• Theorem 3.2: A preference relation ≿ can be represented by a utility function only if it is rational.
[Figure 1.1]
1
• Axiom 3: Continuity.
(Mas-colell) The preference relation ≿ on X is continuous if it is preserved under
• Convexity implies “the law of diminishing marginal rate of substitution”.
4
3.D: Utility maximization
• Utility maximization:
• Axiom 5’: Convexity. If x ≿ y, then αx + (1−α ) y ≿ y for all α ∈ [0,1]. • Axiom 5: Strict Convexity. If x ≠ y and x ≿ y, then αx + (1−α ) y f y for all α ∈ [0,1].
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• Theorem 3.3’: If the binary relation ≿ is complete, transitive, continuous, and strictly monotonic, there exists a continuous real-valued function, u : R+n → R , which represents ≿ .
Proof:
Step 1. Let e ≡ (1,1,...1). For every x ∈ R+n , monotonicity implies that x ≿ 0. There is a
u such that ue >> x . Hence, we have ue ≿ x. Monotonicity and continuity can be
• Properties of the consumption set X 1. Φ ≠ X 2. X is closed. 3. X is convex. 4. 0 ∈ X
{ } • Budget Set: Bp,w = x ∈ R+n : Px ≤ w is the set of all feasible consumption bundles for
• Axiom 4: Monotonicity. For all x, y ∈ R+n , if x ≥ y then x ≿ y, while if x >> y then xf y. [Figure 1.5]
( ) • Axiom 4’: Local Nonsatiation. ∀ x0 ∈ R+n , and ε > 0, ∃ x ∈ Bε x0 ∩ R+n such that
{( )} limits. That is, for any sequence of pairs
xn, yn
∞ n=1
with
xn
≿
yn
for all n,
x = lim xn y = lim yn , we have x ≿ y .
n→∞
n→∞
(Reny) For all x ∈ R+n , the upper contour set {y∈X: y ≿ x} and the lower contour set
• Summary: Axiom 1 says that individuals “can” choose. Axiom 2 says that the choice of individuals is “consistent”. Æ They are “rational”. Axiom 3 says that there is no “sudden” preference. Æ We can use a “function” to represent our preference. Axiom 4 says that “more” is better than “less”. Æ The indifference curve can not be “thick” and “bend upward”. Axiom 5 says that “mix” is better than “pure”. Æ The indifference curve is convex-shaped relative to the origin. Æ The principle of diminishing marginal rate of substitution
x1 > y1 or x1 = y1, and x2 ≥ y2 . It is rational. (Proof?)
Is it continuous? Suppose two sequences: xn = ⎜⎛1+ 1 , 1⎟⎞ yn = (1, 2). Then xn ≿ yn .
⎝ n⎠
However, x = lim xn = (1,1) y = lim yn = (1,2), we have y ≿ x.
3.C: The marginal rate of Substitution
MRSij (x)
≡
∂u(x)/ ∂xi ∂u(x)/ ∂x j
.
Additional assumption: differentiability. It says that our preference is “smooth”. Note that the MRS does not depend on the utility function chosen to represent the underlying preference. In other word, the MRS is invariant to a monotonic transformation of utility.
the inverse image under u of every open ball in R is open in R+n . Because open balls
in R are open intervals, this is equivalent to show that u −1((a,b)) is open in R+n for
every a < b .
{ } u−1((a,b)) = x ∈ R+n | a < u(x) < b { } = x ∈ R+n | ae p u(x)e p be { } = x ∈ R+n | ae p x p be
Q.E.D.
• Theorem 3.4: Invariance of the Utility Function to Positive Monotonic Transforms
n→∞
n→∞
• Theorem 3.3: If the binary relation ≿ is complete, transitive and continuous, there exists a continuous real-valued function, u : R+n → R , which represents ≿ .
Chapter 3: The Consumer’s Problem
3.A: Budget set
• Consumption Bundles: x = (x1, x2 ,.....xn )T , where xi , i ∈ N is the quantity of good i . Hence, X is R+n .
the consumer who faces market prices P = ( p1, p2......pn )∈ Rn and has wealth w .
• Remark : market price ( P )
• Budget Set is nonempty ( why ? (0,0,…0) ), closed ( ? ) and bounded ( all prices are strictly positive )
3
• Theorem 3.5: Properties of Preferences and Utility Functions
1) u(x) is strictly increasing iff ≿ is strictly monotonic. 2) u(x) is quasiconcave iff ≿ is convex. 3) u(x) is strictly quasiconcave iff ≿ is strictly convex.
{y∈X: x ≿ y} are both closed in R+n .
[Figure 1.2]
Not all rational preferencesmple: Lexicographic preference relation. Suppose X = R+2 . Define x ≿ y if either
Graphically, MRS measures the “curvature (曲率)” of the indifference curve; economically, MRS implies the rate, at which the consumer is willing to give up x j in exchange for xi , such that she remains indifferent after the exchange.