The effect of risk aversion on a supply chain with postponed pricing

Abstract

We consider a supply chain consisting of a supplier and a risk-averse retailer operating under endogenous demand in retail pricing. The demand potential is uncertain and is revealed at the beginning of the selling season when it is too late to order products. The product price, on the other hand, is not determined in advance and can be postponed until the demand is revealed. The goal is to study the effect of risk-aversion and postponed pricing on both the retailer’s decisions and the overall supply chain. We find that the risk-averse retailer does not necessarily order less than the risk-neutral one and may introduce a bias by choosing a specific demand distribution. We contrast two specific choices. One is symmetric (balanced) with respect to the mean demand potential. The other is skewed (pessimistic) with most observations expected below the mean demand potential. Our numerical results show that the binding downside risk constraint deteriorates the supply chain performance when the forecast is balanced and improves it when the forecast is pessimistic.

Appendix

Proof of Proposition 1

First, differentiating the optimal price (4) with respect to q we have

Thus the optimal price monotonically decreases in q, when 0⩽q<A/2, and constant for q⩾A/2. Since p*| _q=0=A/b, p*|_0⩽q<A/2=(A−q)/b⩾p*| _q⩾A/2=A/2b, (see Figure 1(a)).

Next, consider the retailer’s profit function. According to (1) and (4), we have:

Consequently,

Thus the retailer’s profit monotonically decreases in q, when q⩾A/2 and π(p*, A ²/4wb)=0. On the other hand, when 0⩽q<A/2, we have . Since , the profit function in this case is concave with the maximum π(p*, (A−bw)/2)=(A−bw)²/4b. In addition, π(p*, 0)=0 and π(p*, A/2)=A(A−2bw)/4b, (A−bw)²>A(A−2bw). □

Proof of Lemma 1

Consider case (11(ii)), that is,

and according to (12) function is monotonically decreasing. Then substituting 2q=a _min, we find that equation always has a unique solution , if or with respect to (11) if With simple manipulations, we obtain, which implies that case (ii) is feasible if μ _a >a _min+wb. We thus derive that if μ _a >a _min+wb, then the optimal order belongs to the interval .

Let μ _a ⩽a _min+wb. This implies that the profit decreases for , cases (10(ii)) and (10(i)) and (11) which requires that

can provide the best solution. According to (12), function is monotonically decreasing and any solution we find is better, in terms of profit, than that of (10(ii)). Next we have

ie, μ⩽a _min+wb. Finally, we find that a unique solution of always exists if , or equivalently, . Thus, the case (i) is feasible if wb⩽μ _a ⩽a _min+wb. □

Proof of Lemma 2

Let 0⩽q⩽bw. Then according to (5) a⩾2q, and (13) can be rewritten as P{π(d(q))⩽α}=P{2bw⩽a⩽Θ₂}⩽β, when 2bw<Θ₂, and P{π(d(q))⩽α}=0⩽β, when 2bw⩾Θ₂. It is straightforward to show that 2bw<Θ₂ is equivalent to {0<q<Q ₁ or q>Q ₂}, when α⩽bw ²/4. Otherwise, 2bw<Θ₂ holds for any q, when α>bw ²/4. Therefore, if 0⩽q⩽bw, then

when α⩽(1/4)bw ², P{π(d(q))⩽α}=P(2bw⩽a⩽Θ₂), when α>bw ²/4.

Let q>bw, then with respect to (5), (13) can be rewritten as, P{π(d(q))⩽α}=P{a⩽Θ₁(q)∩2bw<a⩽2q}+P{a⩽Θ₂(q)∩a⩾2q}⩽β. First, consider P{a⩽Θ₁(q)∩2bw<a⩽2q}. It is easy to show that 2bw<Θ₁ for any q>bw. In addition, 2q<Θ₁, when q>Q ₃, and 2q>Θ₁, when bw<q<Q ₃. Consequently,

Next, consider P{a⩽Θ₂(q)∩a⩾2q}. Then 2q<Θ₂, when bw<q<Q ₃ and 2q>Θ₂, when q>Q ₃. Accordingly,

and

The proof completes by taking into account that F(Θ₁(Q ₁))=F(Θ₂(Q ₁))=0 and Θ₂(Q ₃)=Θ₁(Q ₃). □

Proof of Proposition 2

To prove this proposition, we examine the effect of the order quantity on the downside risk constraint (Lemma 2). Specifically, by considering the arguments of the density functions, we have for any q. In addition, , when , and when . Note, , when α⩽bw ²/4 and , when α>bw ²/4. The proof is concluded by accounting for (14), (15). □

Proof of Proposition 3

Based on (14) we conclude, that if and only if α⩽bw ²/4 and Q ₁⩽q⩽Q ₂, then w, b and α do not affect the downside risk. If these conditions do not hold, the effect of w on the downside risk is readily obtained from and ∂Θ₂/∂w=b>0 for any q. Similarly, we examine the effect of the other parameters on the downside risk. □

Proof of Proposition 4

If the downside risk is not binding, the optimal order quantity is defined by (11). The effect of the wholesale price w and of the price sensitivity b on the optimal order quantity, , is readily observed by implicit differentiation of (11).

Specifically, from

it follows that for any , and . Similarly, we examine the effect of the price sensitivity b on the optimal order quantity. □

Proof of Lemma 3

It immediately follows from Lemma 2 and Proposition 2  □

Proof of Lemma 4

The proof is very similar to that of Lemma 3  □

Proof of Proposition 5

Considering (17) and (18), and taking into account (5) which implies that F ⁻¹(β)⩾2bw for any 0<β<1, we have

In addition, by differentiating (16) with respect to w, we obtain that

According to (21) q*=q ₁, if and only if and . Note, , this implies that ∂q ₁/∂w<0.

Similarly, we examine the effect of the other parameters on the optimal order quantity. □

Proof of Corollary 1

Let q*₀ and be found for the wholesale price w ₀, the downside risk constraint is binding, but does not affect the retailer decision, that is, . Consider q*₁ and found for the wholesale price w ₁, w ₀<w ₁. Then, according to Proposition 3, the retailer’s decision is affected by the risk aversion under wholesale price w ₁, that is, (13) doesn’t hold for . Furthermore, according to Proposition 4, . In addition, if condition (i) of Proposition 5 holds for , then q*₀<q*₁, which implies . Finally, when condition (ii) of Proposition 5 holds for , by (A.1), (A.3) and (A.4), we have , that is, . □