Show that Euler's theorem implies that, for a constant returns-to-scale production function \([q=f(k, l)]\) $$q=f_{k} \cdot k+f_{l} \cdot l$$ Use this result to show that, for such a production function, if \(M P_{l}>A P_{l}\) then \(M P_{k}\) must be negative. What does this imply about where production must take place? Can a firm ever produce at a point where \(A P_{l}\) is increasing?

Euler's theorem states that for a homogeneous function of degree \(n\), say \(F(x,y)\), the following holds: $$n \times F(x,y) = x \cdot \frac{\partial F(x,y)}{\partial x} + y \cdot \frac{\partial F(x,y)}{\partial y}$$

Since the production function \(q=f(k,l)\) exhibits constant returns to scale (CRS), it is a homogeneous function of degree 1 (\(n=1\)). Thus, applying Euler's theorem, we get: $$q=f(k,l) = k \cdot \frac{\partial f(k,l)}{\partial k} + l \cdot \frac{\partial f(k,l)}{\partial l}$$ Which can also be written as: $$q= f_k \cdot k + f_l \cdot l$$

We are given that the marginal product of labor (MPL) is greater than the average product of labor (APL): \(MP_l > AP_l\). We need to show that in this case, the marginal product of capital (\(MP_k\)) must be negative. Since \(AP_l = \frac{q}{l}\), we have \(MP_l > \frac{q}{l}\). Multiplying both sides by \(l\), we get: $$MP_l \cdot l > q$$ Using our derived equation from Euler's theorem, we can rewrite this inequality as: $$MP_l \cdot l > f_k \cdot k + f_l \cdot l$$ Now, we can rearrange the inequality to isolate \(MP_k = f_k\) on one side: $$f_k \cdot k < - (MP_l - f_l) \cdot l$$ Since \(l > 0\), we can conclude that: $$MP_k = f_k < 0$$ This means that the marginal product of capital must be negative if the marginal product of labor is greater than the average product of labor.

The results imply that if the firm produces at a point where the marginal product of labor is greater than the average product of labor, it must have a negative marginal product of capital. This means that at such a point, adding more capital to the production process would actually reduce the total production. Firms would not want to produce at such a level, as it would be inefficient and lead to wastage of resources. As for the question whether a firm can ever produce at a point where the average product of labor is increasing, the answer is no. An increasing average product of labor implies that the marginal product of labor is greater than the average product. As we have shown, at such points the marginal product of capital would be negative, and therefore, firms would not choose to produce at such levels.