Theory of Stochastic Optimal Economic Growth

This paper is a survey of the theory of stochastic optimal economic growth.International Development,

Estimating Dynamic Models of Imperfect Competition

We describe a two-step algorithm for estimating dynamic games under the assumption that behavior is consistent with Markov Perfect Equilibrium. In the first step, the policy functions and the law of motion for the state variables are estimated. In the second step, the remaining structural parameters are estimated using the optimality conditions for equilibrium. The second step estimator is a simple simulated minimum distance estimator. The algorithm applies to a broad class of models, including I.O. models with both discrete and continuous controls such as the Ericson and Pakes (1995) model. We test the algorithm on a class of dynamic discrete choice models with normally distributed errors, and a class of dynamic oligopoly models similar to that of Pakes and McGuire (1994).

Reading policies for joins: An asymptotic analysis

Suppose that $m_n$ observations are made from the distribution $\mathbf {R}$ and $n-m_n$ from the distribution $\mathbf {S}$. Associate with each pair, $x$ from $\mathbf {R}$ and $y$ from $\mathbf {S}$, a nonnegative score $\phi(x,y)$. An optimal reading policy is one that yields a sequence $m_n$ that maximizes $\mathbb{E}(M(n))$, the expected sum of the $(n-m_n)m_n$ observed scores, uniformly in $n$. The alternating policy, which switches between the two sources, is the optimal nonadaptive policy. In contrast, the greedy policy, which chooses its source to maximize the expected gain on the next step, is shown to be the optimal policy. Asymptotics are provided for the case where the $\mathbf {R}$ and $\mathbf {S}$ distributions are discrete and $\phi(x,y)=1 or 0$ according as $x=y$ or not (i.e., the observations match). Specifically, an invariance result is proved which guarantees that for a wide class of policies, including the alternating and the greedy, the variable M(n) obeys the same CLT and LIL. A more delicate analysis of the sequence $\mathbb{E}(M(n))$ and the sample paths of M(n), for both alternating and greedy, reveals the slender sense in which the latter policy is asymptotically superior to the former, as well as a sense of equivalence of the two and robustness of the former.Comment: Published at http://dx.doi.org/10.1214/105051606000000646 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Size, openness, and macroeconomic interdependence

The curse of dimensionality, a problem associated with analyzing the interaction of a relatively large number of endogenous macroeconomic variables, is a prevailing issue in the open economy macro literature. The most common practice to mitigate this problem is to apply the so-called Small Open Economy Framework (SOEF). In this paper, we aim to review under which conditions the SOEF is a justifiable approximation and how severe the consequences of violation of key conditions might be. Thereby, we use a multicountry general equilibrium model as a laboratory. ; First, we derive the conditions that ensure the existence of the equilibrium and study the properties of the equilibrium using large N asymptotics. Second, we show that the SOEF is a valid approximation only for economies (i) that have a diversified foreign trade structure and if (ii) there is no globally dominant economy in the system. Third, we illustrate that macroeconomic interdependence is primarily related to the degree of trade diversification, and not to the extent of trade openness. Furthermore, we provide some evidence on the pattern of global macroeconomic interdependence by calculating probability impulse response functions in our calibrated multicountry model using data for 153 economies.International trade

If You're So Smart, Why Aren't You Rich? Belief Selection in Complete and Incomplete Markets

This paper provides an analysis of the asymptotic properties of consumption allocations in a stochastic general equilibrium model with heterogeneous consumers. In particular we investigate the market selection hypothesis, that markets favor traders with more accurate beliefs. We show that in any Pareto optimal allocation whether each consumer vanishes or survives is determined entirely by discount factors and beliefs. Since equilibrium allocations in economies with complete markets are Pareto optimal, our results characterize the limit behavior of these economies. We show that, all else equal, the market selects for consumers who use Bayesian learning with the truth in the support of their prior and selects among Bayesians according to the size of the their parameter space. Finally, we show that in economies with incomplete markets these conclusions may not hold. Payoff functions can matter for long run survival, and the market selection hypothesis fails.Market selection hypothesis, subjective beliefs, general equilibrium, incomplete markets, complete markets

Robustness of Stochastic Optimal Control to Approximate Diffusion Models under Several Cost Evaluation Criteria

In control theory, typically a nominal model is assumed based on which an optimal control is designed and then applied to an actual (true) system. This gives rise to the problem of performance loss due to the mismatch between the true model and the assumed model. A robustness problem in this context is to show that the error due to the mismatch between a true model and an assumed model decreases to zero as the assumed model approaches the true model. We study this problem when the state dynamics of the system are governed by controlled diffusion processes. In particular, we will discuss continuity and robustness properties of finite horizon and infinite-horizon $\alpha$-discounted/ergodic optimal control problems for a general class of non-degenerate controlled diffusion processes, as well as for optimal control up to an exit time. Under a general set of assumptions and a convergence criterion on the models, we first establish that the optimal value of the approximate model converges to the optimal value of the true model. We then establish that the error due to mismatch that occurs by application of a control policy, designed for an incorrectly estimated model, to a true model decreases to zero as the incorrect model approaches the true model. We will see that, compared to related results in the discrete-time setup, the continuous-time theory will let us utilize the strong regularity properties of solutions to optimality (HJB) equations, via the theory of uniformly elliptic PDEs, to arrive at strong continuity and robustness properties.Comment: 33 page

A Note on Asymptotics Between Singular and Constrained Control Problems of One-Dimensional Diffusions

We study the asymptotic relations between certain singular and constrained control problems for one-dimensional diffusions with both discounted and ergodic objectives. In the constrained control problems the controlling is allowed only at independent Poisson arrival times. We show that when the underlying diffusion is recurrent, the solutions of the discounted problems converge in Abelian sense to those of their ergodic counterparts. Moreover, we show that the solutions of the constrained problems converge to those of their singular counterparts when the Poisson rate tends to infinity. We illustrate the results with drifted Brownian motion and Ornstein-Uhlenbeck process