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In linear-quadratic control (LQC) problems with singular control cost matrix and/or singular transition matrix, we derive a reduction of the dimension of the Riccati matrix, simplifying iteration and solution. Employing a novel transformation, we show that, under a certain rank condition, the matrix of optimal feedback coefficients is linear in the reduced Riccati matrix. For a substantive class of problems, our technique permits scalar iteration, leading to simple analytical solution. By duality the technique can also be applied to Kalman filtering problems with a singular measurement error covariance matrix.

The literature exhibits a dearth of comparative static results for n -asset portfolios. This paper shows a necessary and sufficient condition on utility functions for an increase in initial wealth never to decrease expected per-dollar final wealth in optimal n -asset portfolios. For the resulting utility function class the literature provides a measure of relative risk of portfolios; it is shown here that relative risk never decreases in optimal n -asset portfolios as initial wealth rises.

This paper considers portfolio choice when decisions are made for several future time periods all at once. The risky asset share sequence must be precommitted for the entire decision interval, either constrained (as in Samuelson (1991) ) or not constrained (as in Balvers and Mitchell (2000) ) to be constant across time periods within the interval. For a broad, plausible class of dynamic returns processes, contrary to Samuelson, under log utility the decisions for the more distant future are more conservative. This class is exemplified by autocorrelated ARMA(p,q) processes and finite-state Markov processes. The source of Samuelson’s contrary result is elucidated.

This paper analytically solves the Riccati equation of discrete optimal control with two targets and one tool. This is accomplished by reducing the problem to a nonlinear univariate dynamic equation; this can be solved by a suitable transformation of variable. Beyond its direct application to two-target economic problems, the present approach may provide insight toward the eventual solution of the general Riccati equation.

This paper examines intertemporal portfolio plans under autocorrelation in asset returns, and considers whether these plans conform to the common advice that risky assets be bought gradually and then held in decreasing amounts as the investment horizon approaches. Given elliptical returns, optimal portfolio plans with precommitment must be mean–variance efficient. Then, for ARMA (1, 1) parameterizations with negative autocorrelation, the age effect (gradual diminishing of risky holdings as the horizon approaches) is confirmed, as is dollar-cost averaging (gradual entry into the risky asset) for sufficiently distant horizons. For a numerically analyzed alternative bivariate returns process, only the age effect is confirmed.

We consider utility function partial orderings to predict comparative portfolio features with two risky assets. No utility ordering can predict comparative holdings of the riskier asset for any reasonable definition of the latter. We thus reinterpret results of Arrow-Pratt and Ross as predicting comparative mean-seeking behavior. We also present a stronger utility ordering which predicts comparative portfolio means with no joint distribution restrictions. Thus we present a progression of contexts, with successively more relaxed distributional restrictions and hence successively stronger restrictions on utility function pairs, in which comparative mean-seeking behavior (not comparative risk avoidance) is predictable.

This paper considers decision-making in the presence of two additive risk sources, with no restrictions on the relation between the two risks. A utility function is said to exhibit broad DARA if and only if a rise in wealth always decreases the magnitude of the risk premium for one of the risks vis-a-vis the other. A condition on utility functions giving this property is derived: utility must be of the linear plus exponential form. It is shown that certain problems involving portfolios and risk-averse firms give unambiguous comparative statics if and only if utility exhibits broad DARA.

In recent years it has been shown empirically that stock returns exhibit positive or negative autocorrelation, depending on observation frequency. In this context of autocorrelated returns the present paper is the first to derive an explicit analytical solution to the dynamic portfolio problem of an individual agent saving for retirement (or other change of status, like the purchase of a house or starting college). Using a normal ARMA(1,1) process, dynamic programming techniques combined with the use of Stein's Lemma are employed to examine “dollar-cost-averaging” and “age effects” in intertemporal portfolio choice with CARA preferences. We show that with a positive moving average parameter and positive riskfree rates, if first-order serial correlation is nonnegative, then the expected value of the optimal risky investment is increasing over time, while if first-order serial correlation is negative this path can be increasing or decreasing over time. Thus a necessary but not sufficient condition to obtain the conventional age effect of increasing conservatism over time is that first-order serial correlation be negative. Further, dollar-cost averaging in the general sense of gradual entry into the risky asset does not emerge as an optimal policy. Simulation results for U.S. data are used to illustrate optimal portfolio paths.

This paper shows that predominantly aggregate demand- (supply-) driven output fluctuations are consistent with a negative (positive) correlation between the price level and output, but only if aggregate supply is flatter (steeper) than aggregate demand. These results hold even with zero covariance between demand and supply shocks, and these confluences are more likely to occur for algebraically higher values of the correlation between demand and supply shocks. Thus care should be taken in using observed signs of price level-output correlations to draw inferences about the validity of supply- or demand-based macro theories.

It is proven here that with elliptically symmetric returns and two risky assets, a diversifier who is more risk averse in the Arrow-Pratt or Ross sense will hold more of the second-order stochastically dominated asset. Thus, consider any concept of “strictly more risk averse” implying greater Arrow-Pratt risk aversion, and consider a conjecture that if agent A is strictly more risk averse than agent B then A must hold more of the second-order stochastically dominating asset (in diversified two-asset portfolios). This paper shows by counterexample that the conjecture is false.

This paper considers the normative question of how to obtain computationally convenient optimal intertemporal portfolios under assumptions that are as realistic as possible. In each period expected utility of final-period wealth is maximized, possibly subject to linear inequality constraints. Random net portfolio additions are permitted. Return distributions can evolve deterministically. Under certain assumptions the portfolio problem is shown to be computationally straightforward. Intertemporal properties of the portfolio sequence, including the time path of the portfolio fraction held in risky assets collectively, are discussed.

This paper presents a random number generator, for use in generating cryptographic keystreams, which is based on the division algorithm. This generator permits choice of seed values giving a long cycle length which is known a priori and/or costlessly inferred during keystream generation. Advice is given for choosing seed values.

This paper considers the implications of exogenously generated chaotic monetary dynamics for the timepaths of endogenous variables in an otherwise linear rational expectations model. It is shown that chaos in the money stock is transmitted, with an increase in complexity and with noise, to all nominal variables and the real money supply, while output and the real interest rate are, as is to be expected, unaffected. The paper also discusses the difficulty of forming a rational expectation in the presence of chaotic dynamics.

This paper investigates the extent to which a commonly used electoral scheme — voting for n candidates when there are n vacancies to fill — is likely to lead, in practice, to paradoxical results. Two types of paradox are investigated. One occurs when an increase in the number of vacancies causes a candidate to drop from the list of winners. The second occurs when the Condorcet candidate fails to win. Using a Monte Carlo study, we show that the likelihood of paradox can be substantial and depends critically on the degree of homogeneity of voters' preferences and on the number of voters.

This paper presents the results of a simulation study investigating the efficacy of the correlation dimension technique in detecting the presence of deterministic structure in small data sets. The results show that for some very simple structures a mere 100 data points suffice; but for some slightly more complicated structures, which arguably are still simple relative to real-world structures, even 2500 data points are inadequate. These results suggest that substantial caution be used in interpreting negative results on empirical data.

This article shows that an orthogonalization procedure sometimes used in the finance literature does not do what it is intended to do. It does not fight multicollinerity by improving the standard error of the key coefficient estimate, and it does not estimate a separate effect of the variable in question, aside from any effects associated with the other regressors. Instead, the key regression results are invariant to the use of this procedure.

This paper develops and performs a series of non-parametric tests to find which among various hypothetical policy regimes would have been best at controlling the money stock. Non-activist use of the federal funds rate or of one of several versions of the base is compared to observed activist use of the federal funds rate in the 1970s and of the borrowed reserves base after 1982 in the USA. The results indicate that the observed activist use of the federal funds rate in the earlier period was superior to non-activist use of either the federal funds rate or non-borrowed reserves, but inferior to non-activist use of the monetary base; and the actual policy was underactive relative to the optimum. For the latter period the evidence shows that the observed activist use of the borrowed reserves base is superior to non-activist use of borrowed reserves while comparison to a non-activist use of the federal funds rate is inconclusive regarding which policy was superior. Finally, when compared to the optimum, actual policy was deemed to be underactive.

This paper presents a tractable method of risk sensitive optimal control based on approximate Neumann-Morgenstern expected utility. The technique is to first write any expected loss function (defined as the negative of expected utility) exactly in terms of all relevant moments of the additive shocks, and then to approximate expected loss with an expression quadratic in the first moment but also involving all relevant higher moments. Then the atandard quadratic linear control technique can be used to obtain risk sensitive closed loop policy decisions. Requirements for preference revelation by the policy maker are minimal.

Conditions are derived for linear-quadratic control (LQC) problems to exhibit linear evolution of the Riccati matrix and constancy of the control feedback matrix. One of these conditions involves a matrix upon whose rank a necessary condition and a sufficient condition for controllability are based. Linearity of Riccati evolution allows for rapid iterative calculation, and constancy of the control feedback matrix allows for time-invariant comparative static analysis of policy reactions.

Despite discussions about the relationship between returns to scale and economies of scale under assumptions of constant and nonconstant input prices, mistakes continue to be made in textbooks on these issues. This article demonstrates the pointwise relation between returns to scale and economies of scale, with an adaption to a calculus-based intermediate microeconomics class.

It is well-known that if every attainable gamble x can be completely characterized by its mean μ and variance σ2, then expected utility is given by a mean-variance preference function: EU (x) = V (μ, σ2). It is shown here thatV σ 2 can be positive even if U″0. Necessary and sufficient conditions on the family of attainable gambles are given to rule out these possibilities.

Recent papers have shown utility function conditions that are sufficient, in a two-asset context with or without stochastic dependence, for a conditional first-order stochastically dominating shift (or a conditional mean-preserving contraction) of one asset's return distribution never to result in a decline in holdings of that asset. The present paper shows that these conditions are sufficient even when there are an arbitrary number of assets. Copyright 1997 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

This note considers a portfolio problem with a complete set of Arrow-Debreu securities, each of which pays a positive return in only one state. It is shown that an increase in the return to asset i in state i causes an increase (no change; a decrease) in demand for asset i if and only if relative risk aversion evaluated in state i is less than (equal to; greater than) unity. Demands for all other assets change in the opposite direction. Copyright 1994 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

This article uses the correlation-dimension technique, along with the residual and shuffle tests, to look for evidence of nonlinear dynamic structure in various weekly monetary aggregates, their components, the monetary base, and the money multipliers. The results support the Barnett and Chen (1988) and Chen (1988) findings of nonlinearity in Divisia M2, Divisia M3, and simple sum M2. Evidence of nonlinear structure is also found for simple sum M1, currency, demand deposits, all checkable deposits, M3 minus M2, the monetary base, and the multipliers of M1, M2, and M3.

A FREQUENTLY POSED QUESTION in monetary policy formulation concerns the choice of which money supply to focus on. Usually one of two alternative choice critena is advocated. Some economists advocate a particular money supply definition on the grounds that it has a stable velocity function that is, the linkage between the chosen money supply and nominal GNP is very predictable. 1 Others favor choosing the money supply that is most easily controllable that is, which has the most predictable linkage to the directly controlled monetary base.2 What these viewpoints have in common is the assertion that it is appropriate to focus on one "best" version of the money supply. They argue that the base should be set to try to achieve a target value of that money supply and should be adjusted in response to surprise deviations of that money supply in order to restore the original target value. This paper, by contrast, shows that in general it is suboptimal to focus on one money supply alone. Deviations of either M1 or M2 from their expected values

A technique introduced by Friedman is used to analyze the ability of the Federal Reserve to control the money supply. We find that the necessary conditions for control are so restrictive that a constant growth rate for the money stock may not be achievable. We suggest, therefore, a steady growth rate for Federal Reserve Credit.

This paper adds to the literature on the illogical pricing of GNMA pass-through certificates. It is already known that the market incorrectly computes the mean capital gains yield on such a pool of mortgages as the capital gains yield on the mortgage of mean maturity. The present paper shows the strong assumptions about the behavior through time of the mean maturity or duration of the pool inherent in the market's attempt to give identical yields to securities issued at different times in the past. This market behavior implies a very strong assumption about the frequency distribution of prepayment dates of the individual mortgages.

In an interesting recent paper, Merris (1985) presented a modification of my (1979) model of the effect of a binding demand deposit interest ceiling upon a bank's choice of the level of the service charge per check. The model assumes bank profit maximization under certaintyS with a perfectly competitive loan market but finite elasticities of consumer demand for savings accounts and demand deposits. My earlier version assumed that the consumer-determined ratio of checks written (n) to demand deposit balances (D) is an ad hoc linear function of the exogenous demand deposit interest rate (rd ) and the service charge per check (b). That model implied that the bank's response to a change in the rate ceiling, db / drd, is ambiguous in sign regardless of whether the savings account rate (rS ) is fixed or flexible. In contrast, Merris considers two alternative versions of the model. In the first, n rather than n / D is assumed in ad hoc fashion to be linear in rS, rd, and b. This feature results in db / drd being unambiguously positive in the case of a fixed rS (though still ambiguous in the case of a flexible rS). His second, and more interesting, version attempts to establish a sound grounding in the preexisting theory of the transactions demand for money. By applying the Barro-Santomero (B-S) (1972) model of the demands for savings accounts, demand deposits, and currency, which has implications for the number of account transactions, he uses nonlinear functions for the quantities of savings accounts (S) and demand deposits (D), as well as for the number of transactions. Merris asserts that the results here are the same as for his linear model: db / drd is unambiguously positive if rS is exogenously fixed (though still ambiguous if rS is flexible). Section 2 of this comment points out two mistakes in Merris's analysis of the model with consumer-theoretic underpinnings, either of which is sufficient to invalidate his analysis and results. Then section 3 shows that, when these mistakes are corrected, the ambiguity in my original paper regarding the sign of db / drd is restored (except in certain special cases).

The polynomial inverse lag (PIL) of Mitchell and Speaker (1986) is a flexible distributed lag technique which is easy to estimate, because it requires only a brief nested search for the polynomial degree and estimation is otherwise linear. This paper presents an approximation theorem for the PIL, showing that a PIL of sufficient degree can arbitrarily closely approximate any absolutely summable sequence of true lag weights, with distance measured by the L1 metric.

This paper shows the role of macro policy under multiplier uncertainty when expectations are rational and the supply structure is affected by price level variance. Policy less is a function of price level variance and possibly of the expected price level. If the letter argument is omitted from the less function, optimal policy simply minimizes the horizontal variance of the aggregate demand curve. If it is not omitted, the level of policy depends on the responsiveness of the supply structure to price level variance, and there may be multiple local policy optima.

This paper has shown that the LM curve could be downward sloped if the rate of interest on money (specifically liquid deposits) is sufficiently flexible. If so, the momentary equilibrium could be unstable, especially if the LM curve is far from the vertical (i.e., if the interest rate on money is very flexible). Prospects for instability are enhanced if output adjusts slowly or if the central bank varies the money supply strongly over time in response to the general interest rate. If stability obtains with a downward sloped LM curve, fiscal policy has an unconventional direction of effect on income.