TRANSPORTATION DEMAND ANALYSIS DEMAND THEORY 3

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

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

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

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

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

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

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

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

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 .

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

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

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

Chapter 2: DEMAND THEORY Example Chapter 2: DEMAND THEORY

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

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

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

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

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

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

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

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