Multisector Optimal Growth Models Research Articles

Optimal growth in the Robinson-Shinkai-Leontief model: the case of capital-intensive consumption goods

Abstract We study the two-sector Robinson-Shinkai-Leontief (RSL) model of discrete-time optimal economic growth for the case of capital-intensive consumption goods. We frame the model in the context of Nishimura’s oeuvre, and more specifically, relate it to its neoclassical cousin: the Uzawa-Srinivasan continuous-time version studied in Haque, W. 1970. “Sceptical Notes on Uzawa’s ‘Optimal Growth in a Two-Sector Model of Capital Accumulation’, and a Precise Characterization of the Optimal Path.” The Review of Economic Studies 37: 377–394. We take Fujio’s identification of the marginal rate of transformation ζ of capital goods today into capital goods tomorrow (Fujio, M. 2006. Optimal Transition Dynamics in the Leontief Two-Sector Growth Model. Ph.D. thesis, The Johns Hopkins University), and using a mix of the Bellman-Blackwell methods of dynamic programming and the value-loss approach of Brock-Mitra, pin down the optimal policy for a specific subset of the parameter space. We discern a bifurcation pattern with respect to the discount factor that echoes the results of Benhabib, J., and K. Nishimura. 1979. “The Hopf Bifurcation and the Structure and Stability of Closed Orbits with Heterogeneous Steady States in Multi-Sector Models of Optimal Growth.” Journal of Economic Theory 21: 421–444 for the neoclassical model, and Khan, M. A., and T. Mitra. 2005. “On Choice of Technique in the Robinson-Solow-Srinivasan Model.” International Journal of Economic Theory 1: 83–110 for the RSS model. We also study the optimal policy correspondence for a general parameter set.

Weak concavity properties of indirect utility functions in multisector optimal growth models

Studies of optimal growth in a multisector framework are generally addressed in reduced‐form models. These are defined by an indirect utility function which summarizes the consumers’ preferences and the technologies. Weak concavity assumptions of the indirect utility function allow one to prove differentiability of optimal solutions and stability of the steady state. This paper shows that if the consumption good production function is concave‐, and the instantaneous utility function is concave‐, then the indirect utility function is weakly concave, and its curvature coefficients are bounded from above by a function of and .

DOES THE CAPITAL INTENSITY MATTER? EVIDENCE FROM THE POSTWAR JAPANESE ECONOMY AND OTHER OECD COUNTRIES

The capital intensity takes an important role in two-sector and multisector growth models. Surprisingly very few empirical studies have been conducted so far except by Kuga (1967). This fact implies that few people have ever tried to perform any empirical research to study whether the two-sector and multisector optimal growth models could explain the economic development properly based on the empirical data. Although we witnessed fairly active theoretical research on two-sector and multisector growth models in the 1990s and recent years, R. M. Solow has thrown doubt on the capital intensities [in Philippe Aghion and Steven Durlauf (eds.),Handbook of Economic Growth, Vol. 1A (2005, pp. 3–10)]. Our purpose is to measure the capital intensities of the consumption good and the investment good sectors mainly in the postwar Japanese economy, and also in other OECD countries. By so doing, we will demonstrate that the capital intensity does matter and our empirical evidence will strongly support the common assumption that the consumption goods sector is more capital-intensive than the capital goods sector.

Optimal and competitive programs in reachable multi sector models

We show that in multi-sector optimal growth models, where the technology satisfies a simple reachability condition, infinite horizon programs which satisfy the competitive conditions are optimal. We provide examples of a variety of production models where the reachability condition is satisfied. An example is also provided where the reachability condition is not satisfied and there are competitive programs which are not optimal. The results of the paper are of interest from the standpoint of decentralization in intertemporal economies.

Strong Concavity Properties of Indirect Utility Functions in Multisector Optimal Growth Models

Studies of optimal growth in a multisector framework are generally addressed in reduced form models, defined by an indirect utility function which summarizes the consumer's preferences and the technologies. Strong concavity assumptions allow to obtain some results about differentiability of optimal solutions and stability of steady states. However there does not exist any information about the conditions that have to be placed on the fundamentals to obtain such a property. This paper shows that if the consumption good production function isα-concave and if the capital goods technologies are Lipschitz-continuous then the indirect utility function is strongly concave.Journal of Economic LiteratureClassification Numbers: C61, O41.

Endogenous cycles with small discounting in multisector optimal growth models: Continuous-time case

This paper discusses the role of the trade-off between the level of discounting and the curvature of preferences and technology in establishing the existence of endogenous cycles. Using the equivalence between the Euler-Lagrange equations and a modified Hamiltonian dynamic system, in a general continuous-time multisector optimal growth model, it is proved that, if the indirect utility function is weakly concave (i.e., concave-γ, with γ>0 arbitrarily close to 0), then the discount rate values compatible with endogenous fluctuations are arbitrarily low. We show by numerical simulations that our result explains and generalizes to the multisector case the recent contribution of Benhabib and Rustichini in this field.

Endogenous cycles in discrete symmetric multisector optimal growth models

This paper discusses the asymptotic stability of the steady state in discrete symmetric multisector optimal growth models. Using a variational method, we provide a new proposition which gives some conditions ensuring the local saddle point property. A characterization of the bound above which the steady state is locally unstable is also proposed in terms of the indirect utility function concavity properties. On this basis, some sufficient conditions for the existence of competitive cycles are stated. We thus prove the existence of a Flip bifurcation.

On uniqueness of a steady state and convergence of optimal paths in multisector models of optimal growth with a discount rate

On uniqueness of a steady state and convergence of optimal paths in multisector models of optimal growth with a discount rate

On the Uniqueness of Steady States in an Economy with Heterogeneous Capital Goods

It well known that in multisector models of optimal growth, optimal paths converge to a unique steady state when future utilities are not discounted. Sutherland [1970], Kurz [1968], and Liviatan and Samuelson [1969] gave examples of multiple steady states when future utilities are discounted. Beals and Koopmans [1969] and Iwai [1972] used intertemporal utility functions, which also yield multiple steady states. The uniqueness of steady states with multiple consumption goods when future utility discounted has been studied by Brock [1973]. He showed tlhat if we assume that none of the goods are inferior in consumption (he calls this the normality condition for the utility function) the uniqueness of the steady state assured. Brock did not allow for pure consumption goods. Later Brock and Burmeister [1976] generalized Brock's result to allow pure consumption goods as well (Morishima [1974] type). Brock also formulated an alternative approach where uniqueness assured under the of a non-vanishing Jacobian for every non-negative discount rate. He writes however that the non-singularity of the Jacobian is an obscure assumption and that it would be worthwhile to relate it to the normality condition of the utility function. We first propose to weaken Brock's of non-vanishing Jacobian for every discount rate. Then in Section 3, Theorem 2, we show that the normality condition on the utility function implies a non-vanishing Jacobian. In Theorem 3 we weaken the normality condition for the uniqueness of the steady state by investigating the conditions for a non-vanishing Jacobian. We then clarify the economic content of our weaker conditions. In the final section we observe that the normality theorem can be proved for the joint production case (Mirrlees [1969] type) using a technique due to McKenzie [1963, 1973].

Global asymptotic stability results for multisector models of optimal growth under uncertainty when future utilities are discounted

Global asymptotic stability results for multisector models of optimal growth under uncertainty when future utilities are discounted

Some Results on the Uniqueness of Steady States in Multisector Models of Optimum Growth When Future Utilities are Discounted

IT IS WELL KNOWN that in multisector models of optimal growth that optimal paths converge to a unique steady state when future utilities are not discounted. See Gale [8], McKenzie [18], and Brock [3] for results of this type. Then Sutherland [32], in the case when future utilities are discounted, produced examples of multiple steady states in Gale's [8] model. Thus the qualitative behavior of optimal growth when future utilities are discounted may be quite different than the Ramsey case when future utilities are not discounted. Also Kurz [11] and Liviatan and Samuelson [13] have produced multipLe equilibria in the discounted case by introducing wealth effects and joint production effects respectively. What we shall do here is give conditions for several multisector models that give uniqueness of steady states. In the case of no joint production and one primary factor of production we use a nonsubstitution theorem to determine relative prices and the choice of technique independently of the utility function. We then assume of the utility function and show that there is at most one steady state for that case. We examine a model with no joint production, no primary factor and no depreciation of capital. In this case we show that summability of the utility function and a type of normality condition on each production function yields uniqueness of steady state. New techniques have to be invented to handle this case. The techniques may be independently interesting. A very general model that handles cases of joint production is examined in Section 3. All that is required here is that the steady state equations can be written in the form G(k, o) = 0 where p is the discount. It turns out that if