Short-cut demand elasticities
by� Ken Clements*
Economics, UWA
August 2025
Abstract
Suppose you don’t know the price elasticity of demand but need some value. The rule of ½ states that the elasticity is equal to -½. This presentation
(i) highlights the precise conditions under which the rule holds,
(ii) indicates its practical importance, and
(iii) points to related results on simplifications of demand elasticities.
When nothing much is known about the elasticity, -½ is likely to be a good starting point for certain types of goods.
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*This is mostly based on the book by K. W. Clements, H. Liu, J. M. Mariano, E. A. Selvanathan, S. Selvanathan and G. Verikios (2025), Short-cut demand elasticities and other convenient approaches to consumer demand (Springer). In addition to my co-authors of the book, for helpful comments and support, I also thank Simon Chang, Shawn Xiaoguang Chen, Tom Hertel, Haiyan Liu, Anda Nugroho and seminar participants at UWA.
Booze consumption
Price elasticities of demand, imported booze, USA:
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Source: A. Muhammad (2025). “Trump Tariffs 2.0: Assessing the Impacts on US Distilled Spirits Imports.” Agribusiness, 1-7
Own-price elasticities around - ½
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Small cross-price elasticities
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Trouble-makers
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Simplifications in 3 steps
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Step II
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Step III
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Trouble-makers undisturbed
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Payoff from simplification
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Rule of ½�Price elasticities = -½
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Utility and demands
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More familiar elasticity form
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Preference independence
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Three (very) different economists�What’s the link?
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Arthur Pigou
1877-1959
Milton Friedman
1912-2006
Henry Schultz
1893-1938
Early contributors to �preference independence
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Early contributors to �preference independence
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Early contributors to �preference independence
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Pigou’s law, �price and income elasticities linked
Salt – essential (few substitutes) and viewed as a necessity (low income elasticity)
Foreign vacation -- not essential and not a necessity
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Elasticity = ½�in two more steps
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Summary of rule of ½
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Preference independent Rotterdam
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Preference independent Rotterdam
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Mean = - 0.59
Median = - 0.46
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Centre of gravity not too far from -½, but note dispersion
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Possible objections
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Objection 1: �No necessities, no luxuries
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Response 1:�Allowing for necessities & luxuries
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Objection 2: �Micro goods not preference independent
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Response 2:�Aggregate micro goods into groups
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Direct evidence
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Price elasticities, �broad aggregates
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Sources of elasticities
Beer, wine and spirits
Clements, Mariano, Verikios and Wong (2022). “How Elastic is Alcohol Consumption?” Economic Analysis and Policy
Elder, Lawrence, Ferguson, Naimi, Brewer, Chattopadhyay, Toomey and Fielding (2010). “The Effectiveness of Tax Policy Interventions for Reducing Excessive Alcohol Consumption and Related Harms.” American Journal of Preventive Medicine
Fogarty (2010). “The Demand for Beer, Wine and Spirits: A Survey of the Literature.” Journal of Economic Surveys
Gallet (2007). “The Demand for Alcohol: A Meta-Analysis of Elasticities.” Australian Journal of Agricultural and Resource Economics
Nelson (2013a). “Robust Demand Elasticities for Wine and Distilled Spirits: Meta-Analysis with Corrections for Outliers and Publication Bias.” Journal of Wine Economics
Nelson (2013b). “Meta-Analysis of Alcohol Price and Income Elasticities – with Corrections for Publication Bias.” Health Economics Review
Nelson (2014). “Estimating the Price Elasticity of Beer: Meta-Analysis of Data with Heterogeneity, Dependence, and Publication Bias.” Journal of Health Economics
Selvanathan and Selvanathan (2005). The Demand for Alcohol, Tobacco and Marijuana: International Evidence
Sornpaisarn, Shield, Cohen, Schwartz and Rehm (2013). “Elasticity of Alcohol Consumption, Alcohol-Related Harms, and Drinking Initiation in Low- and Middle-Income Countries: A Systematic Review and Meta-Analysis.” International Journal of Alcohol and Drug Research
Wagenaar, Salois and Komro (2009). “Effects of Beverage Alcohol Price and Tax Levels on Drinking: A Meta‐Analysis of 1003 Estimates from 112 Studies.” Addiction
Water
Dalhuisen, Florax, de Groot, and Nijkamp (2003). “Price and Income Elasticities of Residential Water Demand: A Meta-Analysis.” Land Economics
Espey, Espey and Shaw (1997). “Price Elasticity of Residential Demand for Water: A Meta-Analysis.” Water Resources Research
Cigarettes
Gallet and List (2003). “Cigarette Demand: A Meta-Analysis of Elasticities.” Health Economics
Petrol
Espey (1996). “Explaining the Variation in Elasticity Estimates of Gasoline Demand in the United States: A Meta-Analysis.” Energy Journal
Espey (1998). “Gasoline Demand Revisited: An International Meta-Analysis of Elasticities.” Energy Economics
Goodwin, Dargay and Hanly (2004). “Elasticities of Road Traffic and Fuel Consumption with Respect to Price and Income: A Review.” Transport Reviews
Graham and Glaister (2002). “Review of Income and Price Elasticities of Demand for Road Traffic.” Working Paper, Imperial College
Electricity
Espey and Espey (2004). “Turning on the Lights: A Meta-Analysis of Residential Electricity Demand Elasticities.” Journal of Agricultural and Applied Economics
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Micro products more elastic
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Source: Tellis (1988). “The Price Elasticity of Selected Demand: A Meta-analysis of Econometric Models of Sales.” Journal of Marketing Research
Wrap up, rule of ½
(i) Applicable to broad aggregates only
(ii) ½ a guideline, not a law
(iii) Based on approximations
“Approximations are the soul of science” (von Neumann)
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Linear expenditure system
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Under-appreciated problem:�LES and food
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Under-appreciated problem:�LES and food
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Food luxury for rich!
Warning �LES income elasticities�
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One solution:�Piecewise LES
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Food and income �quintile LES, Australia HES
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Food and income �quintile LES, Australia HES
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Food and income �quintile LES, Australia HES
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Food and income �quintile LES, Australia HES
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CGE applications
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Thanks for your attention
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References
Below is a list of references for the presentation, as well useful sources on applied demand analysis.
Barten, A. P. (1964). “Consumer Demand Functions under Conditions of Almost Additive Preferences.” Econometrica 32: 1-38.
Bewley, R. A. (1986). Allocation Models: Specification, Estimation and Applications. Ballinger.
Baumol, W. J. (1952). “The Transactions Demand for Cash: An Inventory Theoretic Approach.” Quarterly Journal of Economics 66: 545-56.
Brown, A., and A. Deaton (1972). “Surveys in Applied Economics: Models of Consumer Behaviour.” Economic Journal 82: 1145-236.
Clements, K. W. (2008). “Price Elasticities of Demand are Minus One-Half.” Economics Letters 99: 490-93.
Clements, K. W., H. Liu, J. M. Mariano, E. A. Selvanathan, S. Selvanathan and G. Verikios (2025). Short-cut Demand Elasticities and other Convenient Approaches to Consumer Demand. Springer.
Clements, K. W., and J. Si (2017). “Notes on the Pattern of OECD Consumption.” Unpublished working paper, UWA Business School.
Clements, K. W., and J. Si (2018). “Divisia and Frisch are Friends.” UWA Economics Discussion Paper 18-05.
Clements, K. W., J. Si, E. A. Selvanathan and S. Selvanathan (2020). “Demand Elasticities for 9 Goods in 37 Countries.” Applied Economics 52: 2636-55.
Dawkins, C., T. Srinivasan and J. Whalley (2001). “Calibration.” In J. J. Heckman and E. Leamer (Eds.), Handbook of Econometrics, vol. 5. Elsevier.
Deaton, A. S. (1974). “A Reconsideration of the Empirical Implications of Additive Preferences.” Economic Journal 84: 338-48.
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References (cont’d)
Deaton, A. S., and J. Muellbauer (1980). Economics and Consumer Behaviour. Cambridge University Press.
Friedman, M. (1935). “Professor Pigou's Method for Measuring Elasticities of Demand from Budgetary Data.” Quarterly Journal of Economics 50: 151-63.
Frisch, R. (1959). “A Complete Scheme for Computing All Direct and Cross Demand Elasticities in a Model with Many Sectors.” Econometrica 27: 177-96.
Frisch, R. (1932). New Methods of Measuring Marginal Utility. J. C. B. Mohr.
Goldberger, A. S. (1987). Functional Form and Utility: A Review of Consumer Demand Theory. Westview Press.
Gorman, W. M. (1959). “Separability Utility and Aggregation.” Econometrica 27: 469-81.
Houthakker, H. S. (1957). “An International Comparison of Household Expenditure Patterns, Commemorating the Centenary of Engel’s Law.” Econometrica 25: 532-51.
Houthakker, H. S. (1960). “Additive Preferences.” Econometrica 28: 244-57.
Lluch, C., A. A. Powell and R. A. Williams (1977). Patterns in Household Demand and Savings. Oxford University Press.
Muhammad, A. (2025). “Trump Tariffs 2.0: Assessing the Impacts on US Distilled Spirits Imports.” Agribusiness: 1-7.
Pigou, A. C. (1910). “A Method of Determining the Numerical Value of Demand Elasticities.” Economic Journal 20: 636-40.
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References (cont’d)
Pollak, R. A., and T. J. Wales (1992). Demand System Specification and Estimation. Oxford University Press.
Powell, A. A. (1974). Empirical Analytics of Demand Systems. DC Heath.
Selvanathan, S. (1993). A System-Wide Analysis of International Consumption Patterns. Kluwer.
Stone, R. (1954). “Linear Expenditure Systems and Demand Analysis: An Application to the Pattern of British Demand.” Economic Journal 64: 511-27.
Tellis, G. J. (1988). “The Price Elasticity of Selected Demand: A Meta-analysis of Econometric Models of Sales.” Journal of Marketing Research 25: 331–41.
Theil, H. (1965). “The Information Approach to Demand Analysis.” Econometrica 33: 67-87.
Theil, H. (1975/76). Theory and Measurement of Consumer Demand. Two Volumes. North-Holland.
Theil, H. (1980). The System-Wide Approach to Microeconomics. The University of Chicago Press.
Tobin, J. (1956). “The Interest Elasticity of the Transactions Demand for Cash.” Review of Economics and Statistics 38: 241-47.
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This contains elaborations of the material in the main slides and some further results. Optional reading!
Appendix
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Glossary and key results, demand analysis and ½ | |
Symbol/Result | Meaning |
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| Utility function |
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| Income flexibility, the reciprocal of the income elasticity of the marginal utility of income |
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Table 1. SUMMARY OF MAJOR CONCEPTS
Glossary and key results, LES | ||
Symbol/Equation | Meaning/Name | Constraints |
| ||
| |
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| Price of good i |
|
| Total expenditure, “income” for short |
|
| Budget share of i | |
| ||
| |
|
| Cost of subsistence of i | |
| Marginal share of i | |
| Supernumerary income | |
(Continued next page)
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Glossary and key results, LES (cont’d) | ||
Symbol/Equation | Meaning/Name | Constraints |
| ||
| Income elasticity of i | |
| | |
| | |
| ||
| Uncompensated own-price elasticity | |
| Uncompensated cross-price elasticity | |
(Continued next page)
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Glossary and key results, LES (cont’d) | ||
Symbol/Equation | Meaning/Name | Constraints |
| ||
| Compensated own-price elasticity | |
| Compensated cross-price elasticity | |
| ||
| Stone-Geary utility | |
| Marginal utility of income | |
| Frisch parameter -- income elasticity of marginal utility of income | |
Measurement of income flexibility 𝜙
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Linear expenditure system
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Lluch, Powell and Williams (1977),� influential book
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LES and Frisch
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Source: Lluch et al. (1977, p. 77).
Source: Lluch et al. (1977, p. 77).
Some weaknesses with LES
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½ and monetary economics
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Price stabilisation and ½
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References for appendix
Acland, D., and D. H. Greenberg (2023). “The Elasticity of Marginal Utility of Income for Distributional Weighting and Social Discounting: A Meta-analysis.” Journal of Benefit-Cost Analysis 14: 386-405.
Arrow, K. J., H. B. Chenery, B. S. Minhas and R. M. Solow (1961). “Capital-Labor Substitution and Economic Efficiency.” Review of Economics and Statistics 43: 225-50.
Brown, A., and A. Deaton (1972). “Surveys in Applied Economics: Models of Consumer Behaviour.” Economic Journal 82: 1145-236.
Chen, D. L. (1999). World Consumption Economics. World Scientific.
Clements, K. W., H. Liu, J. M. Mariano, E. A. Selvanathan, S. Selvanathan and G. Verikios (2025). Short-cut Demand Elasticities and other Convenient Approaches to Consumer Demand. Springer.
Clements K. W., M. J. Mariano and G. Verikios (2022). “Expenditure Patterns, Heterogeneity, and Long-term Structural Change.” Economic Modelling. 113: 105888.
DeJanvry, A., J. Bieri and A. Nunez (1972). “Estimation of Demand Parameters Under Consumer Budgeting: An Application to Argentina.” American Journal of Agricultural Economics 54: 422-30.
Evans, D. J. (2005). “The Elasticity of Marginal Utility of Consumption: Estimates for 20 OECD Countries.” Fiscal Studies 26: 197-224.
Frisch, R. (1959). “A Complete Scheme for Computing All Direct and Cross Demand Elasticities in a Model with Many Sectors.” Econometrica 27: 177-96.
Gao, G. (2012). The Economic Behaviour of Seven Billion Consumers: Food, Functional Form and Quality. PhD thesis, The University of Western Australia.
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References for appendix (cont’d)
Groom, B., and D. Maddison (2019). “New Estimates of the Elasticity of Marginal Utility for the UK.” Environmental and Resource Economics 72: 1155-82.
Hjertstrand, P. (2025). “The Marginal Utility of Income and Homogeneous Demand Systems.” Journal of Economic Behavior and Organization 229: 106853.
Layard, R., G., Mayraz and S. Nickell (2008). “The Marginal Utility of Income.” Journal of Public Economics 92: 1846-57.
Lluch, C., A. A. Powell and R. A. Williams (1977). Patterns in Household Demand and Savings. Oxford University Press.
Newbery, D. M. G., and J. E. Stiglitz (1981). The Theory of Price Stabilisation: A Study in the Economics and Risk. Oxford University Press.
Selvanathan, S. (1993). A System-Wide Analysis of International Consumption Patterns. Kluwer.
Stern, N. (1977). “The Marginal Valuation of Income.” In M. Artis and R. Norbay (eds), Studies in Modern Economic Analysis. Blackwell.
Theil, H. (1987). “Evidence from International Consumption Comparisons” Chapter 2 in H. Theil and K. W. Clements Applied Demand Analysis: Results from System-Wide Approaches. Ballinger.
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