Lecture 5: Consumer Demand
Advanced Microeconomics I
Yosuke YASUDA
National Graduate Institute for Policy Studies
October 22, 2013
Convex Sets and Concave Functions
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Def S ⊂ R✆ n is a convex set if for all x1, x2∈ S, tx1+ (1 − t)x2 ∈ S ∀t ∈ [0, 1].
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Def Let D be a convex set and x✆ t= tx1+ (1 − t)x2. f : D → R is a concave function if for all x1, x2∈ D,
f(xt) ≥ tf (x1) + (1 − t)f (x2) ∀t ∈ [0, 1].
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Fg Figures A1.5 and A1.27 (see JR, pp.502 and pp.534)✆ The points below the graph of all concave regions appear to be convex. Formally, the next theorem holds.
Thm Let A := (x, y)|x ∈ D, f (x) ≥ y be the set of points “on and below” the graph of f : D → R, where D ⊂ Rn is a convex set. Then,
f is a concave function ⇐⇒ A is a convex set.
Convex Preference (1)
There are three equivalent definitions of convex preferences. Def The preference relation % satisfies convexity if
1. x% y and α ∈ (0, 1) implies that αx + (1 − α)y % y. 2. For all x, y and z such that z = αx + (1 − α)y for some
α∈ (0, 1), either z % x or z % y.
3. For all y, the set AsGoodAs(y) := {z ∈ X|z % y} is convex. The notion of convex preferences captures the following intuitions: 1. If x is preferred to y, then going part of the way from y to x
is also an improvement upon y.
2. If z is between x and y then it is impossible that both x and y are better than z.
3. If both x1 and x2 are better than y, then the average of x1 and x2 is definitely better than y.
Convex Preference (2)
Convexity has a stronger version.
Def The preference relation % satisfies strict convexity if a % y, b% y, a 6= b and λ ∈ (0, 1) imply that λa + (1 − λ)b ≻ y.
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Q How is the convexity of preferences translated into properties✆ of the utility function?
Def A function u(·) : X → R is quasi-concave if for all x, y ∈ X, u(αx + (1 − α)y) ≥ min[u(x), u(y)]
holds for all α ∈ (0, 1). u(·) is strictly quasi-concave if for all x6= y in X,
u(αx + (1 − α)y) > min[u(x), u(y)] holds for all α ∈ (0, 1).
Properties of Convex Preference
Convex preferences expressed by utility functions.
Thm Suppose % is represented by utility function u(·). Then, 1. u(x) is quasiconcave if and only if % is convex.
2. u(x) is strictly quasiconcave if and only if % is strictly convex. Convexity induces a simple solution structure.
Thm Convex preference satisfies the following properties. 1. If % is convex, then the set of solutions for a choice from
B(p, ω) is convex.
2. If % is strictly convex, then every consumer problem has at most one solution.
Proof
We give the proof for each property in the second theorem.
1. Assume that both x and y maximize % given B(p, ω). By the convexity of the budget set B(p, ω) we have
αx+ (1 − α)y ∈ B(p, ω) and, by the convexity of %, αx+ (1 − α)y % x % z for all α ∈ [0, 1] and z ∈ B(p, ω). Thus, αx + (1 − α)y is also a solution to the consumer problem.
2. Assume that both x and y (where x 6= y) are solutions to the consumer problem B(p, ω). By the convexity of the budget set B(p, ω) we have αx + (1 − α)y ∈ B(p, ω) and, by the strict convexity of %, αx + (1 − α)y ≻ z for all α ∈ (0, 1) and z∈ B(p, ω), which is a contradiction of x being % optimal in B(p, ω).
(Marshallian) Demand Function
Thm The demand function x(p, ω) satisfies the following: 1. Homogeneous of degree zero:
x(p, ω) = x(λp, λω) for any λ > 0.
2. Walras’s Law: If the preferences are monotonic, then any solution x to the consumer problem B(p, ω) is located on its budget line, i.e., px(p, ω) = ω.
3. Continuity: If % is a continuous preference, then the demand function is continuous in p and in ω.
Proof We give the sketch of the proof for 1 and 2.
1. The budget sets are identical, i.e, B(p, ω) = B(λp, λω). 2. If px(p, ω) < ω, there must exist some consumption bundle
x′ with x ≪ x′ and px(p, ω) ≤ ω. By monotonicity, x′ must be strictly preferred to x, which contradicts x being a solution of the consumer problem.
Indirect Utility Function
There are several properties that the indirect utility function possesses.
Thm If u(x) is continuous and strictly increasing on Rn+, then
v(p, ω) is
1. Continuous in p and ω.
2. Homogeneous of degree zero in (p, ω). 3. Strictly increasing in ω.
4. Decreasing in p. 5. Quasiconvex in (p, ω).
6. Roy’s identity: If v(p, ω) is differentiable at (p0, ω0), then
xi(p0, ω0) = −
∂v(p0,ω0)
∂pi
∂v(p0,ω0)
∂ω
, i = 1, ..., n.
More on Roy’s Identity
Roy’s identity says that the consumer’s Marshallian demand for good i is simply the ratio of the partial derivatives of indirect utility with respect to pi and ω after a sign change.
By the Envelope theorem and Lagrangian method,
∂v(p0, ω0)
∂pi
= −λxi(p0, ω0) and
∂v(p0, ω0)
∂ω = λ,
which implies
−
∂v(p0,ω0)
∂pi
∂v(p0,ω0)
∂ω
= xi(p0, ω0).
Expenditure Function
There are several properties that the expenditure function possesses.
Thm If u(x) is continuous and strictly increasing on Rn+, then
e(p, u) is
1. Continuous in p and ω.
2. Homogeneous of degree 1 in p. 3. Strictly increasing in u, for all p ≫ 0. 4. Increasing in p.
5. Concave in p.
If, u(·) is strictly quasi-concave, we have
6. Shephard’s lemma: e(p, u) is differentiable in p at (p0, u0) with p0≫ 0, and
∂e(p0, u0)
∂pi = x
hi(p0, u0), i = 1, ..., n.
More on Shephard’s Lemma
To prove Shephard’s Lemma, we appeal to the Envelope theorem. Note that EMP and expenditure function can be rewritten as:
−e(p, u) = max
x∈Rn+
−px s.t. u(x) ≥ u,
Applying the Envelope theorem to this maximization problem,
∂− e(p0, u0)
∂pi =
∂L
∂pi = −x
h
i(p0, u0).
where L is the corresponding Lagrangian function: L = −px + λ(u(x) − u).
Canceling out negative signs, we obtain the Shephard’s lemma.
