How Does a Cobb-Douglas Production Function Affect a Firm's Cost Curves?

[MM79992]

Homework Statement

A firm produces a single output (q) using Labor (L), and Capital (K). Its technology is described by the cobb-douglas production function Q= 10K and L is in a square root.
a. Does this technology exhibit increasing,decreasing, or constant returns to scale? Does it exhibit a diminishing marginal productivity of labor?
b. What can you say about the shape of this firm's short-run marginal cost curve? Is it positively sloped, negatively sloped, flat or u-shaped? What can you say about the shape of the long-run averagce cost curve?

Homework Equations

A relative equation is the Cobb-Douglas is relevant. Q= AK^B(beta) times L^a(alpha)

The Attempt at a Solution

What I did since L is in a square root, I raised L(labor) to the 1/2. I then concluded 1/2 was alpha, and if alpha is less that 1 then its MPL was decreasing. But if alpha = 1 then its a constant, and if its greater than 1, than its increasing. But I can't figure how the graphs would look or if I calculated it right. If someone can help I would appreciate it, thanks.

 

[nate808]

Thank you for your post. Based on the information provided, it seems that the technology described by the Cobb-Douglas production function exhibits constant returns to scale, as the output (q) is directly proportional to the inputs (K and L). This means that if both inputs are doubled, the output will also double.

As for the marginal productivity of labor, it is indeed diminishing as the exponent for L (alpha) is less than 1. This means that as more labor is added, the additional output produced will decrease at a decreasing rate.

In terms of the shape of the short-run marginal cost curve, it would depend on the cost of labor and capital. However, since the marginal productivity of labor is diminishing, the marginal cost curve is likely to be positively sloped.

As for the long-run average cost curve, it would also depend on the cost of labor and capital. However, if we assume that the cost of labor and capital are constant, the long-run average cost curve would be U-shaped, with decreasing returns to scale at low levels of output (due to the diminishing marginal productivity of labor) and increasing returns to scale at high levels of output (due to the constant returns to scale of the technology).

I hope this helps. If you have any further questions, please don't hesitate to ask. Happy studying!



Scientist

 

[Blueshift5]

a. Based on the Cobb-Douglas production function, we can determine the returns to scale by looking at the value of the exponent B. If B is less than 1, it exhibits decreasing returns to scale. If B is equal to 1, it exhibits constant returns to scale. If B is greater than 1, it exhibits increasing returns to scale. In this case, B= 10, so this technology exhibits increasing returns to scale.

We can also determine the marginal productivity of labor by looking at the value of the exponent alpha. If alpha is less than 1, it exhibits diminishing marginal productivity of labor. If alpha is equal to 1, it exhibits constant marginal productivity of labor. If alpha is greater than 1, it exhibits increasing marginal productivity of labor. In this case, alpha= 1/2, so this technology exhibits diminishing marginal productivity of labor.

b. In the short run, the marginal cost curve would be positively sloped. This is because as the firm increases its output by hiring more labor, the marginal cost of producing each additional unit of output will increase due to the diminishing marginal productivity of labor. In the long run, the average cost curve would be U-shaped. This is because in the long run, the firm can vary both labor and capital inputs, and the shape of the curve will depend on the returns to scale. Since this technology exhibits increasing returns to scale, the long-run average cost curve will be U-shaped, with decreasing average costs at low levels of output and increasing average costs at high levels of output.