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Learning and Technological Progress in Dynamic Games

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Abstract

We study investment and consumption decisions in a dynamic game under learning. To that end, we present a model in which agents not only extract a resource for consumption but also invest in technology to improve the future stock. At the same time, the agents learn about the stochastic process governing the evolution of public capital, including the effect of investment in technology on future stock. Although the characterization of a dynamic game with Bayesian dynamics (and without the assumption of adaptive learning) is generally intractable, we characterize the unique symmetric Bayesian-learning recursive Cournot–Nash equilibrium for any finite horizon and for general distributions of the random variables. We also show that the limits of the equilibrium outcomes for a finite horizon exist. The addition of learning to a stochastic environment is shown to have a profound effect on the equilibrium.

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Notes

  1. There is a two-way interaction between decision-making and learning. On the one hand, decision-making may have an effect on learning, which is referred as experimentation. On the other hand, the presence of learning adds risk which affects future payoffs and thus behavior.

  2. In a different context, Mirman and To [12] have addressed the issue of investing in capital in an overlapping generation model.

  3. See Bernhardt and Taub [3] for a recent paper on learning in oligopoly when the firms learn from prices.

  4. There is no experimentation in our model. For the literature on single-agent experimentation with capital accumulation, see Freixas [8], Bertocchi and Spagat [4], Datta et al. [5], El-Gamal and Sundaram [6], Huffman and Kiefer [9], and Beck and Wieland [2].

  5. There is no adaptive learning. Under adaptive learning, agents are bounded because they assume that beliefs will not change over time, i.e., they do not anticipate learning. See Evans and Honkapohja [7] for a detailed exposition of adaptive learning.

  6. One popular approach is to rely on the fact that the family of normal distributions with an unknown mean is a conjugate family for samples from a normal distribution.

  7. A tilde sign is used to distinguish a random variable from its realization.

  8. Since the investment goods are perfectly substitutable, there exist many equilibrium points in which only total investment can be determined. However, for any equilibrium and for any horizon, total investment remains the same whereas individual investment changes.

  9. Plugging (31) into (21) (ignoring the constant) and taking limits yields (52).

  10. Evaluating (52) at the true distribution (i.e., θ ∗ is known) yields (54).

References

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Acknowledgements

We thank two anonymous referees for their very helpful comments.

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Authors and Affiliations

  1. Department of Economics, University of Virginia, Charlottesville, USA

    Leonard J. Mirman

  2. Institute of Applied Economics and CIRPÉE, HEC Montréal, Montreal, Canada

    Marc Santugini

Authors
  1. Leonard J. Mirman
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  2. Marc Santugini
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Correspondence to Marc Santugini.

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Mirman, L.J., Santugini, M. Learning and Technological Progress in Dynamic Games. Dyn Games Appl 4, 58–72 (2014). https://doi.org/10.1007/s13235-013-0089-4

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