The Loss-Averse Retailer’s Ordering Policy under Yield and Demand Uncertainty - Intelligent Computing for Sustainable Energy and Environment - page 209

∂v ∂N

Proof. (i) Using the implicit function theorem we have ∂Q ∗
∂v

∂N

= −

∂Q ∗ ,where

1) 1
0

yg ( y ) F [ ( c−v ) yQ ∗
s

∂N
∂v =( λ

− ] dy

− v

1

c ) Q ∗ v ) yQ ∗

+ ( λ

− 1)( c − v )( s −

y 2 g ( y ) f [ ( c

−

(14)

] dy

v ) 2 ( s − s − v

0

+ 1
0

yg ( y ) F ( yQ ∗ ) dy > 0 .

∂Q ∗ < 0, then Q ∗ is increasing in v .
(ii)-(iv) It is easy to calculate that

∂N

Since

1

v ) 2 Q ∗ v ) yQ ∗ ∂N

∂s = ( λ

− 1)( c −

y 2 g ( y ) f [ ( c

−

] dy

( s

− v ) 2 s − v

0

1) 1
0

yg ( y ) F [(1 + s

− c

) yQ ∗ ] dy

−

( λ

−

h

(15)

1

c + h ) Q ∗

( λ

−

1)( s

−

y 2 g ( y ) f [(1 + s

− c

) yQ ∗ ] dy

−

h h

0

1

yg ( y ) F ( yQ ∗ ) dy + λμ, −

0

∂c = − ( λ− 1) 1
0

v ) yQ ∗

∂N

yg ( y ) F [ ( c

−

] dy

s − v

1

v ) Q ∗ v ) yQ ∗ ( λ − 1)( c −

y 2 g ( y ) f [ ( c

−

− ] dy

s − v s − v

0

(16)

1) 1
0

yg ( y ) F [(1 + s

− c

) yQ ∗ ] dy +( λ −

h

1

+ ( λ

−

1)( s

− c + h ) Q ∗

y 2 g ( y ) f [(1 + s

− c

) yQ ∗ ] dy

− λμ,

h h

0

and

1) 1
0

∂N
∂h =

yg ( y ) F [(1 + s

− c

) yQ ∗ ] dy

− ( λ −

h

1

c ) Q ∗

+ ( λ

− 1)( s − c + h )( s −

y 2 g ( y ) (17)

h 2

0

1

f [(1 + s

− c

) yQ ∗ ] dy yg ( y ) F ( yQ ∗ ) dy + λμ.

× −

h

0

Then we can prove these three results in a similar way.

It follows from this theorem that the loss-averse retailer's optimal order quantity
may be decreasing in shortage cost and selling price, and increasing in purchasing
cost. These will never occur in the risk-neutral case.