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Chapter 9: Problem 11

Suppose that a production function \(f\left(x_{1}, x_{2}, \ldots, x_{n}\right)\) is homogeneous of degree \(k\). Euler's theorem shows that \(\sum_{i} x_{i} f_{i}=k f,\) and this fact can be used to show that the partial derivatives of \(f\) are homogeneous of degree \(k-1\) a. Prove that \(\sum_{i=1}^{n} \sum_{j=1}^{n} x_{i} x_{j} f_{i j}=k(k-1) f\) b. In the case of \(n=2\) and \(k=1\), what kind of restrictions does the result of part (a) impose on the second-order partial derivative \(f_{12} ?\) How do your conclusions change when \(k>1\) or \(k<1 ?\) c. How would the results of part (b) be generalized to a production function with any number of inputs? d. What are the implications of this problem for the parameters of the multivariable Cobb-Douglas production function \(f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\prod_{i=1}^{n} x_{i}^{\alpha_{i}}\) for \(\alpha_{i} \geq 0 ?\)

Short Answer

Expert verified
Question: Prove that if a production function f(x1, x2, ..., xn) is homogeneous of degree k, then: $$ \sum_{j=1}^{n} \sum_{i=1}^{n} x_{i} x_{j} f_{ij} = k(k-1) f $$ where \(f_{ij}\) is the second derivative of f with respect to \(x_i\) and \(x_j\). Additionally, discuss the case when n = 2 and k = 1. Finally, analyze the implications for the multivariable Cobb-Douglas production function.

Step by step solution

01

Deriving the Partial Derivatives

To prove the statement in part (a), we first need to find the mixed partial derivatives for the function \(f\). The second derivative of \(f\) with respect to \(x_i\) and \(x_j\) is denoted by \(f_{ij}\). This can be calculated as follows: $$ f_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j} $$
02

Using Euler's Theorem

Now, let's use Euler's theorem \(\sum_{i} x_{i}f_{i}=k f\) for the homogeneous function \(f\) of degree \(k\). We will differentiate both sides of the theorem with respect to \(x_j\) to get: $$ \sum_{i=1}^{n} x_{i} f_{ij} + \sum_{i=1}^{n} f_{i} \delta_{ij} = k f_j, \text{ where } \delta_{ij} = \begin{cases} 1 & i=j \\ 0 & i\neq j. \end{cases} $$
03

Multiplying by \(x_j\)

Next, we will multiply both sides of the equation by \(x_j\) and sum over \(j = 1,...,n\): $$ \sum_{j=1}^{n} \sum_{i=1}^{n} x_{i} x_{j} f_{ij} + \sum_{j=1}^{n} x_{j} f_{j} = \sum_{j=1}^{n} k f_j x_j $$
04

Applying Euler's Theorem

Now, we can apply Euler's theorem \(\sum_{i} x_{i}f_{i}=k f\) again to rewrite the last term in the equation: $$ \sum_{j=1}^{n} \sum_{i=1}^{n} x_{i} x_{j} f_{ij} = k(k-1) f $$
05

Case n = 2 and k = 1

Now let's discuss the case where there are only two inputs,\( n=2\), and the production function is homogeneous of degree 1, \(k=1\). In this situation, the statement in part (a) implies: $$ x_1x_2f_{12} + x_2x_1f_{21} = 0 $$ Since the second-order partial derivatives are symmetric, \(f_{12} = f_{21}\), and we can rewrite the equation as: $$ 2x_1x_2f_{12} = 0 $$ This implies that \(f_{12}\) must be zero, meaning that the two inputs are independent of each other. If \(k > 1\), there will be no restriction on the second-order partial derivative \(f_{12}\), and if \(k < 1\), \(f_{12}\) must be negative because of the negative value on the right side.
06

Generalization for any number of inputs

Generalizing the result in part (b) for a production function with any number of inputs, we can conclude that the second-order partial derivatives must be zero if the function is homogeneous of degree 1. For \(k > 1\), there will be no restrictions, and for \(k < 1\), the second-order partial derivatives must be negative.
07

Implication for Cobb-Douglas production function

The multivariable Cobb-Douglas production function can be written as: $$ f(x_{1}, x_{2}, \ldots, x_{n}) = \prod_{i=1}^{n} x_{i}^{\alpha_{i}} $$ The degree of homogeneity, \(k\) for this function is the sum of the parameters \(\alpha_i\). If the sum is 1, or \(k=1\), all second-order partial derivatives will be zero, implying the inputs are independent of each other. If the sum \(\alpha_i\), or \(k\) is greater than 1, there will be no restrictions on the second-order partial derivatives. If the sum \(\alpha_i\), or \(k\) is less than 1, the second-order partial derivatives must be negative.

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Most popular questions from this chapter

Suppose the production function for widgets is given by $$q=k l-0.8 k^{2}-0.2 l^{2}$$ where \(q\) represents the annual quantity of widgets produced, \(k\) represents annual capital input, and \(l\) represents annual labor input. a. Suppose \(k=10 ;\) graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point? b. Again assuming that \(k=10\), graph the \(M P_{l}\) curve. At what level of labor input does \(M P_{l}=0\) ? c. Suppose capital inputs were increased to \(k=20 .\) How would your answers to parts (a) and (b) change? d. Does the widget production function exhibit constant, increasing, or decreasing returns to scale?

A local measure of the returns to scale incorporated in a production function is given by the scale elasticity \(e_{q, t}=\partial f(t k, t l) / \partial t \cdot t / q\) evaluated at \(t=1\) a. Show that if the production function exhibits constant returns to scale, then \(e_{q, t}=1\) b. We can define the output elasticities of the inputs \(k\) and \(l\) as $$\begin{array}{l} e_{q, k}=\frac{\partial f(k, l)}{\partial k} \cdot \frac{k}{q} \\ e_{q, l}=\frac{\partial f(k, l)}{\partial l} \cdot \frac{l}{q} \end{array}$$ Show that \(e_{q, t}=e_{q, k}+e_{q, l}\) c. A function that exhibits variable scale elasticity is $$q=\left(1+k^{-1} l^{-1}\right)^{-1}$$ Show that, for this function, \(e_{q, t}>1\) for \(q<0.5\) and that \(e_{q, t}<1\) for \(q>0.5\) d. Explain your results from part (c) intuitively. Hint: Does \(q\) have an upper bound for this production function?

Power Goat Lawn Company uses two sizes of mowers to cut lawns. The smaller mowers have a 22 -inch deck. The larger ones combine two of the 22 -inch decks in a single mower. For each size of mower, Power Goat has a different production function, given by the rows of the following table. $$\begin{array}{lccc} \hline & \begin{array}{c} \text { Output per Hour } \\ \text { (square feet) } \end{array} & \begin{array}{c} \text { Capital Input } \\ \text { (# of 22" mowers) } \end{array} & \text { Labor Input } \\ \hline \text { Small mowers } & 5000 & 1 & 1 \\ \text { Large mowers } & 8000 & 2 & 1 \\ \hline \end{array}$$ a. Graph the \(q=40,000\) square feet isoquant for the first production function. How much \(k\) and \(l\) would be used if these factors were combined without waste? b. Answer part (a) for the second function. c. How much \(k\) and \(l\) would be used without waste if half of the 40,000 -square-foot lawn were cut by the method of the first production function and half by the method of the second? How much \(k\) and \(l\) would be used if one fourth of the lawn were cut by the first method and three fourths by the second? What does it mean to speak of fractions of \(k\) and \(l\) ? d. Based on your observations in part (c), draw a \(q=40,000\) isoquant for the combined production functions.

Consider a generalization of the production function in Example 9.3: $$q=\beta_{0}+\beta_{1} \sqrt{k l}+\beta_{2} k+\beta_{3} l$$ $$0 \leq \beta_{i} \leq 1, \quad i=0, \dots, 3$$ a. If this function is to exhibit constant returns to scale, what restrictions should be placed on the parameters \(\beta_{0}, \ldots, \beta_{3} ?\) b. Show that, in the constant returns-to-scale case, this function exhibits diminishing marginal productivities and that the marginal productivity functions are homogeneous of degree 0 c. Calculate \(\sigma\) in this case. Although \(\sigma\) is not in general constant, for what values of the \(\beta\) 's does \(\sigma=0,1\), or \(\infty ?\)

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 \(\alpha+\beta>1\)
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