ISSN 1403-2473 (print) ISSN 1403-2465 (online) SCHOOL OF BUSINESS, ECONOMICS AND LAW, UNIVERSITY OF GOTHENBURG. Whether or not f is concave is then a question of “technology”. • We make the assumptions about preferences that utility functions require. Draw a strictly quasiconcave curve )z =f(x which is (a) also quasiconvex (d) not concave (b) not quasiconvex (e) neither concave nor convex (c) not convex (f) both concave and convex Ans: Examples of acceptable curves are: 2. • A symmetric matrix A is negative semidefinite iff all of the nonzero k'th principal minor determinants (not just the leading ones) have the same sign as (−1)k. We define Conditional quasi concave Performance Measures (CPMs), on random variables bounded from below, to accommodate for additional information. In the picture that is neither, there are local maxima. is strictly quasi-concave. On the bell, it is admissible (though not required) to have both concave and convex segments. This video introduces widely used concepts of quasiconcavity and quasiconvexity in economics through a mathematical as well as graphical explanation. I owe my vivid understanding of this topic to. 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. A convex or a concave function is always continuouson the relative interior of its domain. Is it homogeneous with respect to L & K? If so, of what degree? (d) Is the profit function concave in L &K? Is it quasiconcave? Show your work. University. (e) Give an example of a function of one variable (not two or more variables) which is quasiconvex but not convex. make concave. A cutting plane method for solving concave minimization problems with linear constraints has been advanced by Tui. Example ex is quasiconcave but not concave. It is concave (and quasiconcave; all concave functions are. A Note on Concavity, Homogeneity and Non-Increasing Returns to Scale∗ Juan David Prada-Sarmiento [email protected] , increasing or decreasing) of one variable are both quasiconvex and quasiconcave WARNING - Quasiconcave is not the same as concave; quasiconcave is talking about the level. If not, provide a counterexample. $\begingroup$ No, what I ask is that if it is meaningful to impose a shape restriction when estimating demand functions in the form of concavity of the utility function. variables) which is quasiconcave but not quasiconvex. Definition 1(Hypographandepigraph ofafunction) The hypograph (orsubgraph)and epigraph of. (b) Is the production concave? Is it quasiconcave? Show your work. Quermassintegrals of quasi-concave functions and generalized Pr ekopa-Leindler inequalities S. The left-hand side of the inequality is not strongly smaller than the right-hand side. Despite this obvious relation, applying the interface of these two areas has been limited to date. function-and therefore of a concave function-is quasi-concave. March 2008. CONVEXITY AND OPTIMIZATION 5 2. Midterm Examination: Economics 210A October 2011 The exam has 6 questions. It is not quasiconcave or concave. In general, such a sum is not guaranteed to be quasiconcave. This property does not hold for quasi–concave functions, in general. We illustrate our results on weighted inequalities of Hardy type, on weighted inequalities of Hardy type involving supremum, and on reverse. Lagrange Multiplier Theorem applies, the constrained maximizer may not be an unconstrained maximizer of the Lagrangean. If not, provide a counterexample. quasiconcave. 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 $). Isn't the function y=1/x restricted to the positive orthant quasi convex because the sublevel curves are simply intervals and therefore convex?. As far as I understand a function can be both at a certain point, but is not clear to me why sometimes is said that the function is quasiconcave rather than just convex. 1 Definition 1: Concave Function The function f is concave on if for any and any Definition 2: Concave function The differentiable function f is concave on X if for any x x X01, and any. are quasi-concave. The bivariate normal joint density is quasiconcave. The Hessian of f which is not positive or negative semidefinite- Therefxe, f is not convex or concave. Optimality conditions and dual formulations for programming problems over cone domains Bradley Owen Skarpness Iowa State University Follow this and additional works at:https://lib. edu Northwestern University and Banco de la República de Colombia September 2010 Abstract 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. The following result is the analog of Theorem 4. Therefore, the folfowing implication diagram holds: concave + CF-concave =+ C-concave. But it is a quasiconcave function because if we take any two points in. 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!). pseudoconcave (not comparable) (mathematics) (said of a function) differentiable and decreasing in any direction where it has a negative directional derivative. The Hessian of f is v2f(r) = 2/r3 Therefore, f is corwex and quasiconvex. Why or Why not? Quasi Concave:. Convex Optimization — Boyd & Vandenberghe 3. If f : Rn!R is concave, then f is quasiconcave. 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. Quasiconvex and quasiconcave expressions can be constructed using convex and concave atoms, using the curvature rules given above. The robust choice function is increasing and quasi-concave but not necessarily translation invariant, a key property of monetary risk measures. Convex & Concave Function; Quasiconvex and Quasiconcave functions. is zero (but not always). Our method can be applied to any quasi-concave density, which covers a broad class of models. When is the sum of two quasiconcave functions quasiconcave? Economist that is why you have not seen general conditions. Fromnow on we will assume thatX is aconvex subset of Rn. Measuring efficiency with quasiconcave production frontiers Measuring efficiency with quasiconcave production frontiers Chang, Kuo-Ping 1999-06-16 00:00:00 Comparing with the frontiers derived from the data envelopment analysis (DEA) and Banker–Charnes–Cooper (BCC) methods, the frontier derived from the free disposal hull (FDH) method is the closest to observed decision making units (DMUs. datasets, but they do not. For the rst category, our starting result considers saddle functions that are locally convex-concave on the set of saddle points. concave, then a critical point x is a global maximum. Therefore, the folfowing implication diagram holds: concave + CF-concave =+ C-concave. If f : Rn!R is concave, then f is quasiconcave. 1) A) Must every quasi-concave function must be concave? If so, prove it. One mode is around 30 and the other is around 60. Proof: It is easy to see that the domain A is a convex set on Rn, f(a) and g(a) are differentiable on A. Bayesian preferences and each ex post game is compact, quasiconcave and better payoff secure, even if we do not require the private information measurability for any player’s strategy, Reny (1999)’s theorem is still not applicable to conclude the existence of an equilibrium in the ex ante game. More importantly, a concave function is not preserved under an increasing transformation of f (·), for example, (p x)4. Concave functions are quasi-concave (not conversely). 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. Consider, for example, the function f ( x , y ) = x y defined on the set of pairs of nonnegative real numbers. of a function is a sufficient condition for this property, but not a necessary one. Every convex function is quasiconvex but the converse is not true. Every concave function is quasiconcave, but not every quasiconcave function is concave. It is quasiconvex and quasiconcave i. −Werner Fenchel We limit our treatment of multidimensional functions3. If f is concave and differentiable then. pseudoconcave (not comparable) (mathematics) (said of a function) differentiable and decreasing in any direction where it has a negative directional derivative. Geometry of convex functions The link between convex sets and convex functions is via the epigraph: A function is convex if and only if its epigraph is a convex set. 2 and, if in addition, we drop the quasiconcavity requirement on g and replace the concavity requirement on H in Theorem 3. Erik Balder, Mathematical Institute, University of Utrecht, Netherlands Fall 2012 1 Introduction In 2010 a sudden departure at the Economics group of Utrecht University caused one of their courses to be without its regular teacher. This is too long for a comment, but I was not able to access the translations, so I can not provide details. If not, provide a counterexample. If we repeat the same analysis for the second function, we see that it could be concave and therefore it could be quasiconcave. f 11 ,f 22 0,f 12 0 Strictly quasiconcave c. The Hessian of f is v2f(r) = 2/r3 Therefore, f is corwex and quasiconvex. Convexity and concavity for functions of a real variable 2. Heconsidered a broader class of functions than the bilinear formsh. 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. The robust choice function is increasing and quasi-concave but not necessarily translation invariant, a key property of monetary risk measures. If X is an interval, one can replace "U(x) is convex" by "U(x) is an interval" Also: a function f is quasi-concave if, for any two a,b in the domain,. To estaglish that this is the case, we check that whenever first deriv. For f, then q = the sum of exponents is 0. f 11 f 22 f 2 0 Not strictly quasiconcave. non-decreasing), then g(f(x)) = h(x) composite function will remain a quasiconcave function Note: this is weaker requirement than preserving concavity which requires that the g function be both concave and increasing. This also means that if a monotonic transformation of f is concave, then f is concave. Quasiconcave and Quasiconvex Functions. Ziemba (New York: Academic Press, 1981). Slope well-defined everywhere. For example, we saw that the Cobb-Douglas function production f(x;y) = xayb (a;b>0) is not concave unless a+b 1. (a) Prove that every concave function on R^n is quasi-concave. 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. Example ex is quasiconcave but not concave. And illustrate one example that is a quasiconcave function but not a concave function. In other words, (x) is the set consisting of all the consumption bundles on the indifference curves through x, or on better indifference curves. In general, such a sum is not guaranteed to be quasiconcave. Lecture Note Microeconomic Theory 1 − To obtain results that may not be available through verbal arguments is quasiconcave iff is convex. 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. Hence the ratio of a nonnegative convex function and a positive concave function is quasiconvex, and the ratio of a nonnegative concave function. Level sets of convex, concave, quasiconvex, and quasiconcave functions. quasiconcave. Œ It is with these economic applications in mind that Arrow and Enthoven developed the theory of quasi-concave programming. Upon re-reading the exam, I do not think this exam was unreasonably difficult (except question 2(c)). (b) Is the production concave? Is it quasiconcave? Show your work. The bivariate normal joint density is quasiconcave. 16 For each of the following functions determine whether it is convex, concave, quasiconvex, or quasiconcave. More generally, a function which decreases up to a point and increases from that point on is quasiconvex (compare unimodality). If f() is quasi-concave, then -f() is quasi-convex. However it is not increasing, not concave and not strictly quasiconcave. • May increase voluntary contributions, but does not solve the free‐rider problem. 1 Definition 1: Concave Function The function f is concave on if for any and any Definition 2: Concave function The differentiable function f is concave on X if for any x x X01, and any. They are general-ized concave functions, since it is easy to show that every concave function is quasiconcave, but not conversely. We denote income by M,as usual, with M>0. This course is about. Introduction Most of utility theory depends on the assumption that the preference relation is convex, i. In fact it is also convex and quasiconvex. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. A key result is a geometric characterization of the value of cheap talk, described by the quasiconcave envelope of the sender’s value function. homogeneous of degree 2 D. Aguiar† PerHjertstrand‡ RobertoSerrano§ Thisversion: June2019 Abstract. You can check this by looking at the second derivative of exp(x). 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?. The quasiconcavity is obvious from the graph: For the extension of the local-global theorem we would need F to be strictly quasiconcave, which it is not. (d) Give an example of a function of one variable (not two or more variables) which is quasiconcave but not concave. They are general-ized concave functions, since it is easy to show that every concave function is quasiconcave, but not conversely. How to cite: Fuleky, P. This doesn't change the preference, whereas concave functions are easier to use than quasi-concave functions). You do not have to choose functions which are defined over. is a hyperplane Both "above" and “below" sets are convex So f is both concave. So it’s a quasiconcave function. The constant updates and the ability to contact the writer and vice versa has been a phenomenal and reassuring service. Any monotonic transformation of concave function is quasi-concave. It is not quasicot1"X- (c) on Solution. But then it's immediate that it's quasiconcave. They are also important in optimization, since many properties of concave functions can be extended to quasiconcave functions. Consider, for example, the function f(x, y) = xy defined on the set of pairs of nonnegative real numbers. which is also compact and concave if the game Gis compact and concave. However, even if we do not assume that the. For example, we saw that the Cobb-Douglas function production f(x;y) = xayb (a;b>0) is not concave unless a+b 1. It is concave (and quasiconcave; all concave functions are. Sydsaeter [4] points out the following simple example. −Werner Fenchel We limit our treatment of multidimensional functions3. Utility maximization. quasiconcave said of a function, if the inverse image of any set of the form (a,∞) for that function is a convex set. Concave and quasiconcave utility function Thus, ᩤ and ≠ implies ≿ and not ≿. Comments on some chapters in "Mathematics for Economics" (third edition, 2011) by M. This definition of the word Quasiconcave is from the Wiktionary dictionary, where you can also find the etimology, other senses, synonyms, antonyms and examples. Also strictly quasi-concave. However, convexity is a very restrictive assumption, which is important when we come to applications. (e) f(x 1;x 2) = x2=x 2 with domf= R2 ++. CHAPTER 5: Concave andQuasiconcave Functions 1 ConcaveandConvexFunctions 1. In the case of unconstrained optimization, we knew that there were various conditions regarding convexity that helped us out - for example, if a func-tion is concave, we know that any point satisfying the first order conditions is a global minimum. (mathematics) said of a function, if the inverse image of any set of the form (a,∞) for that function is a convex set. All convex functions are also quasiconvex, but not all. Quasi Concave & Convex • Definition (algebraically). quasi-concave utility function can be transformed into a concave utility function. Several results characterizing the extreme values of generalized concave functions are given. But it is a quasiconcave function because if we take any two points in. Носител на Нобелова награда за икономика през 1983 г. Note that χ s is not a concave function with respect to P t, thus is not a concave fractional programming problem. −Werner Fenchel We limit our treatment of multidimensional functions3. where u, v and k are concave and increasing. Con v ex log-concave functions not alw a ys (but sum of log-convex functions is log. There is no useful calculus for. In general, such a sum is not guaranteed to be quasiconcave. A log of a Cobb-Douglas is concave, being the sum of concave logs; take exp(ln(Cobb-Douglas)) and get a quasiconcave. 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 $). b) Prove that a pseudoconcave function is quasiconcave but the converse is not always true (show a counter- example) c) Prove that the sum of pseudoconvex functions need not be pseudoconvex. If f is a quasiconcave function and g is a weakly increasing (i. This paper presents a branch and bound algorithm for globally solving the sum of concave-convex ratios problem (P) over a compact convex set. In words, if f is such that it is always true that a point directly between two other points does not give a higher a value of the function than do both of the other points, then f is quasiconvex. (d) = on R2 Solution. Ronaldo CARPIO Advanced Microeconomic Analysis, Lecture 2. It is quasi-convex and quasiconcave (i. The production set Y is not affected by re-scaling all prices by the same amount, hence the optimal solution of the PMP does not change. We use these properties to propose greedy algorithms to combinatorially optimize some diversity (low statistical dependence) promoting functions of distance covariance. is zero (but not always). Relationship between concavity and quasi-concavity: All concave functions are quasi-concave. The best-selling, best known text in Mathematical Economics course, Chiang teaches the basic mathematical methods indispensable for understanding current economic literature. Every convex function is quasiconvex but the converse is not true. Product of convex and concave functions. But is not concave for > 1. Concave is a related term of cavity. convex sets is convex, while the union of two convex sets is not necessarily convex. Definition and properties. Concave and quasiconcave functions Correspondences and hemicontinuity. otherwise it is weakly concave or simply \concave". Question: Answer the following true/false and justify your answer. However, if f is explicitly quasiconcave, then we have the following. We first verify the domain of f. There is no useful calculus for. quasiconcavity. pseudoconvex (though functions can be both) Derived terms. If a function f is concave, and f(0) ≥ 0, then f is subadditive. It is not quasiconcave or concave. edu Northwestern University and Banco de la República de Colombia September 2010 Abstract 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. Œ One result provides conditions under which a point satisfying the Kuhn-Tucker con-. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Translation for 'concave' in the free French-English dictionary and many other English translations. (c) Rewrite the profit function as a function of L and K. • Payoffs functions are not quasiconcave →Can’t use Kakutani’s FPT. Economics > Production Possibility Frontier. Figure1(a) illustrates this de nition, which says that the secant line must lie below the function. 2 on local maxima of concave functions does not hold for quasiconcave functions. Instead of using the Lagrange multiplier method or some other method based on differential calculus,. In fact it is also convex and quasiconvex. We define a family of functions by the convexity of their upper-level sets. Example ex is quasiconcave but not concave. A continuous function on C is concave if and only if for any x and y in C. Background and theorem statements 2. Are the following functions quasiconcave? Strictly so? First check. Dalal* Abstract This paper makes both a methodological as well as a substantive contribution to the literature on the. It is well known that quasiconcavity may not be preserved under sum-mation or integration. Here is a very short glossary of some economic concepts and their explanations in mathematics (calculus). Osborne (University of Toronto): books on game theory, research and courses on game theory and political economy, math tutorial for economists, LaTeX style files, short guide to BibTeX, conversion of references to BibTeX format. 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. , increasing). Anatomy of a Cobb-Douglas Type Production/Utility Function in Three Dimensions. As a adjective concave is curved like the inner surface of a sphere or bowl. Even if we only consider cases where xy , both of the own second order partials are ambiguous and therefore the function is not necessarily strictly quasiconcave. 5: the upper contour sets of a quasi-concave function are convex). [4] 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. To explain the prevailing levels of output. non-decreasing), then g(f(x)) = h(x) composite function will remain a quasiconcave function Note: this is weaker requirement than preserving concavity which requires that the g function be both concave and increasing. If f is continuous in the interval I and twice differentiable in the interior of I (denoted I0) then we say 1: f is convex on I ⇔ f”(x) ≥ 0 for all x in I0. In this paper, we find the solution of a quasiconcave bilevel programming problem (QCBPP). From a layperson's perspective (a mathematician has a more rigorous way of expressing it), a quasiconcave function includes all concave functions and also all functions that overall are concave but that may have sections that are actually convex. Preliminary Examination in Microeconomics, June 13, 2018 1. What's the intuitive difference between quasi-concavity and concavity? Can you give an example of a quasi-concave function that is not concave?. * A function that is both concave and convex, is linear (well, affine: it could have a constant term). Every concave function is quasiconcave, but some quasiconcave functions are not concave. Since fis concave, it is also quasiconcave. But a quasi-concave function may be discontinuous, and then it may not be differentiable, in the interior of its domain. Concave function of more than one variable 7 4. Let each D. For example, any monotonic function is both quasiconcave and quasiconvex, but only linear functions are both concave and convex. quasiconvex; Related terms. Adjective. If f is a quasiconcave function and g is a weakly increasing (i. But the converse is not true. Introduction Most of utility theory depends on the assumption that the preference relation is convex, i. But a quasi-concave function may be discontinuous, and then it may not be differentiable, in the interior of its domain. quasiconcave and concave B. concave is also concave. 1 Definitions and Properties Let f : X ˆRn! R be a real valued function. De ne f: R2 +!R by f(x) = x 1x 2. value function. This function is quasi-concave. Any monotonic function is both quasiconvex and quasiconcave. All convex functions are also quasiconvex, but not all. Are the following functions quasiconcave? Strictly so? First check. Investment Levels Production function y = F (k, λ), CRS and strictly quasiconcave. Concave and convex are used in many lenses, mirrors, etc. (but not on R2), where Rn ++ and R n denote the sets of positive and negative real n-vectors, respectively. Answer as many as you can. Utility maximization. in the (convex-set) domain f , and for 0 1, f (u ) f (v) f (u ) f ( u (1 )v) f (v. I If f is concave, then it is quasi-concave, so you might start by checking for concavity. Mimeo, University of Washington, October 2006. Quasiconcave functions • 6. This means the. The main results concerning aggregation of generalized concave functions are presented and some extremal properties of compromise decisions by adopting aggregation operators are derived and discussed. 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 set is illustrated in red in the following figure. This function is not continuous (and thus. As will be seen later all concave func-tions are quasiconcave, but not all quasicon-cave functions are concave. Almost any su ciently regular quasiconcave function can be concavi ed this way. On the bell, it is admissible (though not required) to have both concave and convex segments. I wonder if the following is true:. function-and therefore of a concave function-is quasi-concave. They are, however, often quasiconcave or concave in some generalized sense. A continuous function on C is concave if and only if for any x and y in C. Remember that concavity requires that connecting any two points on or below the graph of \( f \) must produce a line below the graph. Indifference curve follows axis here. Let C be a con-vex set. 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?. some quasiconcave probability measure with density is Lipschitz continuous. See the solutions attached below. Still, grades in graduate classes are mostly A's and B's. Remember that concavity requires that connecting any two points on or below the graph of \( f \) must produce a line below the graph. Strict Quasiconcavity Just as there are strictly concave vs weakly concave functions or just concave functions, there are also. We plot the function values along thedashed line labeled I. Dasgupta and Maskin [1986] established the existence of pure and mixed strategy Nash equilibria in games where the strategy sets are convex and compact, and pay-. John Riley minor corrections 25 July 2016 Concave functions in economics 1. A concave lens is thinner in the middle and thicker at the edges. For Hessian it makes sense and I am OK with but not for the BH. Why or Why not? Quasi Concave:. Note, that since for any antimatroid the operator Γ is not-empty for each X ∈ Fk−1 , the definition (5) is correct. I wonder if the following is true:. However it is not increasing, not concave and not strictly quasiconcave. If its second derivative is negative then it is strictly concave, but the opposite is not true, as shown by f(x) = -x 4. A continuous function on C is concave if and only if for any x and y in C. 1 Concave and convex functions 1. Kam Yu (Lakehead) Chapter 2 Duality and Revealed Preferences Winter 2019 10 / 29. That is because min(f(x0);f(x00)) tf(x0)+(1 00t)f(x). The -firm Cournot model with general technologies is reviewed to derive generalized and unified conditions for existence of a pure strategy Nash equilibrium. stances where we do not have to check them. I have been struggling trying to understand the difference between a quasiconcave and a convex utility function. Concave and Quasi-concave func-tions In this section, we define quasi-concave functions and extreme points, and state the problem of quasi-concave minimization and its hardness. (e) f(x 1;x 2) = x2=x 2 with domf= R2 ++. Indeed, He and Yannelis (2015a). 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). The proof is given for the logconcave distribution but the case of a quasiconcave prob-. 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?. Translation for 'concave' in the free French-English dictionary and many other English translations. Concavity implies, but is not implied by quasiconcavity. I owe my vivid understanding of this topic to. If a function f is concave, and f(0) ≥ 0, then f is subadditive. That is because min(f(x0);f(x00)) tf(x0) + (1 t)f(x00). Wiskundige Economie 1 6012B0290Y. b) Must every concave function be quasi-concave? If so, prove it. Consider, for example, the function f ( x , y ) = x y defined on the set of pairs of nonnegative real numbers. Thehypograph andepigraph of f are definedas follows. The Hessian of f is v2f(r) = 2/r3 Therefore, f is corwex and quasiconvex. Upon re-reading the exam, I do not think this exam was unreasonably difficult (except question 2(c)). Convexity: convex sets in Rn, concave/convex functions, quasiconcave/ quasiconvex functions, separation theorems 6. Lagrange Multiplier Theorem applies, the constrained maximizer may not be an unconstrained maximizer of the Lagrangean. Arti: almost but not quite Sinonim: pseudo (semu), would-be (seolah-olah) Quasiconcave (quasiconvex) merupakan konsep yang lebih lemah (weaker) daripada konsep. Indifference curves with flat sections make sense. Note that if i is the identity function, then j is R-concave but not strictly R-concave, and quasi-increasing but not strictly quasi-increasing˚ of course the uniqueness result fails spectacularly for this function. This is too long for a comment, but I was not able to access the translations, so I can not provide details. quasiconcave definition: Adjective (not comparable) 1. * Initial scaffolding for dqcp * Initial implementation of dqcp2dcp * Prototype of bisection routine * Solve DQCPs with problem. (e) f(x 1;x 2) = x2=x 2 with domf= R2 ++. Indeed, He and Yannelis (2015a). datasets, but they do not. Example ex is quasiconcave but not concave.