Question

As we have seen in many places, the general Cobb-Douglas production function for two inputs is given by $$q=f(k, l)=A k^{\alpha} l^{\beta}$$ where \(0<\alpha<1\) and \(0<\beta<1 .\) For this production function: a. Show that \(f_{k}>0, f_{1}>0, f_{k k}<0, f_{l l}<0,\) and \(f_{k l}=f_{l k}>0\) b. Show that \(e_{q, k}=\alpha\) and \(e_{q, l}=\beta\) c. In footnote \(5,\) we defined the scale elasticity as$$e_{q, t}=\frac{\partial f(t k, t l)}{\partial t} \cdot \frac{t}{f(t k, t l)}$$ where the expression is to be evaluated at \(t=1 .\) Show that, for this Cobb-Douglas function, \(e_{q, t}=\alpha+\beta .\) Hence in this case the scale elasticity and the returns to scale of the production function agree (for more on this concept see Problem 9.9 ). d. Show that this function is quasi-concave. e. Show that the function is concave for \(\alpha+\beta \leq 1\) but not concave for \(\alpha+\beta>1\)

Short answer

Question: Prove that the Cobb-Douglas production function is quasi-concave and determine the conditions under which it is concave. Answer: The Cobb-Douglas production function is quasi-concave because its level curves are convex due to its multiplicative nature, and its value at a convex combination of two points is greater than or equal to the minimum of the values at those points. The function is concave when the sum of its exponents, α and β, is less than or equal to 1, which is indicated by the non-negative determinant of its Hessian matrix.

Step-by-step solution

Partial Derivatives

Let's calculate the first and second order partial derivatives: $$f_k = \frac{\partial f(k,l)}{\partial k} = A\alpha k^{\alpha -1}l^{\beta}$$ $$f_l = \frac{\partial f(k,l)}{\partial l} = A\beta k^{\alpha}l^{\beta -1}$$ $$f_{kk} = \frac{\partial^2 f(k,l)}{\partial k^2} = A\alpha(\alpha-1)k^{\alpha -2}l^{\beta}$$ $$f_{ll} = \frac{\partial^2 f(k,l)}{\partial l^2} = A\beta(\beta-1)k^{\alpha}l^{\beta -2}$$ $$f_{kl} = f_{lk} = \frac{\partial^2 f(k,l)}{\partial k \partial l} = A\alpha\beta k^{\alpha-1}l^{\beta-1}$$ As \(0 < \alpha < 1\) and \(0 < \beta < 1\), we have: \(f_{k}>0, f_{l}>0, f_{kk}<0, f_{ll}<0,\) and \(f_{kl}=f_{lk}>0.\) #b.#

Output Elasticities

To find the elasticity of output with respect to capital and labor inputs, we will use the following formula for elasticity: $$e_{q,x} = \frac{\partial q}{\partial x} \cdot \frac{x}{q}$$ For capital (k), we have: $$e_{q, k} = f_k \cdot \frac{k}{f(k, l)} = \frac{A\alpha k^{\alpha -1}l^{\beta}}{A k^{\alpha} l^{\beta}} = \alpha$$ For labor (l), we have: $$e_{q, l} = f_l \cdot \frac{l}{f(k, l)} = \frac{A\beta k^{\alpha}l^{\beta -1}}{A k^{\alpha} l^{\beta}} = \beta$$ #c.#

Scale Elasticity

Using the definition of scale elasticity: $$e_{q, t} = \frac{\partial f(tk, tl)}{\partial t} \cdot \frac{t}{f(tk, tl)}$$ Let's first find the partial derivative of \(f(tk, tl)\) with respect to t: $$\frac{\partial f(tk, tl)}{\partial t} = A\alpha(tk)^{\alpha-1}(tl)^{\beta}k + A\beta(tk)^{\alpha}(tl)^{\beta-1}l$$ Now we substitute t=1 and simplify: $$e_{q, t} = \frac{A\alpha k^{\alpha} l^{\beta} + A\beta k^{\alpha} l^{\beta}}{A k^{\alpha} l^{\beta}} = \alpha + \beta$$ Hence, the scale elasticity and the returns to scale of the production function agree. #d.#

Quasi-Concavity

A function is quasi-concave if for any points x and y, the value of the function at a convex combination of x and y is greater than or equal to the minimum of the values of the function at x and y: $$f(\lambda x + (1-\lambda) y) \geq \min(f(x), f(y))$$ The Cobb-Douglas function is multiplicative, which implies that its level curves are convex given its exponents are positive: $$f(\lambda k + (1-\lambda) k', \lambda l + (1-\lambda) l') = A[(\lambda k + (1-\lambda) k')^{\alpha}(\lambda l + (1-\lambda) l')^{\beta}] \geq \min(f(k, l), f(k', l'))$$ Therefore, the Cobb-Douglas function is quasi-concave. #e.#

Concavity and Convexity

A function is concave if its second partial derivatives are negative (or the Hessian matrix is negative semi-definite). We already found that \(f_{kk}<0\) and \(f_{ll}<0\). Now we examine the determinant of the Hessian matrix, represented by \(D\): $$D = f_{kk}f_{ll} - f_{kl}^2 = A^2\alpha(\alpha-1)\beta(\beta-1)k^{\alpha -2}l^{\beta-2} - A^2\alpha^2\beta^2k^{2(\alpha -1)}l^{2(\beta -1)}$$ If \(\alpha + \beta \leq 1\), then: $$D = A^2\alpha(\alpha-1)\beta(\beta-1)k^{\alpha -2}l^{\beta-2}(1 - \alpha\beta) \geq 0$$ When \(\alpha + \beta \leq 1\), the function is concave. However, if \(\alpha + \beta > 1\), the determinant of the Hessian matrix becomes negative, indicating that the function is not concave in this scenario.

Key concepts

Microeconomic Theory

Microeconomic theory is a branch of economics that studies how individuals, households, and firms make decisions to allocate limited resources, typically in markets where goods or services are being bought and sold. The Cobb-Douglas production function is a popular model within microeconomic theory that describes the relationship between two or more inputs—commonly labor and capital—and the quantity of output produced. It's a fundamental concept to understand the nature of production and how different factors of production contribute to the overall output.

Partial Derivatives

In the context of the Cobb-Douglas production function, partial derivatives are crucial for measuring how the output responds to changes in either input, while the other input is held constant. By taking the first and second-order partial derivatives of the production function with respect to capital and labor, economists and students can analyze the marginal effects on production. The first-order partial derivatives indicate the direction of change in output with an infinitesimal increase in capital or labor, and the second-order derivatives provide insights on the curvature of the production function and whether it is increasing at a decreasing rate.

Output Elasticities

Output elasticities measure the responsiveness of output to a change in inputs. In the Cobb-Douglas production function, the output elasticity with respect to an input like capital or labor is the exponent on that variable. This concept helps in understanding how a percentage change in an input leads to a percentage change in output. For example, if the elasticity of output with respect to capital is 0.3, this means that a 1% increase in capital will increase output by 0.3%. These elasticities are used in resource allocation decisions and in the analysis of production efficiency.

Scale Elasticity

Scale elasticity refers to the responsiveness of output to a proportional change in all inputs. It is a measure of returns to scale. If the scale elasticity is equal to one, the production function exhibits constant returns to scale. If it's greater than one, there are increasing returns to scale, and if less than one, decreasing returns to scale. For the Cobb-Douglas production function, the scale elasticity is the sum of the exponents on the inputs, reflecting how output changes with a proportional change in both labor and capital.

Quasi-Concavity

Quasi-concavity is a weaker form of concavity which guarantees that the level sets of the function are convex. For a production function, quasi-concavity implies that by mixing different combinations of inputs, producers can achieve at least as much output as they could by using those inputs in isolation. The Cobb-Douglas production function is quasi-concave, indicating that it adheres to the principle of diminishing marginal rates of technical substitution. This property is essential for analyzing consumer preferences and firm production behaviors in microeconomics.