, No. 698, Biele eld Uni e si y, Cen e
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698
Janua y 2024
Ou pu Unce ain y Mi iga ion in
Compe i i e Ma ke s
Bingbing Li and Yan Long
Cen e o Ma hema ical Economics (IMW)
Biele eld Uni e si y
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Ou pu unce ain y mi iga ion in compe i i e
ma ke s
Bingbing Li∗
, Yan Long†
Janua y 22, 2024
Abs ac
Ou pu unce ain y is a majo conce n o indus ies p one o ex-
ogenous, pe sis en and la ge fluc ua ions in ou pu , such as ag i-
cul u e, wind and sola powe gene a ion, while echnology adop ion
aimed a mi iga ing ou pu unce ain y can imp o e social wel a e.
This pape cons uc s a compe i i e ma ke model wi h andom ou -
pu fluc ua ions o examine he scale o echnology adop ion a he
long- e m equilib ium and i s compa ison wi h he social op imum.
We show ha he Fi s Wel a e Theo em no longe holds in gene al,
and depending on he cha ac e is ics o he demand unc ion, he scale
o echnology adop ion in he compe i i e ma ke may be g ea e o
less han he socially op imal scale.
Keywo ds: echnology adop ion, unce ain y mi iga ion, long- e m
equilib ium, non-op imal scale
JEL: D50, D61, D62
∗School o Economics, Huazhong Uni e si y o Science and Technology, Wuhan, China.
Email: bbingli@hus .edu.cn.
†School o Economics, Huazhong Uni e si y o Science and Technology, Wuhan, China.
Email: ylong@hus .edu.cn. We a e g a e ul o he Minis y o Educa ion P ojec o Hu-
mani ies and Social Sciences (G an 21YJC790080) and he Na ional Na u al Science
Founda ion o China (G an 7210030643) o unding. We hank Kaige Hu o he excel-
len RA wo k.
1
1 In oduc ion
Unce ain y is an inescapable eali y ha many indus ies ha e o conside
in p oduc ion ac i i ies. Ho (1989) di ided p oduc ion unce ain y in o wo
ca ego ies: (i) en i onmen al unce ain y and (ii) sys em unce ain y. En i-
onmen al unce ain y e e s o unce ain ies caused by ac o s ou side he
p oduc ion p ocess, such as demand unce ain y and unce ain y caused by
supplie s. Sys em unce ain y is ela ed o unce ain ies in he p oduc ion
p ocess, such as unce ain y in ope a ion yield, quali y, and ailu e o he
p oduc ion sys em. This pape conside s he la e ca ego y o unce ain y.
Specifically, his pape ocuses on p oduc ion ac i i ies ha depend g ea ly
on na u al condi ions such as wea he and empe a u e, including ag icul u e,
aquacul u e, and he apidly de eloping sola and wind powe gene a ion in
ecen yea s. The ou pu unce ain y o such p oduc ion ac i i ies has he
ollowing cha ac e is ics: (1) The unce ain y in ou pu is de e mined by ex-
ogenous ac o s a he han he unce ain y in inpu s such as aw ma e ials;
(2) The unce ain y in ou pu is pe sis en a he han caused by acciden al
e en s such as sudden in e up ions o inexpe ience; (3) The fluc ua ions in
ou pu caused by exogenous ac o s a e la ge and a e among he undamen-
al easons de e mining he final ou pu . Wi h he in ensifica ion o global
wa ming and he la ge-scale applica ion o ola ile enewable ene gy, such
ou pu unce ain y issues become inc easingly se e e.
While ou pu unce ain y may cause huge nega i e impac s, he e does
exis a se ies o echnologies ha can educe ou pu fluc ua ions. Fo ex-
ample, ag icul u e can use echnologies such as p ecision i iga ion and in-
elligen e iliza ion o educe he impac o na u al en i onmen al ac o s
such as clima e and soil on ou pu (Koundou i e al., 2006). In wind and
sola powe gene a ion, pumped s o age and elec ochemical ene gy s o age
a e effec i e ways o mi iga e ou pu fluc ua ions (Schmid e al., 2017).
Howe e , whe he fi ms in a compe i i e ma ke ha e enough incen i es o
adop hese echnologies is s ill an open ques ion, since echnology adop ion
2
among fi ms will affec each o he , ha is, he effo s o one fi m o educe
ou pu fluc ua ions will affec he effo s o o he fi ms h ough he impac
on p ices. The Fi s Wel a e Theo em no longe holds au oma ically due o
he fi ms’ enla ged ac ion space. Whe he compe i i e ma ke equilib ium
can achie e he social op imum is he main esea ch ques ion s udied in his
pape .
To answe his ques ion, we cons uc a compe i i e ma ke model wi h
he abo e p oduc ion cha ac e is ics o examine how he in e ac ion be ween
fi ms de e mines he scale o echnology adop ion. While ma ke p ices a e
de e mined by supply and demand, as in a s anda d compe i i e ma ke
model, fi ms’s effo s o adop echnology a e de e mined in he ollowing
way: each fi m’s effo choice depends on p ice, which depends on he o al
effo o he indus y; bu each fi m, whose ou pu is oo small compa ed
wi h he o al ou pu o he indus y, igno es he effec i s own effo choice
has on p ice.1Hence he equilib ium concep we define, hough consis ing o
mo e a iables han he s anda d compe i i e equilib ium, is compe i i e in
sp i . And we find ha he Fi s Wel a e Theo em no longe holds in gene al,
and depending on he cha ac e is ics o he consume demand unc ion, he
scale o echnology adop ion in he compe i i e ma ke may be g ea e o
less han he social op imal scale. Hence go e nmen s need o de e mine he
op imal in e en ion policy acco ding o he demand cha ac e is ics.
We end he in oduc ion wi h a b ie e iew o he li e a u e. The ex-
is ence o unce ain y is well acknowledged in economics, and i has been
in eg a ed in o gene al equilib ium models (e.g., Bo ch (1962) and A ow
and Nea e (1978)) and pa ial equilib ium models (e.g., Joskow and Ti ole
(2006) and Joskow and Ti ole (2007)). I is well-known ha when ma -
ke p ice can be con ingen on he ealized s a e, compe i i e equilib ium
achie es social op imum. Howe e , o ou bes knowledge, he p oblem o
1The equilib ium de e mina ion o effo s has some fla o o Nash equilib ium in la ge
games, see Schmeidle (1973) and Khan and Sun (2002).
3
mi iga ing unce ain y in compe i i e ma ke has a ely been add essed in
he li e a u e o economics.
On he o he hand, he e a e many s udies in he li e a u e o ope a ion
ha ocused on p e en ing isks caused by ex e nal ac o s such as na u al
disas e s h ough supply chain isk managemen (SCRM) (e.g. Singhal e
al. (2011) and Ri chie and B indley (2007)) o mi iga ing unce ain y in he
p oduc ion p ocess h ough p oduc ion planning (e.g., Be and and Ru en
(1999), G a es (2011)). Fo he p oduc ion ac i i ies s udied in his pape ,
which ea u e pe sis en and la ge ou pu fluc ua ions, he ole o SCRM
and p oduc ion planning a e bo h limi ed. Fu he mo e, he a o emen ioned
li e a u e has mainly cen e ed on mi iga ing unce ain y o indi idual fi ms,
whe eas his pape discusses how he scale o fi ms’ echnology adop ion
o educe unce ain y in e ac wi h each o he , and whe he compe i i e
ma ke s necessi a e policy in e en ion.
2 Model Se ing
On he supply side, he model consis s o wo indus ies, one wi h de e -
minis ic ou pu , and he o he wi h s ochas ic fluc ua ing ou pu s as well as
echnology ha can mi iga e fluc ua ions. And hey a e e e ed o as de e -
minis ic indus y and s ochas ic indus y, espec i ely. The demand side is
comp ised o ep esen a i e consume s who own capi al ha can be in es ed
in he wo indus ies and who consume p oduc s om bo h indus ies.
We fi s desc ibe he p oduc ion o each indus y. Fo he de e minis-
ic indus y, one uni o capi al can p oduce one uni o p oduc . Fo he
s ochas ic indus y, he ou pu a e pe uni o capi al is i∈[0,1], whe e i
is a uni o m andom a iable whose ealiza ion only depends on exogenous
ac o s like wea he o empe a u e. And we u he assume he ealiza ion
o ou pu a e is iden ical o each uni o capi al. Tha is, when he capi al
s ock o all fi ms in he s ochas ic indus y is K, he o al ou pu le el is iK.
4
The a iable cos o p oduc ion in each indus y is assumed o be ze o. Each
indus y has a la ge numbe o fi ms, and hey a e all p ice ake s.
Fi ms in he s ochas ic indus y can adop some kind o echnology ha
educes ou pu fluc ua ions.2Le θ∈[0,1] be he measu e o echnology
adop ion scale. Fixed θ, he ou pu a e pe uni o capi al becomes j=
(1 −θ)i+θ
2. No e ha j∈[θ
2,1−θ
2], and E[j] = (1 −θ)E[i] + θ
2=E[i],
V a [j] = (1 −θ)2V a [i]. When θis la ge , he ange o jis na owe and
he a iance is smalle . Le h(θ)be he amoun o capi al ha is needed
o achie e echnology adop ion scale θ o each uni o capaci y in es men ,
whe e h(0) = h′(0) = 0,h′(θ)>0 o θ > 0,limθ→1h′(θ) = ∞, and h′′(θ)>0.
Consume s own a fix amoun o capi al ¯
K. And he u ili y unc ion o
he ep esen a i e consume is U(q, x), whe e qis he p oduc s o he in-
dus y wi h s ochas ic ou pu , and xis he p oduc s o he indus y wi h
de e minis ic ou pu . We u he assume ha he u ili y unc ion is sepa a-
ble, ha is U(q, x) = S(q) + W(x), whe e S′(q)>0,S′′(q)<0,W′(x)>0,
W′′(x)≤0,limq→0S′(q)> W′(¯
K)and limx→0W′(x)> S′(¯
K). No e ha
consume s p e e lowe ou pu ola ili y. Tha is, S(E[˜q]) > E[S(˜q)], o
any ˜qbeing a andom consump ion amoun o p oduc s om he s ochas ic
indus y.
We will fi s look a he socially op imal echnology adop ion scale. Then
we will define and cha ac e ize he long- e m ma ke equilib ium. Finally, we
will compa e he ma ke equilib ium esul s wi h he socially op imal esul s.
3 Social op imum
Le Kbe he capi al in es men le el and θbe he echnology adop ion scale
in he s ochas ic indus y. The social op imal p oblem is he ollowing.
2In gene al, echnology adop ion will affec bo h ou pu ola ili y and ou pu le el.
He e we only conside he impac on ola ili y. Some echnologies can be app oxima ed
as only affec ing ola ili y hough, such as powe s o age echnologies.
5
max
θ,K Ej[S(jK)] + W(¯
K−(1 + h(θ)) ·K).
Since iis uni o mly dis ibu ed in [0,1] and j= (1−θ)i+θ
2, we ha e ha
jis uni o mly dis ibu ed in [θ
2,1−θ
2]. And we can hence w i e he p oblem
as
max
θ,K ∫1−θ
2
θ
2
1
1−θS(jK)dj+W(¯
K−(1 + h(θ)) ·K).
Since limq→0S′(q)> W′(¯
K)and limx→0W′(x)> S′(¯
K), in he op imal
solu ion, 0< K < ¯
K. Since limθ→1h′(θ) = ∞, in he op imal solu ion, θ < 1
. Suppose in he op imal solu ion, θ > 0. The he op imal solu ion sa isfies
he ollowing fi s o de condi ions:
Fo K,
Ej[S′(jK)·j] = W′(¯
K−(1 + h(θ)) ·K)(1 + h(θ)).(1)
Fo θ,
1
(1 −θ)2∫1−θ
2
θ
2
S(jK)dj−1
2(1 −θ)[S(θ
2K)+S((1−θ
2)K)] = W′(¯
K−(1+h(θ))·K)·h′(θ)·K.
(2)
Equa ion (1) indica es ha he ma ginal benefi s o capi al in es men in
he wo indus ies a e equal. Equa ion (2) indica es ha he ma ginal benefi
om inc easing he scale o echnology applica ion θequals he ma ginal cos .
No e ha he ma ginal benefi om inc easing θconsis s o wo pa s: one
is he loss o consume u ili y a he wo endpoin s o θ
2and 1−θ
2, and he
o he is he inc ease in consume u ili y o e he en i e in e al (θ
2,1−θ
2),
since he p obabili y densi y a each posi ion in he in e al has inc eased.
Since S′′ <0, we ha e ha
∫1−θ
2
θ
2
S(jK)dj
1−θ>S(θ
2K) + S((1 −θ
2)K)
2∀θ∈[0,1),
6
ha is, he a e age alue o S(jK)o e he en i e in e al [θ
2K, 1−θ
2K]is
g ea e han he a e age alue o S(jK)a he wo end poin s. The e o e,
he ma ginal benefi om inc easing θis g ea e han ze o o any θ∈[0,1).
Since h′(0) = 0, he op imal scale o echnology adop ion is indeed g ea e
han ze o.
4 Compe i i e equilib ium
In his sec ion, we conside he long- e m equilib ium o he compe i i e
ma ke . Like he s anda d compe i i e ma ke model, he ou pu o each fi m
in each indus y is e y small ela i e o he o al ou pu o he indus y, so i
is a p ice ake . The ma ke equilib ium p ices o p oduc s o bo h indus ies
a e de e mined by he supply o fi ms and he demand o consume s. The
ma ke has ee en y, so in he long- e m equilib ium, he p ofi s o all fi ms
in bo h indus ies a e ze o.
How do fi ms in he s ochas ic indus y de e mine he scale o echnology
adop ion in he long- e m equilib ium? We assume ha each fi m will de-
e mine i s own scale o echnology adop ion acco ding o he ma ke p ices,
while he ma ke p ices a e aken as gi en. Tha is, any single fi m igno es
he impac o i s own choice o echnology adop ion scale on ma ke p ices.
Gi en ha he ou pu o each fi m is e y small ela i e o he o al ma -
ke ou pu , his assump ion is easonable. And we only conside symme ic
equilib ium, ha is, all fi ms choose he same echnology adop ion scale θa
equilib ium. Specifically, θde e mines he ange o ou pu a e j. The ma -
ke o ms diffe en p ices pjbased on diffe en ealiza ions o he ou pu a e
j. Each fi m chooses i s echnology adop ion scale o maximize i s expec ed
p ofi gi en he p ices.
No e ha since he fi ms’ capi al in es men in bo h indus ies ha e he
cha ac e is ics o cons an e u ns o scale, we only need o conside he
o al capi al in es men le el o he whole indus ies a equilib ium. Fo
7
Case 3. In Figu e 1(c), a= 104,b= 0.001,I= 50,k∈[0,15000]. Hence
when j > 0.0001,V′′
j<0. In his case, he scale o echnology adop ion
in he compe i i e equilib ium equals ze o, smalle han he socially op i-
mal adop ion scale. The op imal in es men inc eases wi h he inc ease o
echnology adop ion cos s, while he op imal scale o echnology adop ion
dec eases wi h he inc ease o echnology applica ion cos s, indica ing sub-
s i u abili y be ween capaci y in es men and echnology adop ion.
Since V′′
j(j, K) = −KS′′(jK)−jK2S′′′(jK)and S′′ <0, we ha e ha i
S′′′(jK)≤0o S′′′(jK)>0small enough, hen V′′
j>0. We hence ob ain
he ollowing co olla y.
Co olla y 1. I W′′(x) = 0, and S′′′(jK)≤0o S′′′(jK)>0small enough,
hen he scale o echnology adop ion in he compe i i e equilib ium is g ea e
han he socially op imal scale o echnology adop ion.
The e o e, when he in e se demand unc ion, ha is, S′(q), is conca e
o weakly con ex, he go e nmen should weaken he ma ginal e u ns o
echnology adop ion in he compe i i e ma ke o co ec he endency o
o e -in es men in he compe i i e ma ke . When S′(q)shows s ong con-
exi y, he go e nmen needs o inc ease he ma ginal e u ns o echnology
adop ion in he compe i i e ma ke o co ec he endency o unde in es -
men in he compe i i e ma ke .
Appendix
Le (K∗, θ∗)be he socially op imal solu ion, and (Ke, θe)be he compe i i e
equilib ium solu ion.
14
A.1 P oo o P oposi ion 1
P oo . Suppose (Ke, θe)=(K∗, θ∗). Then he le hand o Equa ion (6) is
equal o he le hand o Equa ion (5). Tha is,
1
1−θ∗[∫1−θ∗
2
θ∗
2
V(j, K∗)dj
1−θ∗−V(1 −θ∗
2, K∗) + V(θ∗
2, K∗)
2] = 0.
Hence Ej[V(j, K∗)] = V(1−θ∗
2,K∗)+V(θ∗
2,K∗)
2.
Suppose he condi ion holds. Then (K∗, θ∗)sa is y Equa ion (3), Equa-
ion (4), and Equa ion (5). Tha is, (K∗, θ∗)is he same as he compe i i e
equilib ium solu ion.
A.2 P oo o P oposi ion 2
P oo . Wi hou loss o gene ali y, assume ha W′(x) = 1 o any x. We fi s
p o e (i). We conside he ollowing wo cases.
Case 1: θe= 0.
Since θeis he op imal choice o fi ms, Ei[S′(iKe)·i]> Ej[S′(jKe)·j]−
h(θ∗), whe e j=i(1 −θ∗) + θ∗
2. Suppose o he sake o con adic ion ha
K∗≥Ke. Since T′
K(j, K) = S′′(jK)·j2<0, we ha e ha Ej[T(j, Ke)] >
Ej[T(j, K∗)], ha is, Ej[S′(jKe)·j]−h(θ∗)> Ej[S′(jK∗)·j]−h(θ∗). Since
Ej[S′(jK∗)·j]−h(θ∗) = 1 ( his is Equa ion (1)), we ha e ha Ej[S′(jKe)·j]−
h(θ∗)>1. Howe e , Ei[S′(iKe)·i] = 1 ( his is Equa ion (4)). Con adic ion!
Hence K∗< Ke.
Case 2: θe>0.
Le
F(K, θ) := Ej[S′(jK)·j]−(1 + h(θ)),
15
and
G(K, θ) := 1
(1 −θ)2∫1−θ
2
θ
2
S(jK)
Kdj−1
2(1 −θ)[S(θ
2K)
K+S((1 −θ
2)K)
K]−h′(θ),
H(K, θ) := 1
(1 −θ)2∫1−θ
2
θ
2
S′(jK)·jdj−1
2(1 −θ)[S′(θ
2K)·θ
2+S′((1−θ
2)K)·(1−θ
2)]−h′(θ).
No e ha h′′(θ)is assumed o be la ge enough such ha G′
θ<0and
H′
θ<0; in addi ion,
F′
K(K, θ) = Ej[S′′(jK)·j2]<0,
and
F′
θ(K, θ) = H(K, θ).
Acco ding o he fi s o de condi ions o he social op imal p oblem, we
ha e ha F(K∗, θ∗)=0( his is Equa ion (1)), and G(K∗, θ∗)=0( his is
Equa ion (2)).
Acco ding o he condi ions o compe i i e equilib ium, we ha e ha
F(Ke, θe) = 0 ( his is Equa ion (4)) and H(Ke, θe) = 0 ( his is Equa ion
(5)).
Since F′
θ(Ke, θe) = H(Ke, θe) = 0 and F′′
θ(Ke, θe) = H′
θ(K, θ)<0, we
ha e ha 0 = F(Ke, θe) = maxθ∈[0,1] F(Ke, θ). Hence F(Ke, θ∗)≤0. Since
F′
K<0and F(K∗, θ∗) = 0, we ha e ha K∗≤Ke.
We now p o e (ii).
Since V′′
j=−KS′′(jK)−jK2S′′′(jK)>0, we ha e ha jK2S′′′(jK)<
−KS′′(jK)<−2KS′′(jK). Hence T′′
j(j, K)=2KS′′(jK) + jK2S′′′(jK)<
0. Acco ding o he analysis in Sec ion 4, θe>0.
16
Since
G′
K(K, θ) = 1
(1 −θ)2∫1−θ
2
θ
2
∂S(jK)
K
∂K dj−1
2(1 −θ)[∂S(θ
2K)
K
∂K ·+∂S((1−θ
2)K)
K
∂K ],
we ha e ha i ∂S(jK)
K
∂K is conca e in j, hen G′
K(K, θ)≥0. Tha is, i
∂3S(jK)
K
∂K∂2j≤0,G′
K(K, θ)≥0.
No e ha ∂3S(jK)
K
∂K∂2j=S′′′(jK)jK +S′′(jK) = −V′′
j
K. Since V′′
j>0, we ha e
ha ∂3S(jK)
K
∂K∂2j<0and G′
K(K, θ)>0.
No e ha
G(K, θ)−H(K, θ) = 1
1−θ[∫1−θ
2
θ
2
V(j, K)dj
1−θ−V(1 −θ
2, K) + V(θ
2, K)
2].
Since V′′
j>0,G(K, θ)< H(K, θ).
Suppose o he sake o con adic ion ha θ∗≥θe. We hen ha e
0 = G(K∗, θ∗)≤G(Ke, θ∗)< H(Ke, θ∗)≤H(Ke, θe) = 0 .
No e ha he fi s inequali y holds because G′
K>0and K∗≤Ke, and he
second inequali y holds because G(K, θ)< H(K, θ), and he hi d inequali y
holds because H′
θ(K, θ)<0and acco ding o (i), θ∗≥θe. Con adic ion!
Hence θ∗< θe.
Now we p o e (iii). I θe>0, hen we can p o e (iii) using simila
a gumen s as in he p oo o (ii). I θe= 0, hen θ∗>0 = θe.
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