1. TRANSPORTATION DEMAND ANALYSIS
DEMAND THEORY 3

2. Consumer demand function Chapter 2: DEMAND THEORY
Looking back we can see quantity of X1 depends on location of point M this is in turn:
Budget level
Price of all goods
Exact shape of indifference curves (or U(x))
Chapter 2: DEMAND THEORY

3. Chapter 2: DEMAND THEORY
Demand function?
Demand function is any relationship that would give quantity of Xi of any good i
from theory that must be satisfied:
Chapter 2: DEMAND THEORY

4. Consumer demand function Chapter 2: DEMAND THEORY
Presuming sufficient of having prices in demand function and eliminating explicit reference to U(x),
a set of parameters that stands for the utility function of the individual consumer:
Chapter 2: DEMAND THEORY

5. Consumer demand function Chapter 2: DEMAND THEORY
We can derive a cost consumption curve from indifference map by varying the cost
of the good which shows relation between
Xi and Pi, for fixed values of all other variables, for a normal good
Chapter 2: DEMAND THEORY

6. Consumer demand function Chapter 2: DEMAND THEORY
Locus of all points, M

7. Chapter 2: DEMAND THEORY
Elasticity
The ratio of the percentage changes of demand and a variable in question,
Only for small percentage changes.
Chapter 2: DEMAND THEORY

8. Chapter 2: DEMAND THEORY
Elasticity
When ν is cost: price elasticity
As the price of normal good increase by 1 percent quantity consumed will decrease by ei
When ν is income: income elasticity
Chapter 2: DEMAND THEORY

9. Chapter 2: DEMAND THEORY
Elasticity
Normal good:
income elasticity: positive
price elasticity: negative
Inferior good:
income elasticity: may be negative
price elasticity: may be positive
Chapter 2: DEMAND THEORY

10. Elastic Behavior Elastic behavior: Relatively elastic behavior:
when the absolute elasticity is greater than unity.
Relatively elastic behavior:
when the absolute elasticity is less than unity .

11. Chapter 2: DEMAND THEORY
Cross Elasticity
When the demand function for a good contains explicit reference to the another good.
Elasticity of the demand for good i with respect to the unit cost of good j
In transportation cross elasticity, not limited to price.
Chapter 2: DEMAND THEORY

12. Chapter 2: DEMAND THEORY
Cross Elasticity
It can be expected positive due to the substitution effect.
If the price of one increases then the consumption of the other will also increase.
Chapter 2: DEMAND THEORY

13. Chapter 2: DEMAND THEORY
Example
Consider:
Utility function U=x1α1x2α2
Quantities,x1 and x2.
α1 andα2 , constant parameters.
p1and p2 are the unit prices.
B budget limit
Chapter 2: DEMAND THEORY

14. Chapter 2: DEMAND THEORY
Example
Chapter 2: DEMAND THEORY

15. Chapter 2: DEMAND THEORY
Example
These two demand functions exhibit unitary price elasticity and no cross-elasticity.
They do satisfy the budget constraint.
At optimality the ratio of the prices is equal to the ratio of the marginal utilities: P1/P2=(a1/x1)/(a2/x2).
Chapter 2: DEMAND THEORY

16. Chapter 2: DEMAND THEORY
Notations
Not all demand functions are derived by maximizing utility functions subject to budget constraints because it is not in general possible to specify a utility function (i.e., this function is quantifiable only on an ordinal scale).
Specifying a priori forms of demand models and using empirical analysis to verify their validity is an ad hoc alternative approach to demand modeling.
Chapter 2: DEMAND THEORY

17. Empirical demand functions Chapter 2: DEMAND THEORY
Three most commonly in transportation demand, are:
the linear,
the multiplicative, and
the exponential forms
Hybrid forms combining any of these three are also to be found.
Empirical evidence is necessary for its validation.
The choice must be based on a logical postulation of the causal relationships involved.
Chapter 2: DEMAND THEORY

18. Linear demand function Chapter 2: DEMAND THEORY
All the factors that affect traffic, such as income, cost, and travel time, have independent additive effects.
For example: T=α0+ α1p+ α2I (α0 ,α1 ,α2 are coefficients)
a linear demand function relating trip T to cost P and income I
Effect of P on T (ðT/ ðP) is constant and equal to α1, independent of I.
Proportional effects (elasticities): ep=(ðT/T)/(ðP/P)=(P/T) α1
Chapter 2: DEMAND THEORY

19. Multiplicative demand function Chapter 2: DEMAND THEORY
For example: T=α0Pα1Iα2
interaction between the effect of its variable ,
Effect of P on T :
this effect is dependent on the value of T and
hence is not independent of the value of I
Thus the effects of p and I are interacting.
the function can be linearized by logarithmic transformation
Chapter 2: DEMAND THEORY

20. Multiplicative demand function Chapter 2: DEMAND THEORY
Elasticity with respect to any of variables, is constant. For example, elasticity of T with respect
To I:
Elasticity of T respect to P is α1
elasticities are constant and independent
this property led to prevalent use in transportation
This is simply a matter of convenience. I
Chapter 2: DEMAND THEORY

21. Exponential demand function Chapter 2: DEMAND THEORY
combines features of the two previous types.
It implies interaction between effects, and results in variable demand elasticities
Example:
The effect of I:
The elasticity of demand with respect to a variable, p is:
Chapter 2: DEMAND THEORY

22. Exponential demand function Chapter 2: DEMAND THEORY
The postulation of demand elasticities that are proportional to the variables concerned is not uncommon in transportation applications.
The pure exponential function arises in transportation demand modeling when postulating certain properties of travel.
A more common functional form, however, is the combined multiplicative and exponential function:
Chapter 2: DEMAND THEORY