1. Demand Theory II Meeghat Habibian Transportation Demand Analysis

2. 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

3. 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

4. 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

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

6. 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

7. 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

8. Transportation Demand Analysis – Demand Theory II

9. ?
Transportation Demand Analysis – Demand Theory II
Marginal rate of substitution between two goods is the inverse ratio of their marginal utilities

10.  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

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

12. 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

13. 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

14. 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

15. 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

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

17. 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

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

19. 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

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

21. 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

22. 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

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

24. 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*)

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

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

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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. Transportation Demand Analysis – Demand Theory II
Finish