Demand Theory II Meeghat Habibian Transportation Demand Analysis

Outline Preference and indifference Substitution between goods Substituting n goods Choice under constraints Two goods example Change in budget Change in price Transportation Demand Analysis – Demand Theory II

Preference and Indifference Analysis of preference deals with the quantities of goods. 5 kg oranges with one pair of shoes? 6 shopping trips by bus with 1 trip to a movie by car? Specific units can be converted into monetary terms (multiplying each by the unit price) Transportation Demand Analysis – Demand Theory II

Preference and Indifference Any point is a combination of two goods x, y. Any point which represent more x, y is preferred: P>M>N M´ is indifferent with M Consumption field For two goods Transportation Demand Analysis – Demand Theory II

Same utility to the consumer Preference and Indifference Indifference curve Same utility to the consumer Transportation Demand Analysis – Demand Theory II

Projection of u(x,y) on consumption field Preference and Indifference 3 dimensional utility function U(x,y), (a surface) Projection of U is an Isoutility curve (convex to origin) Projection of u(x,y) on consumption field Transportation Demand Analysis – Demand Theory II Increasing preference

U is constant therefore: dU(x,y)=0 Substitution Between Goods The curve slope is “marginal rate of substitution” Number of units of X that consumer is willing to give up in order to receive one unit of Y or vice versa Slope: dx/dy or dy/dx Transportation Demand Analysis – Demand Theory II U is constant therefore: dU(x,y)=0

Transportation Demand Analysis – Demand Theory II

? Transportation Demand Analysis – Demand Theory II Marginal rate of substitution between two goods is the inverse ratio of their marginal utilities

 Marginal rate of substitution is negative Marginal Utility of a Good ( ) is the rate of change of utility with quantity consumed Insatiability assumption: Marginal utility of any good is always nonnegative  Marginal rate of substitution is negative Consumer will always give up some of one good for some of the other but never both at the same time! Transportation Demand Analysis – Demand Theory II

Substituting N Goods Consumption Vector X : Transportation Demand Analysis – Demand Theory II

Substituting N Goods In particular case holding all x fixed except two:  Same as the result for two goods (x, y) Transportation Demand Analysis – Demand Theory II

Substituting N Goods The second derivative may have any sign Insatiability assumption: Second assumption: for all i The second derivative may have any sign a common assumption: it is negative (marginal utility from consumption is decreasing) Transportation Demand Analysis – Demand Theory II

Choice Under Constraints 1-Money 2-Time available 3-Space available … It is common to consider all limitations as a monetary budget constraint. Transportation Demand Analysis – Demand Theory II

Choice Under Constraints Only consumption vectors can be chosen that satisfy: P= Vector of unit costs X= Consumption vector B= Total budget Transportation Demand Analysis – Demand Theory II

Choice Under Constraints In n-dimensional preference space Ω Space is divided into two regions: Feasible vs. Infeasible Transportation Demand Analysis – Demand Theory II

Restate the Principle of Consumer Choice Under a Budget Constraint Consumer will chose X* which will maximize U(x) subject to constraint PX=B. Given: X: vector of good, U(x): utility function, P: vector of unit costs, B: budget. Transportation Demand Analysis – Demand Theory II

The analytical formulation: “λ” is a Lagrange multiplier derivatives of L must vanish at X*: Transportation Demand Analysis – Demand Theory II

For a consumption vector X to be optimal from the consumer point of view: Marginal utility of each good is proportional to its unit cost. Transportation Demand Analysis – Demand Theory II

Ratio of marginal utilities Marginal rate of substitution for all i and k Transportation Demand Analysis – Demand Theory II Marginal rate of substitution

Conclusion Optimal consumption of a consumer between any pair of goods: marginal rate of substitution = inverse ratio of unit costs. Transportation Demand Analysis – Demand Theory II

Infeasible region Two Goods Example For two goods x1 and x2: U(x1,x2): utility function P(p1,p2): cost vector (or price vector) B: total budget p1 x1 +p2 x2 =B Budget constraint Transportation Demand Analysis – Demand Theory II Infeasible region slope= p1/p2 Feasible region

Two Goods Example Combining indifference map and budget line for two goods Transportation Demand Analysis – Demand Theory II

I1 is the highest indifference curve can be reached in feasible region Two Goods Example I1 is the highest indifference curve can be reached in feasible region Transportation Demand Analysis – Demand Theory II (x1*, x2*)

Will result in two situations Change in Budget Will result in two situations Transportation Demand Analysis – Demand Theory II

Situation 1: Increasing both x1* and x2* Transportation Demand Analysis – Demand Theory II

Situation 2: Increasing one of x1* and x2* and decreasing another Transportation Demand Analysis – Demand Theory II

Marginal utility of x2 is higher than that of x1 Consumer will give up small amount of x1 for a considerable amount of x2 X1 is an inferior good X2 is a normal good Transportation Demand Analysis – Demand Theory II

transportation as their income increase! Consumption of some good might decrease if income of the consumer increases. In this case X1 could represent bus trip during a period of time and X2 auto trip People reduce their utilization of public transportation as their income increase! Transportation Demand Analysis – Demand Theory II

Changing Prices 1-prices of commodities: In such a way that their relative amount remain same. Consumers income and budget remain unchanged, which is the slope of budget line. Effect exactly similar to that of income change will take place.   Transportation Demand Analysis – Demand Theory II  

Changing Price of one good The slope of the budget line changes B/p1 increasing as p1 declines Transportation Demand Analysis – Demand Theory II P2 is constant

Changing Price of one good Combination of income and substitution effects As the price X1 falls, the budget line RS rotates to RS’ Consumer achieve a higher utility level Consumption of at Least one of the goods increases Transportation Demand Analysis – Demand Theory II

Changing Price of one good Resolving M to M’ into two components: 1- Substitution effect On the same indifference Curve (MM”) Consumer has same Purchasing power shift from X1 to X”1 (p1 price fall) Transportation Demand Analysis – Demand Theory II

Changing Price of one good resolving M to M’ into two components 2- Income effect: Translation of the budget line Causes an increase In both X1 and X2 (Both goods are normal) Transportation Demand Analysis – Demand Theory II

Changing Price of one good If X1 is inferior: i) Substitution effect larger than income effect Net increase in X1 Sub effect: (X´´-X)>0 Income effect: (X´-X´´)<0 Transportation Demand Analysis – Demand Theory II

Changing Parameters Giffen paradox If X1 is inferior: ii) Substitution effect less than income effect Net decrease in X1 consumption decrease despite price has fallen Giffen paradox Transportation Demand Analysis – Demand Theory II

Giffen Paradox When a good is strongly inferior to another The consumption of a good decreases with a lower price or increases with higher price Sometimes reducing price of transit travel cause decrease in ridership Transportation Demand Analysis – Demand Theory II

Conclusion Change in budget Change in price Normal and inferior goods Proportionate changes in all prices and income will result no change in optimal consumption. No proportionate increasing in price and income will change in optimal consumption. Normal good Inferior good Giffen good Transportation Demand Analysis – Demand Theory II

Transportation Demand Analysis – Demand Theory II Finish