The turnpike of dynamic general equilibrium paths and its insensitivity to initial conditions

Asymptotic turnpike theorems in the case where the future is discounted are traditionally derived under the assumptions that a utility function is twice differentiable and that is Hessian is negative definite. Replacing these assumptions by a weaker one. I give a proof of the theorem simpler than existing ones.     

The primal route to the turnpike and asymptotic stability

Asymptotic turnpike theorems in the case where the future is discounted are traditionally derived under the assumptions that a utility function is twice differentiable and that is Hessian is negative definite. Replacing these assumptions by a weaker one. I give a proof of the theorem simpler than existing ones.     

An analytical solution for the robust investment-reinsurance strategy with general utilities

In financial markets, different investors have different attitudes or preferences on the investment policies and reinsurance problems. For investors with different investment utilities, how to provide an optimal investment strategy is not only a very hard problem, but also an urgent problem to be solved. In this paper, we derive an analytical solution for the optimal allocation problem of investment-reinsurance with general-form utility function. The general utility function allows for varying relative risk aversion coefficient, which is an important feature in finance theory. However, obtaining analytical solutions for general utility function has been difficult or impossible. The solution presented in this paper is constructed through the homotopy analysis method (HAM) and written in the form of a Taylor series expansion. The fully nonlinear Hamilton–Jacobi–Bellman (HJB) equation is decomposed into an infinite series of linear PDEs, which can be solved analytically. In the end, three examples are presented to illustrate the convergence and accuracy of the method, it also demonstrates that different risk reference investors have different investment-reinsurance strategies.     

Implicitly additive utility and the nature of optimal economic growth

The solution of an optimization problem frequently depends only on the constraint set and on the indifference set through the optimum. This observation is spelled out and applied to the problem of optimal economic growth. The validity of the turnpike property of optimal growth plans is investigated when the intertemporal utility function is not additive, but has the property that each of its indifference sets is also an indifference set for some other utility function which is additive and has a constant rate of discount. A characterization is provided for such utility functions, which are called implicitly additive.     

Strategic Ramsey equilibrium dynamics

We study strategic Ramsey equilibria. First, we give necessary and sufficient conditions for a stationary equilibrium point to exist where only the most patient household owns capital. We develop the dynamics of strategic Ramsey equilibrium trajectories when the turnpike property holds and there is a stationary equilibrium with only the most patient household owning capital. We conclude by showing there is a class of economies for which the turnpike property obtains.     

Optimal strategies in linear multisector models: Value function and optimality conditions

In this paper we study an optimal control problem with mixed constraints related to a multisector linear model with endogenous growth. The main aim is to establish a set of necessary and a set of sufficient conditions which are the basis for studying the qualitative properties of optimal trajectories. The presence of possibly degenerate mixed constraints, the unboundedness and non-strict convexity of the Hamiltonian, make the problem difficult to deal with. We develop first the dynamic programming approach, proving that the value function is a bilateral viscosity solution to the associated Hamilton–Jacobi–Bellman (HJB) equation. Then, using our results, we give a set of sufficient and a set of necessary optimality conditions which involve so-called co-state inclusion: this can be interpreted as the existence of a dual path of prices supporting the optimal path.     

A continuous-time portfolio turnpike theorem

We prove a continuous-time portfolio turnpike theorem. The proof uses the theory of martin-gales and is more intuitively appealing than the usual discrete-time mode of proof using dynamic programming. When the interest rate is strictly positive, the present value of any contingent claim having payoffs bounded from above can be made arbitrarily small when the investment horizon increases. Thus an investor concentrates his wealth in buying contingent claims that have payoffs unbounded from above at the very beginning of his horizon. As a consequence, it is the asymptotic property of his utility function as wealth goes to infinity that determines his optimal investment strategy at the very beginning of his horizon.     

Equilibrium turnpike theory with time-separable utility

The purpose of this paper is to show that under reasonably general conditions intertemporal competitive equilibrium has a turnpike property. The model is general because it permits (1) time-variant, time-separable utility functions, (2) heterogeneous rates of discount across consumers, and (3) matching flatness in utilities and production possibilities. The results rely on bounded marginal utility for all consumers and aggregate stationarity of utilities of the set of most patient consumers. Under these assumptions, the neighborhood turnpike result holds with (1) and (2) because of the eventual unimportance of the impatient consumer. Matching flatness requires the use of a two-sided Liapounov function and the growth theoretic methods of McKenzie.     

A study on the incentive compensation structure with payroll tax: A continuous-time principal-agent model

This paper studies a case when the government levies a payroll tax on the employee (agent) of an enterprise. We use a continuous-time principal-agent framework to analyze the impact of the tax on the employee’s working strategy and derive an incentive compensation scheme. The agent is supposed to be aware of his pre-tax and after-tax salary. Under the theory of behavioral economics, loss caused by taxation is taken into consideration. The Hamilton-Jacobi-Bellman (HJB) equation of principal’s profits is derived. By exploiting the HJB equation, we get several properties of the optimal contract. We also perform comparative statics to show our results. The model suggests that the agent’s utility loss enlarges as the tax rate increases. However, an increase in the tax rate does not always decrease principal’s profits.     

On the smoothness of optimal paths

Abstract The aim of this paper is to study the differentiability property of optimal paths in dynamic economic models. We address this problem from the point of view of the differential calculus in sequence spaces which are infinite-dimensional Banach spaces. We assume that the return or utility function is concave, and that optimal paths are interior and bounded. We study the C ^r differentiability of optimal paths vis-à-vis different parameters. These parameters are: the initial vector of capital stock, the discount rate and a parameter which lies in a Banach space (which could be the utility function itself). The method consists of applying an implicit function theorem on the Euler–Lagrange equation. In order to do this, we make use of classical conditions (i.e., the dominant diagonal block assumption) and we provide new ones. Mathematics Subject Classification (2000): 90A16, 49K40, 93C55 Journal of Economic Literature Classification: C161, D99, O41     

Conditions for incentive compatibility in models with multidimensional allocation functions and one-dimensional types

We extend the conditions for incentive compatibility in mechanism design problems to a more general structure of preferences than that found in the literature, for the case where type is one-dimensional but the outcome function becomes multidimensional. This is so, at least, as long as preferences can be represented by means of sub-utility functions, it is adopted a weak single-crossing property and direct mechanisms turn out to be differentiable. When direct mechanisms are not differentiable, local incentive conditions still remain fully incentive compatible, provided utility is weakly separable in the outcome function, or else, it exhibits linearity in the type.     

Explicit solution of a general consumption/portfolio problem with subsistence consumption and bankruptcy

This paper solves a general continuous-time single-agent consumption and portfolio decision problem with subsistence consumption in closed form. The analysis allows for general continuously differentiable concave utility functions. The model takes into consideration that consumption must be no smaller than a given subsistence rate and that bankruptcy can occur. Thus the paper generalizes the results of Karatzas, Lehoczky, Sethi, and Shreve (1986).     

Structure of the demand function and Pareto optimal set with natural boundary conditions

We consider utility functions defined on P?, the closure of the positive orthant of R^l, that satisfy the differentiable monotonicity and differentiable convexity conditions. We show that the demand function derived from such a utility function is piecewise smooth in a strong sense. We use this result to show that the Pareto optimal subset of a pure exchange economy derived from the kind of utility functions we consider is generically the union of a finite number of compact manifolds-with-corners.     

On the demand generated by a smooth and concavifiable preference ordering

It is shown that if a consumer's preference ordering is strictly convex and is representable by means of a concave, twice continuously differentiable utility function, then the partial derivative of a demanded commodity with respect to its price is bounded from above in a neighborhood of a price vector at which the demand fails to be differentiable. In the case of two commodities, if the demand does not possess finite derivatives with respect to prices at a certain point, then the partial ‘derivative’ of a commodity with respect to its price is equal to minus infinity. The same result holds for n commodities under ‘almost every’ choice of coordinates in the commodity space. If preferences are weakly convex but the same representation assumption holds, demand may not be single-valued but own-price difference quotients are still bounded from above.     

Property tax and urban sprawl: Theory and implications for US cities

We develop a model that adopts a log-linear utility function with a variable elasticity of substitution greater than one and show that increasing the property tax reduces city size unambiguously. We then test this result using a dataset of effective property tax rates we developed using GIS methods for 448 urbanized areas. The empirical analysis estimates a regression equation relating an urbanized area's size to the property tax rate measure and other control variables such as population, income, agricultural rent, and transportation expenditure. We find that higher property taxes indeed result in smaller cities.     

On the smoothness of optimal paths II: some local turnpike results

We study the local turnpike property for two classes of infinite-horizon discrete-time deterministic maximization problems which have common applications, e.g., optimal growth theory. We follow a functional-analytic approach and rely on an implicit function theorem for the space of the sequences which converge to zero. We shall assume the existence of an optimal path which is not necessarily a steady-state. Relying on material developped in Blot and Crettez (Decis Econo Finance 27:1–34, 2004), “On the smoothness of optimal paths” Decis Econ Finance, 21:1–34, 2004), we provide conditions under which a variation in the initial conditions (i.e., capital stock and discount rate) yields an optimal solution which converges toward a reference solution when time becomes infinite. We also provide new results on bounded solutions of difference equations. We gratefully thank the editor, Silvano Holzer, and two anonymous referees for remarks and advices on a previous version of this paper.     

Rationalizing investors’ choices

Assuming that agents’ preferences satisfy first-order stochastic dominance, we show how the Expected Utility paradigm can rationalize all optimal investment choices: the optimal investment strategy in any behavioral law-invariant (state-independent) setting corresponds to the optimum for an expected utility maximizer with an explicitly derived concave non-decreasing utility function. This result enables us to infer the utility and risk aversion of agents from their investment choice in a non-parametric way. We relate the property of decreasing absolute risk aversion (DARA) to distributional properties of the terminal wealth and of the financial market. Specifically, we show that DARA is equivalent to a demand for a terminal wealth that has more spread than the opposite of the log pricing kernel at the investment horizon.     

A Turnpike Theorem in the Closed Dynamic Leontief Model with a Singular Matrix of Capital Coefficients

The paper refers to the well-known Tsukui turnpike theorem on convergence of optimal growth trajectories in the closed dynamic Leontief model to the maximum balanced growth trajectory, called turnpike. In the original proof of this theorem, the assumption that the matrix B of capital coefficients is non-singular plays an essential role. For many reasons this assumption, very convenient for theoretical analysis, is not always satisfied in input-output systems built for empirical purposes. This paper fills the gap between theory and empirical studies, presenting a proof that convergence of optimal trajectories towards the turnpike is also a characteristic feature of the closed Leontief model in the case when matrix B is singular. The general idea of the proof is based on the approximation of a singular matrix B by an infinite sequence of non-singular matrices.     

On the nature of certainty equivalent functionals

We explore connections between the certainty equivalent return (CER) functional and the underlying utility function. Curvature properties of the functional depend upon how utility function attributes relate to hyperbolic absolute risk aversion (HARA) type utility functions. If the CER functional is concave, i.e., if risk tolerance is concave in wealth, then preferences are standard. The CER functional is linear in lotteries if utility is HARA and lottery payoffs are on a line in state space. Implications for the optimality of portfolio diversification are given. When utility is concave and non-increasing relative risk averse, then the CER functional is superadditive in lotteries. Depending upon the nature of association among lottery payoffs, CERs for constant absolute risk averse utility functions may be subadditive or superadditive in lotteries. Our approach lends itself to straightforward experiments to elicit higher order attributes on risk preferences.     

On the nature of certainty equivalent functionals

We explore connections between the certainty equivalent return (CER) functional and the underlying utility function. Curvature properties of the functional depend upon how utility function attributes relate to hyperbolic absolute risk aversion (HARA) type utility functions. If the CER functional is concave, i.e., if risk tolerance is concave in wealth, then preferences are standard. The CER functional is linear in lotteries if utility is HARA and lottery payoffs are on a line in state space. Implications for the optimality of portfolio diversification are given. When utility is concave and non-increasing relative risk averse, then the CER functional is superadditive in lotteries. Depending upon the nature of association among lottery payoffs, CERs for constant absolute risk averse utility functions may be subadditive or superadditive in lotteries. Our approach lends itself to straightforward experiments to elicit higher order attributes on risk preferences.