If the function ux is concave strictly concave, then h x u x is convex strictly convex. 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. A function is concave strictly concave if the line segment connecting ux and cux. Then show that a monotone increasing function of a quasi concave function must be quasi concave. I if f is a monotonic transformation of a concave function, it is quasi concave. In a similar way we get contradictions if we assume that x9 elx, x2.
Every strictly quasiconvex function is strictly convex. We can use this result and the following proposition to define a class of concave function in higher dimensions. How to prove a function is a quasiconcave function. Concave function a twice continuously differentiable function f is concave if and only if 2 1 0 ii f x x w t w in the one variable case a function is concave if the derivative of the function is decreasing.
K, fx quasi concave functions and concave functions. This doesnt change the preference, whereas concave functions are easier to use than quasiconcave functions. Technical note on constant returns to scale production. Concavity implies that the production function is quasi concave 1. We oftenapproximatea production function using polynomials. For the case of a single input, a cubic production function would take the followingform. Thus, the c function represents the minimum cost necessary to produce output q with fixed input prices. Quasi concavity and maximization quasi concave functions have nice properties for maximization. This video introduces widely used concepts of quasiconcavity and quasiconvexity in economics through a mathematical as well as graphical explanation.
Answer to what is the difference between a quasi concave production function and a strictly concave production function. Near a local maximum in the interior of the domain of a function, the function must be concave. The bivariate normal joint density is quasiconcave. We now show that if in addition, the production function exhibits constant returns to scale crs then the production function must be concave. Note that just monotonic transformation of a concave function is not necessarily concave. Proving that a cobbdouglas function is concave if the sum of. Since 0 is upper semicontinuous, if and only 0 is lower semicontinuous, it also follows that every strictly quasi concave and upper semicontinuous function is quasi concave. I can see that both conclusions arise from an if and only if statement about strictly concave functions, however for some reason i am not able to derive this statement. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Strictly quasiconvex concave functions and duality in. If for many arbitrary constants the function is quasi concave then on the interval of a which is checked, the function is quasi concave. Rby f x p x if 0 apr 10, 2019 it turns out that a concave function is just a specific instance of a quasiconcave functionone without the bumps. With a production function that shows constant returns to scale homogeneous of degree 1, or linear homogeneous, c will be linear with fixed input prices. The cost function does not exist it there is no technical way to produce the output in question.
Fromnow on we will assume thatx is aconvex subset of rn. Continuity ofa concave function mayfailin the boundaryof x. If u represents the relation 2, then u represents the convexed relationk. C represents the minimum isocost line for any level of q. Concave andquasiconcave functions 1 concaveandconvexfunctions 1. This also means that if a monotonic transformation of f is concave, then f is concave.
There is a similar characterization for quasiconcavity. Pseudoconvex function is a strictly quasiconvex function. Quasiconcavity and maximization quasi concave functions have nice properties for maximization. Proving that a cobbdouglas function is concave if the sum. The negative of a quasiconvex function is said to be quasiconcave. Obara ucla preference and utility october 2, 2012 20 20. The probability density function of the normal distribution is quasiconcave but not concave. The definition says that a quasiconcave function of a single variable is strictly quasiconcave if its graph has no horizontal sections. Chapter 8 cost functions done university of tennessee. There is a similar characterization for quasi concavity. I if f is concave, then it is quasiconcave, so you might start by checking for concavity.
Strictly quasiconvex function does not imply quasiconvexity. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i. K, fx strictly quasi concave function may as well be strictly quasi convex. 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. Thus the concavity of a function is not ordinal, it is cardinal property. If someone knows of a result for convex functions that would obviously help as well. Answer to what is the difference between a quasiconcave production function and a strictly concave production function. Function q is said strictly quasiconvex 17, 7,33,8 as it. Any strictly increasing function is quasiconcave and quasiconvex check this. For example, if you prefer vanilla icecream to chocolate, you would assign greater utility to vanilla icecream than to the same quantity of.
Recall that a c2 function f is concave i d2fx is negative semide nite for all x2c. For x between a and c, the value of fx is negative, and for x between c and b, it is positive x a c b f x concave production function z input, f z output. Let 0 be strictly quasi convex strictly quasi concave on a convex set c c e. Quasiconcavity of sum of strictly increasing concave and. If a function is increasing, then is it both quasiconcave. C cv, w, q minimum total cost is a function of input prices and output quantity. Lk or qfk, l strictly quasi concave or k fl,q 0 strictly. The indifference curves of the quasiconcave closure of a quasiconvex function are intersections of r. Strictly quasiconvex function may not be strongly quasiconvex.
Technical note on constant returns to scale production functions. Johnson october 30, 2007 1 introduction utility refers to the perceived value of a good and utility theory spans mathematics, economics and psychology. 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. Show that the indirect utility function is quasiconvex in p. A function f is quasiconcave if hx gfx is a concave function for some strictly increasing function gfrom rby f x p x if 0 a c b f. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. Quasi concave utility functions get that name because quasi concavity is a weaker property than. Concavity implies that the production function is quasiconcave 1. Economic example suppose production function fx is concave and the cost function cx is convex. Suppose utility function u represents preferences t. Domv,w0, 2 or in the case of a single output cw, y min x wx. I if f is concave, then it is quasi concave, so you might start by checking for concavity.
A function f is quasi concave if hx gfx is a concave function for some strictly increasing function gfrom concave, then h must also be quasi concave. I if f is a monotonic transformation of a concave function, it is quasiconcave. X r is a utility representation of if, for every x,x. Convex preferences get that name because they make upper contour sets convex. So for equal increases of the single input and given fixed unit cost, say unit per unit, production increases less and less, while cost increases equally.
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