20, Feb 2002
Edward R. Morey
I.
Curvature Properties*
*Thanks to Laura Langhoff for initial layout, typing and graphics.
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I.
Curvature Properties
What do we mean by curvature properties? A mathematical function (or relation) can be described by a set of points in (N+1) dimensional space. That set of points will have a certain shape (form). For example, is a sphere of radius 1. The sphere has a certain shape: that is, it has a certain curvature. All functions that have the same basic shape are said to have the same curvature property. For example, all linear functions have a certain shape. The same is true of functions that are convex, concave, quasiconvex, quasiconcave, etc. What do we care about curvature properties? There are two basic reasons 1) Economic theory imposes curvature properties on mathematical functions.
For Example: Economic theory dictates that all utility functions are quasiconcave and dictates that all cost functions are concave in input prices. Therefore a cost function that is not concave in input prices is not a cost function. 2) If the objective function in our constrained min or max problem has certain curvature properties we can be sure that the critical point is in fact the point we have been looking for.
For Example: The first-order solution (K0, L0, 80) to the Lagrangian < = wL +rK - 8[y - f(K, L)] will be a min (rather than a max) if f(K, L) is increasing and quasiconcave in K and L. It will be a max (rather than a min) if f(K, L) is increasing and quasiconvex in (K, L). If we impose certain curvature properties on the objective function and constraint function, then there is no need to check second-order sufficient conditions for a constrained max or min (the imposed curvature properties will guarantee that the appropriate second order conditions are fulfilled).
Minimization & Maximization
Edward R. Morey - feb 20 2002
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Another Example Assume a consumer wants to max her utility u = u(x1, x2) subject to the budget constraint y = p1x1 ! p2x2 (note that the constraint function is linear). Find the stationary point x0, x0, and 80. If u(x1, x2) is increasing and quasiconcave in x1 and x2, x0 1 2 1 and x0 will be those quantities that max utility subject to the constraint. 2 However, if u(x1, x2) is increasing and quasiconvex in x1 and x2 then x0 and x0 will min utility 1 2 subject to the constraint.
Geometric Example Find the stationary combination of labor and capital that solves the production manager's problem, K0, L0. If f(K,L) is increasing and quasiconcave in K and L, the stationary point is demonstrated with the left graph. If f(K,L) is increasing and quasiconvex in K and L, the stationary point is demonstrated with the right graph.
In the first graphical example K0, L0 min costs in the second graphical example K0, L0 max costs Before we proceed to define some different curvature properties lets define the concept of a convex set. The concept of a convex set will prove useful in the definitions of different curvature properties for functions and for our intuitive understanding of those properties.
Minimization & Maximization
Edward R. Morey - feb 20 2002
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Definition:
A set of points S in N-dimensional space is convex if for every two points x1 and x2 1 2 belonging to S, where x1 = (x1, x1, .., xN) and x2 = (x2, x2..., xN), the straight line 1 2 1 2 segment joining the two points also belongs to S. That is, a set S is convex if and only if for every x1, x2 0 S, and scalar 8 such that 0 # 8 # 1 we have 8x1 + (1!8) x2 0 S.
Minimization & Maximization
Edward R. Morey - feb 20 2002
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Examples of convex sets in two-dimensional space.
Examples of non-convex sets in two-dimensional space.
In three dimensions a sphere or 3 dimensional rectangle floating motionless in space is a convex set, a donut is not a convex set. Another useful concept is that of a strictly convex set. Definition: A set of points S in N-dimensional space is strictly convex if for every two points x1 and x2 belonging to S such that x1 ... x2, the straight line segment joining the two points belongs to S but does not belong to the boundary of S (i.e. all the points on the interior of the straight line must be strictly in the interior of S): that is, a set S is strictly convex if and only if for every x1, x2 0 S, and scalar 8 such that 0 < 8 < 1 we have 8x1 + (1 - 8)x2 0 the interior of S.
Examples of strictly convex sets in two-dimensional space.
Examples of sets in two-dimensional space that are convex but not strictly convex.
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Edward R. Morey - feb 20 2002
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Note that:
ALL strictly convex sets are convex but ALL convex sets are not strictly convex.
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Now Let's Formally Define Some Curvature Properties As we proceed, I will try to describe what the properties imply about the shape of different economic functions such as the production function and the utility function. Definition: Concave and Convex Functions A function f(x) is a concave function over a convex set S if and only if x1 0 S, x2 0 S, 0 # 8 # 1 implies f(8x1 + (1-8)x2) $ 8f(x1) + (1-8)f(x2), convex if and only if f(8x1 + (1!8)x2) # 8f(x1) + (1-8) f(x2) Geometric Example of concavity and convexity when N=1
assume S / {x : x > 0} N = 1 implies x is a scalar
Minimization & Maximization
Edward R. Morey - feb 20 2002
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A Geometric Example of a Concave Function when N=2.
The graph of a concave function (n = 1) always lies on or below its tangent line. In higher dimensions (N > 1), the graph of a concave function always lies on or below its tangent hyperplane. Things to note about concave functions. a) b) If y = f(x) is concave, then the set Lu(y) / {x : f(x) $ y} is a convex set (that is, all the upper level sets of a concave function are convex sets) However, the fact that Lu(y) is a convex set for all y does not imply that y = f(x) is a concave function.
Example: what does concavity mean in terms of the utility function u(x1, x2)? If a utility function u(x1, x2) is concave and increasing in x1 and x2, this implies i) The upper level set L(u) / {(x1, x2) in R+ : u(x1, x2) $ u} is a convex, but not necessarily 2 strictly convex, set, and the indifference curves don't look like1
In the graph, the continuous line is the indifference curve. Note that it is the boundary of the upperlevel set. The shading indicates the continuation of the upper-level set.
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Edward R. Morey - feb 20 2002
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but could look like
Fat indifference curves are disallowed by the assumption that u(x1, x2) is 8 in x. ii) iii) In terms of marginal utilities (if you believe in such a thing) they are all strictly positive but not necessarily declining over there entire range. Note the restriction that u(x1, x2) is concave is sufficient, but not necessary, for all the upper level sets to be convex sets.
The graph of a convex function (n = 1) always lies on or above its tangent line. In higher dimensions (N > 1), the graph of a convex function always lies on or above its tangent hyperplane. Things to note about convex functions. a) b) If y = f(x) is convex, then the set Ll(y) / {x : f(x) # y} is a convex set (that is, the lower level sets of all convex functions are convex sets) However, the fact that Ll(y) is a convex set for all y does not imply that y = f(x) is a convex function.
Minimization & Maximization
Edward R. Morey - feb 20 2002
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Definition: Strictly Concave and Strictly Convex Functions A function f(x) is a strictly concave function over a convex set S if and only if x1 0 S, x2 0 S, 0 < 8 < 1 implies f(8x1 + (1-8)x2) > 8f(x1) + (1-8)f(x2), and strictly convex if and only if f(8x1 + (1!8)x2) < 8f(x1) + (1-8) f(x2) Things to note about strictly concave functions. a) b) If y = f(x) is strictly concave, then the set / Lu(y) {x : f(x) $ y} is a strictly convex set (i.e. all the upper level sets of a strictly concave function are strictly convex sets) However, the fact that Lu(y) is a strictly convex set for all y does not imply that y = f(x) is a strictly concave function.
Things to note about strictly convex functions. a) b) If y = f(x) is strictly convex, then the set Ll(y) / {x : f(x) # y} is a strictly convex set (i.e. the lower level sets of all convex functions are convex sets) However, the fact that Ll(y) is a strictly convex set for all y does not imply that y = f(x) is a strictly convex function.
Can the production function,
x = f ( k , l ) be convex in the input levels? Strictly convex? x = f ( k , l ) is strictly convex in k and l? How will a firm
What will the isoquants look like if with such inputs produce x?
Minimization & Maximization
Edward R. Morey - feb 20 2002
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Assume a production function x = f(K,L) where f(K,L) is twice differentiable in K and L. Further assume f(K,L) is nondecreasing in K and L, i.e., and
a. What does it mean to say this production function is concave in K and L? Specifically, what does it mean in terms of the marginal products of labor and capital, and what does it mean in terms of the shapes of the isoquants? b. What does it mean to say this production function is strictly concave? Specifically, what does it mean in terms of the marginal products of labor and capital? What do we call this property in principles classes? c. What does strict concavity mean in terms of the shapes of the isoquants? d. Would you expect all production functions to be concave in K and L? Yes or No and explain.
Minimization & Maximization
Edward R. Morey - feb 20 2002
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Now let's discuss the weaker curvature properties quasiconcavity and quasiconvexity. Definition: Quasiconcave and Quasiconvex functions
A function f(x) is a quasiconcave function over a convex set S if and only if x1 0 S, x2 0 S, f(x1) # f(x2), 0 # 8 # 1, implies f(x1) # f(8x1 + (1!8) x2) quasiconvex if f(x2) $ f(8x1 + (1!8) x2) Basically, the definition says that a function is quasiconcave if and only if all of its upper level sets are convex sets and a function is quasiconvex if, and only if, all of its lower level sets are convex sets. All concave functions are quasiconcave but all quasiconcave functions are not concave.
Minimization & Maximization
Edward R. Morey - feb 20 2002
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A Graphical Example of Quasiconcavity and Quasiconvexity when N=1. Assume that
This function is quasiconcave but not concave. We can see that it violates the conditions for concavity and convexity. in the range x = 2 to 10 it is not concave in the range x = 10 to 15 it is not convex However, for any two points belonging to S, the conditions for quasiconcavity will be fulfilled. For example, if x1 = 2 and x2 = 10 then f(2) # f(10) and all the values of f(x) 2 # x # 10 are greater than or equal to f(2). Check that all the upper level sets of the graphed function, f(x), are convex sets. The upper level sets are L(y) / {x : f(x) $ y} if y = 0 the upper level set consists of all nonnegative values of x (a convex set) if y = 6 the upper level set consists of all values of x $ 5 (again obviously a convex set) all of the upper level sets of this function will be convex sets
Is this function quasiconvex? Yes...Why?
Now assume that
Is this function quasiconcave?
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Edward R. Morey - feb 20 2002
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How about this one?
Or this one?
Or this one?
a), b), and d) are quasiconcave. c) is not quasiconcave. Why? Which ones are quasiconvex? c) and d) are quasiconvex. a) and b) are not quasiconvex. Why?
A Geometric Example of Quasiconcavity and Quasiconvexity when N=2.
This is supposed to look like a bell. It's quasiconcave but not concave. The upper level sets of this function are all convex sets.
Minimization & Maximization
Edward R. Morey - feb 20 2002
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If all of the upper level sets of a function f(x1, x2) are of the following sort, f(x1, x2) is quasiconcave.
Why e) and f)? If all of the upper level sets of f(x1, x2) are of the following sort f(x1, x2) is not quasiconcave.
So, with all this in mind, what are the implications of assuming that the utility function u = u(x1, x2) is increasing and quasiconcave in x1 and x2. i) ii) iii) iv) The assumption of quasiconcavity alone means all the bundles that provide at least u utils are a convex set (but not necessarily a strictly convex set). Quasiconcavity alone doesn't eliminate the possibility of thick indifference curve. The assumption of that u(@) is increasing in x1 and x2 eliminates the possibility of thick indifference curves. The assumptions in no way guarantee that the consumer will consume so...