Roughly speaking concavity of a function means that the tangent is above graph. Note we say a function f is strictly increasingdecreasing if the above weak in. A concave function that has no linear parts is said to be strictly concave. You can see some convexity and concavity of functions,mathematical methods in economics 2 sample questions with examples at the bottom of this page. The profit function is the convex conjugate of the cost function. The next result states this observation, and the similar one for convex functions, precisely. Social equilibrium theorems with strict quasi concavity. Tesi di laurea in mathematics, luiss guido carli, relatore marco dallaglio, pp. In many such cases, one nds it easier to defend the assumption that these functions are quasiconcave. Such functions have frequently been employed in microeconomic analysis. Whats the difference between quasiconcavity and concavity.
The negative of a quasiconvex function is said to be quasiconcave. Math prep notes uc san diego department of economics. Abstract concavity of increasing coradiant and quasiconcave. X the notbetterthanx set and the notworsethanx set are closed in x. The proof of strict quasiconcavity follows from a generalisation of minkowskis inequality proved in hardy et al. In mathematics, a quasiconvex function is a realvalued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form.
On the origin of rconcavity and related concepts econstor. Dynamic economics is about explaining economic behaviors through time. However, h f bracketleftbigg 0 2 x 2 x 2 bracketrightbigg so. In this paper, we first prove the strict quasiconcavity of maximizing functions, and next, using a generalization of the kkm theorem, we prove two maximum theorems without assuming the upper. John riley minor corrections 25 july 2016 ucla econ. Robinson 1933 in her comparably classic book on the economics of imperfect competition. Concavity is easier to understand than quasiconcavity, and concave functions on compact sets also have a single, global maximum, but concavity is a much stronger assumption. Can you give an example of a quasi concave function that is not concave. The twist is that while concavity requires that v0d2fxvbe negative for all v, quasiconcavity requires only that v0d2fxvbe negative for vsuch that rfx v 0. But we have gone through the theory of the firm first for a good reason.
Some treatments of micro economics handle consumer analysis first. Oct 11, 2015 economics convexity and concavity of functions,mathematical methods in economics 2 summary and exercise are very important for perfect preparation. In this paper, we study nonnegative, increasing, coradiant and quasiconcave functions over real locally convex topological vector spaces. That is, whether or not a function is concave depends on the numbers which the function assigns to its level curves, not just to their shape. Let a twice continuous differentiable, c2 function f defined on some open domain n t 22be given, and. Maximum theorems with strict quasiconcavity and applications. Economics convexity and concavity of functions,mathematical methods in economics2 summary and exercise are very important for perfect preparation. Quasiconcave, but not strictly quasiconcave and not strictly concave.
This theorem is the theoretical ground behind the mechanical di. My underlying idea is that whit an inferior good, there exist a point where the utility consuming less of one good is better and then its derivative negative at that point, moreove, i think that this behaviour would break the concavity shape. Quasi concavity again we begin with some geometrically suggestive definitions. The twist is that while concavity requires that v0d2fxvbe negative for all v, quasi concavity requires only that v0d2fxvbe negative for vsuch that rfx v 0. If you want to change your field in more fundamental ways, then obtain your primary motivation from life, and use it to look for fundamental shortcomings of previous thinking in the field. Now, if we assume the additional assumption on the constraint multimap c i in theorem 3. New maximum theorems with strict quasiconcavity request pdf. Property, which is always satisfied in the particular case where h is a bilinear form, is nothing but the requirement on the function h of quasi concavity in the variable x and of quasi convexity in the variable y. This volume collects many of his early studies of behavior that departs from the predictions of rational choice economics.
Roughly speaking concavity of a function means that the graph is above chord. Convex analysis is the standard tool for analyzing textbook economics. The lay reader should also be warned that quasi rational economics is, at beast, only quasiaccessible. To understand the significance of the terms applications in economics, it is useful to begin with a brief consideration of the origins and meaning of the term in mathematics. Abstract concavity of increasing coradiant and quasi. It is used to show the important result that for a concave differentiable function f every point x for which f x 0 is a global maximizer, and for a convex differentiable function every such point is a global minimizer.
Concavity and quasiconcavity in economics luissthesis. Some treatments of microeconomics handle consumer analysis first. Arbitrage pricing arbitrage profit average cost balance of payments budget constraint call option concave function consumer surplus consumption function convex function deadweight loss demand curve econometrics economic agent economic model economics economics textbook elasticity endogenous endogenous technical change. The most important is probably that strict concavity implies that consumption growth depends on the level of wealth, which is serially correlated, so the implication of hall 1978 that consumption should follow a random walk at the household level no longer. I wonder how i can show that this production function in capital k and labour l is quasiconcave but not strict concave.
We next characterize the abstract concavity, the upper support set and the superdifferential of this class of functions by applying a type of duality, which is used in microeconomic. If you graph a mathematical function and the graph looks more or less like a badly made bowl with a few bumps in it but still has a depression in the center and two ends that tilt upward, that is a quasiconcave function. This book is quite comprehensive, but not quite readable. In general, a strictly concave funcon must bestrictlyquasiconcave, although the converse is not true. Fromnow on we will assume thatx is aconvex subset of rn. Uncertainty aversion is often modelled as strict quasiconcavity of preferences over uncertain acts.
Can you give an example of a quasiconcave function that is not concave. Lecture notes on elasticity of substitution ted bergstrom, ucsb economics 210a march 3, 2011 todays featured guest is \the elasticity of substitution. Pigou 1934 also contributed to the understanding of the concept at its genesis. Thaler was awarded the 2017 nobel prize in economics. It is with these economic applications in mind that arrow and enthoven developed the theory of quasiconcave programming.
In all answers where you provide a counterexample, you must show that your example is really a counterexample. This is having a y set that is strictly convex along the same as having strictly convex its upper boundary. Social equilibrium theorems with strict quasiconcavity. Research methodology in applied economics don ethridge. When i say intuition i mean what property of the strategic interaction in the game is captured by the strict concavity condition. This paper provides such a characterization for two leading representations of uncertainty averse preferences. Quasiconcave utility functions get that name because quasiconcavity is a weaker property than concavity.
Quasiconcave is a topological property that includes concavity. I if f is concave, then it is quasiconcave, so you might start by checking for concavity. What you write is essentially one of the results in the original rosens paper. The origins of the notion of quasiconcave function are considered, with special. Apr 10, 2019 quasiconcave is a mathematical concept that has several applications in economics. Metric spaces a linear or vector space x is a set of elements together with addition and scalar multiplication operators. If we model the surface of the mountain as a function f of its longitude and latitude x, y, then a contour is a level curve of f. Characterizing uncertainty aversion through preference for. Concavifying the quasiconcave kelley school of business. Fundamental preference axioms completeness transitivity continuity greed strict quasiconcavity smoothness. Whats the intuitive difference between quasi concavity and concavity. Other results of this work are a strict converse duality theorem for mathematical programming. It is frequently plausible in economic applications that a function fx being max. We next characterize the abstract concavity, the upper support set and the superdifferential of this class of functions by applying a type of duality, which is used in.
Inferior goods, monotonic utility and strict concavity duplicate ask question asked 3 years. A strictly quasiconvex function has strictly convex lower contour sets, while a. Economics 210a october 2011 the exam has 6 questions. This also means that if a monotonic transformation of f is concave, then f is concave. Inferior goods, monotonic utility and strict concavity. A function with the property that for every value of a the set of points x, y such that f. In the rush of rstsemester graduate economics, every economist encounters con.
Quasiconcavity is a property of functions which, if strict, guarantees. Concavity and quasiconcavity in economics sciencedirect. At equilibrium prices, the budget hyperplane supports the best attainable indifference curve. Concavity and quasiconcavity 601 if strict inequality holds, namely fx fv, then we can find a point p on the line segment vy such that fx fp and derive a contradiction as before. One problem with concavity and convexity which well encounter again when we look at homogeneity is that they are cardinal properties. A theory of uncertainty aversion may be characterized by the pairs of acts for which strict preference for a mixture between them is permitted. John riley minor corrections 25 july 2016 concave functions in economics 1. The main focus of this paper are the properties quasiconcavity and strict quasiconcavity which describe a particular way in which a convex combination of two given income distributions compares in terms of inequality with these given two income distributions.
In mathematics, a quasiconvex function is a realvalued function defined on an interval or on a. Eric rasmusen, department of business economics and public policy. I if f is a monotonic transformation of a concave function, it is quasiconcave. In this paper, we first prove the strict quasi concavity of maximizing functions, and next, using a generalization of the kkm theorem, we prove two maximum theorems without assuming the upper. Property, which is always satisfied in the particular case where h is a bilinear form, is nothing but the requirement on the function h of quasiconcavity in the variable x and of quasiconvexity in the variable y. I wonder how i can show that this production function in capital k and labour l is quasi concave but not strict concave. In the arrowdebreu model of general economic equilibrium, agents have convex budget sets and convex preferences. Walgreen distinguished service professor of behavioral science and economics at the university of chicagos booth school of business, where he is the director of the center for decision research. R is concave convex if and only if its restriction to every line segment of rn is concave convex function of one variable. These notes are the starting point for a mathpreparation book, primarily for use by ucsd students enrolled in econ 205 potentially for use by folks outside ucsd as well. Concavity, second order conditions, and so on october 5, 2005. Inequality and quasiconcavity by kristof bosmans center. As such, this volume is more of an anthology than a coherent wholeand reads like it.
Find all the books, read about the author, and more. We quantify the size of strict improvements that imposing shape restrictions bring. Concave andquasiconcave functions 1 concaveandconvexfunctions 1. Quasiconvex functions have applications in mathematical analysis, in mathematical optimization, and in game theory and economics. Let a twice continuous differentiable, c2 function f defined on some open domain n t 22be given, and let xt, let ax t, tn 1,, be the 11tt u x x x xw w w w. These notes are the starting point for a mathpreparation book, primarily for use by. We shall demonstrate thjs more formally in the paragraphs that follow. You can see some convexity and concavity of functions,mathematical methods in economics2 sample questions with examples at the bottom of this page. Strict convexity and concavity the inequalities in the definition of concave and convex functions are weak. Necessary and sufficient conditions for a maximum 10 5. Notes on microeconomic theory these are summaries from. Elasticity of a function of a single variable before we meet this guest, let us spend a bit of time with a slightly simpler notion, the elasticity of a a function of a single variable. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. Economics stack exchange is a question and answer site for those who study, teach, research and apply economics and econometrics.
Textbook expositions of quasiconcavity can be found in kreps 1990 p. Quasiconcavity is a property of functions which, if strict, guarantees that a. In this paper, we study nonnegative, increasing, coradiant and quasi concave functions over real locally convex topological vector spaces. Since strict quasiconcavity implies diminishing marginal rate of substitution. This strict version of quasiconcave differences thus parallels strict quasiconcavity in securing a unique optimizerbut as was just explained the conditions are logically independent of each other neither implies the other. Quasiconcave is a mathematical concept that has several applications in economics. The lay reader should also be warned that quasi rational economics is, at beast, only quasi accessible. Quasiconvexity and quasiconcavity extend to functions with multiple. Convex preferences get that name because they make upper contour sets convex. Some level curves of two functions that are not strictly quasiconcave, although. We will say that the solution is interior if x i0, 8i.
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