Abstract
The paper develops a general equilibrium endogenous growth model involving heterogeneous consumption by an age-structured population with uncertain but limited life span and balanced life-time budget without bequests. The heterogeneity is introduced via weights which the individuals attribute in their utility function to consumption of different goods depending on the vintage of the good. The goods are produced by monopolistically competitive firms and the variety of available goods/technologies is determined endogenously through R&D investments. A competitive bank sector provides financial resources for investments, secured by agents’ savings and future firms profits. The general equilibrium is characterized by a system of functional equations and is analytically or numerically determined for several particular weight functions. It is shown that the investments by agents alone may be insufficient to sustain growth, while additional investments provided by the bank sector may lead to growth. The resulting imbalance between agents’ assets and the total value of firms can grow unboundedly in the case of homogeneous consumption. The results exhibit the qualitative difference between the dynamics of the model with heterogeneous versus homogeneous consumption. In particular heterogeneous consumption (when old goods are discounted) reduces the additional investments by the financial sector so that the values of firms become balanced by the assets of agents in the long run.
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Notes
The wage w(t) ≡ 1 is assumed to be equal for all jobs, which is reasonable in a model where qualification is not taken into account.
The equal saving/debt redistribution of deceased agent only within her cohort satisfies the balance of fare insurance, because agents from the same cohort have the same assets and probability of death. This redistribution is possible due to the assumption that there are always some people in the cohort until it becomes ω years old (n(τ, t) > 0 for 0 ≤ t − τ < ω).
The functional form \(m\!\left (\tau ,t,q,Q(t)\right )\) is given exogenously, thus it differs from the quality in quality-adjusted Dixit-Stiglitz consumption index used in some models with vertical innovations (e.g., Dinopoulos & Thompson 1998)
The net profit of the firm is its operating profit π minus taxes (that will be introduced later) and minus interest on debt.
We do not require the aggregated assets of agents to be equal to the total debts of firms like it is done in Sorger 2011, because in our model the optimal consumption profile and investments of a finitely living agent are completely defined via initial condition (6) and terminal conditions (7), like in Cass and Yaari 1967; d’Albis and Augeraud-Véron 2011, which is not the case for infinitely living households in Sorger (2011), where he needs additional condition V(t) ≡ A(t) to specify general equilibrium.
The case of γ = 0 corresponds to homogeneous consumption studied above.
We consider exponentially heterogeneous preferences, where in the limit \(\tilde {M}(\infty )\) is finite. Although, in the excluded cases of not exponential discounting of old goods, \(\tilde {M}(Q)\) could be unbounded. For example, when m = q/Q(t), we would have \(\tilde {M}(Q) = \frac {1-\alpha }{2-\alpha } Q \rightarrow \infty \) as \(Q \rightarrow \infty \), as follows from the definition of \(\tilde {M}\) and Eq. 58. Then, the total value of patents would be \(V(t) = \frac {1-\alpha }{2-\alpha }\frac {(Q(t))^{1-\varphi }}{\beta }\), which is close to Eq. 67 in the homogeneous case.
Note that for φ < 0 even small tax rate δ > 0 improves agents’ utilities in the long run, because the derivative \(\left .\frac {\text {d}}{\text {d} \delta }\frac {\left . u(\tau )\right |_{\delta = 0}}{\left . u(\tau )\right |_{\delta > 0}}\right |_{\delta = 0} = \alpha \varphi \frac {1-\alpha }{1-\varphi }\) becomes negative. A value φ < 0 means that past inventions make it more difficult to find new ideas, which we think to be unlikely.
References
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The authors would like to thank for helpful comments Alain Venditti, David de la Croix, and all participants of Workshop Overlapping Generations Days, Marseille, May 14-16, 2012. We also want to express our gratitude for fruitful discussions to Michael Kuhn and Klaus Prettner. This research was partly financed by Wiener Wissenschafts-, Forschungs- und Technologiefonds (WWTF) under grant No MA07-002 and by the Austrian Science Foundation (FWF) under grant No I 476-N13.
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Belyakov, A.O., Haunschmied, J.L. & Veliov, V.M. Heterogeneous consumption in OLG model with horizontal innovations. Port Econ J 13, 167–193 (2014). https://doi.org/10.1007/s10258-014-0105-7
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DOI: https://doi.org/10.1007/s10258-014-0105-7