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1. T HE CONSUMER CHOICE PROBLEM
1.1. Unit of analysis and preferences. The fundamental unit of analysis in economics is the economic agent. Typically this agent is an individual consumer or a firm. The agent might also be the manager of a public utility, the stockholders of a corporation, a government policymaker and so on.
The underlying assumption in economic analysis is that all economic agents possess a preference ordering which allows them to rank alternative states of the world.
The behavioral assumption in economics is that all agents make choices consistent with these underlying preferences.
1.2. Definition of a competitive agent. A buyer or seller (agent) is said to be competitive if the agent assumes or believes that the market price of a product is given and that the agent’s actions do not influence the market price or opportunities for exchange.
1.3. Commodities. Commodities are the objects of choice available to an individual in the economic sys- tem. Assume that these are the various products and services available for purchase in the market. Assume that the number of products is finite and equal to L ( =1, ..., L). A product vector is a list of the amounts of the various products:
x =
x 1 x 2 .. . xL
The product bundle x can be viewed as a point in RL.
1.4. Consumption sets. The consumption set is a subset of the product space RL, denoted by XL^ ⊂ RL, whose elements are the consumption bundles that the individual can conceivably consume given the phys- ical constraints imposed by the environment. We typically assume that the consumption set is X = R L + = {x ∈ RL: x ≥ 0 for = 1, ... , L}.
1.5. Prices. We will assume that all L products are traded in the market at dollar prices that are publicly quoted. How they are determined will be discussed later. The prices are represented by a price vector
p =
p 1 p 2 .. . pL
∈ RL
For now assume that all prices are strictly positive, i.e. p >> 0. We will also assume that all consumers are price takers in the sense that they cannot influence the price at which they buy or sell a product.
Date : October 10, 2005. 1
1.6. Income or wealth. Assume that each consumer has wealth equal to mi or the representative consumer has wealth m.
1.7. Affordable consumption bundles. We say that a consumption bundle x is affordable for the represen- tative consumer if
p x = p 1 x 1 + p 2 x 2 + · · · + pL xL ≤ m (1) If x is also an element of RL + , then the set of feasible consumption bundles is x RL + : px ≤ m. This is called a Walrasian budget set and is denoted Bp,m.
1.8. Preferences. We assume a preference relation over products with the following properties
1: complete in that for all x 1 , x 2 ∈ X, we have x 1 x 2 or x 2 x 1 (or both) 2: transitive in that ∀ x 1 , x 2 , x 3 X, if x 1 x 2 and x 2 x 3 then x 1 x 3. 3: locally nonsatiated in that for every x 1 ∈ X and every ε > 0 , there is x 2 ∈ X such that ||x 2 - x 1 ≤ ε|| and x 2 x 1. 4: continuous in that for any sequence of pairs
{ (xn 1 , xn 2 ) }∞ n=1 with xn 1 xn 2 ∀ n, x 1 = lim n → ∞ xn 1 , and x 2 = lim n → ∞ xn 2 ,
we have x 1 x 2.
1.9. Existence of a utility function. Based on the preferences defined in 1.8, there exists a continuous utility function v(x) that represents in the sense that x 1 x 2 iff v(x 1 ) ≥ v(x 2 ).
1.10. Convexity. We often assume that preferences are convex in the sense that if x 1 x 2 , then for 0 ≤ λ ≤ 1, λx 1 + (1-λ)x 2 x 1. This implies that indifference curves are convex to the origin. If the utility function is quasi-concave, then the indifference curves will be convex and vice versa.
- T HE UTILITY MAXIMIZATION PROBLEM
2.1. Formal Problem. The utility maximization problem for the consumer is then as follows
max x ≥ 0
u = v(x)
s.t. px ≤ m
where we assume that p >> 0, m > 0 and X = RL +.
This is called the primal preference problem. If we have smooth convex indifference curves and an inte- rior solution, then the problem can be solved using standard Lagrangian techniques. Alternatively, Kuhn- Tucker methods can be used. The Lagrangian function is given by
L = v(x) − λ (Σni=1 pi xi − m ) (3) The first order conditions are
∂ v ∂ xi − λ pi = 0, i = 1, 2 ,... , n
− Σni=1 pi xi + m = 0
The value of λ is the amount by which L would increase given a unit relaxation in the constraint (an increase in income). It can be interpreted as the marginal utility of expenditure. This units of this are of
x 2 =
α 2 (α 1 + α 2 )
m p 2
Note that demand for the kth good only depends on the kth price and is homogeneous of degree zero in prices and income. Also note that it is linear in income. This implies that the expenditure elasticity is equal to 1. This can be seen as follows.
x 1 =
α 1 (α 1 + α 2 )
m p 1
∂ x 1 ∂ m
m x 1
[
α 1 (α 1 + α 2 )
p 1
] [
m α 1 (α 1 + α 2 )
m p 1
]
We can find the value of the optimal u by substitution
u = xα 1 1 xα 22
m p 1
[
α 1 α 1 + α 2
])α 1 ( m p 2
[
α 2 α 1 + α 2
])α 2
= mα^1 +α^2 p− 1 α^1 p− 2 α^2 αα 1 1 αα 2 2 (α 2 + α 2 )−α^1 −α^2
This can also be written
u = xα 1 1 xα 22
[
m p 1
α 1 α 1 + α 2
)]α 1 [ m p 2
α 2 α 1 + α 2
)]α 2
α 1 α 1 + α 2
)α 1 ( α 2 α 1 + α 2
)α 2 ( m p 1
)α 1 ( m p 2
)α 2
For future reference note that the derivative of the optimal u with respect to m is given by
u = mα^1 +α^2 p− 1 α^1 p− 2 α^2 αα 1 1 αα 2 2 (α 2 + α 2 )−α^1 −α^2
∂u ∂m
= (α 1 + α 2 )mα^1 +α^2 −^1 p− 1 α^1 p− 2 α^2 αα 1 1 αα 1 2 (α 2 + α 2 )−α^1 −α^2
= mα^1 +α^2 −^1 p− 1 α^1 p− 2 α^2 αα 1 1 αα 1 2 (α 2 + α 2 )^1 −α^1 −α^2
We obtain λ by substituting in either the first or second equation as follows
α 1 xα 1 1 −^1 xα 2 2 − λp 1 = 0
⇒ λ =
α 1 xα 1 1 −^1 xα 22 p 1 α 2 xα 1 1 xα 2 2 −^1 − λp 2 = 0
⇒ λ =
α 2 xα 1 1 xα 22 −^1 p 2
If we now substitute for x 1 and x 2 , we obtain
λ =
α 1 xα 1 1 −^1 xα 22 p 1
x 1 =
m p 1
[
α 1 α 1 + α 2
]
x 2 =
m p 2
[
α 2 α 1 + α 2
]
⇒ λ =
α 1
m p 1
[
α 1 α 1 + α 2
])α 1 − 1 ( m p 2
[
α 2 α 1 + α 2
])α 2
p 1
α 1 mα^1 +α^2 −^1 p^11 − α^1 p− 2 α^2 αα 1 1 −^1 αα 2 2 (α 1 + α 2 )^1 −α^1 −α^2 p 1
= mα^1 +α^2 −^1 p− 1 α^1 p− 2 α^2 αα 1 1 αα 2 2 (α 1 + α 2 )^1 −α^1 −α^2
Thus λ is equal to the derivative of the optimal u with respect to m.
To check for a maximum or minimum we set up the bordered Hessian. The bordered Hessian in this case is
HB =
∂^2 L(x∗^ , λ∗) ∂x 1 ∂x 1
∂^2 L(x∗, λ∗^ ) ∂x 1 ∂x 2
∂g(x∗^ ) ∂x 1
∂^2 L(x∗^ , λ∗) ∂x 2 ∂x 1
∂^2 L(x∗, λ∗^ ) ∂x 2 ∂x 2
∂g(x∗^ ) ∂x 2
∂g(x∗) ∂x 1
∂g(x∗^ ) ∂x 2
We compute the various elements of the bordered Hessian as follows
2 p 1 p 2 α 1 α 2 xα 1 1 −^1 xα 2 2 −^1 − p^21 (α 2 )(α 2 − 1)xα 1 1 xα 2 2 −^2 − p^22 (α 1 )(α 1 − 1)xα 1 1 −^2 xα 2 2 > 0 (21) We can also write it in the following convenient way
2 p 1 p 2 α 1 α 2 xα 1 1 −^1 xα 22 −^1
+α 2 p^21 xα 1 1 xα 2 2 −^2 − α^22 p^21 xα 1 1 xα 22 −^2
+α 1 p^22 xα 1 1 −^2 xα 2 2 − α^21 p^22 xα 1 1 −^2 xα 2 2 > 0
To eliminate the prices we can substitute from the first-order conditions.
p 1 =
α 1 xα 1 1 −^1 xα 22 λ
p 2 = α 2 xα 1 1 xα 22 −^1 λ This then gives
α 1 xα 1 1 −^1 xα 22 λ
α 2 xα 1 1 xα 22 −^1 λ
α 1 α 2 xα 1 1 −^1 xα 22 −^1
+α 2
α 1 xα 1 1 −^1 xα 22 λ
xα 1 1 xα 2 2 −^2 − α^22
α 1 xα 1 1 −^1 xα 22 λ
xα 1 1 xα 22 −^2
+α 1
α 2 xα 1 1 xα 22 −^1 λ
xα 1 1 −^2 xα 2 2 − α^21
α 2 xα 1 1 xα 22 −^1 λ
xα 1 1 −^2 xα 2 2 > 0
Multiply both sides by λ^2 and combine terms to obtain
2 α^21 α^22 x^31 α 1 −^2 x^32 α^2 −^2
+α^21 α 2 x^31 α 1 −^2 x^32 α 2 −^2 − α^22 α^21 x^31 α 1 −^2 x^32 α^2 −^2
+α 1 α^22 x^31 α 1 −^2 x^32 α 2 −^2 − α^21 α^22 x^31 α 1 −^2 x^32 α 2 −^2 > 0
Now factor out x^31 α 1 −^2 x^32 α 2 −^2 to obtain
x^31 α 1 −^2 x^32 α^2 −^2
2 α^21 α 22 + α^21 α 2 − α^22 α^21 + α 1 α^22 − α^21 α^22
⇒ x^31 α 1 −^2 x^32 α^2 −^2
α^21 α 2 + α 1 α^22
With positive values for x 1 and x 2 the whole expression will be positive if the last term in parentheses is positive. Then rewrite this expression as
( α^21 α 2 + α 1 α 22
Now divide both sides by α^21 α^22 (which is positive) to obtain ( 1 α 2
α 1
3. T HE EXPENDITURE ( COST ) MINIMIZATION PROBLEM
3.1. Basic duality formulation. The fundamental (primal) utility maximization problem is given by
max x ≥ 0
u = v(x)
s.t. px ≤ m
Dual to the utility maximization problem is the cost minimization problem
min x ≥ 0
m = px
s.t. v(x) = u
3.2. Marshallian and Hicksian demand functions. The solution to equation 29 gives the Hicksian demand functions x = h(u, p). The Hicksian demand equations are sometimes called ”compensated” demand equa- tions because they hold u constant. The solutions to the primal and dual problems coincide in the sense that
x = g (p, m) = h (u, p) (30)
3.3. Indirect objective functions. We can substitute the optimal levels of the decision variables as functions of the parameters back into the objective functions to obtain the indirect objective functions. For the primal problem this gives
u = v (x 1 , x 2 , · · · , xn ) = v [g 1 (m, p), g 2 (m, p),... , gn (m, p) ] = ψ(m, p) (31)
This is called the indirect utility function and specifies utility as a function of prices and income. We can also write it as follows
ψ (m, p) = max x [v(x) : px = m] (32)
The indirect utility function for the Cobb-Douglas utility function is given by
u = xα 1 1 xα 22
[
m p 1
α 1 α 1 + α 2
)]α 1 [ m p 2
α 2 α 1 + α 2
)]α 2
α 1 α 1 + α 2
)α 1 ( α 2 α 1 + α 2
)α 2 ( m p 1
)α 1 ( m p 2
)α 2
For the Cobb-Douglas utility function with multiple inputs, the indirect utility function is given by
3.5.1. C.1. The cost function is nondecreasing in p, increasing in u, and increasing in at least one p.
Let p^1 ≥ p^2. Let x^1 be the cost minimizing input bundle with p^1 and x^2 be the cost minimizing input bundle with p^2. Then p^2 x^2 ≤ p^2 x^1 because x^1 is not cost minimizing with prices p^2. Now p^1 x^1 ≥ p^2 x^1 because p^1 ≥ p^2 by assumption so that
C(p^1 , y) = p^1 x^1 ≥ p^2 x^1 ≥ p^2 x^2 = C(p^2 , y)
Nonsatiation guarantees that the function will be increasing in u. Let V(u 0 ) be the set of all bundles that are equivalent to or preferred to bundles that provide utility level u 0. Now let u 1 ¿ u 2. Because V(u 1 ) is a subset of V(u 2 ) if u 1 ≥ u 2 then
C(u 1 , p) = min x {px : x ∈ V (u 1 )} ≥ min x {px : x ∈ V (u 2 )} = c(u 2 , p)
The point is that if we have a smaller set of possible x’s to choose from then cost must increase.
3.5.2. C.2. Positively linearly homogenous in p
C(u, θp) = λ C(u, p), p > 0.
Let the cost minimization problem with prices p be given by
C(u, p) = min x {px : xV (u)}, u ∈ Dom V, p > 0 , (37)
where
Dom V = {u ∈ R^1 + : V (u) = ∅}
The x vector that solves this problem will be a function of u and p, and is usually denoted h(u,p). This is the is Hicksian demand function. The cost function is then given by
C(u, p) = p h(u, p) (38)
Now consider the problem with prices tp (p >0)
C^ ˆ(y, tp) = min x {tpx : x ∈ V (u)}, u ∈ Dom V, p > 0
= t min x {px : x ∈ V (y)}, y ∈ Dom V, p > 0
The x vector that solves this problem will be the same as the vector which solves the problem in equation 37, i.e., h(u,p). The cost function for the revised problem is then given by
C^ ˆ(p, tp) = tp h(u, p) = tC(u, p) (40)
3.5.3. C.3. C is concave and continuous in w
To demonstrate concavity let (p, x) and (p’, x’) be two cost-minimizing price-consumption combinations and let p”= tp + (1-t)p’ for any 0 ≤ t ≤ 1. Concavity implies that C(u, p”) ≥ tC(u, p) + (1-t) C(u, p’). We can prove this as follows.
We have that C(u, p”) = p”· x”= tp · x”+ (1-t)p’ · x” where x” is the optimal choice of x at prices p” Because x” is not necessarily the cheapest way to obtain utility level u at prices p’ or p,we have p · x”≥ C(u, pw) and p’· x” ≥ C(u, p’) so that by substitution C(u, p”) ≥ tC(u, p) + (1-t) C(u, p’). The point is that if p · x” and p’· x” are each larger than the corresponding term is the linear combination then C(u, p”) is larger than the linear combination.
Rockafellar [11, p. 82] shows that a concave function defined on an open set (p > 0) is continuous.
3.6. Shephard’s Lemma.
3.6.1. Definition. If indifference curves are convex, the cost minimizing point is unique. Then we have
∂C(u, p) ∂pi = hi (u, p) (41)
which is a Hicksian Demand Curve
3.6.2. Constructive proof using the envelope theorem. The cost minimization problem is given by
C(y, w) = min x px : v(x) − u = 0 (42) The associated Lagrangian is given by
L = px − λ(v(x) − u) (43) The first order conditions are as follows
∂L ∂xi
= pi − λ
∂v ∂xi
= 0, i = 1,... , n (44a)
∂L ∂λ
= − (vx − u) = 0 (44b)
Solving for the optimal x’s yields
xi(u, p) = hi(u, p) (45) with C(u,p) given by
C(u, p) = px(u, p) = ph(u, p) (46) If we now differentiate 46 with respect to p (^) i we obtain
∂C ∂pi
= Σnj=1 pj
∂xj (u, p) ∂pi
From the first order conditions in equation 44a (assuming that the constraint is satisfied as an equality) we have
pj = λ
∂v ∂xj
where we treat the first good asymmetrically and solve for each demand for a good as a function of the first. Now substituting in the utility function we obtain
v =
∏^ n
j=
x αj j
∏^ n
j=
αj x 1 p 1 α 1 pj
)αj (58)
Because x 1 , p 1 and α 1 do not change with j, they can be factored out of the product to obtain
u =
x 1 p 1 α 1
)Σnj=1 αj (^) ∏n
j=
αj pj
)αj (59)
We then solve this expression for x 1 as a function of u and the other x’s. To do so we divide both sides by the product term to obtain
x
Σnj=1 αj 1
p 1 α 1
)Σnj=1 αj
u ∏n j=
αj pj
)αj (60)
We now multiply both sides by
α 1 p 1
)Σnj=1 αj to obtain
x Σnj=1 αj 1 =
α 1 p 1
)Σnj=1 αj u ∏n j=
αj pj
)αj (61)
If we now raise both sides to the power (^) Σn^1 j=1 αj^ we find the value of x 1
x 1 =
α 1 p 1
⎝ (^) ∏ u n j=
αj pj
)αj
1 Σnj=1 αj (62)
Similarly for the other xk so that we have
xk =
αk pk
⎝ (^) ∏ u n j=
αj pj
)αj
Σn^1 j=1 αj (63)
Now if we substitute for the ith x in the cost expression we obtain
C = Σni=1 pi
αi pi
⎝ u ∏n j=
αj pj
)αj
Σn^1 j=1αj
= ( Σni=1 α (^) i )
⎝ (^) ∏ u n j=
αj pj
)αj
Σn^1 j=1 αj
= ( Σni=1 α (^) i ) u Σn^1 j=1 αj
∏^ n
j=
pj αj
)αj
1 Σnj=1 αj
3.9. The indirect utility function and Hicksian demands. If we substitute C(u,p) in the Marshallian de- mands, we get the Hicksian demand functions
xi = xi(m, p) = gi(m, p) = gi[C(u, p), p] = hi(u, p) = xi(u, p) (65)
3.10. Roy’s identity. We can also rewrite Shephard’s lemma in a different way. First write the identity
ψ (C(u, p), p] = u (66) Then totally differentiate both sides of equation 66with respect to p (^) i holding u constant as follows
∂ ψ[C(u, p), p] ∂m
∂ C(u, p) ∂ pi
∂ ψ[C(u, p), p] ∂ pi
Rearranging we obtain
∂ C(u, p) ∂ pi
− ∂ ψ^ [C ∂ p(u,pi),p] ∂ ψ [C(u,p),p ] ∂ m
= gi(m, p) (68)
where the last equality follows because we are evaluating the indirect utility function at income level m. Figure 2 makes these relationships clear.
4. P ROPERTIES OF INDIRECT UTILITY FUNCTION
The indirect utility function has the following properties.
4.1. ψ .1. ψ(m,p) is nonincreasing in p, that is if p’ ≥ p, ψ(m,p’) ≤ ψ(m,p).
4.2. ψ .2. ψ(m,p) is nondecreasing in m, that is if m’ ≥ m, ψ(m’,p) ≥ ψ(m,p).
4.3. ψ .3. ψ(m,p) is homogeneous of degree 0 in (p, m) so that ψ(tm,tp) = ψ(m,p) for t > 0.
4.4. ψ .4. ψ(m,p) is quasiconvex in p; that is {p: ψ(m,p) < α } is a convex set for all α.
4.5. ψ .5. ψ(m,p) is continuous for all p > 0, m > 0.
B = {x : px < m}
B′^ = {x : p′x < m}
B′′^ = {x : p′x < m} We can show that any x in B’ must be in either B or B’; that is that B ∪ B’ ⊃ B”. Assume not; then x is such that tpx + (1-t)p’x |leq m, but Px > m and p’x >m. These two inequalities can be written as
tpx > tm
(1 − t)p′x > (1 − t)m
Summing the two expressions in equation 69 we obtain
tpx + (1 − t)p′x > m But this contradicts the original assumption that x is in neither B or B’.
Now note that
ψ(m, p′′) = max x v(x), such that x ∈ B′′
≤ max x v(x), such that x ∈ (B ∪ B′); because B ∪ B′^ ⊃ B′′
≤ α because ψ(m, p) ≤ α and ψ(m, p′) ≤ α
5.5. ψ .5. ψ(m,p) is continuous for all p > 0, m > 0.
By the theorem of the maximum (given below) ψ(m,p) is continuous for p > 0, m > 0. In the utility maximization problem, f(x,λ) in the theorem of the maximum is the utility function. It does not depend on λ. The constraint set is those values of x that are in the budget set as parameterized by p and m. So λ in this case is (p,m). The indirect utility function ψ(m,p) is M(λ) while the ordinary demand functions x(m,p) are m(λ). The utility function is continuous by assumption. The constraint set is closed. If p > 0 and m > 0, the constraint set will be bounded. If some price were zero, the consumer might want to consume infinite amounts of this good. We rule that out.
Theorem 1 (Theorem of the Maximum). Let f(x, λ ) be a continuous function with a compact range and suppose that the constraint set γ ( λ )is a non-empty, compact-valued, continuous correspondence of λ. Then
(i) The function M( λ ) = maxx {f(x, λ) : x ∈ γ(λ) } is continuous (ii) The correspondence m( λ ) = {x ∈ γ(λ) : f(x, λ) = M (λ)} is nonempty, compact valued and upper semi-continuous. Proof : See Berge [1, p. 116].
- M ONEY M ETRIC U TILITY F UNCTIONS
6.1. Definition of m(p,x). Assume that the consumption set X is closed, convex, and bounded from below. The common assumption that the consumption set is X = R L + = {x ∈ RL: x ≥ 0 for = 1, ... , L} is more than sufficient for this purpose. Assume that the preference ordering satisfies the properties given in section 1.8. Then for all x ∈ X, let BT(x) = {y∈BT | y x}. For the price vector p, the money metric m(p,x) is defined by
m(p, x) = min y ≥ 0 py
s.t. y ∈ BT (x)
If p is strictly greater than zero and if x is a unique element of the least cost commodity bundles at prices p, then m(p,x) can be viewed as a utility function for a fixed set of prices. It can also be defined as follows.
m(p, x) = C(u(x), p) (71) The money metric defines the minimum cost of buying bundles as least as good as x. Consider figure 3
F IGURE 3. Utility Maximization and Cost Minimization
xi
xj
z
x
mp, x
p 2
All points on the indifference curve passing through x will be assigned the same level of m(p,x), and all points on higher indifference curves will be assigned a higher level.
6.2. Example of a money metric utility function. Consider the Cobb-Douglas utility function
v(x 1 , x 2 ,... , xn) = A
∏^ n
i=
xα i i = A xα 1 1 xα 2 2 xα 3 3 · · · (72)
The cost function associated with this utility function is given in equation 64, which we repeat here.
C = ( Σni=1 αi ) v Σn^1 j=1 αj
∏^ n
j=
pj αj
)αj
1 Σnj=1 αj
⇒ m(p, x) = ( Σni=1 αi )
[
m Σnj=1 αj
] (^) Σnk=1 αk (^) n ∏
i=
[
αi p^0 i
] (^) αi
Σn^1 j=1 αj
∏^ n
j=
pj αj
)αj
Σn^1 j=1 αj
Σnj=1 αj
[
m Σnj=1 αj
] (^) n ∏
j=
( [
αj p^0 j
] (^) αj ) (^) Σn^1 j=1 αj
∏^ n
j=
pj αj
)αj
Σn^1 j=1 αj
= m
∏^ n
j=
( [
αj αj
]αj ) (^) Σn^1 j=1 αj
∏^ n
j=
pj p^0 j
)αj^ ⎞ ⎠
Σn^1 j=1 αj
= m
∏^ n
j=
pj p^0 j
)αj^ ⎞ ⎠
Σn^1 j=1 αj
8. P ROPERTIES OF DEMAND FUNCTIONS
Demand functions have the following properties
8.1. Adding up or Walras law.
∑^ n
i=
pih(u, p) =
∑^ n
i=
pi gi(m, p) = m (79)
8.2. Homogeneity.
hi (u, θ p ) = hi (u, p ) = gi (θ m, θ p ) = gi (m, p) (80) The Hicksian demands are derivatives of a function that is homogeneous of degree one, so they are homogeneous of degree zero. Euler’s theorem then implies that
∑^ n
j=
∂ hj (u, p) ∂ pj
pj = 0 (81)
If all prices and income are multiplied by a constant t > 0, the budget set does not change and so the optimal levels of x(m,p) do not change. We can also write this in differential form using the Euler equation.
∑^ n
j=
∂ gj (m, p) ∂ pj
pj + ∂ gj (m, p) ∂ m
m = 0
∑^ n
j=
∂ gj (m, p) ∂ pj
pj =
−∂ gj (m, p) ∂ m
m
8.3. Symmetry. The cross price derivatives of the Hicksian demands are symmetric, that is, for all i = j
∂ hj (u, p) ∂ pi
∂ hi(u, p) ∂ pj
This is clear from the definition of the Hicksian demands as derivatives of the cost function. Specifically,
∂C(u, p) ∂pi
= hi(u, p) (84)
so that
∂^2 C(u, p) ∂ pj ∂ pi
∂ hi(u, p) ∂ pj
and
∂^2 C(u, p) ∂ pi ∂ pj
∂ hj (u, p) ∂ pi
by Young’s theorem on the equality of cross-partials.
8.4. Negativity. The nxn matrix formed by the elements ∂ h ∂ pi^ (u,pj )is negative semi-definite, that is, for any
vector z, the quadratic form
∑^ n
i=
∑^ n
j=
zi zj
∂ hi(u, p ) ∂ pj
If the vector z is proportional to p, then the inequality becomes an equality and the quadratic form is zero. This means the matrix is negative semi-definite. This follows from the concavity of the cost function.
If we denote ∂ h ∂ pi^ (u,pj )by sij , then we can write the entire matrix of cross partial derivatives as S = sij. This
then implies that
z′Sz ≤ 0 (87) By the properties of a negative semi-definite matrix, this means that s (^) ii ≤ 0, or that the Hicksian demand functions have a slope which is non-positive. This follows from concavity of cost, and does not require convex indifference curves.
8.5. The Slutsky equation. If we differentiate equation 65 with respect to pj and then substitute from
Shephard’s lemma for ∂ C ∂ p(u,pj ),, we obtain
xi = xi(m, p) = gi[C(u, p), p] = hi(u, p) = xi(u, p)
⇒ xi(u, p) = hi(u, p) = gi[C(u, p), p]
∂ xi(u, p) ∂ pj
= sij = ∂ hi(u, p) ∂ pj
∂ gi(m, p) ∂ m
∂ C(u, p) ∂ pj
∂ gi(m, p) ∂ pj
∂ gi(m, p) ∂ m
xj +
∂ gi(m, p) ∂ pj
∂ xi(m, p) ∂ m
xj +
∂ xi(m, p) ∂ pj
The last term in equation 88 is the uncompensated derivative of xi with respect to pj. To compensate for this, an amount, xi, times ∂ g ∂ mi must be added on. We can also write equation 88 as follows
∂ xi(m, p) ∂ pj
∂ xi(u, p) ∂ pj
∂ xi(m, p) ∂ m
xj (m, p) (^) (89)
