1. ECON6021 Microeconomic Analysis
Consumption Theory I

2. Topics covered Budget Constraint Axioms of Choice & Indifference Curve
Utility Function
Consumer Optimum

3. Bundle of goods
A is a bundle of goods consisting of XA units of good X (say food) and YA units of good Y (say clothing).
A is also represented by (XA,YA) Y X YA YB XA XB A B

4. Convex Combination y
(xA, YA) A C
(xA, YB) B x

5. Convex Combination  C is on the st. line linking A & B
Conversely, any point on AB can be written as

6. Slope of budget line
(market rate of substitution)
Unit:

7. Example: jar of beer Px=$4 loaf of bread Py=$2 Both Px and Py double,
feasible consumption set
|Slope|=
Example:
jar of beer Px=$4
loaf of bread Py=$2
Both Px and Py double,
No change in market rate of substitution

8. Tax: a $2 levy per unit is imposed for each good
 Slope of budget line changes y x
after levy is imposed
After doubling
the prices

9. Axioms of Choice
& Indifference Curve

10. Axioms of Choice Nomenclature: Completeness (Comparison)
: “is preferred to”
: “is strictly preferred to”
: “is indifferent to”
Completeness (Comparison)
Any two bundles can be compared and one of the following holds: AB, B A, or both ( A~B)
Transitivity (Consistency)
If A, B, C are 3 alternatives and AB, B C, then A C;
Also If AB, BC, then A C.

11. Axioms of choice Continuity Strong Monotonicity (more is better)
AB and B is sufficiently close to C, then A C.
Strong Monotonicity (more is better)
A=(XA , YA), B=(XB , YB) and XA≥XB, YA≥YB with at least one is strict, then A>B.
Convexity
If AB, then any convex combination of A& B is preferred to A and to B, that is, for all 0 t <1,
(t XA+(1-t)XB, tYA+(1-t)YB)  (Xi , Yi), i=A or B.
If the inequality is always strict, we have strict convexity.

12. Indifference Curve
When goods are divisible and there are only two types of goods, an individual’s preferences can be conveniently represented using indifference curve map.
An indifference curve for the individual passing through bundle A connects all bundles so that the individual is indifferent between A and these bundles.

13. Properties of Indifference CurvesX Y A I II
Not preferred bundles
Preferred bundles
Negative slopes
ICs farther away from origin means higher satisfaction

14. Properties of Indifference Curves
Non-intersection
Two indifference curves cannot intersect
Coverage
For any bundle, there is an indifference curve passing through it. X Y A Q P

15. Properties of Indifference Curves
Bending towards Origin
It arises from convexity axiom
The right-hand- side IC is not allowed Y X

16. Utility Function

17. Utility Function
Level of satisfaction depends on the amount consumed: U=U(x,y)
U0 =U(x,y)
All the combination of x & y that yield U0 (all the alternatives along an indifference curve)
y=V(x,U0), an indifference curve
U(x,y)/x, marginal utility respect to x, written as MUx.

18. (if strong monotonicityX Y A B YA YB XA XB U0
(by construction)
(if strong monotonicity
holds)
Slope:

19. Y A B X
The MRS is the max amount of good y a consumer would willingly forgo for one more unit of x, holding utility constant (relative value of x expressed in unit of y)

20. Marginal rate of substitution
DMRS:

21. Measurability of Utility
V=100
V=200
V=2001
An order-preserving re-labeling of ICs does not alter the preference ordering.

22. Positive monotonic (order-preserving) transformation
They are called positive monotonic transformation

23. Positive Monotonic Transformation
What is the MRS of U at (x,y)?
How about U’?  

24. Positive Monotonic Transformation
IC’s of order-preserving transformation U’ overlap those of U.
However, we have to make sure that the numbering of the IC must be in same order before & after the transformation.

25. Positive Monotonic Transformation
Theorem: Let U=U(X,Y) be any utility function. Let V=F(U(X,Y)) be an order-preserving transformation, i.e., F(.) is a strictly increasing function, or dF/dU>0 for all U. Then V and U represent the same preferences.

26. Proof
Consider any two bundles and
Then we have:
Q.E.D.

27. Consumer Optimum

28. Constrained Consumer Choice Problem
Preferences: represented by indifference curve map, or utility function U(.)
Constraint: budget constraint-fixed amount of money to be used for purchase
Assume there are two types of goods x and y, and they are divisible

29. dI0= Pxdx+Pydy=0 (by construction)
Consumption problem
Budget constraint
I0= given money income in $
Px= given price of good x
Py= given price of good y
Budget constraint: I0Pxx+Pyy
Or, I0= Pxx+Pyy (strong monotonicity)
dI0= Pxdx+Pydy=0 (by construction)
Pxdx=-Pydy

30. Psychic willingness to substituteD B YB YD XB XA XD C A
At B, my MRS is very high for X. I’m willing to substitute XA-XB for
YB-YD. But the market provides me more X to point D! 

31. Consumer Optimum Normally, two conditions for consumer optimum:
MRSxy = Px/Py (1)
No budget left unused (2)

32. Both A & C satisfy (1) and (2)
Problem: “bending toward origin” does not hold. U1 U0 A C X

33. Special Cases coffee coffee Generally low MRS Generally high MRS teaU0 U1 U2
Generally low MRS

34. Quantity Control
Max U=U(x,y)
Subject to (i) I ≥Pxx+Pyy
(ii) R≥x

35. Interior solution 0<x<R “corner” at Ry x
(1)
(2)
(3)
(4)
Corner at x=0
Interior solution 0<x<R
“corner” at R

36. An Example: U(x,y)=xy

37. A satisfies (1) but not (2) B, C satisfy (2) but not (1) D B A
A satisfies (1) but not (2)
B, C satisfy (2) but not (1)
Only D satisfies both (1) &(2)

38. Other Examples of Utility Functions

39. An application: Intertemporal Choice
Our framework is flexible enough to deal with questions such as savings decisions and intertemporal choice.

40. Intertemporal choice problem
Income in period 2 C2
u(c1,c2)=const
1600
500
Slope = -1.1 C1
1000
Income in period 2

41. Substituting (1) into (2), we have 500+(1000-C1)(1+r)=C2
1000-C1=S (1)
500+S(1+r)=C2 (2)
Substituting (1) into (2), we have
500+(1000-C1)(1+r)=C2
Rearranging, we have
r-(1+r) C1=C2 > C
Using C1=C2=C, we finally have
r   C  (S )