The Loss-Averse Retailer’s Ordering Policy under Yield and Demand Uncertainty - Intelligent Computing for Sustainable Energy and Environment - page 207

Q =0 =

dE [ U ( π ( Q,X,Y ))]
dQ

Then E [ U ( π ( Q,X,Y ))] is concave in Q . We also have

dE [ U ( π ( Q,X,Y ))]
dQ

λμ ( s − c + h ) > 0 and lim

Q−→ ∞

− λμ ( c − v ) < 0. Thus there exists

=

a unique Q ∗ that satisfies dE [ U ( π ( Q,X,Y ))]
dQ

| Q = Q ∗ = 0, i.e., expression (5).

We have mentioned above that the retailer is risk-neutral when λ =1,then
from (5) the risk-neutral retailer's optimal order quantity Q 0 satisfies

v ) 1
0

( s + h − yg ( y ) F ( yQ 0 ) dy = μ ( s − c + h ) . (8)

Next we will investigate the impact of loss aversion on the retailer's optimal
order quantity. Let

1

M ( h,Q )= c

− v

yg ( y ) F [ ( c

− v ) yQ

] dy

s − c + h s − v

0

(9)

+ 1
0

yg ( y ) F [(1 + s

− c

) yQ ] dy − μ,

h

and Q = Q ∗

N ( Q ∗ ,λ,c,s,v,h )= dE [ U ( π ( Q,X,Y ))]
dQ

. (10)

Note that N ( Q ∗ ,λ,c,s,v,h )=0and ∂N

∂Q ∗ < 0. Then we have the following

theorem:
Theorem 2. For any λ> 1 ,if M ( h,Q 0 ) < 0 ,then Q ∗ >Q 0 and ∂Q ∗

∂λ > 0 ;if

M ( h,Q 0 )=0 ,then Q ∗ = Q 0 and ∂Q ∗
∂λ

=0 ;otherwise, Q ∗ <Q 0 and ∂Q ∗

∂λ < 0 .

Proof. Plugging Q 0 into (6) and combining (8), then
dE [ U ( π ( Q 0 ,X,Y ))]
dQ

c + h ) c

1

v
s−c + h

−

yg ( y ) F [ ( c

− v ) yQ 0

s−v

−

( λ

−

1)( s

−

] dy =

(11) 0

μ

+ 1
0

yg ( y ) F [(1 + s

− c

) yQ 0 ] dy −

h

= − ( λ − 1)( s − c + h ) M ( h,Q 0 ) .

If M ( h,Q 0 ) < 0, then dE [ U ( π ( Q 0 ,X,Y ))]

dQ > 0, which implies that Q ∗ >Q 0 .Fur-
thermore, since N ( Q ∗ ,λ,c,s,v,h ) = 0, using the implicit function theorem we
can obtain ∂Q ∗
∂λ

∂λ ∂N

∂N

= −

∂Q ∗ ,where

v ) 1
0

v ) yQ ∗ ∂N

∂λ =

yg ( y ) F [ ( c

−

− ( c − ] dy

s − v

(12)

c + h ) 1
0

yg ( y ) F [(1 + s

− c

) yQ ∗ ] dy + μ ( s − ( s − − c + h ) .

h