This paper shows how standard arguments supporting the imposition of price caps break down in the presence of demand uncertainty. In particular, though in the deterministic case the introduction or lowering of a price cap (above marginal cost) results in increased production, increased total welfare, decreased prices, and increased consumer welfare, we show that all of the above comparative statics predictions fail for generic uncertain demand functions. For example, for price caps sufficiently close to marginal cost, a decrease in the price cap always leads to a decrease in production and total welfare under certain mild conditions. Under stronger regularity assumptions, all of the monotone comparative statics predictions from the deterministic case also do not hold for a generic uncertain demand if we restrict attention to price caps in an arbitrary fixed interval (as long as the price caps are binding for some values in that interval).

The purpose of this paper is to examine the two-fund separation paradigm in the context of an infinite-horizon general equilibrium model with dynamically complete markets and heterogeneous consumers with time and state separable utility functions. With the exception of the dynamic structure, we maintain the assumptions of the classical static models that exhibit two-fund separation with a riskless security. In addition to a security with state-independent payoffs agents can trade a collection of assets with dividends following a time-homogeneous Markov process. We make no further assumptions about the distribution of asset dividends, returns, or prices. Agents have equi-cautious HARA utility functions. If the riskless security in the economy is a consol then agents' portfolios exhibit two-fund separation. But if agents can trade only a one-period bond, this result no longer holds. Examples show this effect to be quantitatively significant. The underlying intuition is that general equilibrium restrictions lead to interest rate fluctuations that destroy the optimality of two-fund separation in economies with a one-period bond and result in different equilibrium portfolios.

Transaction costs on financial markets may have important consequences for volumes of trade, asset pricing, and welfare. This paper introduces an algorithm for the computation of equilibria in the general equilibrium model with incomplete asset markets and transaction costs. We show that economies with transaction costs can be analyzed with differentiable homotopy techniques and thus in the same framework as frictionless economies despite the existence of non-differentiabilities of agents’ asset demand functions and the existence of locally non-unique equilibria. We introduce an equilibrium selection concept into the computation of economic equilibria that picks out a specific equilibrium in the presence of a continuum of equilibria.

In a comment Peter Bossaerts and William R. Zame [2006, Asset Trading Volume in Infinite-Horizon Economies with Dynamically Complete Markets and Heterogeneous Agents: Comment, Finance Research Letters, this issue] claim that the main result of our paper [Judd, Kenneth L., Felix Kubler, and Karl Schmedders, 2003, Asset Trading Volume with Dynamically Complete Markets and Heterogeneous Agents, The Journal of Finance 58, 2203-2217], namely the no-trade theorem for the dynamic Lucas infinite horizon economy with heterogeneous agents, is an artifact of the assumption that asset dividends and individual endowments follow the same stationary finite-state Markov process. In this reply, we clarify our assumptions and contrast them with the examples in Bossaerts and Zame.

This paper develops theoretical foundations for an error analysis of approximate equilibria in dynamic stochastic general equilibrium models with heterogeneous agents and incomplete financial markets. While there are several algorithms that compute prices and allocations for which agents' first-order conditions are approximately satisfied ("approximate equilibria"), there are few results on how to interpret the errors in these candidate solutions and how to relate the computed allocations and prices to exact equilibrium allocations and prices. We give a simple example to illustrate that approximate equilibria might be very far from exact equilibria. We then interpret approximate equilibria as equilibria for close-by economies; that is, for economies with close-by individual endowments and preferences. We present an error analysis for two models that are commonly used in applications, an overlapping generations (OLG) model with stochastic production and an asset pricing model with infinitely lived agents. We provide sufficient conditions that ensure that approximate equilibria are close to exact equilibria of close-by economies. Numerical examples illustrate the analysis.

In a three-period finite exchange economy with incomplete financial markets and retrading, we study the effects of the degree of incompleteness and of changes in the financial structure on asset price volatility. In what are essentially no aggregate risk economies, asset price volatility is a sunspot-like phenomenon. If markets are completed by financial innovation, asset price volatility reduction is generic. With aggregate risk, changes in the financial structure affect asset price volatility through a pecuniary externality. Financial innovation which decreases equilibrium price volatility can be crafted under conditions of sufficient market incompleteness. Numerical examples illustrate the role of risk aversion for volatility changes and show that, with or without aggregate risk, reducing the degree of incompleteness per se is not necessarily associated with a volatility reduction.

We quantitatively explore how asset market structure affects risk-sharing, welfare, and trading volume in stylized rational expectations models. We examine five market structures: perfect asset trading, only bonds and equity, only equity, only bonds, and autarchy. We find a variety of results. First, welfare gains from asset trading are very small in most of our examples. Second, the value of adding new assets falls rapidly as we add assets beyond a single asset. Third, if there is already equity trading, then bonds add little welfare in most of our examples. Fourth, even in our simple model, the addition of a new asset may harm many traders. Fourth, adding a new asset may increase trading volume in old assets.

Trading volume of infinitely-lived securities, such as equity, is generically zero in Lucas asset-pricing models with heterogeneous agents. More generally, the end-of-period portfolio of all securities is constant over time and states in the generic economy. General equilibrium restrictions rule out trading of equity after an initial period. This result contrasts the prediction of portfolio allocation analyses that portfolio rebalancing motives produce nontrivial trade volume. Therefore, other causes of trade must be present in asset markets with large trading volume.

We consider a Lucas asset-pricing model with heterogeneous agents, exogenous labor income, and a finite number of exogenous shocks. Although agents are infinitely lived, endowments and dividends are time-invariant functions of the exogenous shock alone and are thus restricted to lie in a finite-dimensional space; genericity analysis can be conducted on sets of zero Lebesgue measure. When financial markets are incomplete, that is, there are fewer financial securities than shocks, we show that generically in individual endowments all competitive equilibria are Pareto inefficient.

We consider an infinite-horizon exchange economy with incomplete markets and collateral constraints. As in the two-period model of Geanakoplos and Zame (2000), households can default on their liabilities at any time, and financial securities are only traded if the promises associated with these securities are backed by collateral. We examine an economy with a single perishable consumption good, where the only collateral available consists of productive assets. In this model, competitive equilibria always exist and we show that under the assumption that all exogenous variables follow a Markov chain there also exist stationary equilibria. These equilibria can be characterized by a mapping from the exogenous shock and the current distribution of financial wealth to prices and portfolio choices. We develop an algorithm to approximate this mapping numerically and discuss ways to implement the algorithm in practice. A computational example demonstrates the performance of the algorithm and shows some quantitative features of equilibria in a model with collateral and default.

Purpose: The motion path of the digits follows the path of an equiangular spiral in which a constant angle is formed by all radial vectors along the curve. This implies that the lengths of the metacarpals, proximal, middle, and distal phalanges approximate a Fibonacci sequence in which the ratio of any 2 consecutive numbers approaches the number 1.61803 (phi). This study tested the hypothesis that the metacarpal and phalangeal bone lengths follow the Fibonacci relationship. Methods: Standardized x-rays were taken of the hands of 100 healthy volunteers. The proximal phalanx length was subtracted from the sum of the lengths of the middle and distal phalanges and the metacarpal length was subtracted from the sum of the lengths of the middle and proximal phalanges. Confidence intervals for the quotients of the measured lengths of the adjacent bones of the hand also were used for statistical analysis. Results: Only 1 of 12 bone length ratios contained the ratio phi in the 95% confidence interval, that of the small finger metacarpal and proximal phalanx. The largest variability was seen in the small finger phalangeal relationships. Conclusion: The application of the Fibonacci sequence to the anatomy of the human hand, although previously accepted, is a relationship that is not supported mathematically. The difference between individual bone lengths as measured at the joint line and the center of rotation of the joints may explain our finding

We examine minimal sufficient state spaces for equilibria in a Lucas asset pricing model with heterogeneous agents and incomplete markets. It is clear that even if all fundamentals of the economy follow a first-order Markov process, equilibrium prices and allocations will generally not only depend on the current exogenous shock but also on the distribution of wealth among the heterogeneous agents. The main contribution of this paper is to give an example of an infinite-horizon economy with Markovian fundamentals where the joint process of equilibrium asset holdings and exogenous shocks does not constitute a sufficient state space either.

Although equilibrium allocations in models with incomplete markets are generally not Pareto-efficient, it is often argued that quantitative welfare losses from missing assets are small when time horizons are long and shocks are transitory. In this paper we use a computational analysis to show that even in the simplest infinite horizon model without aggregate uncertainty welfare losses can be substantial. Furthermore we show that in this model welfare losses from incomplete markets do not necessarily disappear when one considers calibrations of the model in which agents become very patient. We argue that when the economic model is calibrated to higher frequency data, the period persistence of negative income shocks must increase as well. In this case the welfare loss of incomplete markets remains constant even as agents' rate of time preference tends to one.

The purpose of this paper is to analyze endogenous asset innovation by an entrepreneurial exchange owner in a general equilibrium model of incomplete security markets with financial transaction fees. A monopolistic market maker has the technology to introduce a new option into the economy and charge investors proportional transaction fees if they trade on the exchange. The market maker's objective is to choose the security and transaction fee that maximize revenues when opening the exchange. A computational analysis of this problem is necessary since there are no interesting models with closed-form solutions. We compute the price and welfare effects of the option introduction.

Motivated by the computation of equilibria in economic models with incomplete asset markets, a cellation of the Grassmann manifold is constructed by restricting a common atlas. The Grassmann manifold ofm-planes inn-dimensional space is shown to be a union ofn choosem congruentm(n−m)-dimensional topological disks whose interiors are disjoint.

We present an intuitive homotopy algorithm for the computation of equilibria in the general equilibrium model with incomplete asset markets. The central concept is the introduction of utility maximization problems for all but one agent with penalties for transactions on the asset markets. We compute equilibria with homotopy path-following techniques using the first-order conditions of the agents' optimization problems and gradually lifting the penalty restriction as the algorithm proceeds. Finally, we present computational results from an implementation of the algorithm, showing convincingly that the algorithm is very reliable in general and suitable for large-scale computations.

This paper examines the equilibrium correspondence in exchange economies with semi-algebraic preferences. We develop the foundation for a systematic analysis of multiplicity and for robust calibration in applied general equilibrium. Given a class of semi-algebraic exchange economies parametrized by individual endowments and possibly other exogenous variables such as preference parameters, asset-payoffs or tax-rates there exists a semi-algebraic correspondence that maps parameters to positive numbers such that for generic parameters each competitive equilibrium can be associated with an element of the correspondence and each endogenous variable (i.e. prices and consumptions) is a rational function of that value of the correspondence and the parameters. This correspondence can be characterized as zeros of a univariate polynomial equation that satisfy additional polynomial inequalities. This polynomial as well as the rational functions that determine equilibrium can be computed using versions of Buchberger's algorithm which is part of most computer algebra systems. The computation is exact whenever the input data (i.e. preference parameters etc.) are rational.

This paper examines investors' portfolios in a dynamic (Lucas-style) general equilibrium model with heterogeneous agents. The celebrated two-fund separation theorem from static portfolio analysis generalizes, under the classical preference assumptions, to our dynamic model only when a consol is present or if the endogenous prices for bonds of different maturity allow for a trading strategy that replicates the consol. We give sufficient conditions on stock-payoffs that ensure that replication is possible but also argue that these conditions are quite restrictive. If the conditions do not hold, if all bonds have finite maturity and do not span the consol, then we show that equilibrium will deviate, often significantly, from two-fund separation. Nevertheless, the bonds play an important role in that they may complete the financial market. However, the optimal portfolios often imply unrealistically large trades in bonds of long maturity. The main result of the paper demonstrates that investors choosing two-fund-like portfolios with bond ladders that approximately replicate consols do almost as well as traders with equilibrium investment strategies -- our model may give a rationale for bond-ladders as a popular bond-portfolio management strategy.

This paper examines investors' portfolios in a dynamic (Lucas-style) general equilibrium model with heterogeneous agents. The celebrated two-fund separation theorem from static portfolio analysis generalizes, under the classical preference assumptions,

In this paper we examine non-parametric restrictions on counterfactual analysis in a simple dynamic stochastic general equilibrium model. Under the assumption of time-separable expected utility and complete markets all equilibria in this model are stationary, the Arrow-Debreu prices uniquely reveal the probabilities and discount factor and the equilibrium correspondence defined as the map from endowments to stationary (probability-free) state prices, is identical to the equilibrium correspondence in a standard Arrow-Debreu exchange economy with additively separable utility. We examine observable restriction on this correspondence and give necessary as well as sufficient conditions on profiles of individual endowments that ensure that associated equilibrium prices cannot be arbitrary. While often there are restrictions on possible price changes we also show that in most cases results from a single agent economy do not carry over to a setting with heterogeneous agents.

This paper analyzes optimal patent policy in a dynamic multi-stage innovation model where firms compete to receive a patent and its associated prize. Firms control their independent R&D project and invest to reach the stage at which a patent is awarded. At this stage, the laggard firms are forced to leave the race and the winner continues to invest in R&D until the innovation process has achieved its goal. Firms are assumed to possess perfect information about each others' innovation state and cost structures. A planner, who cannot distinguish between the firms, chooses the stage at which the patent is awarded and the magnitude of the prize to the winner to maximize either a ''social'' or a ''consumer'' surplus. We study the evolution of competition along the race path, explore the socially optimal patent policy and the sensitivity of R&D investment and competition to the prize level and the degree of heterogeneity between the firms.

Static and dynamic games are often used to analyze strategic interactions. While existence of equilibrium can often be proved by conventional methods, uniqueness is much more difficult to establish. If a game reduces to solving a system of polynomial equations, then one could use algorithms for finding all solutions to such systems to establish uniqueness of equilibrium. We first illustrate this for a static game. While most dynamic games are far too large for a direct application of this approach, we study a common type of dynamic games where equilibrium can be analyzed as a sequence of small games and apply an all solutions algorithm to each such game. We apply this to an R&D race, a cost-reducing investment game, and a learning curve game to show that this approach is practical given current computational technology.

Standard oligopoly theory suggests that price caps will tend to constrain the price of a good over the short-term and increase production. Firms become price-takers when the amount produced is less than where the price cap intersects the demand function. Recently imposed price caps in California, however, have resulted in anecdotal evidence that suggests that this might not always be the case. Typical explanations for increases in price and decreases in production are sociological and psychological in nature. While these lines of reasoning may go a long way in explaining the observed fact, the absence of an economic explanation is rather unsatisfying. In this note we give such an economic explanation by examining a simple economic model. We enhance a standard Cournot model through the introduction of demand uncertainty and agents' risk aversion. Multiple examples show that the introduction of a price cap in this model may indeed lead to higher prices and lower production quantities. Very interestingly, even a price cap set above the equilibrium prices obtained with no price cap, can result in lower output and higher prices.

Optimization problems are ubiquitous in economics. Many of these problems are sufficiently complex that they cannot be solved analytically. Instead economists need to resort to numerical methods. This article presents the most commonly used methods for both unconstrained and constrained optimization problems in economics; it emphasizes the solid theoretical foundation of these methods, illustrating them with examples. The presentation includes a summary of the most popular software packages for numerical optimization used in economics, and closes with a description of the rapidly developing area of mathematical programs with equilibrium constraints, an area that shows great promise for numerous economic applications.

Applied Computational Economics and Finance by Mario Miranda and Paul Fackler

The Arbor City Community Foundation (ACCF) was a medium-sized endowment established in the late 1970s through the hard work of several local families. The vision of the ACCF was to be a comprehensive center for philanthropy in the greater Arbor City region. ACCF had a fund balance (known collectively as “the Fund”) of just under $240 million. The ACCF board of trustees had appointed a committee to oversee investment decisions relating to the foundation assets. The investment committee, under the guidance of the board, pursued an active risk-management policy for the Fund. The committee members were primarily concerned with the volatility and distribution of portfolio returns. They relied on the Value-at-Risk (VaR) methodology as a measurement of the risk of both short- and mid-term investment losses.

The questions in part (A) of the case direct the students to analyze the risk inherent in both one particular asset and the entire ACCF portfolio. For this analysis the students need to calculate daily VaR and monthly VaR values and interpret these figures in the context of ACCF’s risk management.

In part (B) the foundation receives a major donation. As a result the risk inherent in its portfolio changes considerably. The students are asked to evaluate the risk of the fund’s new portfolio and to perform a portfolio rebalancing analysis.

The Arbor City Community Foundation (ACCF) was a medium-sized endowment established in the late 1970s through the hard work of several local families. The vision of the ACCF was to be a comprehensive center for philanthropy in the greater Arbor City region. ACCF had a fund balance (known collectively as “the Fund”) of just under $240 million. The ACCF board of trustees had appointed a committee to oversee investment decisions relating to the foundation assets. The investment committee, under the guidance of the board, pursued an active risk-management policy for the Fund. The committee members were primarily concerned with the volatility and distribution of portfolio returns. They relied on the Value-at-Risk (VaR) methodology as a measurement of the risk of both short- and mid-term investment losses.

The questions in part (A) of the case direct the students to analyze the risk inherent in both one particular asset and the entire ACCF portfolio. For this analysis the students need to calculate daily VaR and monthly VaR values and interpret these figures in the context of ACCF’s risk management.

In part (B) the foundation receives a major donation. As a result the risk inherent in its portfolio changes considerably. The students are asked to evaluate the risk of the fund’s new portfolio and to perform a portfolio rebalancing analysis.

The financial success of dairy farms depends critically on the price of their main output, milk. Large volatility in the price of milk poses a considerable business risk to dairy farms. This is particularly true for family-run dairy farms. The question then arises: how can a farm owner hedge the milk price risk? The standard approach to establish a price floor for a commodity such as milk is to purchase put options on commodity futures. At the Chicago Mercantile Exchange, farmers can buy put options on the price of a variety of milk products. However, the price a farm receives for its milk depends on many factors and is unique to the farm. Thus, a farmer cannot directly buy put options on the price he receives for the milk his farm produces. Instead the farmer needs to determine which of the options available for trade at the Chicago Mercantile Exchange offer the best hedge for his own milk price. The assignment in this case is to examine historical data on several prices of milk products and the milk price received by a family-run dairy farm in California. Students need to find the price that is most closely correlated to the farm's milk price and to then choose options with the appropriate strike price that serve as the best hedge for the farm's price risk.

Spiegel Online (www.spiegel.de) is the leading news Web site in Germany. The site was first designed to accompany Der Spiegel, one of Europe’s largest and Germany’s most influential weekly magazine, which has a weekly circulation of around one million. The site’s content is produced by a team of more than fifty journalists writing in several categories: politics, business, networld, panorama, arts and entertainment, science, university, school, sports, travel, weather, and automobiles. The original content is complemented by articles purchased from news agencies and selected articles from the print edition. Spiegel-Verlag is a major contributor to the Hamburg Media School, which offers professional master’s degree programs in Media Management (MBA), film, and journalism. In their second year, MBA students typically engage in consulting projects with major media companies. In a recent assignment, Spiegel Online posed two questions to the MBA team: are there any chances for an economically successful entry into the market for interactive classifieds? And if so, what should the business model look like in detail? A student team analyzed markets for classified ads and found one market segment that appeared to be particularly promising: the market for art objects. During the development of a business plan for a new venture in this market it became apparent that there is much uncertainty about the key input parameters to the business plan. As a result, it is very difficult to assess the viability of the business idea. How can the team properly account for the uncertain input parameters? What is the impact if this uncertainty on the bottom line? Will a Web site for art objects earn or lose money? How can the team communicate this uncertainty to a group of high-level decision makers who want a simple “go or no-go” recommendation?

EuroPet S.A. was a multinational company operating gas stations in many European countries. There was a growing propensity for supermarkets to attach gas stations to their retail operations, which was developing into a major threat to EuroPet. As a result, in the mid-1990s, the company began to develop and brand its own convenience stores co-located with its gas stations. However, the company was spending much more on advertising the convenience stores than its competitors did. Management now had to decide if the increase in sales attributed to advertising efforts justified the advertising spend by analyzing the market data from one large metropolitan area: Marseille, France.

An asset management company must replace the manager of its two signature mutual funds, who is about to retire. Two candidates have been short-listed. The management team is divided and cannot decide which of the two candidates would make the better mutual fund manager. The retiring manager presents a linear regression model to examine success factors of mutual fund managers. This linear regression is the starting point for the subsequent analysis.

The decision maker is in charge of procurement auctions at the department of transportation of Orangia (a fictitious U.S. state). Students are asked to assist him in estimating the winning bids in various auctions concerning highway repair jobs using data on past auctions. The decision maker is faced with various professional, statistical, and ethical dilemmas.

In Case (B) models for computing optimal bids in highway procurement auctions are developed from the perspective of the bidders.