Individual Demand Functions 
The demand function is a relationship between the quantity of a good/service that an individual will consume at different prices, holding other prices and income constant. Every point on the demand function is a utility maximizing point. In effect, the demand curve is a translation from utility metric space into dollar metric space. Thus, point 'e' in the diagram above is a point on the demand curve.
To construct the demand curve simply vary the price of one good holding the price of other goods and income constant. In graphical terms this is represented as in the diagrams below. Note that the equilibrium points in the upper diagram have their counterparts in 'quantity space' in the lower diagram. Therefore, this shows that prices or expenditure information provides a measure of people's preferences and can be used in making assessments with respect to valuation.
An Engel curve is also associated with the development of the demand curve from the utility maximizing framework. An Engel curve is the locus of combinations of goods that an individual would consume if they were faced with changes in income holding all prices constant. Pictorially this would mean a parallel shift in the budget constraint either up or down if income rises or falls, respectively. This Engel curve is also known as an income-consumption curve.
- normal goods: the Engel curve is upward sloping
- inferior goods: Engel curve is downward sloping
- perfect substitutes: Engel curve is positively slope with a slope value of
- perfect complements: Engel curve is positively sloped with a slope equal to
Homothetic Preferences depend only on the ratio of goods in the consumption bundle. This means that homothetic preferences will yield straight line Engel curves which pass through the origin. This has the interpretation that if income goes up by a factor , the demand bundle goes up by a factor . Log-linear preferences are an example of homothetic preferences but not all homothetic preferences are log-linear.
The demand curve is defined as the relationship between price and quantity in which the quantity demanded is the unknown and the price is the exogenously given variable. The relationship is represented as:
The inverse demand curve is simply the monotone transformation of the 'ordinary' demand curve. The inverse demand curve indicates, for each level of demand for good 1, the price which would have to be charged for the consumer to consume a given amount. The inverse demand curve is represented as:
Aggregate Demand 
Moving from individual to aggregate demand requires that we sum individual demands in some way. The level of aggregation is determined by the nature of the issue at hand. Demand functions can be defined over socio-economic groups, cities, states and economy wide. There are numerous issues of 'aggregation' not least of which is how one handles the diversity of consumer preferences while aggregating.
One of the interesting issues is how to aggregate given the nature of the good. This is an issue in transportation since some people consider transportation infrastructure 'quasi-public' goods.
Private goods: if I increase my consumption I reduce the amount available for anyone else, the aggregation from individual to aggregate is to sum horizontally. (Left) This reflects the scarcity of the good.
Public goods: if I increase my consumption, the amount available remains, the aggregation from individual to aggregate should be vertical. (Right) P=society’s willingness to pay
Input Demand Theory 
To date we have looked at demand for consumers. Demand for firms applies similar ideas. For instance, a firm may need to choose a trucking company to ship its goods. It can either approach the problem as cost minimization or profit maximization, which are called Duals of each other, and when solved will produce the same answer.
Cost minimization: given the output level Q', minimize costs.
In a competitive market, and a whole set of associated assumptions, firms maximize profits by producing when
Marginal Cost = Marginal Revenue.
The utility function is a representation of consumer preferences and a demand function is the mapping of utility (and hence preferences) into quantity space. The elasticity is a summary measure of the demand curve and it is therefore influenced to a great extent by the underlying preference structure. Elasticity is defined as a proportionate change in one variable over the proportionate change in another variable. It, therefore, provides a measure of how sensitive one variable is to changes in some other variable.
- How sensitive are people to purchasing transit tickets if the fare went up 5%, 10% or 50%?
- How would the demand for housing change if mortgage rates fell by 30%
- How would the demand for international air travel change if airfares went up 15%?
All of these questions are really asking, "what is the elasticity of demand with respect to some variable"?
Own-price Elasticity 
the price elasticity of demand (own price elasticity) is defined as:
Note that dq/dp is the slope of the demand function so unless there is a very particular type of demand function the slope is not the same as the elasticity.
In general, own price elasticity is negative. An increase in should increase the consumption of , (all else equal). However it is often referred to as positive, this is just confusing. All goods have a price elasticity, however, if the elasticity is less than -1, than the good is called elastic and if the elasticity is between 0 and -1, then the good is inelastic.
This is important when looking at the effect of fuel prices on travel demand.
Cross-price Elasticity 
Cross price elasticity examines how the quantity of good i consumed changes as the price of j changes:
If increases and increases, then and are substitutes. If decreases and increases, then and are complements
This is important in examining modal competition.
Income Elasticity 
If Y increases and Qi increases, then Qi is a normal good. If Y increases and Qi decreases then Qi is an inferior good.
Examples are auto ownership, and the difference between new and used cars.
- Parkin; Powell; Matthews (2002). pp.74-5.
- Gillespie, Andrew (2007). p.43.
- Gwartney, James D.; Stroup, Richard L.; Sobel, Russell S. (2008). p.425.
- Gillespie, Andrew (1 March 2007). Foundations of Economics. Oxford University Press. ISBN 9780199296378. http://books.google.com/books?id=9NoT4gnYvPMC. Retrieved 28 February 2010.
- Gwartney, James D.; Stroup, Richard L.; Sobel, Russell S.; David MacPherson (14 January 2008). Economics: Private and Public Choice. Cengage Learning. ISBN 9780324580181. http://books.google.com/books?id=yIbH4R77OtMC. Retrieved 28 February 2010.
- Parkin, Michael; Powell, Melanie; Matthews, Kent (2002). Economics. Harlow: Addison-Wesley. ISBN 0-273-65813-1.