Question

Use bordered determinants to check the following functions for quasiconcavity and quasiconvexity: (d) \(z=-x^{2}-y^{2} \quad(x, y>0)\) \((b): z=-(x+1)^{2}-(y+2)^{2} \quad(x, y>0)\)

Short answer

Both functions are quasiconcave, not quasiconvex.

Step-by-step solution

Introduction to Bordered Determinants

To check for quasiconcavity and quasiconvexity, we can use bordered Hessian determinants. The given function's second-order partial derivatives are needed to form this matrix.

Calculate First-order Partial Derivatives

For (d), calculate the first-order partial derivatives: \[ \frac{\partial z}{\partial x} = -2x, \quad \frac{\partial z}{\partial y} = -2y. \] For (b), calculate the first-order partial derivatives: \[ \frac{\partial z}{\partial x} = -2(x+1), \quad \frac{\partial z}{\partial y} = -2(y+2). \]

Calculate Second-order Partial Derivatives

For (d), calculate the second-order partial derivatives: \[ \frac{\partial^2 z}{\partial x^2} = -2, \quad \frac{\partial^2 z}{\partial y^2} = -2, \quad \frac{\partial^2 z}{\partial x \partial y} = 0. \] For (b), calculate the same: \[ \frac{\partial^2 z}{\partial x^2} = -2, \quad \frac{\partial^2 z}{\partial y^2} = -2, \quad \frac{\partial^2 z}{\partial x \partial y} = 0. \]

Form the Bordered Hessian Matrix

The bordered Hessian for function (d) is: \[B_h = \begin{bmatrix} 0 & -2x & -2y \ -2x & -2 & 0 \ -2y & 0 & -2 \end{bmatrix} \]And similarly for (b).

Calculate Determinants of Bordered Hessian

Calculate the determinant of the bordered Hessian matrices. Both matrices are exactly similar, hence:\[\text{Det}(B_h) = \begin{vmatrix} 0 & -2x & -2y \ -2x & -2 & 0 \ -2y & 0 & -2 \end{vmatrix} = -8(x^2 + y^2) \]

Analyze Determinant Signs

For quasiconcavity, we check if all leading principal minors of the bordered Hessian change sign, starting with a positive sign. In our case, as \(-8(x^2 + y^2) < 0\), this indicates quasiconcavity. The function is not quasiconvex as the sign does not fit the criteria.

Key concepts

Quasiconcavity

Quasiconcavity is an important concept in optimization and economics. It's about functions that are somewhat like concave functions, but with a bit more flexibility. The key idea is how the function's value changes across its domain. For a function to be quasiconcave, it needs to satisfy the condition that if you pick any two points within its domain, the function's value at any point on the line segment joining these two points should not exceed the minimum value at these points.

When using bordered determinants to determine quasiconcavity, the Hessian matrix comes into play. You primarily look at the signs of the leading principal minors. Starting with a positive sign, if these signs alternate (positive, negative, positive, etc.), the function exhibits quasiconcavity. This helps in identifying functions that curve downwards even if they aren't perfectly concave. This property is useful in economics where maximizing utility functions or profit functions that aren't perfectly concave is common.

Quasiconvexity

Quasiconvexity is somewhat the opposite of quasiconcavity. A function is quasiconvex if along any line segment between two points in the domain, the maximum value of the function at these points is always at least as great as the function's value at any point on the segment between these points.

In mathematical terms, the concept is a bit trickier because it reflects a property of non-linear functions or curves that do not curve upwards entirely, unlike convex functions. If the bordered Hessian determinant reveals a consistent pattern of signs starting with a negative sign (negative, positive, negative, etc.), the function may turn out to be quasiconvex. While quasiconcavity is about ensuring maxima along lines, quasiconvexity focuses on the smooth progression towards them, a useful property in finding constraints in optimization.

Hessian Matrix

The Hessian Matrix is a fascinating tool in the realm of calculus and optimization, particularly when dealing with functions of multiple variables. Basically, it's a square matrix consisting of second-order partial derivatives. In simpler terms, it shows how the curvature of a function behaves around a particular point.

For any function of two variables, like in the given exercise, the Hessian matrix would generally have elements \( \frac{\partial^2 z}{\partial x^2} \), \( \frac{\partial^2 z}{\partial y^2} \), and \( \frac{\partial^2 z}{\partial x \partial y} \). If you add a row and column involving the first derivatives, you get a bordered Hessian matrix, which is useful for determining both quasiconcavity and quasiconvexity.

The matrix essentially helps in understanding how the surface of the function curves, and its determinant tells us a lot about the nature of the function's local extremes. Whether a function is likely to be concave or convex in nature, how the function responds to variable changes, and its shape—all of this can be inferred using the Hessian Matrix.