Geometrically stopped Markovian random growth processes and Pareto tails

Many empirical studies document power law behavior in size distributions of economic interest such as cities, firms, income, and wealth. One mechanism for generating such behavior combines independent and identically distributed Gaussian additive shocks to log-size with a geometric age distribution. We generalize this mechanism by allowing the shocks to be non-Gaussian (but light-tailed) and dependent upon a Markov state variable. Our main results provide sharp bounds on tail probabilities, a simple equation determining Pareto exponents, and comparative statics. We present two applications: we show that (i) the tails of the wealth distribution in a heterogeneous-agent dynamic general equilibrium model with idiosyncratic investment risk are Paretian, and (ii) a random growth model for the population dynamics of Japanese municipalities is consistent with the observed Pareto exponent but only after allowing for Markovian dynamics

Optimal taxation and the Domar-Musgrave effect

This article concerns the optimal choice of flat taxes on labor and capital income, and on consumption, in a tractable economic model. Agents manage a portfolio of bonds and physical capital while subject to idiosyncratic investment risk and random mortality. We identify the tax rates which maximize welfare in stationary equilibrium while preserving tax revenue, finding that a very large increase in welfare can be achieved by only taxing capital income and consumption. The optimal rate of capital income taxation is zero if the natural borrowing constraint is strictly binding on entrepreneurs, but may otherwise be positive and potentially large. The Domar-Musgrave effect, whereby capital income taxation with full offset provisions encourages risky investment through loss sharing, explains cases where it is optimal to tax capital income. In further analysis we study the dynamic response to the substitution of consumption taxation for labor income taxation. We find that consumption immediately drops before rising rapidly to the new stationary equilibrium, which is higher on average than initial consumption for workers but lower for entrepreneurs

Archimedean Copulas and Temporal Dependence

We study the dependence properties of stationary Markov chains generated by Archimedean copulas. Under some simple regularity conditions, we show that regular variation of the Archimedean generator at zero and one implies geometric orgodicityof the associated Markov chain. We verify our assumptions for a range of Archimedean copulas used in applications

Optimal Measure Preserving Derivatives

We consider writing a derivative contract on some underlying asset in such a way that the derivative contract and underlying asset yield the same payo� distribution after one time period. Using the Hardy-Littlewood rearrangement inequality, we obtain an explicit solution for the cheapest measure preserving derivative contract in terms of the payo� distribution and pricing kernel of the underlying asset. We develop asymptotic theory for the behavior of an estimated optimal derivative contract formed from estimates of the pricing kernel and underlying measure. Our optimal derivative corresponds to a direct investment in the underlying asset if and only if the pricing kernel is monotone decreasing. When the pricing kernel is not monotone decreasing, an investment of one monetary unit in our optimal derivative yields a payo� distribution that �rst-order stochastically dominates an investment of one monetary unit in the underlying asset. Recent empirical research suggests that the pricing kernel corresponding to a major US market index is not monotone decreasin

Copulas and Temporal Dependence

An emerging literature in time series econometrics concerns the modeling of potentially nonlinear temporal dependence in stationary Markov chains using copula functions. We obtain sucient conditions for a geometric rate of mixing in models of this kind. Geometric beta-mixing is established under a rather strong sucient condition that rules out asymmetry and tail dependence in the copula function. Geometric rho-mixing is obtained under a weaker condition that permits both asymmetry and tail dependence. We verify one or both of these conditions for a range of parametric copula functions that are popular in applied work

Distributional Replication

Suppose that X and Y are random variables. We define a replicating function to be a function f such that f(X) and Y have the same distribution. In general, the set of replicating functions for a given pair of random variables may be infinite. Suppose we have some objective function, or cost function, defined over the set of replicating functions, and we seek to estimate the replicating function with the lowest cost. We develop an approach to estimating the cheapest replicating function that involves minimizing the cost function over an estimate of the set of replicating functions. Our estimated set of replicating functions is obtained by considering the functions f in some sieve space for which the empirical distributions of f(X) and Y are close. Under suitable conditions, we show that our estimated function comes close to achieving distributional replication, and close to achieving the minimum cost among replicating functions. We discuss the relevance of our results to the financial literature on hedge fund replication; in this case, X is the market return, Y is the return from a hedge fund or other asset, and our estimation procedure amounts to choosing the cheapest portfolio of options on X such that the returns from our portfolio have the same distribution as the hedge fund returns