Gerhard Tintner, Elasticities of Expenditure in the Dynamic Theory of Demand (1939)

Gerhard Tintner’s theoretical journal article extends his earlier analysis of dynamic demand from quantities consumed to expenditures. Its question is how expected income, prices, and interest rates affect spending on particular commodities and at particular dates within an individual’s consumption plan. The central argument is that expenditure responses can be derived systematically from quantity-demand responses, provided that the direct effects of prices and discounting are distinguished from changes in consumption. Tintner expresses these relationships as elasticities to make comparisons independent of the units of measurement:

It seems desirable to express the relationships in form of elasticities, since this makes them independent of the scale.

The article develops this argument through four sections: a definition of the intertemporal framework, followed by expenditure elasticities for one commodity at one date, one commodity across all dates, and all commodities at one date. This progression shows how the same underlying demand relationships generate different expenditure measures. The contribution is a formal extension of utility-based demand theory, connecting the allocation among commodities with the allocation of consumption over time.

Section A places the consumer at time zero, planning consumption over a finite sequence of future dates. Expected incomes, prices, and interest rates are given, while quantities are chosen to maximize utility over the entire horizon. Crucially, the objects entering utility are dated quantities of individual commodities:

The utility of the individual depends upon all the quantities of all commodities which he expects to consume over the whole period.

This assumption makes consumption of a commodity at one date analytically distinct from consumption at another. Substitution can consequently occur across both goods and dates. Tintner discounts prices and incomes by cumulative accumulation factors, defining total discounted income as the sum of discounted future incomes. Because the model assumes no net saving over the complete horizon, that total also equals total discounted expenditure. This terminal constraint does not require income and expenditure to coincide at each intermediate date.

The notation identifies expenditure shares in several different totals: spending on a commodity-date pair relative to the whole budget, spending at a date relative to lifetime spending on that commodity, and spending on a commodity relative to all spending at that date. These shares are substantive components of the derivation, not merely bookkeeping. They weight substitution effects and income responses, and later permit the aggregation of individual expenditure elasticities. The budget identities impose consistency conditions: expenditure-share-weighted substitution elasticities sum to zero, while the corresponding weighted income elasticities of demand sum to one.

Section B establishes the elementary distinction between quantity and expenditure responses. For a particular commodity at a particular date, the income elasticity of expenditure equals the income elasticity of demand because its discounted price is held constant. A change in another discounted price likewise affects expenditure through the quantity demanded. A change in the commodity’s own discounted price, however, has an additional direct effect: its expenditure elasticity equals its quantity elasticity plus one. Thus a quantity response alone cannot establish whether spending rises or falls when the price changes. Tintner expresses price responses through expenditure shares, substitution elasticities, and income elasticities.

Accumulation-rate changes require a parallel distinction. When the altered rate falls after the consumption date, it does not directly change that expenditure’s discount factor. When it falls at or before the date, expenditure elasticity acquires an additional minus-one term. Beyond this direct effect, the formula incorporates substitution and the difference between the shares of discounted income and expenditure accumulated before the rate change. Tintner identifies that difference as earlier discounted saving relative to the total budget:

This saving may of course be positive or negative.

The qualification matters because the framework allows income and consumption to diverge within the planning horizon even though total net saving is zero. The interest-related response therefore depends partly on whether the individual has planned to accumulate resources or spend ahead of income. The article does not assign a universal direction to that response; its formulas retain the relevant income, substitution, and saving components.

Sections C and D extend these results through the theorem that the elasticity of a sum is the share-weighted average of its components’ elasticities. For expenditure on one commodity over the whole horizon, the weights are each date’s share of expenditure on that commodity. An own-price change at a particular date adds that date’s expenditure share to the weighted quantity response. An accumulation-rate change subtracts the share of the commodity’s expenditure occurring at or after the affected date. For expenditure on all commodities at one date, the weights instead represent each commodity’s share of that date’s spending. The same-date price correction is therefore the affected commodity’s share, while the direct discounting correction remains minus one when applicable.

The closing comparison with Irving Fisher clarifies the significance of this machinery. Tintner’s dated expenditure totals approach Fisher’s analysis of expenditure streams, but the underlying preferences remain defined over every planned commodity quantity, rather than merely over aggregate expenditure at successive dates. Aggregation summarizes expenditure responses without replacing the more differentiated utility structure:

The time preference, if any, should be expressed in the form of the utility function.

Time preference is thus located within the specification of preferences. The article’s lasting conceptual move is to connect commodity choice, temporal substitution, and discounting in a single expenditure framework while preserving the distinction between behavioral adjustments and the direct arithmetic effects of prices and accumulation rates.