Quasiconcavity is a property of functions which, if strict, guarantees that a function. For strict quasiconcavity it holds as a strict inequality. However, quasiconcavity is preserved under increasing transformation, while concavity is not. Perhaps the most important things to know for assignments and the exam is that.
Convex, concave, strictly convex, and strongly convex functions first and second order characterizations of convex functions optimality conditions for convex problems 1 theory of convex functions 1. Social equilibrium theorems with strict quasi concavity. Economics 210a october 2011 the exam has 6 questions. Dynamic economics is about explaining economic behaviors through time. Delivered in 1014 days or select twodayexpedited shipping. Tesi di laurea in mathematics, luiss guido carli, relatore marco dallaglio, pp. As drawn, both surfaces depict increasing fubctions, as they contain only the ascending portions of a dome and abell, respectively. In the arrowdebreu model of general economic equilibrium, agents have convex budget sets and convex preferences. The central development is a condition called quasi concave differences, which implies concavity of the policy function in optimization problems without imposing differentiability or quasi concavity conditions. Now, if we assume the additional assumption on the constraint multimap c i in theorem 3. Concavity and continuity, mathematics for economists economics 392m. If you are taking this course, please check this site regularly.
On the origin of rconcavity and related concepts econstor. Roughly speaking concavity of a function means that the graph is above chord. This paper provides such a characterization for two leading representations of uncertainty averse preferences. Absolute advantage the ability of a country or region to produce a good or service at a lower. Notes on microeconomic theory these are summaries from. Eric rasmusen, department of business economics and public policy, kelley. His most famous work was the wealth of nations 1776 a study of the progress of nations where people act according to their own selfinterest which improves the public. The production technology encompasses the details of how workers interact. In words, quasiconcavity of a continuous function means that the function does not have a ushaped section. In mathematics, a quasiconvex function is a realvalued function defined on an interval or on a. The negative of a quasiconvex function is said to be quasiconcave. 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. If u is quasiconcave and g is increasing then the function f defined by f x g u x for all x is quasiconcave. 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.
Then, for any initial distribution of the stock of goods between consumers, there is a price vector in which general competitive equilibrium is achieved. The central development is a condition called quasiconcave differences, which implies concavity of the policy function in optimization problems without imposing differentiability or quasiconcavity conditions. The term economics comes from the ancient greek from oikos, house and nomos, custom or law. Quasiconcavity is a property of functions which, if strict, guarantees that a.
Uncertainty aversion is often modelled as strict quasiconcavity of preferences over uncertain acts. So in terms of getting a unique or at least convex set of solutions to the foc, concavity is global, whereas quasiconcavity is only local. Composition results for quasi concave functions concavity and quasi concavity of cobbdouglas and c. In mathematics, a quasiconvex function is a real valued 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. Note we say a function f is strictly increasingdecreasing if the above weak in. However, quasi concavity is preserved under increasing transformation, while concavity is not. The origins of the notion of quasiconcave function are considered, with special interest. I if f is concave, then it is quasiconcave, so you might start by checking for concavity.
Convex preferences get that name because they make upper contour sets convex. It makes no sense to impose a stronger condition, like concavity, on this function, because the only significant property of the function is the character of its level curves, not the specific numbers assigned to these curves. It is supposed that the assumptions about the continuity and strict quasi concavity of utility functions, as well as the continuity and strict concavity of production functions are satisfied. Economics is a social science concerned with the factors that determine the production, distribution, and consumption of goods and services.
Strict quasiconcavity and strict quasiconvexity rule out flat segments. Quasiconcave is a topological property that includes concavity. I understand your explanation of expected utility but in the previous comment i suggested that the convexity of preferences has as an implication that there is a concave utility representation. R is concave convex if and only if its restriction to every line segment of rn is concave convex function of one variable. In all answers where you provide a counterexample, you must show that your example is really a counterexample. New maximum theorems with strict quasiconcavity request pdf. Inequality and quasiconcavity by kristof bosmans center. Your purchase helps support the african childrens educational trust acet. At equilibrium prices, the budget hyperplane supports the best attainable indifference curve.
These notes are the starting point for a mathpreparation book, primarily for use by. This is having a y set that is strictly convex along the same as having strictly convex its upper boundary. This also means that if a monotonic transformation of f is concave, then f is concave. Fundamental preference axioms completeness transitivity continuity greed strict quasiconcavity smoothness.
In this sense, it has something in common with psychology, sociology, and political science. Paul madden concavity and optimization in microeconomics. A theory of uncertainty aversion may be characterized by the pairs of acts for which strict preference for a mixture between them is permitted. 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. Social equilibrium theorems with strict quasiconcavity. Strict quasiconcavity of theproduction function requires. Joel sobel and joel watson have revised parts of these notes and added material, but the document is still rough and disorganized. The more important property for utility functions is quasiconcavity. X the notbetterthanx set and the notworsethanx set are closed in x. Note that concavity is not what we are interested in. See graphs in notes for concavity and quasiconcavity pictures. To check for quasiconcavity, select and xand y such that ux uy and then consider the convex combination.
Textbook expositions of quasiconcavity can be found in kreps 1990 p. The profit function is the convex conjugate of the cost function. Concavity and quasiconcavity in economics luissthesis. Complete economics dictionary complete economics dictionary a abnormal profit the surplus of revenue over costs enjoyed by a monopoly that is in excess of profit the same firm could expect to earn if it faced competition for its market. 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. Complete economics dictionary complete economics dictionary. Whats the difference between quasiconcavity and concavity. Concavity and quasiconcavity in economics sciencedirect.
This cannot be the case if f is a concave function. 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. The walrasian general equilibrium theory publish your. In microeconomics, convexity plays a fundamental role in general equilibrium theory. For strict quasiconcavity replace the second inequality with a strict inequality, but not the first.
Quasiconvexity and quasiconcavity extend to functions with multiple. I will use this site for posting announcements about assignments. Exlibrary, so some stamps and wear, but in good overall condition. Quasiconcave utility functions get that name because quasiconcavity is. Economics 210a graduate economic theory fall 2016 ted bergstrom economics department, ucsb. 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. The strict quasiconcavity assumption plays a crucial role in economics as it. The relationship between strict quasiconcavity and this sections main.
This pdf is a selection from an outofprint volume from the. Math has become a common language for most economists. The bivariate normal joint density is quasiconcave. So quasiconcavity is a statement about underlying preferences, while concavity is not. Math prep notes uc san diego department of economics. Economics stack exchange is a question and answer site for those who study, teach, research and apply economics and econometrics. Strict quasiconcavity also rules out flat portions, so that the function must be either hump shaped or strictly monotone. So quasi concavity is a statement about underlying preferences, while concavity is not. 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. It is frequently plausible in economic applications that a function fx being max imized is. The origins of the notion of quasiconcave function are considered, with special. In the rush of rstsemester graduate economics, every economist encounters con.
In section 1 we analyse how the agent chooses among a number of competing alternatives, investigating when preferences can be represented by a utility function. Characterizing uncertainty aversion through preference for. Strict convexity and concavity the inequalities in the definition of concave and convex functions are weak. First, economics is a social science, so it seeks to explain something about society. From a laypersons 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. It turns out that a concave function is just a specific instance of a quasiconcave functionone without the bumps. But economics is different from these other social sciences, because of what economists study and how they study it. Functions of a single variable the definitions above apply to any function, including a function of a single variable. You should check that the theorem of the maximum, along with the strict qconcavity assumption ensures that these are all continuous functions. 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. A concave function that has no linear parts is said to be strictly concave. The proof of strict quasiconcavity follows from a generalisation of minkowskis inequality proved in hardy et al.
If a function f is semistrictly quasiconcave then it is quasiconcave. Function 2 is homogeneous of degree v and has a variable elasticityof substitution given by. These material here is incomplete and contain many mistakes. Convex analysis is the standard tool for analyzing textbook economics. In this paper, we study nonnegative, increasing, coradiant and quasiconcave functions over real locally convex topological vector spaces. Such functions have frequently been employed in microeconomic analysis. This proves one of the four cases cited in theorem i. Inequality and quasiconcavity by kristof bosmans center for.
I if f is a monotonic transformation of a concave function, it is quasiconcave. If u is quasiconcave and g is decreasing then the function f defined by f x g u x for all x is quasiconvex. Inferior goods, monotonic utility and strict concavity. Thaler is an american economist who was awarded the 2017 nobel prize in economics. Abstract concavity of increasing coradiant and quasiconcave. The probability density function of the normal distribution is quasiconcave but not 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. New maximum theorems with strict quasiconcavity article in bulletin of the korean mathematical society 38.
Economics books goodreads meet your next favorite book. For theorem ii, we shall only prove that concavity implies quasi concavity. Quasiconcave utility functions get that name because quasiconcavity is a weaker property than concavity. In this paper, we study nonnegative, increasing, coradiant and quasi concave functions over real locally convex topological vector spaces. 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. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. These lectures examine the preferences of a single agent. A competitive equilibrium is a price vector and an allocation such that. Maximum theorems with strict quasiconcavity and applications. The syllabus that you see is a bit like the weather report. The general takeaway is that many distributional questions in economics which cannot be solved by direct calculations or the implicit.
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