Abstract
In this chapter we present the discrete time version of some of the issues discussed in the previous chapter. We introduce a government in the economy, and define and characterize the competitive equilibrium. The intertemporal government budget constraint, the relationship between the competitive equilibrium allocation and that of the benevolent planner mechanism, and the Ricardian doctrine, can be all analyzed in discrete-time in a similar fashion as we have done in the continuous time version of the model. Dealing with all the details of the discrete time version of the Cass–Koopmans economy is very instructive in order to be able to formulate alternative, more complex growth models, as well as to perform policy analysis, as we do towards the end of the chapter. It is particularly important to get familiar with the formulation and use of the transversality condition and with the characterization of stability conditions. As we will see below, stability conditions are crucial to generate a numerical solution for this model in the form of a set of time series for the endogenous variables.
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Notes
- 1.
Which, by redefining the discount factor as \(\beta = \frac{1} {1+\theta }\), can be written,
$$\displaystyle{ \frac{\lambda _{t+1} -\lambda _{t}} {\lambda _{t}} = \frac{n +\theta +\delta + n\theta - f^{{\prime}}(k_{t+1})} {f^{{\prime}}(k_{t+1}) + 1-\delta }, }$$in terms of the rate of change of the Lagrange multiplier, so that it can be compared to the similar condition in the continuous time model.
- 2.
The g(k t , c t ) function is obtained after using the budget constraint to eliminate k t+1.
- 3.
Remember the equivalence: \(\beta = \frac{1} {1+\theta }\).
- 4.
The proof is analogous to that in Sect. 3.4.
- 5.
Where it can be seen that, at a difference of the planner’s problem, government expenditures do not appear.
- 6.
As a consequence, the competitive equilibrium allocation will not be efficient.
- 7.
The rate of growth along the solution is clearly related to the absolute values of the μ 1, μ 2 roots. The critical rate of growth below which the solution is stable is model-specific. The requirement for a well-defined solution to exist is that the objective function remains bounded, which will require upper bounds on its variable arguments. Those bounds will depend on the functional form of the objective function. Sometimes, as in the Cass–Koopmans model, transversality conditions take care of that. In other cases, transversality conditions may be needed for feasibility or optimality even when the objective function is bounded, so that extra upper bounds on growth rates will then need to be added, to guarantee that transversality conditions hold. Note that a linear approximation to the set of first order conditions for the representative agent problem amounts to a linear-quadratic approximation to that problem. Hence, given a quadratic approximation to the objective function (i.e., \(\mathop{\sum }\limits _{ t=0}^{\infty }\beta ^{t}U(c_{t}) \simeq \mathop{\sum }\limits _{t=0}^{\infty }\beta ^{t}(a\tilde{c}_{t}^{2} + b\tilde{c}_{t} + d)\), where \(\tilde{c}_{t} = c_{t} - c_{\mathit{ss}}\)), it is clear that the sum will converge for solutions of the type \(\tilde{c}_{t} =\mu ^{t}\tilde{c}_{0}\), only if \(\left \vert \mu \right \vert < 1/\sqrt{\beta }\).
- 8.
After using equations that involve only contemporaneous values of decision variables (as it may be the case with some identities) to eliminate some of these decision variables from the problem.
- 9.
Initial consumption, in the Cass–Koopmans economy considered in this chapter. We will get back to this issue in the Mathematical Appendix.
- 10.
Even though we did not compute it that way. Rather, we obtained the linear approximation to the model, and obtained the exact stability condition for this approximated model.
References
Benhabib, J., and R. Perli. 1994. Uniqueness and indeterminacy: Transitional dynamics with multiple equilibria. Journal of Economic Theory 63: 113–142.
Lucas, R.E. 1987. Models of business cycles. Oxford: Blackwell.
Xie, D. 1994. Divergence in economic performance: Transitional dynamics with multiple equilibria. Journal of Economic Theory 63: 97–112.
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Novales, A., Fernández, E., Ruiz, J. (2014). Optimal Growth: Discrete Time Analysis. In: Economic Growth. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54950-2_4
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DOI: https://doi.org/10.1007/978-3-642-54950-2_4
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