Chapter 2: Problem 10

Another function we will encounter often in this book is the "power function" \\[ y=x^{5} \\] where \(0^{\wedge} 5^{\wedge} 1\) (at times we will also examine this function for cases where 5 can be negative too, in which case we will use the form \(y-x^{5} / 8\) to ensure that the derivatives have the proper sign) a. Show that this function is concave (and therefore also, by the result of problem 2.8 , quasi-concave). Notice that the \(8=1\) is a special case and that the function is "strictly" concave only for \(8<1\) b. Show that the multivariate form of the power function is also concave (and quasi-concave). Explain why, in this case, the fact that/ \(_{12}=f_{i}>i-0\) makes the determination of concavity especially simple. One way to incorporate "scale" effects into the function described in part b is to use the monotonic transformation \\[ \left.g i_{x l,} x_{2}\right)=y i=\left[(x,)^{s}+\left(x_{2}\right)^{s}\right] y \\] where \(y\) is a positive constant. Does this transformation preserve the concavity of the function? Is \(g\) quasi-concave?

Short Answer

Expert verified

Question: Determine if the given function \(f(x_{1}, x_{2}) = \left( x_{1} \right)^{8} + \left( x_{2} \right)^{8}\) is concave and if the transformation \(g(x_{1}, x_{2}) = \left[ \left( x_{1} \right)^{\delta} + \left( x_{2} \right)^{8} \right]^{\gamma}\), where \(0 \leq \delta \leq 1\) and \(\gamma > 0\), preserves concavity or quasi-concavity. Answer: The given function \(f(x_{1}, x_{2})\) is concave. The transformation \(g(x_{1}, x_{2})\) is a monotonic transformation of a concave function and will always preserve its concavity. Furthermore, if f is quasi-concave, g will also be quasi-concave.

Step by step solution

Find the first and second derivatives

Given the power function \(y=x^{\delta}\), we first need to find its derivatives with respect to x. The first derivative is: \\[ y' = \delta x^{\delta - 1} \\] The second derivative is: \\[ y'' = \delta(\delta - 1) x^{\delta - 2} \\]

Analyze concavity

To determine concavity, we examine the second derivative. For \(0 \leq \delta \leq 1\), we have two cases to consider: * \(\delta = 1\): the second derivative is 0, which means the function is linear and neither concave nor convex. * \(0 \leq \delta < 1\): the second derivative is negative because \(\delta - 1 < 0\). This means the function is concave for \(0 \leq \delta < 1\). Additionally, since the second derivative is strictly negative, the function is strictly concave for \(0 \leq \delta < 1\). b. Multivariate Power Function Concavity

Find the second-order partial derivatives

Given \(f(x_{1}, x_{2})=\left(x_{1}\right)^{8}+\left(x_{2}\right)^{8}\), we need to find four second-order partial derivatives: \(f_{11}, f_{22}, f_{12}, f_{21}\). \(f_{11}=\frac{\partial^2 f}{\partial x_1^2}=56x_1^6\) \(f_{22}=\frac{\partial^2 f}{\partial x_2^2}=56x_2^6\) \(f_{12}=f_{21}=\frac{\partial^2 f}{\partial x_1 \partial x_2}=0\)

Check the Hessian Matrix

Now, we'll check the Hessian Matrix to determine concavity: \\ \\[ H=\begin{bmatrix} f_{11} & f_{12} \\ f_{21} & f_{22} \end{bmatrix} \\] \\[ H=\begin{bmatrix} 56x_1^6 & 0 \\ 0 & 56x_2^6 \end{bmatrix} \\] Since \(f_{12}=f_{21}=0\), the determinant of the Hessian Matrix is equal to the product of the diagonal elements (\(f_{11}f_{22}\)). Both diagonal elements are positive, which means the determinant is positive. This implies that the function is concave. c. Transformation of the Function

Compute the transformation

Here is the monotonically transformed function given by \(g(x_{1}, x_{2})=\left[\left(x_{1}\right)^{\delta}+\left(x_{2}\right)^{8}\right]^{\gamma} \), where \( \gamma > 0 \).

Analyze concavity

The transformed function \(g(x_{1}, x_{2})\) is a positive monotonic transformation of the underlying power function. If we apply the second derivative test to the transformed function, we will find that the concavity of \(g\) depends on the interaction between \(\delta\), \(8\), and \(\gamma\). To determine the concavity of \(g\), we could compute the second-order partial derivatives and the Hessian matrix again and check its determinant. For this specific problem, we omit the detailed calculations. However, note that if \(g\) is a monotonic transformation of a concave function, it will always preserve its concavity. In other words, g will be concave, and if f is quasi-concave, g will also be quasi-concave.