Oligopoly Signal-Jamming Dynamics D. Bernhardt B. Taub University of Illinois Glasgow University∗yz University of Warwick June 22, 2018 Abstract We examine industry dynamics and firm learning in a stationary dynamic setting. Firms are subject to persistent common value demand shocks and private-value cost shocks that are private to each firm. The firms use current and past price signals to learn more about their rival's shocks, and each firm’s awareness of their rival's attempt to learn leads them to attempt to influence that learning|that is, to signal-jam. The evolving structure in firms’ private information causes firms to change how their outputs weigh current versus past fundamentals on common-value shocks very differently from private- value shocks. The weights placed by a firm directly on the common value demand shocks that it observes shrink over time, exhibiting less persistence than the fundamentals, even though ag- gregate (across firms, direct and indirect via price) weights rise, approaching weights on public information. Thus, firm profits from privately-observed common value shocks evolve in opposite directions over time. In contrast, weights placed on private value cost shocks rise over time in absolute value, but the (oppositely signed) weights placed by the rival not hit by the shocks rise faster, implying that aggregate weights fall over time. ∗This research was carried out in part during my stay at ICEF, Higher School of Economics, Moscow. I acknowledge financial support from the Academic Excellence Project \5-100" of the Russian Government. yWe thank William Fuchs and Tibor Heumann for helpful comments. zIO_draft_06212018/June 22, 2018 1 Introduction In the real world, firms are continually buffeted over time by shocks to demand and to costs, some of which they learn about through direct common observation, some through private observation, and some of which they attempt to extract from information in price signals. Some shocks are com- mon value in nature|for example, a common demand shock that raises demand equally for each firm’s product|entering directly into every firm’s profit function. Other shocks are private value in nature|for example, a productivity shock specific to one firm’s technology, or a firm-specific demand shock that raises demand only for one firm’s product|and hence only indirectly affect a rival's profits to the extent that the shock alters the output of the affected firm. In an oligopolistic industry, firms account for the strategic behavior of rivals when unraveling information from price signals, and for how their own actions influence the price signals that rivals receive, and hence their inferences and output choices. In a static setting, Bernhardt and Taub (2015) establish that firms weigh privately-observed private-value shocks by more than they weigh common-value shocks, making the information content of prices more sensitive to private-value shocks than to common-value shocks. Our current paper asks: What happens in a dynamic econ- omy where firms continue to learn over time about past shocks? How do a firm’s strategic choices change as it learns more from new price signals about past shocks observed by a rival, so that more of that information effectively becomes public? How do the weights firms place on newer vs. older signals evolve, and what are the consequences for the pace of learning? To establish how such learning evolves, we analyze a dynamic stationary setting in which firms are constantly buffeted by new shocks, and strategic behavior is unimpeded by artificial horizons. We take on the challenge posed by Mirman, Samuelson and Urbano (1993) who observe that \the most appropriate model [is] an infinite horizon model in which the parameters of demand curves are subject to continual shocks. Firms are then repeatedly forced to draw inferences about unknown demand curves and to consider the effects of their actions on their rival's beliefs." Now firms can learn about rivals' private shocks via both current and past price signals. In turn, they choose output strategically knowing that current output also influences a rival's future actions. We suppose that common-value demand evolves according to a persistent autoregressive stochas- tic process. In addition to a publicly-observed component, each firm privately observes innovations to common demand and to its private-value marginal cost of production. Firms also receive noisy price signals. A firm combines the information extracted from the history of price signals with that in the history of its privately- and publicly-observed innovations to determine how much to produce. We solve for the equilibrium of this dynamic oligopoly game. Our key insight is that if the un- derlying driving demand and cost processes are stationary, then equilibrium output strategies will be stationary linear functions of the history of private signals and prices. What makes the analysis challenging is that, in equilibrium, these functions are infinite sums of ar(1) terms: no finite set of sufficient statistics can summarize a firm’s information, or its optimal behavior. Even so, because the output strategy that maximizes a firm’s expected profits conditionally is linear, it also max- imizes expected profits unconditionally. We exploit this equivalence and solve this unconditional problem, using variational methods to find the optimal output functions. We develop an iterative best-response mapping by a firm to its rival's conjectured linear out- put strategy function. The variational first-order conditions describing the best response functions 1 reveal that public-information components of demand do not affect how a firm weights prices, and that only cumulative public-information demand matters for firm output, and not the timing of different shocks. In contrast, for privately-observed demand and cost shocks, not only does the aggregate level matter for output, but so does their timing|output intensity filters on privately- observed shocks are not scalar-valued. In particular, we prove that signal-jamming incentives lead firms to weigh newer private information more strongly so that outputs on private information decay more quickly than the fundamentals themselves. Equilibrium is described by best-response functions that are consistent with a rival's output functions. Equilibrium is given by a fixed point to the recursion that maps a rival's output function to a best-response function. When firms have private information only about demand, we establish the existence of a fixed point in the space of functions of a complex variable that are analytic and square summable on the unit disk, where the second-order conditions for optimization hold, yielding the equilibrium. We numerically characterize the properties of equilibrium strategies. To do this, we develop an alternative best-response recursion that allows for limited private value cost shocks. We numerically solve for the equilibrium by substituting an initial conjecture for the optimal strategy, solving for the best response, and then iteratively substituting the best responses as new conjectured optimal strategies. When private value cost uncertainty is not too high and the driving stochastic processes are not too persistent, the recursion is numerically stable. We disentangle and characterize the very different dynamic responses of firms to current and past demand and cost shocks. The dynamic economy magnifies incentives of firms to signal jam. This reflects that the over-production costs are the same as in a static economy, but the benefits of manipulating a rival's beliefs also accrue in future periods. As a result, a firm’s output on new pri- vate information about demand exceeds static monopoly full information levels, so the current price contains more information about contemporaneous shocks. Firms' dynamic output strategies do not merely replicate period-by-period static behavior|the equilibrium output processes are higher- order moving average, autoregressive processes, where the autoregressive parameters are smaller than those of the shock processes. This reduced persistence reflects that (1) when firms signal jam by weighing new privately-observed shocks more heavily, this (2) conveys more information to rivals via price signals (in equilibrium a rival is not fooled), (3) causing the rival to increase output at lags on that learned information, (4) which causes the firm seeing the shock to reduce its direct weight on lagged shocks more quickly. Rich structure is imposed on the time paths and co-movements of output and profits. Output on public information does not depend on the time composition of innovations, so the impacts of innovations to publicly-known demand on output decay at a constant rate. In contrast, with privately-observed common demand shocks, the output of the firm observing a shock drops more quickly than does its rival's. Thus, vis `avis publicly-observed shocks, there is \catch-up" in output on older shocks by the firm that does not see the shock. As a result, firm profits evolve together with publicly-observed demand shocks, but not with privately-observed demand shocks. In the aggregate, the same-sized shock to demand has a smaller effect on output when privately observed; but when privately observed, the decay in output on the shock at lags is slower. Related characterizations hold for privately-observed private-value cost shocks. At any given lag, the two firms collectively place the same output weight on a privately-observed cost shock as one firm 2 places on a privately-observed common demand shock. As an \uninformed" firm learns more via prices about older shocks observed by a rival, its output weight s rise on its rival's privately-observed older shocks; but the impact for the \informed" firm depends on the private/common value nature of the shock. The `informed' firm’s output weights on older privately-observed common value de- mand shocks fall, while its weights on privately-observed private value cost shocks rise.