Bundling and Platforms

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Bundling: What, Why, and How

Bundling is the idea of offering two or more products at a discounted price relative to buying each of those items separately (or of offering large-quantity packs at a lower price than buying equal quantity in smaller packs). It can be practiced in multiple ways,

Bundling is employed widely in business. It improves profits in multiple ways, by delivering economies of scope in selling, distribution or consumption (e.g., paying once for a mix of goods saves on transactions costs), by making heterogeneous demand more homogeneous, by delivering strategic advantage in multi-product competition, or by creating network effects advantages. Here are some examples - let’s see which of these forces apply.

Some motivating examples

Would you bundle these?

  • At a ski shop ski boots skit jacket

  • At a sports store ski boots ski boots

  • At a magazine publisher ski boots ski boots

  • Online Subscriptions ski boots ski boots

Economies of Scope in Selling Bundles

Suppose that production and distribution costs for two goods 1 and 2 are e and d respectively, with e+d = c. So, if the two goods are sold separately their unit variable costs are c for each. Suppose that there are M customers u in the market and their valuations for these goods are both distributed uniformly in [0, a]. Then optimal prices for selling separately are (a+c1)/2 and (a+c2)/2 respectively, and profits are M((a-c)^2)/(4 a) and M ((a-c)^2)/(4 a).

Now suppose that bundling the two goods allows for full economies in distribution, so that the bundle has a unit cost (2e+d). Bundle demand will depend on the correlation or dependence between customers’ valuations for product 1 vs product 2. For illustration, suppose that each customer values each individual good identically (say v), and the bundle at the sum of these valuations (2v). Then, bundle demand is distributed uniformly between [0,2a]. Optimal bundle price is (2a+2e+d)/2, and bundle profit is M((2a-2e-d)^2)/(8a). Let’s see which profit is larger.

\[\begin{align*} \mbox{bundle profit\ vs.}\quad & \mbox{Separate selling profit} \\ M \frac{(2a-2e-d)^2}{8 a} &> 2 M \frac{(a-c)^2)}{4 a} \\ \equiv \quad \frac{(2a-2e-d)^2}{4}&> \frac{(a-c)^2)}{1} \\ \equiv \quad \frac{(2a-2e-d)^2}{4}&> \frac{(a-e-d)^2)}{1} \\ \equiv \quad (2a-2e-d)^2 &> 4 (a-e-d)^2 \\ \equiv \quad (2a-2e-2d + d)^2 &> (2a-2e-2d)^2. \end{align*}\]

So, that was the case where there was positive correlation in reservation prices for products 1 and 2. Consider the other extreme now, where reservation prices have perfect negative correlation.

Demand-Smoothing Mechanics of bundling

We’ll explore bundling of two products, 1 and 2. Suppose there is a total of M customers in the market for either product. Suppose that demand for each product is uniformly distributed, in \([0, a_1]\) and \([0, a_2]\) respectively. A uniform distribution implies that reservation prices are quite heterogeneous - no two customers (or customer segments) are alike. We can visualize the distribution of reservation prices (uniform) and the consequent demand curve (linear).

Suppose that a consumer’s value for one product is not affected by whether they purchase the second item or not, hence the value from purchasing both products is the sum of their individual product valuations. Suppose that each product has constant marginal cost per unit, c1 and c2 respectively.

https://blog.revolutionanalytics.com/2016/02/multivariate_data_with_r.html

Base Case: Selling two products separately

If the two products are sold independently, priced at p1 and p2, then the demand and profit functions are as follows.

\[\begin{align} D_1(p_1) = \frac{(a_1-p_1)}{a_1} M & \quad\qquad D_2(p_2) = \frac{(a_2-p_2)}{a_2} M \\ \Pi_1(p_1) = (p_1 - c_1) \frac{(a_1-p_1)}{a_1} & \quad\qquad \Pi_2(p_2) = (p_2 - c_2) \frac{(a_2-p_2)}{a_2} M \end{align}\]

The optimal price \(\frac{a_{1} + c_{1}}{2}\) and \(\frac{a_{2} + c_{2}}{2}\) is obtained by maximizing the profit function above, yielding optimal profits \((\frac{a_1-c_1}{2})^2 \frac{M}{a_1}\) and \((\frac{a_2-c_2}{2})^2 \frac{M}{a_2}\), which might be more conveniently written as \[\begin{align} \Pi^{\ast}_{sep} = \frac{M}{4} \left(\frac{(a_1-c_1)^2}{a_1} + \frac{(a_2-c_2)^2}{a_2} \right) \quad \qquad \mbox{profit from separate sales}. \end{align}\]

Data Generation

For purposes of experimentation and illustration, let’s fix the parameter values as follows.

Let’s generate reservation prices for the \(M\) customers by sampling from a uniform distribution.

rp.1 <- runif(M, min=0, max=a1)
rp.2 <- runif(M, min=0, max=a2) # a2 > a1

# the "real" demand functions (sort the rp decreasing)
dem.1 <- data.frame(sales = 1:M, price = sort(rp.1,decreasing=TRUE))
dem.2 <- data.frame(sales = 1:M, price = sort(rp.2,decreasing=TRUE))

# and the theoretical demand functions (linear)
demand.1 <- as_tibble(Hmisc::bezier(c(0,M), c(a1,0)))
demand.2 <- as_tibble(Hmisc::bezier(c(0,M), c(a2,0)))

p1.opt <- (a1+c1)/2 # standard for linear demand
p2.opt <- (a2+c2)/2

We can visualize the demand curves (empirical one from the data - in black - and theoretical one in red), optimal pricing, and profit.

Given the initial parameters ((a1, a2, c1, c2) = (100, 150, 6, 7)), we get the (theoertical) optimal prices for the two products as 53 and 78.5 respectively, yielding profit 22.09 and 34.08 for the two products.

Selling Just a Bundle

Now suppose that instead of selling each item separately the firm sells a bundle comprising both items at a single price \(p_B\). So, everyone whose v1+v2 valuations exceed \(p_{B}\) would buy the bundle, others would not. Here is what the buying decision looks like.

Demand when selling component products (i) separtely and (ii) bundled.

In a sense, this is taking away flexibility - perhaps there are some consumers who might buy item 1 (they have high value for it) but not the bundle (because their total value might be below the bundle price). So, one might expect the bundle to perform poorly? Let’s examine this.

Bundle Demand

Note that we drew these reservation prices rp.1 and rp.2 independently. So, here’s what the bundle demand looks like.

rp.plot <- ggplot(data.frame(rp.1, rp.2), aes(x=rp.1, y=rp.2)) + geom_point(shape = ".") + labs(title="Joint distribution of reservation prices for products (1,2)") 

bundle.dem.plot <- ggplot(data.frame(dem = 1:M, price = sort(rp.1 + rp.2,decreasing=TRUE)), aes(x=dem, y=price)) + geom_line() + geom_hline(yintercept=c1+c2, linetype="dashed", color = "red") + labs(title="Demand curve for bundle") # (descending sort rp1+rp2)  

grid.arrange(rp.plot, bundle.dem.plot, ncol=2)

This is not quite a linear demand curve, so we cannot use the previous formula for optimal pricing and profit.

To compute bundle demand, pricing and profit (following Bhargava (2012)), start with the bundle demand function \(D_{b}(p) = 1-G(p)\) where \(G\) is the \(cdf\) of reservation prices for the bundle for which the probability density function \(g(p)\) is defined below, then integrate this distribution of reservation prices (from p to \(a_1+a_2\)) to get the bundle demand function.

\[\begin{align} g(p) = \begin{cases} \frac{p}{a_1 a_2} & p \in [0, a_1] \\ \frac{1}{a_2} & p \in [a_1, a_2] \\ \frac{a_1+a_2-p}{a_1 a_2} & p \in [a_2, a_1+a_2] \end{cases}, and \end{align}\] \[\begin{align} D_{b}(p) = \begin{cases} 1 - \frac{p^2}{2 a_1 a_2} & p \in [0, a_1] \\ 1 - \frac{2p - a_1}{2 a_2} & p \in [a_1, a_2] \\ \frac{(a_1 + a_2 - p)^2}{2 a_1 a_2} & p \in [a_2, a_1+a_2] \end{cases}. \end{align}\]

This is what the theoretical density function and demand function for the bundle looks like. Unlike the demand for individual products (uniformly distributed reservation prices), where every price-point was equally likely to occur, we see that reservation prices for the bundle are more commonly in the middle range than in the extremes! The market looks more homogeneous when selling the bundle, and this leads to higher profit through bundle pricing.

This demand-smoothing effect becomes even more pronounced when the number of items being bundled increases. As a consequence of the central limit theorem when the number of items bundled is extremely large (quite common with information goods, for example streaming video providers such as Netflix newspapers that can be thought of as a bundle of multiple articles), the demand gets rather flat as reservation prices become relatively similar across previously-heterogeneous consumers.

Pure Bundle Pricing and Profits

The profit function when selling the pure bundle at price \(p_b\) is \(\Pi_{b}(p) = (p-(c_1+c_2)) D_{b}(p)\). The optimal bundle price and profit (from Eckalbar (2010)) is \[\begin{align} p^{\ast}_{b} &= \frac{1}{4} \left(2 (c_{1}+c_{2}) + 2 a_1 + a_{2} \right) \\ \Pi^{\ast}_{b} &= (p^{\ast}_{b} - c_1 - c_2)\ D_{b}(p^{\ast}_{b}) % \frac{M}{16 a_1} \left(2 a_1 + a_2 - 2 (c_{1}+c_{2}) \right)^2. \qquad \mbox{(Pure Bundle Profit)} \end{align}\]

Compare this profit expression for the bundle against the profit for separate selling, the difference or profit advantage from selling a pure bundle is \[\begin{align} Adv_{PB} &= \frac{M}{4} . \left(\frac{(a_1 + a_2/2 - (c_{1}+c_{2}))^2}{a_1} - \left(\frac{(a_1-c_1)^2}{a_1} + \frac{(a_2-c_2)^2}{a_2} \right) \right) \\ &= \frac{M}{4} . \left(\frac{(a_2/2 - c_2)^2}{a_1} + \frac{(a_1-c_1) (a_2 - 2 c_2)}{a_1} - \frac{(a_2-c_2)^2}{a_2} \right). \end{align}\]

With the initial parameter values, \(\Pi_{sep}\) is 28085.83, while \(\Pi_{PB}\) is 28571.4, so that the additional profit from pure bundling is 485.57 (or an improvement of 1.73 % over the profit from separate selling).

Bundling theory establishes that this profit advantage is highest when marginal costs are low (hey, information goods!) and when the two items being bundled are of comparable demand profiles (e.g., two arbitrary movies in a movie archive, or two mortgages in a mortgage bundle). We’ll see later how bundling plays a role when the two items have hugely different demand profile. For now, here is a visualization of the extent of bundle profit relative to the parameters. To show the visualization along the 4 dimenions, let’s normalize two parameters and examine the changes along the other two.

First, suppose the two marginal costs are identical are in absolute sense (so they both equal c, and we can reduce one parameter). Next, normalize a1 to 100, and vary a2 (and vary c).

## Warning in regularize.values(x, y, ties, missing(ties)): collapsing to unique
## 'x' values

Next, suppose the two marginal costs are an identical fraction h of the highest reservation price (so, \(c1 = h*a1\) and \(c2 = h*a2\)). Again, normalize a1 to 100, and vary a2 (and vary c).

## Warning in regularize.values(x, y, ties, missing(ties)): collapsing to unique
## 'x' values

Mixed Bundling

We’ve seen the basis for why bundling can increase profit. However, bundling tactics can be used in more sophisticated ways than pure bundling (i.e., selling just the bundle of all goods, and not individual products or subsets). We can generally do even better by combining both bundle and unbundled sales, what is called “mixed bundling” where you offer both the individual products and a bundle of both (at a discount to the sum of the individual prices).

Mixed Bundle Demand

A mixed bundling strategy essentially is a way to implement second-degree price discrimination by having a wider product line from which customers can pick the option that works best for them. With mixed bundling, customers with extreme preferences (high value for one item, low value for other) pick the individual products, while those with moderate (or high) values are steered towards the bundle, which is offered at a discount relative to the sum of prices of individual products. The picture below (from Bhargava, 2013) depicts demand for different items in the product line under mixed bundling (and contrasts against separate selling and pure bundle selling).

Demand under mixed bundling strategy.

Converting the geometry into equations, we get \[\begin{align} Q_{1} &= (a_1-p_1)*(p_B-p_1) \\ Q_{2} &= (a_2-p_2)*(p_B-p_2) \\ Q_{B} &= (a_2-p_B+p_1) (a_1-p_B+p_2) - \frac{1}{2}(p_1+p_2-p_B)^2. \end{align}\]

Partial Mixed Bundling

We’ll see shortly that mixed bundling is even better for profitability than pure bundling. This is generally the case - when the goods have relatively similar demand profiles. In contrast, when item commands much higher valuations, overall, than the other (say, item 2 has higher demand profile than item 1), then the optimal strategy is to sell the bundle and ONE of the items. Guess which one?

The condition for partial bundling to be optimal (using Eq. 6 from Bhargava (2012)) is that c1=0 and a1 < a threshold (multiplier of A2). \[\begin{align} a_{1} < \bar{A_{1}} &= \frac{1}{48 a_{2}} \left(28 a_{2}^2 - 12 a_{2} w_{2} - w_{2}^2 - (2 a_{2}+w_{2}) \sqrt{4 a_{2}^2+20 a_{2}w_{2} +w_{2}^2} \right) \\ \approx \quad \bar{A_{1}} &= \frac{(1 - 0.976 w_{2}) a_{2}}{2}. \end{align}\]

Mixed Bundle Pricing and Profits

The optimal implementation of a mixed bundle pricing strategy is to set the prices as \[\begin{align} p_{B} &= 0.5742 c1 + 0.5695 c2 + 0.35616 a1 + 0.4889 a2 + 0.0054 c_{1} \frac{a_{2}}{a_{1}} - 0.0201 c_{2} \frac{a_{1}}{a_{2}} \\ p_{1} &= \frac{1}{6}\left(3 p_B + (2 a_1+c_{1}-c_{2}) - \sqrt{((3 p_{B}-2 a_1+(c_{2}-c_{1}))^2) + 12 c_{2} (a_1-p_B)} \right) \\ p_{2} &= \frac{1}{6}\left(3 p_B + (2 a_2+c_{2}-c_{1}) - \sqrt{((3 p_{B}-2 a_2+(c_{1}-c_{2}))^2) + 12 c_{1} (a_2-p_B)} \right) \end{align}\]

Profit Advantage from Mixed Bundling

For c1 = c2 = 0, mixed bundling raises the firm’s profit 9.84%.

Bundling by a Retailer

Bundling products from multiple producers.

Optimal bundling and sales (under zero marginal costs). In the second case - a retailer who bundles goods from multiple producers (where the manufacturer’s marginal costs are 0) - manufacturer prices are w1 = w2 = 0.5067, then the retailer deploys a mixed bundle strategy with pB = 1.04271, and p1 = p2 = 0.7778. Each manufacturer earns 0.1317, the retailer’s profit is 0.1248, and the combined firms’ profit is 0.3882. Compared with component selling, mixed bundling in the mutlifirm case delivers only a modest 3.5% increase in industry profits from 0.375 to 0.3882, as against a 9.84% gain for the integrated firm.

How effective is bundling? Sales for zero cost goods.

Bundling and Platform Envelopment, Foreclosure, Exclusion

one platform provider moving into another one’s market, combining its own functionality with the target’s, to form a multi-platform bundle.

Bundling and Value Co-Creation

Value cocreation markets. ## Motivation for Value Aggregators

Value cocreation model

Value cocreation equilibrium

Appendix

Debugging

a1 = 100
a2 = 160
c1 = 15
c2 = 15

pi.sep <- (M/4)*(((a1-c1)^2)/a1 + ((a2-c2)^2)/a2)

popt.PB <- function(a1,a2,c1,c2) {
    (1/4)*(2*(c1+c2) + 2*a1 + a2)
}

pi.PB <- M*(popt.PB(a1,a2,c1,c2) - c1-c2) * (1 - ((popt.PB(a1,a2,c1,c2))^2)/(2*a1*a2))

pi.sep
## [1] 25457
pi.PB
## [1] 24580
100*(pi.PB - pi.sep)/pi.sep
## [1] -3.4
# mixed bundling
c(p1(100,150,6,7), p2(100,150,6,7), pB(100,150,6,7), profit.MB(100,150,6,7))
## [1]      67      97     116 1012564
c(p1(100,220,6,7), p2(100,220,6,7), pB(100,220,6,7), profit.MB(100,220,6,7))
## [1]      68     137     150 1976015
c(p1(100,220,0,7), p2(100,220,0,7), pB(100,220,0,7), profit.MB(100,220,0,7))
## [1]      66     147     147 2048983