Question

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?

Short answer

Explain your answer. Answer: No, a firm can never produce at a point where the average product of labor is increasing. This is because an increasing average product of labor implies that the marginal product of labor is greater than the average product. As shown in the solution, at such points, the marginal product of capital would be negative, implying that adding more capital to the production process would reduce the total output. Firms would not choose to produce at such levels, as it would be inefficient and lead to wastage of resources.

Step-by-step solution

Write down Euler's theorem for homogeneous functions

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

Apply Euler's theorem to the production function

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: $$q= f_k \cdot k + f_l \cdot l$$

Analyze the relationship between MPL, APL, and MPK

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.

Discuss the implications of these results for production decisions

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.

Key concepts

Constant Returns to Scale

A production function with constant returns to scale (CRS) means that if you multiply all inputs by a certain factor, the output will be multiplied by the same factor too. For instance, doubling both labor and capital will double the output.
This property can be formally captured by homogeneous functions of degree 1. CRS is important in economics because it suggests that the size of the firm doesn't affect its efficiency.
Such functions maintain proportionality, ensuring that expanding production doesn't inherently increase or decrease average costs. Euler's theorem is often used to prove properties of CRS, demonstrating how output is distributed among inputs. This balance in scaling assures predictability and stability in production processes.

Marginal Product of Labor

The marginal product of labor (MPL) is the additional amount of output produced when one more unit of labor is added, keeping other inputs constant. To find MPL mathematically, you take the derivative of the production function with respect to labor.

• MPL = \(\frac{\partial f(k,l)}{\partial l}\)

Understanding MPL helps businesses decide how much labor to employ. If MPL is greater than the wage rate, the company benefits by hiring more labor. However, as more workers are added, MPL usually decreases because less capital and resources are available per worker—this is known as diminishing marginal returns.

Average Product of Labor

The average product of labor (APL) represents the output produced on average per unit of labor. It is calculated by dividing the total output by the number of labor units used. It acts as a measure of productivity and indicates how well resources are utilized.

• APL = \(\frac{q}{l}\)

Comparing APL with MPL can provide insights into productivity trends. If MPL is greater than APL, then APL is increasing, suggesting potential efficiency gains. However, if MPL drops below APL, adding more workers reduces efficiency, indicating an overuse of labor relative to other inputs.

Marginal Product of Capital

The marginal product of capital (MPK) describes the additional output obtained from increasing the capital input by one unit, while holding labor constant. It is computed as the derivative of the production function with respect to capital.

• MPK = \(\frac{\partial f(k,l)}{\partial k}\)

Understanding MPK is crucial for firms when deciding on investments in capital. Like MPL, MPK also typically exhibits diminishing returns, meaning each additional unit of capital contributes less to output than the previous one.
In the context of Euler's theorem, negative MPK indicates that further increases in capital can actually decrease total output, guiding firms in their capital allocation decisions.

Homogeneous Functions

Homogeneous functions play an essential role in understanding economic principles like constant returns to scale. A function is homogeneous of degree \(n\) if, when all inputs are scaled by a constant \(t\), the output scales by \(t^n\). For CRS, this degree is typically 1, as output scales directly with inputs.
Homogeneous functions allow economists to use Euler's theorem to decompose the contributions of different inputs. For instance:

• If a production function is homogeneous of degree 1, the function can always be expressed as a sum of its partial derivatives, multiplied by their respective inputs, emphasizing the output's reliance on each input's combined effect.

Recognizing a function's homogeneity aids in predicting how shifts in resources affect productivity and helps in formulating efficient resource allocation strategies.