Optimal Acquisition Prices

In this section, we investigate how acquisition prices have to be chosen to maximize the firm’s expected profit. For this purpose, the previously determined equilibria and the probabilities for submitted products having certain qualities are taken into account.

First, consider the firm’s profit in one quality class i. By Proposition 3.1 (compare also with Figure 3.2), the firm’s equilibria payoffs depending on viare as follows: if vi ∈ [min{v_i^∗, ˆa}, vi], then the firm gains 0 because the holder does not hand in; if v_i ∈ (min{co^∗_i, ˆa}, min{v_i^∗, ˆa}), then the firm has payoff −c because the holder triggers a counteroffer, which he rejects; and if v_i∈ [v_i, min{co^∗_i, ˆa}], then the firm has payoff m_i− min{co^∗_i, ˆa} because either the product holder is paid ˆa or he accepts the made counteroffer co^∗_i, depending on how ˆa and co^∗_i relate to one another.

Note that the specific quality-dependent acquisition prices do not play a role in equilibrium.

The firm’s only decision variable that impacts its profit is the value of the greatest acquisition price, ˆa. Therefore, after the optimal level of the largest price is found, one can arbitrarily set the quality-dependent prices while making sure that all of them are not greater than ˆa and that at least one price coincides with ˆa. One convenient choice is to set a_i = min{co^∗_i, ˆa}.

This choice also ensures that there are no counteroffers for true quality statements.

With regard to the above interval boundaries which specify conditions for product holders’

behavior depending on v_i, we define the upper bound for acceptance of the counteroffer, ˆ

co^∗_i(ˆa) := min{co^∗_i, ˆa}, the upper bound for handing in, wi(ˆa) := min{v_i^∗, ˆa}, and an auxiliary variable, w_i(ˆa) := min{v_i, ˆa} that ensures no profit or losses in quality i if ˆa ≤ v_i.

Drawing on the above observations, the firm’s expected profit as a function of ˆa considering only quality i is

Πⁱ_ACP(ˆa) = P^{(vi≤ min{co∗}i, ˆa})(m_i− min{co^∗_i, ˆa})

+ P^(min{co∗i, ˆa} < vi < min{v_i^∗, ˆa})(−c) +P^(min{vi^∗, ˆa} ≤ vi)0

= 1

v_i− v_i[( ˆco^∗_i(ˆa) − w_i(ˆa))(m_i− ˆco^∗_i(ˆa)) + (w_i(ˆa) − ˆco^∗_i(ˆa))(−c)]

Note that Πⁱ_ACP is equal to zero for ˆa < v_i, quadratic concave in ˆa for v_i < ˆa < co^∗_i, linearly decreasing in ˆa for co^∗_i ≤ ˆa < v_i^∗, and constant for ˆa > v_i^∗. Furthermore, Πⁱ_ACP is differentiable except in v_i and v_i^∗.

The following lemma presents the optimal largest price under negligence of all other quality classes and the corresponding profit in quality class i.

Lemma 3.1 (Optimal Acquisition Price - Single Quality). Considering only quality class i, it holds that ˆa^∗= min{(m_i+ v_i)/2, v_i} maximizes Πⁱ_ACP and always paid for a quality i product submission, which implies that there are no losses due to rejection.

In order to illustrate the single-quality case, we present Figure 3.3(a), which depicts the profit and corresponding hand-in, sales, and rejection volumes in quality class 1 depending on ˆa, given the stated parameter setting comprising three quality classes.

The classical price-volume trade-off reveals itself for ˆa less than co^∗₁. As the price increases, the hand-in and sales volumes increase. If ˆa is greater than co^∗₁, the sales volume stays constant, but the hand-in volume further increases. Here, product holders with residual value greater than co^∗₁ are incentivized to hand in due to the high price, but they receive a counteroffer that they reject. Product holders with v₁ less than co^∗₁ accept the counteroffer.

As a next step, we investigate how to choose the highest price in order to maximize the firm’s overall profit considering all quality classes. It is already intuitive that the choice of ˆ

a is related to the trade-off between the achievable profits in different quality classes. The firm’s total expected profit is just the weighted sum of all the quality-specific profits, where the weights correspond to the market shares. The total profit is given by

Π_ACP(ˆa) =

n

X

i=1

δ_iΠⁱ_ACP(ˆa)

Due to the relation between Π_ACP and the quality-specific profits, Πⁱ_ACP, the total profit exhibits similar properties, i.e., it consists of constant, quadratic concave, and linear parts and is differentiable except at finitely many (2n) points.

Figure 3.3.: Firm’s Profit under ACP

(a) Quality 1 Profit and Volumes (b) Profit ΠACP

Parameters: c = 8; for quality 1: δ1 = 0.2 , m1 = 10, m1 = 14, m1 = 8, v1 = 12, v1= 1; quality 2:

δ₂ = 0.2, m₂ = 25, m₂ = 32, m₂ = 20, v₂ = 30, v₂ = 12; quality 3: δ₂ = 0.6, m₃ = 30, m₃ = 34, m3 = 28, v3 = 32, v3 = 21.

In summary, the firm’s profit maximization problem is as follows:

P 1 : max

ˆ

a Π_ACP(ˆa)

=

n

X

i=1

δ_i vi− vi

[( ˆco^∗_i(ˆa) − w_i(ˆa))(m_i− ˆco^∗_i(ˆa)) + (w_i(ˆa) − ˆco^∗_i(ˆa))(−c)]

(3.1)

s.t. ˆco^∗_i(ˆa) = min{co^∗_i, ˆa} ∀i ∈ {1, ..., n} (3.2) w_i(ˆa) = min{v_i^∗, ˆa} ∀i ∈ {1, ..., n} (3.3) wi(ˆa) = min{vi, ˆa} ∀i ∈ {1, ..., n} (3.4)

The following proposition characterizes the highest acquisition prices ˆa that locally solve optimization problem P 1.

Proposition 3.2 (Locally Optimal Acquisition Prices). ˆa^∗ locally maximizes Π_ACP

There are at most 3n − 2 local maxima.

Note that j, k, l determine how ˆa relates to co^∗_i, v_i^∗ and v_i for all quality classes i. The profit-relevant intervals for ˆa are [v_i, v^∗_i] for all qualities i. Holders only hand in and cause either losses or profits if the residual values belong to those intervals. If some of these quality-specific intervals overlap, then the essential trade-off concerning the choice of ˆa within such an overlap comprises balancing the profits of the corresponding quality classes.

The following corollary presents a simple procedure for determining the globally optimal largest acquisition price ˆa by applying Proposition 3.2.

Corollary 3.2 (Finding the Global Optimum). Due to the profit function’s character-istics, there has to be at least one global optimum. Considering Proposition 3.2, for every triple (j, k, l) with k 6= l, determine ˆa^∗ according to (i). Select all ˆa^∗ for which conditions (ii)-(v) are satisfied, and determine Π_ACP(ˆa^∗). The one that yields the maximum profit is the globally optimal one.

To illustrate Proposition 3.2, we present Figure 3.3(b), which depicts the total profit and the quality-dependent profits times their market shares depending on ˆa for the indicated setting comprising three quality classes. The profit concerning a single quality class depends

on how all corresponding parameters relate to one another. No single parameter that drives the optimal highest price can be identified. For example, there are identical market shares for classes 1 and 2, but how losses and profits through sales relate to one another is quite different when considering the shape of quality specific profit-curves. Contrarily, the market shares of qualities 1 and 3 deviate from one another, but the distances between the remaining parameters coincide. Therefore, the profits follow the same “shape”, with the difference that the slope is steeper for quality 3, and therefore, the profits are magnified.

Considering the globally optimal largest price, we observe the mentioned trade-off of bal-ancing the profits of quality classes 2 and 3. We see that the profit in class 2 has a negative slope in the optimal price, whereas the slope of the class 3 profit is positive here. Consider-ing Proposition 3.2, the globally optimal price is equal to ˆa^∗ = (δ₂(−c)/(v₂− v₂) + δ₃(m₃+ v₃)/(v₃− v₃))/(2δ₃/(v₃− v₃)) = 24.69. Hence, one optimal choice for the acquisition prices is to set a3 = ˆa^∗, a2 = co^∗₂ = 22.5, and a1 = co^∗₁ = 9.5. By this choice, product holders with quality-1 and quality-2 products request ˆa^∗, thereby triggering corresponding counterof-fers that coincide with the respective quality-specific acquisition prices. Quality-3 product holders are paid acquisition price a₃ = ˆa^∗. Hence, the firm incurs losses through rejection in quality class 1 caused by product holders with v₁ ∈ (co^∗₁, v₁^∗) = (9.5, 11.5) and in class 2 caused by holders with v₂∈ (co^∗₂, ˆa^∗) = (22.5, 24.69) .

Finally, we remark that ACP leads to higher profit than the acquisition without quality differentiation. Acquisition without quality differentiation means offering only one fixed acquisition price independent of the condition of the product. To this end, assume that the optimal acquisition price, which does not depend on a product’s quality, is given. Then, for ACP, ˆa can be set equal to this price. If it is profitable to make counteroffers for submissions of products with certain qualities, the firm can do so if ACP is applied. This is not the case for having only one fixed quality-independent acquisition price. Thus, basically, the benefit is that reacting to low-quality product submissions is feasible when ACP is applied. Therefore, ACP is always preferable to the acquisition without quality differentiation.