SME Activities and Platform

We consider an SME as pursuing two activities in order to make a sale (besides of course production): i) discovering buyers by acquiring likely buyers (through advertising and promotions etc) and ii) fulfillment (or e-commerce). Each activity might be conducted by the SME (a DIY approach) or with the help of a ``platform’’ provider.

SME Economic Assumptions and Operational Parameters

As an SME, we assume that the firm is a price-taker, or at least that a (small) change in scale of operation will not impact selling price.

An SME’s operation is defined by its “characteristic operational parameters” with regard to the above two activities, the triplet \((\alpha, \gamma, \phi)\), where \(\phi\) is a unit fulfillment cost, and \((\alpha, \gamma)\) together reflect the cost of acquiring buyers through discovery and promotion activities.

  • Net revenue (unit price - unit fulfillment cost) from \(N\) sales is linear in \(N\): SME’s unit price \(p\) does not change with scale (this is reasonable at least in the short term) and fulfillment incurs a linear cost. So, revenue net of fulfillment can be captured by a function such as \((p-\phi) N\) where \(p\) is price and \(\phi\) is fulfillment cost.

  • Discovery-related activities have diminishing returns. Conversion of potential customers (visitors) into actual sales incurs a cost that is convex in number of sales, for each single channel or mechanism that brings these visitors (as more visitors arrive, their conversion into sales drops). We use a discovery cost function \(\alpha N^{(1+\gamma)}\)m where \(\alpha\) can is similar to a unit cost and \(\gamma\) is (the inverse of) how well the firm can leverage opportunities for growth - or degree of diminishing returns.

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 \(\gamma\).

DIY Benchmark for SME

Let the triplet \((\alpha_j, \gamma_j, \phi_j)\) be the SME’s operational parameters under a DIY operations for the two activities. With these operational parameters, we get the optimal scale and profit below.

\[\begin{align} N_j^{\ast} &= \arg \max \Pi_j = \left((p-\phi_j) N - \alpha_j N^{(1+\gamma_j)} \right) \qquad (\#eq:Nstar0) \\ &= \left(\frac{p - \phi_j}{\alpha {(1+\gamma_j)}} \right) ^ \frac{1}{\gamma_j} \qquad (\#eq:Sales0) \\ \Pi_j^{\ast} &= \alpha_j \gamma_j \left(\frac{p - \phi_j}{\alpha_j {(1+\gamma_j)}} \right) ^ \frac{1+\gamma_j}{\gamma_j} \qquad (\#eq:Pistar0) \end{align}\]

SME Heterogeneity and Boundary Conditions

For a particular SME with capacity \(\bar{N}\), the above equations are valid if they produce \(N_j^{\ast} \leq \bar{N}\). For a “smaller firm” (with maximum possible scale \(\bar{N} < N_j^{\ast}\)), the operating scale is \(\bar{N}\) and profit is simply \(\Pi_j^{\ast} = \left((p-\phi_j) \bar{N} - \alpha_j \bar{N}^{(1+\gamma_j)} \right)\).

But varying parameters is an alternate way to think about maximum scale. Higher \(\phi\) for a smaller SME. And regarding discovery, small size is proxied by a high \(\gamma\). Small, but highly efficient, SME, would be low \(\alpha\) combined with high \(\gamma\). So, there are two ways to think about how the SMEs vary.

  1. Same operational parameters \((\alpha, \gamma, \phi)\), but different (exogenously given) maximum operating scale, say \(\bar{N}_i\) for SME \(j\).

  2. No explicit maximum scale \(\bar{N}\), but captured through worse (higher) operational parameters: a smaller firm suffers higher costs due to economies of scale effect.

SME working with a Platform

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:

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.

SME partnering with a Single Platform

Now suppose that the SME can partner with a platform that offers discovery and fulfillment services, with the characteristic triplet \((\alpha_1, \gamma_1, \phi_1)\). On the surface, this setup fits a platform that offers both services. The model can easily be applied to a platform that offers just one service, simply by setting the characteristic value of the missing service to the SME’s DIY value (i.e., set either \(\phi_1 = \phi_j\), or \(\alpha_1 = \alpha_j\) and \(\gamma_1 = \gamma_j\)).

Optimal Scale (with one Platform)

The potential partnership is meaningful only if the triplet dominates the SME’s DIY triplet, in other words at least some of the operational parameters are lower and produce a higher optimal operating scale \(N^{\ast}\) and profit.

\[\begin{align} N^{\ast} &= \arg \max \Pi_1 = \left((p-\phi_1) N - \alpha_1 N^{(1+\gamma_1)} \right) \qquad (\#eq:Nstar1) \\ &= \left(\frac{p - \phi_1}{\alpha {(1+\gamma_1)}} \right) ^ \frac{1}{\gamma_1} \qquad (\#eq:Sales1) \\ \Pi_1^{\ast} &= \alpha_1 \gamma_1 \left(\frac{p - \phi_1}{\alpha_1 {(1+\gamma_1)}} \right) ^ \frac{1+\gamma_1}{\gamma_1} \qquad (\#eq:Pistar1) \end{align}\]

DIY or Partner with Platform?

The SME would partner with such a platform if \(\Pi_1^\ast > \Pi_j^\ast\) (which would also entail \(N_1^\ast > N_j^\ast\)). If there is a fixed cost \(C\) involved in partnering with the platform, then the requirement becomes \((\Pi_1^\ast - \Pi_j^\ast) > C\).

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

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).

Discovery-only Platform

So, now suppose that a platform gives all SMEs the same (\(\alpha\)) parameter, and each SME’s new \(\gamma_j\) becomes the midpoint between its own \(\gamma_j\) and the \(\gamma\) offered by the platform.
*** Need to compute and show this.

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:

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

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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.