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Duality Relationship, Utility Function, Roys' Identity, Monet metric, utility functions
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1.1. Utility Function. The utility maximization problem for the consumer is as follows
max x ≥ 0 v(x)
s.t. px ≤ m
where we assume that p >> 0, m > 0 and X = RL +. The solution to 1 is given by x(p,m) = g(p,m). These functions are called Marshallian demand equations. Note that they depend on the prices of all good and income. This is called the primal preference problem. If we substitute the optimal values of the decision variables x into the utility function we obtain the indirect utility function. For the utility maximization problem this gives
u = v (x 1 , x 2 , · · · , xn ) = v [x 1 (m, p), x 2 (m, p),... , xn (m, p) ] = ψ(m, p) (2) The indirect utility function 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] (3)
Given that the indirect utility function is homogeneous of degree zero in prices and income, it is often useful to write it in the following useful fashion.
ψ (m, p) = max x [v(x) : px = m]
= max x [v(x) :
( (^) p m
x = 1]
= max x [v(x) : qx = 1], q =
p m
{ (^) p 1 m
p 2 m
pn m
= ψ (q)
We can obtain the utility function from the indirect utility function as follows.
u(x) = min q ≥ 0 ψ(q)
s.t. qx ≤ 1
We can obtain the utility function from the cost function as follows.
u(x) = max u
s.t. c(u, p) ≤ px, ∀p ∈ Rn ++
Date : October 4, 2005. 1
1.2. The expenditure (cost) minimization problem. The fundamental (primal) utility maximization prob- lem 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
The solution to equation 8 gives the Hicksian demand functions x = h(u, p). The Hicksian demand equa- tions are sometimes called ”compensated” demand equations 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) (9) For the dual problem the indirect objective function is
m = Σnj=1 pj hj u, p ) = c(u, p) (10) This is called the cost (expenditure) function and specifies cost or expenditure as a function of prices and utility. We can also write it as follows
c(u, p) = min x [p x : v(x) = u] (11) Because c(u, p) = m, we can rearrange or invert it to obtain u as a function of m and p. This will give ψ(m, p). Similarly inversion of ψ(m,p) will give c(u, p). These relationships between the utility maximization cost minimization problems are summarized in figure 1
FIGURE 1. Utility Maximization and Cost Minimization
1.5. Money Metric Utility Functions. Assume that the consumption set X is closed, convex, and bounded from below. The common assumption that the consumption set is X = RL + = {x ∈ RL: x≥ 0 for
= 1, ... , L} is more than sufficient for this purpose. Assume that the preference ordering satisfies the normal properties. 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) (19) The money metric defines the minimum cost of buying bundles as least as good as x. Consider figure 3
FIGURE 3. Utility Maximization and Cost Minimization
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.