Question

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?

Short answer

Answer: When the average product of labor (AP_L) is increasing, it means that the marginal product of labor (MP_L) is greater than the average product of labor (AP_L). In such a case, the marginal product of capital (MP_K) must be negative, implying that increasing capital would lead to a decrease in total output. This indicates that the firm is operating inefficiently, using a non-optimal combination of labor and capital.

Step-by-step solution

1. Using Euler's theorem for the production function

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 \\]

2. Relating marginal product and average product

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} \\]

3. Proving that if \(MP_L > AP_L\), then \(MP_K\) must be negative

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).

4. Implications of negative \(MP_K\)

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.

5. Analysis of a firm producing at a point where \(AP_L\) is increasing

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.

Key concepts

Constant Returns to Scale

The idea behind constant returns to scale in production functions is that when you increase the input by a certain percentage, the output increases by the same percentage. For example, if you double both labor and capital in a production process, the output will also double. This scenario is mathematically represented by Euler's theorem, which states that if a production function is homogeneous of degree one (as with constant returns to scale), then the sum of its partial derivatives—multiplied by their respective inputs—will equal the output. The formula from Euler's theorem, \( q = f_K \cdot K + f_L \cdot L \), illustrates how the contributions of each input (capital \(K\) and labor \(L\)) sum up to match the total output \(q\). This ensures no economies or diseconomies of scale distortion in the production process.

Constant returns to scale suggest efficiency, where changes in input sizes do not affect the average cost of production, making it predictable for firms to scale up or down without loss of efficiency.

Marginal Product

The marginal product measures how much additional output is generated by adding one more unit of a particular input, keeping other inputs constant. Two common measures are the marginal product of labor (\(MP_L\)) and the marginal product of capital (\(MP_K\)). They are calculated as the derivative of the production function with respect to labor or capital.

Mathematically, this is expressed as:

• \(MP_L = \frac{\partial q}{\partial L} = f_L\)
• \(MP_K = \frac{\partial q}{\partial K} = f_K\)

These expressions show how each additional worker or unit of capital affects total output respectively.

In our exercise context, if \(MP_L\) is greater than the average product of labor (\(AP_L\)), the implication for \(MP_K\) is crucial. The exercise concludes that \(MP_K\) would be negative, signifying less efficiency when adding more capital. This entails that production would be better with a focus on labor rather than capital expansion.

Average Product

Average product is the measure of output per unit of input, specifically labor in most cases. It is calculated by dividing the total output \(q\) by the quantity of labor \(L\) used, formally expressed as \(AP_L = \frac{q}{L}\). This provides a general indication of how productive the labor force is on average.

The relationship between marginal and average product is key in determining production efficiency. When \(MP_L > AP_L\), it suggests that adding more labor increases productivity beyond the current average, thus has a growing effect on total production. However, if we continue in this direction, the pressure on capital causes \(MP_K\) to potentially become negative, which signifies an overload in the capital capacity within that production process.

Production Efficiency

Production efficiency occurs when the firm produces output at the lowest possible cost, using an optimal mix of inputs. In our example, achieving production efficiency involves understanding the balance between labor and capital inputs, as determined by movements in \(MP_L\) and \(MP_K\).

If a firm finds itself in a position where \(MP_L > AP_L\), while \(MP_K < 0\) (marginal product of capital is negative), it is a signal that the current use of capital is inefficient. Increasing labor while constraining capital input can improve output efficiency.

Additionally, a point of increasing \(AP_L\) signifies that each additional unit of labor yields more than the average. Although initially beneficial, sustained operations at this point can lead to diminishing returns elsewhere. Hence, true production efficiency is often about finding the sweet spot where neither capital nor labor is under or over-utilized, maximizing output without unnecessary expenditure.