Linear-quadratic Approximation Research Articles

Linear quadratic approximation of rationally inattentive control problems

Abstract This paper proposes a linear quadratic approximation approach to dynamic nonlinear rationally inattentive control problems with multiple states and multiple controls. An efficient toolbox to implement this approach is provided. Applying this toolbox to five economic examples demonstrates that rational inattention can help explain the comovement puzzle in the macroeconomics literature.

Linear Quadratic Approximation of Rationally Inattentive Control Problems

Linear Quadratic Approximation of Rationally Inattentive Control Problems

A dynamic model of an insurer: loss shocks, capacity constraints and underwriting cycles

PurposeThis paper aims to revisit the assumption of the cyclicality of the property-liability insurance market and identify a scenario in which the so-called underwriting cycles are unpredictable, according to a dynamic cash flow model which generates non-cyclical output dynamics.Design/methodology/approachThis paper is on the intersection of real business cycle models and financial cycles. The authors construct a dynamic model of an insurer’s cash flows with stochastic loss shocks and capacity constraints, in which loss shocks have a dual impact on both underwriting profits and access to external capital. They simulate the insurer’s optimal output responses to loss shocks, including output movements in underwriting coverage and external capital, to explore the source of unpredictable underwriting cycles through linear quadratic approximation in the model economy.FindingsThe authors find that the effect of loss shocks on the insurer’s cash flows could spread out and amplify over time because of the dynamic interaction between its underwriting capability and ability to raise external capital. This dynamic interaction can generate a non-cyclical pattern of changes in underwriting coverage and access to external capital in the benchmark economy. Applied to different experimental economies, the simulation results reveal that the determinants of the level of output fluctuations include the size of loss shocks, the sensitivity of capital market to loss shocks and the tightness of capital market.Originality/valueTo the best of the authors’ knowledge, there has been no attempt to study insurance output cyclicality with a dynamic cash flow model based upon the real business cycle literature, in which the dynamic interaction between underwriting and access to external capital because of loss shocks has an amplifying effect on output markets. This paper contributes to the current body of research by being able to simulate and show the insurance output dynamics resulting from the amplifying effect under capacity constraints.

The choice of a growth path under a linear quadratic approximation

There is a recent strand of literature which suggests that second order approximations of linear quadratic objective functions in the steady state vicinity, namely when assuming stochastic scenarios, lead to very interesting and useful results. For example, applications in monetary policy resort to such technique. In this paper we find that, for a specific optimal control problem under a purely deterministic setup, a second-order approximation of the objective function may lead to inaccurate results, particularly when one considers exogenous variables as arguments of the objective function. These results are related to the stability conditions, which in the present case can be written as constraints to a discount rate associated with future outcomes. We designate the proposed model as an ‘optimal growth control’ model, from which we compute general conditions about stability and analyse the application of such a framework to a fertility - human capital problem.

Constructive <inline-formula> <tex-math notation="TeX">$\epsilon$</tex-math></inline-formula>-Nash Equilibria for Nonzero-Sum Differential Games

In this paper, a class of infinite-horizon, nonzero-sum differential games and their Nash equilibria are studied and the notion of $\epsilon_{\alpha} $ -Nash equilibrium strategies is introduced. Dynamic strategies satisfying partial differential inequalities in place of the Hamilton–Jacobi–Isaacs partial differential equations associated with the differential games are constructed. These strategies constitute (local) $\epsilon_{\alpha} $ -Nash equilibrium strategies for the differential game. The proposed methods are illustrated on a differential game for which the Nash equilibrium strategies are known and on a Lotka–Volterra model, with two competing species. Simulations indicate that both dynamic strategies yield better performance than the strategies resulting from the solution of the linear-quadratic approximation of the problem.

Linear-quadratic approximation of optimal policy problems

We consider a general class of nonlinear optimal policy problems with forward-looking constraints, and show how to derive a problem with linear constraints and a quadratic objective that approximates the exact problem. The solution to the LQ approximate problem represents a local linear approximation to optimal policy from the “timeless perspective” proposed in Benigno and Woodford (2004, 2005) [6] , [7] , in the case of small enough stochastic disturbances. We also derive the second-order conditions for the LQ problem to have a solution, and show how to correctly rank alternative simple policy rules, again in the case of small enough shocks.

A Novel Solution for Stochastic Dynamic Game of Water Allocation from a Reservoir Using Collocation Method

In this study, a continuous model of stochastic dynamic game for water allocation from a reservoir system was developed. The continuous random variable of inflow in the state transition function was replaced with a discrete approximant rather than using the mean of the random variable as is done in a continuous model of deterministic dynamic game. As a result, a new solution method was used to solve the stochastic model of game based on collocation method. The collocation method was introduced as an alternative to linear-quadratic (LQ) approximation methods to resolve a dynamic model of game. The collocation method is not limited to the first and second degree approximations, compared to LQ approximation, i.e. Ricatti equations. Furthermore, in spite of LQ related problems, consideration of the stochastic nature of game on the action variables in the collocation method would be possible. The proposed solution method was applied to the real case of reservoir operation, which typically requires considering the effect of uncertainty on decision variables. The results of the solution of the stochastic model of game are compared with the results of a deterministic solution of game, a classical stochastic dynamic programming model (e.g. Bayesian Stochastic Dynamic Programming model, BSDP), and a discrete stochastic dynamic game model (PSDNG). By comparing the results of alternative methods, it is shown that the proposed solution method of stochastic dynamic game is quite capable of providing appropriate reservoir operating policies.

Prudence and Robustness as Explanations for Precautionary Savings: An Evaluation

This paper evaluates approximation methods to make manageable the numerical solution of overlapping generation models with aggregate risk. The paper starts with a model in which households maximize expected utility over their life cycle. Instantaneous utility is characterized by constant relative risk aversion. Prudence, a characteristic of the utility function, leads to precautionary saving. The first-order conditions include expectations. One source of uncertainty is not prohibitive for numerical integration of the expectation term. Because of its accuracy numerical integration results are used as a bench mark. Taylor series approximations can lead to the same results dependent on the linearization point. A linear quadratic approximation of the household model is evaluated subsequently. Alternatively, precautionary saving effects can be the result of robust decision making. This approach leads to linear policy functions and gives a rather good approximation of the bench mark model, although not as good as the Taylor series approximation.

Fiscal and monetary rules for a currency union

This paper addresses the optimal joint conduct of fiscal and monetary policy in a two-country model of a currency union with staggered price setting and distortionary taxes. A tractable linear-quadratic approximation permits a representation of the optimal policy plan in terms of targeting rules. In the optimal equilibrium, monetary policy should achieve aggregate price stability following a flexible inflation targeting rule. Fiscal policy should stabilize idiosyncratic shocks allowing for permanent variations of government debt but should abstain from creating inflationary expectations at the union level. Simple policy rules can approximate the optimal commitment benchmark through a mix of strict inflation targeting and flexible budget rules. Conversely, the welfare costs of balanced budget rules are at least one order of magnitude higher than conventional estimates of the costs of business cycle fluctuactions.

Increasing Returns to Scale and Welfare: Ranking the Multiple Deterministic Equilibria

We consider a real business cycle model with a productive externality and an aggregate non- convex technology set µa la Benhabib and Farmer embodying capacity utilization, which exhibits indeterminacy of the steady state and multiplicity of deterministic equilibria under plausible values of the increasing returns to scale. The aim of the paper is to rank these different equilibria according to the initial value of consumption using both a linear-quadratic approximation, extensively explained by Benigno and Woodford [2006a, 2006b], and simulation methods. We study the implications of such a ranking in terms of smoothness of the welfare-maximizing trajectory and show that the welfare- maximizing consumption and labor paths are all the smoother since the level of increasing returns is low. At last, we show that this solution provides a good benchmark for judging the desirability of the stabilization policy proposed by Guo and Lansing [1997].

Linear-Quadratic Approximation to Unconditionally Optimal Policy: The Distorted Steady-State

This paper establishes that one can generally obtain a purely quadratic approximation to the unconditional expectation of social welfare when the steady-state is distorted. A specific example is provided employing a canonical New Keynesian model. Unlike in the non-distorted steady state case, the approximate loss function is not defined simply over terms in inflation and output. Furthermore, optimal steady state inflation and the nominal interest rate are positive.

METHOD TO SOLVE THE NONLINEAR INFINITE HORIZON OPTIMAL CONTROL PROBLEM WITH APPLICATION TO THE TRACK CONTROL OF A MOBILE ROBOT

We present a numerical method to solve the infinite time horizon optimal control problem for low dimensional nonlinear systems. Starting from the linear-quadratic approximation close to the origin, the extremal field is efficiently calculated by solving the Euler–Lagrange equations backward in time. The resulting controller is given numerically on an interpolation grid. We use the method to obtain the optimal track controller for a mobile robot. The result is a globally asymptotically stable nonlinear controller, obtained without any specific insight into the system dynamics.

Linear-quadratic approximation, external habit and targeting rules

Abstract We examine the linear-quadratic approximation of nonlinear dynamic stochastic optimization problems. A discrete-time version of Magill [1977a. A local analysis of N-sector capital accumulation under uncertainty. Journal of Economic Theory 15(2), 211–219] is generalized to models with forward-looking variables paying special attention to second-order conditions. This is the ‘large distortions’ case in the literature. We apply the approach to monetary policy in a DSGE model with external habit in consumption. We then develop a condition for ‘target-implementability’, a concept related to ‘targeting rules’. Finally, we extend the approach to a comparison between cooperative and non-cooperative equilibria in a two-country model and show that the ‘small distortions’ approximation is inappropriate for this exercise.

Linear-Quadratic Approximation, External Habit and Targeting Rules

We examine the linear-quadratic (LQ) approximation of non-linear stochastic dynamic optimization problems in macroeconomics, in particular for monetary policy. We make four main contributions: first, we draw attention to a general Hamiltonian framework for LQ approximation due to Magill (1977). We show that the procedure for the ‘large distortions’ case of Benigno and Woodford (2003, 2005) is equivalent to the Hamiltonian approach, but the latter is far easier to implement. Second, we apply the Hamiltonian approach to a Dynamic Stochastic General Equilibrium model with external habit in consumption. Third, we introduce the concept of target-implementability which fits in with the general notion of targeting rules proposed by Svensson (2003, 2005). We derive sufficient conditions for the LQ approximation to have this property in the vicinity of a zero-inflation steady state. Finally, we extend the Hamiltonian approach to a non-cooperative equilibrium in a two-country model. JEL Classification: E52, E37, E58

Optimal taxation in an RBC model: A linear-quadratic approach

We reconsider the optimal taxation of income from labor and capital in the stochastic growth model analyzed by Chari et al. [1994. Optimal fiscal policy in a business cycle model. Journal of Political Economy 102, 617–652; 1995. Policy analysis in business cycle models. In: Cooley, T.F. (Ed.), Frontiers of Business Cycle Research. Princeton University Press, Princeton], but using a linear-quadratic (LQ) approximation to derive a log-linear approximation to the optimal policy rules. The example illustrates how inaccurate ‘naive’ LQ approximation – in which the quadratic objective is obtained from a simple Taylor expansion of the utility function of the representative household – can be, but also shows how a correct LQ approximation can be obtained, which will provide a correct local approximation to the optimal policy rules in the case of small enough shocks. We also consider the numerical accuracy of the LQ approximation in the case of shocks of the size assumed in the calibration of Chari et al. We find that the correct LQ approximation yields results that are quite accurate, and similar in most respects to the results obtained by Chari et al. using a more computationally intensive numerical method.

Optimal Monetary and Fiscal Policy under Discretion in the New Keynesian Model: A Technical Appendix to 'Great Expectations and the End of the Depression'

This paper details the microfoundations of the model presented in Staff Report no. 234, Great Expectations and the End of the Depression. It defines the Markov perfect equilibrium formally in the nonlinear model, discusses in some detail the approximation method used and the order of accuracy of this approximation, and gives proofs of two propositions not proved in Staff Report no. 234. In addition, this paper states a proposition that shows the equivalence between the linear quadratic approximation in Staff Report no. 234 and a first order approximation to the exact nonlinear conditions of the government in the Markov perfect equilibrium defined here.

Inflation Stabilization and Welfare: The Case of a Distorted Steady State

This paper considers the appropriate stabilization objectives for monetary policy in a microfounded model with staggered price-setting. Rotemberg and Woodford (1997) and Woodford (2002) have shown that under certain conditions, a local approximation to the expected utility of the representative household in a model of this kind is related inversely to the expected discounted value of a conventional quadratic loss function, in which each period's loss is a weighted average of squared deviations of inflation and an output gap measure from their optimal values (zero). However, those derivations rely on an assumption of the existence of an output or employment subsidy that offsets the distortion due to the market power of monopolistically-competitive price-setters, so that the steady state under a zero-inflation policy involves an efficient level of output. Here we show how to dispense with this unappealing assumption, so that a valid linear-quadratic approximation to the optimal policy problem is possible even when the steady state is distorted to an arbitrary extent (allowing for tax distortions as well as market power), and when, as a consequence, it is necessary to take account of the effects of stabilization policy on the average level of output. We again obtain a welfare-theoretic loss function that involves both inflation and an appropriately defined output gap, though the degree of distortion of the steady state affects both the weights on the two stabilization objectives and the definition of the welfare-relevant output gap. In the light of these results, we reconsider the conditions under which complete price stability is optimal, and find that they are more restrictive in the case of a distorted steady state. We also consider the conditions under which pure randomization of monetary policy can be welfare-improving, and find that this is possible in the case of a sufficiently distorted steady state, though the parameter values required are probably not empirically realistic.

Optimal Control and Spatial Heterogeneity: Pattern Formation in Economic-Ecological Models

This paper extends Turing analysis to standard recursive optimal control frameworks in economics and applies it to dynamic bioeconomic problems where the interaction of coupled economic and ecological dynamics under optimal control over space creates (or destroys) spatial heterogeneity. We show how our approach reduces the analysis to a tractable extension of linearization methods applied to the spatial analog of the well known costate/state dynamics. We explicitly show the existence of a non-empty Turing space of diffusive instability by developing a linear-quadratic approximation of the original non-linear problem. We apply our method to a bioeconomic problem, but the method has more general economic applications where spatial considerations and pattern formation are important. We believe that the extension of Turing analysis and the theory associated with the dispersion relationship to recursive infinite horizon optimal control settings is new.

Does Precommitment Raise Growth? The Dynamics of Growth and Fiscal Policy

We develop an endogenous growth model driven by externalities from both private and public capital. The government levies distortionary taxation to finance a publicly provided consumption good and public infrastructure. Firms face adjustment costs. We compare the optimal and time‐consistent policies in a linear‐quadratic approximation of the model. Although the time‐consistent equilibrium is sub‐optimal in terms of ex‐ante intertemporal welfare, it yields higher long‐run growth and welfare, through an accumulation of assets by the state and a cut in government consumption.

A thermodynamic noise model for nonlinear resistors

A Gaussian white noise model for passive nonlinear resistors, valid in the linear quadratic current-voltage (I-V) approximation at zero bias, is suggested. Although approximative, the model, which has originally been developed by R.L. Stratonovich, is thermodynamically well founded. The model is applied to the shot noise of exponential diodes and tunnel junctions as well as to the thermal noise of JFET's and MOSFET's. Within the range of the linear-quadratic approximation, the results are in full agreement with the well-known device specific standard low frequency thermal and shot noise models for these devices. The model can he considered as a linear-quadratic short circuit noise current generalization of Nyquist's formula.