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Quasi concave programming

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QUASI-CONCAVE PROGRAMMING
781
function can always be so extended.) In this paper, we shall usually deal with functions quasi-concave over nonnegative values of the variables. It is clear from (1.1) that all concave functions are quasi-concave. It also can be shown that any monotonic nondecreasingfunction of a quasi-concave function-and thereforeof a concave function-is quasi-concave.5However, not every quasi-concave function can be expressed as a monotonic nondecreasingfunction of a concave function.6Thus quasi-concavityis a generalization of the notion of concavity. In terms of traditional economic theory, a concave function is one that satisfies the second order conditions for a maximum, that is, (1.8)
(1.7)
f(X) > t(x?)
implies
t?(x
xO) > O
for differentiablequasi-concavefunctions.4 In speaking of a quasi-concavefunction, some specificdomainof definition, taken to be a convex set, is assumed. Thus, the function x1x2is quasi-concave for nonnegative xi, x2, but not for all x. A function which is quasi-concave for a convex domain of definition cannot necessarily be extended to a quasiconcave function over the entire space. (In the same circumstances,a concave
f(x) > (xo)
f(x) > C implies t[Ox+(1O- 0)X1 > (xo)
is convex. That is, f(x) is quasi-concave if
forO < 0 < 1. Now, for any x satisfying (1.5), let (1.6) F(0) /[Ox + (1 0)x0] ? /(xo) = F(O) . Therefore, F'(O) > 0. Thus, differentiating F(0) and setting 0 equal to zero, we have
3 See [4, pp. 485-486]. A function, f(x), of several variables is differentiable if f (x + h) = f(x) + ch + eh, where c is a vector depending on x but not on h and e is a vector which goes to zero with h. If a function is differentiable, then its partial derivatives exist, and f- = c, but the existence of the partial derivatives does not necessarily imply differentiability; see [2, pp. 59-61]. In particular, f[xo + O(x-xo)] = f(xo) + f?O(x-xo) + e0, where e = e(x-xo) goes to zero with 0. Then,
d2/ = n
fx_f
n
dx dxj <?O.
Quasi-concavityis a weaker condition; (1.8) does not have to hold for quasiconcave functions.7 A quasi-concave function is one that has a diminishing marginal rate of substitution if /f > 0, or an increasing marginal rate of transformation if /f < 0, between any pair of variables, or between any distinct composite variables. Let xo and xl be any two nonnegative vectors not zero and not proportionalto each other. Then, if we let (1.9)
(xi,..., x"), subject to the constraints g(x) > 0, where g(x)
/o + ogo < O,
(KTL) x0(/f + 20g?)
0
0,
Aog(xo) =
AO0> 0, is the vector of partial derivatives of /(x) evaluated at where, for example, /ox the point xo.2 Kuhn and Tucker also proved that if /(x) and g(x) are concave functions, (KTL) are sufficient conditions for a constrained maximum. A function is concaveif the chord joining any two points on any plane profile of its graph lies everywhere on or below the function. That is, /(x) is concave if (0 < 0 < 1) (1.1) /[Ox + (1 - )xo] > 01(x) + (1 -0)/(xo) for all points x and xo in the region of definition of /(x). Write (1.1) in the form
Econometrica,
Vol. 29, No. 4 (October 1961)
QUASI-CONCAVEPROGRAMMING
By KENNETHJ. ARROW AND ALAINC. ENTHOVEN
We extend present theorems on conditions for a constrained maximum to the case where the maximand and the constraint functions are quasi-concave (e.g., utility functions). Economic applications are briefly discussed. 1. INTRODUCTION
780
K. J. ARROW AND A. C. ENTHOVEN
*
and take the limit of the left-hand side
tx(X xo) + t(xO) > /(x) (1.3) which is an alternative definition of concavity for differentiable functions. The inequality (1.3) states that if f(x) is concave, it lies everywhere on or below its tangent planes. A function is quasi-concave if, for each real number c, the set x defined by the inequality (1.4) (1.5)
f[xo + O(x- Xo)] -f(xo) =f?(x-xo)+e + -fx(x-X ) as 0
-?0
Here fo(x -xo) is an inner product of the two vectors. 4 The differentiation with respect to 0 is, in effect, taking the directional derivative of f(x) at xo in the direction of the point x. It is clear from the definition of quasiconcavity that this derivative, f0 d s where the terms x x are the direction must be nonnegative. For a definition of directional derivacosines (d - (x[-xo)2]i), tives, see [2, pp. 262-263]. Wold [7] defines a function to be convex towards the origin if (1.7) holds (his terminology is geometrically valid in the case he considers, where fx > 0); since (1.7) can be shown to imply quasi-concavity, the two definitions are equivalent.

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偏好凸向原点的英文

偏好凸向原点的英文

偏好凸向原点的英文## Convex Preferences and the Origin.In economics, a preference relation is a binaryrelation that represents an individual's preferences over a set of alternatives. A preference relation is said to be convex if it satisfies the following condition:> For any three alternatives x, y, and z, if x is preferred to y and y is preferred to z, then x is preferred to any convex combination of y and z.In other words, a preference relation is convex if, for any two alternatives x and y, the individual prefers any weighted average of x and y to any alternative that is not on the line segment connecting x and y.Convex preferences have a number of important implications. First, they imply that the indifference curves of an individual are convex. An indifference curveis a curve that represents all of the alternatives that an individual is indifferent between. If an individual's preferences are convex, then their indifference curves will be bowed outward, as in the following diagram:[Image of a convex indifference curve]Second, convex preferences imply that the demand curve of an individual is downward sloping. A demand curve is a curve that shows the relationship between the price of a good and the quantity of the good that an individual demands. If an individual's preferences are convex, then their demand curve will be downward sloping, as in the following diagram:[Image of a downward sloping demand curve]Third, convex preferences imply that the utility function of an individual is quasi-concave. A utility function is a function that represents an individual's preferences over a set of alternatives. If an individual's preferences are convex, then their utility function will bequasi-concave, which means that the level sets of theutility function will be convex.Convex preferences are a common assumption in economic models. This is because convex preferences are consistent with a number of reasonable assumptions about human behavior. For example, convex preferences are consistent with the assumption that individuals are risk-averse.## Applications of Convex Preferences.Convex preferences have a number of applications in economics. For example, convex preferences can be used to explain why individuals demand more of a good when its price decreases. This is because, when the price of a good decreases, the individual can afford to purchase more of the good without giving up any other goods.Convex preferences can also be used to explain why individuals are willing to pay more for a good when its quality increases. This is because, when the quality of a good increases, the individual's utility from consuming thegood increases.Finally, convex preferences can be used to explain why individuals are more likely to choose a risky option when the expected value of the option is high. This is because, when the expected value of a risky option is high, the individual's expected utility from choosing the option is also high.## Conclusion.Convex preferences are a powerful tool for understanding human behavior. Convex preferences can be used to explain a wide range of economic phenomena, including the shape of indifference curves, the slope of demand curves, and the behavior of individuals under uncertainty.。

Nash均衡存在性证明

Nash均衡存在性证明

Brouwer’s 不动点定理：若 f : X → X 是连续函数，而且 X 是非空、紧的凸集合，则存在
x* = f ( x *) 。
Kakutani’s 不动点定理： X 是 ℜl 空间中非空，紧的凸子集， f : X → X 是一个对应。如
果 f 满足：
(i) 对于所有的 x∈X，集合 f(x)非空且为凸， (ii) f 的图像是闭的，即，如果 xn→x，yn∈f(xn)，yn→y，那么 y∈f(x)（或者说对应 f 是上半
di
s
i
,σ
∗ −i
σ
∗ i
si
= di
si ,σ −i * ,
同乘
ui
si
,σ
∗ −i
− ui
σ
∗ i
, 对 si加总
s i ∈Si
( ( ) ( )) ( ) ( ( ) ( )) ( ) ∑ ( ) ∑ ∑ ⎛
⇒⎜
σ
∗ i
si
ui
si
,
σ
∗ −i
− ui
σ
∗ i
⎞ ⎟
di
s
i
,σ
∗ −i
连续的(upper hemi-continuous，uhc)），
那么则存在 x* 使得 x*∈f(x*)。
( ) ( ) 引理 1.1：σ ∗ 是纳什均衡当且仅当 ui
σ∗
≥ ui
si
,
σ
∗ −i
对所有 i 与 si ∈ Si 均成立。
( ) ( ) 证 明 ： 当 ui
σ∗
≥ ui
si
,σ
∗ −i
( ) 即存在 s∗ 使得 si∗ ∈ b s−∗i 对所有 i 成立。
convex and quasi convex

convex and quasi convex


separable
function.
• Why does Theorem 5 require that each φi be strictly concave to ensure that f is strictly concave, while Theorem 4 requires only one fi be strictly concave to ensure that f is is strictly concave?
i=1
i=1
This establishes that f is concave. If some fi is strictly concave and αi > 0, then the inequality is strict.
Since a constant function is concave, Theorem 4 implies:
V31.0006 Mathematics for Economists
New York University Department of Economics
C. Wilson September 15, 2011
Concave and Quasi-Concave Functions
A set X ⊂ Rn is convex if x, y ∈ X implies λx + (1 − λ) y ∈ X for all λ ∈ [0, 1] . Geometrically, if x, y ∈ Rn, then {z ∈ Rn : z = λx + (1 − λ) y for λ ∈ [0, 1]} constitutes the straight line connecting x and y. So a convex set is any set that contains the entire line segment between any two vectors in the set.
第四章效用最大化和选择

第四章效用最大化和选择


可得：
Y 1 / 4 ，就是说： Y * 1, X * 4 ， U 2, 1 。 X
4 间接效用函数
* * 最大效用= U ( X1* , X 2 ,..., X n ) * * = U [ X1* ( P 1, P 2 ,..., P n , I ), X 2 ( P 1, P 2 ,..., P n , I ),..., X n ( P 1, P 2 ,..., P n , I )]
Y
Px X PyY I
X
2.2 最大化的一阶条件 通过图形中 A、B、C、D 四点说明， 预算限制线的斜率=
Px dY =无差异曲线的斜率= Py dX
=常量
U
或
Px dY ＝－ ＝常量＝MRS(X对Y) Py dX U
Y
B C A D
Px X PyY I
X
消费者选择 A 点的商品是不明智的，因为如果他花费掉剩余的货币，就会达到 更高的效用水平。B 点也是不明智的，因为如果重新安排这两种商品的比率，会 使效用更大。D 点虽然可以有更高的效用，但是消费者没有可能达到。 2.3 最大化的二阶条件 上述的相切原则只是获得最大效用的必要条件， 但如果我们假设无差异曲线 的边际替代率递减，那么相切的条件就成为最大化的充要条件。 2.4 角解 （corner solution） 在这一点上，市场上商品 X 与 Y 地交换比率要比消费者心理上的交换比率 （MRS）低。在现行的市场价格之下，消费者更愿意不断地用 Y 来换取 X。但 实际中，消费 Y 的数量不可能为负，也就是说 X 轴限制了消费者这个不断交换 的过程。
PX X PY Y ( u X 0.5Y 0.5 ) PX 0.5 X 0.5Y 0.5 0