Question

One of the most important functions we will encounter in this book is the Cobb-Douglas function: \\[ y=\left(x_{1}\right)^{\alpha}\left(x_{2}\right)^{\beta} \\] where \(\alpha\) and \(\beta\) are positive constants that are each less than 1 a. Show that this function is quasi-concave using a "brute force" method by applying Equation 2.114 b. Show that the Cobb-Douglas function is quasi-concave by showing that any contour line of the form \(y=c\) (where \(c\) is any positive constant is convex and therefore that the set of points for which \(y>c\) is a convex set. c. Show that if \(\alpha+\beta>1\) then the Cobb-Douglas function is not concave (thereby illustrating again that not all quasiconcave functions are concave). Note: The Cobb-Douglas function is discussed further in the Extensions to this chapter.

Short answer

**Answer**: Yes, the Cobb-Douglas function is quasi-concave. However, it is not concave when the condition α+β > 1 holds.

Step-by-step solution

a. Show that the Cobb-Douglas function is quasi-concave using the "brute force" method by applying Equation 2.114.

To show that the Cobb-Douglas function is quasi-concave, we can use the definition of a quasi-concave function given by Equation 2.114, which states that if for any two points \(x_1, x_2 \in X\) and any \(\lambda \in [0, 1]\), the following inequality holds: \\[ f(\lambda x_1 + (1-\lambda) x_2) \geq \min[f(x_1), f(x_2)] \\] We have the Cobb-Douglas function given by \(y=f(x_1,x_2)=(x_1)^{\alpha}(x_2)^{\beta}\), where \(\alpha\) and \(\beta\) are positive constants that are each less than 1. Let's choose two points \(x_1 = (a_1, a_2)\) and \(x_2=(b_1, b_2)\) and consider \(\lambda \in [0, 1]\). We then have: \(f(x_1) = (a_1)^{\alpha}(a_2)^{\beta}\) and \(f(x_2) = (b_1)^{\alpha}(b_2)^{\beta}\) Now, we need to evaluate the function at the point given by \(\lambda x_1 + (1-\lambda)x_2=(a_1\lambda+b_1(1-\lambda),a_2\lambda+b_2(1-\lambda))\). We have: \(f(\lambda x_1+(1-\lambda) x_2)=((a_1\lambda+b_1(1-\lambda))^{\alpha}((a_2\lambda+b_2(1-\lambda))^{\beta}\). To satisfy the inequality of quasi-concave functions, we need to prove that: \(((a_1\lambda+b_1(1-\lambda))^{\alpha}((a_2\lambda+b_2(1-\lambda))^{\beta} \geq \min[f(x_1), f(x_2)]\) Using the fact that \(A^{\lambda}B^{1-\lambda} \geq \lambda A + (1-\lambda)B\) for positive A, B and \(\lambda \in [0, 1]\) (which can be proved using Jensen's inequality), we get: \(((a_1\lambda+b_1(1-\lambda))^{\alpha}((a_2\lambda+b_2(1-\lambda))^{\beta} \geq (\lambda a_1^{\alpha} + (1-\lambda) b_1^{\alpha})(\lambda a_2^{\beta} + (1-\lambda) b_2^{\beta})\) Now, notice that: \((\lambda a_1^{\alpha} + (1-\lambda) b_1^{\alpha})(\lambda a_2^{\beta} + (1-\lambda) b_2^{\beta}) \geq \min[f(x_1), f(x_2)]\) Therefore, the Cobb-Douglas function is quasi-concave.

b. Show that the Cobb-Douglas function is quasi-concave by showing that any contour line of the form y=c is convex and therefore the set of points for which y>c is a convex set.

This approach requires showing that the contour lines of the Cobb-Douglas function, given by \(y=c\), are convex and the set of points for which \(y>c\) is a convex set. Let's define the contour line of the Cobb-Douglas function as: \(y = (x_1)^{\alpha}(x_2)^{\beta} = c\) To show that this contour line is convex, we can use the Hessian matrix to check the property of the function. The Hessian matrix is given by: \(H = \begin{bmatrix} \frac{\partial^2y}{\partial x_1^2} & \frac{\partial^2y}{\partial x_1\partial x_2} \\ \frac{\partial^2y}{\partial x_2\partial x_1} & \frac{\partial^2y}{\partial x_2^2} \end{bmatrix}\) Taking the necessary second-order partial derivatives, we get: \(\frac{\partial^2y}{\partial x_1^2} = \alpha(\alpha-1)(x_1)^{\alpha-2}(x_2)^{\beta}\), \(\frac{\partial^2y}{\partial x_2^2} = \beta(\beta-1)(x_1)^{\alpha}(x_2)^{\beta-2}\), \(\frac{\partial^2y}{\partial x_1\partial x_2}=\frac{\partial^2y}{\partial x_2\partial x_1}=\alpha\beta(x_1)^{\alpha-1}(x_2)^{\beta-1}\). Thus, the Hessian matrix is: \(H = \begin{bmatrix} \alpha(\alpha-1)(x_1)^{\alpha-2}(x_2)^{\beta} & \alpha\beta(x_1)^{\alpha-1}(x_2)^{\beta-1} \\ \alpha\beta(x_1)^{\alpha-1}(x_2)^{\beta-1} & \beta(\beta-1)(x_1)^{\alpha}(x_2)^{\beta-2} \end{bmatrix}\) The function is convex if the Hessian matrix is positive semidefinite, meaning that its determinant is non-negative: \(\det(H) = \alpha^2\beta^2(x_1)^{2\alpha-2}(x_2)^{2\beta-2}-\alpha\beta(\alpha-1)(\beta-1)(x_1)^{2\alpha-2}(x_2)^{2\beta-2} \geq 0\) Notice that \(\alpha\beta(\alpha-1)(\beta-1) \leq \alpha^2\beta^2\) because \(\alpha, \beta < 1\), so the determinant of the Hessian is non-negative, and therefore, the Cobb-Douglas function is quasi-concave.

c. Show that if \(\alpha+\beta>1\) then the Cobb-Douglas function is not concave.

A function is concave if the negative of the function is convex. Let's check the conditions for the Cobb-Douglas function to be concave by considering the negative of this function: \(-y = -(x_1)^{\alpha}(x_2)^{\beta}\) Using the Hessian matrix of the negative function, we have: \(\frac{\partial^2(-y)}{\partial x_1^2} = \alpha(\alpha-1)(x_1)^{\alpha-2}(x_2)^{\beta}\), \(\frac{\partial^2(-y)}{\partial x_2^2} = \beta(\beta-1)(x_1)^{\alpha}(x_2)^{\beta-2}\), \(\frac{\partial^2(-y)}{\partial x_1\partial x_2}=\frac{\partial^2(-y)}{\partial x_2\partial x_1}=-\alpha\beta(x_1)^{\alpha-1}(x_2)^{\beta-1}\). The determinant of the Hessian matrix for the negative function is: \(\det(-H) = -\alpha^2\beta^2(x_1)^{2\alpha-2}(x_2)^{2\beta-2}+\alpha\beta(\alpha-1)(\beta-1)(x_1)^{2\alpha-2}(x_2)^{2\beta-2}\) Now, if \(\alpha+\beta>1\), then \(\alpha-1\) and \(\beta-1\) are positive values. Therefore, \(\det(-H) > 0\) for \(\alpha+\beta>1\), which means that the negative of the Cobb-Douglas function is not convex, and therefore, the original function is not concave under this condition.

Key concepts

Quasi-Concave Function

The concept of a quasi-concave function is particularly significant in economics, where it describes a preference pattern that is not necessarily equivalent to the standard notion of concavity. In simple terms, a function is quasi-concave if, for any two points within its domain, the function value at any linear combination of these points is at least as great as the minimum value of the function at these two points. This principle can be mathematically stated as:

For any two points \(x_1, x_2\) in the domain and any \(\lambda\) in the interval \([0, 1]\), the inequality\[ f(\lambda x_1 + (1-\lambda) x_2) \geq \min[f(x_1), f(x_2)]\]must hold. Intuitively, the set of all points where the function takes on a value greater than a certain level forms a convex set.

The Cobb-Douglas production function, given by \(y = (x_{1})^{\alpha}(x_{2})^{\beta}\), where \(\alpha\) and \(\beta\) are positive constants less than 1, exhibits quasi-concave properties. This means that within the context of production, the outputs from a set of inputs will at least maintain a minimum output level when inputs are combined in different proportions.

This characteristic of the Cobb-Douglas function reflects the economic principle that consumers or producers can adapt input combinations without falling below a certain utility or production level, reflecting flexibility and adaptability in economic decisions.

Hessian Matrix

In multivariable calculus, the Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function. It is instrumental in examining the local curvature of a function, especially with regard to optimization problems. Economists and mathematicians use the Hessian matrix to study the concavity or convexity of functions.

For a function \(y = f(x_1, x_2)\), the Hessian matrix is defined as:\[ H = \begin{bmatrix} \frac{\partial^2y}{\partial x_1^2} & \frac{\partial^2y}{\partial x_1\partial x_2} \ \frac{\partial^2y}{\partial x_2\partial x_1} & \frac{\partial^2y}{\partial x_2^2} \end{bmatrix}\]A function is considered convex if its Hessian matrix is positive semidefinite across its domain, which means that any leading principal minor of the Hessian is non-negative. In the context of the Cobb-Douglas function, the Hessian matrix helps confirm its quasi-concavity by indicating that its contour lines are convex sets.

Understanding the Hessian matrix is essential for various applications across economics, such as in consumer choice and production theory, where it is often used to verify the curvature properties of utility functions and production functions—information crucial for determining optimal behaviors and outcomes in microeconomic theory.

Concavity in Microeconomic Theory

Concavity plays a pivotal role in microeconomic theory, particularly in the analysis of consumer preferences and production technologies. A concave function indicates that as one adds more of an input, the incremental gain (or marginal return) from that input decreases, which aligns with the principle of diminishing returns.

In a concave function, the line segment between any two points on the function's graph lies below or on the graph, leading to a graphic representation that curves downwards. This property is crucial when examining production functions like the Cobb-Douglas function. However, when the sum of the exponents in a Cobb-Douglas function, such as \(\alpha + \beta\), exceeds 1, the function exhibits increasing returns to scale and is no longer concave. This scenario demonstrates that not all quasi-concave functions are concave, as quasi-concavity is a weaker condition that allows for a constant or increasing marginal rate of substitution or production.

In the realm of economics, concave utility and production functions represent realistic and rational behavior, underpinning many models and theories. The investigation into concavity of functions like Cobb-Douglas provides insights into resource allocation, cost minimization, and profit maximization strategies, all fundamental aspects of economic analysis and decision-making.