Pricing through ambiguity: a flocking model of the inter-dynamics between pricing practices and market uncertainties

ABSTRACT

Pricing practices of firms are an important yet the least studied aspect of the price phenomenon in sociology. This paper answers the question: why do firms, even in the same market, tend to use different pricing practices – value-informed, competition-informed, or cost-informed pricing – to set prices? To that end, this study constructs a formal dynamic flocking model to investigate the inter-dynamics between market uncertainties and the viability of the three pricing practices. The model is a substantial revision and extension of Harrison White’s static W(y) market model by reformulating the latter into a dynamic one and by explicitly incorporating different market uncertainties into the model as variables. The study shows that each kind of pricing practice is only viable under certain distributions of market uncertainties. The theory is then used to explain the distribution of pricing practices among firms in the Burgundy wine market. Theoretical and methodological innovations and the implications for firms and for sociological research on markets and uncertainties are also discussed.

KEYWORDS:

• Uncertainties
• pricing practices
• flocking model
• W(y) market model

Acknowledgments

I gratefully acknowledge funding from the Graduate School of Arts and Sciences at Columbia University, Institute for Humane Studies at George Mason University, and Dickinson College. For helpful comments, I am grateful to Yinon Cohen, Randal Collins, Felix Elwert, Mark Gould, Eric Leifer, Shiding Liu, James Montgomery, Paolo Parigi, Michael Sobel, Yuan Shen, Harrison White, and Josh Whitford.

Notes

¹ These sociological studies have important limitations. Since the models estimated are regressions, they can only be interpreted as price-association rather than price-determination models. Moreover, price is usually isolated inappropriately from other economic variables, such as revenue, cost, and output.

² Pricing practices should also be distinguished from pricing strategies. The latter pertain to price mechanisms; they are the results of market-level forces and usually modeled using game theory. Pricing strategy refers to a firm’s short-term calculative strategy of setting prices to achieve specific market goals (for example, to increase its market share, or to deter the entry of new firms). A price war between two firms is an example of pricing strategy. For a given firm, pricing strategy usually changes often, as it is very responsive to market change. In contrast, pricing practice refers to some more general and stable organizational orientation of the firm; what the concept emphasizes is not the final pricing decision/strategy itself, but what major information (costs, competitors’ decisions, or how consumers think of the product) is routinely used and how such information is usually processed and interpreted within the firm for making pricing decisions (Ingenbleek et al., 2003).

³ It is true that customers’ perception of values and quality of goods typically depends on price (i.e., price often serves as a signal for quality; Stiglitz, 1987). However, this dependence of values/quality on price is not explicitly modeled throughout this study. There are four considerations for this modeling decision. First, it is reasonable to assume that consumers’ perception of quality is not very sensitive to relatively small price changes. For example, if a smartphone manufacturer offers a promotional discount of $25 for its smartphones within a certain period (this is usually how firms adjust pricing in the short run), consumers are unlikely to perceive this as signaling a drop in quality. By contrast, if the company offers a 60% discount for one of its phone models, customers are likely to think that there may be some major problems with that model. It is the former that is considered here (as explained later, the model deals with near-equilibrium dynamics, which means changes in price are relatively small within a short period). Therefore, perceived quality is treated as an “exogenous” variable here. Second, however, this does not mean that perceived quality considered in this study cannot depend on some sense of overall price level. For example, customers may notice that the long-term overall price level of company A’s smartphones is higher than that of company B’s. As a result, customers may perceive that the former has a higher quality than the latter. This is not contradictory to the first point, and perceived quality, even if dependent on long-term overall price level, can be seen as a given exogenous variable if price changes are small in the short run. Third, when competing firms change prices in response to each other’s pricing decisions, consumers are unlikely to see these price changes as signals for quality changes. Fourth, price is not always correlated with quality. For example, if a firm owns a patented technology that can significantly reduce the production cost of its product while greatly improving quality, the firm can consistently offer a price lower than its competitors to increase its market share even if its product has a higher quality. Thus, it makes sense to treat perceived quality as given for the modeling purpose here.

⁴ A real firm’s pricing practice could be a hybrid of the three, but usually with a chief orientation.

⁵ Although Ingenbleek et al. (2003) advocate a “contingency approach” to pricing practices, their study reinforces the universal superiority of value-informed pricing: “Unlike cost-informed and competition-informed pricing, we find no situation in which value-informed pricing can be considered ‘bad practice’ … This confirms conventional marketing wisdom that understanding the customer’s value perception is key to successful pricing” (p. 301).

⁶ In the W(y) model, firms also need their own cost information when making pricing decisions. Yet the predominantly important piece of information is the pricing decisions of competitors. Thus, the W(y) model corresponds to competition-informed pricing.

⁷ Unlike game theoretical models of pricing strategy which are highly sensitive to the specification of game setting, the W(y) model captures nicely the long-term, general, and stable character of the informational aspect of competition-informed pricing. Thus, the model is a good representation of competition-informed pricing.

⁸ The viability conditions of competition-informed pricing to be specified here differ from those already discussed by White. The issue to be addressed here is that not all market conditions judged viable for competition-informed pricing by White’s static theory are really viable under dynamic conditions.

⁹ The demand curve describes the relation between price p and volume y, whereas the revenue curve is about the relation between revenue R and volume y. Note that the three – p, R, and y – are not independent, since R = p*y. Once any two of the three are known, so is the third. Thus, the demand curve and the revenue curve are equivalent.

¹⁰ This assumes the realized revenue and volume of each firm are shared information. This assumption can be relaxed. For example, each firm may only collect information about several, rather than all, competing firms. This would not change the nature of the model. Moreover, the assumption of synchronized behavior among firms is not necessary for the model to hold, but it makes the situation simple.

¹¹ This can be relaxed to allow firms to choose different functional forms to fit the R_t(y) curve.

¹² It is assumed throughout that firms do not keep stocks, i.e., in each run firms would adjust the original asked price (p_it = R_it/y_it*) until the entire volume of output is sold out.

¹³ Firms in reality need not explicitly conduct these calculations. They could just perform some coarse estimation. What matters is: each firm uses the realized outcomes of its peers as reference to locate its own “niche” in the price hierarchy.

¹⁴ Empirical measures of viability, such as the varying prevalence of different pricing practices among the firms in a market, are not usable in the current study, because the individual choices of pricing practices made by firms are not directly modeled. Thus, a composite qualitative criterion of viability is needed. Here, the most stringent one – the “short board” criterion – is followed, that is, the overall viability of a pricing practice is determined by the factor that makes it least viable. Therefore, if a factor renders a pricing practice nonviable, it is nonviable regardless of the value of any other factor.

¹⁵ This means that the fact that the recognitive uncertainty associated with competition-informed pricing is high (or low) does not suggest that the recognitive uncertainty associated with the other two pricing practices is necessarily high (or low).

¹⁶ As the methods used for prediction are different among the three practices, the fact that the predictive uncertainty associated with one practice is high (or low) does not imply that the predictive uncertainty associated with the other two practices are high (or low).

¹⁷ In contrast to the current literature on qualification and valuation which usually emphasizes the multi-dimensionality of valuation (that is, speaking of “values” instead of value, and “qualities” instead of quality), this study focuses only on a single composite index for the overall valuation of a product, which is called “quality.” In other words, “quality” here refers to some generalized quality that subsumes all possible dimensions of valuation (including status). However, the purpose of this simplification is merely practical: since the buyer has to buy or not buy the product as a whole (not just one or several of its many features), s/he has to evaluate the product as a whole, including both the relevant physical and nonphysical features with status being one of these. Such an overall evaluation is the “quality” meant here.

¹⁸ Note that such ambiguity persists even after customers use the product, in contrast to the usual framing that customers reveal the true product quality after purchasing it.

¹⁹ The effects of these factors on perceived quality of products (just like the effect of price on perceived quality) are not explicitly modeled here; rather, these are treated as given exogenous factors, the effects of which are all absorbed into the final realized quality perceived by consumers.

²⁰ An analogy is the two-sample t-test. High absolute quality ambiguity is analogous to saying that the two samples’ estimated standard deviations are big. However, if the distance between the two estimated means of the samples are also big enough, the difference in mean could still be statistically significant. That is, statistically speaking, we can distinguish the two underlying distributions of the two samples, which is analogous to saying that the relative quality ambiguity is low (even though the absolute quality ambiguity is high).

²¹ Without product differentiation, the revenues of firms would be driven down toward their costs due to competitive pressure. Thus, information of cost, rather than that of competitors, would become the most important. As a result, cost-informed pricing, not competition-informed pricing, would prevail. Value-informed pricing is based on the “unique” value of the product perceived by consumers, which also depends on quality differentiation. If relative quality ambiguity is high, quality differentiation is obscured. This means the demand of a firm’s product can be easily influenced by its competitors’ pricing decisions, which makes value-informed pricing nonviable.

²² An issue is how to mathematically model uncertainty. The approach here is to use probability distributions to approximate the properties of uncertainty, since techniques dealing with probabilities are well-developed. However, this does not convert the issue of uncertainty into one of risk, because it is stipulated in the model that all of the parameters and information concerning the assumed probabilistic distributions are unknowable to the main market actors – the competing firms – and cannot be used in their decision making.

²³ There are alternatives that incorporate information from earlier iterations (e.g., Bayesian updating). However, the Markov model is technically simple and substantively reasonable. Due to elusive demand, market information quickly becomes outmoded and useless, and the Markov-like assumption captures this situation.

²⁴ Since the static W(y) model restricts the range of quality n to [0, +∞], letting σ << n_i makes the current model practically consistent with the static one. Another possibility is to assume N_i follows a Log-normal or Gamma distribution, but the normal distribution is simpler.

²⁵ The equation is adapted from White (1981b, p. 521, Equation (2)(2) ${U^i}_{pr.}={\sum }_{t=1}^T{\left(R_{it}^{\prime }-R_{it}\right)}^2/T,\mathrm{o}\mathrm{r}$(2) , and p. 525, Equation (4)(4) ${U^M}_{pr.}={\sum }_{i=1}^I{U}_{pr.}^i/I,$(4) )(4) ${U^M}_{pr.}={\sum }_{i=1}^I{U}_{pr.}^i/I,$(4) . A salient difference between the two is that the latter has an unambiguous quality n, while the current function has a random variable N_i representing the absolute quality ambiguity. Another difference is that Equation (10)(10) $V\left(y,N_i\right)=\frac{r{y}^a{N}_i^b}{\theta },\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}r,a,b>0,\theta \ge 1,\mathrm{a}\mathrm{n}\mathrm{d}{N}_i\in \left(n_i-2\sigma ,n_i+2\sigma \right),$(10) integrates the “discount” θ into the valuation function.

²⁶ The “discount” parameter θ may be time-varying. For run t, the “discount” (now denoted by θ_t) can be modeled as θ_t = max {θ_1t, θ_2t, …, θ_It}, where θ_it = $r{y}_{it}^a{n}_i^b$/R_it, for i = 1, 2, …, I. The numerator is the “undiscounted” (average) value of firm i’s product with volume y_it (see Equation (10)(10) $V\left(y,N_i\right)=\frac{r{y}^a{N}_i^b}{\theta },\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}r,a,b>0,\theta \ge 1,\mathrm{a}\mathrm{n}\mathrm{d}{N}_i\in \left(n_i-2\sigma ,n_i+2\sigma \right),$(10) ), and the denominator R_it is the total price the firm asks for this volume. Thus, θ_it represents the “discount” offered by firm i, and consumers choose the greatest “discount” (from all θ_it’s) to impose uniformly on all firms, which becomes θ_t. This captures consumers’ bargaining power and the price competition among firms. However, as made clear later, the current model focuses on near-equilibrium dynamics, so θ_t is fixed at its equilibrium value θ (to be set as a simulation parameter) for the sake of simplicity. That is, the dispersion among all the “discounts” offered by firms is small enough (relative to relative quality ambiguity) so that consumers cannot discern or do not care about the differences among them.

²⁷ There are alternatives to this stipulation. For example, consumers could pay some random amount distributed between the lower and the upper bounds.

²⁸ Equation (12)(12) $C\left(y,n_i\right)=q{y}^c/n_i^d,\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}q,c>0,$(12) is similar to White (1981b, p. 521, Equation (1)(1) $U_{rc.}=1-R^2,$(1) )(1) $U_{rc.}=1-R^2,$(1) . However, different from White’s original cost function where n stands for an unambiguous quality level, here n_i stands for the mean of the quality distribution N_i.

²⁹ Insofar as costs are prices, there are uncertainties in costs too. However, to simplify the model, it is assumed that the supply side of the firm is steady, that is, no big unexpected changes in upstream supplies and prices of raw materials and parts. White (2002) develops a more sophisticated model integrating markets from upstream of the supply chain; in theory, it is possible to incorporate uncertainties into the supply side in that more complex model.

³⁰ The first order condition is: $d\left[R_t(y)-C_i(y)\right]/dy=0$. After substituting in Equations (13)(13) $C_i\left(y\right)=A_i{y}^{c_i},$(13) and (15(15) $R_t\left(y\right)=B_{0t}{y}^{B1t},\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}:$(15) ), it becomes $d\left(B_{0t}{y}^{B_{1t}}-A_i{y}^{c_i}\right)/dy=0$. One need not worry about the second order condition, since the current market is assumed to satisfy all the static viability conditions derived by White (1981a, 1981b). This means the solutions, if any, to the first order condition is guaranteed to be maxima, rather than minima.

³¹ The simulation settings in this paper are all based on the examples in White (1981a, 1981b), all satisfying the static viability conditions and thus all judged viable by White’s static model.

³² This somewhat arbitrary number results from the trade-off between two considerations. On the one hand, if there are too many firms in the market, it is unreasonable to assume each firm will treat all the other firms as its immediate competitors and keep tracking the information of all of them. On the other hand, if there are too few firms in the market, there would not be enough data points to reasonably assume all firms fit the curve with the same functional form. In fact, the model allows for a market with as few as three firms.

³³ That is, all three viable regions in White’s topology (White, 1981b, p. 527, fig. 3) are actually nonviable under dynamic condition if there is no quality ambiguity.

³⁴ If all firms in the market are using value-informed pricing, a new (equilibrium) market profile, called here the R’(y) curve, may result, and this R’(y) curve is very similar to the W(y) curve in terms of functional form (see detailed derivation of the R’(y) curve in appendix A). Thus, cross-sectional data, such as those used in White (1981b), are not sufficient to distinguish these two pricing practices. This implies that some empirical markets identified by White as following competition-informed pricing might actually use value-informed pricing.

³⁵ Settings 2 and 3 are adapted from the examples in White (1981a, p. 35, p. 25).

³⁶ In reality, the R_t(y) curve would not really collapse to zero, nor explode or resonate toward infinity – firms would abandon the mechanism long before that. The analytical condition for judging if a specific market setting is vulnerable to disturbance is derived in appendix B. It can be shown that many settings judged viable for competition-informed pricing by White (1981b, p. 527, fig. 3) are actually nonviable due to vulnerability to disturbance.

³⁷ That is, it is not the high quality/status per se that enables the firm to perform value-informed pricing (as intuition may tell us). Rather, it is the low absolute and relative quality ambiguities, usually associated with high mean quality, that allow the firm to do so.

³⁸ The authors explain the distribution of pricing practices using appellation and sales structures as explanatory variables, but the current theory offers a deeper explanation for the phenomenon – the effects of appellation and sales structures may themselves be some function of the distribution of market uncertainties.