1 Introduction

We wish to analyze the decisions and payoffs of small and medium enterprises (SMEs) to partner with a platform or with multiple platforms, for obtaining support fortwo core business activities: i) discovering buyers by acquiring likely buyers (through advertising and promotions etc) and ii) fulfillment (or e-commerce). The idea is that platforms can often provide superior capabilities to small enterprises, help them save costs and possibly help them operate at larger scale.

2 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 \(\gamma\).

2.1 DIY Benchmark for SME

An SME \(j\)’s operation is defined by its “characteristic operational parameters” with regard to the above two activities, the 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. Under DIY operations for the two activities, we get the profit and optimal scale below.

\[\begin{align} \Pi_j &= \left((p-\phi_j) N - \alpha_j N^{(1+\gamma_j)} \right) \qquad \tag{2.1} \\ N_j^{\ast} &= \arg \max \Pi_j = \left(\frac{p - \phi_j}{\alpha {(1+\gamma_j)}} \right) ^ \frac{1}{\gamma_j} \qquad \tag{2.2} \\ \Pi_j^{\ast} &= \alpha_j \gamma_j \left(\frac{p - \phi_j}{\alpha_j {(1+\gamma_j)}} \right) ^ \frac{1+\gamma_j}{\gamma_j} \qquad \tag{2.3} \end{align}\]

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.

2.2 Modeling SME Heterogeneity

There are two ways for differentiating between SMEs given the general behavior of economies of scale for discovery and fulfillment activities. We can either think of smaller SMEs as having higher cost parameters, or we can think of smaller SMEs as having a lower maximum scale of operation (or capacity) \(\bar{N}\). In the former approach, a smaller SME has higher \(\phi\) and higher \(\gamma\). A small SME which is somehow better at converting traffic into sales would be low \(\alpha\) combined with high \(\gamma\).

Using the former approach, the equations above apply across all SMEs. With the latter approach, the equations are valid for a particular SME \(j\) with capacity ${N}_j if they produce \(N_j^{\ast} \leq \bar{N}_j\). For a “smaller firm” (with maximum possible scale \(\bar{N} < N_j^{\ast}\)), the operating scale is \(\bar{N}\) and profit is simply \(\Pi_j^{\ast} = \left((p-\phi_j) \bar{N} - \alpha_j \bar{N}^{(1+\gamma_j)} \right)\).

3 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.

3.1 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).

3.2 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.

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

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

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

In this case, we can see that SMEs with higher \(\alpha_j\) have greater preference for the platform because of the efficiency it delivers (notice the \(C\) values for these plots); higher \(\phi\) reduces preference for the platform because greater fulfillment cost makes the SME less able to recoup the fixed cost needed to adjust for discovery partnership.

3.2.2 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.

3.3 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)

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

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

4 OLD STUFF - ignore for now.

4.1 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.

4.2 Some Results and Computational Evidence

5 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.

5.0.1 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

\[\begin{align} (N_1^{\ast}, N_2^{\ast}) &= \arg \max \Pi_2 = \left((p-\phi_1) N_1 - \alpha_1 N_1^{(1+\gamma_1)} + (p-\phi_2) N_2 - \alpha_2 N_2^{(1+\gamma_2)} \right) \qquad \tag{5.1} \\ &= \left(\frac{p - \phi_1}{\alpha_1 {(1+\gamma_1)}} \right) ^ \frac{1}{\gamma_1}, \qquad \left(\frac{p - \phi_2}{\alpha_2 {(1+\gamma_2)}} \right) ^ \frac{1}{\gamma_2} \qquad \tag{5.2} \\ \Pi_2^{\ast} &= \alpha_1 \gamma_1 \left(\frac{p - \phi_1}{\alpha_1 {(1+\gamma_1)}} \right) ^ \frac{1+\gamma_1}{\gamma_1} + \alpha_2 \gamma_2 \left(\frac{p - \phi_2}{\alpha_2 {(1+\gamma_2)}} \right) ^ \frac{1+\gamma_2}{\gamma_2} \qquad \tag{5.3} \\ s.t. & \left(\frac{p - \phi_1}{\alpha_1 {(1+\gamma_1)}} \right) ^ \frac{1+\gamma_1}{\gamma_1} > \frac{k}{\alpha_1 \gamma_1}, \qquad \left(\frac{p - \phi_2}{\alpha_2 {(1+\gamma_2)}} \right) ^ \frac{1+\gamma_2}{\gamma_2} > \frac{k}{\alpha_2 \gamma_2} \end{align}\]

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.

5.0.2 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:

\[\begin{align} (N_1^{\ast}, N_2^{\ast}) &= \arg \max \Pi_2 = \left((p-\phi_1) N_1 - \alpha_1 N_1^{(1+\gamma_1)(1+\delta)} + (p-\phi_2) N_2 - \alpha_2 N_2^{(1+\gamma_2)(1+\delta)} \right) \qquad \tag{5.4} \\ &= \left(\frac{p - \phi_1}{\alpha_1 {(1+\gamma_1)(1+\delta)}} \right) ^ \frac{1}{\gamma_1+\delta(1+\gamma_1)}, \qquad \left(\frac{p - \phi_2}{\alpha_2 {(1+\gamma_2)(1+\delta)}} \right) ^ \frac{1}{\gamma_2+\delta (1+\gamma_2)} \qquad \tag{5.5} \\ \Pi_2^{\ast} &= compute \tag{5.6} \\ s.t. & compute \end{align}\]

6 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 ($p$).

7 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.