1 ECON6021 Microeconomic Analysis Consumption Theory I
2 Topics covered 1.Budget Constraint 2.Axioms of Choice & Indifference Curve 3.Utility Function 4.Consumer Optimum
3 Y X YAYA YBYB XAXA XBXB A B Bundle of goods A is a bundle of goods consisting of X A units of good X (say food) and Y A units of good Y (say clothing). A is also represented by (X A,Y A )
4 Convex Combination B A C (x A, Y A ) (x A, Y B ) x y
5 C is on the st. line linking A & B Conversely, any point on AB can be written as Convex Combination
6 Slope of budget line (market rate of substitution) Unit:
7 Example: jar of beer P x =$4 loaf of bread P y =$2 Both P x and P y double, feasible consumption set |Slope|= 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: : 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=(X A, Y A ), B=(X B, Y B ) and X A X B, Y A Y B 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
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 I 0 = given money income in $ P x = given price of good x P y = given price of good y Budget constraint: I 0 P x x+P y y Or, I 0 = P x x+P y y (strong monotonicity) dI 0 = P x dx+P y dy=0 (by construction) P x dx=-P y dy
30 D B YBYB YDYD XBXB XAXA XDXD C A Psychic willingness to substitute At B, my MRS is very high for X. Im willing to substitute X A -X B for Y B -Y D. But the market provides me more X to point D!
31 Consumer Optimum Normally, two conditions for consumer optimum: MRS xy = P x /P y (1) No budget left unused(2)
32 Y X A C U1U1 U0U0 Both A & C satisfy (1) and (2) Problem: bending toward origin does not hold.
33 coffee tea U0U0 U1U1 U2U2 Generally low MRS tea coffee Generally high MRS Special Cases
34 Quantity Control Max U=U(x,y) Subject to (i) I P x x+P y y (ii) Rx
35 y x (1) (2) (3) (4) (1)Corner at x=0 (2)Interior solution 0
41 1000-C 1 =S(1) 500+S(1+r)=C 2 (2) Substituting (1) into (2), we have 500+(1000-C 1 )(1+r)=C 2 Rearranging, we have 1500+1000r-(1+r) C 1 =C 2 > C Using C 1 =C 2 =C, we finally have r C (S )