Computation of heterogenous agent models: Krusell/Smith vs. backwardinduction
The paper deals with the efficient computation of general equilibrium models with a continuum of heterogenous agents. It compares an improved version of the Krusell-Smith algorithm to a backward-induction algorithm. The Krusell-Smith algorithm I use in the paper modifies the original algorithm in Krusell/Smith (1998) in two ways. First, I do not simulate the cross-sectional distribution over time, but use the exact transition law for the distribution, based on the individual policy function. Second, I do not rely on fixed point iteration, but use a Quasi-Newton algorithm to find the equilibrium. The backward induction algorithm is a method that I proposed in an earlier paper (Reiter 2002), modified in several respects to increase accuracy and efficiency (faster solution of the household problem, better ways to compute the distribution selection function etc.). The methods are applied to two basic versions of the heterogenous agent model. In the first version, there is trade in only one asset, namely the physical capital stock in the economy. In the second version, there are two assets, physical capital and a one-period riskless bond. The model is calibrated so as to give a very small risk premium, which makes it challenging to solve numerically. The paper makes 4 contributions. First, it presents improved implementations of the two algorithms, as described above. Second, it provides a thorough comparison of the two approaches. Most details of the algorithms (solution of the household problem, statistics used to characterize the cross-sectional distribution, finite representation of the cross-sectional distribution, etc.) are kept constant across algorithms, so as to focus the comparison on the essential differences between the two approaches. The key results are the following. The backward induction algorithm converges without any problems. Krusell-Smith converges easily with a Quasi-Newton algorithm and a reasonable starting value, which can be obtained from the steady state of the model. Both algorithms give approximately the same accuracy, as far as fluctuations about a steady state are concerned. Backward induction is generally faster, and can handle transition periods more naturally. The third contribution of the paper is a new statistic to be used as a state variable, in addition to the mean or other moments of the distribution. This statistic is not a moment, but it is a linear functional of the distribution, and it is constructed using information on the household consumption function. It allows to increase the accuracy in the solution by a factor of 10 over the Krusell-Smith "one-moment" solution, much better than what is obtained by including another moment (such as the variance). Fourth, the paper comes with publicly available Matlab codes, supported by compiled C-routines, for both algorithms. These codes can be easily modified by the user to handle new models. Although written in Matlab, they are reasonably fast. The one-asset model can be solved in about 2 minutes on a PC.
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