Customer value is the potential customer’s perception of the worth of a product or service. Value-based pricing (VBP) is a pricing strategy used by some businesses to charge products and services at a rate they believe consumers are willing to pay, as opposed to calculating production costs and applying a standard markup. Artwork, amusement parks and cars are just some of the types of products/services where VBP is used¹.

A business that wishes to price in VBP manner, conducts market research in its entire addressable marketto understand how each of its potential customers values its product and what they would be willing to pay for it (Hinterhuber, 2014). Nevertheless, even after extensive market research at least some uncertainty as to a consumer’s value for that product will remain. Thus, we assume a formation of a subjective probability distribution by the retailer (R) for a consumer’s valuation.

Further, in addition to price, the nature of the operation is such that it must commit to produce a certain quantity of the product, and if demand turns out to exceed it, sales are lost.

Assuming a homogeneous population, in the sense that each customer’s valuation is drawn from the same probability distribution, the number of customers wishing to buy the product is thus binomially distributed. [10] and [7] also addressed the impact of valuations, but while that stream of research considered a newsvendor who bases her decisions on the expectedcustomer’s demand only, we account for its full distribution. Given the price set by the retailer, the demand may exceed the short-term supply, so the retailer faces a price-setting newsvendor problem (surveyed by [16] and [3]), where the probability that a customer will wish to buy the product is that of her valuation exceeding the product’s price ([11]).

Note that our customer-level fine-grained model is distinct from the well known d ( p , ε ) random total demand model, a function of price and noise (mostly additive or multiplicative, e.g., [14]). It also differs from the model of [15] which ascribes uncertainty directly to demand, rather than, as in our model, to relevant population’s size which then determines the demand. However, in examples, we assume, like Schulte and Sachs, that the product is such that the relevant population is sparse, and thus potential stocking quantities are small.

Another stream of somewhat related research is the data-driven pricing literature, where historical observations are utilized to form statistical estimates of the demand distribution, which are then fed into an optimization model to yield the pricing and inventory decision (e.g., [4] and references therein).

In this work we attempt to find the quantity/price combination which maximizes the retailer’s expected profit for linear and non-linear ([6]) production costs. Initially, we assume that the size of the relevant population of potential buyers is known to the retailer. We then consider the implications of unknown relevant population size, possibly Poisson distributed (not directly the demand distributed Poisson as in Schulte and Sachs).

Finally, we introduce a manufacturer (denoted by M) who has a revenue sharing contract with R, where the parties negotiate the revenue shares and wholesale price as a function of the retail price, using the Nash bargaining solution (NBS), possibly asymmetric (ANBS) ([9]). The retailer then selects the retail price which, in turn, determines the revenue shares and wholesale price, and the parties’ expected revenues.

Consider a setting where an expected-profit-maximizing retailer, who upon selecting an order quantity q (integer) and setting the retail price per unit p for the product faces N potential customers. For the retailer the customers’ valuations are random, denoted by V , and follow some subjective distribution F ( p ) on [0, 1]. A customer would wish to buy (one unit of) the product iff its individual valuation exceeds the price, F ¯ ( p ) = 1 − F ( p ) . That valuation is assumed to be independent of other customers’ valuations ([12]; [11]). Thus, the number of customers wishing to buy the product (that is, the number of customers with valuation exceeding the retail price p ), X , is binomially distributedwith N and F ¯ ( p ) .

Specifically, the probability that k out of the N potential customers have valuations exceeding the price p set by the retailer is given by

The retailer (who, for now, is assumed to know N ), sets q ( q ≤ N ) and p ( p ≥ 0 ) , so as to maximize its expected profit, which equals expected revenue minus production costs. We assume that the product/service has no salvage value².

By (1),

where c is the unit production cost (we shall use quadratic production costs in a subsequent section). The first sum is the expected number of units sold when all demand can be met, while the second sum is the expected number of units sold, in multiples of q , when entire demand cannot be met. We assume that E(Revenue) from q units is greater than c q in the relevant price range. For fixed q , p * does not depend on c . However, c is relevant to finding q * , on which p * does depend.

Thus, we use a stylized single-period model, where each customer can buy at most one unit. We assume that the price p is upper-bounded, so wlog it is in [ 0 , 1 ] .

Hence, this setting resembles the traditional price-setting newsvendor model ([16]), though here customers’ valuations are modeled explicitly. [10] (see also [7]), who also explored the consequences of valuations, worked directly solely with the expected sales; So, our model is more detailed as we capture the intricate decisions of the independent individual potential customers.

Our examples will assume that the valuations are distributed as beta on [0, 1]. The standard beta ( α , β ) distribution on [0, 1] has the pdf

0 ≤ x ≤ 1 , α , β ≥ 1 ³, where Γ ( z ) = ∫ 0 ∞ x z − 1 e − x d x . For a positive integer z = n Γ ( n ) = ( n − 1 ) ! (e.g., [8]). If β = 1 , then f ( x ) = α x α − 1 , and F ( x ) = x α , α ≥ 1 0 ≤ x ≤ 1 ; E ( X ) = α α + 1 is increasing in α ⁴. Beta ( α , 1 ) is a family of distributions, often referred to as power ( α ) distribution, whose pdf is convex (for α = 1 the distribution is uniform). Here V a r ( X ) = α ( α + 2 ) ( α + 1 ) 2 , which is decreasing in α .

In Appendix Awe show how to find the ( p , q ) pair which maximizes E ( π ) for any N for a uniform distribution.

Example: N = 3 , so q ∈ { 0 , 1 , 2 , 3 } . We shall use the notation “ E ( m | n ) ” to express

E (profit from m customers and n units). Thus

iff c < p ( 1 − [ F ( p ) ] 3 ) .

iff c < 1 2 p ( [ F ( p ) ] 3 − 3 [ F ( p ) ] 2 + 2 ) .

E ( 3 | 3 ) = p ( 3 − 3 [ F ( p ) ] ) − 3 c > 0 iff c < p ( 1 − [ F ( p ) ] ) .

So 1 ≻ 2 (i.e., expected profit is higher with q = 1 than with q = 2 , so q = 1 is preferred to q = 2 ) iff p ( 1 − [ F ( p ) ] 3 ) − c > p ( [ F ( p ) ] 3 − 3 [ F ( p ) ] 2 + 2 ) − 2 c

i.e., iff c > p ( 2 [ F ( p ) ] 3 − 3 [ F ( p ) ] 2 + 1 ) = p ( 2 p 3 α − 3 p 2 α + 1 ) ≡ A 1 , 2

2 ≻ 3 iff p ( [ F ( p ) ] 3 − 3 [ F ( p ) ] 2 + 2 ) − 2 c > p ( 3 − 3 F ) − 3 c

i.e., iff c > p ( 1 − [ F ( p ) ] ) 3 = p ( 1 − p α ) 3 ≡ A 2 , 3

1 ≻ 3 iff p ( 1 − [ F ( p ) ] 3 ) − c > p ( 3 − 3 [ F ( p ) ] ) − 3 c ,

i.e., iff c > 1 2 p ( [ F ( p ) ] 3 − 3 [ F ( p ) ] + 2 ) = 1 2 p ( p 3 α − 3 p α + 2 ) ≡ A 1 , 3 .

Can show that A 1 , 3 > A 2 , 3 > A 1 , 2 ∀   0 ≤ p ≤ 1 . Thus if c > A 1 , 3 then 1 ≻ 2 ≻ 3 (in particular, q * = 1 ); if A 2 , 3 < c < A 1 , 3 then 1 ≻ 2 , 2 ≻ 3 but 1 ≺ 3 → impossible; if A 1 , 2 < c < A 2 , 3 then 1 ≻ 3 ≻ 2 (in particular, q * = 1 ); if c < A 1 , 2 , then 3 ≻ 2 ≻ 1 (if particular, q * = 3 ).

Here the production costs are assumed to be quadratic: C ( q ) = c 1 q + c 2 q 2 , c 2 > 0 , c 2 > − c 1 (see [6]), i.e., increasing marginal costs. Like Holt et al., we allow c 1 to be negative to make the model more general. Again, for a fixed q , p * does not depend on the cost parameters.

Example: N = 3

Let p α ≡ y (so p = y 1 α ).

Let p i * be the optimal price if q = i . Then it can be shown that p 1 * > p 3 * for all α ;

So

Thus for α = 1 , 1 ≻ 2 iff c 1 + 3 c 2 > 0.54

2 ≻ 3 iff c 1 + 5 c 2 > − 0.48 (if c 1 > 0 , that always holds).

1 ≻ 3 iff c 1 + 4 c 2 > 0.03 .

So if 1 ≻ 2 then 2 ≻ 3

For α = 2 , 1 ≻ 2 iff c 1 + 3 c 2 > − 0.19 (if c 1 > 0 that always holds).

2 ≻ 3 iff c 1 + 5 c 2 > 0.20

1 ≻ 3 iff c 1 + 4 c 2 > 0.005

So if 2 ≻ 3 then 1 ≻ 3 and thus 1 ≻ 2 .

Thus far, we assumed that the newsvendor knows the potential population size N. Rather, now suppose she is not sure and thinks that

The valuation distribution is assumed to be the same for all realizations of N. Then, as a mixture of probabilities,

So

See solution procedure for X ~ U [ 0 , 1 ] in Appendix B.

Example

q = 1

Now specialize to F ( p ) = p α , 0 ≤ p ≤ 1 , α ≥ 1 :

q = 2

Thus 1 is preferred to 2 iff c > 1.573 .

For q = 3

has no solution in [ 0 , 1 ] for θ = 1 2 and α = 1 .

Suppose that the newsvendor thinks that the number of potential customers follows a Poisson ( λ ) distribution,

The probability that a potential customer will attempt to buy is still F ¯ ( p ) . Note the difference of that model from [15] who assumed that the demand is Poisson distributed.

For this mixture of Poisson,

For [ F ( p ) ] ~ U [ 0 , 1 ] , P ( X ) = k equals ( 1 − p p ) k e − λ ∑ n = 0 ∞ ( n k ) n ! .

For any distribution of N we can express the expected profit as follows:

Reorganizing terms and noting that the outer summation of the second term requires at least q customers:

Note that the revenue expression is composed of two terms, each of which consists of a double summation: The first term sums over the total number of customers and over the number of customers whose valuations exceed the price, which is summed up to and including q , the retailer’s order quantity. The second term considers instances where demand exceeds q , and hence it sums over population size realizations that can give rise to demand greater than q , i.e., n ≥ q + 1 . For the Poisson distribution,

For example if V ~ power ( 1 ) , i.e., V ~ U [ 0 , 1 ] , and λ = 10 , c = 0.05 the optimal expected profit is achieved at q = 8 and p = 0.52 , yielding an expected profit of 2.04. For c = 0.25 the optimal expected profit is achieved at q = 4 and p = 0.63 , yielding an expected profit of 0.94.

The retailer (R) and a manufacturer (M) are to set a revenue sharing contract ([2]; [1]). They are going to bargain over the revenue shares ( β , 1 − β ) , 0 ≤ β ≤ 1 and wholesale price ( w ), for the outcome of which we will use (at first) the symmetric Nash Bargaining Solution (NBS), and then possibly an asymmetric one (ANBS) (e.g., [13]). Note that after ( β , w ) are determined R will set the retail price ( p ) and associated order quantity ( q ) to maximize its expected profit. While p will affect the parties’ profits, the bargaining phase takes place before it is selected by R. Meanwhile, M has no direct influence on the retail price.

Valuations are (still) distributed beta ( α , 1 ) , i.e., have a power ( α ) distribution on [ 0 , 1 ] . Note the assumption that the retail price is upper bounded. At first, M will have a unit production cost c (where c < w ). Later we shall consider non-linear production costs.

We assume that population size is N = 3 . Let the demand be a ( p ) ≡ p ( p 2 − 3 p + 3 )

Then if q = 1 , E 3 ( Π R | 1 ) = β p ( p 2 − 3 p + 3 ) − w ≡ β a ( p ) − w , and E 3 M ( Π | 1 ) = ( 1 − β ) a ( p ) + w − c

So, d ( product ) d β = − 2 a ( p ) 2 β + a ( p ) 2 + 2 a ( p ) w − c a ( p ) = 0

d d w = 0 ⇒ same! So, w can be arbitrary⁵, between 0 and p, and β = 1 2 + 2 w − c 2 p ( p 2 − 3 p + 3 ) .

Let E ˜ ( Π R | k ) be the expected profit with ( β * , w * ) , substituted in E * ( Π R | k ) is E ˜ where p * was also substituted for.

q = 2 Let p ( [ F ( p ) ] 3 − 3 [ F ( p ) ] 3 + 2 ) ≡ [ F ^ ( p ) ] . So, for [ F ( p ) ] = p α , [ F ^ ( p ) ] equals

α

1: p 4 − 3 p 3 + 2 p

2: p 7 − 3 p 5 + 2 p

3: p 10 − 3 p 7 + 2 p

d ( product ) d w = same !

So w can be arbitrary and β = 1 2 + 2 w − c [ F ^ ( p ) ] ,

Now substitute F ( p ) = p α : p 3 α + 3 p 2 α + 1 − 3 p 2 α − 16 p α + 1 + 2 = 0 .

α

1: 4 p 3 − 19 p 2 + 2 = 0 ⇒ p ~ 0.34 ⇒ E 3 * ( Π R | 2 ) ~ 0.29 − c

2: p 6 + 3 p 5 − 3 p 4 − 16 p 3 + 2 = 0 ⇒ p ~ 0.49 ⇒ E 3 * ( Π R | 2 ) ~ 0.48 − c

3: p 9 + 3 p 7 − 3 p 6 − 16 p 4 + 2 = 0 ⇒ p ~ 0.59 ⇒ E 3 * ( Π R | 2 ) ~ 0.56 − c .

q = 3

E 3 ( Π M | 3 ) = ( 1 − β ) p ( 1 − [ F ( p ) ] 3 ) + 3 w − 3 c . Denote p ( 1 − [ F ( p ) ] 3 ) by, L > 0 .

So, product = β ( 1 − β ) L 2 − 3 w ( 1 − β ) L + 3 w β L − 9 w 2 − 3 c β L + 9 c w .

⇒ β = 1 2 + 3 ( 2 w − c ) 2 L → same !

So let w be arbitrary in the range [ 1 2 c − p ( 1 − p 3 α ) 6 , 1 2 c + p ( 1 − p 3 α ) 6 ] ,

while β = 1 2 + 3 ( 2 w − c ) 2 L .

So, E ˜ 3 ( Π R | 3 ) = [ 1 2 + 3 ( 2 w − c ) 2 L ] L − 3 w = L 2 − 3 c 2 = E ˜ 3 ( Π M | 3 ) .

d d p = 1 2 [ 1 − [ F ( p ) ] 3 + p ( − 3 [ F ( p ) ] 2 [ f ( p ) ] ) ] = 0 , so for power ( α ) distribution

AsymmetricNBS , N = 3

q = 1 . Expected revenue = p ( 1 − [ F ( p ) ] 3 ) ≡ a ( p )

So

Let λ [ ( 1 − β ) a ( p ) + w − c ] − ( 1 − λ ) [ β a ( p ) − w ] ≡ M .

Assuming that p ≠ 0 , 1 , there are thus three solutions:

1) β a ( p ) − w = 0 ⇒ β = w a ( p ) , w < a ( p ) ;

2) ( 1 − β ) a ( p ) + w − c = 0 ⇒ β = 1 + w − c a ( p ) > 1 so that solution is not valid;

3) { M } = 0 ⇒ β = λ + w − λ c a ( p ) , λ [ c − a ( p ) ] < w < λ c + ( 1 − λ ) a ( p ) . We shall use that solution.

So for a power ( α ) distribution

q = 2