Gerhard Tintner, A Note on Distribution of Income Over Time (1936)

Gerhard Tintner’s journal article develops a variational approach to allocating a consumer’s income across goods and over time. Its central move is to replace utility depending only on quantities with utility depending also on their rates of change. The article proceeds from Hotelling’s static budget-constrained maximization, through a dynamic formulation using Euler equations, to alternative expressions intended to make those equations economically intelligible. Its contribution is a formal extension of consumer theory: intertemporal allocation requires adjusting marginal utility for the changing influence of consumption flows.

Suppose the utility of a certain quantity of goods to depend not only on the amount possessed at any moment but also on the flow of goods in time.

Tintner motivates this dependence through two contrasting experiences. Familiarity with some luxury goods and services may increase the utility of further consumption; more ordinarily, accumulated consumption may diminish a good’s attraction because the consumer desires novelty. These examples establish why timing matters even when prices remain fixed. They are motivations rather than separately specified models of habit or satiation: the mathematical formulation makes utility a function of quantities and their first time derivatives, without introducing an explicit variable for accumulated past consumption.

The ordinary case, however, will be that the individual in question strives for change and novelty: the utility of a given amount of goods will be less the more has been consumed in the past.

The static analysis supplies both the mathematical template and its vocabulary of valuation. Given a known, measurable utility function, prices, and money income, Tintner uses a Lagrange multiplier to derive equality of marginal utility per unit of expenditure across goods. He then eliminates the multiplier and rewrites this condition in several forms. Marginal utility divided by price equals the ratio of total quantity-weighted marginal utilities to total expenditure, or equivalently the ratio of quantity-weighted average marginal utility to a quantity-weighted average price. Invoking Wieser, he interprets quantity multiplied by marginal utility as the utility-value of a stock. At the optimum, each stock’s share in aggregate utility-value therefore equals its share in money expenditure.

These rearrangements matter because the dynamic argument reproduces them with a modified valuation term. Instead of allocating a budget at a single moment, the consumer maximizes integrated utility over a specified interval, subject to a fixed total expenditure over that interval. Prices are constant, while quantities follow time paths. A multiplier incorporates the integral budget constraint, and the Euler equations provide necessary conditions for an optimum. Solving the resulting second-order equations also requires integration constants, which Tintner suggests supplying through quantities fixed at the interval’s endpoints. The budget alone cannot determine the path.

The dynamic counterpart to marginal utility is (\partial\psi/\partial q_i-d(\partial\psi/\partial q_i')/dt). The first term measures utility’s direct response to the quantity of a good; the second measures the change over time in the marginal influence of its rate of change. Dividing this adjusted expression by price produces the counterpart of static marginal utility per monetary unit. The allocation condition equates it across goods and connects it to aggregate expenditure. This is the article’s principal conceptual achievement: an instantaneous valuation acquires a correction reflecting the temporal structure of utility.

The interpretation of (15) is not very easy.

Tintner’s admission introduces a careful attempt to interpret the formalism, rather than a claim that the new term has an immediately familiar economic meaning. If utility is independent of rates of change, the correction disappears and the ordinary weighted marginal-utility condition returns. He subsequently derives dynamic analogues of the weighted-average and stock-value proportions. Quantities multiplied by adjusted marginal utilities stand in the same proportions as expenditures at the corresponding moment. These identities preserve the architecture of the static argument while changing what counts as the relevant marginal valuation. Tintner also suggests that allowing utility to depend on higher time derivatives would produce a similar formal structure.

The final derivation exploits the absence of explicit time dependence in the augmented utility function to obtain a first integral of the Euler equations. Tintner interprets its integration constant through the limiting case in which utility does not depend on consumption flows: there, total utility would equal the constant. He presents the resulting equations as another route to determining quantities through time. These expressions should be distinguished from the article’s clearer central Euler condition: the supplied text contains irregularities in notation and summation, as well as a prose description that reverses the numerator and denominator of the displayed ratio. Those details warrant mathematical checking rather than silent correction.

We should not forget, however, to mention that we only stated the necessary and not the sufficient conditions of a maximum.

This closing qualification limits the article’s claims of determination. Stationarity does not by itself establish a maximum; additional variational conditions would impose restrictions on the utility function. The note’s lasting relevance lies in making the allocation of income through time a problem of optimizing consumption paths, while showing exactly how familiar static valuation conditions must change. Its methodological premise is also consequential: integrating utility over time restricts admissible transformations of the utility index, so the analysis requires more structure than a merely ordinal ranking of instantaneous bundles.