Product safety spillovers and market viability for biologic drugs

Abstract

When a pharmaceutical manufacturer experiences a safety problem, negative impacts on profitability can spread to its competitors. Reduced consumer confidence, product recalls, and litigation are limited to the responsible manufacturer only if that manufacturer can be clearly linked to the safety problem. We analyze the impact of “accountability” for safety problems on manufacturer entry decisions and investments to mitigate risk. Consistent with prior research, we find investment levels increase with accountability in a duopoly market, and that accountability can thus enhance market viability and improve consumer welfare. However, we also analyze the impact of accountability on entry of a competitor, after the originator’s exclusivity has expired. Accountability promotes the development of a robust market by raising expected profits, particularly for an entrant with a relatively low likelihood of a safety problem. Yet entry need not improve consumer welfare, and may benefit the incumbent in our model. In contrast to the traditional entry deterrence mechanism, when accountability is sufficiently low, increased incumbent investment encourages entry. Our analysis has important implications for biologic drugs, insofar as pathways for entry by “biosimilars” have been established in Europe and the United States, and informs pharmacovigilance and other accountability policies for biologics.

Appendices

Appendix 1: Utility framework, competition, and welfare

Recall from “Framework for analysis” section that the utility of consumer k from manufacturer i’s drug is

$$\begin{aligned} u_{ki} =\alpha _k q_i -\beta _k p_i +\epsilon _{ki} , \end{aligned}$$

where $q_i$ is perceived quality of i’s drug, $p_i$ is price, $\alpha _k$ and $\beta _k$ are consumer i’s taste for each attribute, and $\epsilon _{ki}$ captures idiosyncratic taste and is i.i.d. type-1 extreme-valued. This generates a mixed logit demand function.

Before firms compete in prices, perceived qualities and safety investments ($s_i )$ have been determined. Thus, in either of the two periods of market competition, the profit-maximization problem for manufacturer i is

$$\begin{aligned} \mathop {max}\limits _{p_i } \left[ {p_i -c_i \left( {s_i } \right) } \right] d_i \left( {p_i ,p_j ,q_i ,q_j } \right) , \end{aligned}$$

(8)

in which $d_i$ is demand and $c_i$ is marginal cost. As mentioned in the text, we assume that consumers do not anticipate serious safety problems, so safety-enhancing investments do not affect demand before any problems occur. Marginal cost is constant with respect to demand but can differ between the manufacturers. In the main analysis, we assume that safety investments do not affect marginal costs, but we discuss the robustness of our results when this assumption is relaxed. In this latter case, we assume that marginal cost is weakly increasing and convex in safety.

In this setup, a unique price equilibrium $\left\{ {p_{d1}^*,p_{d2}^*} \right\}$ exists (Berry et al. 1995; Anderson et al. 1992; Caplin and Nalebuff 1991). The equilibrium is symmetric if perceived quality levels and the marginal cost functions are identical. In any case, this model of competition generates the duopoly profits $\pi _{di}$ introduced in the preceding subsection.⁹ In the absence of safety problems, these profit levels are the same in both periods of competition.

This model is consistent with our accountability framework. If a manufacturer has a serious safety problem in period 1 and is held accountable, a collapse in its demand in period 2 follows from a decrease in perceived quality; in particular, demand approaches zero as $q_1 \rightarrow -\infty$. In this case, its competitor becomes a monopolist in period 2, with profits corresponding to the monopoly solution to Eq. (8). The monopolist’s pricing behavior is constrained by substitution toward the outside option, $p_{mi}^*>p_{di}^*$ and $\pi _{mi} >\pi _{di}$. If a manufacturer has a safety problem in period 1 and there is no accountability, consumers revise their assessment of the quality of both drugs, and the market collapses in period 2 as $q_i ,q_j \rightarrow -\infty$.

This model also determines consumer surplus. Clearly, in the absence of any safety problems, consumer surplus is higher under a duopoly than a monopoly. When a manufacturer has a safety problem in a period and is held accountable, consumer surplus within the period depends on actual quality $q_j^a$, rather than the perceived quality ($q_j )$ when consumption decisions were made. This discrepancy between ex ante and ex post quality implies that the standard formula expected utility with logit demand (conditional on a consumer’s tastes $\alpha _i$ and $\beta _i )$ is not appropriate for our purposes, as the formula assumes choice behavior reflects complete information (Anderson et al. 1992). For our purposes, we only need to make the reasonable assumption that consumer surplus is higher in a duopoly when more of the competitors have safe products.

Appendix 2: Proof of Proposition 1

Proposition 1

Accountability increases investments by at least one firm. When firm investments are strategic complements or when firms have identical costs, accountability increases investments by both firms.

Consider the case where firms choose $s_i$ simultaneously. From above, the FOCs for firms i and j are

$$\begin{aligned}&s_j \pi _d +\left( {1-s_j } \right) a\pi _m -\frac{\partial F_i \left( {\hbox {s}_i^*} \right) }{\partial s_i }=0 \end{aligned}$$

(9)

$$\begin{aligned}&s_i \pi _d +\left( {1-s_i } \right) a\pi _m -\frac{\partial F_j \left( {\hbox {s}_j^*} \right) }{\partial s_j }=0, \end{aligned}$$

(10)

respectively. Expression (9) provides the reaction function

$$\begin{aligned} \frac{ds_i^*}{ds_j }=\frac{\pi _d -a\pi _m }{\frac{\partial ^{2}F_i \left( {s_i^*} \right) }{\partial s_i^2 }}. \end{aligned}$$

(11)

Note that when a competitor’s likelihood is $s_j =0$, then a firm’s best response is $s_i >0$, as given by

$$\begin{aligned} a\pi _m -\frac{\partial F_i \left( {\hbox {s}_i^*} \right) }{\partial s_i }=0. \end{aligned}$$

The implicit function theorem provides

$$\begin{aligned} \left[ {{\begin{array}{l} {\frac{ds_i^*}{da}} \\ {\frac{ds_j^*}{da}} \\ \end{array} }} \right]= & {} -\left[ {{\begin{array}{ll} {-\frac{\partial ^{2}F_i }{\partial s_i^2 }}&{} {\pi _d -a\pi _m } \\ {\pi _d -a\pi _m }&{} {-\frac{\partial ^{2}F_j }{\partial s_j^2 }} \\ \end{array} }} \right] ^{-1}\left[ {{\begin{array}{l} {\left( {1-s_j } \right) \pi _m } \\ {\left( {1-s_i } \right) \pi _m } \\ \end{array} }} \right] \nonumber \\= & {} -\frac{1}{\frac{\partial ^{2}F_i }{\partial s_i^2 }\frac{\partial ^{2}F_j }{\partial s_j^2 }-\left( {\pi _d -a\pi _m } \right) ^{2}}\nonumber \\&\left[ {{\begin{array}{ll} {-\frac{\partial ^{2}F_j }{\partial s_j^2 }}&{} {-\left( {\pi _d -a\pi _m } \right) } \\ {-\left( {\pi _d -a\pi _m } \right) }&{} {-\frac{\partial ^{2}F_i }{\partial s_i^2 }} \\ \end{array} }} \right] \left[ {{\begin{array}{l} {\left( {1-s_j } \right) \pi _m } \\ {\left( {1-s_i } \right) \pi _m } \\ \end{array} }} \right] \nonumber \\= & {} \frac{1}{\frac{\partial ^{2}F_i }{\partial s_i^2 }\frac{\partial ^{2}F_j }{\partial s_j^2 }-\left( {\pi _d -a\pi _m } \right) ^{2}}\nonumber \\&\left[ {{\begin{array}{l} {\left( {1-s_j } \right) \pi _m \left( {\frac{\partial ^{2}F_j }{\partial s_j^2 }} \right) +\left( {1-s_i } \right) \pi _m \left( {\pi _d -a\pi _m } \right) } \\ {\left( {1-s_j } \right) \pi _m \left( {\pi _d -a\pi _m } \right) +\left( {1-s_i } \right) \pi _m \left( {\frac{\partial ^{2}F_i }{\partial s_i^2 }} \right) } \\ \end{array} }} \right] \end{aligned}$$

(12)

Safety investments are strategic complements. When firms’ investments are strategic complements, $\pi _d -a\pi _m >0$ so the numerators in (12) are positive for both effects. Because each firm’s best response to a competitor choosing $s_j =0$ is to provide $s_i >0$, in the strategic complements case, a necessary condition for the existence of an interior equilibrium is $\frac{ds_i^*}{ds_j }<1$, equivalently $\frac{\pi _d -a\pi _m }{\frac{\partial ^{2}F_i \left( {\hbox {}s_i^*} \right) }{\partial s_i^2 }}<1$. This implies that the denominator is also positive. Thus, increased accountability increases investment for both firms under strategic complements.

Safety investments are strategic substitutes, but firms are identical in costs as well as demand. Assuming firms are identical, Eq. (12) becomes

$$\begin{aligned} \left[ {{\begin{array}{l} {\frac{ds_i^*}{da}} \\ {\frac{ds_j^*}{da}} \\ \end{array} }} \right]= & {} -\frac{1}{\left( {\frac{\partial ^{2}F}{\partial s_i^2 }} \right) ^{2}-\left( {\pi _d -a\pi _m } \right) ^{2}}\left[ {{\begin{array}{ll} {-\frac{\partial ^{2}F}{\partial s_i^2 }}&{} {-\left( {\pi _d -a\pi _m } \right) } \\ {-\left( {\pi _d -a\pi _m } \right) }&{} {-\frac{\partial ^{2}F}{\partial s_i^2 }} \\ \end{array} }} \right] \left[ {{\begin{array}{l} {\left( {1-s_i } \right) \pi _m } \\ {\left( {1-s_i } \right) \pi _m } \\ \end{array} }} \right] \nonumber \\= & {} \frac{\left( {1-s_i } \right) \pi _m }{\left( {\frac{\partial ^{2}F}{\partial s_i^2 }} \right) ^{2}-\left( {\pi _d -a\pi _m } \right) ^{2}}\left[ {{\begin{array}{ll} {\frac{\partial ^{2}F}{\partial s_i^2 }}&{} {\pi _d -a\pi _m } \\ {\pi _d -a\pi _m }&{} {\frac{\partial ^{2}F}{\partial s_i^2 }} \\ \end{array} }} \right] \left[ {{\begin{array}{l} 1 \\ 1 \\ \end{array} }} \right] \nonumber \\= & {} \frac{\left( {1-s_i } \right) \pi _m }{\left( { \frac{\partial ^{2}F}{\partial s_i^2 }+\pi _d -a\pi _m } \right) \left( {\frac{\partial ^{2}F}{\partial s_i^2 }-\left( {\pi _d -a\pi _m } \right) } \right) }\left[ {{\begin{array}{ll} {\frac{\partial ^{2}F}{\partial s_i^2 }}&{} {\pi _d -a\pi _m } \\ {\pi _d -a\pi _m }&{} {\frac{\partial ^{2}F}{\partial s_i^2 }} \\ \end{array} }} \right] \left[ {{\begin{array}{l} 1 \\ 1 \\ \end{array} }} \right] \nonumber \\ \Rightarrow \frac{ds_i^*}{da}= & {} \frac{\left( {1-s_i } \right) \pi _m \left( {\frac{\partial ^{2}F}{\partial s_i^2 }+\pi _d -a\pi _m } \right) }{\left( {\frac{\partial ^{2}F}{\partial s_i^2 }+\pi _d -a \pi _m } \right) \left( {\frac{\partial ^{2}F}{\partial s_i^2 }-\left( {\pi _d -a\pi _m } \right) } \right) } =\frac{\left( {1-s_i } \right) \pi _m }{\frac{\partial ^{2}F}{\partial s_i^2 }-\left( {\pi _d -a\pi _m } \right) } \end{aligned}$$

(13)

Note that the numerator of (13) is always positive. Under strategic substitutes, $\pi _d -a\pi _m <0$. We have already shown above that under strategic complements, both firms improve their investment, regardless of costs.

Accountability increases investments by at least one firm. Given the result in the case of strategic complements, it suffices to show that accountability increases investments by at least one firm in the case of strategic substitutes. Totally differentiating (9) provides

$$\begin{aligned} ds_j \left( {\pi _d -a\pi _m } \right) +da\left( {1-s_j } \right) \pi _m =ds_i \left[ {\frac{\partial ^{2}F}{\partial s_i^2 }} \right] . \end{aligned}$$

(14)

By strategic substitutes, $\pi _d -a\pi _m <0$.

If $da>0$, then the second term on the left-hand side of (14) is positive.

If $ds_j <0$ (so firm j’s safety decreases), then the first term on left-hand side is positive, so overall, the left-hand side increases. Since $\frac{\partial ^{2}F}{\partial s_i^2 }>0$, we must have $ds_i >0$ so that the right-hand side increases as well. Thus, if $ds_j <0$, then $s_i$ must increase.

If $ds_j >0$, we are done.

Appendix 3: Effects of accountability on profit

First, consider the case where firms are identical and choose $s_i$ simultaneously. Differentiating equation (1) provides

$$\begin{aligned} \frac{dE\left[ {{\Pi }_\mathrm{i} } \right] }{da}=\frac{\partial E\left[ {{\Pi }_\mathrm{i} } \right] }{\partial s_i }\left( {\frac{\partial s_i^*}{\partial a}} \right) +\frac{\partial E\left[ {{\Pi }_\mathrm{i} } \right] }{\partial s_j }\left( {\frac{\partial s_j^*}{\partial a}} \right) +\frac{\partial E\left[ {{\Pi }_\mathrm{i} } \right] }{\partial a}. \end{aligned}$$

By the Envelope Theorem, we have $\frac{\partial \mathrm{E}\left[ {{\Pi }_\mathrm{i} } \right] }{\partial s_i }=0$. Thus,

$$\begin{aligned} \frac{dE\left[ {{\Pi }_\mathrm{i} } \right] }{da}=\frac{\partial E\left[ {{\Pi }_\mathrm{i} } \right] }{\partial s_j }\left( {\frac{\partial s_j^*}{\partial a}} \right) +\frac{\partial E\left[ {{\Pi }_\mathrm{i} } \right] }{\partial a}. \end{aligned}$$

By the proof for Proposition 1, we have $\frac{ds_j^*}{da}=\frac{\left( {1-s_j } \right) \pi _m }{\frac{\partial ^{2}F}{\partial s_j^2 }-\left( {\pi _d -a\pi _m } \right) }$, which provides

$$\begin{aligned} \frac{dE\left[ {{\Pi }_\mathrm{i} } \right] }{da}= & {} \left( {s_i \pi _d -s_i a\pi _m } \right) \frac{\left( {1-s_j } \right) \pi _m }{\frac{\partial ^{2}F}{\partial s_j^2 }-\left( {\pi _d -a\pi _m } \right) }+s_i \left( {1-s_j } \right) \pi _m\\= & {} \frac{s_i \left( {1-s_j } \right) \pi _m \left( {\pi _d -a\pi _m } \right) +s_i \left( {1-s_j } \right) \pi _m \left( {\frac{\partial ^{2}F}{\partial s_j^2 }-\left( {\pi _d -a\pi _m } \right) } \right) }{\frac{\partial ^{2}F}{\partial s_j^2 }-\left( {\pi _d -a\pi _m } \right) }\\= & {} \frac{s_i \left( {1-s_j } \right) \pi _m \left( {\frac{\partial ^{2}F}{\partial s_j^2 }-\left( {\pi _d -a\pi _m } \right) +\pi _d -a\pi _m } \right) }{\frac{\partial ^{2}F}{\partial s_j^2 }-\left( {\pi _d -a\pi _m } \right) }\\= & {} \frac{s_i \left( {1-s_j } \right) \pi _m \left( {\frac{\partial ^{2}F}{\partial s_j^2 }} \right) }{\frac{\partial ^{2}F}{\partial s_j^2 }-\left( {\pi _d -a\pi _m } \right) }. \end{aligned}$$

When $\frac{ds_i^*}{ds_j }<1$, the denominator in the last line above is positive, implying $\frac{\mathrm{dE}\left[ {{\Pi }_\mathrm{i} } \right] }{\hbox {da}}>0$.

When the incumbent’s investment is fixed, it is trivial to show that the effect of accountability on the entrant’s expected profit is $\frac{dE\left[ {{\Pi }_2 } \right] }{da}=s_2 \left( {1-s_1 } \right) \pi _m >0$. The effect on an incumbent’s expected profit is

$$\begin{aligned} \frac{dE\left[ {{\Pi }_1 } \right] }{da}= & {} \frac{\partial E\left[ {{\Pi }_\mathrm{1} } \right] }{\partial s_2 }\left( {\frac{\partial s_2^*}{\partial a}} \right) +\frac{\partial E\left[ {{\Pi }_1 } \right] }{\partial a}\\= & {} s_1 \left( {\pi _d -a\pi _m } \right) \frac{\left( {1-s_2 } \right) \pi _m }{\frac{\partial ^{2}F_2 \left( {s_2 } \right) }{\partial s_2^2 }}+s_1 \left( {1-s_2 } \right) \pi _m\\= & {} s_1 \left( {1-s_2 } \right) \pi _m \left( {\frac{\left( {\pi _d -a\pi _m } \right) }{\frac{\partial ^{2}F_2 \left( {s_2 } \right) }{\partial s_2^2 }}+1} \right) . \end{aligned}$$

The sign of the last line above is ambiguous, depending on the sign of $\left( {\frac{\left( {\pi _d -a\pi _m } \right) }{\frac{\partial ^{2}F_2 \left( {s_2 } \right) }{\partial s_2^2 }}+1} \right)$. Note that if we restrict investments in safety to be strategic complements, ($\pi _d -a\pi _m >0)$, then the sign of this term is positive, and accountability increases the incumbent’s expected profit.

Appendix 4: Vertical differentiation

Here we consider the implications of vertical differentiation (potentially with heterogeneous marginal costs) for our results. Except in the case where safety investments are strategic substitutes and fixed safety costs are identical, we are able to demonstrate that Proposition 1 holds with vertical differentiation. To characterize the equilibria in each of the marketing periods, we only need to utilize ex ante utility, since we have assumed consumers are naïve regarding potential safety problems in the first period and that firms producing unsafe products are removed from the market in the second period.

We assume the product of firm 1, the incumbent, is preferred. As specified in “Appendix 1”, this provides $p_{d,1} >p_{d,2}$ and $\pi _{d,1} >\pi _{d,2}$ in equilibrium. That is, the innovator enjoys a higher price and profit than the biosimilar producer. Monopoly profits are also now firm-specific, denoted by $\pi _{m,1}$ and $\pi _{m,2}$.

With respect to Proposition 1, we show below that an increased likelihood of accountability always increases investment by at least one firm, and increases investments by both firms when investments are strategic complements.

Accountability increases investments by at least one firm. The first order condition for safety investments is now:

$$\begin{aligned} s_j \pi _{d,i} +\left( {1-s_j } \right) a\pi _{m,i} =\frac{\partial F_i \left( {\hbox {s}_\mathrm{i}^*} \right) }{\partial s_i }. \end{aligned}$$

Totally differentiating this FOC provides:

$$\begin{aligned} ds_j \left( {\pi _{d,i} -a\pi _{m,i} } \right) +da\left( {1-s_j } \right) \pi _{m,i} =ds_i \left[ {\frac{\partial ^{2}F}{\partial s_i^2 }} \right] . \end{aligned}$$

If safety investments are strategic substitutes, we have $\pi _{d,i} -a\pi _{m,i} <0$ for $i=1,2$. We apply an analogous proof as in “Appendix 2”. If safety investments are strategic complements, we have the stronger result that both firms increase safety investments, as shown below.

Safety investments are strategic complements.

An analogous argument to that in “Appendix 2” holds. The implicit function theorem characterizing the effect of increased accountability on investments is now

$$\begin{aligned} \left[ {{\begin{array}{l} {\frac{ds_i^*}{da}} \\ {\frac{ds_j^*}{da}} \\ \end{array} }} \right] =-\left[ {{\begin{array}{ll} {-\frac{\partial ^{2}F_i }{\partial s_i^2 }}&{} {\pi _{d,i} -a\pi _{m,i} } \\ {\pi _{d,j} -a\pi _{m,j} }&{} {-\frac{\partial ^{2}F_j }{\partial s_j^2 }} \\ \end{array} }} \right] ^{-1}\left[ {{\begin{array}{l} {\left( {1-s_j } \right) \pi _{m,i} } \\ {\left( {1-s_i } \right) \pi _{m,j} } \\ \end{array} }} \right] . \end{aligned}$$

The conditions for complementarity ($\pi _{d,i} -a\pi _{m,i} >0$ for $i=1,2)$ and for the existence of an interior equilibrium ($\pi _{d,i} -a\pi _{m,i} <\frac{\partial ^{2}F_i \left( {s_i^*} \right) }{\partial s_i^2 }$ for $i=1,2)$ provide $\frac{ds_i^*}{da}>0, \frac{ds_j^*}{da}>0$.

To understand the intuition behind the above results, note that the primary change from vertical differentiation is that a firm’s profit differs from that of its competitor. However, the comparative statics with respect to the likelihood of accountability rely on the ordering of a firm’s own monopoly versus duopoly profit, e.g. $\pi _{m,i} >\pi _{d,i} ,$ which remains unchanged with vertical differentiation.

As a result, the consumer welfare result for the duopoly game is also preserved for the strategic complements case. While consumers prefer the reference product to the biosimilar, recall from Eq. (4) that the likelihood of accountability affects consumer welfare through its direct effect on the viability of the second period market subsequent to a safety problem from a single firm and through its indirect effect on safety investments by firms. The former is unchanged by vertical differentiation, since consumers fare better when any single firm serves the market than when there is no market at all. The indirect effect is also preserved under strategic complementarity. In sum, under vertical differentiation, consumers prefer the reference product to the biosimilar, yet the direction of the effect of accountability on consumer welfare does not pivot on which firm serves the market. Instead, accountability determines the number of duopolists producing safe goods in the first period, and how many firms are left to produce in the second period.

Appendix 5: Marginal cost of safety investments

If safety investments involve (identical) marginal costs, firm i’s profit in each period will depend on its chosen likelihood of safety $s_i$. We assume that the order of events remains the same as in the basic model: First, firms simultaneously choose $s_i$ and $s_j$ given the probability of accountability a, then firms simultaneously choose $p_i$ and $p_j$ after observing $s_i$ and $s_j$. Let $\pi _d^*\left( {s_i } \right)$ and $\pi _m^*\left( {s_i } \right)$ denote the equilibrium duopoly and monopoly profits. Thus, at the optimal $s_i$, the following first-order condition holds:

$$\begin{aligned} \frac{\partial E\left( {{\Pi }_i } \right) }{\partial s_i }= & {} s_j \pi _d^*+\left( {1-s_j } \right) a\pi _m^*-\frac{\partial F}{\partial s_i }\nonumber \\&+\left( {1+s_j s_i } \right) \left( {\frac{d\pi _d^*}{ds_i }} \right) +\left( {1-s_j } \right) s_i a\frac{d\pi _m^*}{ds_i }=0. \end{aligned}$$

(15)

Totally differentiating (15) provides

$$\begin{aligned} ds_j \left[ {\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }} \right] +da\left[ {\left( {1-s_j } \right) \left( {\pi _m +s_i \frac{\partial \pi _m }{\partial s_i }} \right) } \right] =-ds_i \left[ {\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }} \right] . \end{aligned}$$

(16)

Accountability increases investments by at least one firm when investments are strategic substitutes

First consider the case of strategic substitutes. Assuming strategic substitutes is equivalent to the following condition:

$$\begin{aligned} -\frac{\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }}{\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }}<0. \end{aligned}$$

By the second-order condition for an optimum, we have $\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }<0$. Thus, strategic substitutes requires $\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }<0$. Now assume $\pi _m +s_i \frac{\partial \pi _m }{\partial s_i }>0$. Suppose that $ds_j <0$. Then the left-hand side of (16) will be positive, and so we must have $ds_i >0$. Thus, at least one firm will increase investments in safety when firms are strategic substitutes.

However, if $\pi _m +s_i \frac{\partial \pi _m }{\partial s_i }<0$, then it is possible for both firms to reduce safety when $da>0$, since now the left-hand side of (16) may be positive or negative when $da>0$ and $ds_j <0$.

Accountability increases investments by both firms when investments are strategic complements Now consider the case where firms are strategic complements. Note that

$$\begin{aligned} \frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }= & {} \pi _d^*-a\pi _m^*+s_i \left( {\frac{\partial \pi _d^*}{\partial s_i }} \right) +\left( {1+s_j s_i } \right) \left( {\frac{\partial ^{2}\pi _d^*}{\partial s_i \partial s_j }} \right) \\&-s_i a\frac{\partial \pi _m^*}{\partial s_i }+\left( {1-s_j } \right) s_i a\frac{\partial ^{2}\pi _m^*}{\partial s_i \partial s_j } \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }= & {} s_j \frac{\partial \pi _d^*}{\partial s_i }+\left( {1-s_j } \right) a\frac{\partial \pi _m^*}{\partial s_i }-\frac{\partial ^{2}F_i }{\partial s_i^2 }+s_j \frac{\partial \pi _d^*}{\partial s_i }+\left( {1+s_j s_i } \right) \frac{\partial ^{2}\pi _d^*}{\partial s_i^2 }\\&+\left( {1-s_j } \right) a\frac{\partial \pi _m^*}{\partial s_i }+\left( {1-s_j } \right) s_i a\frac{\partial ^{2}\pi _m^*}{\partial s_i^2 }<0 \end{aligned}$$

where the sign of the second term is given by the second-order condition $\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }<0$. The implicit function theorem provides

$$\begin{aligned} \left[ {{\begin{array}{l} {\frac{ds_i^*}{da}} \\ {\frac{ds_j^*}{da}} \\ \end{array} }} \right]= & {} -\left[ {{\begin{array}{ll} {\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }}&{} {\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }} \\ {\frac{\partial ^{2}E\left( {{\Pi }_j } \right) }{\partial s_i \partial s_j }}&{} {\frac{\partial ^{2}E\left( {{\Pi }_j } \right) }{\partial s_j^2 }} \\ \end{array} }} \right] ^{-1}\left[ {{\begin{array}{l} {\left( {1-s_j } \right) \left( {\pi _m^*+s_i \frac{\partial \pi _m^*}{\partial s_i }} \right) } \\ {\left( {1-s_i } \right) \left( {\pi _m^*+s_j \frac{\partial \pi _m^*}{\partial s_j }} \right) } \\ \end{array} }} \right] \\= & {} \frac{1}{\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }\frac{\partial ^{2}E\left( {{\Pi }_j } \right) }{\partial s_j^2 }-\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }\frac{\partial ^{2}E\left( {{\Pi }_j } \right) }{\partial s_i \partial s_j }}\\&\left[ {{\begin{array}{ll} {-\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }}&{} {\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }} \\ {\frac{\partial ^{2}E\left( {{\Pi }_j } \right) }{\partial s_i \partial s_j }}&{} {-\frac{\partial ^{2}E\left( {{\Pi }_j } \right) }{\partial s_j^2 }} \\ \end{array} }} \right] \left[ {{\begin{array}{l} {\left( {1-s_j } \right) \left( {\pi _m^*+s_i \frac{\partial \pi _m^*}{\partial s_i }} \right) } \\ {\left( {1-s_i } \right) \left( {\pi _m^*+s_j \frac{\partial \pi _m^*}{\partial s_j }} \right) } \\ \end{array} }} \right] . \end{aligned}$$

Again, the denominator is positive by the necessary condition for the existence of an interior equilibrium $\left( {\frac{ds_i^*}{ds_j }<1\hbox { or }\frac{\pi _d -a\pi _m }{\frac{\partial ^{2}F_i \left( {s_i^*} \right) }{\partial s_i^2 }}<1} \right)$. Note that the terms in the first matrix above are all positive. Then the comparative statics are positive if $\pi _m^*+s_i a\frac{\partial \pi _m^*}{\partial s_i }>0$. Further note that this is a weaker assumption than the condition utilized in the case of strategic substitutes (because $a\le 1)$.)

Accountability increases investments by both firms when firms have identical fixed costs and safety investments are strategic substitutes

The condition $\pi _m^*+s_i \frac{\partial \pi _m^*}{\partial s_i }>0$ also ensures positive comparative statics in this case, using a similar proof as in “Appendix 2”. Specifically, the implicit function theorem simplifies to

$$\begin{aligned} \left[ {{\begin{array}{l} {\frac{ds_i^*}{da}} \\ {\frac{ds_j^*}{da}} \\ \end{array} }} \right]= & {} \frac{\left( {1-s_i } \right) \left( {\pi _m^*+s_i a\frac{\partial \pi _m^*}{\partial s_i }} \right) }{\left( {\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }} \right) ^{2}-\left( {\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }} \right) ^{2}}\left[ {{\begin{array}{ll} {-\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }}&{} {\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }} \\ {\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }}&{} {-\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_j^2 }} \\ \end{array} }} \right] \left[ {{\begin{array}{l} 1 \\ 1 \\ \end{array} }} \right] \\ \frac{ds_i^*}{da}= & {} \frac{-\left( {1-s_i } \right) \left( {\pi _m^*+s_i a\frac{\partial \pi _m^*}{\partial s_i }} \right) \left( {\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }-\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }} \right) }{\left( {\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }+\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }} \right) \left( {\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }-\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }} \right) }\\= & {} \frac{-\left( {1-s_i } \right) \left( {\pi _m^*+s_i a\frac{\partial \pi _m^*}{\partial s_i }} \right) }{\left( {\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i^2 }+\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }} \right) }. \end{aligned}$$

Again, strategic substitutes implies $\frac{\partial ^{2}E\left( {{\Pi }_i } \right) }{\partial s_i \partial s_j }<0$ so the denominator is negative overall, and thus $\frac{ds_i^*}{da}>0$.

To understand the intuition behind the conditions $\pi _m^*+s_i a\frac{\partial \pi _m^*}{\partial s_i }>0$ (complements) and $\pi _m^*+s_i \frac{\partial \pi _m^*}{\partial s_i }>0$ (substitutes), note that since $\frac{\partial \pi _m }{\partial s_i }<0$, the condition essentially requires that the impact of $s_i$ on monopoly profit be small relative to monopoly profit. Consider the role of this assumption in relation to Proposition 1. Intuitively, accountability improves a firm’s likelihood of the monopoly outcome subsequent to a competitor’s safety problem only if the firm itself does not have a safety problem. Accountability increases the firm’s willingness to raise $s_i$ and thereby achieve monopoly profit, so long as the associated marginal cost does not deplete monopoly rents.

The impact of accountability on consumer welfare, as described in Proposition 2, directly follows since the welfare analysis depends on the number of firms and the realization of safety problems in the market. That is, when accountability improves safety investments, it improves welfare. Thus, Proposition 2 holds when $\pi _m^*+s_i a\frac{\partial \pi _m^*}{\partial s_i }>0$ or $\pi _m^*+s_i \frac{\partial \pi _m^*}{\partial s_i }>0$ holds.

Appendix 6: Proof of Proposition 4

Proposition 4

When accountability is high, increased investment by the incumbent deters entry. When accountability is low, increased investment by an incumbent encourages entry.

Proof

The entry cost threshold $E^{*}$ such that the potential entrant enters so long as $E\le E^{*}$ is given by

$$\begin{aligned} E^{*}= & {} \left( {1+s_1 s_2^*(s_1)} \right) \pi _d +s_2^*\left( {s_1 } \right) \left( {1-s_1 } \right) a\pi _m -F_2 \left( {s_2^*\left( {s_1 } \right) } \right) . \end{aligned}$$

The effect of the incumbent’s safety on the threshold is:

$$\begin{aligned} \frac{dE^{*}}{ds_1 }= & {} \frac{\partial E^{*}}{\partial s_1 }+\frac{\partial E^{*}}{\partial s_2 }\frac{ds_2^*}{ds_1 }\\= & {} \frac{\partial E^{*}}{\partial s_1 }\\= & {} s_2 \pi _d -s_2 a\pi _m \end{aligned}$$

where the second line utilizes the Envelope Theorem. The sign of the above effect depends on the value of a. Specifically, the safety threshold is increasing in $s_1$, and therefore the likelihood of entry is increasing in the incumbent’s investment, when

$$\begin{aligned} a<\frac{\pi _d }{\pi _m }. \end{aligned}$$

Thus, when accountability is sufficiently low, increased investment in safety promotes entry. Similarly, when accountability is higher, an incumbent that invests more to achieve a higher $s_1$ will decrease the likelihood of entry for a given type of entrant. $\square$

Appendix 7: Example for Proposition 5

Proposition 5

Entry reduces consumer surplus as $s_2 \rightarrow 0$ and $CS\left( {2,N_s } \right) \rightarrow CS\left( {1,N_s } \right)$ for $N_S \le 1.$

Period 1 surplus with entry approaches

$$\begin{aligned} E\left[ {CS^{1}\left( {a|entry} \right) } \right] \rightarrow s_1 CS({1,1})+\left( {1-s_1 } \right) CS({1,0}) \end{aligned}$$

which is equivalent to the consumer surplus under no entry. Thus, it suffices to show that period 2 consumer surplus is lower under entry than under no entry.

In period 2, the condition $s_2 \rightarrow 0$ is sufficient for period 2 consumer surplus without entry to dominate period 2 surplus with entry:

$$\begin{aligned}&E\left[ {CS^{2}\left( {a|\textit{no entry}} \right) } \right] =s_1 CS({1,1}),\\&E\left[ {CS^{2}\left( {a|entry} \right) } \right] \rightarrow as_1 CS({1,1})\\&\quad \Rightarrow E\left[ {CS^{2}\left( {a|entry} \right) } \right] \le E\left[ {CS^{2}\left( {a|\textit{no entry}} \right) } \right] \hbox { for }a\ne 0,s_1 \ne 0. \end{aligned}$$

Appendix 8: Effect of accountability on incumbent profit is ambiguous in the entry model

In the model with entry, the incumbent’s profit is a function of the probability of entry $G\left( {E^{*}} \right)$. Specifically, the incumbent’s expected profit is:

$$\begin{aligned} \hbox {E}\left[ {{\Pi }_1 } \right] =G\left( {E^{*}} \right) \left[ {\left( {1+s_2 s_1 } \right) \pi _d +\left( {1-s_2 } \right) s_1 a\pi _m } \right] +\left( {1-G\left( {E^{*}} \right) } \right) \left( {1+s_1 } \right) \pi _m -F_1 \left( {s_1 } \right) . \end{aligned}$$

The first term above provides the probability-weighted profit under entry, depending upon accountability and each firm’s likelihood of safety. The second term above provides the probability-weighted profit without entry, which is only a function of the incumbent’s own safety.

The effect of accountability on the incumbent’s profit is given by

$$\begin{aligned} \frac{d\hbox {E}\left[ {{\Pi }_1 } \right] }{da}=\frac{\partial \hbox {E}\left[ {{\Pi }_1 } \right] }{\partial a}+\frac{\partial \hbox {E}\left[ {{\Pi }_1 } \right] }{\partial s_1 }\frac{d\hbox {s}_1^*}{da}+\frac{\partial \hbox {E}\left[ {{\Pi }_1 } \right] }{\partial s_2 }\frac{d\hbox {s}_2^*}{da}. \end{aligned}$$

The middle term falls to zero regardless in both the entry scenarios described in “Accountability and entry” section. First, when the incumbent’s investment is determined in a regime in which entry is not possible and a biosimilar pathway is then introduced (Proposition 3), $\frac{d\hbox {s}_1^*}{da}=0$ by assumption. Second, when the incumbent’s investment is chosen optimally in response to accountability (Proposition 4), then $\frac{\partial \mathrm{E}\left[ {{\Pi }_1 } \right] }{\partial s_1 }=0$ by the Envelope Theorem.

Analyzing the other terms provides

$$\begin{aligned} \frac{\partial \hbox {E}\left[ {{\Pi }_1 } \right] }{\partial a}= & {} \frac{dG\left( {E^{*}} \right) }{dE^{*}}\frac{dE^{*}}{da}\left[ {\left( {1+s_2 s_1 } \right) \pi _d +\left( {1-s_2 } \right) s_1 a\pi _m -\left( {1+s_1 } \right) \pi _m } \right] \\&+G\left( {E^{*}} \right) \left[ {\left( {1-s_2 } \right) s_1 \pi _m } \right] \\ \frac{\partial \hbox {E}\left[ {{\Pi }_1 } \right] }{\partial s_2 }= & {} \frac{dG\left( {E^{*}} \right) }{dE^{*}}\frac{dE^{*}}{ds_2 }\left[ {\left( {1+s_2 s_1 } \right) \pi _d +\left( {1-s_2 } \right) s_1 a\pi _m -\left( {1+s_1 } \right) \pi _m } \right] \\&+G\left( {E^{*}} \right) \left[ {s_1 \left( {\pi _d -a\pi _m } \right) } \right] . \end{aligned}$$

Note that $\frac{\partial E^{*}}{\partial s_2 }=0$ by the Envelope Theorem. We therefore have

$$\begin{aligned} \frac{d\hbox {E}\left[ {{\Pi }_1 } \right] }{da}= & {} \frac{dG\left( {E^{*}} \right) }{dE^{*}}\frac{dE^{*}}{da}\left[ {\left( {1+s_2 s_1 } \right) \pi _d +\left( {1-s_2 } \right) s_1 a\pi _m -\left( {1+s_1 } \right) \pi _m } \right] \nonumber \\&+G\left( {E^{*}} \right) \left[ {\left( {1-s_2 } \right) s_1 \pi _m +s_1 \left( {\pi _d -a\pi _m } \right) \frac{d\hbox {s}_2^*}{da}} \right] . \end{aligned}$$

(17)

Suppose $s_1 \rightarrow 1$. Applying this to Eq. (5) provides $\left. {\frac{d\varPi _2^*}{da}} \right| _{s_1 \rightarrow 1} =0$, implying an increased likelihood of accountability has no effect on profit for a potential entrant when the incumbent is likely to produce safe products. Note that the threshold entry cost $E^{*}$ is determined by setting ${\Pi }_2^{*} (a)=0$, which provides

$$\begin{aligned} E^{*}=\left( {1+s_1 s_2^*(a)} \right) \pi _d +\left( {1-s_1 } \right) s_2^*(a)a\pi _m -F_2 \left( {s_2^*(a)} \right) . \end{aligned}$$

As a direct consequence, $\left. {\frac{dE^{*}}{da}} \right| _{s_1 \rightarrow 1} =\left. {\frac{d\varPi _2^*}{da}} \right| _{s_1 \rightarrow 1} =0$. The first term of (17) therefore approaches zero as $s_1 \rightarrow 1$. Thus,

$$\begin{aligned} \left. {\frac{d\hbox {E}\left[ {{\Pi }_1 } \right] }{da}} \right| _{s_1 \rightarrow 1} =G\left( {E^{*}} \right) \left[ {\left( {1-s_2 } \right) \pi _m +\left( {\pi _d -a\pi _m } \right) \frac{d\hbox {s}_2^*}{da}} \right] . \end{aligned}$$

To determine the sign of the above expression, we must determine the sign of $\frac{d\mathrm{s}_2^*}{da}$. Note that the entrant’s optimal investment is given by

$$\begin{aligned} \frac{\partial {\pi }_2 }{\partial s_2 }=s_1\pi _d +(1-s_1)a \pi _m-\frac{\partial F_2 }{\partial s_2 }=0. \end{aligned}$$

Differentiating provides

$$\begin{aligned}&ds_2^*\left[ {-\frac{\partial ^{2}F_2 }{\partial s_2^2 }} \right] +da[(1-s_1)\pi _m]=0\\&\Rightarrow \frac{ds_2^*}{da}=\frac{(1-s_1)\pi _m}{\frac{\partial ^{2}F_2 }{\partial s_2^2 }}\ge 0. \end{aligned}$$

In this case, if accountability is sufficiently low so that $\pi _d -a\pi _m \ge 0$, then overall $\left. {\frac{d\mathrm{E}\left[ {{\Pi }_1 } \right] }{da}} \right| _{s_1 \rightarrow 1} \ge 0$. In sum, the incumbent’s profit may increase in accountability when $s_1 \rightarrow 1$.

Now consider the case where $s_1 \rightarrow 0$. Then (17) becomes

$$\begin{aligned} \left. {\frac{d\hbox {E}\left[ {{\Pi }_1 } \right] }{da}} \right| _{s_1 \rightarrow 0} =\frac{dG\left( {E^{*}} \right) }{dE^{*}}\frac{dE^{*}}{da}\left[ {\pi _d -\pi _m } \right] . \end{aligned}$$

Here, $\frac{dE^{*}}{da}=\pi _m s_2 >0$, while $\pi _d -\pi _m <0$, so that $\left. {\frac{d\mathrm{E}\left[ {{\Pi }_1 } \right] }{da}} \right| _{s_1 \rightarrow 0} <0$. Thus, accountability reduces incumbent profit when the incumbent’s safety is low.