# Quasiconcave But Not Concave

Economics > Production Possibility Frontier. 1012–1016, ©2010 INFORMS 1013 models with concave ordering cost, which may be partly. It is this approach to optimization under uncertainty that is extensively used and studied in the second part of this book. CHAPTER 5: Concave andQuasiconcave Functions 1 ConcaveandConvexFunctions 1. Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. Define B(1,2) mathematically. Similarly, f is quasiconvex if the lower level set is a convex set for every real number a. In Example 1 below we will show that Theorem 1 is strictly more general than The-orem 2. But such critical points need not exist - and even if they do, they are not necessar-ily maximizers of the function - consider f(x)=x3. , there is a diminishing marginal rate of substitution between f 1 and fp), a property shared by most utility indicator functions arising in practice, it is shown how to sub-. This paper presents a branch and bound algorithm for globally solving the sum of concave-convex ratios problem (P) over a compact convex set. If a function f is concave, and f(0) ≥ 0, then f is subadditive. of the utility function? Why or why not? (c) Drop the assumptions that u is differentiable and concave Œ assume only that u is strictly in-creasing and a utility maximizing choice exists. Theorem 14 Definition 1 and Definition 2 are equivalent. Start studying ECON Chapter 6. A rm has produced x>2 units of (divisible) output at cost c. R is concave, then f is quasiconcave. We deﬁne a family of functions by the convexity of their upper-level sets. On the bell, it is admissible (though not required) to have both concave and convex segments. (d) Give an example of a function of one variable (not two or more variables) which is quasiconcave but not concave. This shows that a concave function is also quasiconcave, but not vice versa. I Example: Check whether the f(x;y) = xy. In the picture that is neither, there are local maxima. characterization of all the additively separable functions whose quasi concave closure is a transformation of an additively separable function. Chen, Zhang, and Zhou: Preservation of Quasi-K-Concavity and Its Applications Operations Research 58(4, Part 1 of 2), pp. some quasiconcave probability measure with density is Lipschitz continuous. They are also integral parts of most analyses in macroeconomics. Quasiconcave. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Concave and Quasiconcave Functions The proof is most easily accomplished through the use of the matrix algebra of quadratic forms. Still, grades in graduate classes are mostly A's and B's. Quasiconvexity replaces the convex combination of two function endpoints with the maximum of the two endpoints; $f(\lambda x_1+(1-\lambda)x_2)\le \lambda f(x_1. 3 Existence for weakly Ky Fan’s points of set-valued mappings Lemma. This allows to treat more general operators, but doesn't permit to. Quasiconcave functions • 6. Every concave function is quasiconcave, but not every quasiconcave function is concave. also show that this dynamics does not converge in general if the topologies are directed. The below code specifies and solves the QCP, using DQCP. Flavia Giammarino Indi erence Pricing with Uncertainty Averse Preferences. More generally, a function which decreases up to a point and increases from that point on is quasiconvex (compare unimodality). Besides f s quasiconcave f and only if f is quasiconvex. Again, picture a badly made bowl with a few bumps and protrusions in it. Deﬁnitions of convex and concave functions. Quasiconvexity replaces the convex combination of two function endpoints with the maximum of the two endpoints; [math]f(\lambda x_1+(1-\lambda)x_2)\le \lambda f(x_1. Œ It is with these economic applications in mind that Arrow and Enthoven developed the theory of quasi-concave programming. University at Albany, State University of New York Department of Economics Ph. The function f: R2 +! R given by f(x;y) = xy is A. Джерард Дебрю е американски икономист от френски произход. concave functions can considerably differ from concave functions, for instance, quasi-concave functions can be discontinuous in the interior of their domain, not every local maximum is a global maximum, local maxima that. This implies that the cone sklutions not contain an entire line. ] The curves are not consistent with the function's being concave: notice that (2, 2) is the. Utility maximization. In order along the light path followed by light from the reticle to the wafer substrate, the system includes a dominantly hyperbolic convex mirror, a dominantly elliptical concave mirror, spherical convex mirror, and spherical concave mirror. In general, such a sum is not guaranteed to be quasiconcave. ex is quasiconcave but not concave. We use these properties to propose greedy algorithms to combinatorially optimize some diversity (low statistical dependence) promoting functions of distance covariance. Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. That is because min(f(x0);f(x00)) tf(x0) + (1 t)f(x00). quasiconcave. The below code specifies and solves the QCP, using DQCP. Reading: [Simon], Chapter 21, p. 3) Consider the utility funcion (,). some quasiconcave probability measure with density is Lipschitz continuous. Adjective. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We study whether we can weaken the conditions given in Reny  and still obtain existence of pure strategy Nash equilibria in quasi-concave normal form games, or, at least, existence of pure strategy ε−equilibria for all ε> 0. f is quasi concave in the domain X if the set U(x) = {t in X : u(t) ≥ u(x)} is convex for any x in X. concave functions are also a quasi-concave function. But such critical points need not exist - and even if they do, they are not necessar-ily maximizers of the function - consider f(x)=x3. 6 Thus quasi-concavity is a generali- zation of the notion of concavity. However, even if we do not assume that the. But T is not a "density" so ∂ 2 f / ∂ T 2 v ≤ 0 for local stability, meaning that f is a concave function of T. It is not quasiconvex. Kam Yu (Lakehead) Chapter 2 Duality and Revealed Preferences Winter 2019 10 / 29. University at Albany, State University of New York Department of Economics Ph. Properties of Expected Demand Functions. is a plane The graph in. However, even if we do not assume that the. Still, grades in graduate classes are mostly A's and B's. This paper presents a new, non-calculus approach to solving the utility maximization problem with CES utility function, as well as with Cobb-Douglas utility function in case of n≥2 commodities. We deﬁne a family of functions by the convexity of their upper-level sets. For example is concave, and it is quasiconvex. Besides providing an introduction to Financial economics, these notes. Lam Tran Convex Optimization. the economics of the good society. convex sets is convex, while the union of two convex sets is not necessarily convex. It is not quasicot1"X- (c) on Solution. a) Prove that a concave function is pseudoconcave. The classic text in advanced microeconomic theory, revised and expanded. is a straight line Draw the set "below" the function f Graph in. concave, and the solution pair (I*, P*) must satisfy First order condition (2) characterizes the optimum provided that U(I, P=flZ)) is a quasiconcave function of I and U(I* , P*) > U(0,O) , (3) as entering into no contract remains an option of the consumer. Mimeo, University of Washington, September 2006. However, even if we do not assume that the. Quasi-concave functions and concave functions. Chapter1 SimpleRepresentativeAgent Models Thischapterdealswiththemostsimplekindofmacroeconomicmodel, whichabstractsfromallissuesofheterogeneityanddistributionamong. , Eugene Silberberg MATLAB Documentation, MathWorks The Anatomy of a Production/Utility Function in 3D, or any part thereof, may not be used as part of a document distributed. Our notion encompasses a wide variety of cases, from conditional expected utility and certainty equivalent to conditional acceptability indexes. Convex Optimization — Boyd & Vandenberghe 3. If not, provide a counterexample. We deﬁne a family of functions by the convexity of their upper-level sets. (2) Suppose j+s, is a system of excess demand functions that satis¿es the. However it is not increasing, not concave and not strictly quasiconcave. −Werner Fenchel We limit our treatment of multidimensional functions3. It is not quasiconcave or concave. Definition and properties. Every convex function is quasiconvex but the converse is not true. After excluding a certain small class of exceptions, the paper shows that, for arbitrary (perhaps not differentiable) functions, concave implies pseudoconcave, pseudoconcave implies strictly quasiconcave, and strictly quasiconcave implies quasiconcave. If the BH is negative definite than your function is quasiconcave, and you can only find a maximum over a convex set. They are, however, often quasiconcave or concave in some generalized sense. If we repeat the same analysis for the second function, we see that it could be concave and therefore it could be quasiconcave. We have seen that convexity has powerful implications for optimization prob-lems. Every convex function is quasiconvex but the converse is not true. Deﬁnition 1(Hypographandepigraph ofafunction) The hypograph (orsubgraph)and epigraph of. Bich philippe ∗ Abstract In a recent but well known paper, Reny has proved the existence of Nash equilibria for compact and quasiconcave games, with possibly discontinuous payoﬀ functions. 2007 is a concave function. whatsoever, and therefore not only of every action of a Quasi-concave utility functions get that name because quasi-concavity is a weaker property than concavity. 1 DeﬁnitionsandProperties Let f:X ⊂Rn −→Rbe a realvalued function. Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. The level sets are convex, therefore this is quasiconcave. Let C be a con-vex set. LECTURES AND EXERCISES But this approach does not make it clear exactly Afunctionf iscalledquasiconcave if,foranytwopointsxandx initsdomain,thefollowingproperty. some quasiconcave probability measure with density is Lipschitz continuous. It is not quasicot1"X- (c) on Solution. Weighted Inequalities of Hardy-Type on Amalgams Jain, Pankaj and Kumar, Suket, Real Analysis Exchange, 2009; Bilinear weighted Hardy inequality for nonincreasing functions Křepela, Martin, Publicacions Matemàtiques, 2017. A Level-Set Hit-And-Run Sampler for Quasi-Concave Distributions Abstract We develop a new sampling strategy that uses the hit-and-run algorithm within level sets of a target density. 1 to ﬁnite-dimensional Euclidean space. GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together. 4) by inﬁmum. Note that quasi-concavity does not imply that f ii ≤ 0. See the solutions attached below. where u, v and k are concave and increasing. Vazirani† Yinyu Ye ‡ Abstract Eisenberg and Gale (1959) gave a convex program for. More generally, a function which decreases up to a point and increases from that point on is quasiconvex (compare unimodality). But is not concave for > 1. 1 by the following assumption: For each t [0,T], if we define A1(t) = {x|u, g(x,u,t) 0 for some u}, then H0(x,(t),t) is concave on A1(t), if A1(t) is convex. This property does not hold for quasi-concave functions, in general. (d) = on R2 Solution. Kam Yu (Lakehead) Chapter 2 Duality and Revealed Preferences Winter 2019 10 / 29. It is concave (and quasiconcave; all concave functions are. The answer is simple if the carve is to be convace (strictly concave) then the opitamal point must be unique. Anatomy of a Constant Elasticity of Substitution Type Production/Utility Function in Three Dimensions. Quasiconvexity replaces the convex combination of two function endpoints with the maximum of the two endpoints; [math]f(\lambda x_1+(1-\lambda)x_2)\le \lambda f(x_1. 5 The approach taken here centers around a new and elementary method for constructing a. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i. It is this approach to optimization under uncertainty that is extensively used and studied in the second part of this book. We put forward that condition (c) should also. Long known for careful development of complex theory, together with clear, patient explanation, this. Claim: no other point belongs to U and. f 11 0,f 22 0,f 12 0 Strictly quasiconcave d. It is not quasiconvex. Consider, for example, the function f(x, y) = xy defined on the set of pairs of nonnegative real numbers. quasi-concave functions is not necessarily quasi-concave. If f : Rn!R is convex, then f is quasiconvex. Similarly, convexity implies quasiconvexity, but the converse does not hold. i) is strictly concave 8i, and therefore P n i=1 R i (Q i) is strictly concave (sum of strictly concave functions is str. Besides providing an introduction to Financial economics, these notes. 2007 is a concave function. quasiconcave. Similarly if f is convex, then f is quasiconvex. Adjective. Theorem 1, using a set valued map from X into the set of all USC quasiconcave functions on X, shows that we can have both quasiconcave functions and X paracompact. convex sets is convex, while the union of two convex sets is not necessarily convex. makes concave. The intuition for the theorem. A convex or a concave function is always continuouson the relative interior of its domain. Any monotonic transformation of concave function is quasi-concave. If f is concave and differentiable then  A continuous function on C is concave if and only if for any x and y in C. Let C be a con-vex set.  present Particle Swarm Optimization (PSO) algorithm for solving the bilevel programming problem with multiple linear objectives at the lower level while supposing the objective function at the upper level quasiconcave. Similarly, convexity implies quasiconvexity, but the converse does not hold. When is a function concave 11 6. Lecture Note Microeconomic Theory 1 − To obtain results that may not be available through verbal arguments is quasiconcave iﬀ is convex. Concave and convex functions ⊳𝑓is concave if 𝑓>0 and homogenous of degree Suppose to the contrary that 𝑓is regular and quasiconcave but not. Points where concavity changes (between concave and convex) are inflection points. We define Conditional quasi concave Performance Measures (CPMs), on random variables bounded from below, to accommodate for additional information. April 2018. 1 to ﬁnite-dimensional Euclidean space. also show that this dynamics does not converge in general if the topologies are directed. Econ 205 Sobel. Aguiar† PerHjertstrand‡ RobertoSerrano§ Thisversion: June2019 Abstract. For example is concave, and it is quasiconvex. As a verb concave is to render , or increase the degree of concavity. An equivalent description of a quasiconcave function is the following de-nition: f : Rn! R is quasiconcave if for all y, the upper contour set U, de-ned as follows U (y) = fx 2 X jf (x) yg (0. Convex Preferences Take one particular consumption bundle x : the at least as good as set (x) is the set of all consumption bundles which the person ﬁnds at least as good as the “reference bundle” x. According to the definition of convexity, we thus know function is convex, but not strongly convex. [math]f(x) = \sqrt{|x|}$ It's quasiconvex because any horizontal line segment connecting two points on the graph of the function, contains no points that lie beneath the graph of the function. → Definition and anagrams of quasiconcave. Positive monotonic transformations of a concave (convex) function do not preserve concavity (convexity) necessarily, but they do preserve quasiconcavity (quasiconvexity). The constant updates and the ability to contact the writer and vice versa has been a phenomenal and reassuring service. is zero then the second deriv. This paper presents a new, non-calculus approach to solving the utility maximization problem with CES utility function, as well as with Cobb-Douglas utility function in case of n≥2 commodities. But as you were saying "a function is quasiconcave if it is monotone" and although the dotted line is convex (see the. Quasi-concave density estimation. CONVEXITY AND OPTIMIZATION 5 2. Concave and Quasiconcave Functions The proof is most easily accomplished through the use of the matrix algebra of quadratic forms. Tufts University EE109 - Convex Optimization Department of Electrical and Computer Engineering Professor Mai Vu Homework 3 Due: March 15, 2019 1. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i. In this paper, we find the solution of a quasiconcave bilevel programming problem (QCBPP). A cutting plane method for solving concave minimization problems with linear constraints has been advanced by Tui. otherwise it is weakly concave or simply \concave". Therefore, we call a function quasilinear if it is both quasiconcave and quasiconvex. If it is helpful, you may use the result of. expand_more Nous ne supposons pas que f(z) Context sentences for "concave" in French. But then it's immediate that it's quasiconcave. Note: Concavity implies quasiconcavity, but the converse does not hold. It is definitely not concave or quasiconcave because the superlevel sets arenot convex. Antwoorden geselecteerde opgaven. JOURNAL of ECONOMIC THEORY 1, 291-314 (1969) Some Properties of Concave Functions PETER NEWIVIAN The Johns Hopkins University If V and W are two vector spaces over the same field, then a function f from V to W is linear if and only if it is additive and homogeneous (see e. This video introduces widely used concepts of quasiconcavity and quasiconvexity in economics through a mathematical as well as graphical explanation. Given: UK = {x, f (x) K} is a convex set. 16 For each of the following functions determine whether it is convex, concave, quasiconvex, or quasiconcave. Джерард Дебрю е американски икономист от френски произход. (1) completed but at a level of achievement that is not worthy of credit or (2) was not completed and there was no agreement between the instructor and the student that the student would be awarded an I. Kam Yu (Lakehead) Chapter 2 Duality and Revealed Preferences Winter 2019 10 / 29. If x 1 and x 2 solve: maxf (x) subject to x 2[a;b] and f is quasi-concave, then so does x 1 + (1 )x 2 for 2(0;1). If a random vector y has a logconcave (quasi-concave) probability distribution in Rq and x = Ay+b,whereA is a constant k×q matrix and b is a constant k-component vector, then x has a logconcave (quasi-concave) probability distribution. Theorem 14 Definition 1 and Definition 2 are equivalent. Qua-siconcavity requires the function merely not to dip down and back up between x0and x00, but concavity requires it to rise faster than linear from the lower point to the upper one. Let f(x,y) = xy and g(x,y) = 1−(x+y), so the Lagrangean is L(x,y;π) = xy +π(1−x−y). We note that the ﬁrst. We illustrate our results on weighted inequalities of Hardy type, on weighted inequalities of Hardy type involving supremum, and on reverse. Concave function of one variable 4 3. Determine if a given function is convex (strictly convex, concave, strictly concave). fis convex, as mentioned on page 72 and worked out in class. In this paper, we find the solution of a quasiconcave bilevel programming problem (QCBPP). be continuous. NO NO NO You are mixing up quasiconcavity with concavity. We don't want to assume utility functions are concave, because the only properties of utility functions that matter are the properties of their level curves, not the actual numbers a function assigns to the vectors in its domain. It is quasi-convex and quasiconcave (i. Show that the general quadratic form of a 11x 2 1 + a 12x 1x 2 + a 22x 2 2 can be written as ( x 1 2) a 11 a 12 0 a 22! x 1 x 2!:. functions will be a maximum, just as is the case with a concave function. convex sets is convex, while the union of two convex sets is not necessarily convex. PDF | This paper provides a simple proof of the result that if a production function is homogeneous, displays non-increasing returns to scale, is increasing and quasiconcave, then it is concave. Concerning the second statement of the last corollary, we emphasize that in general even the existence of a bounded and continuous density does not imply the Lipschitz continuity of the distribution function (for a counterexample see Henrion and Römisch 1999, Ex. In what follows, we cover the elements of this theory by focusing on its two main results. This module provides significant rigour in mathematics that I have not seen out of Department of Mathematics, and illustrates how these mathematical theorems is used to obtain results that make economic sense. Homework 2 F14 Solutions Documents. It's not convex because there do exist lines segments (not horizontal) which connect two points on the graph in which some points of the line segment lie beneath the graph of the function. However, we do not know the inlpl~cations between the qu~~iconcavity and the C-concavity. Recall that a C2 function f is concave i D2f(x) is negative semi-de nite for all. I emphasize that it is crucial. Meaning of Quasiconcave: said of a function, if the inverse image of any set of the form (a,∞) for that function is a convex set. The key is that both have a unique maximum, but that quasiconcave allows more functional forms (also note that a concave function is quasiconcave, but that a quasiconcave function may not be concave). of a function is a suﬃcient condition for this property, but not a necessary one. Econ 205 Sobel. 2 for quasi-concave functions, it characterizes quasi-concave functions via this class of "hockey stick" support functions. Chapter1 SimpleRepresentativeAgent Models Thischapterdealswiththemostsimplekindofmacroeconomicmodel, whichabstractsfromallissuesofheterogeneityanddistributionamong. Concerning the second statement of the last corollary, we emphasize that in general even the existence of a bounded and continuous density does not imply the Lipschitz continuity of the distribution function (for a counterexample see Henrion and Römisch 1999, Ex. You can check this by looking at the second derivative of exp(x). Typically, the membership functions of fuzzy sets involved in such problems are neither concave nor convex. Project Euclid - mathematics and statistics online. The answer is simple if the carve is to be convace (strictly concave) then the opitamal point must be unique. Many of these new atoms are integer-valued. Lars-Göran Larsson. Analogously, fis said to be quasiconcave on Sif, for all x,y∈ S, f(λx+(1 r-concave analog of the above theorem by replacing supremum in (1. Slope well-defined everywhere. Quasi-concave density estimation. Universiteit van Amsterdam. quasiconcave. A good test of whether you understand quasi concavity or not is to look at the concept in one dimension. Thus, a +(1 )a 2Sa and f is quasiconcave. But T is not a "density" so ∂ 2 f / ∂ T 2 v ≤ 0 for local stability, meaning that f is a concave function of T. A function f is defined to be quasi-concave on R^n if the set S = {x ∈ R^n : f(x) ≥ a} (x is a vector) is a convex set for every real number a. In terms of traditional economic theory, a concave function is one that. We shall show that these latter two classes of generalized concave functions coincide with the class of strictly pseudoconcave functions in the case of quadratic functions, at least on open convex sets C. What's the intuitive difference between quasi-concavity and concavity? Can you give an example of a quasi-concave function that is not concave?. University at Albany, State University of New York Department of Economics Ph. (but not on R2), where Rn ++ and R n denote the sets of positive and negative real n-vectors, respectively. is a straight line Draw the set "below" the function f Graph in. Quasiconvexity replaces the convex combination of two function endpoints with the maximum of the two endpoints; [math]f(\lambda x_1+(1-\lambda)x_2)\le \lambda f(x_1. Anatomy of a Constant Elasticity of Substitution Type Production/Utility Function in Three Dimensions. otherwise it is weakly concave or simply \concave". A minimal feasible subset of E that includes a set X is called a cover of X. You can check this by looking at the second derivative of exp(x). Schaible and W. One mode is around 30 and the other is around 60. that a concave function deﬁned on an open and convex set is continuously differentiable everywhere on this set, except possibly at a set of points of Lebesgue measure zero. described by the quasiconcave envelope of the sender’s value function. Microeconomic Theory Week 9: Topics in Consumer Theory (Jehle and Reny, Chapter 2)November 15, 2015 6 / 29 What if the utiltiy function is not strictly increasing and not quasiconcave?. he only cares about football and doesn't care about tennis except as a tiebreaker between bundles with the same amount of football time. It is not quasicot1"X- (c) on Solution. Thus, a concave function is automatically a quasiconcave function. Ronaldo CARPIO Advanced Microeconomic Analysis, Lecture 2. Frank Cowell: Consumption Basics. 3) is that. Quasiconcave programming Notes by Janos Mayer, 16. I (Incomplete): Assigned at the discretion of the instructor when, due to ex-. indifference. However, it's clear that exp(x) is a convex function. In the limit −∞,concave functions satisfy the condition f(p0x0 +p1x1)≥min{f(x0),f(x1)}, so they are (and consequently for all ρ-concave functions) quasi-concave. Determine whether or not two sets can be strictly separated. Any monotonic function is both quasiconvex and quasiconcave. ] (c) Show that a monotonically increasing function of a concave func­ tion is quasiconcave. CHAPTER 5: Concave andQuasiconcave Functions 1 ConcaveandConvexFunctions 1. Concave and convex functions ⊳𝑓is concave if 𝑓>0 and homogenous of degree Suppose to the contrary that 𝑓is regular and quasiconcave but not. Point 4) If the optimal point was not a singleton, then since the preferences are strictly convex, two points which are optimal will have a convex combination which is strictly prefered to either of the points. Formulate the following problem as a convex Feb 13, 6 pages. quasiconcave. Every concave function is quasiconcave, but some quasiconcave functions are not concave. Therefore, we call a function quasilinear if it is both quasiconcave and quasiconvex. As a adjective concave is curved like the inner surface of a sphere or bowl. f 11 ,f 22 0,f 12 0 Strictly quasiconcave c. Isn't the function y=1/x restricted to the positive orthant quasi convex because the sublevel curves are simply intervals and therefore convex?. I Example: Check whether the f(x;y) = xy. Using this observation, we examine the possibility of valuable communication, assess the value of commitment, and explicitly solve for sender-optimal equilibria in several examples. How to prove a function is a quasi-concave function? It may be not too early to "worry about" it, since around every 10 years there comes a newer generation of wireless communication. ex is quasiconcave but not concave. If its second derivative is negative then it is strictly concave, but the opposite is not true, as shown by f(x) = -x 4. Every concave function is quasiconcave, but some quasiconcave functions are not concave. It's easy to verify that this function is quasi-concave (actually, all monotonically increasing functions are quasi-concave). We define a decision maker as decreasing (resp. Universiteit van Amsterdam. stances where we do not have to check them. A function f is concave over a convex set if and only if the function −f is a convex function over the set. This peculiarity arises because the local stability condition for an intensive variable such as T is derived from a Legendre transformation, rather than splitting a system into parts having different values of T , as was. But we’re generally willing to assume utility functions are quasiconcave. NO NO NO You are mixing up quasiconcavity with concavity. Proof Part I Let U be the utility possibility set for Eand given a Pareto optimal allocation (^x;^y) let be de-ned as = v 2RI: v v^ where ^v = u(^x;^y) Clearly, ^v belongs to U and. I-lint 2: this is probably the hardest part of this question, so you might want to skip over it and answer the remaining parts first. Generally speaking, a quasiconcave function that is not also concave has a graph roughly shaped \ike a bell, or a portion thereof, and a quasiconvex function has a graph shaped like an inverted bell, or a portion thereof. The indifference curve associated with this is convex, while the function itself is quasi concave (because it satisfies $f_{xx} f_x^2 - 2 f_{12} f_1 f_2 + f_{yy} f_y^2$). (a)Suppose f(x) is an increasing function of one variable. f is quasi concave in the domain X if the set U(x) = {t in X : u(t) ≥ u(x)} is convex for any x in X. Not Every Quasi-Concave function is Concave. JOURNAL of ECONOMIC THEORY 1, 291-314 (1969) Some Properties of Concave Functions PETER NEWIVIAN The Johns Hopkins University If V and W are two vector spaces over the same field, then a function f from V to W is linear if and only if it is additive and homogeneous (see e. But f is quasiconcave and quasiconvex. Tufts University EE109 - Convex Optimization Department of Electrical and Computer Engineering Professor Mai Vu Homework 3 Due: March 15, 2019 1. I wonder if the following is true:. Midterm Examination: Economics 210A October 2011 The exam has 6 questions. (e) neither (quasiconcave if restricted to a single orthant) (f) both (g) quasiconcave (ex is a non-decreasing transformation, -x12 - 2x22 is concave). 1 Concave and convex functions 1. Downloadable! This paper provides a simple proof of the result that if a production function is homogeneous, displays non-increasing returns to scale, is increasing and quasiconcave, then it is concave. It is not quasicot1"X- (c) on Solution. They are also important in optimization, since many properties of concave functions can be extended to quasiconcave functions. I wonder if f(x) = -x 2 + 10 is quasiconvex. More importantly, a concave function is not preserved under an increasing transformation of f (·), for example, (p x)4. (a) Prove that every concave function on R^n is quasi-concave. When is the sum of two quasiconcave functions quasiconcave? Economist that is why you have not seen general conditions. Hence the ratio of a nonnegative convex function and a positive concave function is quasiconvex, and the ratio of a nonnegative concave function. The constant updates and the ability to contact the writer and vice versa has been a phenomenal and reassuring service. he only cares about football and doesn't care about tennis except as a tiebreaker between bundles with the same amount of football time. function is quasiconcave. The level sets are convex, therefore this is quasiconcave. Thus, a concave function is automatically a quasiconcave function. For (1), think of z=x3-x, we know that both x3 and –x are both monotone in R, so they are quasiconcave, but z, a sum of quasiconcave functions is neither. Convex and concave function. Other types of IC: not strictly quasiconcave. Theorem (Glicksberg). is a straight line Draw the set "below" the function f Graph in. If f : Rn!R is concave, then f is quasiconcave. strategies (dropped ui is quasiconcave and Si is convex) Proof - (a) can try to apply Nash by showing that all mixed strategies result in convex strategy space and quasiconcave expected payoffs, but that's difficult because there are lots of distributions to try (# of distributions not bounded!). We present a simple fully polynomial time approximation scheme (FPTAS) for minimizing a fairly gen-eral class of low-rank quasi-concave functions. We consider the problem of minimizing a class of quasi-concave functions over a convex set. They are general-ized concave functions, since it is easy to show that every concave function is quasiconcave, but not conversely. quasiconcave. Tight conditions are formulated alternatively (i) in terms of concavity of two-sided transforms of inverse demand, or (ii) as. quasi-concave functions is not necessarily quasi-concave. There exists w 6= ^ v that belongs to. Chapter 21 Concave and Quasiconcave Functions 110 2 MATHEMATICS FOR ECONOMISTS c) We know m but need to compute b. A convex ora concave function is always continuous on the relative interior of its domain.