DEMAND THEORY III Meeghat Habibian Transportation Demand Analysis Lecture note Meeghat Habibian

Consumer Demand Function Quantity of X1 depends on location of point M, this is in turn: Budget level Price of other goods (e.g., X2, X3, …, Xn) Exact shape of indifference curves (i.e., U(x))

Demand Function? Any relationship that would give quantity of Xi, in terms of: Budget level Price of other goods Exact shape of indifference curves

Form of Demand Function From the theory: Explicit reference to U(x) could be eliminated Assuming the prices in demand function is suffice

Cost consumption curve Shows relation between Xi and Pi, for fixed values of all other variables It can derive from indifference map by varying the cost of a normal good

Consumer Demand Function Locus of all points, M

Elasticity 𝐸 𝑣 𝑋 = 𝑉 𝑋 𝜕𝑋 𝜕𝑉 Problem: Derivative value depends on units in which V and X are measured. Solution: A dimensionless measure of change Elasticity of X with respect to V, Ev(X): The percentage change in X for a 1 percent change in V. 𝐸 𝑣 𝑋 = 𝑉 𝑋 𝜕𝑋 𝜕𝑉 Transportation Demand Analysis - Title

Elasticity The sensitivity of demand with respect to a variable v: 𝑒 𝑣 = 𝜕𝑋/𝑋 𝜕𝑉/𝑉 = 𝜕 ln 𝑋 𝜕 ln 𝑉 Graphically: The slope of the demand curve if it were drawn on logarithmic scales Valid only for small percentage changes.

Elasticity When ν is cost: Price elasticity As price of a normal good increases by 1 percent, quantity consumed will decreases by ei When ν is income: income elasticity

Elasticity Normal good: Income elasticity: Positive Price elasticity: Negative Inferior good: Income elasticity: May be negative Price elasticity: May be positive

Elastic Behavior (definition) When the absolute elasticity is greater than unity. Relatively elastic behavior: When the absolute elasticity is less than unity .

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

Example (Direct vs. Cross elasticity ) Direct elasticity Ekk Elasticity of the volume choosing mode k with respect to a change in the level of service of mode k. Cross elasticity Ekr Elasticity of the volume choosing mode k, Vk, with respect to a change in the level of service of competing mode r, Sr. 𝑬 𝒌𝒓 ≡ 𝑺 𝒓 𝑽 𝒌 𝝏𝑽 𝒌 𝝏𝑺 𝒓 Transportation Demand Analysis - Title

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.

Example Consider: Utility function U=x1α1x2α2 Quantities: x1 and x2 α1 and α2: parameters p1and p2 : unit prices B budget limit

Example

Example P1/P2=(α1/x1)/(α2/x2). 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=(α1/x1)/(α2/x2).

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 alternative approach to demand modeling.

Empirical Demand Functions The three most common forms in transportation demand: linear multiplicative exponential Hybrid forms combining any of these three are also to be found. Empirical evidence is necessary for validation. The choice must be based on a logical postulation of the causal relationships involved.

Linear Demand Function Factors have independent additive effects: T=α0+ α1p+ α2I α0 ,α1 ,α2: coefficients T: trip, P: cost, I: income 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

Multiplicative Demand Function For example: T=α0Pα1Iα2 interaction between the effect of its variables Effect of P on T : This effect depends on value of T, hence it is not independent of value of I  p and I are interacting on it The function can be linearized by logarithmic transformation

Multiplicative Demand Function Elasticity with respect to any of variables, is constant. For example, elasticity of T with respect to I: eI=(ðT/T)/(ðI/I)=(ðT/ðI)/(T/I) =α2 Similarly, elasticity of T with respect to P is α1 ep=(ðT/T)/(ðP/P)= α1 Elasticities are constant and independent This property led to widespread use in transportation Therefore, this form is applied for simplification

Exponential Demand Function 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:

Exponential Demand Function The postulation of demand elasticities proportional to the variables concerned is not uncommon in transportation applications (e.g., Entropy model in trip distribution) A more common functional form, however, is the combined multiplicative and exponential function:

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