Gerhard Tintner, Monopoly Over Time (1937)

Gerhard Tintner’s journal article develops a mathematical bridge between classical monopoly theory and the analysis of monopoly through time. Its central move is to replace demand depending solely on the current price with demand also depending on the price’s rate of change and, subsequently, its higher time derivatives. The simplified models serve a methodological purpose: to discover whether dynamic profit maximization can be expressed through economically intelligible concepts comparable to those used in static theory. Tintner situates this project between Chamberlin and Robinson’s work on imperfect competition and Evans and Roos’s dynamic economics, while reserving bilateral monopoly and imperfect competition over time for further investigation.

The economic motivation is that a price cannot always be understood independently of its trajectory:

The economic significance of this latter situation lies in the fact that the previous prices influence the buyers' reactions to the present price.

This dependence matters particularly for goods that can be stored. Tintner does not develop a theory of inventories or expectations; instead, he assumes that buyers’ reactions are represented by the demand curve or surface. He nevertheless sketches a utility-theoretic foundation: expected prices may enter utility functions, and expectations may depend on price tendencies. Time derivatives then become parameters of the demand functions derived from those utilities. Dynamic demand thus gives formal expression to anticipation without resolving how expectations are formed.

The article proceeds through three nested cases: Cournot’s demand depending on present price, Evans’s demand depending on price and its first time derivative, and a generalized demand function incorporating higher derivatives. Each case is formulated successively in terms of profits, revenue, and demand. Tintner acknowledges that this hierarchy does not exhaust possible dynamic models: demand might involve integrals, with a functional representing the most general case.

The Cournot analysis supplies the benchmark. With profit equal to revenue minus costs and revenue equal to price times quantity, the monopolist varies price until its marginal effect on profit vanishes. Tintner rewrites this condition through revenue, marginal cost, and the response of demand to price. In the Evans case, however, optimization concerns a price path over a specified interval, maximizing the integral of profit rather than profit at an isolated moment. The calculus of variations supplies the Euler equation as a necessary, explicitly insufficient, condition.

Dependence on price movement does not determine a single behavioral response:

For instance, a rising tendency of price may lead people to expect the price rise to continue or to expect that it may end soon.

Continued increases encourage present purchases, whereas an anticipated reversal reduces demand. Although Tintner considers the first response more frequent, the model leaves the demand surface to specify the relationship. Its mathematical structure accommodates contrasting expectations rather than establishing their empirical prevalence.

Because time does not enter the profit function explicitly, Tintner obtains a first integral of the Euler equation, written as (\pi-p'\partial\pi/\partial p'=\pi_0). He identifies the integration constant (\pi_0) with the profit obtainable under simple Cournot monopoly and uses it as a benchmark throughout the subsequent comparisons. A specified demand surface and boundary conditions determine the price trajectory. Introducing higher price derivatives produces analogous variational equations, but also adds terms whose economic interpretation becomes increasingly difficult.

The second half translates these results into logarithmic derivatives. This is more than a change of notation: elasticities allow comparisons independent of arbitrary measurement units.

These have the advantage that they are pure numbers of dimension zero and therefore invariant with respect to a change in the scale.

In the static case, profit elasticity with respect to price is zero at the optimum. In the Evans formulation, profit elasticity with respect to the rate of price change equals ((\pi-\pi_0)/\pi). Tintner thereby relates sensitivity to price movement to the difference between current profit and his Cournot benchmark. The generalized formulation includes sums of elasticities and additional time-derivative terms; these resist the relatively direct interpretation available in the simpler cases.

The revenue analysis introduces “weighted marginal costs,” marginal cost divided by price. Tintner distinguishes revenue elasticity with respect to price from marginal revenue, which takes quantity as its independent variable. At the static monopoly point, revenue elasticity equals demand elasticity multiplied by weighted marginal cost. The dynamic counterpart adds a term relating the profit difference to total revenue. These parallel expressions make the static and dynamic problems comparable through recognizable economic magnitudes.

The demand formulation similarly connects elasticity to price and marginal cost. Tintner interprets marginal cost multiplied by monopoly output as the payment consumers would make if that same quantity were sold at a competitive, marginal-cost price. This is a fixed-quantity comparison, not a prediction that competition would produce monopoly output. He emphasizes that it avoids interpersonal utility comparisons. In the Evans case, demand elasticity with respect to price movement is related to the profit difference divided by revenue above this hypothetical competitive payment.

The conclusion preserves the distinction between formal generality and economic understanding:

In the general case however the solution was not so simple, and it is doubtful if much economic meaning can be attached to the results reached.

The article’s contribution is therefore strongest in its systematic translation between static monopoly and the simpler dynamic model. It offers a vocabulary for intertemporal optimization while recognizing the interpretive limits of higher-order extensions. Tintner also leaves second-order conditions untreated, so the derived conditions do not themselves establish a maximum. The work’s lasting conceptual interest lies in treating the price trajectory as part of the demand problem and asking how far familiar economic parameters can make that dynamic structure intelligible.