On the origin of rconcavity and related concepts department of. Recently some results have been obtained for a larger class of functions. 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 0 is strictly quasi convex and lower semicontinuous1 on c strictly quasi concave and upper semicontinuous on c then it is quasi convex quasi concave on c, respectively. Fx,y xy is therefore not quasiconvex and only quasiconcave because the upper contour set is a convex set. Technical note on constant returns to scale production. It is well known that convex and lipschitz functions can be minimized e ciently using stochastic gradient descent sgd. A local maximum of a quasiconcave function need not also be a global maximum of the function. Intuitively, the hypograph of a function is the area lying below the graph of the function, while the epigraph is thearealying above thegraph. Composition of a convex function and a convex decreasing function is quasiconcave 0 is the ratio of a decreasing function and an increasing function, a quasiconcave function. Strictly quasiconvex concave functions and duality in.
The fact that such a production function is increasing means that more input generates more output. Technical note on constant returns to scale production functions. The profit function is the convex conjugate of the cost function. Quasicon vexity has been of particular interest in economics, where it. Cost functions come directly from the production function and prices. Convex preferences get that name because they make upper contour sets convex. Then check the conditions of the above theorem for. Proving that a cobbdouglas function is concave if the sum.
We contribute with a new characterization of uniformly quasi concave functions that allows for easier veri cation and provide more straightforward insight. Quasiconcavity of the utility function has, therefore, become the standard and less restrictive assumption in the study of demand theory. Working paper the analysis and optimization of probability functions. Quasiconcave production function x y if the function f is quasiconcave the upper contour or isoquants are convex. Convex analysis is the standard tool for analyzing textbook economics. C the upper contour sets of quasiconcave quasiconvex functions. Mathematically, this is the assumption that the firms production function, f.
The function does, however, share a property with a perfect dome and a cone. The following theoremgives acharacterizationof concave convex functions. In the theory of the firm it is almost always postulated that there are gains to input diversification. Pdf many theorems involving convex functions have appeared in the. Fx,y xy is therefore not quasiconvex and only quasiconcave because the upper contour set is a. However, strictly quasi concave function may as well be strictly quasi convex. Levyy shai shalevshwartzz may 2014 abstract stochastic convex optimization is a basic and well studied primitive in machine learning. Production function the firms production function for a particular good q. In section 3 we apply the results for convex sets to convex and quasiconvex functions and show. Homogeneous functions homogeneous of degree k applications in economics. Thus the function defined by the surface of the mountain is not concave. In the arrowdebreu model of general economic equilibrium, agents have convex budget sets and convex preferences. Notes on microeconomic theory these are summaries from. Chapter 8 cost functions done university of tennessee.
We now present some results which indicate the importance of convexity for. If the function f is quasiconcave the upper contour or isoquants are convex. Vy is a closed, nonempty set for all y 0 3 nonemptiness. A function is homogeneous if it is homogeneous of degree. Vaziraniy yinyu ye z abstract eisenberg and gale 1959 gave a convex program for computing market equilibrium for. Suppose production function fx is concave and the cost function cx is convex. Proving that a cobbdouglas function is concave if the sum of.
Property of monotonicity is a property of functions and not of level curves. If the function f is quasi concave the upper contour or isoquants are convex. Characterization of quasiconcave functions and its optimality conditions in n efor, t. Examples of such a function for a firm that uses a single input are shown in the next two figures. Additional results on quasiconvex optimization are obtained in 19. A quasiconcave utility function shares with a concave function the fundamental property of representing convex preferences. Additional results on quasi convex optimization are obtained in 19. Similarly, a function is quasiconvex if its lower contour sets are convex sets. An explicit expression for such a solution can be obtained by convolution with a friedrichs mollifier function 8. Informally, the input requirement set should include its own boundary. For a function from reals to reals, if f 0 0 and f 00 convexity. That is, a function is both concave and convex if and only if it is linear or, more properly, affine, taking the form fx.
To avoid trivial situations, we assume that y can be produced. Production functions the technology that is available to a. So a function is concave iff the function s value at a convex combination of any two points is at least as great as the same convex combination of the function s values at each point. Under the assumption of monotonicity, quasiconcavity says that isoquants are convex. Note that since f is a function that is it assigns a single number in the. Wilson mathematics for economists may 7, 2008 homogeneous functions for any r, a function f. The production functions most widely used in the economic literature are quasiconcave ones and in the case of increasing returns to scale. To check its properties mainly quasiconcavity and quasi convexity, level curves are used.
A function f is quasi concave if hx gfx is a concave function for some strictly increasing function gfrom aug 16, 2014 this video introduces widely used concepts of quasiconcavity and quasiconvexity in economics through a mathematical as well as graphical explanation. A diagrammatic proof that indirect utility functions are. Concavity implies that the production function is quasiconcave 1. Thus, so in stage ii isoquants are strictly convex and rts is diminishing. K consists of convex functions in d and c consists of closetoconvex function in d, respectively. Concave production function implies convex cost function.
Vazirani yinyu ye abstract eisenberg and gale 1959 gave a convex program for computing market equilibrium for fishers model for linear. Pdf in the first chapter of this book the basic results within convex and. Market equilibria for homothetic, quasiconcave utilities. Suppose utility function u represents preferences t. Many theorems involving convex functions have appeared in the literature since the pioneering work of jensen. This video introduces widely used concepts of quasiconcavity and quasiconvexity in economics through a mathematical as well as graphical explanation. Economists often assume that a firms production function is increasing and concave. More significantly, first order conditions are not in general sufficient to identify global of quasi. Market equilibria for homothetic, quasiconcave utilities and.
R is concave convex if and only if its restriction to every line segment of rn is concave convex function of one variable. Vazirani yinyu ye abstract eisenberg and gale 1959 gave a convex program for computing market equilibrium for fishers model for linear utility functions, and eisenberg 1961 generalized. More generally, we refer to an optimization problem as a convex optimization. When a utility function is quasiconcave, the associated indifference curves are.
Fromnow on we will assume thatx is aconvex subset of rn. Concave and convex functions 1concaveandconvexfunctions 1. Rna function is homogeneous if it is homogeneous of. C cv, w, q minimum total cost is a function of input prices and output quantity. Quasiconvexity, strictly quasiconvexity and pseudoconvexity of.
Note that this loss function f x, w is convex with respect to the pair of. Market equilibria for homothetic, quasiconcave utilities and economies of scale in production kamal jain. Note that, since a quasiconcave function may have concave as well as convex sections, there exist no definition of quasiconcavity in terms of the property of the set of points lying on or. Let 6 be a scalar function defined on a convex set c c e. A characterization of quasiconvex functions springerlink. This means that the mrts decreasing in z1 along the isoquant. As is the case with concave and convex functions, it is also true for quasicon cave. Pdf introduction to convex and quasiconvex analysis. Composition of a convex function and a convex decreasing function is quasi concave 0 is the ratio of a decreasing function and an increasing function, a quasi concave function. Moreover, product of nonnegative uniformly quasiconcave functions is a quasiconcave function. Afunctionfis linearly homogenous if it is homogeneous of degree 1. A function f is quasiconcave if hx gfx is a concave function for some strictly increasing. Convex sets sometimes, we know the second order conditions of a optimization problem are satised because the function has a particular shape.
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. Homogeneous functions ucsbs department of economics. If 0 is strictly quasiconvex and lower semicontinuous1 on c strictly quasi concave and upper semicontinuous on c then it is quasiconvex quasiconcave on c, respectively. Derivative of an implicit function youngs theorem if numerator coradiant and quasiconcave functions. At equilibrium prices, the budget hyperplane supports the best attainable indifference curve. The study of quasiconvex functions is several decades old 10,24,25.
Moreover, product of nonnegative uniformly quasi concave functions is a quasi concave function. Homogenous, homothetic, concave, quasiconcave this lecture note is based on chapter 20, 21 and 30 of mathematics for economists by simon and blume. Each of the assumptions, upon examination, makes realworld sense. To show this, we make use of the fact that any monotone increasing transformation of a concave function is quasiconcave. Stochastic quasiconvex optimization elad hazan k r y. For assignment helphomework help in economics, mathematics and statistics please visit this video explains what. A local maximum of a quasiconcave function need not also be a global maximum of the. We contribute with a new characterization of uniformly quasiconcave functions that allows for easier veri cation and provide more straightforward insight. As observed above, the set k of quasi convex functions is a closed cone but not convex. Convex lower boundary ofinput requirement set x1 x2 2. The problem with this is that a monotonic transformation of a concave or convex function need not be concave or convex. If the graph of the technology gror t, is convex, cy,w is convex in y, w 0. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. Thus, the c function represents the minimum cost necessary to produce output q with fixed input prices.
C represents the minimum isocost line for any level of q. Characterization of uniformly quasiconcave functions. Concave andquasiconcave functions 1 concaveandconvexfunctions 1. Useful properties of convex functions 11 this implies fy fx. In addition, concavity implies decreasing returns to scale. If a function is increasing, then is it both quasiconcave. This is usefulin problems of costminimizationas can be seen in. To prove that the indirect utility function is quasiconvex, it is thus necessary and sufficient to show that the priceindifference curves are convex to the origin. Disciplined quasiconvex programming stanford university. To show this, we make use of the fact that any monotone increasing transformation of a concave function is quasi concave. Generalized concavity in optimization and economics, pages 109. T est called locally upper signcontinuouson k if, for any x. Aug 14, 2012 assignment helphomework help in economics, mathematics and statistics please visit video explains what are.
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