Intermediary Commissions in a Regulated Market with Heterogeneous Customers Pilar Alcalde∗ and Bernardita Vial† January, 2018
Abstract We analyze competition among firms within a regulated industry, where intermediaries’ commissions are paid by customers and set exogenously. Firms set their prices taking into account the impact of commissions over the intermediaries’ advice and consumers’ choices. In this setting, the effect of a change in fees on the equilibrium prices and quantities is nontrivial: Under unbiased advice, the reduction of the commission for one product induces consumers to purchase this product more frequently. Under biased advice, however, intermediaries will recommend this product less often and may reduce the demand for it. The model sheds light on the analysis of the retirement market in Chile, where the annuitization rate decreased after a reform that lowered commissions paid by retirees who choose an annuity.
In several industries, the complexity of products forces customers to rely on intermediary’s advice to make informed decisions. Many people seek for financial advice to make investment decisions, choose among insurance policies, or plan their retirement, while they ask for medical advice when they face a health issue, and for legal advice when they deal with a legal problem. While expert advice may be convenient in most cases, the possibility that conflicts of interest cause biased advice is a concern when the adviser’s payoff depends on the alternative chosen by the customer. Inderst and Ottaviani (2009) and Inderst and Ottaviani (2012b) analyze how different compensation schemes induce misselling from financial advisers and sales agents in general, and they study the effect of various policy interventions usually discussed in the literature. The existing empirical evidence suggests that the likelihood of using an intermediary and the impact of the advice received are correlated with the customer’s wealth and financial literacy. Additionally, the literature that studies the extent and policy relevance of supplyinduced demand in health economics is vast and controversial (see, for instance, Rossiter and Wilensky (1984) and Rossiter and Wilensky (1987)). Intermediary’s advice may affect market outcomes not only through its direct effect on customers’ choices but also through the competition among firms. Inderst and Ottaviani (2012a) analyzes an industry where competition occurs only through intermediary’s commissions: firms set commissions to intermediaries taking into account its effect over the advice received by customers ∗ †
Universidad de Los Andes, Chile.
[email protected] Pontificia Universidad Católica de Chile.
[email protected]
1
and the choices they make. They find that regulations created to protect consumers may have unintended consequences on efficiency in equilibrium. What we learn from this literature is that policy interventions may induce different firms’ reactions, affecting equilibrium prices, market shares and consumers’ welfare in nontrivial ways. Even if commissions are set exogenously, they can still affect the reaction of firms and equilibrium prices and behavior. We analyze a regulated industry where commissions are paid by customers and set exogenously, either by a regulator or association. Firms determine prices considering the level of commissions and its impact on the intermediary’s advice and consumers’ choices. In this setting, the effect of a change in commissions in the equilibrium prices and quantities is also nontrivial. On the one hand, when customers must choose between two products with different commissions, a reduction in the commission paid for one of them makes this product relatively cheaper and more attractive for the customer. On the other hand, the reduction in the commission makes this product less profitable and less attractive for the intermediary. If prices were constant, the final effect would depend on the fraction of the population that follows the intermediary’s advice in equilibrium. Customers realize that advice may be biased, and therefore they may follow the intermediary’s advice less frequently when the incentive to recommend one firm increases: there are limits for rational persuasion (see Kamenica and Gentzkow (2011) and Benoît and Dubra (2011)). Additionally, firms will react to the reduction in the commission and the shift in demand changing their prices; hence, the final effect is ambiguous. We consider a continuum of customers who must choose between two different products with productindividual specific match unknown: the value for a customer of choosing one product may be high or low depending on the customer’s type (where the type is a state variable unobserved by customers, firms, and intermediaries). Each customer has private information about her type, which determines her prior belief. Each intermediary cares about his commissions but also about the suitability of the product for his client–for example, because of genuine concerns or future recommendations. The intermediary also receives private information about the customer’s type and sends her a message in the form of advice; hence, the customer may learn about her type upon observing the advice received from the intermediary. We show that customers with prior beliefs within an interval follow the advice in equilibrium: they choose the product recommended by the intermediaries. In contrast, customers with extreme priors (outside the bounds of the interval) do not follow advice. The reason is that a customer with an extreme prior is almost convinced that one product fits her needs better than the other; a message recommending her to purchase another product is not enough to convince her to switch. Moreover, the lower and upper bounds of this interval depend on the price and commission differentials between products. The effect of a cut in a commission for one product depends on how this reduction affects these bounds, changing the likelihood of following advice, in addition to the effect on the intermediary’s advice and firm prices. We illustrate the use of the model by analyzing the retirement market in Chile. The market for pension products is a highly regulated market, where customers (retirees) must choose among two basic types of products: a programmed withdrawal or an annuity. Customers may receive the advice of an intermediary in exchange for a commission (which is determined as a fraction of the retiree’s pension fund). We analyze the consequences of a reform that changed the commissions paid for both products (reducing the commissions for annuities in relative terms) and introduced a cap in the year 2008. We show that the decline in the market share of annuities after the reform
2
is consistent with biased advice from intermediaries: even though annuities became less expensive for customers, it dampened the incentive for intermediaries to recommend this product.
1
The Model
We consider a market with two firms, A and B, and a continuum of customers and intermediaries. Firm n ∈ {A, B} provides product n charging a price pn . Each customer (she) must select a single product. She hires an intermediary (he) who provides advice regarding product selection and charges an exogenous and publicly observed commission fn , according to the product selected. The type of the customer is θ ∈ {A, B}. As in Inderst and Ottaviani (2012a) and Inderst and Ottaviani (2012b) we define that product n is suitable for the customer if n = θ; hence type θ summarizes all customer’s characteristics and circumstances that affect product suitability for her. The type θ for each customer is unobservable to everyone, but the customer has a prior belief about it: even though she might be aware of some determinants of her type, she is not able to learn in advance (or fully understand) which product is suitable for her with certainty. Each intermediary may be concerned about the suitability of the product selected by his client and his monetary payoff fn . We assume that the intermediary cannot observe the customer’s prior belief. Nonetheless, he observes a private signal s about her type θ, and after updating his prior belief he sends a message m; the message (advice) consists on a recommendation to buy product A or product B. The probability of sending a message with a recommendation for product n instead of product n0 may depend on the difference between fn and fn0 ; we define ∆f ≡ fA − fB as the commission differential. Definition 1 (Biased advice). Advice is biased if the distribution of messages depends on the level of commissions fA and fB in equilibrium. The customer is aware of the potential bias of the message, but she still may want to follow the recommendation under certain circumstances. Firms are also aware of this potential bias and its consequences over customers’ choices, and take this information into account when setting prices pA and pB .
Payoffs and timing. The customer’s utility with product n (gross of intermediary’s commission and firm’s prices) is ( vh if n = θ Vn (θ) ≡ vl if n 6= θ, where ∆v ≡ vh − vl > 0. The adviser’s concern about the suitability of the product for the customer is modeled similarly: (
Wn (θ) ≡
wh if n = θ wl if n 6= θ,
where ∆w ≡ wh − wl > 0.
3
After meeting his client, the intermediary observes a private signal s about her type; he updates his belief to a posterior q = Pr(As) and sends message m. We will denote by mA the message that recommends product A and mB that of B.1 The expected utility of the adviser conditional on s if the customer chooses product n is given by wn (s) =
X
Pr(θs)Wn (θ) + λfn ,
(1)
θ∈{A,B}
where λ ∈ {0, 1} captures the intermediary’s concern about his own monetary payoff. Let q0 denote the prior probability of type A for the customer (i.e., the customer’s belief about her own type, conditional on her prior information). After observing the message m the customer updates her prior q0 to Pr(Am, q0 ). The customer’s net expected utility with product n conditional on m and q0 is given by un (m, q0 ) =
X
Pr(θm, q0 )Vn (θ) − fn − pn .
(2)
θ∈{A,B}
When the message is uninformative, the customers’ posterior probability is equal to her prior; i.e., in this case Pr(Am, q0 ) = q0 . We assume for simplicity that the population distribution of q0 is uniform in [0, 1]. Even though the adviser cannot observe q0 , he knows the population distribution; his prior belief is therefore Pr(θ = A) = 0.5. Finally, the payoff of firm n if the customer purchases its product is simply the price pn minus the unitary cost cn (and zero otherwise), where the cost differential ∆c ≡ cA − cB is possibly different from zero. Summarizing, the timeline of the game is as follows: • both firms set their prices pn simultaneously (pricing game) • each intermediary observes firms’ prices and the private signal s about the type for his customer, and after updating his beliefs he sends a message m • each customer observes commissions fn , prices pn and the message m, and after updating her beliefs she chooses a product n ∈ {A, B}
2
Equilibrium
An equilibrium is a set of firm choices (prices pA and pB ), intermediary strategies (a message policy m(s) conditional on the private signal), customer strategies (a product selection policy Pr(nm, q0 ) conditional on the message m and the prior q0 ), and beliefs (Pr(θs) for the intermediaries and Pr(θm, q0 ) for customers) such that: • given intermediary and customer strategies, pn is optimal for each firm n ∈ {A, B} 1
As commissions are exogenous there is no role for competition among intermediaries in the model. Hence, we could also assume that there is a unique intermediary who receives a signal for each customer and sends personalized messages to all of them accordingly.
4
• given customer strategies and prices (pA , pB ), the intermediaries’ message policy m(s) is optimal for any feasible signal s (according to Pr(θs)), • given prices (pA , pB ), the customers’ product selection policy Pr(nm, q0 ) is optimal for any feasible message m and prior q0 ∈ [0, 1] (according to Pr(θm, q0 )), • Pr(θs) is derived by Bayes’ rule, • Pr(θm, q0 ) is derived by Bayes’ rule given the message policy m(s) whenever possible. In an uninformative equilibrium customer choices depend only on their prior beliefs, in addition to price and commission differentials. In an informative equilibrium, in contrast, customers update their prior beliefs about their type upon observation of the personalized message m according to Bayes rule given the adviser’s equilibrium strategy m(s).
Customer choices: Each customer observes the personalized message m chosen by the adviser given his private signal s about the customer’s type θ. The customer neither observes the type, nor the signal, but the message could be informative about s in equilibrium: Even though the customer realizes that the message may be biased, she is still able to learn something about s upon the observation of m. A customer with prior q0 chooses product A after message mA if uA (mA , q0 ) ≥ uB (mA , q0 ) (and B otherwise), and she chooses message B after message mB if uA (mB , q0 ) ≤ uB (mB , q0 ) (and A otherwise). Because Pr(Am, q0 ) ∈ [0, 1] for any q0 , the price differential ∆p ≡ pA − pB must satisfy ∆p ∈ [−∆v − ∆f, ∆v − ∆f ] for an interior solution, otherwise all customers would choose A or B regardless of advice. Given these rules, there are three cutoff levels for the prior q0 that define the equilibrium policy functions. Let us define
q0∗
=
1 1 2
if ∆p > ∆v − ∆f ∆v+∆f +∆p ∆v
if ∆p ∈ [−∆v − ∆f, ∆v − ∆f ]
(3)
if ∆p < ∆v + −∆f
0
while q0 and q0 denote the thresholds that after each message solve uA (mA , q0 ) = uB (mA , q0 )
(4)
uA (mB , q0 ) = uB (mB , q0 ).
(5)
if ∆p ∈ [−∆v − ∆f, ∆v − ∆f ]. In other words, q0 defines the cutoff such that the customer chooses A after message mA if q0 ≥ q0 . Similarly, q0 is the cutoff such that the customer chooses B after message mB if q0 ≤ q0 . If the message is informative and ∆p ∈ [−∆v − ∆f, ∆v − ∆f ], they are defined as: Pr(mA B)q0∗ Pr(mA B)q0∗ + Pr(mA A)(1 − q0∗ ) Pr(mB B)q0∗ q0 = Pr(mB B)q0∗ + Pr(mB A)(1 − q0∗ ) q0 =
otherwise, they are defined as q0 = q0 = q0∗ .
5
≤ q0∗
(6)
≥ q0∗
(7)
Intermediary choices: The adviser updates his prior belief upon the observation of a private signal s about his client’s type θ, with distribution Pr(sθ); the posterior Pr(As) = q˜(s) is given by Bayes rule. If advice is followed in equilibrium, the intermediary recommends A if wA (s) > wB (s) and this happens when q˜(s) > q ∗ , where the cutoff is given by:2 ∆f 1 1−λ . q = 2 ∆w ∗
(8)
The commission differential must satisfy λ∆f ∈ [−∆w, ∆w], otherwise the intermediary sends always the same message regardless of the signal, and advice is not informative. We will restrict attention to values of ∆f and ∆w such that this condition always holds in what follows. Notice that for those levels of q0 that the advice is not followed in equilibrium the message becomes uninformative, and hence we can assume without loss of generality that the intermediary’s strategy is ( mA if q˜(s) > q ∗ , m(s) = (9) mB if q˜(s) ≤ q ∗ .
Firm choices:
Firm’s n expected payoff is Πn = (pn − cn ) Pr(n),
where Pr(n) denotes either the unconditional probability that the customer chooses its product or the market share of firm n. From the firstorder condition for an interior solution we obtain firmn’s best response p∗n (pn0 ), that satisfies:3 ∂ Pr(n) −1 ∗ pn (pn0 ) − cn + Pr(n) = 0, (10) ∂pn where Pr(n) depends on prices pn and pn0 , and on parameters ∆v, ∆w and ∆f .
2.1
Uninformative equilibrium
When the message is uniformative Pr(Am, q0 ) = q0 and therefore uA (m, q0∗ ) = uB (m, q0∗ ) for any message m. Hence, q0 = q0 = q0∗ . Figure 1 shows the expected utility without any message; as the expected utility with product A is increasing in q0 while the expected utility with product B decreases with q0 , customers with q0 > q0∗ choose product A, while those with q0 < q0∗ choose B. As the distribution of prior beliefs is uniform in [0, 1], the probability that a given customer chooses B is: Pr(n = B) = 1 − Pr(n = A) = q0∗ . 2
As the intermediary does not observe the customer’s prior belief, the cutoff does not depend on q0 . Notice that the firm’s n expected profit is pn −cn when Pr(n) = 1 (while its profit is zero when Pr(n) = 0). Hence, the unrestricted maximization of ΠA is equivalent to maximizing ΠA subject to pA ∈ [pB − ∆v − ∆f, pB + ∆v − ∆f ] (the optimum is always attained within that set). If cA > pB + ∆v − ∆f , then p∗A = pB + ∆v − ∆f is indeed a best response from firm A; otherwise, p∗A ∈ [pB − ∆v − ∆f, pB + ∆v − ∆f ). Similarly, the unrestricted maximization of ΠB is equivalent to maximizing ΠB subject to pB ∈ [pA − ∆v + ∆f, pA + ∆v + ∆f ], and p∗B = pA + ∆v + ∆f is a best response from firm B only if cB > pA + ∆v + ∆f . 3
6
𝑢𝑛
𝑢𝑛 𝑢𝐵
𝑢𝐴
𝑢𝐴 𝑚𝑎 , 𝑞
𝑢𝐵 𝑚𝑏 , 𝑞0
𝑢𝐵 𝑚𝑎 , 𝑞0
𝑞0 *
1
𝑢
𝑞0
Figure 1: Expected utility with product A and B without any message.
always B
It follows that firms’ best response functions are:
𝑞0
𝑞0 * A if 𝑚𝑎 , B if 𝑚𝑏
1 (cA + ∆v − ∆f + pB ) , 2 1 p∗B = (cB + ∆v + ∆f + pA ) , 2 in an interior solution (i.e., when ∆p ∈ [−∆v − ∆f, ∆v − ∆f ]). p∗A =
As best response functions are increasing (i.e., prices are strategic complements, as in Hotelling’s ∂pn < 1 in the interior solution, there is a unique Nash equilibrium (p∗A , p∗B ) in the model) and ∂p n0 pricing game; hence, the uninformative equilibrium is also unique.4 Proposition 1. [Uninformative equilibrium] The informative equilibrium always exists. If there is an uninformative equilibrium with two active firms, it is the unique uninformative equilibrium. In this equilibrium firms set prices (p∗A , p∗B ) in a first stage, and if the customer’s prior q0 is lower than q0∗ he chooses firm B, while he chooses firm A otherwise. If both firms are active, equilibrium prices are ∆c + ∆f , 3 ∆c + ∆f p∗B = cB + ∆v + . 3
p∗A = cA + ∆v −
1 Then, q0∗ = 2∆v ∆v + ∆c+∆f . Notice that q0∗ ∈ (0, 1) only if −∆v < 3 expected payoffs (if both are active in equilibrium) are:
∆c+∆f 3
< ∆v. Firms’
1 ∆c + ∆f 2 ∆v − , = 2∆v 3 1 ∆c + ∆f 2 ∗ πB = ∆v + . 2∆v 3
∗ πA
4
This result can be generalized to any distribution of prior beliefs G0 with density g0 (q0 ) > 0 over [0, 1] such that g0 (q0 ) g0 (q0 ) the hazard rate 1−G is increasing and the reverse hazard G is decreasing. 0 (q0 ) 0 (q0 )
7
𝑞0
a
If ∆c+∆f = 0, then p∗n = cn +∆v. Therefore, q0∗ = 0.5 and both firms obtain the same expected payoff πn = 0.5∆v.5 We can think of this case as “equal advantage” between the firms. Instead, if ∆c+∆f > 0, in contrast, q0∗ > 0.5 (firm B has an advantage considering costs and commissions, and attracts more customers in equilibrium) even though ∆p < ∆c; similarly, q0∗ < 0.5 if ∆c + ∆f < 0 even though ∆p > ∆c. Intermediation does not play an informative role in this equilibrium, but it affects customer choices and firm strategies: as ∆f increases the expected payoff of firm A decreases and that of firm B increases in the uninformative equilibrium with two active firms. Eliminating intermediaries would benefit the firm with the highest commission (i.e., it would benefit firm A if ∆f > 0).
2.2
Informative Equilibrium
The informative equilibrium is characterized by a set Q0 ⊂ [0, 1] such that only customers with q0 within that set follow the intermediary’s advice in equilibrium. We assume that the distributions of intermediary’s signals Pr(sA), Pr(sB) satisfy the monotone Pr(sA) likelihood ratio property (MLRP), where the likelihood ratio is defined as r(s) ≡ Pr(sB) under a discrete signal and as r(s) ≡ ggBA (s) (s) under a continuous signal with a density gθ (s) conditional on type θ ∈ {A, B}. From Bayes rule, the intermediary’s posterior belief q˜(s) ≡ Pr(As) can be written in terms of the likelihood ratio, q˜(s) ≡
r(s) Pr(A) . r(s) Pr(A) + Pr(B)
(11)
We assume (without loss of generality) that r(s) is monotonically increasing. In other words, a “higher signal” means a signal that is more likely to be observed under type A. This in turn implies that q˜(s) is increasing. The customer’s posterior belief Pr(Am, q0 ) is defined as: Pr(Am, q0 ) =
Pr(mA)q0 , Pr(mA)q0 + Pr(mB)(1 − q0 )
(12)
where Pr(Am, q0 ) is increasing in q0 . The MLRP implies that Pr(AmA , q0 ) ≥ q0 ≥ Pr(AmB , q0 ). It follows that Pr(AmA , 0) = Pr(AmB , 0) = 0 and Pr(AmA , 1) = Pr(AmB , 1) = 1. In other words, when the customer is convinced about her type, the message becomes irrelevant as her posterior belief is equal to the prior, and hence she doesn’t follow the advice in equilibrium. The following lemma shows that, indeed, it is not possible that messages are followed in equilibrium for all q0 ∈ [0, 1]. 5 Moreover, under symmetric costs (i.e., ∆c = 0) if ∆f  < ∆v both firms are active in the (unique) uninformative equilibrium. In other words, equilibrium prices satisfy p∗A − p∗B ∈ (−∆v − ∆f, ∆v − ∆f ) and hence q0∗ ∈ (0, 1). Indeed, p∗A = pB + ∆v − ∆f is a best response from firm A only if c ≥ pB + ∆v − ∆f , which in turn implies that pB ≤ c − ∆v + ∆f < c. Similarly, p∗B = pA + ∆v + ∆f is a best response from firm B only if c ≥ pA + ∆v + ∆f , meaning that pA ≤ c − ∆v − ∆f < 0. In contrast, only one firm is active if the cost differential or the commission differential is too large. For instance, if ∆f < −∆v (i.e., A is the firm with the lowest commission), then there is an uninformative equilibrium where p∗B = c and p∗A = c − ∆v − ∆f > c under symmetric costs.
8
Lemma 1. In any informational equilibrium advice is followed only by customers whose priors are in a subset Q0 ( [0, 1]. See the Appendix for proofs. Figure 2 shows the expected utilities with products A and B under message m ∈ {mA , mB }. Notice that a higher probability of type A–either the prior or posterior belief–increases uA (m, q0 ) and reduces uB (m, q0 ). Therefore, the gain from choosing A after any message increases with q0 . In addition, as Pr(AmA , q0 ) ≥ q0 ≥ Pr(AmB , q0 ), the gain from choosing A after message mA is higher than after message mB (and for B the relationship is the opposite).
𝑢𝑛
𝑢𝐵
𝑢𝐴
𝑢𝐴 𝑚𝑎 , 𝑞0
𝑢𝐵 𝑚𝑏 , 𝑞0
𝑢𝐵 𝑚𝑎 , 𝑞0
𝑞0 *
1
𝑢𝐴 𝑚𝑏 , 𝑞0
𝑞0 𝑞0
𝑞0
𝑞0 * A if 𝑚𝑎 , B if 𝑚𝑏
always B
1
𝑞0
always A
Figure 2: Expected utility with product A and B conditional on message m ∈ {mA , mB }. By definition from the previous section, a customer with q0 = q0 is indifferent between choosing A or B after receiving message mA ; similarly, a customer with q0 = q0 is indifferent between choosing A or B after receiving message mB . As the gain from choosing A increases with q0 , it follows that a customer with q0 > q0 strictly prefers A over B after message mA . Similarly, q0 < q0 implies that she strictly prefers B over A after message mB . In other words, a customer with prior q0 ∈ [q0 , q0 ] always follows the intermediary’s advice. On the other hand, if q0 < q0 the customer prefers B after any message, and if q0 > q0 she prefers A after any message. In other words, Q0 = [q0 , q0 ] and the product selection policy Pr(nm, q0 ) can be summarized as: (
Pr(n = Bm, q0 ) =
1 if q0 < q0 or (q0 ∈ [q0 , q0 ] and m = mB ), 0 if q0 > q0 or (q0 ∈ [q0 , q0 ] and m = mA ).
(13)
In the pricing game of the first stage, given the customer policy (conditional on m and q0 ) described in equation (13), the unconditional probability that the customer chooses firm B–or B’s market share–is Pr(n = B) = Pr(q0 < q0 ) + Pr(mB ) Pr(q0 ∈ [q0 , q0 ]). 9
As the distribution of prior beliefs is uniform, we obtain Pr(n = B) = q0 Pr(mB ) + q0 Pr(mA ).
(14)
Market shares depend on the price and commission differentials ∆p and ∆f through q0 and q0 : when product n becomes less expensive relative to n0 , the fraction of customers q0 who always choose n increases, while the fraction (1 − q0 ) who always choose n0 decreases. This intuition is formalized in the following lemma: Lemma 2. As q0 and q0 are increasing in q0∗ , they are increasing in the price and commission differentials, ∆p and ∆f . Given the intermediary policy described in equation (9), the unconditional probability of message mA is Pr(mA ) = 0.5(Pr(mA) + Pr(mB)). Notice that the probability Pr(mA ) does not depend on prices (pA , pB ), and it only depends on q ∗ (and hence, on the commission differential ∆f ). Let H0 : [0, 1] −→ [0, 1] denote Pr(n = B) as a function of the cutoff q0∗ : H0 (x) ≡ q0 (x) Pr(mB ) + q0 (x) Pr(mA ),
(15)
where H0 (x) is a well defined distribution function with density h0 (x). Let us define the hazard ∂q0∗ h0 (x) h0 (x) = rate HR(x) ≡ 1−H and the reverse hazard RH(x) ≡ . Using equation (10) and ∂p (x) H (x) 0 0 A ∂q ∗
− ∂pB0 =
1 2∆v
we conclude that best response functions in the interior solution must satisfy p∗A (pB ) − cA −
2∆v HR(q0∗ )
= 0,
(16)
p∗B (pA ) − cB −
2∆v RH(q0∗ )
= 0.
(17)
Using the implicit function theorem, the slope of the bestresponse p∗n (pn0 ) is obtained as ∂p∗A = ∂pB ∂p∗B = ∂pA
∗) HR0 (q0 HR(q ∗ )2 0 HR0 (q ∗ ) 0 1+ HR(q ∗ )2 0 ∗) −RH 0 (q0 RH(q ∗ )2 0 RH 0 (q ∗ ) 0 1− RH(q ∗ )2 0
=
HR0 (q0∗ ) , HR(q0∗ )2 +HR0 (q0∗ )
(18)
=
−RH 0 (q0∗ ) , RH(q0∗ )2 −RH 0 (q0∗ )
(19)
where the denominator is always positive in both cases (from the second order conditions for the firms’ problem). It follows that the slope of the best response function is less than one for both firms. Hence, if there exists a Nash equilibrium of the pricing game with ∆p ∈ (−∆v − ∆f, ∆v − ∆f ), this equilibrium is unique, and therefore the informative equilibrium is also unique. Proposition 2. [Informative equilibrium] If there exists an informative equilibrium with two active firms, it is unique. In this equilibrium firms set prices (p∗A , p∗B ) in a first stage. For each customer the intermediary observes a private signal s about his type θ, and he recommends him to purchase product A whenever his posterior belief q˜(s) is above q ∗ . If the customer’s prior belief is below q0 he chooses firm B, while he chooses firm A if his prior is above q0 ; only if the prior is in the interval [q0 , q0 ] he follows intermediary’s advice in equilibrium. 10
Moreover, if there exists an informative equilibrium with two active firms, then at least one of the best response functions has a positive slope at the equilibrium price levels p∗A and p∗B . Indeed, HR0 (q0∗ ) =
h00 (q0∗ )(1 − H0 (q0∗ )) + h0 (q0∗ )2 (1 − H0 (q0∗ ))2
(20)
h00 (q0∗ )H0 (q0∗ ) − h0 (q0∗ )2 . H0 (q0∗ )2
(21)
and RH 0 (q0∗ ) =
Then, it is not possible to obtain HR0 (q0∗ ) < 0 and RH 0 (q0∗ ) > 0 simultaneously. It follows that (using the second order conditions) 1+
HR0 (q0∗ ) RH 0 (q0∗ ) − >0 HR(q0∗ )2 RH(q0∗ )2
(22)
in any informative equilibrium with two active firms. Under symmetric costs (i.e., cA = cB ≡ c) and if ∆f  < ∆v there exists an informative equilibrium with two active firms, where equilibrium prices satisfy ∆p ∈ (−∆v − ∆f, ∆v − ∆f ) and hence 0 < q0 < q0∗ < q0 < 1. Only customers with q0 < q0 choose with certainty product B, and only those with q0 > q0 choose A; customers with q0 ∈ [q0 , q0 ] choose the product recommended by their intermediaries. In contrast, under large cost differentials or commission differentials, only an uninformative equilibrium may be possible, where ∆p ∈ / (−∆v − ∆f, ∆v − ∆f ) and all customers select the firm with the lowest cost and/or the lowest commission in equilibrium.
3
Characterizing the informative equilibrium
In the analysis that follows we focus on parameter values that sustain an informative equilibrium with two active firms. We distinguish first between scenarios where the intermediary’s signal is discrete or continuous, where advice is biased or unbiased, and then we analyze comparative statics in both cases.
3.1
Discrete or continuous signal
If the distribution of the signal for the intermediary Pr(sθ) is discrete, let us define the support s ∈ {sa , sb } with sa > sb , so that Pr(sa A) = α and Pr(sb B) = β. Then, the likelihood ratio is Pr(sA) r(s) ≡ Pr(sB) and the MLRP implies that α > 1 − β and that in equation (11), q˜(sa ) > 0.5 > q˜(sb ). If q˜(sa ) > q˜(sb ) > q ∗ , then the intermediary always recommends product A; similarly, if q˜(sb ) < q˜(sa ) < q ∗ he always recommends product B. It follows that under a discrete signal, the informative equilibrium is feasible only if q˜(sb ) < q ∗ < q˜(sa ).
(23)
We will restrict attention to values of ∆f, ∆w and λ such that this condition is always satisfied in what follows.
11
The probability of receiving message mA conditional on type θ ∈ {A, B} under a discrete signal s ∈ {sa , sb } is therefore Pr(mA θ) = Pr(sa θ). and the unconditional probability of message mA is Pr(mA ) = 0.5α + 0.5(1 − β). Notice that Pr(mA θ) and Pr(mA ) do not depend on the level of q ∗ , and therefore on the commissions differential ∆f . Instead, if the distribution of the signal for the intermediary Pr(sθ) is continuous, let us define s ∈ (0, 1) with distribution Gθ (s) and density gθ (s). Then, the likelihood ratio is r(s) ≡ ggBA (s) (s) and the MLRP implies that GA stochastically dominates GB according to the likelihood ratio order–and consequently, also according to the firstorder stochastic dominance order. The probability of receiving message mA conditional on type θ ∈ {A, B} under a continuous signal is Pr(mA θ) = 1 − Gθ (˜ s(q ∗ )), where the inverse function s˜(q) = q˜−1 denotes the signal required to obtain a posterior q; i.e., such that q˜(˜ s(q)) = q. As q˜0 > 0, then s˜ exists and s˜0 > 0. The unconditional probability of message mA is now Pr(mA ) = 0.5(1 − GA (˜ s(q ∗ )) + 0.5(1 − GB (˜ s(q ∗ ))). Notice that Pr(mA θ) and Pr(mA ) now depend on the commission differential ∆f .
3.2
Biased or unbiased advice
Given intermediary equilibrium strategies, the previous section showed that the distribution of messages does not depend on q ∗ under a discrete signal, provided that the condition q˜(sb ) < q ∗ < q˜(sa ) holds and the informative equilibrium exists. Hence, the message is unbiased if the signal is discrete in this case. ∗
∂q λ A) Under a continuous signal, ∂ Pr(m < 0. As ∂∆f = − 2∆w , the message is also unbiased if ∂q ∗ λ = 0 (i.e., if the intermediary is concerned only about product suitability). If λ = 1, in contrast, a change in fn affects the fraction of customers who receive messages mA and mB as advice.
Moreover, when customers update their beliefs after receiving advice under a continuous signal, the effect of the message m ∈ {mA , mB } depends on the level of q ∗ : ∂ Pr(AmA ,q0 ) ∂q ∗ ∂ Pr(BmB ,q0 ) ∂q ∗
= s˜0 (q ∗ ) Pr(AmA , q0 )(1 − Pr(AmA , q0 ))
gB (˜ s(q ∗ )) 1−GB (˜ s(q ∗ ))
= −˜ s0 (q ∗ ) Pr(AmB , q0 )(1 − Pr(AmB , q0 ))
gA (˜ s(q ∗ )) GA (˜ s(q ∗ ))
−
gA (˜ s(q ∗ )) 1−GA (˜ s(q ∗ ))
−
gB (˜ s(q ∗ )) GB (˜ s(q ∗ ))
,
(24) (25)
where the assumption that GA stochastically dominates GB according to the likelihood ratio order implies that GA stochastically dominates GB according to the hazard rate order and to the reverse gA (s) gB (s) gB (s) hazard rate order. Hence, 1−G < 1−G and GgAA(s) (s) > GB (s) . A (s) B (s) A ,q0 ) B ,q0 ) It follows that ∂ Pr(Am > 0 and ∂ Pr(Bm < 0. When q ∗ increases, customers realize ∂q ∗ ∂q ∗ that message mA is sent less frequently (and message mB more frequently). Then, message mA becomes more informative about type A, while message mB becomes less informative about type B. Hence, the level of q ∗ affects the expected utility of products A and B conditional on message m ∈ {mA , mB }, and also the fraction of customers that follow the intermediary’s advice.
12
3.3
Costs, commissions and equilibrium prices
If both firms are active in equilibrium, then the price differential ∆p and the cost differential ∆c must satisfy 1 1 ∆p − ∆c − 2∆v = 0. (26) − HR(q0∗ ) RH(q0∗ ) As q0∗ depends only on the price differential ∆p, not on price levels, the lefthandside of equation (26) depends on ∆p, ∆c, ∆f, ∆v and ∆w. Using equations (3), (6), (7), and (15) we can conclude that q0∗ = 0.5, q0 =
Pr(mA B) Pr(mA ) 0.5,
B B) ∗ ∗ q0 = Pr(m Pr(mB ) 0.5 and H0 (q0 ) = 1 − H0 (q0 ) = 0.5 if and only if ∆p + ∆f = 0. In other words, both firms attract the same fraction of customers if and only if the price differential fully compensates the commission differential (i.e., ∆p = −∆f ). As this means that HR(q0∗ ) = RH(q0∗ ), using equation (26) we obtain ∆p = ∆c; we conclude that ∆p = −∆f in equilibrium if and only if ∆c = −∆f .
3.4
Comparative statics
We analyze the effect of a change in the commission differential on equilibrium outcomes. A change in ∆f affects the demand for product B through two different channels: if ∆f increases, it becomes relatively cheaper for the customer to choose product B (q0∗ increases), and it becomes relatively less attractive for the intermediary to recommend product B (q ∗ is reduced). This induces firms to change their prices, and therefore ∆p changes (and it further affects the demand for product B). The final effect of ∆f on the market share of product B is ∗ ∂q ∗ ∂H0 (q0∗ ) ∂q ∗ d Pr(n = B) ∗ ∂q0 d∆p = h0 (q0∗ ) 0 + + h (q ) 0 0 d∆f ∂∆f ∂q ∗ ∂∆f ∂∆p d∆f ∂H0 (q0∗ ) λ 1 d∆p 1 − + h0 (q0∗ ) , (27) = h0 (q0∗ ) ∗ 2∆v ∂q 2∆w 2∆v d∆f ∂q ∗
∂q ∗
∗
∂q 1 λ 0 0 because ∂∆f = ∂∆p = 2∆v > 0, and ∂∆f = − 2∆w < 0. The first term (which is positive) is the direct effect of ∆f on the demand for product B. The second term accounts for the effect of ∆f over customer choices through the recommendation, which can be positive or negative: under biased advice intermediaries recommend product A more frequently when ∆f increases, but the ∂H0 (q0∗ ) fraction of customers following advice may also change. If ∂q > 0 the first and second terms ∗ in equation (27) have opposite signs. Finally, the third term adds the indirect effect of ∆f on the demand for product B through its effect on ∆p: firms’ best response functions and equilibrium prices change.
The effect of ∆f on firms’ best response functions can be obtained applying the implicit function
13
theorem to equations (16) and (17): ∂p∗A ∂∆f ∂p∗B ∂∆f
=−
1 HR(q0∗ )2
1+ =−
1 RH(q0∗ )2
∂HR(q0∗ ) λ∆v ∂q ∗ ∆w ∗ 0 HR (q0 ) HR(q0∗ )2
HR0 (q0∗ ) −
∂RH(q0∗ ) λ∆v ∂q ∗ ∆w ∗ 0 RH (q0 ) RH(q0∗ )2
RH 0 (q0∗ ) − 1−
,
(28)
.
(29)
Again, best response functions change as a result of the sum of two effects: the effect through q0∗ (which is similar to the effect of ∆p, as product A becomes relatively more expensive when ∆f increases), and the additional effect through q ∗ , as shown in the following lemma. Lemma 3. The effect of a change in ∆f on firms’ best responses is: ∂HR(q ∗ )
0 HR0 (q0∗ ) λ∆v ∂p∗A ∂q ∗ =− + , ∂∆f HR(q0∗ )2 + HR0 (q0∗ ) HR(q0∗ )2 + HR0 (q0∗ ) ∆w
(30)
∂RH(q ∗ )
0 ∂p∗B RH 0 (q0∗ ) λ∆v ∂q ∗ =− . + ∂∆f RH(q0∗ )2 − RH 0 (q0∗ ) RH(q0∗ )2 − RH 0 (q0∗ ) ∆w
(31)
∂p∗
The first term in equation (30) is equal to − ∂pBA , while the first term in equation (31) is equal
to
∂p∗B ∂pA .
On the other hand, the effect of q ∗ on the hazard rate is ∂HR(q0∗ ) = ∂q ∗
∂h0 (q0∗ ) ∂q ∗ (1
− H0 (q0∗ )) + h0 (q0∗ )
∂H0 (q0∗ ) ∂q ∗
(1 − H0 (q0∗ ))2
,
(32)
while the effect on the reverse hazard is ∂RH(q0∗ ) = ∂q ∗
∂h0 (q0∗ ) ∗ ∂q ∗ H0 (q0 )
− h0 (q0∗ )
(H0 (q0∗ ))2
∂H0 (q0∗ ) ∂q ∗
.
(33)
The reduction of q ∗ induces intermediaries to recommend product A more frequently under biased advice, affecting the belief updating process of customers and also the fraction of them who follow ∂H0 (q0∗ ) < 0), this in turn induces firm advice. If the demand for product A is reduced (i.e., if ∂q ∗ A to reduce pA and firm B to increase pB . In contrast, if the demand for product B is reduced ∂H0 (q0∗ ) (i.e., if ∂q > 0), this induces firm B to reduce pB and firm A to increase pA . Additionally, ∗ the reduction in q ∗ also affects h0 (q0∗ ), which measures the responsiveness of customers to price ∂h0 (q0∗ ) is positive (negative), both firms are induced to increase and commission differentials. If ∂q ∗ (reduce) prices. In equilibrium, pn changes as a result of the effect of ∆f on the bestresponse function for firm n, and also of the effect pn0 . The final effect can be written as:
∗
∂p∗
∗
∂pn ∂pn n0 dpn ∂pn0 ∂∆f + ∂∆f = ∗ ∂p∗ d∆f 1 − ∂pn n0 ∂pn0 ∂pn
14
The change in ∆p is
dpA d∆f
−
dpB d∆f .
Replacing an rearranging we thus obtain:6
RH 0 (q0∗ ) RH(q0∗ )2 0 HR0 (q0∗ ) RH 0 (q ∗ ) − RH(q∗0)2 HR(q0∗ )2 0
HR0 (q ∗ )
− HR(q∗0)2 +
d∆p = d∆f 1+
∂HR(q0∗ ) ∂RH(q0∗ ) 1 1 − ∗ 2 2 ∗ ∗ ∂q ∂q ∗ RH(q0 ) HR(q0 ) HR0 (q0∗ ) RH 0 (q0∗ ) 1 + HR(q∗ )2 − RH(q∗ )2 0 0
+
λ∆v , ∆w
(34)
and ∂HR(q0∗ ) ∂RH(q0∗ ) 1 1 − RH(q ∗ ∗ )2 ∗ )2 ∂q ∂q ∗ HR(q 0 0 HR0 (q ∗ ) RH 0 (q ∗ ) 1 + HR(q∗0)2 − RH(q∗0)2 0 0
d∆p + d∆f = d∆f 1+
1 HR0 (q0∗ ) HR(q0∗ )2
−
RH 0 (q0∗ ) RH(q0∗ )2
+
λ∆v ,
∆w
(35)
where the second term in equation (34) and equation (35) can be written as ∂HR(q0∗ ) ∂RH(q0∗ ) 1 1 − ∗ 2 2 ∗ ∗ ∂q ∂q ∗ RH(q0 ) HR(q0 ) ∗ ∗ 0 0 RH (q0 ) HR (q0 ) 1 + HR(q∗ )2 − RH(q∗ )2 0 0
λ∆v ∆w
=
∂h0 (q0∗ ) ∂q ∗
∂H0 (q0∗ ) 1 RH(q0∗ ) + 2 ∂q ∗ RH 0 (q0∗ ) HR0 (q0∗ ) 2 − ∗ HR(q0 ) RH(q0∗ )2
1 HR(q0∗ )
1+
−
1 h0 (q0∗ )
λ∆v . (36) ∆w
Using equation (22), the first term in equation (35) is positive. Then, when the second term is zero (i.e., when ∆f affects only through q0∗ , not through q ∗ ), ∆p + ∆f increases after an increase in ∆f in equilibrium. It follows that Pr(n = B) = H0 (q0∗ ) increases in this case. This happens under unbiased advice, and may also happen under biased advice if ∆c + ∆f = 0: H0 (q0∗ ) = q0∗ = 0.5 for ∂H0 (q0∗ ) any level of q ∗ , and thus HR(q0∗ ) = RH(q0∗ ) and ∂q = 0. Equation (36) implies that the second ∗ 0 term in equations (34) and (35) disappears in this case, and therefore ∆p + ∆f and Pr(n = B) increase. In general, however, the final effect may be positive or negative.
3.4.1
Unbiased advice
Under unbiased advice ∆f does not affect the intermediaries’ recommendation (either because ∂ 2 H (q ∗ ) ∂H0 (q0∗ ) B) = 0, implying that ∂q∗0∂q∗0 = ∂q = 0). Hence, ∆f affect firms’ best λ = 0 or because ∂ Pr(m ∗ ∂q ∗ 0 response functions only through its direct effect on customers’ choices. It follows that
∂p∗
1 − ∂pBA dpA ∂p∗ , =− A ∗ ∗ d∆f ∂pB 1 − ∂pA ∂pB ∂pB ∂pA and
∂p∗
1 − ∂pBA dpB ∂p∗ . = B ∗ ∗ d∆f ∂pA 1 − ∂pA ∂pB ∂pB ∂pA If the slope of the best response function is positive at the equilibrium price levels, then p∗A increases and p∗B is reduced. But in any case ∆p + ∆f increases, and therefore the final effect over q0∗ and over H0 (q0∗ ) is still positive. We summarize this result in the following lemma. 6
The effect on ∆p can also be obtained by applying the implicit function theorem in equation (26).
15
∂p∗
∂p∗
A Lemma 4. Under unbiased advice ∂∆f = − ∂pBA and and therefore Pr(n = B) increases.
∂p∗B ∂∆f
=
∂p∗B ∂pA .
In equilibrium ∆p+∆f increases,
This finding implies that in a setting where a change in ∆f only affects customers’ choices through commissions and equilibrium prices, and not through the intermediary recommendation, an increase in ∆f should imply an increase in firm’s B market share: product B becomes relatively less expensive and hence more customers choose this product instead of product A.
3.4.2
Biased advice
If we consider a continuous signal s ∈ (0, 1) and λ = 1, however, the effect of an increase in ∆f over firms’ best responses includes the additional effect of the reduction in q ∗ : the intermediary provides biased advice and thus recommends firm A more frequently when ∆f increases, affecting customers’ B) > 0. choices. The direct effect of such change in q ∗ on B’s market share is (q0 − q0 ) ∂ Pr(m ∂q ∗ But q ∗ also affects the belief updating process of customers and the fraction of them who follow advice. Taking the derivative of the cutoffs points defined in equations (6) and (7) we get: ∂q0 gB (˜ s(q ∗ )) gA (˜ s(q ∗ )) 0 ∗ = s ˜ (q )q (1 − q ) − 0 0 ∂q ∗ GB (˜ s(q ∗ )) GA (˜ s(q ∗ ))
and
< 0,
∂q0 gA (˜ s(q ∗ )) gB (˜ s(q ∗ )) 0 ∗ = −˜ s (q )q − (1 − q ) 0 0 ∂q ∗ 1 − GB (˜ s(q ∗ )) 1 − GA (˜ s(q ∗ ))
The derivatives
∂q0 ∂q ∗
and
∂q0 ∂q ∗
(37)
< 0.
(38)
are negative because GA stochastically dominates GB according to
the hazard rate order and to the reverse hazard rate order (implying that gA (s) GA (s)
gB (s) gA (s) 1−GA (s) < 1−GB (s) and q ∗ decreases and inter
> GgBB(s) (s) ). Customers realize that after an increase in ∆f the cutoff mediaries recommend product A more frequently, and they take this into account when updating A ,q0 ) beliefs. Indeed, Pr(AmA , q0 ) decreases (because ∂ Pr(Am > 0); therefore, the gain from choos∂q ∗ ing product A after message mA decreases. Similarly, as product B is recommended less frequently B ,q0 ) Pr(BmB , q0 ) increases (because ∂ Pr(Bm < 0) and the gain from choosing product B after ∂q ∗ message mB increases. As a consequence, given prices pA and pB , when ∆f increases the fraction of customers who always select product A decreases, and the fraction of them who always choose B increases. It follows that the sign of
∂H0 (q0∗ ) ∂q ∗
is ambiguous:
∂q0 ∂H0 (q0∗ ) ∂ Pr(mB ) ∂q0 + = (q − q ) Pr(m ) + Pr(mA ) . 0 0 B ∂q ∗ ∂q ∗ ∂q ∗ ∂q ∗ 
{z
>0
}

{z
0; therefore, q0∗ > 0.5 in the initial equilibrium, as expected. Figure 5 shows the best response functions and equilibrium prices under the initial differential ∂H0 (q0∗ ) > 0, the effect of the increase in commission ∆f 0 = 1 and the final one, ∆f 1 = 1.2. As ∂q ∗ q0∗ and the effect of the reduction in q ∗ on firms’ best responses have opposite signs. Moreover, ∂h0 (q0∗ ) as ∂q > 0 both firms are induced to increase prices due to the effect of ∆f on customers’ ∗ responsiveness to price and commission differentials. The final effect on prices is positive in both cases: both p∗A and p∗B increase after the change in ∆f . Figures 6a and 6b show the expected utility for a customer with prior q0 with products A and B in the initial and the final equilibrium respectively, given message m ∈ {mA , mB }. Both p∗A + fA and p∗B + fB increase; then, the unconditional expected utility (i.e., the expected utility without any message, continuous line) decreases for product A and for B. After the change in ∆f both q ∗ and Pr(mB ) decrease, and message mB becomes more informative about type B. Hence, the effect of conditioning on mB on the expected utility (with respect to the unconditional expected utility) increases both for product A and B. In contrast, message mA becomes less informative and therefore the effect of conditioning on mA decreases. 19
Figure 5: Best response fuctions and equilibrium, example 2
(a) Initial equilibrium
(b) Final equilibrium
Figure 6: Expected utility with products A and B, example 2
20
Even though the fraction of customers who follow advice (q0 − q0 ) increases, as intermediaries send message B less frequently the fraction of customers who choose product B following advice is reduced after the change in ∆f . The fraction of customers who always choose product B increases, but the first effect dominates. Then, the market share for product B is reduced in equilibrium.
4
Application: The Chilean market for pension products
We apply the model to the retirement market in Chile. In this highly regulated market, retirees must choose between two retirement products: a programmed withdrawal–product A–or an annuity– product B. Those products depend on two different sets of firms–retirement funds administrators (AFP, Administradoras de Fondos de Pension) and insurance companies respectively. Both types of firms are independent of each other, face different regulation, and depend on separate regulatory agencies. Even though most countries show low annuitization rates, the market in Chile is relatively large: in our data, 38.6% of retirees choose immediate annuities. Retirees must enter a mandatory and centralized retirement process to receive quotes for both products from the different firms, and must necessarily choose one of the quotes to retire.7 Retirees can enter this process either independently, with a sales agent from an insurance company, or with a certified financial adviser. Sales agents and financial advisers receive different commissions for the products chosen and thus will face different incentives according to the model. Indeed, some policymakers believe that the considerable annuitization rate in Chile is explained–among other things–by biased advice from intermediaries, as they obtain a higher payoff when their clients choose an annuity instead of a programmed withdrawal. A reform changed the commissions level in 2008. Before, both intermediary types would receive 2.5% of the retiree’s retirement fund if she chooses an annuity (the sales agent would only get paid if the annuity is from his firm, the adviser in any event) and no commission if the retiree chooses a programmed withdrawal. After the reform, the commission for annuities decreased to 2% of the retiree’s retirement fund, with a cap of approximately US$2,500. This change took place starting November 2008. Additionally, the reform introduced a commission for advisers if the retiree chooses a programmed withdrawal, of 1.2% of the retiree’s fund, with a cap of approximately US$1,500. This change took place starting April 2009. In other words, after the reform, the commission differential ∆f increased, with a more substantial effect for advisers. Sections 3.4.1 and 3.4.2 above show that the theoretical predictions are different under the assumption of unbiased and biased advice. Under the first assumption, the change in ∆f does not affect the likelihood that the intermediary recommends product A; thus, the sum of the commission differential and the equilibrium price differential ∆f + ∆p should reduce, and product’s B market share should increase after the reform. In other words, if advice is unbiased we should observe a reduction in annuity rates and an increase in the market share of annuities after the reform. Instead, under the second assumption, the fact that an increase in ∆f makes product B relatively less expensive for the customers is compensated by the fact that the intermediary recommends more customers to buy product A. The final effect may be positive or negative. To compare the predictions of the model with the effect of the 2008 reform in commissions, we 7
Chile’ social security system works as a defined contribution plan: workers save mandatorily on during their work years, with individual accounts administered by AFPs.
21
use the data from the Chilean mandatory system for quoting retirement products called SCOM P (Sistema de Consultas y Ofertas de Montos de Pensión in Spanish). SCOMP centralizes all information about the pension process and shares it with the relevant parties: Retirees submit requests for quotations, pension providers receive information about the retiree and her beneficiaries, providers give quotes, and retirees choose their preferred alternative. The dataset contains all requested and received quotes for all retirement processes in the period from August 2004 (the beginning of the SCOMP system) to April 2013. For more information on the system or details of the data, see Alcalde and Vial (2017) and Larrain and Morales (2010). Retirees that enter the system for quotes may face different incentives. For this exercise, we keep a subsample of retirees with similar incentives to choose between both products: retirees on standard retirement (no early retirement), that enter the system for the first time (no change in product), quotes with no lumpsum withdrawal, and quotes above the minimal pension.8
1
4
0.9
3.8
0.8
3.6
0.7
3.4
0.6
3.2
0.5
3
0.4
2.8
0.3
2.6
0.2
2.4
0.1
2.2 2 200408 200412 200504 200508 200512 200604 200608 200612 200704 200708 200712 200804 200808 200812 200904 200908 200912 201004 201008 201012 201104 201108 201112 201204 201208 201212 201304
0
Share of Annuities by month
Mean Annuity Rate by month
Figure 7: Market share of annuities and annuity rates
Figure 7 shows the evolution of the annuity market between 2004 and 2013, with the vertical lines indicating the two commission changes. The solid line shows the average annuity rate offered to retirees in that period; an increase in the annuity rate is equivalent to a decrease in the price charged.9 The price of the programmed withdrawal, measured as the monthly fee charged by 8
(a) Retirees on early retirement choose programmed withdrawal more frequently than retirees on standard retirement, and this may depend on different preferences in both groups but also on tax incentives. Furthermore, the 2008 reform increased the requirements for early retirement, so we eliminate this group from our sample to avoid a composition problem. (b) Retirees that choose a programmed withdrawal are allowed to enter the system again to buy an annuity with the remaining funds; therefore, retirees that enter the system a second time can only choose annuities, and we eliminate this group from our sample. (c) Quotes with lumpsum withdrawal fix the monthly payment at a given level and compete by withdrawal amount, and thus are slightly different to regular quotes. Moreover, the requirements for a lumpsum withdrawal are the same than for early retirement and were increased by the 2008 reform, and we eliminate these quotes–not the individuals–to avoid a composition problem. (d) Retirees with small fund sizes that only get quotes under the minimum pension are forced by law to choose a programmed withdrawal, and we eliminate this group from our sample. 9 The annuity rate is defined as the internal rate of return on the annuity contract, and is determined by the insurance companies given the retiree’s fund size and demographic characteristics. Higher annuity rates imply that
22
retirement funds administrators, is almost constant over the entire period at 1.26%. The dotted (gray) line shows the market share of annuities. Before the reform, the evolution of the annuities’ market share follows the evolution of quoted prices closely: increases in the quoted annuity rates (so that annuities are more convenient relative to programmed withdrawals) are followed by increases of the market share of annuities. But after the reform, there is a gap. Annuity rates decline until the end of 2010, increase at the beginning of 2011, and remain fairly constant until the end of the sample period. But in contrast, the annuities’ market share falls more sharply until the end of 2010 and then rises slowly. This evidence suggests that the reduction in the commission paid for annuities affected the market for retirement products–in particular the demand–in a nontrivial way: consumers’ choices seem to be affected by other variables beyond prices and commissions differentials.
3 2.5 2 1.5 1 0.5
200408 200412 200504 200508 200512 200604 200608 200612 200704 200708 200712 200804 200808 200812 200904 200908 200912 201004 201008 201012 201104 201108 201112 201204 201208 201212 201304
0
Sales agent  Annuities
Adviser  Annuities
Sales agent  PW
Adviser  PW
Figure 8: Intermediation commissions
Figure 8 shows the evolution of commissions for each intermediary type. As described above, sales agents are less affected by the reform: They receive a lower fee for annuities after 2008 but no fee for programmed withdrawal. Instead, advisers are more affected by the reform: They receive a lower fee for annuities after 2008 and start receiving a positive fee for programmed withdrawal, although smaller in size. During the whole period, average fees are lower for sales agents than advisers because agents only get paid if the annuity is from his firm, but the advisers get paid in any event. After the reform, average fees are lower than the limit of 2% and 1.2% respectively because of the cap. From Figure 7 we see that prices–annuity rates– move over time according to costs (market interest rates and requirements of the regulatory agency). Interestingly, by design of the system, firms ignore the intermediary type of each retiree. To confirm this, Figure 9a shows that the average annuity rate is practically the same for every intermediary. At the same time, Figure 9b shows that before the reform, both sales agents and advisers have a similar share of annuities, close to 90%. After the reform, the annuity share for sales agents drops as annuity rates decline, but the annuity annuitants are obtaining higher expected payments in exchange for their premiums paid in advance. Rocha et al. (2008) find that the market for annuities in Chile shows high levels of annuity rates (relative to the riskfree rate), which leads them to conclude that this market is very competitive.
23
1 0.9
3.6
0.8
3.4
0.7
3.2
0.6
3
0.5
2.8
0.4
2.6
0.3
2.4
0.2
2.2
0.1
2
0 200408 200412 200504 200508 200512 200604 200608 200612 200704 200708 200712 200804 200808 200812 200904 200908 200912 201004 201008 201012 201104 201108 201112 201204 201208 201212 201304
200408 200412 200504 200508 200512 200604 200608 200612 200704 200708 200712 200804 200808 200812 200904 200908 200912 201004 201008 201012 201104 201108 201112 201204 201208 201212 201304
4 3.8
All
Direct
Sales Agent
All
Adviser
(a) Annuity rates by intermediary.
Direct
Sales Agent
Adviser
(b) Share of annuities by intermediary.
Figure 9: Annuity rates and market share of annuities by intermediary type. share for advisers drops much sharply. Therefore, we will focus on the changes in incentives for independent advisers, and use the retirees with sales agents as a control group to abstract from changes in other market conditions.
4.1
Predictions using prereform data
As a first exercise, we estimate the probability of choosing an annuity. We use a binary logit specification, using as regressors the prices of each product–annuity rates and fees charged by retirement funds administrators–10 and individual demographic characteristics–gender, age, fund size, and beneficiary characteristics. To capture some of the observed heterogeneity in preferences, we include squares and interactions of prices and individual characteristics. We estimate this logit specification separately for retirees that use advisers and sales agents, using only prereform data–up to September 2008. Then, we use the estimated coefficients to predict the market share for annuities after the reform, using the actual prices and demographic characteristics after 2008. From Figure 7 we see that annuity rates and mean individual characteristics vary, and thus we focus on how the response to a change in annuity rates is different before and after the reform, for each group. Figure 10a shows that, for independent advisers, the predicted annuity share using prereform coefficients drops relative to the actual share–more than the decline in annuity rates would suggest– so that the difference between the actual and predicted share is significant after the reform. In contrast, this decrease is absent for sales agents, our control group; Figure 10b shows that the predicted annuity share using prereform coefficients is not significantly different from the actual share after the reform. Interestingly, for advisers, the difference between actual and predicted shares becomes significant after 2009–after the fee for programmed withdrawal was established– even though the last period of 2008 is also an outofsample prediction–after the legal commission for annuities decreased. 10 Each retiree faces a varying number of annuity and programmed withdrawal quotes; for a logit specification, we use the maximum annuity rate and the minimum fee charged by retirement funds administrators.
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Figure 10: Annuity market share by intermediary type, true vs predicted (prereform). This evidence suggests that intermediaries’ recommendations do affect the likelihood of choosing the different products. If the change in the commission differential ∆f affected customers’ choices only through its direct effect and the effect of the change in the price differential (as in section 3.4.1), we should observe an increase in the market share of annuities after the reform. In contrast, we observe a decline in annuities’ market share after the reform. We interpret this evidence as indicative of biased advice.
5
Final Remarks
The model shows that customers’ choices may be affected by the intermediary’s advice, but only when their prior beliefs are moderate; a customer who is almost convinced that one product fits better her needs than the other chooses that product independently of the adviser’s opinion. The group susceptible to advice is that of customers with priors within an interval, which bounds depend on the price and commission differential between products. When the commission differential increases, the product which fee increases becomes relatively more expensive for the customers, reducing the demand for that product. But this change also affects the intermediary’s recommendation, as it becomes more attractive to recommend the product with a higher commission. This second effect increases the demand for that product among customers who follow the intermediary’s advice. Hence, the effect of a change in the commissions over firms’ strategies and payoffs depends on the fraction of the population that follows the intermediary’s advice in equilibrium. When we apply the model to the retirement market in Chile, we find that the relative reduction in the annuities’ market share after a decrease in commissions paid for annuities is consistent with 25
biased advice from intermediaries. There are many avenues to extend the analysis. We are particularly interested in modeling the choice of intermediary. The fact that some customers do not follow intermediary’s advise suggests that they would prefer to avoid paying the commission. Moreover, an interesting effect of the reform we look at is an increase in the fraction of retirees that choose a programmed withdrawal without intermediation (i.e., they enter the retirement process independently, without an intermediary). We are working on incorporating this feature to the model.
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References Alcalde, P. and B. Vial, “Implicit Tradeoff in Replacement Rates: Consumer Preferences for Firms, Intermediaries, and Annuity Attributes,” SSRN working paper (available at http://ssrn.com/abstract=2720444) (2017). Benoît, J.P. and J. Dubra, “Apparent Overconfidence,” Econometrica 79 (2011), 1591–1625. Inderst, R. and M. Ottaviani, “Misselling through Agents,” American Economic Review 99 (June 2009), 883–908. ———, “Competition through Commissions and Kickbacks,” American Economic Review 102 (April 2012a), 780–809. ———, “Financial Advice,” Journal of Economic Literature 50 (June 2012b), 494–512. Kamenica, E. and M. Gentzkow, “Bayesian Persuasion,” American Economic Review 101 (October 2011), 2590–2615. Larrain, G. and M. Morales, “The Chilean Electronic Market for Annuities (SCOMP): Reducing Information Asymmetries and Improving Competition,” Technical Report, 2010. Rocha, R., M. Morales and C. Thorburn, “An empirical analysis of the annuity rate in Chile,” Journal of Pension Economics and Finance 7 (3 2008), 95–119. Rossiter, L. F. and G. R. Wilensky, “Identification of PhysicianInduced Demand,” The Journal of Human Resources 19 (1984), 231–244. ———, “Health EconomistInduced Demand for Theories of PhysicianInduced Demand,” The Journal of Human Resources 22 (1987), 624–627.
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Appendix A.1 Proof of Lemma 1 Proof. Let ∆u(m, q0 ) ≡ uA (m, q0 ) − uB (m, q0 ) denote the difference between uA and uB given a message m and a prior q0 . Then, ∆u(m, q0 ) = (2 Pr(Am, q0 ) − 1) ∆v − ∆f − pA − pB . The MLR property implies that ∆u(mA , q0 ) > ∆u(mB , q0 ) for any q0 ∈ (0, 1); but ∆u(mA , 1) = ∆u(mB , 1) = ∆v − ∆f − pA − pB and ∆u(mA , 0) = ∆u(mB , 0) = −∆v − ∆f − pA − pB . For a given level of pA − pB , Q0 is defined as the set of all q0 such that the customer chooses A if the intermediary sends message mA and chooses B otherwise; i.e., as the set Q0 = {q0 ∈ [0, 1] : ∆u(mA , q0 ) ≥ 0 and ∆u(mB , q0 ) ≤ 0}. 1 ∈ Q0 only if ∆v − ∆f − ∆p = 0 but this would imply that −∆v − ∆f − pA − pB < 0 and hence 0 6∈ Q0 . Similarly, if 0 ∈ Q0 then 1 6∈ Q0 . In other words, it is not possible that messages are followed in equilibrium for all q0 ∈ [0, 1].
A.2. Proof of Lemma 2 Proof. The cutoffs points q0 and q0 are defined as Pr(mA B)q0∗ Pr(mA B)q0∗ + Pr(mA A)(1 − q0∗ ) Pr(mB B)q0∗ q0 = Pr(mB B)q0∗ + Pr(mB A)(1 − q0∗ ) q0 =
≤ q0∗
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≥ q0∗ ,
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where Pr(mA A) = α and Pr(mB B) = β in the discrete case, while Pr(mB θ) = Gθ (˜ s(q ∗ )) if the signal is continuous. Taking the derivative of q0 and q0 and rearranging we obtain: q0 0 (q0∗ ) =
q0 (1 − q0 ) > 0, q0∗ (1 − q0∗ )
q0 0 (q0∗ ) =
q0 (1 − q0 ) > 0. q0∗ (1 − q0∗ )
and
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