Introduction

Multi-sided platforms offer specialized services to small and medium enterprises (SMEs), potentially enabling them to achieve greater scale and efficiency than would be possible through independent operations. This study examines SME decisions to partner with platforms for two critical business activities: discovery (customer acquisition through advertising and promotion) and fulfillment (order processing and delivery).

Our research addresses three key questions: 1. Under what conditions do SMEs prefer platform partnerships over do-it-yourself (DIY) approaches? 2. How do SME characteristics (scale, operational efficiency, product type) influence platform adoption decisions? 3. When might SMEs benefit from multi-platform strategies rather than single-platform partnerships?

The analysis focuses on the trade-off between platform partnership costs (both fixed setup costs and operational fees) and the operational efficiencies platforms can provide through superior scale, technology, or network effects. # SME Activities

We treat the SME as a price-taker, or at least that a (small) change in scale of operation will not impact selling price. So, the SME’s unit price \(p\) does not change with scale (this is reasonable at least in the short term), and its revenue from \(N\) units sold is \(p N\). On the cost side, the two activities’ costs change with scale of operations as follows.

To understand this better, imagine that \(\gamma=0\) so that returns are linear rather than diminishing, then \(\alpha\) is a unit cost of acquiring 1 buyer. The buyer acquisition cost function is illustrated below, with \(\alpha=1\) and several values of \(\gamma\).

Notation: SME DIY Approach vs Partnering with Platform

SME Operational Characteristics \((\alpha_j, \gamma_j, \phi_j)\)

Our model covers SMEs with different levels of economic capabilities. Let \(j\) index SMEs, so that SME \(j\)’s operation is defined by its “characteristic operational parameters” over the discovery and fulfillment activities. This is represented as a triplet \((\alpha_j, \gamma_j, \phi_j)\), where \(\phi_j\) is a unit fulfillment cost, and \((\alpha_j, \gamma_j)\) together reflect the cost of acquiring buyers through discovery and promotion activities. Generally, smaller SMEs will have inferior parameters.

Generally, smaller SMEs will have a higher \(\gamma\) relative to a larger one, representing that the smaller scale will mean that more sales will imply more rapidly increasing costs. Similarly, a smaller SME will generally have greater fulfillment cost, although it is possible that it has lower \(\phi\) when it fulfills only in a closer local area.

At any rate the idea is that we can think of different SMEs as points in a 3-dimensional \((\alpha_j, \gamma_j, \phi_j)\) space.

Changing Operational Characteristics when Partnering with Platform

An SME might partner with a platform for either discovery or fulfillment or both. Since our goal is to study multi-homing, let’s suppose that there are two platforms in the market. Define platform 1’s characteristic parameters as \((\alpha_1, \gamma_1, \phi_1)\) and those of platform 2 as \((\alpha_2, \gamma_2, \phi_2)\). This reflects that each platform conducts both activities. However, if a platform conducts only fulfillment, then the \(\alpha\) and \(\gamma\) parameters would vanish; likewise, if a platform only conducted discovery, then the \(\phi\) parameter would not be relevant.

First, consider an SME which uses platform 1 for fulfillment only. Since fulfillment is mainly a logistics activity, we will assume that an SME \(j\)’s partnership with the platform would transfer the platform’s fulfillment parameter \(\phi_1\) to SME \(j\) (while keeping \(\alpha_j, \gamma_j\) unchanged), so that \(j\)’s characteristic parameters are now \((\alpha_j, \gamma_j, \phi_1)\). Notice that in this scenario, all SMEs become homogeneous on fulfillment cost because now it is essentially “outsourced” to the platform.

Second, if SME \(j\) uses platform 1 for discovery only, then our empirical observations suggest two possibilities, discussed below.

  1. This partnership helps the SME reduce its scale-independent cost parameter \(\alpha_j\) (e.g., by having a more efficient way to discover customers than perhaps the less sophisticated and more expensive methods used by the SME). Then, SME \(j\)’s new characteristic parameters upon partnering with platform 1 (who has its \(\alpha_1\)) will be \((\alpha_1, \gamma_j, \phi_j)\). Like in the case of fulfillment, all SMEs become homogeneous on this scale-free discovery cost because now discovery activities are now “outsourced” to the platform. However, they remain heterogeneous on \(\gamma_j\) the scaling parameter.

  2. This partnership helps SME \(j\) lower its \(\gamma_j\) parameter: the platform’s wider reach and/or better algorithmic targeting capability would improve the SME’s ability to “scale up” and its costs of making a sale would increase less rapidly as it made more sales. Specifically, SME \(j\)’s new characteristic parameters upon partnering with platform 1 (who has its \(\gamma_1\)) will be \((\alpha_j, \frac{\gamma_1 + \gamma_j}{2}, \phi_j)\). Notice that in this case, SMEs remain heterogeneous on the scaling rate for discovery cost because those costs still depend partially on the SME’s own scaling capabilities.

Third, if SME \(j\) uses platform 1 for both discovery and fulfillment then its new characteristic parameters will be \((\alpha_1, \gamma_j, \phi_1)\) or \((\alpha_j, \frac{\gamma_1 + \gamma_j}{2}, \phi_1)\) as discussed above.

Fourth, and finally, consider that SME \(j\) multi-homes over both platforms 1 and 2. If the multi-homing is for fulfillment, then each unit of fulfillment done in partnership with platform 1 will have a parameter \(\phi_1\) while each unit of fulfillment done in partnership with platform 2 will have a parameter \(\phi_2\). Similarly, if multi-homing is for discovery, then each unit of discovery done in partnership with platform 1 will have a parameter \(\frac{\gamma_1 + \gamma_j}{2}\) while each unit of fulfillment done in partnership with platform 2 will have a parameter \(\frac{\gamma_2 + \gamma_j}{2}\). We will see that even if, say, \(\gamma_1 < \gamma_2\) (i.e., platform 1 is a more attractive partner) the SME might have a motive to use both platforms because of the diminishing returns aspect of discovery costs.

DIY Benchmark for SME

An SME \(j\) adopting a “do it yourself” DIY approach with regard to the above two activities will have the characteristic parameter triplet \((\alpha_j, \gamma_j, \phi_j)\). Under DIY operations for the two activities, we get the profit and optimal scale below.

\[\begin{align} \Pi_j &= \left((p-\phi_j) N - \alpha_j N^{(1+\gamma_j)} \right) \qquad (\#eq:Pi0) \\ N_j^{\ast} &= \arg \max \Pi_j = \left(\frac{p - \phi_j}{\alpha {(1+\gamma_j)}} \right) ^ \frac{1}{\gamma_j} \qquad (\#eq:Sales0) \\ \Pi_j^{\ast} &= \alpha_j \gamma_j \left(\frac{p - \phi_j}{\alpha_j {(1+\gamma_j)}} \right) ^ \frac{1+\gamma_j}{\gamma_j} \qquad (\#eq:Pistar0) \end{align}\]

It is obvious from the last equation that any thing that lowers \(\alpha_j\) and/or \(\phi_j\) will enable the SME to achieve larger scale at better efficiency, hence improve profit; whether the gain exceeds any fixed costs will determine whether or not to employ such an opportunity.

SME partnering with a Single Platform

Now suppose that the SME can partner with a platform (platform 1) that offers discovery and fulfillment services, with the characteristic triplet which we will denote by \((\alpha_1, \gamma_1, \phi_1)\).

Partnering with a platform might also involve a fixed cost \(C\) borne by the SME (this need not be a “price” charged by the platform, it could also be internal transformation or other costs). We can think of \(C\) as \(C_d + C_f\), reflecting costs relative to the two activities.

To compute the SME’s optimal scale when partnering with a platform whose operational parameters are \((\alpha_1, \gamma_1, \phi_1)\), we therefore apply Eq. 2a-2c to \((\alpha_1, \gamma_1, \phi_1)\), with the idea that (a) for fulfillment only platform, replace \((\alpha_1, \gamma_1)\) with \((\alpha_j, \gamma_j)\) in the optimal scale equation, and (b) for a discovery-only platform replace \(\phi_1\) with \(\phi_j\) and \(\gamma_1\) with \(\frac{\gamma_j+\gamma_1}{2}\).

The SME would partner with such a platform if (after making suitable adjustments as above), \((\Pi_1^\ast - \Pi_j^\ast) > C\) (which would also entail \(N_1^\ast > N_j^\ast\)).

\[\begin{align} \left(\alpha_1 \gamma_1 \left(\frac{p - \phi_1}{\alpha_1 {(1+\gamma_1)}} \right) ^ \frac{1+\gamma_1}{\gamma_1} - \alpha_j \gamma_j \left(\frac{p - \phi_j}{\alpha_j {(1+\gamma_j)}} \right) ^ \frac{1+\gamma_j}{\gamma_j} \right) > C \end{align}\]

Let’s take each activity at a time.

Fulfillment-only Platform

Suppose a platform offers only fulfillment support. So, the \((\alpha_1, \gamma_1)\) in the left-hand term would be replaced by \((\alpha_j, \gamma_j)\), and the participation condition now is \(\left(\alpha_j \gamma_j \left(\frac{p - \phi_1}{\alpha_j {(1+\gamma_j)}} \right) ^ \frac{1+\gamma_j}{\gamma_j} - \alpha_j \gamma_j \left(\frac{p - \phi_j}{\alpha_j {(1+\gamma_j)}} \right) ^ \frac{1+\gamma_j}{\gamma_j} \right) > C\). From this expression it is clear that

  • if two SMEs had the same discovery capabilities (same \((\alpha_j, \gamma_j)\)) then the one with worse (higher) \(\phi_j\) would benefit more from the platform partnership, and would have a higher fixed-cost threshold to participate.

  • But if the worse-fulfillment SME also were worse on discovery, then it would benefit less (and might find the \(C\) to be too high to participate).

Looking across all SMEs with a variety of \((\alpha, \gamma, \phi)\) values, the following plots depict which SMEs would engage with platform vs go with a DIY approach. Plots are drawn for \(\phi_1 = 1\), with the upper row being \(\gamma_j = 0.22\) and the lower row being \(\gamma_j = 0.25\). Each contour represents the profit gain that SME \(j\) gets (for all \((\alpha_j, phi_j)\) points for that contour) from join the platform, after considering the fixed cost \(C\) and better fulfillment cost \(\phi_1\).

The shaded region represent SMEs who would prefer to go it alone vs partnering with the platform. Naturally, SMEs with higher \(\phi_j\) prefer the platform (this is obvious moving from bottom to top within a plot). And an SME with higher \(\alpha\) is LESS attracted to the platform (move from left to right within a plot), because it is unable to efficiently take advantage of the better fulfillment cost that the platform offers – it will not generate enough fulfillment orders. Preference for platform becomes LESS likely with higher \(\gamma\) (comparing bottom vs top panels).

Discovery-only Platform

So, now suppose that an SME partners with a platform for discovery service. This means that the SME’s \(\phi_j\) value remains the same (because it does fulfillment by itself) but it takes on the platform’s \(\alpha_1\) value vs its own \(\alpha_j\). For \(\gamma_j\), the first case we analyze is that it remains the same (i.e., \(\gamma_1\) is simply \(\gamma_j\)) because that depends on how well the SME can convert the traffic that arrives to its store.

Simple Case: No Change in \(\gamma_j\)

Then, the condition for partnering with the platform for discovery is

\[\begin{align} \left(\alpha_1 \gamma_j \left(\frac{p - \phi_j}{\alpha_1 {(1+\gamma_j)}} \right) ^ \frac{1+\gamma_j}{\gamma_j} - \alpha_j \gamma_j \left(\frac{p - \phi_j}{\alpha_j {(1+\gamma_j)}} \right) ^ \frac{1+\gamma_j}{\gamma_j} \right) > C \end{align}\]

^ Need to add an explanation for this correct plot, removed the older incorrect one.

Case: Improvement in \(\gamma_j\)

Now, consider the case that an SME can improve its own operations in terms of return to scale from discovery efforts, when it partners with the platform. So, an SME \(j\) has parameter \(\gamma_j\) when it does discovery on its own, but improves this parameter to \(\frac{\gamma_j + \gamma_1}{2}\) when it partners with the platform.

\[\begin{align} \left( \alpha_1 \gamma' \left( \frac{p - \phi_j}{\alpha_1(1+\gamma')} \right)^{\frac{1+\gamma'}{\gamma'}} - \alpha_j \gamma_j \left( \frac{p - \phi_j}{\alpha_j(1+\gamma_j)} \right)^{\frac{1+\gamma_j}{\gamma_j}} \right) > C \end{align}\] \[ \gamma' = \frac{\gamma_j + \gamma_1}{2} \]

Number 1 : HeatMap

This is a heatmap of SME profitability gains from partnering with the platform across varying levels of discovery difficulty (αⱼ) and fulfillment cost (φⱼ). Each subplot corresponds to a unique combination of SME efficiency (γⱼ) and fixed cost of partnering (C), with the fill color representing the value of Dgain. Dgain is calculated as:

\[ \text{Dgain} = P^*(\phi_j, \alpha_1, \gamma′) - P^*(\phi_j, \alpha_j, \gamma_j) - C \quad \text{where} \quad \gamma′ = \frac{\gamma_j + \gamma_1}{2} \]

Green regions indicate that partnering leads to a net profit increase (Dgain > 0), while red regions indicate that DIY remains more favorable. Yellow points highlight SMEs that are approximately at the break-even point (|Dgain| < 0.1). We observe that SMEs with high discovery difficulty (αⱼ) and moderate to low fulfillment cost (φⱼ) tend to gain the most from partnering, especially when γⱼ is low or the fixed cost C is small. As γⱼ increases or C rises, the platform needs to offer stronger operational support (lower γ₁) to maintain the same level of attractiveness.

Number 2 : Line Plot of Dgain vs Alpha (for fixed φ)

This line plot shows how the net profit gain (Dgain) from partnering with a platform changes as an SME’s discovery difficulty (αⱼ) increases, while keeping fulfillment cost (φⱼ) fixed at two levels: 1.2 (red) and 1.6 (blue). Each panel represents a different combination of fixed cost (C = 20 or 100) and the SME’s return-to-scale parameter (γⱼ = 0.22 or 0.25). The overall trend across all panels is that Dgain increases with αⱼ, meaning SMEs who face more difficulty acquiring buyers on their own stand to gain more from platform discovery services.

Additionally, SMEs with lower φⱼ (more efficient fulfillment) consistently show higher profit gains, since they can convert new customers into profit more effectively. While increasing the fixed cost from 20 to 100 lowers the overall gain across both lines, it does not significantly alter the pattern — SMEs with high αⱼ and low φⱼ continue to benefit the most. Similarly, a lower γⱼ (worse scaling efficiency) slightly reduces the gain but the comparative trend remains consistent. This confirms that SMEs with high discovery difficulty and cost-efficient fulfillment are the most likely to benefit from platform partnerships, especially when platform fees are modest.

This is the same plot but using C = 2 and 5 like we did in our previous analysis. The value differ by such a less factor that we are unable to notice a small change between the two in this.

Number 3 : 3D Surface Plot (Interactive using plotly)

The 3D surface plot illustrates how the net gain (Dgain) from partnering with a discovery-only platform varies with an SME’s discovery difficulty (αⱼ) and fulfillment cost (φⱼ). The upward slope along the αⱼ axis indicates that as it becomes harder for an SME to attract customers on its own, the value of the platform’s discovery support increases significantly. Conversely, the downward slope along the φⱼ axis shows that SMEs with higher fulfillment costs benefit less from this increased demand, as the cost of servicing new orders erodes the net gain. The plot clearly reveals that SMEs with high discovery difficulty and low fulfillment cost derive the most value from platform partnership, while those with low αⱼ and high φⱼ experience minimal benefit. This highlights the interplay between visibility and operational efficiency in determining the effectiveness of platform collaboration.

Number 4 : SME who prefer partnering

This is a summary of how the platform’s operational support (γ₁) influences SMEs’ willingness to partner. The X-axis represents the platform’s scaling efficiency (γ₁), with lower values indicating more effective discovery scaling. For each value of γ₁, we simulate Dgain for a full grid of SME types (αⱼ, φⱼ), holding SME efficiency fixed at γⱼ = 0.22. We then calculate the percentage of SMEs for which Dgain > 0 — i.e., where partnering provides a higher net return than operating independently.

As the plot shows, when the platform is highly efficient (γ₁ ≤ 0.18), nearly all SMEs find value in partnering. However, as γ₁ increases — meaning the platform becomes less efficient — the share of SMEs preferring partnership declines steadily. This trend confirms that platform-side efficiency plays a critical role in determining the breadth of SMEs the platform can effectively serve. In strategic terms, the plot highlights how even modest improvements in γ₁ can dramatically expand platform adoption, especially among marginal or undecided SMEs.

Number 5 : Conversion Map

This is a behavioral “conversion map” that highlights SME types (defined by αⱼ and φⱼ) who change their decision from DIY to Partnering when the platform provides a modest improvement in operational efficiency. The logic compares two scenarios: one where the SME’s internal efficiency (γⱼ) remains unchanged (Section 3.2.1), and another where γⱼ improves to γ′ due to platform support (Section 3.2.2), modeled as:

\[ \gamma′ = \frac{\gamma_j + \gamma_1}{2} \]

For each SME, we compute their net gain from partnering in both cases:

\[ \text{Dgain}_{\text{no-γ}} = P^*(\phi_j, \alpha_1, \gamma_j) - P^*(\phi_j, \alpha_j, \gamma_j) - C \] \[ \text{Dgain}_{\text{γ′}} = P^*(\phi_j, \alpha_1, \gamma′) - P^*(\phi_j, \alpha_j, \gamma_j) - C \]

We then identify SMEs who initially preferred DIY (when Dgain ≤ 0), but switch to Partnering once γ improves (Dgain > 0). These “swing SMEs” represent the marginal cases that are most responsive to operational improvement, and are shown in shaded cells. This approach helps isolate the tipping-point SMEs who are most influenced by modest platform enhancements, making them ideal targets for strategic platform outreach or investment.

Number 6 : For which SMEs is partnering more profitable than going DIY?

Each cell represents a specific SME type defined by its discovery difficulty (αⱼ) and fulfillment cost (φⱼ), and is color-coded based on whether partnering with the platform leads to a higher net profit than operating independently. The decision is based on the value of Dgain, defined as:

\[ Dgain = \text{Profit}_{\text{partner}} - \text{Profit}_{\text{DIY}} - C \]

C is the fixed cost of partnering. A blue cell indicates that Dgain is positive — i.e., partnering yields a higher net profit than DIY even after accounting for the fixed cost — and thus partnering is preferred. A gray cell indicates that Dgain is zero or negative — meaning that partnering offers no additional gain or is strictly worse than operating independently. It’s important to note that SMEs in gray regions may still be profitable; however, they achieve equal or greater profit by continuing with the DIY model. In essence, the plot does not show profitability in isolation, but rather the relative profitability of partnering versus staying independent. The color transition across the map thus visually marks the decision boundary where SMEs are indifferent between the two modes, helping identify the conditions under which platform participation becomes economically advantageous.


Similar to previous plots, contour :

Plotting the same values with correct parameter arguments plots the graph above. This is at a very large scale and does not show any negative Dgain.

Different Types of Platforms

A platform can facilitate an SME’s discovery and fulfillment (either, or both) activities, enabling the SME to have lower (better) values for corresponding operational parameter(s). With higher parameter value, there is no point in the partnership, unless the platform offers both activities and ties them.

For fulfillment, the conventional setting has linear fulfillment cost, so the expectation would be a lower \(\phi\) parameter.

For discovery activities, the parameters can reflect different types of platforms:

  • Small platform is represented by high \(\gamma\). High volume of traffic (potential buyers) from platform to SME implies a very high and rapidly increasing cost.

  • Niche (so, small but high-fit) platform would be high \(\gamma\) but low \(\alpha\). In other words it can deliver a small amount of high-quality traffic (i.e., leads that are high likelihood of converting to actual buyers) at low cost to the SME; but the costs go up rapidly for additional traffic (because it is a small platform)ß

  • Large but low-fit mass-market untargeted platform would have high \(\alpha\), low \(\gamma\).

  • Large and high-fit platform would have low \(\alpha\), low \(\gamma\).

OLD STUFF - ignore for now.

Firm Size vs Likelihood of Partnering with Platform

In other words, a necessary condition for partnership with the platform is that the platform enables it to function with superior operational parameters (intuitive). And when there is a fixed cost, the incremental superiority needs to be higher to overcome the fixed cost.

For an SME which was previously operating below its maximum capacity, partnering with a platform (which has superior operational parameters) would enable the SME to increase its scale, thereby reaping two advantages, i) higher margin from lower unit cost of selling and ii) higher volume. An SME which was already operating at maximum capacity in the DIY scenario would only reap a cost advantage (in the short term; in the longer term this cost advantage should drive it to increase its operational resources and increase scale).

In the current setup where firms differ only on \(\bar{N}\) but initially have the same characteristic operational parameters, this model suggests that firms with larger capacity are more likely to partner with the platform. That is, they need a smaller “incremental improvement” in the operational parameters (the operational parameters).

If we want the reverse result – then we need to go with the first approach outlined above – that “smaller” firms have worse operational parameters.

Some Results and Computational Evidence

SME Partnering with Two Platforms

Now suppose that the SME has access to two platforms, 1 and 2, whose characteristic values are \((\alpha_1, \gamma_1, \phi_1)\) and \((\alpha_2, \gamma_2, \phi_2)\) respectively. Once again, for the moment, we will assume that the platform’s two services are tied together, so that the SME must utilize both services if it wants to use a particular platform. The analysis below will also cover the case of specialized platforms (those that offer only-discovery or only-fulfillment, because then the other characteristic parameter can be interpreted as the SME’s own values for it).

A platform \(i\) is more attractive than platform \(j\) along each characteristic if, \((\alpha_i < \alpha_j)\), \((\gamma_i < \gamma_j)\), or \((\phi_i < \phi_j)\), respectively. The analysis below allows for each direction of the inequality, and even that one platform \(i\) is superior to the other along all 3 characteristics.

Zero Overlap

First, assume that the two platforms reach non-overlapping sets of customers. Now, if the SME serves \(N_1\) customers through platform 1 and \(N_2\) through platform 2, its profit function is $_2 = \(\left((p-\phi_1) N_1 - \alpha_1 N^{(1+\gamma_1)} + (p-\phi_2) N_2 - \alpha_2 N^{(1+\gamma_2)} \right)\). Therefore its optimal allocation of effort across the two platforms (under the zero-overlap setting) is

\[\begin{align} (N_1^{\ast}, N_2^{\ast}) &= \arg \max \Pi_2 = \left((p-\phi_1) N_1 - \alpha_1 N_1^{(1+\gamma_1)} + (p-\phi_2) N_2 - \alpha_2 N_2^{(1+\gamma_2)} \right) \qquad (\#eq:Nstar2) \\ &= \left(\frac{p - \phi_1}{\alpha_1 {(1+\gamma_1)}} \right) ^ \frac{1}{\gamma_1}, \qquad \left(\frac{p - \phi_2}{\alpha_2 {(1+\gamma_2)}} \right) ^ \frac{1}{\gamma_2} \qquad (\#eq:Sales2) \\ \Pi_2^{\ast} &= \alpha_1 \gamma_1 \left(\frac{p - \phi_1}{\alpha_1 {(1+\gamma_1)}} \right) ^ \frac{1+\gamma_1}{\gamma_1} + \alpha_2 \gamma_2 \left(\frac{p - \phi_2}{\alpha_2 {(1+\gamma_2)}} \right) ^ \frac{1+\gamma_2}{\gamma_2} \qquad (\#eq:Pistar2) \\ s.t. & \left(\frac{p - \phi_1}{\alpha_1 {(1+\gamma_1)}} \right) ^ \frac{1+\gamma_1}{\gamma_1} > \frac{k}{\alpha_1 \gamma_1}, \qquad \left(\frac{p - \phi_2}{\alpha_2 {(1+\gamma_2)}} \right) ^ \frac{1+\gamma_2}{\gamma_2} > \frac{k}{\alpha_2 \gamma_2} \end{align}\]

From the above, if the two platforms have non-overlapping segments, then the SME’s decision to employ them is essentially two independent decisions. That is, even an ``inferior platform’’ (indeed, which has worse characteristic values under every component) will capture some utilization from the SME despite the presence of a superior platform. However, if the inferior platform’s characteristic values are too unattractive – high – then the SME will find alternate options (e.g., a DIY approach) and only use the superior platform.

Overlapping User Segments

Suppose now that the two platforms have overlapping user bases. For exposition, and without loss of generality, assume that platform 2 is the smaller one, or more specifically that \(\gamma_2 > \gamma_1\), i.e., it starts running out of ``good customers’’ (those likely to convert into buyers) more quickly than platform 1. We will allow that platform 2 might have a lower cost parameter (or not) and charge a lower fulfillment cost (or not).

An elegant way of capturing this overlap into the model is to recognize that when there are overlapping segments, both platforms will convert clickers into buyers at a lower rate than before. That is, i) their diminishing returns parameter (presently \(\gamma\)) will be higher than if they were the SME’s sole partner, and ii) the incremental (negative) effect will be worse for platform 2. We capture this with the following trick: let \(\delta\) be the degree of overlap. Then, a platform’s diminishing returns parameter adjusts by a multiplier \((1+\delta)\). That is, the cost of acquiring \(N\) actual buyers through discovery activities on platform \(i\) now becomes \(\alpha N^{(1+\gamma)(1+\delta)}\).

The effort-allocations to platforms 1 and 2 under a \(\delta\) degree of overlap therefore become:

\[\begin{align} (N_1^{\ast}, N_2^{\ast}) &= \arg \max \Pi_2 = \left((p-\phi_1) N_1 - \alpha_1 N_1^{(1+\gamma_1)(1+\delta)} + (p-\phi_2) N_2 - \alpha_2 N_2^{(1+\gamma_2)(1+\delta)} \right) \qquad (\#eq:Nstar2o) \\ &= \left(\frac{p - \phi_1}{\alpha_1 {(1+\gamma_1)(1+\delta)}} \right) ^ \frac{1}{\gamma_1+\delta(1+\gamma_1)}, \qquad \left(\frac{p - \phi_2}{\alpha_2 {(1+\gamma_2)(1+\delta)}} \right) ^ \frac{1}{\gamma_2+\delta (1+\gamma_2)} \qquad (\#eq:Sales2o) \\ \Pi_2^{\ast} &= compute (\#eq:Pistar2o) \\ s.t. & compute \end{align}\]

Old

Digital platforms provide a variety of services that facilitate participation and business transactions by a variety of businesses, small and big. Among other things, some platforms enable help businesses to expand their market reach through discovery of potential buyers. Some platforms provide support for fulfillment, covering delivery logistics, payment, and trust in transactions. Some platforms provide both categories of services.

This note will build a model to explore a small firm’s use of a marketplace platform to support discovery and/or fulfillment: whether, for each service, it should use the platform or not, and specifically the likelihood that the business will single-home with one platform vs multi-home and leverage both platforms. You will see that the answers will depend on

  • The distinctive customer reach of each platform (\(\frac{M}{M+2S}\) below).

  • The relative efficiency that the platform provides over the business’s do-it-yourself approach (\(a\) relative too \(\alpha\) for discovery, … and \(t\) vs \(\tau\) for fulfillment).

  • The (fixed) cost of joining the platform, hence cost of multi-homing, for the business.

  • The business’s efficiency in converting potential buyers to actual buyers ($p$).


Model of the Firm: Discovery and Fulfillment Flow

Assume, without loss of generality, that each “customer” represents at most single-unit demand. For a business who aims to sell to these customers, its actions follow the conversion funnel and flow depicted below. The node “discovery” represents that the firm reached or interacted with a potential buyer, and some fraction of buyers thus reached convert to actual sales. A naive and simple way of connecting discoveries and sales is a linear model, in which \(N\) reached customers convert into \(p \times N\) sales.