Steffen Jørgensen

Equilibrium Pricing and Advertising Strategies in a Marketing Channel

This paper is concerned with conflict and coordination in a two-member channel of distribution. We propose a differential game model that includes carryover effects of advertising, expressed by a retailer-specific stock of advertising goodwill. Pricing and advertising strategies for both firms are identified under channel conflict as well as coordination. Dynamic advertising policies are designed as stationary Markov perfect strategies. In a symmetric case, these strategies can be determined in closed form, taking into consideration explicitly nonnegativity constraints on advertising rates. We establish a global result for the relationship between the advertising strategies of the two firms under conflict and coordination.

Production, inventory, and pricing under cost and demand learning effects

Abstract The paper considers a monopolist firm that plans its production, inventory, and pricing policy over a fixed and finite horizon. The problem is represented by an optimal control model which combines elements from three streams of literature. The first of these is a classical OR area and deals with optimal production and inventory under exogenously given demand conditions. The second is an area of marketing science which studies dynamic pricing under demand learning effects. Demand learning refers to the situation where current demand for a product is influenced by past demand. The third area belongs to microeconomics and industrial organization and is concerned with the effects of learning-by-doing in a firms production process. Learning is reflected in a unit production cost that decreases with cumulative production. Using a path-synthezising procedure we obtain closed-form characterizations of optimal production, pricing, and inventory policies.

Optimal dynamic investment policies under concave-convex adjustment costs

The paper considers a dynamic investment model of a monopolistic firm facing adjustment costs that are concave-convex in the rate of gross investment. The firm wishes to maximize the discounted stream of dividends over a finite planning horizon, plus the terminal value of the stock of equity. The problem of finding an optimal investment path is solved by combining results from a (nonstandard) model with concave adjustment costs with results from the (standard) model with convex adjustment costs. The results are presented in phase diagrams and are economically interpreted. The solution involves the use of a chattering control, and we discuss various approaches to avoid the occurrence of such a control policy.

Optimal investment and finance in renewable resource harvesting

Abstract The paper studies a dynamic optimization problem in renewable resource harvesting, capital investment, and financing. It combines two different approaches, viz., renewable resource harvesting with capital investments, and investment policy under a borrowing/lending constraint according to which the interest rate on outstanding debt/lending increases with cumulative debt/lending. The problem is set up as an optimal control model, having two state variables (resource stock and stock of equity) and two controls (effort rate and dividend payout rate). The solution is identified by the maximum principle and a synthesizing procedure.

New product advertising in dynamic oligopolies

The success of the introduction of a new product in a market is very sensitive to the marketing decision variables adopted by the firm. In the present paper we are concerned with the question of new product advertising in a heterogeneous oligopoly market consisting of N firms. A dynamic game is formulated to model strategic as well as sales interactions in such a market. Optimal advertising strategies are identified as open-loop Nash solutions.

Petrarch's canzoniere: Rational addiction and amorous cycles

The paper is concerned with a celebrated collection of love poems, the 14th century Italian poet Francis Petrarchs Canzoniere. A striking feature of these poems is the emotional ups and downs experienced by Petrarch and his platonic mistress Laura. Recently, attempts have been made to model these emotional swings by catastrophe theory or nonlinear differential equations. This paper takes a different approach. Starting with a pair of differential equations that model the dynamics of the emotions of the two individuals, we formulate an optimal control problem. A key hypothesis of this problem is that Petrarch was a rational addict of his desire for Laura. With specific functional forms and parameter values we identify a stable limit cycle that gives a representation of the oscillating emotions of Laura and Petrarch.

Price subsidies and guaranteed buys of a new technology

Abstract We consider a problem of a government that wishes to stimulate the adoption of a new technology in order to replace an older, environmentally less desirable, technology. The new technology is manufactured by a monopolist firm which has learning-by-doing in its production process. The firm sells the new product to both private households and government institutions and wishes to determine an optimal pricing policy. The government has at its disposal two instruments: subsidizing the consumer price and making purchases of the new technology from the firm. We assume profit maximization on the part of the firm. The government wishes to maximize the cumulative number of units of the new technology sold to private households by the terminal date of the government program. The problem is set up as a Stackelberg differential game in which we identify an open-loop equilibrium, supposing that the government can credibly precommit to its subsidy and buying program.

Dynamic investment policy with installation experience effects

This paper analyzes the consequences of incorporating a learning-by-doing effect in the firms adjustment cost function. The hypothesis is that, the larger the existing capital stock, the larger the installation experience gained, and therefore the smaller the cost of installing an additional unit of capital stock. The implications of the hypothesis are investigated in an optimal control model for the determination of the firms optimal investment policy over an infinite planning period.

Optimal investment, financing, and dividends: A stackelberg differential game☆

Abstract Over a finite planning period, a film has two shareholders who trade shares at a fixed price. External transactions are disregarded. The total amount of shares is fixed and the majority shareholder decides on the rate of dividend payout. Each shareholder maximizes a profit functional comprised of total earnings from share transactions plus dividends, and capital gains at the horizon date. The shareholders are subject to personal taxation on dividends and capital gains. Decisions on investments and borrowing/lending are made by a manager who maximizes accumulated profits after corporate taxation. The problem is modelled as an open-loop Stackelberg differential game such that the manager is the leader; the shareholders are followers, playing a Nash game. The latter game is analyzed by using standard techniques of optimal control theory. The analysis of the managers problem is done by using a path-connecting procedure.

Government Price Subsidies to Promote Fast Diffusion of a New Consumer Durable

We consider a market in which a single supplier sells a new product characterized by diffusion effects on the demand side. In this setting we analyze the problem whether or not the government should subsidize the diffusion of this innovation. We assume that the government is a Stackelberg leader and decides about the subsidy (price or cost subsidy) before the supplier sets his optimal price. The objective of the government is to choose a subsidy policy such that the number of adopters at the horizon date is maximized. The firm chooses a pricing strategy so as to maximize the present value of profits.