It is frequently plausible in economic applications that a function fx being max. Quasiconcavity is a property of functions which, if strict, guarantees. In the arrowdebreu model of general economic equilibrium, agents have convex budget sets and convex preferences. Strict convexity isnt needed to have an indifference curve, but without it, we are assuming that the two goods are perfect substitutes, which isnt likely. In footnote 7 we showed that for a utility function for two. Note that concavity is not what we are interested in. 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. Concavity and convexity, inflection points of a function. Characterizing uncertainty aversion through preference for. Concavity and quasiconcavity in economics luissthesis. Apr 10, 2019 it turns out that a concave function is just a specific instance of a quasiconcave functionone without the bumps. Some level curves of two functions that are not strictly quasiconcave, although.
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. For example, economic theory dictates that the true underlying. This paper provides such a characterization for two leading representations of uncertainty averse preferences. If, for example, someone prefers one slice of bread and half a glass of milk than either a whole bread or a whole glass of milk, he or she has convex preference. In footnote 7 we showed that for a utility function for two goods to have a strictly diminishing mrs i. Concavity and quasiconcavity in economics sciencedirect. The fiat segments on b and c would be ruled out by strict quasi concavity. Inequality and quasiconcavity by kristof bosmans center for. An alternative way see introduction of defining a quasiconvex function is to require that each sublevel set.
Inferior goods, monotonic utility and strict concavity. Functions of a single variable the definitions above apply to any function, including a function of a single variable. Notes on microeconomic theory these are summaries from. Quasiconcave utility functions get that name because quasiconcavity is a weaker property than concavity. For theorem ii, we shall only prove that concavity implies quasi concavity. Uncertainty aversion is often modelled as strict quasi concavity of preferences over uncertain acts. Situation in which a combination of two or more items is preferable to any one of the individual items. The true function and the estimated function jstor. It turns out that a concave function is just a specific instance of a quasiconcave functionone without the bumps. Maximum theorems with strict quasiconcavity and applications. Perhaps the most important things to know for assignments and the exam is that. As lau 1978, 410 notes, checking the curvature properties of estimated functions. Quasi concavity quasi convexity monotonic function. The other two possible cases are handled similarly.
But if rationality is such a reliable factor, why do economic models so often fail to predict market behavior accurately. A theory of uncertainty aversion may be characterized by the pairs of acts for which strict preference for a mixture between them is permitted. Concave andquasiconcave functions 1 concaveandconvexfunctions 1. Eric rasmusen, department of business economics and public policy. In the onedimentional case, the hessian of fis simply one number. Almost because no conclusions can be drawn if the hessian is semide. Use this condition to check the convexity of the indifference curves for each of the utility functions in problem 3. Recall that a c2 function f is concave i d2fx is negative semide nite for all x2c. Math prep notes uc san diego department of economics. Fromnow on we will assume thatx is aconvex subset of rn. Economics 210a october 2011 the exam has 6 questions. Now, if we assume the additional assumption on the constraint multimap c i in theorem 3. Uncertainty aversion is often modelled as strict quasiconcavity of preferences over uncertain acts. In all answers where you provide a counterexample, you must show that your example is really a counterexample.
See graphs in notes for concavity and quasiconcavity pictures. Convex preferences get that name because they make upper contour sets convex. However, strictly quasi concave function may as well be strictly quasi convex. Lecture notes on elasticity of substitution ted bergstrom, ucsb economics 210a october 26, 2015 todays featured guest is \the elasticity of substitution.
The profit function is the convex conjugate of the cost function. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The lay reader should also be warned that quasi rational economics is, at beast, only quasi accessible. 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. There is a similar characterization for quasi concavity. This proves one of the four cases cited in theorem i. Price competition with differentiated goods and incomplete. Convex analysis is the standard tool for analyzing textbook economics. However, quasi concavity is preserved under increasing transformation, while concavity is not. I wonder how i can show that this production function in capital k and labour l is quasiconcave but not strict concave. The more important property for utility functions is quasiconcavity.
Concavity, second order conditions, and so on october 5, 2005. Social equilibrium theorems with strict quasiconcavity. Similarly, if it is easier to make a product using two. This is because of the linear nature of a budget constraint. In footnote 7 we showed that for a utility function for.
This is having a y set that is strictly convex along the same as having strictly convex its upper boundary. The origins of the notion of quasiconcave function are considered, with special. According to richard thaler, the shortcomings of the standard approach arise from its failure to take into account systematic mental biases that color all human. Differentiable functions we can evaluate quasi concavityconvexity using the first derivative. Strict convexity and concavity the inequalities in the definition of concave and convex functions are weak. 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. A differentiable function of one variable, f x, quasiconcave is iff, for any pair of distinct points u and v in the domain, quasiconvex f u v u f v f u 0 f v v. The fiat segments on b and c would be ruled out by strict quasiconcavity. Dynamic economics is about explaining economic behaviors through time. Social equilibrium theorems with strict quasi concavity.
Price competition with differentiated goods and incomplete the uninformed consumers know about each of the two goods, so we posit that these consumers end up allocating themselves equally across the two. We assume throughout that dipi, pj and dipi are twice. The study of the concavity and convexity is done using the inflection points. Thaler is an american economist who was awarded the 2017 nobel prize in economics. To check for quasiconcavity, select and xand y such that ux uy and then consider the convex combination. So quasi concavity is a statement about underlying preferences, while concavity is not.
Textbook expositions of quasiconcavity can be found in kreps 1990 p. Inequality and quasiconcavity by kristof bosmans center. 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. 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. For strict quasiconcavity replace the second inequality with a strict inequality, but not the. Economics stack exchange is a question and answer site for those who study, teach, research and apply economics and econometrics. 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. A concave function that has no linear parts is said to be strictly concave. Unlike winners curse, which collects a relatively accessible a series of articles thaler wrote for the journal of economic perspectives, qre collects his more technical literature. Additionally, tangency can only be achieved when preferences are wellbehavedstrictly convex. Quasiconcavity is a property of functions which, if strict, guarantees that a.
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. If a function f is semistrictly quasiconcave then it is quasiconcave. At equilibrium prices, the budget hyperplane supports the best attainable indifference curve. Quasiconcave utility functions get that name because quasiconcavity is. Find all the books, read about the author, and more. 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. Whats the difference between quasiconcavity and concavity. Lecture notes based mostly on chiang and wainwright, fundamental. 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. Inflection points an inflection point is defined as the point in which the function changes from being convex to concave or vice versa. 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. Standard economics theory is built on the assumption that human beings act rationally in their own self interest.
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