*** Introduction
** Economic Rationality
Consider a decision-maker facing a particular economic situation.
Let x = (x1, …, xn) be a (n1) vector of decision variables
f(x, ) be the (direct) objective function,
= (1, …, m) be a (m1) vector of parameters representing the economic environment
g(x, ) = 0 be a constraint on the choice of x, reflecting technological feasibility and/or resource scarcity (e.g. budget constraint).
Then, economic rationality is defined by the following optimization problem
L*() = maxx {f(x, ): subject to g(x, ) = 0},
where L*() is the indirect objective function.
Let x*() be the decision rule obtained as the solution of the optimization problem. By definition, we have L*() = f(x*(), ).
In a given situation , assume that the decision-maker chooses xa.
- We may want to investigate whether or not xa = x*(). In the case where xa x*(), this corresponds to a lack of economic rationality. If in addition f(xa, ) < f(x*(), ) = L*(), then a lack of economic rationality implies a welfare loss for the decision maker. In this context, we may want to know the welfare implications of actual behavior. This corresponds to normative economic analysis.
- If we assume that the decision-maker is “rational”, then xa = x*(). Then the properties of actual observed behavior xa are the same as the properties of x*(). In this context, the investigation of actual economic behavior corresponds to positive economic analysis.
** Measurement
In any empirical economic analysis, it is necessary to obtain measurements on f(x, ) and g(x, ). We will present the arguments using the “utility function” f(x, ) = f(x, ) representing the preferences of the decision maker. However, keep in mind that similar arguments can be presented with respect to the function g(x, ) = g(x, ).
* Ordinal Measurement
The function f(x, ) can be measured based on an ordinal scale. This is a situation where f(x, ) is defined up to a monotonic increasing transformation
fo(x, ) = F(f(x, )), where F(f) is an increasing function.
Then, fo and f give representations of the same preference structure.
Note that, under differentiability,
fo/x = (F/f)(f/x),
implying that sign[fo/x] = sign[f/x], since F/f > 0.
Also, in the case where x is a scalar,
2fo/x2 = (F/f)(2f/x2) + (2F/f2)(f/x)2.
Since (2F/f2) can be either positive or negative, it follows that sign[2f/x2] may differ from sign[2fo/x2]. This implies that sign[2f/x2] is not meaningful.
Example: Consumer theory, where “diminishing marginal utility is not meaningful”.
* Cardinal Measurement
- Interval scale:
The function f(x, ) can be measured based on an interval scale. This is a situation where f(x, ) is defined up to a positive linear transformation
fo(x, ) = a + b f(x, ), where b > 0.
Then, fo and f give representations of the same preference structure.
Note that, with a 0, f(x, ) = 0 does not imply fo(x, ) = 0. This implies that “zero is not meaningful”.
Under differentiability,
fo/x = b (f/x),
implying that sign[fo/x] = sign[f/x], since b > 0.
Also,
2fo/x2 = b (2f/x2).
Since b > 0, it follows that sign[2f/x2] = sign[2fo/x2]. This implies that sign[2f/x2] is meaningful.
Examples:
Temperature measurement: C = -160/9 + (5/9) F, where C = degree Celsius and F = degree Fahrenheit.
Utility function under risk in the expected utility model, where “diminishing marginal utility” is meaningful.
- Ratio scale:
The function f(x, ) can be measured based on a ratio scale. This is an interval scale measurement with a = 0.
fo(x, ) = b f(x, ), where b > 0.
Then, fo and f give representations of the same preference structure.
Note that, with a = 0, f(x, ) = 0 implies fo(x, ) = 0. This implies that “zero is meaningful”.
And as in the interval scale measurement, sign[fo/x] = sign[f/x], and sign[2f/x2] = sign[2fo/x2]. This implies that sign[2f/x2] is meaningful.
Examples:
prices, quantities
revenue, cost, profit
production theory, where “diminishing marginal productivity is meaningful”.
** Basic Structure
In any empirical economic analysis, it is often useful to make some specific assumptions about the functions f(x, ) and g(x, ). We will present the arguments using the objective function f(x, ) = f(x, ) representing the preferences of the decision maker. However, keep in mind that similar arguments can be presented with respect to the function g(x, ) = g(x, ).
Assume that the function f(x, ) is twice continuously differentiable. Let
f/x = [f/x1, …, f/xn] be a (1n) vector
2f/x2 = be a (nn) symmetric matrix.
This implies that 2f/xixj = 2f/xjxi, for all i, j = 1, …, n.
* Concavity of f(x, ):
The function f(x, ) is said to be (strongly) concave in x if, for any x,
v’ [2f/x2] v < 0, for all (n1) vectors v satisfying v 0.
Note: f(x, ) is said to be convex in x if [-f(x, )] is concave in x.
* Quasi-Concavity of f(x, ):
The function f(x, ) is said to be (strongly) quasi-concave in x if, for any x,
v’ [2f/x2] v < 0, for all (n1) vectors v satisfying (f/x) v = 0 and v 0.
Note: f(x, ) is said to be quasi-convex in x if [-f(x, )] is quasi-concave in x.
Note: A concave (convex) function is always quasi-concave (quasi-convex). However, a quasi-concave (quasi-convex) function is not necessarily concave (convex). As a result, concavity (convexity) is a more restrictive assumption than quasi-concavity (quasi-convexity).
** Optimal Behavior
Let x*() be the solution to maxx{f(x, ): subject to g(x, ) = 0}.
Consider the Lagrangean
L(x, , ) = f(x, ) + g(x, ),
where is a Lagrange multiplier.
Assume that
g/x 0 (rank condition -- R)
and
(v1’ v2’) < 0, for all scalar v2 and (n1) vector v1 satisfying
[g/x] v1 = 0, v1 0 (second order condition -- SOC).
Under the rank condition (R) and the second order condition (SOC), the bordered hessian H = = is a (n+1)(n+1) matrix that is invertible.
Under (R) and (SOC), and assuming an “interior solution”, the solution x*() necessarily satisfies the first order condition – (FOC)
L/x = 0 (n equations)
and
L/ = 0 (1 equation).
This is a system of (n+1) equations that can be solved for the (n+1) unknown: x*() and *().
* Comparative Statics
Differentiate the first order condition (evaluated at x*(), *()) with respect to . This yields
2L/x + (2L/x2)(x*/) + (2L/x)(*/) = 0
and
2L/ + (2L/x)(x*/) + (2L/2)(*/) = 0.
This can be written as
= 0,
or
, where H = = is the bordered hessian, or
.
This gives the properties of the optimal decision rule x*().
Note: Differentiate g(x, ) = 0 (evaluated at x = x*()) with respect to . This yields
(g/x)(x*/) + g/ = 0.
These are “adding-up restrictions” that the optimal decision rule x*() must satisfy.
* Envelop Theorem
Note that
L*() = f(x*(), )
= f(x*(), ) + *() g((x*(), ) (since g(x*(), ) = 0)
= L(x*(), *(), ).
Differentiating this expression with respect to gives
L*/= (L/x)(x*/) + (L/)(*/) + L/
= L/ since (L/x) = 0 and (L/) = 0 from (FOC).
This gives the envelope theorem:
L*/= L/, where L(x, , ) is evaluated at x*()and *().
* Primal-Dual Approach
Consider the second order condition (SOC):
(v1’ v2’) H < 0, for all scalar v2 and (n1) vector v1, such that [g/x] v1 = 0, v1 0.
Let v1 = (x*/) w, and v2 = (*/) w, where w is a (m1) vector. Note that (g/x)(x*/) = -g/ from the adding-up restriction. It follows that [g/x] v1 = 0 implies [g/] w = 0. This gives
w' [(x*/)' (*/)'] H w < 0, for all vectors w such that [g/] w = 0, (x*/) w 0.
Using the transpose of the comparative static results yields
w' w > 0, for all w such that [g/] w = 0, (x*/) w 0.
This implies
w' w 0, for all vectors w satisfying [g/] w = 0.(A1)
Differentiating the envelope theorem L*/ = L/ (where L is evaluated at x*, *) with respect to gives
2L*/2 = 2L/2 + (2L/x)(x*/) + (2L/)(*/),
or
2L*/2 - 2L/2 = .(A2)
Since the left-hand side of (A2) is a (mm) symmetric matrix, it follows that the right-hand side of (A2) is also a (mm) symmetric matrix.
Combining (A1) and (A2) yields the primal-dual result:
[2L*/2 - 2L/2] = is a (mm) symmetric, positive semi-definite matrix "subject to constraint" ([g/] w = 0).
- Case where L(x, , ) is linear in : The linearity of L in implies 2L/2 = 0. It follows that the indirect objective function L*() has the following property
[2L*/2] is a (mm) symmetric, positive semi-definite matrix "subject to constraint"
([g/] w = 0).
- Case 1: if g/ = 0, then [2L*/2] is a (mm) symmetric, positive semi-definite matrix, implying that L* is convex in . Example: the indirect profit function in production theory.
- Case 2: if g/ 0, then [2L*/2] is a (mm) symmetric, positive semi-definite matrix subject to constraint, implying that L* is quasi-convex in . Example: the indirect utility function in consumer theory.
* Some Applications
1- Production Theory
Let f(x, ) = ' x
x = (n1) vector of netputs (i.e., outputs if positive, and inputs if negative)
= (n1) vector of prices for x
'x = firm profit
g(x) = 0 is a representation of the production function (or production frontier).
We have
L*() = maxx {'x: g(x) = 0}.
where
x*() are the profit maximizing output supply (input demand) functions when positive (negative)
L(x, , ) = 'x + g(x)
L*() is the indirect profit function.
Note that 2L/x = In, and 2L/ = 0. It follows from the primal-dual result that [x*/] is a symmetric, positive semi-definite matrix. This means that
xi*/j = xj*/i for all i j, i, j = 1, ..., n (generating (n2 - n)/2 symmetry restrictions)
and
xi*/i 0 for all i = 1, ..., n, implying that profit maximizing output supply (input demand) functions are upward-sloping (downward sloping).
The envelope theorem gives
L*/ = x*(), (Hotelling lemma).
Finally, L(x, , ) being linear in with g/ = 0, we have seen above that the indirect profit function L*() is convex in .
2- Consumer Theory
Let f(x, ) = f(x) is the utility function representing household preferences
x = (n1) vector of consumer goods
g(x, ) = 0 - 1'x, where ' = (0, 1')
0 > 0 is household income
1 = (n1) vector of prices for x
g(x, ) = 0 - 1'x = 0 is the household budget constraint.
We have
L*() = maxx {f(x): 0 - 1'x = 0}.
where
x*() are the utility maximizing (Marshallian) consumer demand functions
L(x, , ) = f(x) + [0 - 1'x]
L*() is the indirect utility function.
Note that 2L/0x = 0, 2L/1x = - In, and 2L/ = 1. It follows from the primal-dual result that
is a (n+1)(n+1) symmetric, positive semi-definite matrix "subject to constraint" ([g/] w = 0). (A3)
Note that [g/] w = 0 implies w0 - x' w1 = 0, or (w1', w0) = w1' [In, x], where w' = (w1', w0), w0 being a scalar, and w1 being a (n1) vector. Pre-multiplying the matrix in (A3) by [In, x] and post-multiplying it by [In, x]' gives
- [x*/1 + (x*/0) x'] is a (nn) symmetric, positive semi-definite matrix,
or, given > 0,
[x*/1 + (x*/0) x'] is a (nn) symmetric, negative semi-definite matrix (A4)
The matrix [x*/1 + (x*/0) x'] is called the Slutsky matrix, measuring "compensated (or Hicksian) price effects" Equation (A4) states that the Slutsky matrix is symmetric, negative semi-definite. This means that
xi*/1j + (xi*/0) xj = xj*/1i + (xj*/0) xi for all i j; i, j = 1, ..., n (generating (n2 - n)/2 symmetry restrictions)
and
xi*/1i + (xi*/0) xi 0 for all i = 1, ..., n, implying that compensated (Hicksian) demand functions are downward sloping.
The envelope theorem gives
L*/0 = *(),
L*/1 = -*() x*().
Assuming that *() > 0, this implies that
x*() = -[L*/1]/[L*/0] (Roy's identity).
Finally, L(x, , ) being linear in with g/ = x, we have seen above that the indirect utility function L*() is quasi-convex in .
** Additional Structure
Often, one is interested in imposing additional structure in economics, particularly if it simplifies empirical analysis. One issue then is whether the additional structure may possibly become "too restrictive" as a representation of the real word.
Again, these restrictions may involve either the function objective function f(x, .) and/or the constraint g(x, ). For simplicity, we will focus on the function f(x, ), where x = (x1, ..., xn)' is a (n1) vector.
* Homogeneity
A function f(x) is said to be homogenous of degree k in x if
tk f(x) = f(t x) for any t > 0.
If k = 1, the function is said to be linear homogenous.
If a differentiable function f(x) is homogenous of degree k, then
k f(x) = [(f/xi) xi] (Euler equation).
Proof: Differentiating tk f(x) = f(t x) with respect to t gives
k tk-1 f(x) = [f/(t xi) xi].
Evaluated at t = 1, this yields the desired result.
If a differentiable function f(x) is homogenous of degree k, then f/x is homogenous of degree
(k-1).
Proof: Differentiating tk f(x) = f(t x) with respect to x gives
tkf/x = f/(t x) t
or
tk-1f/x = f/(t x),
showing that f/x is homogenous of degree (k-1).
* Homotheticity
A function f(x) is homothetic if
f(x) = F[h(x)],
where F[h] is an increasing function, and h(x) is linear homogenous in x.
If a differentiable function f(x) is homothetic, then MRSij = (f/xi)/(f/xj) is homogenous of degree zero in x, where MRSij = (f/xi)/(f/xj) is the marginal rate of substitution between xi and xj. In other words, MRSij(x) = MRSij(t x) for all t > 0. This means that, for a homothetic function, the marginal rate of substitution is constant along ray through the origin.
Proof: We have MRSij = (f/xi)/(f/xj). Given f(x) = F[h(x)], it follows that
MRSij = [(F/h)(h/xi)]/[(F/h)(h/xj)]
= (h/xi)/(h/xj).
But h(x) being linear homogenous in x implies that h/x is homogeneous of degree zero in x. This implies that MRSij = (h/xi)/(h/xj) is also homogenous of degree zero in x, i.e. that
MRSij(x) = MRSij(t x) for all t > 0.
Note: A homogenous function is always homothetic. However, a homothetic function is not necessarily homogenous. Thus, homogeneity is a stronger restriction than homotheticity.
* Weak Separability
Let x = (x11, ..., x1,n1; x21, ..., x2,n2; ...; xK1, ..., xK,nK).
A function f(x) is said to be weakly separable into K groups if
f(x) = F[X1(x11, ..., x1,n1), X2(x21, ..., x2,n2), ..., XK(xK1, ..., xK,nK)],
where F/Xk 0 for k = 1, ..., K
Xk(xk1, ..., xk,nk) is an "aggregator function" for the k-th group, k = 1, ..., K.
If a function f(x) is weakly separable, then
MRSki,kj = (f/xki)/(f/xkj) is independent of xk'm for all k' k.
Proof: We have MRSki, kj = (f/xki)/(f/xkj). The definition of weak separability implies
(f/xki)/(f/xkj) = [(F/Xk)(Xk/xki)]/[(F/Xk)(Xk/xkj)]
= (Xk/xki)/(Xk/xkj), which is independent of xk'm for all k' k.
It follows that MRSki, kj is independent of xk'm for all k' k.
This implies that
MRSki,kj/xk'm = 0 for all k' k, and for all i, j and m.
In other words, under weak separability, the marginal rate of substitution among commodities within a group is independent of the commodities outside this group.
* Strong Separability
Let x = (x11, ..., x1,n1; x21, ..., x2,n2; ...; xK1, ..., xK,nK).
A function f(x) is said to be strongly separable into K groups if
f(x) = F[Xk(xk1, ..., xk,nk)],
where F/Xk 0
Xk(xk1, ..., xk,nk) is an "aggregator function" for the k-th group, k = 1, ..., K.
If a function f(x) is strongly separable, then
MRSki,k'j = (f/xki)/(f/xk'j) is independent of xk"m for all k" (k or k').
Proof: We have MRSki, k'j = (f/xki)/(f/xk'j). The definition of strong separability implies
(f/xki)/(f/xk'j) = [(F/Xk)(Xk/xki)]/[(F/Xk')(Xk'/xkj)]
= (Xk/xki)/(Xk'/xk'j), (since F/Xk = F/Xk'), which is independent of xk"m for all k" (k or k').
It follows that MRSki, k'j is independent of xk"m for all k" (k or k').
This implies that
MRSki,k'j/xk"m = 0 for all k" (k or k'), and for all i, j and m.
In other words, under strong separability, the marginal rate of substitution among commodities within any two groups is independent of the commodities outside these two groups.
** Duality
* Let f(x) be an increasing, quasi-concave function in x (e.g., a production function or a utility function).
1/ Let V(0, 1) = maxx≥0{f(x): subject to 1'x 0}, where 1 is the vector of prices for x.
When f(x) is a utility function and 0 is consumer income, then V(0, 1) is an indirect utility function. In general, V(0, 1) is homogenous of degree zero and quasi-convex in = (0, 1). And it satisfies Roy's identity x*() = -(V/1)/(V/0).
2/ Let C(1, y) = minx≥0{1'x: subject to y f(x)}.
When f(x) is a production function and y is an output, then C(1, y) is a cost function. When f(x) is a utility function and y is a utility level, then C(1, y) is an expenditure function. In either case, C(1, y) is linear homogenous and concave in 1. And it satisfies the envelope result x*() = C/1 (Shephard's lemma).
3/ Let D(x, y) = 1 be the implicit solution of f(x/D) = y (and vice versa). The function D(x, y) is called a radial distance function. It is linear homogenous in x.
4/ Let B(x, y) = Maxβ {β: f(x – β g) ≥ y: subject to x – β g ≥ 0}, where g is a reference bundle satisfying g ≥ 0, g ≠ 0. When f(x) is a production function and y is an output, then B(x, y) is a directional distance function: it measures the distance (in the direction g) between x and the frontier isoquant associated with y. When f(x) is a utility function and y is a utility level, then B(x, y) is a benefit function: it measures the willingness-to-pay (in number of units of the reference bundle g) to reach utility y starting from bundle x. In either case, B(x, y) is concave in x.
5/ Note that C(1, y) = 0 is the implicit solution of V(C, 1) = y. And V(0, 1) = y is the implicit solution of C(1, y) = 0.
6/ Note that
f(x) = min1≥0{V(0, 1): subject to 1’ x = 0}.
7/ Finally, note that
C(1, y) = minx≥0 {1’ x – B(x, y) (1’ g)}
B(x, y) = min1≥0 {1’ x – C(1, y): subject to 1’ g = 1}
It follows that, when f is increasing and quasi-concave, the functions f, V, C, B and D are dual to each others in the sense that knowing one is equivalent to knowing the others.
* Let f(x) be an increasing, concave function in x (e.g., a production function).
1/ Let (p, 1) = maxx≥0,y{py - 1'x: subject to y f(x)} (profit maximization).
Then (p, 1) is an indirect profit function. It is homogenous of degree one and convex in prices (p, 1). And it satisfies Hotelling's lemma: /p = y*(p, 1), and /1 = -x*(p, 1).
2/ Finally, note that
f(x) = min1≥0 {(p, 1)/p + 1'x/p}, where (p, 1)/p is the "normalized profit function".
It follows that, when f is increasing and concave, the functions , f, V, C, B and D are dual to each other in the sense that knowing one is equivalent to knowing the others.
Note: Production theory makes use of the functions , f, C, B and D. For example, by duality, technology can be represented by the production function f, by the cost function C, by the distance functionsB or D, or by the profit function . And consumer theory makes use of the functions f, V, C, B and D.
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