Optimal growth under discounting in the two-sector Robinson–Solow–Srinivasan model: a dynamic programming approach†

^†This essay is dedicated to Kazuo Nishimura, dear friend and admired colleague, on the occasion of his sixtieth birthday. We are grateful to the Center for Analytic Economics at Cornell University and to the Center for a Livable Future at Johns Hopkins University for research support. We thank a referee for useful comments on an earlier version of this paper.

Abstract

We use a version of a two-sector model to provide a strong form of a “folk-theorem” on the existence of a threshold discount factor such that: (i) for discount factors above this threshold value, optimal behavior is qualitatively similar to that in the corresponding undiscounted optimization problem, (ii) for discount factors below this threshold value, optimal behavior is qualitatively different from that in the undiscounted case. In the process, we provide an explicit solution of a non-linear optimal policy function for all discount factors above the threshold value. Our bifurcation analysis is conducted by using the dynamic programming approach, and we exploit the convex structure of our model to develop a variation of the standard method in dynamic programming used to identify the optimal policy correspondence.

Keywords:

• Dynamic programming
• Value function
• Optimal policy correspondence
• Bifurcation analysis
• Threshold discount factor

Keywords:

• C61
• D90
• O41

Notes

¹ Email: [email protected]

^†This essay is dedicated to Kazuo Nishimura, dear friend and admired colleague, on the occasion of his sixtieth birthday. We are grateful to the Center for Analytic Economics at Cornell University and to the Center for a Livable Future at Johns Hopkins University for research support. We thank a referee for useful comments on an earlier version of this paper.

This is admittedly a rather loose statement, but that is the nature of “folk theorems”. For more precise versions, see especially Refs. [3,7,14].

See Refs. [12,15,17]. A continuous-time version of the general RSS model, under discounting, was analyzed by Stiglitz [18]. An analysis of the discrete-time version of the general RSS model, when future utilities are not discounted, is contained in Ref. [4].

See Ref. [5] for the complete analysis of the corresponding undiscounted optimization problem in the two-sector RSS model, and an explicit solution of the optimal policy function in that case.

In particular, the first part of this statement implies that we obtain an explicit solution for the optimal policy correspondence when . For the second part of this statement, it is not important to actually obtain an explicit solution of the optimal policy correspondence for and we do not. See, however, our concluding remarks in the final section of the paper for some hints about what it might look like.

The best known instance of an explicit solution of a non-linear optimal policy function is the Weitzman example, reported in Ref. [13], and discussed extensively in Refs. [1,7,9].

Our exposition is deliberately brief. For a comprehensive account of the dynamic programming approach to optimal growth models, see Ref. [6].

On the other hand, explicit solutions of non-linear optimal policy functions (such as the tent map) have been obtained, given a discount factor, by choosing the transition possibility set and the utility function appropriately (dependent on the given discount factor). Such constructs, though useful in understanding other issues, are ill-suited to conducting bifurcation analysis with respect to the discount factor, given the transition possibility set and the utility function. For a full discussion of this point, and for the relevant literature, the reader is referred to Mitra and Nishimura [9].

For applications of this standard method in solving for value functions, see Ref. [16], and the references cited there.

Additional information

Notes on contributors

M. Ali Khan

1 1 Email: [email protected]