Abstract
The model with capital from Chap. 4 has a number of plausible properties, but there is still room for improvement. For example an undesirable feature of the model is that consumption and output per capita fall over time with population growth. This is due to the fact that the total amount of capital was initially given. However, if we extend the model to multiple time periods, it is possible to balance population growth with a higher savings rate and thereby to hold the stock of capital per capita constant over time.
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- 1.
Note that there are two interpretations for the time preference rate: over shorter time horizons it compares the utilities of a household obtained in different time periods; over longer time horizons it compares the utilities of different generations.
- 2.
However, the initial stock of capital, K 0, is still exogenously given.
- 3.
See Exercise 5.1.
- 4.
If t = 0, the lower bound of the index τ exceeds the upper bound, and the product is a so-called empty product and defined to be 1.
- 5.
Note the difference between K t and \(K_t^d\). K t is the stock of physical capital that the firm uses to produce Y t in period t. \(K_t^d\) is the firm’s demand for financial capital, i.e., the issuance of bonds in period t.
- 6.
See Magill and Quinzii (1996, Chapter 6) for a thorough discussion.
- 7.
See Exercise 5.1.
- 8.
The first order conditions with respect to capital and the no-Ponzi-game conditions are sufficient conditions for optimality. Appendix B shows that any solution to these conditions solves the optimization problem of the household. Similar arguments can be made for the firm and the central planner.
- 9.
An example with a specific utility function is given in Exercise 5.3.
- 10.
Section 4.4 explains why the utility function of the representative agent takes this form.
- 11.
This is the case because with each additional unit of labor more of the good can be produced, but the consumer has no disutility from working more.
- 12.
Note that we can drop \(\bar L_{0,n}\) since it is a constant. Multiplying the objective function with a constant does not affect the optimal allocation of the maximization problem.
- 13.
To have a stationary solution from the beginning, the initial endowment needs to be \(K_{0,n}=k^* \bar {L}_{0,n}\). If this does not apply, then in the long run the economy will still move toward the stationary equilibrium, but in the short run, consumption per capita is not constant.
- 14.
See for example Marx (1968) for Marx’s writings on capital.
- 15.
Note that we can find parameter values so that the share of income from capital in total income becomes greater than one but this corresponds to equilibrium capital in labor-efficiency units being negative.
- 16.
One of the Modigliani–Miller Theorems states that the firm is indifferent between financing investments with retained earnings or with debt. Here we assume that the amount of debt is zero in every period (see Sect. 5.2).
- 17.
Note that r needs to be bigger than g for prices to be finite.
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Hens, T., Elmiger, S. (2019). Extension of the Model to an Infinite Horizon. In: Economic Foundations for Finance. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-030-05427-4_5
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