Gerhard Tintner, The Maximization of Utility Over Time (1938)

Gerhard Tintner’s journal article develops a mathematical theory of consumption planned across time. Its four sections move from discrete consumption dates to continuous consumption streams, reformulate the resulting equilibrium conditions through marginal rates of substitution, and identify unresolved questions about maximum conditions and empirical verification. The central argument extends the familiar equalization of marginal utility per unit of expenditure: when consumers can transfer purchasing power between dates through saving, the corresponding marginal utilities of money must be related by expected interest rates. Intertemporal choice thus combines substitution among commodities with substitution among dates.

The discrete model begins with three goods and a finite planning horizon, while allowing extension to any number of commodities. Utility depends jointly on the quantities expected to be consumed at all dates; Tintner does not impose an additive decomposition into separate period utilities. Prices, incomes, and interest rates are likewise expected quantities. The model concerns a plan formed at an initial date, rather than a sequence of decisions revised as information arrives. Its distinction between commodity holdings and financial saving is explicit:

He does not accumulate any commodity stocks.

Goods enter the model as dated consumption quantities, while saving carries purchasing power forward with interest. Final-period saving is set to zero, so the plan exhausts available resources over its horizon. Each period’s expenditure and saving must equal its income plus the proceeds of previous saving. These linked accounting constraints provide the economic structure of the maximization problem.

Tintner attaches a Lagrange multiplier to each period’s constraint. Differentiation with respect to consumption equates each commodity’s marginal utility to its price multiplied by that period’s multiplier. Differentiation with respect to saving links successive multipliers through the interest rate. Interpreting the multipliers as marginal utilities of money gives the article’s central temporal relationship:

The expected marginal utility of money at the point in time j equals the marginal utility of money at point 1 discounted by the expected interest rates in the interval 1 · · · j.

This is a condition governing the allocation of resources across dates, not an independently imposed psychological discount factor. Within each date, marginal utilities divided by prices are equal across goods; across dates, their common value follows the discounting relationship generated by saving. The interest rate therefore connects otherwise separate commodity-allocation decisions. Tintner also collapses the sequence of period constraints into a single budget equation equating discounted income with discounted expenditure. Maximizing subject to this present-value constraint yields the same equilibrium conditions, showing the equivalence of the sequential and consolidated formulations.

Section B carries this argument into continuous time. A utility function of finitely many quantities becomes a functional of entire consumption paths. The budget equation becomes an equality between discounted income and expenditure integrals, with the force of interest replacing discrete interest rates. Functional derivatives express the marginal effect of varying consumption at a particular instant while utility remains dependent on the full paths.

The variational argument requires utility’s first variation to vanish for changes in consumption that preserve the discounted budget. A single constant Lagrange multiplier then yields the continuous counterpart of the discrete result: each functional marginal utility divided by its current price equals the initial marginal utility of money multiplied by the continuous discount factor. The passage to a functional therefore preserves the economic logic while enlarging the mathematical object being optimized. It accommodates preferences over consumption streams without requiring utility to take the form of an integral of independent instantaneous satisfactions.

Tintner next separates the equilibrium conditions from the absolute numerical measurement of utility:

Our ultimate results do not involve the marginal utilities or functional marginal utilities themselves, but only their ratios.

He introduces a transformed utility index as yielding the same results under the same constraint. The conceptual emphasis is on relative valuations rather than utility levels. Section C makes this explicit by defining marginal rates of substitution between dated commodities as ratios of partial derivatives. In continuous time, analogous ratios of functional derivatives compare consumption at a given instant with a reference consumption at the initial instant. These substitution rates are themselves functionals of the consumption paths. Rewriting equilibrium in their terms retains the price and interest relationships while expressing the argument through relative willingness to substitute.

The concluding section carefully limits what the derivation establishes. Tintner suggests extending Hicks–Allen elasticity, complementarity, and independence concepts to consumption through time. For the continuous case, such developments would involve limiting determinants constructed from second-order derivatives of the utility functional. These are also relevant to the distinction between a stationary allocation and an actual maximum:

These conditions would secure a true maximum instead of a minimum or minimax of the utility functional and they involve certain inequalities on the determinants mentioned above.

The article thus supplies first-order equilibrium conditions without completing the second-order analysis needed to guarantee maximization. It also leaves statistical verification open, citing earlier utility measurement and family-budget studies as encouraging precedents rather than evidence testing its own intertemporal model. Its contribution is a compact bridge between consumer theory and mathematical analysis of time-dependent choice: expected interest rates organize transfers of purchasing power, functional methods extend choice to continuous streams, and substitution ratios preserve the economically relevant content without relying on absolute utility magnitudes.