Pricing decisions in a dual supply chain of organic and conventional agricultural products

Abstract

We analyze a dual-channel supply chain comprising two suppliers that offer vertically-differentiated agricultural products; specifically, one offers an organic version of an agricultural product and the other offers a conventionally-grown version of the same product. Each supplier distributes his product through two channels: directly to consumers and via a single retailer who sells both product versions. Consumers are assumed to be heterogeneous in their valuations of the benefits associated with organic products and of the benefits of purchasing from a retailer (reflected, e.g., in extra services). We also assume that the agricultural products can depreciate in value (e.g., due to deterioration and spoilage). We study market competition in the case where all supply chain members (the two suppliers and the retailer) determine their prices simultaneously in order to maximize their respective profits. First, we analytically solve the cases where organic and conventional value depreciations are equal in each channel or where the direct and the retailer value depreciations are equal for each product. Under these assumptions, we show that each supplier sets a wholesale price that is equal to the direct price. Moreover, the price margins set by the retailer for both the organic and conventional versions are identical. The more general case is analyzed numerically; this analysis reveals the relationships between the model parameters, the equilibrium pricing and the profits of the supply chain members. We identify the conditions under which the wholesale prices of each product version are higher than the direct prices.

Appendix

Proof of Proposition 3

Let $r_{R} \equiv r_{OR} = r_{CR}$, $r_{D} \equiv r_{OD} = r_{CD}$ and $\widetilde{r} = r_{R} - r_{D}$.

Solving the set of 6 equations obtained from the first-order conditions, i.e.$\frac{{\partial \prod_{R} }}{{\partial M_{O} }} = \frac{{\partial \prod_{R} }}{{\partial M_{C} }} = \frac{{\partial \prod_{OS} }}{{\partial W_{O} }} = \frac{{\partial \prod_{OS} }}{{\partial f_{O} }} = \frac{{\partial \prod_{CS} }}{{\partial W_{C} }} = \frac{{\partial \prod_{CS} }}{{\partial f_{C} }} = 0$, for the six decisions variables yields $M_{O}^{*} = M_{C}^{*} = \frac{{s - \widetilde{r}}}{2}$, $W_{O}^{*} = \frac{{2k + C_{C} + 2C_{O} }}{3},\,\,\,\,W_{C}^{*} = \frac{{k + 2C_{C} + C_{O} }}{3}$ and $f_{O}^{*} = f_{C}^{*} = 1.$

When $- s < \widetilde{r} < s$ and $- k < C_{O} - C_{C} < 2k$ hold, the second-order conditions hold at equilibrium. That is:

$$ \begin{aligned} & \frac{{\partial^{2} \prod_{R} }}{{\partial M_{O}^{2} }} = \frac{{3(\tilde{r} - s) + 4(C_{O} - C_{C} - 2k)}}{6sk} < 0 \\ & \frac{{\partial^{2} \prod_{R} }}{{\partial M_{C}^{2} }} = \frac{{ - 3r_{D} + 3r_{R} - 3s - 4C_{O} - 4k + 4C_{C} }}{6sk} = \frac{{3(\tilde{r} - s) + 4(C_{C} - C_{O} - k)}}{6sk} < 0 \\ & \det (H)\prod_{R} = - \frac{{4(C_{C} - C_{O} )^{2} - k^{2} ) + 4k(C_{C} - C_{O} - k)\, + 9k(\,\tilde{r} - s)}}{{9s^{2} k^{2} }} > 0 \\ & \frac{{\partial^{2} \prod_{OS} }}{{\partial W_{O}^{2} }} = - \frac{2}{k} < 0 \\ & \frac{{\partial^{2} \prod_{OS} }}{{\partial f_{O}^{2} }} = - \frac{{(C_{C} + 2C_{O} + 2k)^{2} (C_{C} - C_{O} + 2k + 3(s - \tilde{r}))}}{27sk} < 0 \\ & \det (H)\prod_{OS} = \frac{{(2s(C_{C} - C_{O} + 2k) - 3(\tilde{r}^{2} - s^{2} ))(2k + C_{C} + 2C_{O} )^{2} }}{{27k^{2} s^{2} }} > 0 \\ & \frac{{\partial^{2} \prod_{CS} }}{{\partial W_{C}^{2} }} = - \frac{2}{k} < 0 \\ & \frac{{\partial^{2} \prod_{CS} }}{{\partial f_{C}^{2} }} = \frac{{(C_{O} + k + 2C_{C} )^{2} (C_{C} - C_{O} - k - 3(s - \tilde{r})}}{27sk} < 0 \\ & \det (H)\prod_{CS} = \frac{{\left( {2s(C_{O} - C_{C} + k) + 3(s^{2} - \tilde{r}^{2} )} \right)(C_{O} + k + 2C_{C} )^{2} }}{{27k^{2} s^{2} }} > 0 \\ \end{aligned} $$

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Proof of Proposition 4

Let $r_{O} \equiv r_{OD} = r_{OR}$, $r_{C} \equiv r_{CR} = r_{CD}$ and $\overline{r} = r_{O} - r_{C}$.

Solving the set of 6 equations obtained from the first-order conditions, i.e.$\frac{{\partial \Pi_{R} }}{{\partial M_{O} }} = \frac{{\partial \Pi_{R} }}{{\partial M_{C} }} = \frac{{\partial \prod_{OS} }}{{\partial W_{O} }} = \frac{{\partial \prod_{OS} }}{{\partial f_{O} }} = \frac{{\partial \prod_{CS} }}{{\partial W_{C} }} = \frac{{\partial \prod_{CS} }}{{\partial f_{C} }} = 0$, for the six decisions variables yields $M_{O}^{*} = M_{C}^{*} = \frac{s}{2}$, $W_{O}^{*} = \frac{{2k + C_{C} + 2C_{O} + \overline{r} }}{3},\,\,\,\,W_{C}^{*} = \frac{{k + 2C_{C} + C_{O} + \overline{r} }}{3}$ and $f_{O}^{*} = f_{C}^{*} = 1.$

When $- k < C_{O} - C_{C} + \bar{r} < 2k$ holds, the second-order conditions hold at equilibrium. That is:

$$ \begin{aligned} & \frac{{\partial \prod_{R} }}{{\partial M_{O}^{2} }} = \frac{{ - 3s - 4(C_{C} - C_{O} - \bar{r} + 2k)}}{6ks} < 0 \\ & \frac{{\partial \prod_{R} }}{{\partial M_{C}^{2} }} = \frac{{ - 3s - 4(C_{O} - C_{C} + \bar{r} + k)}}{6sk} < 0 \\ & \det (H)\prod_{R} = \frac{{9s^{2} + 18s^{2} k + 16(C_{C} - C_{O} + r_{C} - r_{O} )^{2} + 12s(C_{C} - C_{O} + \bar{r} + k) + 16k(C_{C} - C_{O} - \bar{r} + 2k)}}{{36k^{2} s^{2} }} > 0 \\ & \frac{{\partial \prod_{OS} }}{{\partial W_{O}^{2} }} = - \frac{2}{k} < 0 \\ & \frac{{\partial \prod_{OS} }}{{\partial f_{O}^{2} }} = - \frac{{(C_{C} + 2C_{O} + r_{C} - r_{O} + 2k)^{2} (C_{C} - C_{O} - \bar{r} + 2k + 3s)}}{27sk} < 0 \\ & \det (H)\prod_{OS} = \frac{{(C_{C} + 2C_{O} + r_{C} - r_{O} + 2k)^{2} (3s + 2(C_{C} - C_{O} - \bar{r} + 2k))}}{{27k^{2} s}} > 0 \\ & \frac{{\partial \prod_{CS} }}{{\partial W_{C}^{2} }} = - \frac{2}{k} < 0 \\ & \frac{{\partial \prod_{CS} }}{{\partial f_{C}^{2} }} = \frac{{(2C_{C} - r_{C} + r_{O} + C_{O} + k)^{2} (C_{C} - C_{O} - \bar{r} - k - 3s)}}{27sk} < 0 \\ & \det (H)\prod_{CS} = \frac{{(2C_{C} - r_{C} + r_{O} + C_{O} + k)^{2} \left( {3s + 2(C_{O} - C_{C} + \bar{r} + k)} \right)}}{{27k^{2} s}} > 0 \\ \end{aligned} $$

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