Search Fatigue

Abstract

Consumer search is not only costly but also tiring. We characterize the intertemporal effects that search fatigue has on monopoly and oligopoly prices, the product lines offered by firms, and the provision of consumer assistance (i.e., advice). These effects vary based on the type of search that occurs among stores and within them. In contrast to standard search models, accounting for fatigue leads to product proliferation and time-varying prices, cyclical price dispersion, and consumer assistance. We analyze the welfare implications of search fatigue and discuss the conditions under which product proliferation and cyclical price dispersion are most pronounced.

Appendices

Appendix A

Proof of Proposition 1

The proof proceeds by backward induction within each period and by checking whether the firm or the consumer has a profitable deviation.

Suppose that $\ell =1$. Then, the firm has no incentive to provide assistance. Therefore, $a=0$ is consistent with an equilibrium. At $\tau =2$, the firm chooses $p=\bar{q}$ since: if $p>\bar{q}$, the consumer does not purchase; and if $p<{q}$, the firm earns strictly less. At $\tau =1$, the consumer does not search, since $\ell =1$. Searching after periods of rest is not a profitable deviation even though $c(0)=0$.

Now, we can check if choosing $\ell =1$ at $t=0$ is optimal for the firm. Since the firm’s profit function is

$$\begin{aligned} \varPi ^*=\frac{1}{1-\delta }\bar{q}- \kappa \ell , \end{aligned}$$

choosing any $\ell >1$ is not a profitable deviation. $\square$

Proof of Proposition 2

The proof proceeds by backward induction within each period. There are L products offered in the market, of which n are special products.

Suppose that the consumer becomes a shopper in period t and selects the product with the highest utility. Bertrand competition between the firms that offer identical products leads to $p_{j}^{m}=0$ for all j and m. To show this, suppose that $p_{j}^{m}=0$ for all j and m, and that firm j deviates by setting $p_{j}^{m}>0$ for any m. For $m\ge 2$, the payoff to the consumer from purchasing such a product from firm j would be negative which would lead to no sale. For $m=1$, if $p_{j}^{1}>0$, the consumer would purchase the special product from another firm. Therefore, increasing prices is not a profitable deviation. In this case, the consumer selects one of the special products from one of the n firms and earns a surplus of $\bar{q}-c(x_{t-1})$. All firms earn zero profits.

Suppose that the consumer does not become a shopper in period t. The consumer is willing to buy the product to which she was allocated as long as it offers non-negative utility. If the consumer is randomly allocated to firm j and $\ell _j>1$, the firm optimally chooses $a_{j}=1$. For any $\ell _{j}$, each firm j optimally sets prices $p^{1}=\bar{q}$ and $p^{j}>0$ for $j \ne 1$, and its profits are $\bar{q}$. All other firms earn zero profits. Each firm’s expected per period profit is equal to $\frac{\bar{q}}{n}$. The consumer earns zero surplus.

Hence, the consumer becomes a shopper in period t if and only if $\bar{q}>c(x_{t-1})$. If the consumer did not become a shopper in period $t-1$ (i.e., $x_{t-1}=0$), she becomes a shopper in period t because $\bar{q}>0$. Let $\bar{L}$ be the smallest integer such that $\bar{q} \le c(\bar{L})$. If $L < \bar{L}$, the consumer becomes a shopper in every period, and each firm earns zero discounted expected profits. If $L \ge \bar{L}$, the consumer becomes a shopper in all odd-numbered periods ($t=1, t=3,\dots$) and does not search in all even-numbered periods ($t=2, t=4,\dots$). In this case, each firm j earns discounted expected profits equal to

$$\begin{aligned} \varPi ^*_{j}=\frac{\delta \bar{q} }{n(1-\delta ^{2})} - \ell _j \kappa , \end{aligned}$$

(A1)

and the consumer’s expected discounted surplus is

$$\begin{aligned} U^*=\frac{\bar{q}}{1-\delta ^{2}}. \end{aligned}$$

(A2)

Consider that search is all-or-nothing. We now show that $L^*=\bar{L}$ is unique in equilibrium and that $a_j=0$ in odd-numbered periods. Suppose that all firms except j choose to produce a total of $x<\bar{L}$ products. It is straightforward to show that firm j prefers to produce $\bar{L}-x$ products to deter search in all even-numbered periods. Now, suppose that the total number of products that are offered in the market is equal to $\bar{L}$. If a particular firm j produces $\ell _j^*+1$ instead of $\ell _j^*$ products, it incurs an extra cost $\kappa$ and thus reduces its expected discounted profits. If it instead produces only the special product, it avoids paying $(\ell _j^*-1)\kappa$ in product-line costs; but its (expected) per-period profit drops to zero. Because $\kappa$ is small, this is never a profitable deviation. Finally, when the consumer searches in odd-numbered periods, firm j has a strict incentive to set $a_j=0$ since $c(\bar{L}-1)<\bar{q}$.

Now consider that search is sequential. Since the consumer will visit only one firm when rested, each firm has a strict incentive to produce $\bar{L}$ products. If $\ell _j < \bar{L}$ and the consumer visits firm j, she will become a shopper in the next period, and firm j will lose $\frac{\delta \bar{q}}{n}$. Because $\kappa (\bar{L})<\frac{\delta \bar{q}}{n}$, the firm has no incentive to deviate from producing $\bar{L}$. Likewise, if firm j produces $\bar{L}+1$ instead of $\bar{L}$ products, it incurs an extra cost $\kappa$ and reduces its expected discounted profits. This is not a profitable deviation. $\square$

Proof of Corollary 1

The logic of Proposition 2 for sequential search across firms holds here as well, except that each firm has an incentive to produce more than $\bar{L}$ products. To see this, suppose each firm produces exactly $\ell _j^*=\bar{L}$ and the consumer sorts products one-by-one. Assume that the consumer randomly chooses a product each time she decides to examine a subsequent one. Define $P(\ell )$ as the probability that the consumer chooses the special product after at least $\bar{L}$ draws. By construction, $P(\ell )$ is increasing in $\ell$ for $\ell \ge \bar{L}$. With probability $1-P(\ell )$, the consumer becomes a shopper again at $t+1$, and each firm loses an expected surplus of $\frac{\delta \bar{q}}{n}$. Because $\kappa < [P(\bar{L}+1)-P(\bar{L})] \frac{\delta \bar{q}}{n}$, each firm has a strict incentive to produce more than $\bar{L}$ products. $\square$

Proof of Proposition 3

Suppose the consumer searches sequentially with variable cost $c(0)=0$ and selects the product with the highest utility. Bertrand competition between the firms that offer identical products leads to $p_{j}^{m}=0$ for all j and m. The logic follows the same way as in the proof of Proposition 2 above. In this case, the consumer selects one of the special products from one of the n firms and earns a surplus of $\bar{q}-c(x_{t-1})$. All firms earn zero profits.

Suppose the consumer has a variable search cost of $c(x_{t-1})>0$. Then, following Diamond (1971), it is an equilibrium for each firm j with $\ell _{j}$ to set prices $p^{1}=\bar{q}$ and $p^{j}>0$ for $j \ne 1$. In such case, each firm has an incentive to provide advice $a_j=1$ if the consumer is randomly paired with one of its non-special products. Taking this into account, there is no profitable deviation for the consumer to pay a positive search cost. In this case, the firm that attracts the consumer in this period earns $\bar{q}$, and all other firms earn zero profits. Each firm’s expected per period profit is equal to $\frac{\bar{q}}{n}$. The consumer earns zero surplus.

Hence, the consumer searches when $c(0)=0$ and does not search otherwise. Therefore, it is optimal for each firm j to produce two products. If firm j deviates and produces only the special product, and the consumer visits this store, firm j loses an expected surplus of $\frac{\delta \bar{q}}{n}$ next period. Since $\kappa <\frac{\delta \bar{q}}{n}$, the firm has no incentive to deviate from producing two products. Likewise, if firm j produces three products instead, it incurs an extra cost $\kappa$ and reduces its expected discounted profits. This is not a profitable deviation.

Given this, the consumer searches in all odd-numbered periods ($t=1, t=3,\dots$) and does not search in all even-numbered periods ($t=2, t=4,\dots$). Each firm j earns discounted expected profits equal to

$$\begin{aligned} \varPi ^*_{j}=\frac{\delta \bar{q} }{n(1-\delta ^{2})} - \ell _j \kappa , \end{aligned}$$

(A3)

and the consumer’s expected discounted surplus is

$$\begin{aligned} U^*=\frac{\bar{q}}{1-\delta ^{2}}. \end{aligned}$$

(A4)

$\square$

Proof of Proposition 4

Outline of proof The proof proceeds by backward induction within each period. In each period, we first consider consumer buying behavior and the consumer assistance that is offered by each firm conditional on identifying which consumers are searching. Following that, we consider the firms’ pricing strategies. Working backward, we then consider the search decision by consumers. Finally, we show the existence of an equilibrium $(\ell ^*, F^*_t(p),a_t^*)$.

Step One: Buying behavior and consumer assistance

At any time t, $\mu _t$-type consumers identify all products and prices in the market and choose the one that gives the highest payoff. By construction, $x_t=L$, so that their next period search cost is c(L). When a firm identifies a searching consumer, it chooses $a_j=0$ as there is a cost to reducing future search costs and no benefit to giving assistance. At time t, $(1-\mu _t)$-type consumers are randomly paired with a firm. In this case, the firm will offer $a_j=1$ and direct the consumer to the product that is most profitable for the firm. As we will show shortly, in equilibrium this is the special product.

Step Two: Pricing

First, let us consider the price of the special product and assume that the firm always directs $(1-\mu _t)$-types to this product. Eventually, we will show that this is indeed always optimal in equilibrium. Define $J^{*}$ as the set of firms that quote the lowest price for the special product and $n_{j^{*}}$ as the number of firms in $J^{*}$. Then, the payoff function for each firm $j \in N$ is

$$\begin{aligned} \max _{p_j \in [0,\bar{q}]} \pi _j(p_j)=p_j Q_j, \end{aligned}$$

(A5)

where the expected demand $Q_j$ is calculated as

$$\begin{aligned} Q_j = \frac{\mu _t {\large \text{1 }}_{\{j \in J^{*}\}}}{n_{j^{*}}} + \frac{1-\mu _t}{n} . \end{aligned}$$

Given this, the payoff to each firm is continuous, except when its price is the lowest and equal to at least one of its competitors.

We prove existence of a symmetric mixed-strategy equilibrium by appealing to Theorem 5 in Dasgupta and Maskin (1986). Using their notation, let $A_j=[0,\bar{q}]$ be the action space for firm j and let $a_j \in A_j$ be a price in that space. As such, $A_j$ is non-empty, compact, and convex for all j. Define $A= \times _{j \in N} A_j$ and $a= (a_1, \ldots ,a_n)$. Let $U_j:A \rightarrow \mathbb {R}$ be defined as the profit function in (A5). Define the set $A^{*}(j)$ by

$$\begin{aligned} A^{*}(j)=\{(a_1,\ldots ,a_n) \in A | \exists i \ne j s.t. p_j=p_i\} \end{aligned}$$

and the set $A^{**}(j) \subseteq A^{*}(j)$ by

$$\begin{aligned} A^{**}(j)=\{(a_1,\ldots ,a_n) \in A | \exists i \ne j s.t. p_j=p_i=p_{min}>0\}. \end{aligned}$$

Therefore, the payoff function $U_j$ is bounded and continuous, except over points $\bar{a} \in A^{**}(j)$. The sum $\sum _{j \in N} U_j(a)$ is continuous since discontinuous shifts in demand from informed consumers between firms at points in $A^{**}= \times _{j \in N} A^{**}(j)$ occur as transfers between firms that have the same low price in the industry. Finally, it is straightforward to show that $U_j(a_j,a_{-j})$ is weakly lower semi-continuous. Since any time $p_i=p_j=p_{min}$, firm i and j share the demand, there exists a $\lambda \in [0,1]$ large enough such that

$$\begin{aligned} \lambda \left[(p_j-\epsilon ) \mu _t +\frac{(p_j-\epsilon )(1-\mu _t)}{n}\right] +(1-\lambda )\frac{(p_j+\epsilon )(1-\mu _t)}{n} \ge \frac{p_j \mu _t}{2} +\frac{p_j(1-\mu _t)}{n}, \end{aligned}$$

(A6)

for $\epsilon$ arbitrarily small. Rearranging and letting $\epsilon \rightarrow 0$ yields

$$\begin{aligned} \lambda p_j \mu _t \ge \frac{p_j \mu _t}{2}, \end{aligned}$$

(A7)

which is true for all $\lambda \ge \frac{1}{2}$. Therefore by Theorem 5 in Dasgupta and Maskin (1986), there exists a symmetric mixed-strategy equilibrium for this subgame, conditional on the firms always directing non-searching consumers to the special product.

We can now prove properties about $F^*(p)$, again conditional on the firm always directing consumers to the special product:

• 1. Continuity Suppose that there did exist a countable number of mass points in the distribution of $F^*(p)$. Then, for any $p>0$, we can find a mass point $p'$ and an $\epsilon >0$ such that $f^*(p')=a>0$ and $f^*(p' -\epsilon )=0$. Now consider a deviation by firm j to choose $\hat{F}(p)$ such that $\hat{f}(p')=0$ and $\hat{f}(p'-\epsilon )=a$. Since $E[\pi _j(p)]$ using $F^*(p)$ is strictly less than using $\hat{F}(p)$, this would be a profitable deviation. Last, there can never be a mass point at $p=0$ because it is a dominated strategy to choose $p=0$. Therefore, in equilibrium, no mass points can exist.

• 2. Strict monotonicity (Increasing) Suppose that there exists an interval $[p_a,p_b]$ within $[0,\bar{q}]$ such that $F(p_b)-F(p_a)=0$ and that $F(p_a)>0$. Then, for any $\hat{p}$ such that $p_a<\hat{p}<p_b$, $[1-F(\hat{p})]^{n-1}= [1-F(p_a)]^{n-1}$. Since $\hat{p}[1-F(\hat{p})]^{n-1}> p_a [1-F(p_a)]^{n-1}$ and $\hat{p}[1-(1-F(\hat{p}))^{n-1}]> p_a [1-(1-F(p_a))^{n-1}]$, then there exists a profitable deviation. Thus, $F(p_b)-F(p_a)\ne 0$ for any interval $[p_a,p_b]$ within $[0,\bar{q}]$.

Given continuity and strict monotonicity, we can write the symmetric F(p) explicitly. For any price p that a firm may choose,

$$\begin{aligned} \pi _j(p)= p \mu _t [1-F(p)]^{n-1} +\frac{p(1-\mu _t)}{n}. \end{aligned}$$

(A8)

Since each firm needs to be indifferent between setting an price over a support $[p^*,\bar{q}]$, we can write

$$\begin{aligned} p \mu _t [1-F(p)]^{n-1} +\frac{p(1-\mu _t)}{n}= \frac{\bar{q}(1-\mu _t)}{n}. \end{aligned}$$

(A9)

Rearranging yields the expression in (1). We can then solve

$$\begin{aligned} p^* \mu _t +\frac{p^*(1-\mu _t)}{n}= \frac{\bar{q}(1-\mu _t)}{n} \end{aligned}$$

(A10)

for $p^*$ which yields (2). Finally, inspecting (2), it is clear that $p^*>0$ for any $\mu _t <1$. Therefore, the firms will always direct non-searching consumers to the special product because they do not make positive profits by selling alternative products.

The comparative statics in Proposition 4 regarding $\mu _t$ and n are derived by straightforward differentiation. Taking the limit of $1-F_t(p)$ yields

$$\begin{aligned} \lim _{n \rightarrow \infty } \Bigg [\frac{(\bar{q}-p)(1-\mu _t)}{np \mu _t} \Bigg ]^{\frac{1}{n-1}} = \lim _{n \rightarrow \infty } \Bigg [\frac{(\bar{q}-p)(1-\mu _t)}{p \mu _t} \Bigg ]^{\frac{1}{n-1}} \lim _{n \rightarrow \infty } \Bigg [\frac{1}{n} \Bigg ]^{\frac{1}{n-1}} \rightarrow 1 \end{aligned}$$

which implies that as $n \rightarrow \infty$, $F_t(p) \rightarrow 0$ for all p.

Step Three: Consumer Search Decision

For the consumers with $x_{t-1} >0$, they will search if and only if

$$\begin{aligned} \bar{q}- E[p_{min}|F(p)]> c(x_{t-1}). \end{aligned}$$

(A11)

For now, let us suppose that the firms choose their product lines so that (A11) does not hold. Indeed, we will show this to be the case in step four below.

Given this, any consumer with $x_{t-1}=0$ has the option to search since $c(0)=0$ or to rest again and search in the next period with those currently with $x_{t-1} >0$. Let us consider this choice when the proportion of currently rested consumers is $\mu _t=r$. When the consumer decides to follow the equilibrium strategy to search this period, her expected discounted payoff is

$$\begin{aligned} q - E[\min \{p\} \vert r] + \delta (q - E[p \vert 1-r]) + \delta ^2 (q - E[\min \{p\} \vert r]) + \delta ^3 (q - E[p \vert 1-r]) + \dots , \end{aligned}$$

(A12)

where $E[\min \{p\} \vert r]$ denotes the expected minimum price when a proportion r of consumer searches. If she deviates and rests again, her payoff is

$$\begin{aligned} q - E[p \vert r] + \delta (q - E[\min \{p\} \vert 1-r]) + \delta ^2 (q - E[p \vert r]) + \delta ^3 (q - E[\min \{p\} \vert 1-r]) + \dots . \end{aligned}$$

(A13)

Note that the cost of search does not enter her payoffs because $c(0)=0$. Thus, from Eqs. (A12) and (A13) the consumer has an incentive to deviate if and only if

$$\begin{aligned} \varDelta (\delta , r) = - (E[p \vert r] - E[\min \{p\} \vert r]) + \delta (E[p \vert 1-r] - E[\min \{p\} \vert 1-r]) > 0. \end{aligned}$$

(A14)

By definition, we have $E[p \vert r] > E[\min \{p\} \vert r]$ and $E[p \vert 1-r] - E[\min \{p\} \vert 1-r]$. Thus, for sufficiently small $\delta$ this inequality is not satisfied, and the consumer does not find it profitable to deviate from the equilibrium strategy. Therefore, there exists a $\bar{\delta }$ such that if $\delta < \bar{\delta }$, $\mu _t=r$ for all odd periods $t \in \{1,3,\ldots \}$ and is equal to $1-r$ otherwise.

Step Four: Firms’ Choice of Product Lines

Given that $c(\cdot )$ is strictly increasing in its argument, there exists an $\bar{L}$ such that $\bar{q}- E[p_{min}|F(p)]< c(\bar{L})$, so the consumer does not search. With the condition that $\kappa$ is small, proving that any $\ell ^*$ that induces $L=\bar{L}$ follows the same logic as in Proposition 2. $\square$

Proof of Proposition 5

The proof follows the exact same logic as the proof of Proposition 4. The only difference is the computation of $F_t(p)$.

In any period t, for any price p that a firm may choose,

$$\begin{aligned} \pi _j(p)= p \mu _t \lambda [1-F_t(p)]^{n-1} +\frac{p \lambda (1-\mu _t)}{n} +\frac{p (1-\lambda )}{n}. \end{aligned}$$

(A15)

Since each firm needs to be indifferent between setting an price over a support $[p^*,\bar{q}]$, we can write

$$\begin{aligned} p \mu _t \lambda [1-F_t(p)]^{n-1} +\frac{p \lambda (1-\mu _t)}{n} +\frac{p (1-\lambda )}{n}= \frac{\bar{q}(1-\mu _t)}{n} +\frac{\bar{q} (1-\lambda )}{n}. \end{aligned}$$

(A16)

Rearranging yields the expression in (4). We can then solve

$$\begin{aligned} p^* \mu _t +\frac{p^*(1-\mu _t)}{n} +\frac{p^* (1-\lambda )}{n}= \frac{\bar{q}(1-\mu _t)}{n} +\frac{\bar{q} (1-\lambda )}{n} \end{aligned}$$

(A17)

for $p^*$ which yields (5).

The comparative statics with regard to $\lambda$ in Proposition 5 are derived by straightforward differentiation. $\square$

Appendix B

Let us reconsider the all-or-nothing search model that was posed in Sect. 2 with the following differences: To make the analysis easier, suppose that in each period, every firm chooses prices from $[0,\infty )$ for each of its products. More interestingly, let us relax the assumption with regard to random allocation of the consumer to products in the market when she does not search. Let us suppose that for any $t \ge 2$, if the consumer does not search, she does not incur a cost and remains with the firm with whom she previously transacted. We call this the consumer’s incumbent firm.

Proposition B1

(Inertia) Suppose that n firms compete in a dynamic all-or-nothing search setting with inertia. Then, there exists an equilibrium with$L^*=\bar{L}$and

• 1. In all odd-numbered periods ($t=1, t=3,\ldots$), the consumer searches,$a^*_j=0$for all j, and $p_j^{m,*}=0$for all $j\in N,m \in \ell _j$.

• 2. In all even-numbered periods ($t=2, t=4,\ldots$), no consumer search occurs, $a_j=1$for all j,$p_j^{1,*}=\bar{q}$for all j, and $p_j^{m,*}>0$for all $j \in N, m \in \ell _j$such that$m > 1$.

Each firm earns discounted expected profits equal to

$$\begin{aligned} \varPi ^*_{j}=\frac{\delta \bar{q} }{n(1-\delta ^{2})} - \ell ^*_j\kappa , \end{aligned}$$

(B1)

and the consumer’s expected discounted surplus is

$$\begin{aligned} U^*=\frac{\bar{q}}{1-\delta ^{2}}. \end{aligned}$$

(B2)

Proof of Proposition B1

Suppose that there are $L>n$ products that are offered in the market and that the consumer does not search in period $t=1$. The consumer is willing to buy the product to which she was allocated as long as it offers non-negative utility. If the consumer is randomly allocated to firm j and $\ell _j>1$, the firm optimally chooses $a_{j}=1$. For any $\ell _{j}$, each firm j optimally sets prices $p^{1}=\bar{q}$ and $p^{j}>0$ for $j \ne 1$, and its profits are $\bar{q}$. All other firms earn zero profits during that period, and the consumer earns zero surplus.

Suppose that the consumer does not search in period $t \ge 2$ and remains with her incumbent firm. Again, the consumer is willing to buy a product as long as it offers non-negative utility. If $\ell _j>1$, the firm optimally chooses $a_{j}=1$. For any $\ell _{j}$, each firm j optimally sets prices $p^{1}=\bar{q}$ and $p^{j}>0$ for $j \ne 1$, and its profits are $\bar{q}$. All other firms earn zero profits in that period, and the consumer earns zero surplus.

Now, consider that the consumer searches in period t and selects the product with the highest utility. Bertrand competition between the firms that offer identical products leads to $p_{j}^{1}= 0$ for all j. To see this, assume that $p_{j}^{1}=0$ for all j, and that firm k deviates by setting $p_{k}^{1}>0$. Since $p_{k}^{1}>0$, the consumer would purchase the special product from another firm. Therefore, increasing prices is not a profitable deviation.

Hence, the consumer searches in period t, if and only if $\bar{q}>c(x_{t-1})$. If the consumer did not search in period $t-1$ (i.e., $x_{t-1}=0$), she searches in period t because $c(0)<\bar{q}$. Recall that $\bar{L}$ is the smallest integer such that $\bar{q} \le c(\bar{L})$. If $L < \bar{L}$, the consumer searches in every period, and each firm earns zero discounted expected profits. If $L \ge \bar{L}$, the consumer searches in all odd-numbered periods ($t=1, t=3,\ldots$) and does not search in all even-numbered periods ($t=2, t=4,\ldots$). In this case, each firm j earns discounted expected profits equal to

$$\begin{aligned} \varPi ^*_{j}=\frac{\delta \bar{q} }{n(1-\delta ^{2})} - (\ell _j-1)\kappa , \end{aligned}$$

(B3)

and the consumer’s expected discounted surplus is

$$\begin{aligned} U^*=\frac{\bar{q}}{1-\delta ^{2}}. \end{aligned}$$

(B4)

Suppose that in equilibrium all other $n-1$ firms choose to produce a total of $x<\bar{L}$ products. We now show that firm j prefers to produce $\bar{L}-x$ products to deter search in all even-numbered periods. The assumption that $\kappa$ is small assures this to be the case. Further, when the consumer searches in odd-numbered periods, firm j has a strict incentive to set $a_j=0$ since $c(\bar{L}-1)<\bar{q}$.

Suppose that the total number of products offered in the market is equal to $\bar{L}$. If a particular firm j produces $\ell _j^*+1$ instead of $\ell _j^*$ products, it incurs an extra cost $\kappa$ and thus reduces its expected discounted profits. Now, suppose that the firm only produces the special product. It avoids paying $(\ell _j^*-1)\kappa$ in product line costs, but its (expected) per-period profit drops to zero. Because $\kappa$ is small, this is never a profitable deviation.

Hence, there is a unique equilibrium number of products $L=\bar{L}$. $\square$