Tsvetomir Tsachev

Prices Versus Quantities in a Vintage Capital Model

The heterogeneity of the available physical capital with respect to productivity and emission intensity is an important factor for policy design, especially in the presence of emission restrictions. In a vintage capital model, reducing pollution requires to change the capital structure through investment in cleaner machines and to scrap the more polluting ones. In such a setting we show that emission tax and auctioned emission permits may yield contrasting outcomes. We also show that some failures in the permits market may undermine its efficiency and that imposing the emission cap over longer periods plays a regularizing role in the market.

MAXIMUM PRINCIPLE FOR AGE AND DURATION STRUCTURED SYSTEMS: A TOOL FOR OPTIMAL PREVENTION AND TREATMENT OF HIV

Age and duration since infection are considered in a model of optimal control of the spread of Human Immunodeficiency Virus (HIV) in countries with high prevalence. Prevention and medical treatment are selected so as to maximize an economic objective function.The model extends the classical McKendrick equation. Necessary optimality conditions in the form of Pontryagins global maximum principle and numerical solution based on them are presented. “Critical” initial prevalence is established numerically for which there are two optimal medical treatments: one intense and another less demanding. It is shown that treatment alone can be counterproductive: increase in treatment must be accompanied by increase in prevention.

OPTIMAL EDUCATION IN AN AGE-STRUCTURED MODEL UNDER CHANGING LABOR DEMAND AND SUPPLY

We introduce a model of the optimal education policy at the macro level, allowing for heterogeneity of the workforce with respect to its age and qualification skills. Within this framework we study the optimal education rate in the context of changes in labor demand (as represented by the elasticity of substitution across ages and qualification) and labor supply (as represented by a change in the population growth rates). Applying an age-structured optimal-control model, we derive features of the optimal age-specific education rate. Our results show that the relation between the elasticities of substitution of labor across ages plays a crucial role in the way the demographic changes affect (both in the short and in the long run) the optimal educational policy. We also show that under imperfect substitutability across age and qualification groups, the optimal educational policy is adjusted in advance to any change in the labor supply.

Set-membership estimations for the evolution of infectious diseases in heterogeneous populations

The paper presents an approach for set-membership estimation of the state of a heterogeneous population in which an infectious disease is spreading. The population state may consist of susceptible, infected, recovered, etc. groups, where the individuals are heterogeneous with respect to traits, relevant to the particular disease. Set-membership estimations in this context are reasonable, since only vague information about the distribution of the population along the space of heterogeneity is available in practice. The presented approach comprises adapted versions of methods which are known in estimation and control theory, and involve solving parametrized families of optimization problems. Since the models of disease spreading in heterogeneous populations involve distributed systems (with non-local dynamics and endogenous boundary conditions), these problems are non-standard. The paper develops the needed theoretical instruments and a solution scheme. SI and SIR models of epidemic diseases are considered as case studies and the results reveal qualitative properties that may be of interest.

Pontryagin's Type Optimality Conditions for a Distributed Control Problem Arising in Endogenous Growth Theory

The paper presents an optimality condition for a distributed optimal control problem arising in economics, where the (one-dimensional) spatial domain can be influenced by the control. The result enables analytical and numerical investigation of a class of endogenous growth models with investment-dependent technological frontier and heterogeneous capital stock.

Optimal Bounded Investment Behaviour

We study the investment behaviour of a producer maximizing the present value of his firm over an infinite time horizon and having investments bounded below by zero (irreversibility of capital) and bounded above by his current profit (capital market imperfections). In our model the interest rate, the wage and the price of the unique good, which serves also as capital, are assumed known and held fixed. We find the optimal investment behaviour in the form of a feed-back function depending on the current capital level. We also present some numerical simulations illustrating the effects from changes in each of the problems parameters.