On convergence in endogenous growth models

In this paper we analyze the rate of convergence to a balanced path in a class of endogenous growth models with physical and human capital. We show that such rate depends locally on the technological parameters of the model. but does not depend on those parameters related to preferences. These results stand in sharp contrast with those of the one-sector neoclassical growth model where both preferences and technologies determine the speed of convergence toward a steady state

On convergence in endogenous growth models.

In this paper we analyze the rate of convergence to a balanced path in a class of endogenous growth models with physical and human capital. We show that such rate depends locally on the technological parameters of the model. but does not depend on those parameters related to preferences. These results stand in sharp contrast with those of the one-sector neoclassical growth model where both preferences and technologies determine the speed of convergence toward a steady state.Neoclassical Growth Model; Endogenous Growth Models; Stability; Speed of Convergence;

Tractable multi-product pricing under discrete choice models

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 199-204).We consider a retailer offering an assortment of differentiated substitutable products to price-sensitive customers. Prices are chosen to maximize profit, subject to inventory/ capacity constraints, as well as more general constraints. The profit is not even a quasi-concave function of the prices under the basic multinomial logit (MNL) demand model. Linear constraints can induce a non-convex feasible region. Nevertheless, we show how to efficiently solve the pricing problem under three important, more general families of demand models. Generalized attraction (GA) models broaden the range of nonlinear responses to changes in price. We propose a reformulation of the pricing problem over demands (instead of prices) which is convex. We show that the constrained problem under MNL models can be solved in a polynomial number of Newton iterations. In experiments, our reformulation is solved in seconds rather than days by commercial software. For nested-logit (NL) demand models, we show that the profit is concave in the demands (market shares) when all the price-sensitivity parameters are sufficiently close. The closed-form expressions for the Hessian of the profit that we derive can be used with general-purpose nonlinear solvers. For the special (unconstrained) case already considered in the literature, we devise an algorithm that requires no assumptions on the problem parameters. The class of generalized extreme value (GEV) models includes the NL as well as the cross-nested logit (CNL) model. There is generally no closed form expression for the profit in terms of the demands. We nevertheless how the gradient and Hessian can be computed for use with general-purpose solvers. We show that the objective of a transformed problem is nearly concave when all the price sensitivities are close. For the unconstrained case, we develop a simple and surprisingly efficient first-order method. Our experiments suggest that it always finds a global optimum, for any model parameters. We apply the method to mixed logit (MMNL) models, by showing that they can be approximated with CNL models. With an appropriate sequence of parameter scalings, we conjecture that the solution found is also globally optimal.by Philipp Wilhelm Keller.Ph.D

The Use of Random-Utility Theory in Building Location-Allocation Models

The most important part of a location-allocation model is the allocation rule, that is, the way clients are assigned to facilities. In the well-known models of the "plant-location" family, the embedded allocation rule is the assignment of the least-travel-cost facility, This allocation rule depends on the assumption that the cost, or more generally utility, associated with each possible facility choice is deterministically known. The simplest way to generalize a plant-location model is to add a random term to travel costs, with a known probability distribution. Such randomness may be shown to arise in many real-life situations, and the resulting choice models constitute the subject of random-utility theory, This paper introduces the use of the random-utility modeling philosophy in location-allocation problems, Some relevant properties of the resulting family of models are derived, Among them, of special importance is the submodularity property, which relates the random-utility-based location models to a recent area of research in combinatorial optimization. Submodularity is exploited to develop simple heuristic algorithms, and the effectiveness of the approach is supported with some numerical results

Productivity analysis and functional specification of Pakistani textile industry

This study deals with the general functional characterization of the Pakistani Textile Industry. Previous studies assumed constant elasticity of substitution functional form, the best-fitted model for general manufacturing industries of Pakistan. There is only one study in which the textile industry is treated separately, but the functional form used is the same, and investigations are done at the provincial level with data pooling for other provinces;The important point in this study is that functional form is not restricted and allowed to be estimated freely by the collected data. We assume that the productivity growth of Pakistani Textiles can be properly specified by a non-homothetic and nonneutral growth flexible functional form. This function is continuous, monotic, concave and twice derivable;The concept of duality of cost function is utilized to evaluate the production function characterized by the above-mentioned properties. To discriminate among the flexible functional forms, the Cox-Box transformation is used, both in production as well as in cost functions. Non-neutral scale effect, biases of technical change, input price changes and output quantity changes are used to analyze the productivity growth;Total factor productivity is used as the growth index. It is calculated by residual as well as by parametric methods. The model used is the four input model with stochastic disturbance term. The method of estimation is maximum likelihood estimation. The error terms are used to account for discrepancy in cost minimizing behavior of the function;The four input model has been used to estimate the Pakistani textile industry, 1965-1989. The inputs are capital, labor, energy and intermediate material;The main hypotheses tested are neutrality of technical change and homothetic shifts of the function;The elasticity results show that energy and capital exhibit complementarity behavior, while both of them are labor substitutable. The scale effects are labor and energy saving and technical change effects are energy and capital-using and labor-saving. Also capital and energy are more own price elastic than labor;The contribution of scale economies to the productivity is almost the same in all the selected models, while the contribution of technical change effects is varying and not significant

The calculus according to S. F. Lacroix (1765-1843)

Silvestre François Lacroix (Paris. 1765 - ibid., 1843) was not a prominent mathematical researcher, but he was certainly a most influential mathematical book author. His most famous book is a monumental Traité du calcul différentiel et du calcul intégral (three large volumes, 1797-1800; a second edition appeared in 1810-1819) - an encyclopaedic appraisal of 18th-century calculus. He also published many textbooks, one of which is closely associated to this large Traité: the Traité élémentaire du calcul différentiel et du calcul intégral (first edition in 1802; four more editions in Lacroix's lifetime; four posthumous editions). Although most historians acknowledge the great influence of Lacroix's large Traité in early 19th-century mathematics it has not been thoroughly studied. This thesis is a contribution for correcting this omission. The focus is on its first edition, but the second edition and the Traité élémentaire, are also addressed. The thesis starts with a short biography of Lacroix, followed by an overview of the first edition of the large Traité. Next corne five chapters where particular aspects are analyzed in detail: the foundations of the calculus, analytic and differential geometry, approximate integration and conceptions of the integral, types of solutions of differential equations (singular/complete/general integrals, geometrical interpretations, and generality of arbitrary functions), and three aspects related to finite differences and series (the use of subscript indices, types of solutions of finite difference equations, and mixed difference equations); for all these aspects Lacroix's treatment is compared to the 18th-century background, and to his likely sources. Then we examine how the large Traité was adapted to a textbook - the Traité élémentaire, we take a look at the second edition of the large Traité, and conclude the body of the thesis with some final remarks

Three essays in demand analysis

In this study, flexible representation of consumer demand systems is investigated within the framework afforded by duality in consumer theory. The first essay examines the market situation where supplies are inelastic and prices must adjust to clear the market, a situation that arises for most agricultural commodities, particularly for perishable commodities. A price dependent or inverse demand system can reflect such a market situation. Two flexible inverse demand systems are proposed. The first of these is a linear inverse demand system (LIDS) derived from a flexible specification of a distance function, and presents a particularly convenient system of linear equations. A simulation exercise reveals that the LIDS performs as well and in fact slightly better than the older inverse Translog system. However, in order to specify market demands, aggregation conditions need to hold. A new system of inverse share dependent equations belonging to the class of quasi-homothetic preferences is also proposed;Between the two polar cases of direct and inverse demands lies the class of \u27mixed\u27 demands where the prices of some goods and the quantities (demanded) of others adjust to clear the market. However, the existing theoretical framework for mixed demands does not allow a convenient transition to an empirical model. The concept of \u27shadow\u27 price is used in developing a theoretical framework that does allow the specification of an empirical mixed demand system. The Slutsky equations are derived, the elasticity forms of which are implemented in a differential approximation of the mixed demand system. The empirical context is provided by the Canadian market for meats where free trade with the U.S. in beef and pork makes Canada a price taker for these commodities, while for poultry, domestic marketing boards restrict the supply, and prices must adjust to clear the markets;The notion of separability is frequently invoked in applied demand analysis. In general, separability is a strong maintained hypothesis and should ideally be tested before it is maintained. However, while the alternative of imposing local separability on a Flexible Functional Form (FFF) exists, imposing global separability parametrically on most of the commonly used FFFs renders them inflexible. In the third essay, the ability of a recently proposed globally separable FFF, as well as that of the locally separable model, in providing correct inferences about separability is examined in a Monte Carlo study