ECON6021 Microeconomic Analysis

1. ECON6021 Microeconomic Analysis Consumption Theory I

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

3. 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 YA A YB B X XA XB Bundle of goods

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

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

9. Axioms of Choice & Indifference Curve

10. Axioms of Choice • Nomenclature: • : “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 • 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. Negative slopes ICs farther away from origin means higher satisfaction Preferred bundles Y I A II X Not preferred bundles Properties of Indifference Curves

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

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. Y A YA B YB U0 X XA XB (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. V=2001 V=200 V=100 U=30 U=20 U=10 Measurability of Utility 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. 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. B YB D A YD C XB XA XD Psychic willingness to substitute 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. Y Both A & C satisfy (1) and (2) Problem: “bending toward origin” does not hold. U1 U0 A C X

33. coffee coffee Generally low MRS Generally high MRS U2 U1 U0 tea tea Special Cases

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

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

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

37. B D A C 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 u(c1,c2)=const C2 1600 500 Slope = -1.1 C1 1000 Income in period 2

41. 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 1500+1000r-(1+r) C1=C2 > C Using C1=C2=C, we finally have r   C  (S )