Problem 2
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 anntal 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?
Problem 4
Suppose that the production of crayons \((q)\) is conducted at two locations and uses only labor as an input. The production function in location 1 is given by \(q_{1}=10 l_{1}^{0.5}\) and in location 2 by \(q_{2}=50 l_{2}^{0.5}\) a. If a single firm produces crayons in both locations, then it will obviously want to get as large an output as possible given the labor input it uses. How should it allocate labor between the locations in order to do so? Explain precisely the relationship between \(l_{1}\) and \(l_{2}\) b. Assuming that the firm operates in the efficient manner described in part (a), how does total output \((q)\) depend on the total amount of labor hired \((l)\) ?
Problem 9
A local measure of the returns to scale incorporated in a production function is given by the scale clasticity \(e_{g, t}=\partial f(t k, t l) / \partial t \cdot t / q\) evaluated at \(t=l\) 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{aligned} 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{aligned} \\] c. A function that exhibits variable scale clasticity is \\[ q=\left(1+k^{-1} l^{-1}\right)^{-1} \\] Show that, for this function, \(e_{p, t}>1\) 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?
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 CobbDouglas production function \(f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\prod_{i=1}^{m} x_{i}^{\alpha_{i}}\) for \(\alpha_{i} \geq 0 ?\)
