Misperception and Cognition in Markets

By inergency On Mar 26, 2024

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3.1. Formal Details

The Environment: There are

N \geq 1

profit-maximizing firms that each provide a finite menu of products to a consumer. A product is a bundle

x \equiv (q, t)

, where

q \geq 0

denotes the quality or level of provision of some good and

t \in R

is the price. Providing a product at quality q costs firm i

C (q)

i = 1, \dots, N

, where C is strictly increasing, strictly convex, differentiable, and

C (0) = C^{'} (0) = 0

. Let h denote the inverse of

C^{'}

(the derivative of C), which exists since C is strictly convex4.

If the consumer chooses product

x = (q, t)

from firm i, firm i earns profits equal to

t - C (q)

. Else, if the consumer does not purchase from firm i, firm i earns zero profits. Let

X_{i}

denote the finite product menu designed by firm i and let

X = ⋃_{i = 1}^{N} X_{i}

denote the entire set of products that the consumer can choose from. We assume that the consumer chooses only one product from

X

Consumer Preferences: The consumer decides whether to select a product from

X

or not. If the consumer has preference parameter

\tilde{θ} \geq 0

, the consumer values product

x = (q, t)

according to

$U (x | \tilde{θ}) \equiv \tilde{θ} q - t .$

(1)

Instead, if the consumer does not purchase a product from $X$ , they receive reservation utility of zero, which is equivalent to selecting the product $(0, 0)$ for all $\tilde{θ}$ . Hence, we assume without loss of generality that $(0, 0) \in X$ .

Sophistication and Naiveté: The consumer will be in one of two cognitive states: they will either be sophisticated or naive. More precisely, assume that the true preference parameter of the consumer is given by $θ > 0$ . If the consumer is sophisticated (denoted by S), they understand that their true preference parameter is $θ$ and, as such, value product x according to $V_{S} (x) \equiv U (x | θ)$ . Instead, if the consumer is naive (denoted by N), they misperceive their preference over products by overvaluing provision of the good relative to its price. Specifically, they perceive that their preference parameter is given by $θ + Δ$ , $Δ > 0$ , so that they value product x according to $V_{N} (x) \equiv U (x | θ + Δ)$ . We call $Δ$ the consumer’s degree of naiveté as it captures the extent to which a naive consumer misperceives their true preference. Notice that $V_{N} (x) > V_{S} (x)$ for any product $x = (q, t)$ with $q > 0$ . Hence, the naive consumer over-values products relative to their sophisticated counterpart.

Endogenous Naiveté: The consumer is initially naive but can endogenously transition from naiveté to sophistication through an investment in cognition. If they do not choose to make this investment, then the consumer utilizes the misperceived preference $θ + Δ$ to make decisions. Formally, let $a \in {S, N}$ denote the cognitive investment decision of the consumer, where $a = S$ implies an investment in cognition (i.e., a decision to become sophistication). We assume that such an investment is costly: the consumer incurs a cognitive cost of $κ > 0$ if and only if they choose $a = S$ .

Suppose that the consumer faces menu of products

X

. Let

x_{k}

denote a product that the consumer in cognitive state

k \in {S, N}

selects; that is,

x_{k} \in arg max_{x \in X} V_{k} (x)

. Then, the consumer’s true utility from choosing to be in cognitive state

a \in {S, N}

when facing menu

X

is given by

$V (a | X) \equiv V_{S} (x_{a}) - κ \times I [a = S],$

(2)

where $I (\cdot)$ is an indicator function. It is optimal for the consumer to invest in cognition if $V (S | X) \geq V (N | X)$ and it is optimal for the consumer not to invest in cognition when $V (S | X) \leq V (N | X)$ .

Cognitive Equilibria: We now define the relevant equilibrium concept that we use to make predictions, which we call a cognitive equilibrium. To do so, let $σ_{k} : X \to X$ denote a selection rule for the consumer in cognitive state $k \in {S, N}$ and $α : X \to {S, N}$ a cognitive-investment strategy from an arbitrary menu $X$ . Let $π_{i} (X_{i}, X_{- i}, σ_{S}, σ_{N}, α)$ denote the expected profit of firm i when offering menu $X_{i}$ , given that the rest of the firms offer jointly menu $X_{- i} \equiv ⋃_{j \neq i} X_{j}$ and the consumer’s selection rules are given by $σ_{k}$ , $k \in {S, N}$ and cognitive strategy is $α$ . A notion of equilibrium in which the consumer’s cognitive state is endogenously determined is now defined.

Definition 1.

The tuple $(X_{1}^{*}, \dots, X_{N}^{*}, σ_{S}^{*}, σ_{N}^{*}, α^{*})$ is a cognitive equilibrium if, for all finite menus of products $X$ ,

(i): $σ_{k}^{*} (X) (x) > 0$ only if $x \in arg max_{\tilde{x} \in X} V_{k} (\tilde{x})$ for $k \in {S, N}$ ; and
(ii): $α^{*} (X) (a^{'}) > 0$ only if $a^{'} \in arg max_{a \in {S, N}} V (a | X)$ ,

and $X_{i}^{*}$ maximizes $π_{i} (X_{i}, X_{- i}^{*}, σ_{S}^{*}, σ_{N}^{*}, α^{*})$ for each firm $i \in {1, \dots, N}$ .

Essentially, a cognitive equilibrium is a subgame perfect equilibrium of a market game in which, first, each firm simultaneously designs a menu of products and, given the menu of products faced, the consumer decides whether or not to invest in cognition and makes a product selection given the realized cognitive state. Condition (i) of Definition 1 requires the consumer to only select products that are optimal from the perspective afforded by each cognitive state that may realize. Instead, condition (ii) requires that the cognitive-investment decision is determined optimally given how the consumer would behave in each cognitive state. The final requirement is that the product-menu provided by a given firm constitutes a best response to other firms’ menus given the consumer’s decision rules.

3.2. Efficient and Exploitative Good Provision

We now define the market outcome variables that are of interest in this paper. Let $W_{\tilde{θ}}$ denote the maximal achievable total surplus when the preference parameter of the consumer is given by $\tilde{θ}$ ; that is, $W_{\tilde{θ}} \equiv max_{q \geq 0} \tilde{θ} q - C (q)$ . The maximizer of this problem is $h (\tilde{θ})$ , so that $h (\tilde{θ})$ is the efficient level of provision of the good if the consumer has preference parameter $\tilde{θ}$ . Since the true preference parameter of the consumer is $θ$ , we have the following natural definitions of both an efficient product and an efficient market.

Definition 2.

A product $x = (q, t)$ is efficient if $q = h (θ)$ . A market is efficient if it generates total surplus equal to $W_{θ}$ .

Note that there always exists an efficient product that the consumer is willing to accept even if they are naive6. As such, the market can be efficient only if the consumer does not invest in cognition. Instead, if the consumer invests in cognition, the highest surplus that such a market can attain is

W_{θ} - κ

Next, we define what is meant by consumer exploitation in our framework. Recall that the consumer can guarantee herself a reservation utility of at least zero by choosing the outside option. This leads to the following natural definition of consumer exploitation in the market.

Definition 3.

The market is exploitative if the true utility of the consumer is strictly lower than zero.

Definition 3 states the market is exploitative if the consumer earns true utility lower than their reservation level. This can happen either because the consumer naively chooses an exploitative product (i.e., one in which they overpay for provision) or because the consumer incurs cognitive costs that exceed the benefit of whichever product is selected under sophistication.

3.3. Discussion of Key Modeling Assumptions

We have modeled the consumer’s misperception in a relatively general way: the consumer simply over-values provision of the good relative to their true preference. This could be for a number of reasons. For example, the consumer may be subject to framing or advertising effects that make them over-value the product [15], the consumer may not be paying attention to certain factors that affect their preference for the good [25], or may have alternative self-serving motivations to hold biased beliefs (e.g., thinking that they are more likely to go to the gym as it allows them to feel motivated to get fit). We do not take a stand on precisely the reason for this misperception and instead focus on the consumer’s incentives to correct this bias through costly cognitive investment.

Regarding cognition, we have assumed the consumer has access to a very simple cognitive technology: if they invest in cognition they learn their true preference, $θ$ , (i.e., become sophisticated), while if they do not make this investment they make decisions using the misperception, $θ + Δ$ (i.e., stay naive). We do this to starkly illustrate how cognitive investment can serve as a bridge between naive decision-making and its sophisticated counterpart.

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