Show that Euler's theorem (see footnote 5 of Chapter 7 ) implies that for a constant returnsto-scale production function \([q=f(K, L)]\) Use this result to show that for such a production function, if \(M P_{L}>A P, M P_{K}\) 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 \(f\) of degree \(n\), the following relationship holds: \\[ f(K, L) = n(f_K K + f_L L) \\] Our production function is constant returns to scale (\(n=1\)), so it's a homogeneous function of degree \(1\). Applying Euler's theorem to our production function \(q = f(K, L)\), we have: \\[ q = 1(f_K K + f_L L) \\] This simplifies to: \\[ q = f_K\cdot K + f_L\cdot L \\]

The marginal product of labor (\(MP_L\)) and capital (\(MP_K\)) are defined as the partial derivatives of the production function with respect to labor and capital, respectively: \\[ MP_L = \frac{\partial q}{\partial L} = f_L\\ \] \\[ MP_K = \frac{\partial q}{\partial K} = f_K \\] The average product of labor (\(AP_L\)) is defined as the ratio of total output \(q\) to the quantity of labor \(L\): \\[ AP_L = \frac{q}{L} \\]

First, let's assume that \(MP_L > AP_L\). This means that \(f_L > \frac{q}{L}\). Now, let's use the result found in step 1: \\[ q = f_K\cdot K + f_L\cdot L \\] Divide both sides by \(L\): \\[ \frac{q}{L} = f_K \frac{K}{L} + f_L \\] But we already know that \(AP_L = \frac{q}{L}\), so: \\[ AP_L = f_K \frac{K}{L} + f_L \\] Now, let's isolate \(f_K\): \\[ f_K = \frac{AP_L - f_L}{\frac{K}{L}} \\] Since we assumed that \(MP_L > AP_L\), we have that \(f_L > AP_L\), which means that the numerator, \(AP_L - f_L\), is negative. But we know that the ratio \(\frac{K}{L}\) is positive (as \(K, L > 0\) by definition), so we can conclude that \(f_K < 0\). Then, remembering that \(MP_K = f_K\), we have proven that if \(MP_L > AP_L\), \(MP_K < 0\) (i.e., must be negative).

A negative marginal product of capital, \(MP_K < 0\), implies that adding more capital would lead to a decrease in total output. Production efficiency is achieved by using more labor and less capital in such a case. This is where the firm should produce, provided all other factors remain constant.

At the point where \(AP_L\) is increasing, this means that the marginal product of labor (\(MP_L\)) is greater than the average product of labor (\(AP_L\)). The previous analysis showed that if \(MP_L > AP_L\), then \(MP_K\) must be negative. Producing at a point where \(AP_L\) is increasing would mean that the firm is operating inefficiently, since it's not using an optimal combination of labor and capital. Therefore, a profit-maximizing firm would avoid producing at such a point under normal circumstances.