Embed Size (px)
Citation preview
Lecture 8: Expenditure Minimization
Advanced Microeconomics I, ITAM
Xinyang Wang∗
In the previous lecture, we stated and studied the consumer problem. In this section,
we provide a helpful detour to the expenditure minimization problem. Our work this this
section will be rewarded when later we study comparative statics and welfare analysis.
1 The Expenditure Minimization Problem
The expenditure minimization problem (EMP) is giving as follows
minx∈Rn
+
p · x
s.t. u(x) ≥ v
That is, we try to find the cheapest consumption bundle x at price p such that it yields a
payoff at least as large as v.
The following graph suggests there is a close relation between the consumer problem and
the expenditure minimization problem: in the consumer problem, we fix a budget set and
look for the highest indifference curve passing through this budget set; in the expenditure
minimization problem, we fix a better than set and look for the lowest budget line passing
through the given better than set. For this reason, the expenditure minimization problem
is sometimes called the dual problem of the consumer problem. We shall not go into more
serious details about the duality of optimization problems. But we will come back to this
close relation in our next lecture.
∗Please email me at [email protected] for typos or mistakes. Version: February 16, 2021.
1
Figure 1: The Consumer Problem and the Expenditure Minimization Problem
2 The Hicksian Demand and Expenditure Function
In this section, we first observe the existence of a solution of the expenditure minimization
problem:
Theorem. Suppose p >> 0, u is a continuous function and there is some x̄ such that
u(x̄) ≥ v, then the expenditure minimization problem has a solution.
Proof. We provide an idea of the proof. The completion of this proof is left as an exercise.
Recall that a continuous function on a compact domain has a minimum. The obstacle
here is that the choice set, the better than set, is not compact. For this reason, we proceed
in two steps.
First, we cut the better than set to be a compact set, by studying the set
S = {x ∈ Rn+ : p · x ≤ p · x̄} ∩ {x ∈ Rn
+ : u(x) ≥ v}
2
Next, we prove that minimizer can not appear elsewhere. �
The solution x = h(p, v) of the expenditure minimization problem is called the Hicksian
demand. And the corresponding value function
e(p, v) = minx:u(x)≥v
p · x
is called the expenditure function.
Remark.
• Given a price system p and a payoff level v, the Hicksian demand h(p, v) is in general
a set containing more than one elements.
• We have the equality
e(p, v) = p · x,∀x ∈ h(p, v)
• In particular, if h(p, v) is a function
e(p, v) = p · h(p, v)
3
3 Properties of the Hicksian Demand
In this section, we study the properties of the Hicksian demand.
First, similar to the Marshallian demand, the Hicksian demand is homogenous of degree
0 in p.
Proposition (Homogeneity). For any λ > 0,
h(p, v) = h(λp, v)
Proof. Exercise. �
Second, similar to the Marshallian demand, the inequality constraint in the expenditure
minimization problem is usually binding. That is, to ensure the expenditure is minimized,
consumer will not try to maintain a payoff level higher than necessary.
Proposition (No Excess Utility). If u is continuous, and v > u(0), then for any x ∈ h(p, v),
u(x) = v
Proof. We prove by contradiction. If u(x) > v. Then, we study the line segment λx, for
λ ∈ [0, 1], connecting 0 and x in the set of consumption bundles. As u(x) > v and u(0) < v,
by the continuity of u and the intermediate value theorem, we have u(λx) = v for some
λ ∈ (0, 1). The value of the consumption bundle λx is clearly less than x. Contradiction. �
Third, similar to the Marshallian demand, the Hicksian demand is convex/ a singleton
whenever the utility function is (quasi-)concave or strictly (quasi-)concave.
Proposition (Convexity/ Uniqueness). Given h(p, v) 6= ∅, when u is concave or quasi-
concave, h(p, v) is convex. In addition, when u is strictly concave or strictly quasi-concave,
h(p, v) is a singleton.
Proof. Exercise. �
Last, the Hicksian demand is downward sloping.
4
Proposition (Downward Sloping). For x ∈ h(p, v) and x′ ∈ h(p′, v),
(p− p′) · (x− x′) ≤ 0
Proof. By definition, u(x), u(x′) ≥ v. Since when the price is p, x is a solution of the
minimization problem, we have
p · x′ ≥ p · x
Similarly,
p′ · x ≥ p′ · x′
Minus these two inequalities, we have
p · x′ − p′ · x′ ≥ p · x− p′ · x
which yields the inequality we need.
�
Remark.
• When pj = p′j for all indices j except for i, this proposition implies
(pi − p′i)(xi − x′i) ≤ 0
5
That is, when the price of commodity i increases, the Hicksian demand of commodity
i will decrease. Therefore, this proposition suggests the Hicksian demand is downward
sloping.
• In contrast, Marshallian demand may not be downward sloping on p. See the discussion
on Giffen goods in the next lecture. Now, I just give a picture to illustrate.
4 Properties of Expenditure Function
In this section, we study the properties of the expenditure function. We suppose u is contin-
uous and locally non-satiated in this section.
First, the expenditure is homogeneous of degree 1 in p.
Proposition (Homogeneity). For any λ > 0,
e(λp, v) = λe(p, v)
Proof. Prove that the minimizers of these two minimization problems are the same. �
Second, e(p, v) is a continuous function.
Proposition (Continuity). e(p, v) is continuous in p and v.
Proof. By Berge’s maximum theorem. �
6
Third, e(p, v) exhibits some monotonicity.
Proposition (Monotonicity). e(p, v) is non-decreasing in p and strictly increasing in v.
Proof. Exercise. �
Last, e(p, v) is a concave function in p. We note this concavity does not depend on the
concavity of the utility function.
Proposition (Concavity). e(p, v) is concave in p
Proof. For any λ ∈ [0, 1] and prices p1, p2, by definitions, we know
e(λp1 + (1− λ)p2, v) = (λp1 + (1− λ)p2) · x,∀x ∈ h(λp1 + (1− λ)p2, v)
Note that u(x) ≥ v by the definition of the Hicksian demand,
p1 · x ≥ e(p1, v)
p2 · x ≥ e(p2, v)
Therefore,
e(λp1 + (1− λ)p2, v) = (λp1 + (1− λ)p2) · x ≥ λe(p1, v) + (1− λ)e(p2, v)
�
5 A Relationship between the Hicksian Demand and
Expenditure Function
In this section, we derive the relationship the Hicksian Demand and Expenditure Function.
For this purpose, we recall the envelope theorem: Let F to be the value function of the
following maximization problem
F (y) = maxx:gk(x,y)≤0,∀k
f(x, y)
7
Then, if the maximizer x∗(y) is differentiable in y, we have
∇F (y) = ∇f(x∗(y), y)−∑k
λ∗k∇gk(x∗(y), y)
Remark.
• When the Lagrangian is defined to be L = −f +∑
k λkgk,
∇F (y) = −∇xL(x∗(y), λ∗(y))
That is, the envelope theorem suggests us that to obtain the derivative of the value
function, we can ignore the maximization operation, take the derivative directly on the
negative of Lagrangian, and plug in (x∗, λ∗) satisfies the KKT condition, provided the
solution of the is differentiable in parameter y.
• The following non-constrained case might provide some insights on the envelope theo-
rem.
F (y) = maxx
f(x, y)
Then, we have F (y) = f(x∗(y), y), where x∗(y) is a maximizer of the function f where
its second coordinate is y. If x∗(y) is differentiable in y, we apply the chain rule:
∂
∂yF (y) =
∂
∂xf(x∗(y), y)(x∗)′(y) +
∂
∂yf(x∗(y), y)
As x∗(y) is a maximizer, by the first order condition, ∂∂xf(x, y) = 0 when x = x∗(y).
Therefore,∂
∂yF (y) =
∂
∂yf(x∗(y), y)
Now, we are ready to state the relationship between the Hicksian demand and expenditure
function.
Proposition (Shephard’s Lemma). Suppose u is continuous, locally non-satiated and strictly
(quasi-)concave, then we have
• e(p, v) is differentiable in p.
8
• ∂∂pie(p, v) = hi(p, v),∀i
Proof. First, we write the expenditure minimization problem as a maximization problem:
−e(p, v) = maxx:v−u(x)≤0
(−p · x)
By the envelope theorem,
− ∂
∂pie(p, v) =
∂
∂pi(−p · x− λ(v − u(x)))|x=h(p,v)
Therefore,∂
∂pie(p, v) = hi(p, v)
�
9
